Cumulative Learning and Schematization in Problem Solving
Inaugural-Dissertation
zur
Erlangung der Doktorwürde
der Wirtschafts-und Verhaltenswissenschaftlichen Fakultät
der Albert-Ludwigs-Universität Freiburg. i. Br.
vorgelegt von
JungMi Lee
aus Daegu
SS 2012
Erstgutachter
Prof. Dr. Norbert M. Seel
Zweitgutachter
PD Dr. Ulrike Hanke
Dekan der Wirtschafts- und Verhaltenswissenschaftlichen Fakultät
Prof. Dr. Dieter K. Tscheulin
Datum des Promotionsbeschlusses
16. Oktober 2012
iv
Table of Contents
List of Tables and Figures ................................................................................................... vi ABSTRACT ........................................................................................................................... 9
1 INTRODUCTION .......................................................................................................... 10
2 THEORETICAL FOUNDATION ................................................................................ 12
2.1 Literature Review of Cumulative Learning .............................................................. 12
2.1.1 Gagné’s cumulative learning model ............................................................. 12
2.1.2 Cumulative nature in learning ....................................................................... 15
2.1.3 Ausubel’s subsumption theory ...................................................................... 17
2.1.4 Schema and Schema theory .......................................................................... 19
2.1.5 Kant’s theory of schematism ...................................................................... 21
2.1.6 Piaget’s process of equilibration ................................................................... 23
2.1.7 Cumulative learning in machine learning ..................................................... 24
2.1.8 Mechanisms in cumulative learning ............................................................. 28
2.1.8.1 Learning as an accumulation of knowledge ........................................ 28
2.1.8.2 Learning as a change in cognitive structures ....................................... 29
2.2 Contemporary Problems in Cumulative Learning .................................................... 34
2.2.1 The contemporary concept of cumulative learning ...................................... 34
2.2.2 The sequences of cumulative learning .......................................................... 40
2.2.3 Types of knowledge in cumulative Learning................................................ 42
2.2.4 Cognitive processes in cumulative learning ................................................. 46
2.2.4.1 Aggregation ......................................................................................... 47
2.2.4.2 Abstraction ......................................................................................... 48
2.2.4.3 Generalization .................................................................................... 50
2.2.5 Structuring processes in cumulative learning ............................................... 52
2.2.6 Concept learning in cumulative learning ...................................................... 57
2.2.6.1 Concept formation ............................................................................... 58
2.2.6.2 Concept assimilation and structural mapping ...................................... 61
2.3 Research Questions ................................................................................................... 65
3 METHOD ........................................................................................................................ 67
3.1 Data Collections ...................................................................................................... 68
3.1.1 College scholastic ability test (CSAT) .......................................................... 68
3.1.2 Procedure ...................................................................................................... 69
3.1.3 Material ......................................................................................................... 71
3.1.4 Participants .................................................................................................... 72
3.2 Data Analysis .......................................................................................................... 75
3.2.1 Stages of the analysis .................................................................................... 75
3.2.2 Validity and reliability .................................................................................. 79
3.2.3 Classification of the coding scheme ............................................................. 81
3.2.4 Descriptions of the coding scheme: Cognitive process ................................ 82
3.2.4.1 Aggregation ......................................................................................... 82
3.2.4.2 Abstraction ......................................................................................... 88
3.2.5 Descriptions of the coding scheme: Learning strategy ................................. 92
4 RESULTS ........................................................................................................................ 96
4.1 Case Studies 1 to 6 .................................................................................................. 97
4.1.1 Case 1 ............................................................................................................ 97
4.1.2 Case 2 .......................................................................................................... 103
4.1.3 Case 3 .......................................................................................................... 110
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4.1.4 Case 4 .......................................................................................................... 116
4.1.5 Case 5 .......................................................................................................... 123
4.1.6 Case 6 .......................................................................................................... 132
4.2 Case Studies 7 to 49 in Brief ................................................................................ 141
4.3 Overall Findings ................................................................................................... 193
4.3.1 Mechanisms of knowledge and skill development ..................................... 193
4.3.2 Cognitive processes .................................................................................... 197
4.3.2.1 Aggregation ....................................................................................... 197
4.3.2.2 Abstraction ....................................................................................... 199
4.3.2.3 Generalization .................................................................................. 201
4.3.3 Learning strategies ...................................................................................... 204
4.3.3.1 Cognitive strategies ............................................................................ 209
4.3.3.2 Metacognitive strategies .................................................................... 213
4.3.3.3 Social and affective strategies ............................................................ 214
4.3.4 Learning in problem solving ....................................................................... 215
5 DISCUSSION ............................................................................................................... 220
5.1 Limitations of the Study ....................................................................................... 220
5.2 Implications .......................................................................................................... 222
5.2.1 Theoretical implications.............................................................................. 222
5.2.1.1 Cumulative nature in learning ............................................................ 222
5.2.1.2 Structural nature in learning .............................................................. 224
5.2.1.3 Cognitive processes in learning ....................................................... 227
5.2.2 Practical implications .................................................................................. 229
5.3 Future Research .................................................................................................... 233
5.4 Conclusions........................................................................................................... 236
REFERENCES .................................................................................................................. 241
NOTES ............................................................................................................................... 247
APPENDIX A .................................................................................................................... 248
Curriculum Vitae ................................................................................................................ 270
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List of Tables and Figures
Table 2.1 Sequences of cumulative learning ........................................................................ 37
Table 3.1 Tests and subjects of CSAT .................................................................................. 69
Table 3.2 Number of items and testing time of CSAT ......................................................... 69
Table 3.3 Descriptive data of small experiments .................................................................. 73
Table 3.4 Descriptive data of participants ............................................................................ 73
Table 3.5 List of coding scheme ........................................................................................... 77
Table 3.6 Sample coding of interview transcription ............................................................. 78
Table 3.7 Classification of categories of cognitive processes .............................................. 81
Table 3.8 Classification of categories of learning strategies ................................................ 82
Table 3.9 Coding list of Theme 1: Aggregation (Category 1) .............................................. 87
Table 3.10 Coding list of Theme 1: Abstraction (Category 2) ............................................. 92
Table 3.11 Coding list of Theme 2: Cognitive strategies (Category 1) ................................ 94
Table 3.12 Coding list of Theme 2: Metacognitive strategies (Category 2) ......................... 95
Table 3.13 Coding list of Theme 2: Social and affective strategies (Category 3) ................ 95
Table 4.1 Coding of interview transcription: Case 1 .......................................................... 102
Table 4.2 Coding of interview transcription: Case 2 .......................................................... 108
Table 4.3 Coding of interview transcription: Case 3 .......................................................... 114
Table 4.4 Coding of interview transcription: Case 4 .......................................................... 122
Table 4.5 Coding of interview transcription: Case 5 .......................................................... 130
Table 4.6 Coding of interview transcription: Case 6 .......................................................... 139
Table 4.7 Frequencies of cognitive strategies ..................................................................... 209
Table 4.8 Frequencies of metacognitive strategies ............................................................. 213
Figure 2.1 Cognitive functions of assimilation and accommodation ................................... 24
Figure 2.2 Illustration of cumulative learning ...................................................................... 36
Figure 2.3 Cumulative interactions between prior knowledge & present knowledge .......... 37
Figure 2.4 Sequences of cumulative learning ....................................................................... 40
Figure 2.5 Sequence of condensation ................................................................................... 42
vii
Acknowledgements
The story of my doctoral studies is shared between two countries: The first half of it
was at the Florida State University, Tallahassee, USA, and the rest of it concluded at the
Albert-Ludwigs-Universität, Freiburg, Germany.
I wish to express my sincere gratitude to Prof. Dr. Norbert M. Seel for his guidance,
understanding, patience, and most importantly his encouragement during my entire doctoral
studies at both universities. His mentorship was enormous in providing a profound
experience consistent with my life goal. He has not been just an academic advisor but ‘doctor
vater’. He encouraged me to not only grow as an educational scientist but also as an
independent thinker. I am not sure how many graduate students are given the opportunity to
develop their individuality and self-sufficiency while being allowed to work with such
independence. He greatly emphasized that we, scientists, should always go back to the past
and learn from there: investigating how the historical background has transcended throughout
time and place. Although his decision to leave Florida State University for the
Albert-Ludwigs-Universität caused a great deal of distress for me at one time, in the long run,
I believe that such action provided me the unique opportunity to gain a new and wider
breadth of experience and perspectives. Additionally, I am very grateful for his guidance in
getting my doctoral career started on the right foot. With the publication of the volume 7, last
winter, my almost three-year term as an Editorial Assistant of the Encyclopedia of the
Sciences in Learning, with Prof. Dr. Seel, the Editor-in-Chief is completed. It has been a
privilege to serve as an editorial assistant and see the range of scientific studies published that
is relevant to Learning, as well as be an instrument of dissemination of the creative efforts of
about 1500 authors. I believe this provided me the foundation for becoming an educational
scientist. Among the many incidents during this journey under his guidance, one moment is
worth mentioning: One day, seeing that I was struggling with readings and had not yet
completed them, he said, “This must be your hobby. It’s time to write and not just read.” This
brought a smile and made me move forward. For everything you have done for me, Prof. Dr.
Seel, I thank you.
I would also like to express my sincere gratitude to PD Dr. Ulrike Hanke, for her
invaluable comments and insights in the completion of this study. I thank her for all her
sincerity and encouragements that lead to a great deal of improvement while completing this
study, which I will never forget. She has been my inspiration as I faced a great many
obstacles in the completion of this study. It gave me a bolt of energy that I apparently needed
viii
during the last stages of my studies.
My acknowledgement also goes to Prof. Dr. Andrei I. Podolskij, Moscow State
University, for reading and commenting on this study, and for his concerns and
encouragement in the completion of this study.
I would also like to thank all my professors and wonderful friends while I was studying
at Florida State University. Special thanks should be directed to Prof. J. Michael Spector, Ph.
D. and Prof. Vanessa Dennen, Ph. D. for their guidance and support during my almost three
years study in Tallahassee.
I would like to thank all of the colleagues of the Institut für Erziehungswissenschaft at
Albert-Ludwigs-Universität Freiburg, especially PD Dr. Pablo Pirnay-Dummer and PD Dr.
Dirk Ifenthaler for their guidance and for sharing their knowledge and experience.
I had the fortunate pleasure of making wonderful friends who showed great support
during my years at Tallahassee and Freiburg. Special thanks should be directed to Mathen
Koshy for his great friendship and help during my entire doctoral study. I thank Elfride Seel
who showed me wonderful love and Martina Griesbaum, and Eun Ju Kim for their great
support and help during my stay in Freiburg. I also thank my wonderful next-door neighbors,
Ulla Wyatt and Stephanie Petter for their kindness and encouragements. Because of all of you,
I have always felt at home.
Finally, I would like to thank my parents, Seok-Yeop Lee and Seon-Cho Ha, and my
family members, for their faith, love and patience in me. I love you all.
The journey with bends in the road was not easy but thinking back, I see that every
experience was truly its own reward. Now, I wonder how the road beyond will be. I guess
here, my old habits come in handy, it is time to read.
August 2012
Sonnhaldestraße, Freiburg
9
Cumulative Learning and Schematization in Problem Solving
Abstract
This study defines the theoretical concept and framework of “cumulative learning,” which
deals with the gradual development of knowledge and skills over time in the course of
learning. In doing this, this study investigates the cognitive processes and learning strategies
inherent to cumulative learning in relation to concept and schema construction in problem
solving. The core assumption underlying cumulative learning is that the learning of humans
and any other animals is cumulative by nature so the learned knowledge that one has obtained
through various experiences are consolidated, reproduced, and exploited for further learning
situations. The study presents 49 cases of the publicly available archival interview record
involving 22 second and third-year high school students, who were preparing to take the
college entrance examination, College Scholastic Ability Test (CSAT), and 27 first and
second-year college/university students who had taken the CSAT in Korea. The interviews
used in this study broadcasted on a TV program from August 2009 to June 2011 by Korea
Educational Broadcasting System (EBS) to introduce effective and efficient learning
processes and strategies in students preparing for the college entrance examination in Korea,
titled “The Royal Road to Learning.” The results show the two categories of cognitive
processes: aggregation (with 8 subcategories) and abstraction (with 5 subcategories) of
knowledge; and the three categories of learning strategies: cognitive strategies (with 13
subcategories), metacognitive strategies (with 7 subcategories), and social/affective strategies
(with 3 subcategories). This study suggests that entire cognitive processes in learning interact
with each other cumulatively and that learning in each sequence thus depends on the previous
one. The gradual development and improvement of complex cognitive structures
accomplished in each part of learning is meaningfully compiled into a comprehensive
learning through schematization during the course of learning. This study aims to provide a
detailed theoretical account of how cumulative learning functions in human learning process.
10
CHAPTER I
INTRODUCTION
Learning has been characterized as the process of knowledge construction (Resnick,
1989) or an enduring change or improvement of a learner’s behavior due to experience (e.g.,
Langley, 1995; Shuell & Lee, 1976). In terms of the learning process, it seems that humans
are naturally able to link their existing knowledge into a continuous flow of new experience
in the environment resulting from the recursive nature of learning processes, thus building up
increasingly complex cognitive structures over time.
At the outset, let me support the preceding notion with theoretical arguments that
briefly foreshadow a detailed discussion given later in chapter II of this paper. Piaget (1952)
argued that learners naturally try to resolve the cognitive dissonance that results from their
exposure to information that conflict with their previous conceptions. This cognitive
disequilibrium1 (i.e., the state in which the balance between one’s knowledge and reality is
broken) regulates one’s learning to construct new knowledge. Ausubel’s (1963b) assimilation
theory of learning assumes that new learning experiences are always integrated into
preexisting knowledge structures, and thus states that new information is incorporated into an
anchoring structure already present in the student.
The above mentioned arguments suggest that humans naturally link new information
into their existing knowledge. If one accepts these arguments, then one would agree on the
point that learning involves continuously relating new information into existing knowledge so
the learned knowledge that they have obtained through various experiences are consolidated,
reproduced, and exploited for further learning situations. This cumulative nature of human
learning is the core assumption of the present study.
The cumulative aspect of learning has been explicitly confirmed by Gagné (1970, 1977)
and others authors (such as Shuell, 1986). Thus, in the field of cognitive and educational
psychology, it has been widely stated and often implicitly accepted that the learning of
humans and other animals is cumulative by nature (e.g., Aebli, 1978; Ausubel, 1960; Bruner,
1960; Freebody, Maton, & Martin, 2008; Gagné, 1962a, 1962b, 1968, 1970; Shuell, 1986;
Wittrock, 1991). However, the term “cumulative learning” has neither been explicitly
specified nor studied in detail since Gagné’s (1970) statements concerning cumulative
learning, and there is comprehensive theoretical foundation of cumulative learning. Also,
1 See section 2.1.6 for more detail on Piaget’s equilibrium and disequilibrium.
11
there is a significant lack of study in the field of cognitive and educational science in the past
two decades to understand the process of cumulative learning while in the field of artificial
intelligence and machine learning (wherein the term “incremental learning” is often used)
there has been sufficient interest. Therefore, the present study aims to provide a new
beginning after two decades of relative inactivity in studying cumulative learning.
As an umbrella term, this study addressed “cumulative learning” with a focus on
forming a unified theory leading to the indication of a theoretical framework. Under the
assumption that learning is a cumulative process wherein the learning in each new sequence
builds upon knowledge acquired in a previous sequence, this study endeavors to construct a
scientifically adequate theoretical framework of cumulative learning by compiling diverse
qualitative empirical results.
The sequences of this paper are as follows: The chapter II, theoretical foundation, 1)
reviews the existing literature related to cumulative learning and its major aspects in relation
to the development of knowledge and skills from diverse perspectives, 2) defines the
theoretical concept and framework of cumulative learning, and then 3) presents research
questions of this study providing a rationale based on the theoretical background. Chapter III
explains the method of the study: procedure, material, participants, and the process of data
analysis along with the descriptions of coding scheme used in data analysis. In Chapter IV,
49 case studies are presented: six described in detail and 43 summarized in brief followed by
the overall findings of the analysis performed on the case studies. The transcription of the six
cases (case studies 1 to 6) along with coding is represented by tables to provide structural
descriptions. The transcription of the 43 case studies (case studies 7 to 49) is given in
Appendix A. Finally, the limitations, implications, conclusions of the study and some
suggestions for future studies are stated in the last chapter.
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CHAPTER II
THEORETICAL FOUNDATION
The purpose of this chapter is to provide the theoretical concept and framework of
“cumulative learning.” It first reviews the early approaches addressing cumulative learning,
and then defines the “contemporary” theoretical concept and framework of cumulative
learning based on the reviews.
The first section reviews: 1) Gagné’s cumulative learning model and its implication, 2)
various arguments that explicitly and implicitly asserts cumulative nature in learning, 3)
Ausubel’s subsumption theory on the notion of cumulative learning, 4) the concept of schema
and schema theory in order to define the function of schema(s) in cumulative learning, 5) the
organizational nature of human learning proposed in Kant’s theory of schematism, 6) the
structural nature of human learning proposed in Piaget’s equilibration process, and 7)
cumulative learning in the field of machine learning. Then the last subsection 8) explains the
two mechanisms of learning derived from the preceding reviews of literature addressing
cumulative learning, namely, learning as an accumulation of knowledge, and learning as a
change in cognitive structures.
Based on these reviews, the second section defines the contemporary concept of
cumulative learning which forms the conceptual base of this study. The section also includes
the theoretical framework of cumulative learning namely; the sequences, the types of
knowledge, the fundamental cognitive processes, and the structuring processes in cumulative
learning. Thereafter, it investigates concept learning in relation to cumulative learning since
concepts serve as the basic components in the hierarchy of cognitive structure. The last
section defines the research questions of this study based on the theoretical foundation
provided herein.
2.1 Literature Review of Cumulative Learning
2.1.1 Gagné’s Cumulative Learning Model
An early proponent of the idea that learning is cumulative by nature was Gagné and he
coined the term “cumulative learning.” Under the assumption that intellectual skills can be
broken down into simpler skills, Gagné argued that learning is cumulative and that human
intellectual development consists in building up increasingly complex structures. That is, the
learning of higher-level skills, such as higher-order rules and principles depends primarily
13
upon the prior mastery of subordinate skills or concepts (Gagné, 1970, 1977).
Thus, Gagné (1968) proposed a “cumulative learning model” in which new learning
builds upon prior learning and is dependent on the combination of previously acquired and
recalled entities as well as their potential for learning transfer. Accordingly, he believed that
learners are capable of learning cumulatively, that this capability provides them with the
foundation for further learning through knowledge transfer, and that this process is
generalized when an individual actively engages in an intellectual activity. Further, he argued
that learning occurs not only in the acquisition of new associations but also−due to the
cumulative effect of learning−because the learners learn the processes of classification,
differentiation, categorization, recall, retention, and learning transfer. Consequently, Gagné
viewed generalization or transfer of knowledge to new tasks as an effect of cumulative
learning, and hence, any learned abilities at any sequence in the course of learning can
mediate learning that was not otherwise intentionally taught. Gagné emphasized the role of
previously acquired subordinate skills in this context, as it may be identified on the basis of
the cumulative learning model since competence attainment is hierarchically organized.
Wiegand (1970) studied whether the performance of a complex science problem similar
to one used in Piaget’s studies as well as a Piaget task could be justified by Gagné’s
cumulative learning model with children, providing needed subordinate capabilities in the
interval between two times of tests. The results indicated that the cumulative effects of
learning of concretely referenced intellectual skills that are more conducive to intellectual
development than the adaptation of structures of intellectual growth. Thus, Wiegand (1970)
described “A cumulative effect of learning within varied stimulus situation is the acquisition
of skills, strategies, and/or learning sets that make the individual to deal more effectively with
environmental contingencies” (p. 16).
Gagné viewed problem-solving capabilities as the result of the cumulative learning of
specific intellectual skills of relevant sets of problems, and he assumed a cumulative
organization of learning events based on pre-existing relationships among learned behaviors
(Gagné, 1968, 1970, 1977). Hence, he posited that instruction should provide a set of
component tasks and a sequence of these tasks to ensure the learners’ mastery of each
component task and the optimal transfer of the final task (Gagné, 1970, 1977). According to
this view, the transfer of learning facilitates the development of cognitive abilities. That is,
the intellectual development of a child proceeds through the learning of ordered sets of
capabilities that build upon each other in progressive fashion by means of the processes of
differentiation, recall, and transfer of learning rather than by means of the increasing numbers
14
of associations formed between stimuli and responses (Gagné, 1968). Gagné thus viewed
such process of generalization is constructed through a combination with other learned
entities as the mechanism of transfer in learning, and hence, he believed that the child can
combine learned entities with new knowledge without repetitive practices by receiving
instructions (Gagné, 1968). Accordingly, Gagné’s early research on the learning hierarchy
was to empirically verify his stance as opposed to the assumption that the best way to learn a
task is to practice it (Derry & Lesgold, 1996).
Gagné’s (1962a, 1962b, 1968) learning hierarchy (originally called “hierarchies of
knowledge”) assumes that the learner can only acquire a piece of knowledge if s/he possesses
certain “subordinate knowledge” (Gagné, 1962a, p. 356), a set of ordered intellectual
knowledge which grows increasingly simple and general as the defining process proceeds. He
believed that the subordinate knowledge can be arranged in a certain particularly effective
sequence to insure positive transfer to the final performance. Thus, he proposed that positive
transfer to a higher-level task and the final performance depends on the identification of
subordinate tasks as well as on their optimally suitable arrangement and successive
attainment and integration of a series of lower-level learning sets into prior knowledge
(Gagné, 1962a, 1962b; Gagné & Paradise, 1961). In other words, basic abilities or factors,
which are the most simple and most general learning sets, then gradually support more and
more complex activities. Such support then facilitates learning and affects other relevant
tasks. Thus, Gagné (1970, 1977) demonstrated that successively higher-level skills are better
learned when their subordinate skills (lower on the hierarchy) are learned first. Gagné called
this principle “cumulative learning” as it suggests “a principle of instruction sequencing that
went beyond the frame-by-frame organization of programmed instruction” (Gagné, 1989, p.
171). Thus, Gagné (1985) suspected that cognitive strategic capabilities evolve over time as
byproducts of a multitude of problem solving situations and hence cannot be trained in a
direct manner.
In the research on the learning hierarchy, Gagné broke down a complex task into
progressively more complex skills that help learners perform it, and then tried to empirically
demonstrate the effect of the cumulative learning model using Piaget’s classical conservation
task (see Gagné, 1970, 1977). He was criticized for his incorrect analysis of the conservation
task example because he used an ambiguous combination of “nonmetric judgment of
volume” and “conservation of identity” in his analysis although his specific task example
presents only the latter, but this did not diminish the force of this example in terms of
demonstrating cognitive development process (see Furby, 1972). Gagné (1962a) noted
15
possible similarities between the entities of the learning hierarchy and various hypothetical
constructs, such as Maltzman’s (1955) habit-family hierarchies, Harlow’s (1949) concept of
learning sets; and Katona’s (1940) organizations, whose role is “to establish or to discover or
to understand an intrinsic relationship” (Katona, 1940, p. 54).
Maizam also stated that a learning hierarchy shows the most thorough path to achieving
the “terminal objective” (Maizam, 2011, p. 1884), the highest expected learning outcome of
given instruction. Maizam (2011) viewed that the learning hierarchy can be represented by a
bottom-up directional flow diagram to show that all sub-objectives are mapping onto the
terminal objective, which is at the highest position in the hierarchy. Accordingly, the learning
hierarchy technique is a top-down analysis technique that identifies the prerequisites for an
expected learning outcome in the intellectual learning domain, which would result in a set of
subordinate intellectual skills that are related to each other in a hierarchical manner (Maizam,
2011). Maizam thus stated that the learning hierarchy technique is derived from Gagné’s
(1968) argumentation of cumulative learning, in which a new skill can only be acquired by
mastering prerequisite subordinate skills: For instance, to learn the highest intellectual skill
(e.g., rule applications), a learner must first master the sub-intellectual skills of concepts, and
discriminations (Maizam, 2011).
2.1.2 Cumulative Nature in Learning
In the field of cognitive and educational psychology, not only Gagné and his disciples
have explicitly stated that the learning of humans and other animals is cumulative by nature,
Shuell also explicitly stated that “learning is cumulative in nature” (Shuell, 1986, p. 416) and
that everything has meaning or is learned in connection. Currently, Freebody, Maton, and
Martin (2008) defined cumulative learning as a process in which learners build knowledge
“over time by integrating and subsuming previous knowledge” (p. 188), and they proposed
that learners have the ability to build knowledge over time by transferring it between
contexts.
The cumulative nature of learning was also emphasized and often implicitly accepted
by the early theorists of cognitive learning as follows. Bruner (1960) viewed learning as an
active process in which learners construct new concepts based upon their present and prior
knowledge by means of selecting and transforming information, constructing hypotheses, and
making decisions based on a cognitive structure. Therefore, Bruner hypothesized that “any
subject can be taught effectively in some intellectually honest form to any child at any stage
of development” (Bruner, 1960, p. 33). Hence, he noted that arranging information in a
16
“spiral” fashion helps children to organize knowledge into a structure that makes it
increasingly usable in other areas beyond the current learning situation. Accordingly, Bruner
(1960) emphasized that the learning situation should help learners to actively reorganize new
information, allowing them to build on existing knowledge in a meaningful way and use the
newly gained knowledge effectively in future tasks. Bruner (1960) thus emphasized the
importance of learning the underlying principles of different concepts of children’s learning
in a way that allows subsequent transfer and consequently expands their knowledge. In other
words, Bruner (1960) proposed that repeatedly presented information should be organized
from the simple to the more complex, from the general to the specific, and be examined in
association with other information in the course of learning situations. Accordingly, Bruner
(1960) believed that as children grow, the curriculum should repeatedly present previously
learned information and expand on it until the children understand the information and its
relations more completely.
Aebli (1973) defined concepts as the basic blocks of any discipline, and consequently,
once these basic concepts are acquired and mastered, learners are ready to move to a higher
level of knowledge. These accumulated concepts lead to the higher level of knowledge.
Additionally, Aebli emphasized that learners have to play an active role in the learning
process by actively constructing new knowledge on the basis of existing knowledge
structures (Aebli, 1978, 1983, 1987). Thus, Aebli stated that knowledge is acquired as
learners actively construct and transform it by integrating newly gained information and
experiences into an earlier knowledge structure through a process of revision and
reinterpretation of existing knowledge in light of the newly gained information (Pauli &
Reusser, 2011).
Ausubel strongly emphasized the role of prior knowledge in learning as he stated that
“the most important single factor influencing learning is what the learner already knows”
(Ausubel, Novak, & Hanesian, 1968, p. vi). Thus, Ausubel (1960) noted that a learner’s
present cognitive structure is the primary factor in the acquisition and retainment of pieces of
information in the next sequence of learning. Ausubel’s (1963b) assimilation theory of
cognitive learning and Wittrock’s (1991) theoretical approach of generative learning, which
is closely related to assimilation theory, basically assume that a learner consistently reviews
information which enhances the learning of new concepts. Assimilation theory views that
new information is integrated into previously acquired knowledge by analogy, thereby
improving the learner’s knowledge. In addition to this, Ausubel’s subsumption theory
17
described in the following section corresponds to a great extent to the proposed idea of
cumulative learning in this study.
2.1.3 Ausubel’s Subsumption Theory
Ausubel viewed knowledge as representing an integrated system. He thus assumed
human cognitive structure is hierarchical in organization from greater to lesser inclusiveness
(Ausubel, 1960; Ausubel & Robinson, 1969). Correspondingly, Ausubel’s (1960)
subsumption theory proposed that new information can be connected into relevant existing
knowledge structure through subsumption process (i.e., superordinate concepts subsume
related subordinate concepts). Therefore, Ausubel (1963b) proposed that learning occurs as
“potentially” meaningful material cognitively interacts with, and is appropriately subsumed
under a relevant and more inclusive conceptual system. This is possible because a learner
activates his/her existing knowledge so that the “facts, concepts, propositions, theories, and
raw perceptual data” (Ausubel & Robinson, 1969, pp. 51) of the existing knowledge can be
assimilated, modified, and restructured into new schemas (see section 2.1.4), where the
learner’s appropriate existing knowledge interacts with new learning (Ausubel, Novak, &
Hanesian, 1978).
In this subsumption theory, Ausubel (1963a) emphasized the function of the learner’s
cognitive structure in the acquisition of new information and argued that present experiences
are always absorbed and incorporated (scaffolding) into what the learner already knows. That
is, when new information is perceived it is classified and incorporated into more inclusive
concepts existing in the learner’s cognitive structure. In this view, Ausubel explicitly
emphasized the importance of having well-organized cognitive structure for he believed that
“it provides better anchorage for new learning and retention than if it is unclear, unstable, and
poorly organized” (Ausubel & Fitzgerald, 1962, p. 244). In line with this view, Ausubel thus
introduced the concept of “advance organizers” to support the learner’s prior knowledge.
Advance organizers are to implement “progressive differentiation and integrative
reconciliation in sequencing subject matter” (Ausubel, 1980, p. 403). They consist of the
most relevant ideational antecedents presented “at a higher levels of abstraction, generality,
and inclusiveness than the new materials to be learned” (Ausubel, Novak, & Hanesian, 1978,
p. 171). Ausubel proposed that providing advance organizers can help a learner to activate
existing knowledge structure (i.e. schema) more effectively and allows the learner to use
his/her pre-existing knowledge in a more effective way as s/he relates new information to
relevant previously learned subsuming concepts (Ausubel, Novak, & Hanesian, 1978).
18
Ausubel (1960) viewed that advance organizers activate relevant subsuming concepts of the
learner’s existing knowledge structure, and hence, new material to be learned become more
familiar and meaningful, and provides optimal anchorage.
Accordingly, Ausubel’s (1968) theory of meaningful learning2
is meant to help the
learner activate his/her pre-existing knowledge so that it can be incorporated into new
cognitive structures (i.e. schemas). The theory of meaningful learning assumes that new
learning experiences cannot be understood until they have been integrated into existing
knowledge structures (Ausubel & Fitzgerald, 1962). Consequently, the theory of meaningful
learning views that “the quantity, clarity, and organization of learners’ present knowledge”
(Ausubel & Robinson, 1969, pp. 51-52) is the most important factor influencing learning.
Not only Ausubel and his disciples, but others also supported the notion of advance
organizers as follows. For instance, Novak, Ring, & Tamier (1971) noted that the organizers
help to associate rotely learned information and/or to provide a subsumption process for
subsequent instruction. Luiten, Ames, and Ackerson (1980) in their meta-analysis also noted
the “facilitative effect on both learning and retention” (p. 211) of advance organizers.
In contrast, some authors (e.g., Anderson, Spiro, & Anderson, 1978; Barnes & Clawson,
1975; Clark & Bean, 1982; Lawton & Wanska, 1977) have remarked that it is not so obvious
what an “advance organizer” is or how it should be distinguished from the new learning itself.
Others (e.g., McEneany, 1990) criticized the lack of empirical evidence for advance
organizers to facilitate learning. Lawton and Wanska (1977) also pointed out that advance
organizers do not relate to cognitive structure and new information but they may induce their
own cognitive structure to which the new information may be related. In relation to this view,
Bransford (1984) argued that advance organizers should be composed differently depending
on whether they are to be used for schema activation or schema construction. He thus stated
that advance organizers can be effective when the learner has already acquired the necessary
schemas for a given problem. Nevertheless, it seems that such criticism do not necessarily
discredit the theoretical principles underlying advance organizers (i.e., superordinate concepts
always subsume related subordinate concepts) as Ausubel (1980) himself noted.
In sum, Ausubel’s theory of subsumption and advance organizers provide plausible
theoretical grounds for explaining how new information is learned in relation to existing
knowledge structure. The following section reviews schema theory which also implicitly
emphasizes the cumulative nature of learning.
2
Ausubel’s theory of meaningful learning is grounded in the concept of Piaget’s assimilation in section 2.1.6.
19
2.1.4 Schema and Schema Theory
Schema theory (Norman & Rumelhart, 1975) assumes that learners instantiate existing
schemas and construct new ones by relating new information to old schemas by analogy, and
thereby implying the cumulative nature of learning. In the field of educational psychology,
Piaget (1926) proposed the concept of schema and developed a schema theory and Anderson
and colleagues (Anderson, Reynolds, Schallert, & Goetz, 1977; Anderson, Spiro, &
Anderson, 1978) expanded the meaning of schema claiming that people use schemas to
organize their current information and knowledge and that these schemas then serve as a
framework (e.g. scripts, prototypes) for future interpretations of new information. They
defined the schema as a unit of mental representation or generic knowledge structures in
which the conceptual ideas, relations, procedures, and structures of a schematic feature of the
perceived world are organized.
Historically, the concept of schema can be traced back to Plato, who believed that a
perfect timeless circle of pure abstract thought exists in the human mind. Plato proposed that
reality is in the eternal abstract idea of an object that exists in our minds rather than in any
particular object we sense, and hence, truth can be reached through our thoughts but not by
our senses. Accordingly, Plato valued abstract reasoning as he believed that one would be
more likely to see the truth by using one’s mind. This idea was developed further by Kant in
his concept of schema and schematism, which is explained in detail in the following section
2.1.5. Johann Gottlieb Fichte (1762–1814) conceived of schemas as the shape or form of an
object, which is consistent with Kant’s schematism of empirical concepts. Under the
influence of Kant, German psychologists, Selz (1913) and Bühler (1918), also applied the
schema concept (Seel, 2011b). Selz (1913) understood a schema as a network of concepts
that guides the thinking process and makes a person capable of useful and meaningful
inferences (i.e., schema anticipation).
Authors defined schema (or frame, script) in various ways: an organization of past
experience (Bartlett, 1932), an abstract knowledge structure (Abelson, 1981; Mandler, 1978;
Rumelhart & Ortony, 1977; Seel, 2003), an active and interrelated knowledge structure
(Rumelhart & Norman (1976), a generic knowledge structure (Anderson et al., 1977, 1978;
Seel, Ifenthaler, & Pirnay-Dummer, 2009), a basic building block of cognition (Rumelhart,
1980; Mandler, 1984), or a “simplified interpretative framework used to understand events”
(D’Andrade, 1992, p. 48). Authors assumed that schema never stops changing or becoming
more refined, and hence, as knowledge structure develops, the schema gradually broadens
and becomes progressively more complex as they become more generalized and
20
differentiated. Schemas allow learners to reason about unfamiliar learning situations and
interpret these situations in terms of their generalized knowledge. The learners actively
interact with the perceived information (i.e., realities) of the world. Hence, as the schema gets
broader, individuals are more likely to find some connections between existing knowledge
and new information perceived in the world, and thereby, expand and enhance their
knowledge. Therefore, schemas can be viewed as cognitive constructs that allow us to treat
multiple elements of information in terms of larger, higher-level units (Ifenthaler, Lee, & Seel,
2009).
Rumelhart and Ortony (1977) conceived of schemas as structures that incorporate
typical facts about a category or an event in a flexible way that allows for some variation,
which can encompass other schemas and may have different levels of abstraction. Rumelhart
(1980) defined schemas as the building blocks of cognition and the fundamental elements
upon which all information processing depends. Schemas help us make interpretations of
events through abstraction and unspecified numbers of slots, which are filled in by particular
experiences or contexts (D’Andrade, 1992, 1993).
Schank and Abelson (1977) introduced “scripts,” which are made up of scenes and are
used to deal with generic knowledge of sequences of actions. Scripts are “rich packages of
information and they can be used for testing general ideas about cognitive schemata”
(Abelson, 1981, p. 727). Scripts are memory representations of frequently performed action
sequences that are causally linked. Some errors occur in recalling an event that is not
consistent with the script. Thus, some events that do not follow the script or schema will be
better recalled shortly afterwards because such atypical events become distinctive in terms of
that script or schema. However, more typical events that follow the scripts will be better
recalled as time passes as the memory becomes more consistent with the script (or schema)
for that type of event over time.
Sweller (1988) proposed schemas as the cognitive structures that compose an
individual’s knowledge, and defined a schema as a “cognitive construct that permits problem
solvers to recognize problems as belonging to a particular category requiring particular
moves for solution” (Sweller, 1989, p. 458). Sweller (1988) assumed that learning consists
primarily of the acquisition of schemas, and the change in performance occurs as the
learners’ schemas are increasingly associated (or activated) with the learning material. Hence,
he proposed that learning requires a change in the schematic structures of long-term memory
and is demonstrated by performance that progresses from slow and difficult (novice-like) to
smooth, fast, and effortless automation (expert-like).
21
The above stated descriptions and interpretations of schemas as well as schema theory
proposes the schematic organization of mental representation in human learning. This in turn
reflects that prior knowledge plays a critical role in human learning as schema theoretical
approaches assume that learners activate existing schemas when confronted with new
information in order to process it effectively. That is, learners interpret and make predictions
about the world using schemas (i.e., synthetic abstract knowledge structures) and actively
construct and modify the schemas as they interact with their experiences, and hence,
progressively develop their knowledge. In line with this view, the following section
investigates the organizational nature of human learning proposed in Kant’s theory of
schematism.
2.1.5 Kant’s Theory of Schematism
Kant contemplated the mechanism of human thinking processes and emphasized the
human intellect’s function of structural organization when he developed the notion of
schematism (B179/A140)3 in his “Critique of Pure Reason.” He introduced the word schema
and developed the notion of “Schematismus” (i.e., schematism) by emphasizing the human
intellect’s function of structural organization.
Kant was concerned with the mechanisms of the human thinking process and
questioned how human intuition can be subsumed under pure concepts (B177/A138). He
stated that the concept of understanding forms an integrated pure synthetic organization of
the manifold. Kant’s theory of schematism follows the notion of schematic construction in
pure intuition. He questioned how mind and body are related and believed that the senses
(possibly close to the idea of body), reason (possibly close to the idea of mind), and
understanding (possibly bridging the senses and reason) all work together. As an example,
Kant stated that the schema of a triangle can exist only in thought because no universal image
can contain all of the different figures we refer to as triangles (e.g. right-angled or
acute-angled): An object of experience or its image can hardly be adequate to the empirical
concept because it always “stands in immediate relation to the schema of imagination, as a
rule for the determination of our intuition, in accordance with some specific universal
concept” (B180/A141). Hence, Kant stated that the schema of a triangle shows a rule of the
synthesis of the imagination of pure figures in space. Additionally, Kant conceptualized a
3 For the Kritik der reinen Vernunft the author followed the customary method of referring to pages of the first
(1781) and second (1787) editions, respectively known as “A” and “B.”
22
“system” as the consolidation of pieces of perception under a single united idea (A832/B860).
Under the same thread of thought, Kant characterized schemas as “innate structures
which organize the world” (as cited in Seel, 2011b, p. 2933). Thus, Kant defined (a) the
schema of the concept as the use of the concept of understanding, which is restricted by
formal and pure condition sensibility, and (b) the schematism of pure understanding as the
procedure of understanding in these schemas (B179/A140). Kant viewed the schema as
always produced by imagination. However, he distinguished it from the image as the
synthesis of imagination requires a unity in determination sensibility (B179/A140). Thus, he
interpreted the schema of the concept as the representation of a general procedure of
imagination which provides an image for a concept. Hence, it is not images of objects but the
schemas which underlie human pure sensible concepts, because no image could ever be
perfectly adequate to the concept of an object in general (B180/A141). As an example, he
suggested a “dog” schema as a rule of mental representation which can be generally
delineated as the figure of a four-footed animal which is not restricted to any particular
experience or any possible image that can be represented in concrete form (A141).
Kant distinguished between a posteriori knowledge (i.e., empirical or experience-based
knowledge) and a priori knowledge (i.e., innate general truth) which exists regardless of
individual experience, such as knowledge of the concepts of time and causality. According to
his synthesis, understanding requires both a posteriori (thesis) and a priori innate concepts
(antithesis). In this way, understanding evolves both through experience and innate
knowledge. This is comparable to the mechanism of cumulative learning in that the process
of synthesizing new information (a posteriori empirically acquired knowledge) into the prior
knowledge structure gained through the accumulation of various experiences over time (a
priori general knowledge) is the formal condition for knowledge construction in the course of
learning.
Kant described the schematism of our understanding (i.e., applying our schema to
appearances and their form) as “an art concealed in the depths of the human soul, whose real
modes of activity nature is hardly likely ever to allow us to discover, and to have open to our
gaze” (A141). While this particular statement suggests that Kant himself might not have yet
been quite clear about the underlying mechanism of the schematism of our understanding, it
also seems to be quite likely that the increasing complexity of the human thinking system can
be hardly defined with enough clarity or certainty. Philosophers have acknowledged Kant’s
work on schematism in epistemology. Heidegger (1809/1971), for instance, commented it as
the first explicit systematic consideration of the essence of reason.
23
The organizational nature of human learning in Kant’s theory of schematism described
above, appears to some extent comparable to Piaget’s (1976) epistemology and the basic
cognitive functions of the equilibration process is explained in the following section.
2.1.6 Piaget’s Process of Equilibration
As previously stated, Piaget conceived of humans as actively interacting with their
environment by means of cognitive structures. Piaget (1947) conceptualized that human
beings interact with their surroundings rather than simply behaving in terms of stimuli and
responses. Hence, they gradually form various meaningful concepts. During these
interactions, they internalize their actions and build up a schema of the actions they perform
in their environment to forming an “equilibrium” (i.e., balance) between their actions with the
environment. The state in which the balance between one’s cognitive structure and a stimulus
in reality is broken is referred to as “disequilibrium.” Piaget defined this as “operations,” the
most general activities that can be applied to any objects of human cognition internally or
externally. Piaget viewed the cognitive operations as “acts of organization and adaptation to
the perceived environment” (as cited in Wadsworth, 1979, p. 9). Through the “operations,”
humans keep changing and transforming their cognitive structures (i.e., internal network),
and therefore, human cognitive structures become more stable as they apply such
“operations” (Piaget, 1947).
Piaget (1952) thus stated that human beings apply their existing schemas (i.e.,
organizational properties of intelligence that adapt and change with mental development) to
their environment through a process of “equilibration,” an internal self-regulating system.
The equilibration involves assimilation and accommodation: 1) in assimilation, new
information is assimilated into existing schemas, and hence, results in the growth of schemas;
2) in accommodation, schemas qualitatively change through (a) the modification of an
existing schema when a stimulus cannot be assimilated into the existing schema or (b) the
creation of a new schema when there are no existing schemas into which a stimulus can fit.
Piaget (1976) proposed that a cognitive structure can adapt itself to its environment and
thus reach the state of equilibrium between its activity and its environment and vice versa.
Therefore, the equilibration process is not separated but rather reciprocal between the activity
of the cognitive structure and environment. The equilibrium is accomplished through the
process of equilibration by incorporating the external environment into its already existing
structure (assimilation) or by modifying the existing structure to fit the external environment
(accommodation). In other words, whenever assimilation occurs, accommodation also occurs
24
because changes in internal components of knowledge (assimilation) lead to a modification of
structure (accommodation) that makes the structure fit with its components. Therefore,
assimilation and accommodation are not separated but rather reciprocal cognitive processes.
Authors (e.g., Pascual-Leone & Goodman, 1979; Seel et al., 2009) retained Piaget’s view of
the human cognitive process as a highly dynamic and self-reflective system which passes
through stages of equilibrium and disequilibrium in the course of cognitive development.
Seel and colleagues (2009) described the cognitive functions of assimilation and
accommodation as follows. In assimilation, new information is integrated into existing
knowledge structures and retained in the existing intellectual structures through the activation
of existing schema. If assimilation into existing schema is not successful to meet the
requirements of a new task, the schema should be modified in the process of accretion or
tuning, or a new schema corresponding to the requirements of the new task should be
constructed by constructing a mental model in order to reorganize and structure individual’s
knowledge (see Figure 2.1).
Figure 2.1: Cognitive functions of assimilation and accommodation
(From Seel, Ifenthaler, & Pirnay-Dummer, 2009, P. 18)
The equilibration described above suggests that learners progressively realize, understand,
and interpret the world by assimilating and accommodating (i.e., accreting, tuning,
(re-)organizing, and (re-)structuring) their cognitive structures through cumulative interaction
with the environment. This in turn reflects the cumulative as well as structural nature inherent
to human learning. In line with this view, the following section briefly reviews the
cumulative aspect of learning in the field of machine learning.
2.1.7 Cumulative Learning in Machine Learning
The idea of cumulative learning is not only discussed in cognitive psychology and
cognitive theories of learning but also in the field of machine learning, where the
25
accumulative aspect of learning is represented as incremental or layered learning which
presupposes the comparison of information and puts it in a framework for use with future
processes or problem solving tasks. Information is compared and put into a framework to be
used for future processes. In both human and machine learning, incremental or sequential
learning of this kind is considered to be an effective cognitive capacity in acquiring
knowledge and skills that are conducive to intelligent behavior and in producing new
foundations for further cognitive development.
Minsky (1975) conceived of a frame as “a data-structure for representing a stereotyped
situation” (p. 212) and frame knowledge as fixed generic information which fills slots that can
be substituted by certain values to cope with reality when details change. This frame
knowledge interacts with new information of the world by comparing information, and the
collections of related frames are linked together into frame-systems to be used for upcoming
processes or problem solving tasks. Thus, Minsky claimed that the units of the frame
knowledge should be larger, more complex, and more structured, which means that their
factual and procedural content must be more intimately connected in order to keep up with the
power and speed of mental activities in a serial process and enable a rapid selection of larger
sub-structures.
Pfeffer (2000) defined a cumulative learning agent as one that learns and reasons as it
interacts with the world using its accumulated knowledge and its observations. Michalski
(1994) viewed learning as “a goal-guided process of modifying the learner’s knowledge by
exploring the learner’s experience” (p. 3) in his Inferential Theory of Learning (ITL).
According to the ITL, the learning process consists of the input facts, the background
knowledge, and the types of inferences (i.e., induction, deduction, and analogy) a learner
makes to generate new knowledge. Thus, the changes in the knowledge content, its
organization, and its certainty are all seen as bringing about a total change in the learner’s
knowledge in the course of learning. Successfully learned knowledge is assimilated into the
learner’s background knowledge and can be used in subsequent learning processes.
Zhou (1990) introduced the CSM (classifier system with memory), an extension of the
classifier system model that includes mechanisms for analogical and cumulative learning, and
tested it in the domains of robot navigation and letter extrapolation. The CSM was designed
in response to the problems of conventional expert systems, that is, to update it to any
substantial extent (i.e., adding and removing knowledge), or to function intelligently beyond
their current knowledge.
The CSM can preserve problem solving expertise, recall similar solutions by searching
26
its long-term memory, construct solutions to similar new situations using analogy (i.e.,
recognizing the similarities between two problems), and adapt them to fit new situations.
Rules created by information exchange are stored in a temporary knowledge base. When a set
of detectors relays external information to the system, eligible rules may be triggered, which
in turn generate new messages leading to the performance of the action. The system’s
behavior can be changed through the deletion, modification, and creation of rules (i.e.,
tuning). While the short-term memory (STM) stores previously accumulated active
knowledge, valuable inactive information is also maintained separately in the long-term
memory (LTM), thus preventing rapid forgetting over time and preserving the information
for future use. In other words, when the system has accumulated sufficient knowledge, it
categorizes and generalizes a set of successful task-independent rules by extracting the
common features from a set of relevant rules and then transfers them from STM to LTM,
where it stores them as chunked building blocks, organized and indexed hierarchically (from
specific to general) for future problem solving situations. Zhou (1990) states that “with the
benefit of the prior experience and accumulation of problem solving expertise, it [the CSM]
constructs its knowledge base incrementally through interaction with its environment and
improves its problem solving ability over time” (p. 404). This is consistent with the principle
of cumulative learning in that new information is cumulatively incorporated into existing
knowledge structures, which then produces new foundations for further learning.
In terms of cognitive processes, Swarup, Lakkaraju, Ray, and Gasser (2006) outlined an
extracting ontology from the process of cumulative learning to solve future related problems:
An agent perceives and formulates (or models) concepts from its environment and generates
an appropriate decision by aggregating different pieces of them in order to construct a
solution. That is, each agent aggregates and accumulates a packet of knowledge that is
extracted from solutions to multiple tasks, and these packets impact new learning tasks
through analogy (i.e., the agent recognizes and applies similarities between the tasks). Thus,
the agents can guide each other’s learning process by grounding symbols of the aggregated
cumulative knowledge, thereby improving learning performance. As the agent aggregates
experiences and builds up its cumulative knowledge to find solutions to new problems, it is
expected that more cumulative knowledge will be found (i.e., reinforcement learning). This
illustrates the point that aggregation is a precondition for cumulative learning.
Easterlin (1986) identified aggregation in which important instances of experiences are
grouped into a set of aggregates when he explains the three components that lead to the
formation of concepts containing functional information in machine learning: aggregation,
27
characterization, and utilization: (a) In aggregation, experiences are aggregated by the
learning system itself for further use based on their contribution to a successful problem
solution and to system performance, (b) in characterization, a description of the essential
information for an aggregate of experiences is generated (or constructed) in terms of
characteristics that are useful to the system based on individual descriptions of each member
of the aggregate, and (c) in utilization, the concept description is integrated with the
performance element of the system and the important aspects of the aggregate are captured.
This idea proposes that aggregated experiences, instances, and entities should be
characterized and utilized structurally to facilitate the cumulative learning.
The phenomenon of structuring information in machine learning is also supported by
Murphy and Medin (1985). They argued that a concept is hierarchically related with other
concepts as well as structurally related, thus enabling inferences. Accordingly, to ensure an
effective and efficient cumulative learning, each aggregated concept should be constructively
related in the complex knowledge network in order for transfer to be successful in future
(learning) situations where this knowledge may be relevant. The packet of collected
knowledge should be stored as a meaningful composition which is flexibly linked into a
knowledge network. That is, the piece of knowledge should be appropriately subsumed under
a relevant and more inclusive category of the knowledge network, so that it can be flexibly
assimilated, modified, and restructured into the knowledge network; otherwise the aggregates
would be loosely placed somewhere in the space as a packet of unrelated pieces, and hence
waste limited memory space. This will make it less likely that the knowledge can be used
again in future learning situations.
The question that should be posed at this point is “how the aggregated knowledge is
structurally organized?” In Pfeffer’s (2000) Integrated Bayesian Agent Language (IBAL), a
learning agent can modify its models based on its collected observations and use them in
future situations. Pfeffer stressed that “a representation language must be modular and
extensible” (Pfeffer, 2000, p. 52) so that the knowledge base can be structurally extended and
accumulated. Swarup et al. (2006) suggested starting with small problems with very few
easy-to-find solutions to find networks which have subgraphs that can be reused to solve
other problems. This seems to suggest that schematizing the aggregated knowledge into a
meaningful module that can be extended, modified, and dynamically updated is an
appropriate means for ensuring the rich process of aggregation for successful cumulative
learning. In sum, the preceding review seems to suggest the cumulative as well as the
structural mechanisms in machine learning.
28
2.1.8 Mechanisms in Cumulative Learning
The preceding reviews of literature seem to suggest the two fundamental mechanisms
inherent to human learning: the cumulative as well as the structural nature: Humans
accumulate and structure their knowledge into their knowledge system, and thereby develop
their knowledge and skills. Hence, this section investigates 1) how humans accumulate
knowledge, and 2) how humans (re-)structure their knowledge in the course of learning. At
the outset, “learning” in this study refers to the process of knowledge construction as
described by Resnick’s (1989).
2.1.8.1 Learning as an Accumulation of Knowledge
As stated in the preceding section, in both human and machine learning, accumulation
of knowledge is considered to be an effective cognitive capacity in acquiring knowledge and
skills that are conducive to intelligent behavior and in producing new foundations for further
cognitive development. Theories of cognitive learning also include the assumption that
intelligent systems, human or artificial, fundamentally accumulate knowledge and abilities
that serve as building blocks for subsequent cognitive development (Shuell, 1986).
However, it seems that learning is not all about mere accumulation of knowledge. Among the
various aspects of learning, the authors agree on the point that accumulation and structuring
of knowledge are ways to ensure learning. For instance, Gagné (1962, 1968, 1970, 1977)
viewed the process of learning as an accumulation of increasingly complex interacting
structures of learned capabilities. These structures are then capable of interacting with each
other in patterns of great complexity, thereby cumulatively generating an increasing
competency level. Thus, he stated that “the learning of any new capability builds upon prior
learning” (Gagné, 1968, p. 4). This argumentation corresponds with Piaget’s (1976)
epistemology that considers learning as accumulating knowledge in a manner that is orderly,
sequential, integrative, and hierarchical. However, while Gagné views external organization
as a necessary condition for optimizing learning, Piaget is of the view that experience is
organized within the context of the internal process of the organism (i.e., assimilation and
accommodation) and that this organism gives structure to the experiences.
Rumelhart and Norman (1976) also argued that learning is not just a matter of
accumulating knowledge but often of giving it a new structure. Norman’s (1980) view on
learning may be summed up in his own words:
There has been remarkably little study of learning ― real learning, the learning of
complex topics, the learning that takes months, even years to accomplish. What goes
29
on during that time? Whatever it is, it is slow, continuous [...] just a lot of slow,
continual exposure to the topic, probably accompanied by several bouts of
restructuring of the underlying mental representations, reconceptualizations of the
concepts, plus many hours of accumulation of large quantities of facts (Norman,1980,
p. 20).
Furthermore, Norman (1982) suggested “five thousand hours” as a reasonable minimum for
learning a complex task (e.g., language, psychology, programming, chess, mathematics,
sports, and performing arts). He calculated that “five thousand hours is the equivalent of two
and a half years of study, eight hours a day, five days a week, and 50 weeks a year” (Norman
1982, p. 1) based on the observations of experts as well as his own experiences. For
intellectual activities (e.g., solving a problem), in particular, he pointed out that acquiring
interrelationships among ideas is likely to be difficult to achieve as it demands time and
mental effort.
Many authors (e.g., Gagné, 1979; Ifenthaler, Lee, & Seel, 2009; Seel, 2003; Siegler,
1996) also pointed out that knowledge usually changes gradually. For instance, Gagné and
Briggs (1979) argued that the learning of cognitive strategies which can lead a learner to
become more efficient in self-management cannot be substantially affected by instruction
over short periods of time but that such capabilities improve over relatively long periods of
time (months and years).
If one accepts the above arguments, then one would agree on the point that acquiring
complex forms of knowledge is a slow and continuous process which requires change in
cognitive structures (e.g., reorganization and restructuring) and not only the mere stacking of
knowledge. The following section investigates the underlying mechanism involved with the
change in cognitive structures.
2.1.8.2 Learning as a Change in Cognitive Structures
In the various fields which are concerned with the study of learning there is a general
agreement that learning is defined as change of behavior, knowledge, and skills (e.g., Dole &
Sinatra, 1998; Lewin, 1982). For instance, Langley and Simon (1981) defined learning as
“any process that modifies a system so as to improve, more or less irreversibly, its subsequent
performance of the same task or of tasks drawn from the same population” (p. 367). Further,
Langley (1995) conceived that learning consists of “the improvement of performance in some
environment through the acquisition of knowledge resulting from experience in the
environment” (p. 1). It begs these questions: How do humans change their knowledge or
skills? What are the underlying mechanisms involved in this change?
30
Historically, the question of how and under what conditions humans change their
knowledge has been addressed by philosophers (e.g., Kant’s theory of schematism stated in
section 2.1.5) for a long time. Understanding the processes and conditions of change in
knowledge has also become a major concern in the field of cognitive psychology as it
pertains to “learning.” As stated previously, Piaget (1976), for instance, viewed learning as
the individual’s construction and modification of cognitive structures in order to deal
successfully with the world. This conception implies that humans change knowledge in the
process of changing their cognitive structures. Likewise, research on cognitive restructuring
is categorized under the heading of conceptual change (Lee & Seel, 2011), which implies
change in knowledge, because this always presupposes a change in cognitive structures.
Many authors define the concept of “conceptual change” in their own terms, and thus
differentiated between various degrees of change in cognitive structures (e.g., weak change
and radical change). For instance, Lewin (1982), one of the most prominent Gestalt
psychologists4, described learning as a change in cognitive structures in his field theory. He
argued that an individual’s knowledge can be changed in any environment of his life space,
“including his psychological future, psychological present, or psychological past” (Lewin,
1982, p. 181). This implies that knowledge gained in any different sequence cumulatively
interact with each other.
Lewin (1982) thus identified the following three forms of a change in cognitive
structures: (a) differentiation of unstructured areas, (b) restructuring, and (c) change in
temporal perspective, psychological reality, and irreality. This means that an initially vague
and unstructured range becomes cognitively structured and specific in the process of
differentiating unstructured areas. This cognitive structure is then extended and refined
through assimilation, or through accommodation. Lewin argued that a person’s psychological
world as a whole becomes more specific and increasingly sophisticated: For instance, a
person who is in a new city learns his way around gradually by way of cognitive organization
and structuring. Lewin’s view of the function of structural orientation in the change of
knowledge corresponds to the Gestalt psychologists’ (e.g., Duncker, Köhler, Koffka,
Wertheimer) conception of learning through insight, in which learning is seen as an
organization process in which perceived objects change their function and are compared with
one another until the whole organization of the situational field is apparent to the learner (as
cited in Seel, 2003, 2011a). Both Lewin and the Gestalt psychologists’ conceptions seem to
4 Wertheimer is generally considered the founder of Gestalt psychology.
31
suggest the cumulative and structural nature inherent to human learning.
Similarly, as mentioned in the previous section 2.1.2, Bruner (1966) also argued that
learning is an active process in which learners construct new ideas or concepts on the basis of
their existing knowledge. This implies that, in any given task, learners actively search, select
and transform information, construct hypotheses, and make decisions based on their cognitive
structures. Bruner thus argued that these cognitive structures help learners to actively
generate meaning for real-world experiences and thus allow the learners to process
information in a meaningful way. This implies that learners actively process new information
not as a locally segregated one but as the part of a whole comprehensive cognitive structure.
Therefore, Bruner’s argument also seems to suggest the structural nature inherent to human
learning.
Carey (1992) also described the following three forms of conceptual change in the
course of knowledge acquisition in accordance with the assumption that human knowledge is
organized by concepts: (a) The concept, which is originally regarded as the core property of
an entity, takes on even more fundamental properties, (b) concepts are subsumed or
reassigned to newly created categories, and (c) concepts are embedded in locally
incommensurable theories (e.g., the concepts of the phlogiston and the oxygen theories of
burning; Kuhn, 1982). Furthermore, Carey (1992) differentiated knowledge acquisition
involving all three forms of conceptual change from that involving only enrichment: In the
latter form of knowledge acquisition, the newly acquired knowledge about entities leads a
person to gain new beliefs, which then influences him/her in choosing entities in the world,
thus providing structure to the known properties of the entities. For example, the newly
acquired belief that “unsupported objects fall” (Spelke, 1991) influences a child’s decisions
about object boundaries. This argument seems to suggest the cumulative nature of learning in
that the prior knowledge affects the present decision. It also seems to suggest that the
structural nature of learning in that the present decision, which is accomplished by the
interaction between the new knowledge and existing knowledge, forms one’s cognitive
structure.
The change of cognitive structures is also a major concern for information processing
theorists. Scholars (e.g., Anderson & Pearson, 1984; Rumelhart & Norman, 1976) assume
mechanisms for change in the structure of knowledge in their investigations of the acquisition
of new schemas and changes in existing schemas. For instance, on the basis of
schema-theoretical explanation approaches, Rumelhart and Norman (1976, p. 7) argued that
schemas can be changed through accretion, tuning, and restructuring: The learner assimilates
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new information into existing schema without making any changes to the overall schema
(accretion), modifies an existing schema when it is inadequate for the new knowledge
(tuning), and creates a new schema and reorganizes knowledge to bridge the gap between the
old schema and the newly acquired information (restructuring).
Seel (2003) also argued that learners create cognitive structures among the numerous
relations between various elements of knowledge by assimilating new information into
existing knowledge structures, thus modifying it continually. The degree to which they
change thus depends on the degree of correspondence between new experiences or thought
products and existing knowledge structures.
With regard to the terminology for describing change in cognitive structures, a
distinction is made between reorganization, restructuring, and reconstruction. Seel (2003)
explained how objects are reorganized with mental model (i.e., subjective cognitive
constructions) theoretical explanation approaches: Mental models help a learner to organize
his/her experience or thinking in order to construct the representation of his/her knowledge in
a systematic way. Upon receiving new information in a given situation, the learner searches
his/her retrievable knowledge to identify similar situations in the past that can offer plausible
solutions to the current situation; s/he then constructs a unique mental representation of an
object of the phenomenon in the situation and progressively links these mental objects to
components in the current phenomenon until the mental models fully “maps” onto the
phenomena of the entire situation. In doing this, the learner examines all of his/her available
cognitive resources (e.g., retrievable knowledge, phenomena) and reorganizes the object
representation. This enables the learner to better understand the phenomenon and to
progressively adapt the object representation to the situation. Seel (2003) viewed this process
of “reorganizing” the representation of an object as being similar to the process of
“restructuring” that takes place in the conception of learning through insight in Gestalt
psychology, in which new structures are integrated into the learner’s field of perception.
With regard to restructuring, Duncker (1935), who based problem solving on insight,
viewed mental restructuring as moments of sudden understanding in which restructuring
leads to unexpected insight into the solution to the problem (Oerter, 1971). This idea is
similar to Archimedes’s “eureka moment” or “aha experience,” which Bühler (1922)
describes as follows: What Archimedes restructured was not his perception of the problem
but rather “the direction of his search for a solution within the domain of previous knowledge
of general solution possibilities” (Bühler, 1922, p. 163; as cited in Seel, 2003, p. 329).
Comparably, Seel (2003) distinguished between restructuring and reconstruction as follows:
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While the restructuring process leads to new knowledge in the sense of Archimedes’s eureka
moment, in the reconstruction process the learner changes objects deliberately and gives
them an alternative interpretation by focusing only on their most essential and salient features
(e.g., change difficult and abstract concepts into more clear and comprehensible concepts).
With regard to the degrees of change in cognitive structures, Vosniadou and Brewer
(1987) stated that a weak restructuring involves “the creation of new, high-order relations
between existing concepts,” (p. 62) while a radical restructuring involves a fundamental
change in schemas. Dole and Sinatra (1998) associated weak restructuring with assimilation
and radical changes in thinking with accommodation. Comparably, Seel (2003) stated that a
radical change of concepts only occurs when issues that previously belonged to a certain
semantic category have to be assigned to a completely different category than the previous
one, as in the case of the replacement of Newtonian physics by quantum physics or the theory
of relativity in the history of science. Another term for describing how a conceptual system is
replaced by another one is Thagard’s (1992) “conceptual revolution” which, however, is a
seldom event.
In conclusion, it seems clear that changes in knowledge occur progressively,
cumulatively, and structurally, since conceptual changes involve many steps of modification
led by an increasing differentiation as well as an interactive reconciliation of cognitive
structures.
In conclusion of section 2.1, it appears that the mechanisms in learning that can result
in the development of knowledge and skills are cumulative and structural in nature. The early
approaches of learning explicitly or implicitly assumed that learning is cumulative in nature
without offering any profound reflection on the aspect of cumulative learning. There is no
theory of cumulative learning defined by a scientific theoretical framework. Therefore,
section 2.2 endeavors to define the “contemporary” problems in cumulative learning. It first
defines the contemporary concept and framework of cumulative learning in an attempt to
construct a scientifically adequate theoretical framework of cumulative learning. Based on
the problems identified in the “contemporary cumulative learning,” section 2.3, research
questions are presented.
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2.2 Contemporary Problems in Cumulative Learning
This section attempts to define “contemporary” problems in cumulative learning. It
starts by defining the contemporary concept of cumulative learning which forms the
conceptual base of this study. It then defines the framework of the contemporary concept of
cumulative learning which includes: the structural sequences, the types of knowledge, the
fundamental cognitive processes, and the structuring processes. Thereafter, it investigates
concept learning in relation to cumulative learning.
2.2.1 The Contemporary Concept of Cumulative Learning
The contemporary concept of cumulative learning deals with the gradual development
of knowledge and skills over time. It assumes the two mechanisms inherent to human
learning – cumulative as well as structural nature: 1) With regard to the cumulative nature, it
assumes that learners actively process and interpret their environment in terms of their
cognitive structures and that the components in the cognitive structures are closely
interrelated; and 2) with regard to the structural nature inherent to human learning, it assumes
that in the course of learning, learners progressively perceive, understand, and interpret a
world by assimilating and accommodating (i.e., accreting, tuning, (re-)organizing, and
(re-)structuring) their cognitive structures in cumulative patterns. Therefore, the theoretical
assumption of contemporary cumulative learning posits that the learning process depends on
the available information, existing knowledge, and schematization process. The cumulative
integration of these three parts gradually constitutes learning. Consequently, the
contemporary concept of cumulative learning views that “cumulative learning” is the
precondition for the development of knowledge and skills over time. Hereafter, the
contemporary concept of cumulative learning is referred to as “cumulative learning” and is
differentiated from the “early approaches” of cumulative learning.
In more detail, 1) when learners acquire new information from their environment, they
try to assimilate the new information into their existing knowledge structures (schemas)
without making any changes to the overall structure of the schemas and try to retain it in their
schemas by activating a part of existing schemas that might be relevant to it through a process
of accretion. 2) If the learners realize that their existing schemas do not correspond to the
requirements of the new information, then they modify their existing schemas accordingly in
a process of tuning, or they create a new schema by constructing mental models to bridge the
gap between the old schemas and the newly acquired information. In the process of
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assimilation and accommodation, the learners examine all of their available cognitive
resources (e.g., retrievable knowledge, similar phenomena in the past) and reorganize the
components (i.e., information/knowledge) in the schemas and thus restructure the
sub-components (i.e., categories, sub-schemas) in the schemas until the newly acquired
information successfully “maps” onto relevant phenomena of all of their available cognitive
resources. This enables the learners to better understand the phenomenon and to
progressively adapt the new information to the schemas. These new cognitive structures then
cumulatively interact with further learning situations.
According to Seel’s (1991) theory of mental models, when the information being
processed cannot be assimilated into structures of previous knowledge due to unavailability
of appropriate schemas, a learner constructs mental models (i.e., an internal conceptual
representation). The mental models are constructed by learners on the basis of their world
knowledge in order to meet the specific requirements of situations and tasks. In an extension
of this argument, Seel (1991) describes the function of mental models in analogical
reasoning: A person makes propositions or predictions for a certain phenomenon (target
domain) by falling back on his/her knowledge about similar phenomena (base domain) and
creating a mental model for both the base and target domains. On the basis of the structural
similarities this person finds between the models of the base and target domains, s/he reaches
a conclusion by analogy, integrates both models into a unified solution model under the
assumption that they are similar, and tests whether it is possible to create an alternative
solution model. The mental models constructed for one situation can be used again as a basis
for making inferences and solving other problems in the future by modifying the models to a
certain degree to meet the conditions of a new situation.
If one accepts the above argumentations, the theoretical argumentation for cumulative
learning in this theoretical framework is fulfilled, wherein, the primary benefit of cumulative
learning is that it reconsolidates the knowledge one has obtained through various experiences
and allows it to be reproduced and exploited for further learning situations due to its
cumulative effect. A person’s cognitive structures keep changing and transforming as they
interact with new information/knowledge and build up more complex internal networks in the
course of learning, leading to the acquisition of knowledge and skills that gradually improve
over time. This can be accomplished in different ways. Throughout the course of learning, a
learner actively aggregates relevant units of information/knowledge. Based on the aggregated
information/knowledge, the learner’s cognitive structure can be refined or a new (form of)
cognitive structure can be constructed by constructing mental models through interaction
36
between the new information and existing knowledge using inferences (induction, deduction,
or analogy). The cognitive structure then becomes (re-)organized by schematization as the
learner continually links and categorizes each part of the information/knowledge in relation to
the whole cognitive structure as a system. As a result, the learner’s concrete and specific
knowledge progressively becomes abstract as it reaches a higher level on the hierarchy of
cognitive structure (see Figure 2.2).
Figure 2.2 Illustration of cumulative learning
Below is an explanation of Figure 2.2:
a) The vertical arrow represents the degree of abstractness of knowledge that can be
facilitated through cognitive processes. At higher levels the knowledge is more abstract,
while at lower levels it is more concrete.
b) The horizontal arrow represents the degree of generality of knowledge that can be
facilitated through cognitive processes. As a learner aggregates more relevant units of
information, the learner’s knowledge becomes more general. Therefore, at higher levels
the knowledge is more general, while at lower levels it is more specific. Information can
be aggregated at any level in the hierarchy of cognitive structure.
c) The spiral arrows represent the fact that information/knowledge cumulatively interacts
with existing knowledge structures as learning proceeds. Hence, knowledge gradually
develops as it reaches a higher level on the hierarchy of cognitive structure.
The concept of schematization in this theoretical framework of cumulative learning
refers to the process of organizing (or subsuming) information/knowledge into the relevant
part of the hierarchical cognitive structure by integrating aggregated information/knowledge
in a more abstract way. While doing this, a learner continuously assimilates, modifies,
(re-)organizes, and (re-)structures his/her cognitive structure by aggregating and abstracting
information/knowledge in order to understand new information. While (a) aggregation deals
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with the units of packets of knowledge acquired that are at the same level in the hierarchy of
cognitive structures, (b) abstraction5 deals with knowledge that are acquired at higher levels
in the hierarchy of cognitive structures. This allows the learners to integrate (or subsume)
newly gained information into previously acquired knowledge structure over time in a
cumulative manner. Therefore, in the course of learning, any part of previously learned
knowledge continually affect the interpretation and understanding of the content, relations,
and structures of present learning as the learner’s cognitive structure continually moves to
create a structured whole (see Figure 2.3). This process is thought to enhance the individual’s
long-term memory and thereby gradually develops and improves his/her knowledge and
skills over time.
Figure 2.3 Cumulative interactions between prior knowledge & present knowledge
Therefore, cumulative learning can also be expressed in symbolic terms as follows (see Table
2.1): S=number of learning sequence, a=prior learning, b=present learning, A=existing
knowledge, B=new knowledge.
S a (prior learning) A (existing knowledge) b B (new knowledge)
1 a1 A1 = {a1} b1 B1 = A1 + b1= {a1, b1}
2 a2 = a1 + b1 A2 = B1 = A1+b1 = {a1, b1} b2 B2 = A2+ b2 = {a1, b1, b2}
3 a3 = a2 + b2 = (a1+b1) + b2 A3 = B2 = A2+b2 = {a1, b1, b2} b3 B3 = A3 + b3 = {a1, b1, b2, b3}
n an = a3 +…+ bn-1 =
(a1+b1+b2...) + b(n-1)
An = B(n-1) = A(n-1) + b(n-1) =
{a1, b1, b2, …b(n-1)} bn
Bn = An + bn = {a1, b1, b2,
b3,...,bn}
Table 2.1 Sequences of cumulative learning
(S1): At the 1st sequence of learning, prior learning (a1) affects present learning (b1) as the
existing knowledge (A1) constructed from the prior learning (a1) actively interacts with
present learning (b1), thereby constructing new knowledge (B1). Technically, the new
knowledge (B1) is constructed from the interaction between two instances of learning (a1
5 The process of aggregation and abstraction is explained more in detail in the section 2.2.4.
38
& b1). In symbolic terms, this can be expressed as B1= {a1, b1}. This represents the
general mechanism of learning in any sequence of cumulative learning.
(S2): At the 2nd
sequence of learning, the new knowledge (B1) constructed from the 1st
learning sequence then becomes the existing knowledge (A2). This existing knowledge
(A2), which is constructed from the prior learning (a2=a1+b1), then actively interacts
with the present learning (b2), thereby constructing a new knowledge (B2). Technically,
this new knowledge (B2) is constructed from the interaction between the two instances of
learning (a2+b2). In symbolic terms, it can be expressed as B2= {a1, b1, b2}.
(Sn): This process repeats itself in cumulative patterns of ever increasing complexity and
competency in the course of learning. Hence, technically, in the nth
sequence of learning,
the present learning (bn) interacts with all of the previous instances of learning (b
1, b
2,
b3,…, b
n-1). Consequently, the components of the new knowledge in the n
th sequence of
learning encompass all previous learning up to the nth
sequence (i.e., Bn =An+bn = {a1,
b1, b2, b3,..., bn}).
As described in the preceding sections, the following three theoretical arguments
closely correspond to the concept of cumulative learning in this study. Firstly, Ausubel’s
(1962) theory of subsumption (see section 2.1.3), which proposes the subsumption of less
inclusive subconcepts into highly inclusive concepts into the hierarchical cognitive structure,
forms the conceptual base of the concept of cumulative learning in this study as it
characterizes learning as a process of cumulative subsumption in which new information is
continually integrated into existing knowledge structures. More precisely, the present study
views that cumulative learning sequences help learners to activate pre-existing knowledge
structures (i.e., schemas) and allow them to use their existing knowledge in the new learning
situation more effectively.
Accordingly, cumulative learning proposes that new information is learned and retained
to the extent that it can be related to existing knowledge structures. In this view, more
meaningfully structured knowledge can be better reserved for future use. It is thus assumed
that this cumulative integration may have “a certain vitality of its own” (Seel, 2003, p. 250)
in the cognitive structuring process as a whole. This is accomplished by strengthening the
power of the schematic links since they have a capacity for larger sub-structures, thereby
generating a certain degree of acceleration force, which would ultimately increase the speed
and the power of the whole schematization process, and hence resulting in an ever increasing
learning power.
39
Secondly, Piaget’s (1976) conception of learning (see section 2.1.6) states that humans
apply their existing schemas (i.e., organizational properties of intelligence that adapt and
change in the course of mental development) to their environment through the process of
equilibration, a basic process of development which contains both assimilation and
accommodation. Piaget also stated that equilibration occurs when people are in a state of
equilibrium, and then they experience a cognitive conflict as they realize the flaws in their
thinking. This then induces people to adopt a more plausible idea that resolves the cognitive
conflict and hence attain a more stable state of equilibrium.
Piaget’s notions correspond to the conception of cumulative learning in this study for it
characterizes that new information is cumulatively integrated into existing knowledge
structures by continually accreting, modifying, reorganizing, and (re-)structuring it in a more
abstract way through schematization in order to understand the new information (i.e., resolve
cognitive dissonance): 1) Accretion refers to the new information that is continually
incorporated into existing schemas, which corresponds to assimilation; 2) modification
refers to schemas change (i.e., modify existing schemas or create a new schema) when new
information cannot be incorporated into the schemas because there are no schemas into which
it fits, which corresponds to accommodation; and 3) (re-)organization and (re-)structuring
refers to (re-)organizing information/knowledge and thus (re-)structuring sub-components in
one’s cognitive structure into a more plausible way to create a structured whole one. This
process enables one to resolve “cognitive conflict” between one’s existing knowledge and
new information. This conception corresponds to Piaget’s process of equilibration.
Thirdly, the conception of cumulative learning in this study also corresponds to the
view from the early approaches of cumulative learning propagated by Gagné (1970, 1977,
1979, 2005), who conceptualized all learning as a function of prior learning, which implies
that a particular concept can only be learned if certain other concepts have already been
learned (see section 2.1.1). Gagné argued that every learning situation generates a hierarchy
in which prerequisite learning can be identified: Lower skills in the hierarchy serve as
prerequisite learning for higher skills, and this prerequisite must be mastered before learning
at the next level of the hierarchy can occur. However, while the concept of cumulative
learning described in this study concurs with Gagné’s view that all learning is a function of
prior learning; it does not concur with his belief that the external hierarchical sequencing of
learning experiences can ensure internal (mental) organization. Rather, it concurs with Piaget
(1976) on this point, who argues that learning lies within the individual and that external
organization cannot ensure internal organization.
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2.2.2 The Sequences of Cumulative Learning
This section defines the structural sequences involved in the theoretical framework of
cumulative learning. As stated above, the process of cumulative learning depends on the
available information, existing knowledge, and schematization process. The cumulative
integration of these three parts gradually constitutes learning. And the knowledge constructed
in this learning process can be stored and made available for further learning processes.
Therefore, cumulative learning depends upon activating pre-existing knowledge, integrating
newly gained information/knowledge into the pre-existing knowledge structure, and then
(re-)organizing and (re-)structuring a cognitive structure through the cumulative interaction
between pre-existing knowledge and new information. Such interaction and integration may
promote the activation of other related areas of knowledge within the domain, because
cumulative learning assumes that learners are disposed to associate various parts of
sub-components (e.g., concepts, categories, sub-schemas) in their cognitive structures, hence
expanding their super-schemas and learning more. This can be characterized as shown in
Figure 2.4.
Figure 2.4 Sequences of cumulative learning
Below is an explanation of the Figure 2.4:
1. The whole horizontal frame represents the stages of cognitive processes inherent to
cumulative learning. As learning proceeds, the level of knowledge moves to higher levels
on the hierarchical structure as the knowledge cumulatively interacts.
2. The vertical arrows represent the degree of abstractness/generality of knowledge that can
be facilitated through cognitive processes:
2.1 The arrow on the right side represents the degree of abstractness of knowledge that
can be facilitated through cognitive processes. The higher the level, the more
41
abstract the knowledge. Hence, the process of abstraction and concretion
transforms descriptions along the more-to-less detail/less-to-more detail direction.
2.2 The arrow on the right side represents the degree of generality of knowledge that
can be facilitated through cognitive processes. The lower the level, the more
concrete the knowledge. Hence, the process of generalization and specialization
transforms descriptions along the set-superset/set-subset direction in the hierarchy
of cognitive structure.
Based on the view stated above that learning process depends on the available information,
existing knowledge, and schematization process, the theoretical concept of cumulative
learning assumes that learning is activated when a learner aggregates units of information
obtained from his/her environment (i.e., external sources) and activates relevant parts of
his/her pre-existing knowledge (i.e., internal sources). The learner then progressively
integrates units of aggregated information into pre-existing knowledge structures. While
doing this, the learner (re-)organizes information/knowledge and thus (re-)structures
sub-components in his/her cognitive structure into a more plausible way to create a structured
whole one in a process of schematization.
In more detail, upon aggregating units of information/knowledge, a learner organizes
and categorizes learned information/knowledge into different categories. Each category (a
group of related concepts from single or multiple phenomena) consists of groups of concepts
(packs of related ideas that share one or more common properties) which are formed by
combination (unification or differentiation) of information obtained through several
experiences. These categories are then subsumed under sub-schemas, which are then
subsumed under a schema. The schema now consists of groups of more generalized concepts
at different levels of abstraction, and hence, the learner can correctly identify that a particular
entity should be categorized under a specific concept category without prior learning. This
can be accomplished because the learner can generate a larger description than the given
specific information by using inference (i.e., inductive, deductive, analogical) as the
sequences of learning proceed. In this process, a packet of aggregated units of (concrete and
specific) information/knowledge is progressively condensed into an abstract cognitive
structure as shown in the Figure 2.5.
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Figure 2.5 Sequence of condensation
Correspondingly, in the cumulative learning processes, it is assumed that most of the
general knowledge is preserved at the end of the continuum of the abstraction process. For
instance, in the process of constructing the schema of “string instrument,” learners perceive
the various sensual descriptions (e.g., size, shape) of an object of the string instrument (e.g.,
guitar, violin, harp) while they understand its functional properties (e.g., play strings on the
instrument). Ideally, a general (or generic) representation of the string instrument (a musical
instrument that produces sound by means of vibrating strings – general and abstract
knowledge – should be constructed by boiling away the sensual elements, so the contents of
the concept are no longer concrete. This independence from sensual features of an abstract
concept makes it the basis for high order cognitive processes. This shows the usefulness of
abstraction in learning. However, researchers have also suggested that in many situations
people use information about specific instances rather than abstract representations to process
information and that specific instances can also be useful for reasoning if the relevant
information in a situation can be found at the time of processing (Markman, 1999).
As learning proceeds, the integrated assimilatory schema becomes progressively
complex and profound that it can extend itself to interpret the phenomena of the world, because
it provides an insightful framework for interpreting new information which can direct
modifiable (or accommodative) investigation in further learning sequences. As these
experiences accumulate in the course of learning, the learner’s cognitive structure becomes
more enhanced and developed, because cumulative learning assumes that any learned
capability at any stage of learning process affects the subsequent learning situation. Hence,
the learner’s knowledge and skills progressively “change” over time.
2.2.3 Types of Knowledge in Cumulative Learning
This section describes the types of knowledge constructed in the course of cumulative
learning. It varies in terms of its abstractness/concreteness, and generality/specificity in the
hierarchical cognitive structure as below (see Figure 2.4 in section 2.2.2):
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a) Abstract knowledge reduces the amount of detail in a description of
information/knowledge in one’s cognitive structure. It may change the description
statement of the information/knowledge to one that uses more abstract concepts and
excludes details. Abstract knowledge can be used to induce less abstract
information/knowledge at a lower level on the cognitive structure. For instance, “all
guitars, violins, and harps are string instruments” can be abstracted into “all guitars,
violins, and harps are musical instruments.” Here, while the contents of the reference set
(all guitars, violins, and harps) have not changed, the level of detail in the descriptor has
been decreased: the recognition of a description of the string instruments in contrast to the
recognition of a description of the musical instruments.
b) The opposite of abstract knowledge is concrete knowledge, which generates more and
more additional details about the entities of information/knowledge in one’s cognitive
structure. For instance, “all guitars, violins, and harps are string instruments” is more
concrete than “all guitars, violins, and harps are musical instruments.” Here, while the
contents of the reference set (all guitars, violins, and harps) have not changed, the level of
detail in the descriptor has been increased: the recognition of a description of the musical
instruments in contrast to the recognition of a description of the string instruments.
When the representation of a certain category of concepts contains too many concrete
details, then it becomes more complicated to connect it with a new learning situation,
whereas when it contains more abstract properties, it can be more easily connected with a
new learning situation.
c) General knowledge extends the reference set of information/knowledge in one’s cognitive
structure. General knowledge includes larger reference entities of the information/
knowledge than the prior level of knowledge in terms of its generality through inference
(i.e., inductive, deductive, analogical). For instance, inferencing “all musical instruments
that produce sound by means of vibrating strings are string instruments” is more general
knowledge than inferencing “all guitars, violins, and harps are string instruments”: the
recognition of a general representation of the string instruments (a musical instrument that
produces sound by means of vibrating strings) in contrast to the recognition of a specific
feature (e.g., size or shape) or of a specific class of objects (e.g., guitar, violin, harp) of it.
Typically, general information/knowledge is built from more specific existing
information/knowledge through inference.
d) The opposite of general knowledge is specific knowledge, which includes specific (or
particular) entities of information/knowledge which are more context-bound. It narrows
44
the reference set of the information/knowledge in one’s cognitive structure. For instance,
inferencing “all guitars, violins, and harps are string instruments” is more specific
knowledge than inferencing “all musical instruments that produce sound by means of
vibrating strings are string instruments”: the recognition of a specific feature (e.g., size or
shape) or of a specific class of objects (e.g., guitar, violin, harp) of the string instrument
in contrast to the recognition of a general representation of it (a musical instrument that
produces sound by means of vibrating strings).
In the process of learning, a learner needs to acquire specific pieces of knowledge relevant to
that particular domain. To do this, the learner should first aggregate a packet of information
which is relevant to the learning task and assimilate it into his/her existing knowledge
structure. When the aggregated pieces of information conflict with each other, the learner can
use analogy to manage them in order to resolve the conflict by modifying (accommodating)
the cognitive structure. In contrast, a general knowledge, which is independent of the specific
topic of learning, should also be acquired: For example, the four stages of a general
problem-solving strategy in mathematics: analysis, planning, execution, and checking (Polya,
1957). A learner could also acquire some reasoning skills (i.e., general knowledge) such as
inferences to connect and structure the components of schemas through the structural mapping
process.
Cumulative learning assumes that the learner acquires these inferences through the
following cognitive processes: With the given information in the learning task, (a) the learner
activates his/her existing schema(s) relevant for the new information, (b) the learner
concretely and/or abstractly generalizes the given information, (c) the learner extracts
commonalities and tries to find causal relationships between the activated schema(s) and the
given information, and then (d) the learner (re-)organizes the information/knowledge and
(re-)structures the sub-components in the cognitive structure accordingly. In doing so, the
learner may acquire a general problem-solving strategy that can be used as topic-independent
knowledge (e.g., Polya’s four stages of general problem-solving strategy in mathematics).
Upon acquiring well-organized information processing structures (i.e., well-organized
schematic links of the information processing), the learner may more quickly activate and
retrieve situation-appropriate schemas in learning situations.
Furthermore, cumulative learning assumes that having abstract as well as simple
cognitive structures through systematic organization of knowledge increases the speed of
searching and activating relevant schemas in future learning situations resulting in efficient and
45
effective learning. For instance, back to the same example in learning the concept of “string
instrument,” the prototypical string instrument would be a guitar, violin or harp while a horn
or flute would be a non-example of it. A learner needs not have seen a single instrument with
precisely this characteristic, as the “prototype” is an abstraction of crucial characteristics of
the instrument that the person has experienced. The learner might easily identify a guitar,
violin, or harp as a string instrument since these instruments externally vibrate the strings to
produce sounds.
However, due to their simple and condensed nature, abstract cognitive structures may not
successfully account for the “details” of elements’ properties in the more complex structure,
even if they still preserve information that is relevant to the learning situation. This can be
critical to learning situations, because there are details that can impact the ongoing learning
sequence due to their connection with the thinking process, and it may result in incorrect
learning. For instance, when the learner does not know that the concept of string instrument
does not distinguish vibrating strings of instrument between outside and inside, then the
learner might not identify a piano as a string instrument because it does not externally use a
string but a keyboard to vibrate the strings which are not externally visible. Since the abstract
knowledge of the concept of string instrument does not account for the detail (e.g., vibrating
strings either inside or outside of the instrument) this abstract knowledge may lead to some
errors concerning details in learning. Such lack of details would be more critical for complex
learning situations dealing with complicated and dissimilar subgroups of content than for less
complex learning situations dealing with compact and similar subgroups of content.
For instance, considering the fact that schemas are assumed to be stored in memory with
certain default values (Seel, 2011b), could result in a learner missing certain small but critical
details in some situations. This can happen because default values and schematic links stored in
the learner’s schema may exert a strong influence on his/her cognitive processes that s/he will
habitually use the default knowledge in interpreting and understanding learning situations and
hence produce errors in connecting existing knowledge to new situations. Nevertheless, in
general, the ability to understand situations easily and quickly seems to be useful in learning
due to its efficiency. Furthermore, in contrast to novices, experts are assumed to have acquired
highly sophisticated and thus automated schematization processes which may be able to extract
underlying details from abstract information as they perceive and understand situations,
because highly developed schematic links in their knowledge system correspond more quickly
(or automatically at the end of the continuum) and more carefully to detailed small parts of the
entire structure of the knowledge system.
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In terms of a thinking process, developing abstract and general knowledge requires a
thinking process that is more sophisticated than a simple assimilation or association of ideas.
Piaget (1976) conceives higher-level thinking as consisting of logical operations of the
processing, reorganization, and application of information at the abstract level. Knowledge is
developed to the higher-level by perceiving reality and then “internalizing” (Vygotsky, 1978)
this perceived reality in thinking. Internalization process can be conceived:
[…] in a broad sense as a systematically organized multidimensional transformation of
human action and also, in a narrow sense, when one is considering only one line of this
transformation, namely the change in the form of the developing action − from external
(material, materialized) via verbal to internal (mental, ideal) (Podolskiy 2011,
1630-1631).
This idea of internalization corresponds to Kant’s (1781, 1787/1933) schema (i.e., innate
structures which organize the world, which is always produced by imagination) and
schematism (innate mental ability of structural organization), which is stated in section 2.1.5.
Kant interpreted the schema of the concept as the representation of a general procedure of
imagination which provides an image for a concept. Hence, it is not images of objects but the
schemas which underlie human pure sensible concepts, because no image could ever be
perfectly adequate to the concept of an object in general (B180/A141). Kant thus
conceptualized a “system” as the consolidation of pieces of perception under a single united
idea (A832/B860). This idea seems to correspond to the process of internalization (Podolskiy,
2011), a systematically organized multidimensional transformation of human action from
external to internal.
If one accepts the arguments regarding internalization and schematism, then one would
agree on the point that developing knowledge depends on how a learner structurally processes
his/her knowledge, for this seems as an innate capability of human learning. The following
section defines the fundamental cognitive processes inherent to cumulative learning.
2.2.4 Cognitive Processes in Cumulative Learning
Cumulative learning assumes that the fundamental cognitive processes inherent to
cumulative learning can be broadly represented as the function of the interaction between
aggregation and abstraction. In terms of the level of knowledge in the hierarchy of cognitive
structure, (a) aggregation deals with the units of packets of knowledge acquired that are at the
same level in the structure, while (b) abstraction deals with knowledge that are acquired at a
higher level in the hierarchy of cognitive structure. It can be expressed in symbolic terms
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with C=cumulative learning, Ag = aggregation, Ab = abstraction:
In addition to aggregation and abstraction, cumulative learning also views generalization as a
critical cognitive process in learning, because generalization makes a learner to transform the
descriptions of information/knowledge along the set-superset direction in one’s cognitive
structure. Cumulative learning views that generalization is closely related to abstraction, and
thus, schematizations of these two processes often occur simultaneously in learning situation.
Therefore, this section defines the three cognitive processes inherent to cumulative learning,
namely, aggregation, abstraction, and generalization.
2.2.4.1 Aggregation
As stated previously, learning first requires an aggregation of information and
knowledge. Given a learning goal, a learner aggregates relevant units (or pieces) of
information/knowledge by identifying and extracting them in terms of elements, relations,
and functions in relation to the learning goal using analogies as learning proceeds. If the
aggregated knowledge satisfies the learning goal, then it is assimilated into the existing
knowledge structure so that it can be used in subsequent learning processes. Accordingly, in
the theoretical framework of cumulative learning, the process of aggregation is assumed to
consist of the following processes: With the given information in a learning task, a learner 1)
aggregates relevant units of information/knowledge from existing knowledge by identifying
their relations and functions of the given information, 2) identifies and extracts perceptual
and superficial surface similarities between the aggregated information/knowledge and
existing schemas, and 3) progressively schematizes the extracted information/knowledge into
coherent cognitive structures (e.g., category, sub-schemas) by identifying the structural
relations between them, and hence construct schemas. The schemas then becomes useful
resources in further learning situations.
In terms of the level of knowledge, in aggregation process, the aggregated units of
information/knowledge are at the same level in the hierarchy of cognitive structure as
learners are yet to define the shared structural features (i.e., structural similarities) of the
aggregated packets and cognitive structures in this process. For instance, in learning the
concept of string instrument, the learner first aggregates the information about each
instrument focusing on the perceptual as well as superficial features but s/he is not yet able to
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define the shared structural features (i.e., a structural schematic link: a musical instrument
that produces sound by means of vibrating strings) of the two instruments (e.g., guitar, violin,
or harp vs. piano). defines a structural schematic link (a super-concept and a sub-concept) of
information/knowledge As learning proceeds, this aggregated information becomes a useful
resource when finding out the structural similarities of the two instruments (guitar, violin, or
harp vs. piano), which results in constructing knowledge at a higher level in the hierarchy of
cognitive structure6, and learning about other musical instruments such as a keyed instrument
or wind instrument, and so on. The knowledge constructed in this learning sequence impacts
the next learning sequence, and this happens in a cumulative manner. This allows the learner
to progressively integrate newly gained information/knowledge into existing knowledge
structure over time, and thereby develop knowledge and skills.
2.2.4.2 Abstraction
As previously stated, learners can construct a higher level of knowledge in the
hierarchy of cognitive structure with an abstraction process. Strittmatter (1990) pointed out
that learning can be considered as an “inductive process,” that is, abstractions are made on
the basis of perceived similarities. This implies that a learner abstracts knowledge through
thinking, analogical reasoning, and organizing cognitive structure. In the theoretical
framework of cumulative learning, abstraction is a structural mapping process of information
and knowledge. It involves classifying various bits of information into a particular category
(i.e., categorization), reducing the amount of information and hence constructs simpler
schematic links in one’s cognitive structure in a way that the information/knowledge relevant
to the learning goal is extracted, while the irrelevant information/knowledge is ignored. The
opposite cognitive process to abstraction is concretion, which generates additional details of
information in one’s cognitive structure.
Learners perceive and process information at different levels of abstractions, and
therefore, as cognitive structures develop, the level of abstraction becomes higher. At lower
levels of abstraction, the components of information/knowledge are connected to the
schematic links of a schema relatively loosely, and therefore they can be replaced relatively
flexibly or modified by new information to better correspond to the learning situation.
In the theoretical framework of cumulative learning, the process of abstraction is
assumed to consist of the following processes: With the aggregated information/knowledge
relevant to learning task, a learner 1) extracts commonalities from the underlying structure as
6 This is assumed to be accomplished in the process of abstraction, which is described in section 2.2.4.2.
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well as the superficial features of information/knowledge, 2) defines a structural schematic
link (a super-concept and a sub-concept) of information/knowledge by integrating the
extracted information/knowledge into existing schemas through assimilation and
accommodation of the existing schemas. While doing this, the learner identifies
deep-structural similarities between them. When the existing schemas cannot be matched to
process the new information, the learner constructs new mental models by analogy and
modifies schemas. These new schemas can then be applied to further learning situations.
As stated in the previous section, this abstraction process presupposes a structural
reorganization of knowledge. Dörner (1982) has described this reorganization of learning as
the “condensation” of an abstract schema by means of a conclusion by analogy. The
condensation involves a search for a known reality domain (as the base of an analogy) which
is in certain respects analogous to the unknown target domain. It is in essence a process in
which “the concrete aspects of the known stock of knowledge [...] are in a sense ‘boiled
away’ and the remaining ‘pure’ structure [...] is filled with the concrete aspects of the
unknown domain” (Dörner, 1982, p. 140). Dörner (1976, p. 82) further argued that this
process of abstraction presupposes the existence of a comprehensive system of abstract
concepts which allows transitions from one concrete aspect to another. He thus defined the
following steps of the transitions: (a) abstraction of certain attributes of the phenomenon in
question, (b) the search for a model (for a second phenomenon) which constitutes
concretization of the abstract phenomenon, (c) transfer of (structural) attributes of the model
back to the original phenomenon, (d) a test as to whether the hypothesized attributes are
actually present in the phenomenon.
If one accepts these argumentations stated by Dörner, then the theoretical
argumentation for cumulative learning in this theoretical framework – learners actively
process and interpret their environment in terms of their cognitive structures and that they
progressively assimilate, modify, (re-)organize, and (re-)structure them by aggregation and
abstraction in cumulative patterns – is fulfilled. In other words, learners continuously
concretize and abstract (or condense) synthetic cognitive structures by assimilating,
modifying, (re-)organizing, and (re-)structuring them through aggregation and abstraction of
information/knowledge and this is done in a cumulative manner. While concretizing the
cognitive structures, the learners can verify whether their abstract knowledge is actually
applied in the specific situation which constitutes concrete phenomena or not. They can also
acquire ever increasing abstract knowledge by continuously condensing concrete phenomena.
Practically, when learning goals are defined, a plausible sequence of learning would be
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(a) to abstract attributes of the given phenomenon of the learning task, (b) to search for a
schema which is relevant to the abstract attributes (i.e., aggregation process), (c) to transfer
the (structural) attributes of the schema back to the given phenomenon, and then (d) to test
whether the hypothesized attributes in the schema are actually applied in the given learning
phenomenon which holds concrete attributes.
2.2.4.3 Generalization
Generalization refers to the cognitive process that extends the size of the description of
information/knowledge and transforms descriptions along the set-superset in one’s cognitive
structure. It can be accomplished because a learner can generate a larger description than the
given specific information by using inference (i.e., inductive, deductive, analogical). The
opposite of generalization is specialization of knowledge, which reduces the size of the
description of given information and it transforms descriptions along the set-subset in one’s
cognitive structure. The schematizations of abstraction and concretion, which explained in
the section 2.2.4.2 is closely related to this generalization and specialization. Thus,
schematizations of these two types (generalization and abstraction) often occur
simultaneously in learning situation, which is explained below.
In concept learning, with generalization, a learner can correctly identify that a particular
entity should be categorized under a specific concept category without prior learning as
follows: The prototypical string instrument (a musical instrument that produces sound by
means of vibrating strings) would be a guitar, violin or harp while a horn or flute would be a
non-example of it. A learner need not have seen a single instrument with precisely this
characteristic because the prototype is an abstraction of crucial characteristics of the
instrument that the person has experienced. It should be noted that the prototypes can be a
useful reference “only if the important elements of a situation are abstracted away”
(Markman, 1999, p. 219). The learner might not identify a piano as a string instrument
because it uses attached keyboards to vibrate the strings instead of directly manipulating the
strings. The learner might also not know that the sound of a piano is produced by strings
inside the piano (i.e., under-generalization): This instrument (piano) differs in size, shape,
and playing technique (either plucking or bowing the strings attached to the instruments)
from more prototypical instruments like harps, violins and guitars. On the contrary, the
learner might mistakenly identify a keyboard as a string instrument due to its physical
similarity to the piano (i.e., over-generalization). While the sound of the piano is produced by
vibrating strings inside the piano (hence, string instrument), the keyboard’s sound is not
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produced by vibrating strings (hence, non-string instrument). Here, the characteristic of
“sound is produced by vibrating strings” is the important element of a learning situation that
should be abstracted according to Markman (1999).
To learn a “string instrument” concept, a learner may generate a set of attributes to
characterize the entities for the concept of it. Thus, the learner may construct some different
examples in terms of these attributes (e.g., violin, harp, guitar, piano, etc.). Then general
components of these examples are constructed through a process of generalization (e.g.,
string, sound, and vibration). Thus, by repeating different types of a generalization (e.g.,
over-generalization or under-generalization), the learner can progressively differentiate a set
of different attributes (size, shape, playing technique, the mechanism of producing sound)
that characterize the concept of string instrument. When these processes are repeated, the
units of information/knowledge that belong to the schema of musical instruments become
increasingly routinized and automated as the learner consistently associates the new
information with his/her existing knowledge through extensive practice. Consequently, the
learner progressively develops his/her performance from slow, conscious, and difficult to
more rapid, accurate, unconscious, and effortless automation.
As stated above, in terms of the level of knowledge in the hierarchy of cognitive
structure, generalization refers to the size of the description of information/knowledge that is
extended and it transforms descriptions along the set-superset in the cognitive structures of
individuals. For instance (take the example stated in section 2.2.3), “all guitars, violins, and
harps are musical instruments” is more abstract than “all guitars, violins, and harps are string
instruments.” Another example, “all piano, clavier, and organ are musical instruments” is
more abstract than “all piano, clavier, and organ are keyed instruments.” Here, the levels of
detail in the descriptors have been increased: the recognition of a description of the musical
instruments in contrast to the recognition of a description of the string and keyed instruments.
A general “musical instrument” concept of these examples can be created in a process of
generalization. By repeating different types of generalization, learners can generate a set of
alternative general concepts of these examples. They may then use this particular schematic
instance in a subsequent learning situation.
Ausubel and Robinson’s (1969) understanding of knowledge was as a hierarchical
structure organized like pyramids, in which the most general ideas are at the top of the
hierarchy, while more particular ideas and specific details are subsumed under them in the
course of learning. New information is incorporated into and (re-)organized under the
learner’s existing higher level concepts, and hence the most inclusive (general) ideas remain
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longer in memory than particular facts or specific details. Such mechanisms reflect
cumulative effects of learning, which means that as learning proceeds to a higher level,
relevant knowledge will increasingly interact with each other (Gagné, 1968, 1970a, 1970b,
1977). Accordingly, it can be said that the capacity for learning depends on the amount of
basic knowledge the learner possesses that is relevant for the learning task as well as the
amount of transferable general knowledge s/he has acquired throughout the course of learning.
Consequently, knowledge should be generalized so that it can be more easily transmitted and
used in subsequent situations.
In conclusion, cumulative learning can be defined as a continuous interaction of
abstraction and concretization of an abstract cognitive structure in cumulative patterns using
analogy. In order to concretize an abstract cognitive structure, learners continuously
aggregate packets of relevant information/knowledge that are at the same level in the
hierarchical cognitive structure. In order to condense an abstract cognitive structure, learners
continuously construct multiple units of packets of relevant information/ knowledge that are
at a higher level in the hierarchical cognitive structure. Generalization refers to the size of the
description of information/knowledge that is extended and it transforms descriptions along
the set-superset in one’s cognitive structure. The generalization process can be used to
acquire abstract knowledge because only the abstract form of knowledge can contain the
essential properties of the reality of objects. However, it should also be noted that to a certain
extent, generalization could also be referred to as abstraction as the boundaries between these
two processes is not as sharp as the description suggests.
2.2.5 Structuring Processes in Cumulative Learning
This section investigates the structuring processes inherent to cumulative learning in
addition to its cumulative nature. As stated in the previous section 2.1.2, it seems that the
human learning system is naturally able to apply acquired knowledge onto subsequent
learning processes. Under the assumption that this phenomenon is natural in human learning
to a certain extent, cumulative learning views that cognitive structuring is a precondition for
cumulative learning.
The cumulative connections (or integration) change various aspects of knowledge so
they can be assimilated into existing knowledge, or that a learner can modify and/or construct
cognitive structure, thereby improving knowledge and skills. Hence, cumulative learning can
be conceived as an accumulation of structural knowledge integration that transforms the
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learner’s existing knowledge to the next sequence of learning. Cumulative learning views that
any learning process is a structuring process in which less inclusive concepts are subsumed
under higher and more inclusive concepts: Aggregated information is put into relevant
concepts, which are then mapped onto the sets of categories in the hierarchy of cognitive
structure as the concepts are sequentially subsumed under the corresponding categories,
sub-schemas (or local schemas), and, finally, a (superior) schema (see Figure 2.4 in section
2.2.2). The entire cognitive structuring process involved in the course of learning is referred
as schematization in cumulative learning.
Authors define the process of cognitive structuring in their own terms. For instance,
Gentner (1983) referred to it as “structural mapping” – a process in which a learner maps the
knowledge from model A to model B by analogy. This corresponds to Bruner’s (1960)
“categorization,” which means categorizing knowledge on the basis of its surface and
structural similarities. For example, in the field of geophysics, the known structure “A” can
be mapped to the new structure “B” through the analogical reasoning of structural mapping.
Taking the example of learning the concept of string instrument mentioned previously, to
categorize different pieces of information into groups, the learner needs to identify and
analyze elements of information and define surface similarities (e.g., shape of guitar, violin,
or harp vs. piano) as well as structural similarities (e.g., the mechanism of producing sound of
guitar, violin, or harp vs. piano).
In doing this, the learner only compares his/her mental models of the two objects but
never compares their reality separately. Accordingly, structural mapping presupposes a
mental model. When the model does not fit with new information (e.g., lute, mandolin; or
clavichord, harpsichord), the learner activates his/her schemas and searches comparable
“abnormal” cases (e.g., shape of guitar, violin, or harp vs. piano) in order to comprehend the
new information. S/he then modifies the schemas accordingly by analogy. Ideally, the learner
might be able to categorize lute, mandolin, guitar, harp (playing by plucking the strings);
viola, cello, double bass (playing by bowing the strings); and piano, clavichord, harpsichord
(instruments that have strings but have attached keyboards that the player uses instead of
directly manipulating the strings) into the concept of string instrument by modifying,
(re-)organizing, and (re-)structuring his/her cognitive structure. In general, learners tend to
rely more on the surface similarities of the information rather than its deep structure and thus
fail to reach a correct analogical conclusion as in the case of guitar, violin, or harp versus
piano, clavichord, or harpsichord.
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In terms of the relationship between new information and existing knowledge in
cognitive structuring processes, they can be differentiated as follows:
a) When a learner identifies that new information can be fully subsumed into some part of
his existing schema, this part of knowledge becomes more firmly placed in the
corresponding schema of his/her cognitive structure.
b) When a learner identifies that new information cannot be fully subsumed into his existing
schema, or if it contradicts his existing schema, the learner tries to find some parts of
components in his entire knowledge system that can possibly be related to the new
information with inferences.
c) When a learner identifies that new information is a special case of the existing schema,
then s/he creates an instant mental model that links the new information with the relevant
part of existing schema so that an appropriate link can be made to it by analogy.
d) When the part of existing schema contradicts the new information, then the learner
attempts to restructure it to accommodate the new information. However, if this requires
too much restructuring of the schema, a new schema is constructed.
e) When a learner identifies that the new information does not fit any part of his/her existing
schema but that there is some degree of similarity between them at some level of
abstraction, then s/he tries to match them at this level of abstraction using more
generalized attributes, relations, or functions of the information. For instance, when the
learner finds that lute, mandolin, guitar and harp do not fit his/her existing schema (in
which only viola, violin, cello, double bass are referred as string instruments) but s/he
finds that there is some degree of similarity between them (e.g., all of them contained
strings on their bodies), then s/he tries to match them at this level of abstraction using this
more generalized attributes of the information.
Schematization. In the theoretical framework of cumulative learning schematization is
referred to as the process of incorporating new information/ knowledge into the relevant part
of existing knowledge by integrating aggregated units of information/knowledge in a more
and more abstract way. As mentioned in previous sections, cumulative learning assumes that
any component of the learner’s knowledge will have its own organizational property.
Presumably, a learner’s cognitive structure contains some (at least potential) super-schemas for
every learning situation that connect sub-schemas, categories, concepts, and units of
information downwards as long as further learning occurs. Based on this assumption,
schematization refers to all of the cognitive structuring processes that are involved in the
course of learning: collecting information/knowledge, categorizing and (re-)organizing it into
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categories, subsuming the categories into sub-schemas, subsuming the sub-schemas into a
schema through aggregation and abstraction processes. It is a process of (re-)organizing and
(re-)structuring information/knowledge by means of continuous interaction between new
information and existing knowledge in a particular learning context (see Figure 2.4 in section
2.2.2). Consequently, cumulative learning sequences construct schemas by multi-layered
schematization processes. Through these schematization processes, some aspects of
knowledge (in terms of elements, relations, and functions) may be deleted, modified,
(re-)organized, and/or (re-)structured.
While schematization refers to the entire cognitive structuring processes involved in the
course of learning; categorization, in this theoretical framework, is when a learner categorizes
a particular information/knowledge into a specific category (a group of related
information/knowledge from a single or multiple phenomena). Categorization occurs when
the learner compares the perceptual, surface, and structural similarities of new information
and knowledge with his/her existing schemas and then puts identified and extracted
information/knowledge into a category. At this level of grouping features into objects, the
degree of connectedness of aggregated information constrains the scope of the process.
Schematization is typically a bottom-up directional operation because pieces of
information and knowledge at the lower level of the hierarchical cognitive structure are
subsumed into the higher level of it. For instance, during each sequence of learning, learners
conduct themselves with their cognitive structures that allow them to make adequate sense of
the world. That is, when new pieces of information are perceived, a learner investigates what
is known and what is unknown by searching for relevant experiences/knowledge which can
possibly be linked to the new information and aggregates them in his/her cognitive structure.
The learner then tries to understand and interpret the new information by applying this
aggregated information/knowledge. This also involves a continuous process of abstraction, in
which the learner (re-)condenses the derived units of information/knowledge in a more
abstract form. When the derived information/knowledge does not reach the defined learning
goal, the learner may try to aggregate additional unit(s) of information/knowledge by analogy
until the learning goal in a particular learning sequence is reached. When the goal is reached,
it may be incorporated with the learner’s existing knowledge, and this then becomes available
for use in a future learning situation.
These sequences of the learning process are said to be cumulative in the conception of
cumulative learning. Accordingly, cumulative learning views that learning occurs when a
learner associates new information/knowledge with existing knowledge and then integrates it
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by assimilating, modifying, (re-)organizing, and (re-)structuring cognitive structure in a
meaningful way using analogy.
How a learner proceeds with this process of schematization depends on what s/he
perceives as the most effective way to utilize new information/knowledge within the context
of his/her prior knowledge. This determines the schematic link the learner uses to integrate
the pieces of segregated information/knowledge at a higher level in the hierarchy of cognitive
structure since the components of the cognitive structure are linked by means of their elements,
relations, and functions. Various schematic links of the cognitive structure are interconnected
to some degree and can be merged into a schema to integrate information gathered from
different sub-schemas and categories. These interconnected structures enable the learner to
find various ways to represent, reproduce, and construct knowledge about given information
and help him/her to understand it. This conception is based on Piaget’s (1952) verdict that
“every schema is […] coordinated with all other schemas and itself constitutes a totality with
differentiated parts” (p. 7). Accordingly, cumulative learning assumes that schemas
continuously subsume all relevant sub-schemas. The schemas thus presumably never stop
changing or becoming more sophisticated.
On the basis of Minsky’s (1975) assumption that larger and more structured units of
framed knowledge increase the power and speed of mental activities, cumulative learning thus
assumes that the creation of more complex, larger, and more structured units of schemas
results in a more efficient and effective learning due to the enlarged power and speed of mental
activities. A “frame,” the basic unit of a representational schema according to Minsky (1975),
contains the essential elements of a situation as well as the relations between elements (e.g., a
frame of “classroom” contains desk, chair, board, and etc) of the situation. With this frame,
one can more rapidly identify the situation and recognize the deviations from the “standard”.
When the sub-structures (e.g., categories, sub-schemas) of the schemas are more structurally
and intimately linked with each other, this increases the power and speed of cognitive
processes (i.e., schematization) because the relevant schemas can be activated more rapidly.
In conclusion, cumulative learning assumes that any component of the learner’s
knowledge will have its own organizational property. Therefore, structuring process is critical
in learning because structural knowledge integration that transforms the learner’s existing
knowledge to the next sequence of learning improves human knowledge and skills.
Consequently, schematization, a process of structuring (or organizing) knowledge by means
of continuous interaction between new information and existing knowledge in a particular
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learning context, is critical to cumulative learning. The entire schematization process is not
loosely detached as a series of unrelated pieces but is closely linked as a system which
connects all parts of it. Although highly sophisticated cognitive structures might be a sort of
“Copernican” structure, we would only expect to acquire this at the end of a slow and long
course of learning, while more simple and compact cognitive structures might be formed in the
early stages.
As stated in the previous section, if a change in the cognitive structures improves the
learner’s behavior (e.g., procedural performance of some problem solving tasks, cognitive
structuring processes), and this improvement is required for the ultimate goal of learning,
then such a change can be viewed as learning. Therefore, cumulative learning assumes that
learning occurs if there is an increase in the learner’s total knowledge (including cognitive
structuring process) in a direction determined by the learning goal. Accordingly, when a
learner increases the certainty of some part of his knowledge, and hence strengthens
corresponding schematic links as a result of repetitive practice or of performing some
inference, cumulative learning views this as a part of the learning process.
2.2.6 Concept Learning in Cumulative Learning
This section describes the cognitive processes involved in concept formation and the
cumulative cognitive structuring processes involved in concept assimilation as presented in
the theoretical framework of cumulative learning.
At the outset, concepts serve as the basic components in the hierarchy of cognitive
structure. A concept refers to a mental representation of a class of objects or events that share
one or more common properties, and it varies in terms of its concreteness/abstractness (e.g.,
concrete as the concept of “dog” or abstract as the concept of “justice”). Merrill and
Tennyson (1977) defined a concept as “a set of specific objects, symbols, or events which are
grouped together on the basis of shared characteristics and which can be referred by a
particular name or symbol” (p. 3). Following this definition, when a learner is only able to
recall the definition of a certain concept (e.g., a musical instrument that produces sound by
means of vibrating strings is called a string instrument), but is not able to correctly categorize
an object, this implies that the learner has not acquired the concept yet. Ausubel and
Robinson (1969) also claimed that concepts can be learned by the correct identification and
realization of content and its extension. Consequently, concept acquisition entails correctly
identifying its central properties and placing them into corresponding categories according to
certain common characteristics which they possess.
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2.2.6.1 Concept Formation
Concepts are the products of thinking as stated above. For instance, Bruner as well as
Piaget assumed that concepts are formed by internalizing empirical experiences (e.g., facts,
ideas, observations, etc.) in the world. In the process of learning, the essential features
extracted (i.e., internalized) from aggregated information of the properties of an entity can be
grouped together by common features and properties and then condensed into a particular
group, namely a concept.
Markman states that “features are symbols that correspond to particular aspects of the
represented world. A feature is an entity or object in the representing world” (Markman, 1999,
p. 60). Thus, features can vary in their level of specificity: While specific features represent a
narrow range of things in the represented world, general features represent a broad range of
things (Markman, 1999, p. 60). For instance, a feature representing a particular piece of
jewelry, a pearl necklace, is a specific feature, whereas “wonderful” is a general concept that
covers various features representing a similar central tendency. A wonderful person is not the
same as a wonderful day or wonderful dinner. Accordingly, the spectrum of general/specific
changes the size of the description and thus transforms descriptions along the line of
supercategory – category in one’s cognitive structure.
Concepts also can be defined as abstract/concrete. This spectrum increases the amount
of detail contained in one’s cognitive structure and transforms descriptions along the line
leading from less detail (e.g., peace, justice, etc.) to more detail (e.g., trees, tables, etc.). In
other words, while concrete concepts contain specific information, abstract concepts contain
more general information. The more abstract a concept is, the higher its level in the
hierarchical structure of a knowledge system will be, as it represents more profound and
comprehensive information. Accordingly, concepts which require a higher level of
abstraction may be more difficult to form than those that require less abstraction.
Ausubel and Robinson (1969) distinguished between the learning and formation of
concepts: While the learning of concepts implies the process of learning the meaning of a
concept by means of its verbal definition, the forming of concepts implies the process of
discovering the general and essential characteristics that identify a class of objects and
integrating these attributes into a complex image by acquiring the definition of the objects
and their relation to other objects. In other words, the learning of concepts is accomplished by
learning the definition while the forming of concepts is accomplished by discovering through
experiences. Furthermore, the process of learning the attributes of a concept by definition
(not by discovering them as in concept formation) is referred to as concept assimilation
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(Ausubel & Robinson, 1969), which is further investigated in the following subsection.
Bruner, Goodnow, and Austin (1956/1986) explained how people form concepts about
abstract items in their psychological study on pure concept formation. They defined concept
learning as “the search for and testing of attributes that can be used to distinguish exemplars
from non exemplars of various categories” (Bruner, Goodnow, & Austin, in the reprinted
version: 1986, p. 233). Consequently, to form a concept, a learner needs to compare and
contrast categories on the basis of the relevance of their features. This in turn suggests that to
form a concept, the learner should be able to identify the systemic relations within the
schematic network where the concept can be placed. This corresponds with Lunzer’s
conception about concept formation.
Lunzer defined the concepts as “systems of relations” which represent the systematic
aspect of the relations in contrast to mere associations between ideas, because relations
become systematic only when a person “possesses a repertory of internalized actions to
translate them” (Lunzer, 1979, p. 118). Accordingly, he concluded that concept formation is
“the process of gaining an understanding of the relations between itself and other related
concepts” (Lunzer, 1979, p. 118). Lunzer (1979) thus stated that “one might say that a
concept is more abstract than a second concept if it can be attained only by establishing a
relation (or a system of relations) between that second concept and other concepts of equal
abstraction to it” (Lunzer, 1979, p. 114). Correspondingly, forming (or mastering) a concept
means that a person should be able to define the relations between ideas or information and
then recognize the relevance of the chosen relations in a given context.
Kant also seemed to emphasize the “relations” or “associations” within the schematic
network where the concept can be placed. Kant (B 178 /A 139) viewed concepts as being
dependent on questions about how they may be given to us, because only when an object is
given to us, a concept can have meaning; hence, these conditions of sensibility comprise the
general condition under which the category can be applied to any object. The conditions of
sensibility (i.e., how) in Kant’s statement seem to emphasize the context within the schematic
network where the concept can be placed. For instance, objectification of reality can be
accomplished only when objects come to be placed in a network of interconnected schemas.
Take an object, desk, as an example; it may be presented as an independent entity with
properties as a function of the network of interconnected concepts or classes in which it can be
assimilated: “four-legged,” “for studying,” and so on. Such associations construct an
increasing number of relationships between the compounds of knowledge which accomplish
the objectification. Thus, the learner’s schematic structure of knowledge develops (i.e.,
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becomes more elaborate and complex) gradually due to a constant cumulative associations
between the compounds of knowledge through assimilation and accommodation. In turn, this
ever more sophisticated network of interconnected schemas gradually improves one’s
knowledge in the course of learning, assuming that there are always active cumulative
associations between any given new information/knowledge and existing knowledge.
The arguments stated above seem to suggest that to form a concept a learner needs to
learn the attributes of the concept and their relations to other objects within the cognitive
structure where a concept can be placed by discovering them through the following
processes: thinks and internalizes (i.e., actions executed in thought) empirical experiences
(e.g., facts, ideas, observations, etc.) in the world, identifies the general and essential
characteristics that identify a class of objects, and integrates these attributes into a complex
image by acquiring the definition of the objects and their relation to other objects.
In terms of cognitive processes, the learner is assumed to go through the following: 1)
aggregates units of information (i.e., objects) obtained from his/her environment (i.e.,
external sources); and 2) internalizes the figurative, functional, and/or operative regularities
and invariants identified from the aggregated information. In more detail, the learner should
identify and extract information from the sources in terms of the following components (i.e.,
internalize): (a) individuals/elements, which requires comprehension of an object by
identifying its characteristics; (b) functions, which requires identification of the function of
an object; and (c) relations, which requires the ability to isolate and compare attributes, to
find (dis-)similarities by comparing the similarities and differences between facts, to
understand the attribute relation, and to construct relations between concepts. 3) While the
learner internalizes the information, s/he activates relevant parts of his/her existing prior
knowledge and associates different parts of the information to each other (reorganization).
This enables the learner to identify the schematic relations within the cognitive structure
where a concept can be placed (restructuring). While doing this, the learner uses available
information to infer the meanings and usage of the information by applying learned or
self-constructed rules by analogy. Hence, the information becomes more sophisticated to
reflect essential properties and phenomena of the objects a concept includes. And the
meaning of the concept becomes more precisely developed through a process of abstraction.
In sum, in the theoretical framework of cumulative learning described in this
dissertation, concept formation is an active process of mental (re-)construction of perceived
reality from the world, because concepts are formed by cognitive (re-)organization and
(re-)structuring. Once relevant units of information have been aggregated, the learner
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differentiates the features of this aggregated information, and extracts and retains the
common features of it, and then abstracts the core attributes of a class of objects by
incorporating and condensing them into a coherent category (i.e., a concept).
2.2.6.2 Concept Assimilation and Structural Mapping
As stated above, Ausubel and Robinson (1969) referred the process of learning the
attributes of a concept by definition (not by discovering them as in concept formation) as
concept assimilation. Cognitive psychologists conceive of assimilation as the expression of a
weak restructuring of cognitive structures which enables a learner to differentiate an
unstructured environment. Accommodation, on the other hand, is seen as bringing about a
radical restructuring of cognitive structures because it involves the creation of new structures,
“either to reinterpret old information or to account for new information” (Vosniadou &
Brewer 1987, p. 52). Accordingly, concept assimilation can be accomplished by assimilating
information into existing concepts which consequently results in “changes” (i.e., weak
restructuring) in cognitive structures.
For instance, in the course of learning, a learner aggregates units of concrete
information and then progressively abstracts the aggregated information in a generalization
process. When the learner finds the relations between the information (i.e., concrete or
abstract) and any part of existing concepts, the information is integrated (i.e., assimilated)
into the corresponding parts of the concepts and its “meanings” become interconnected with
previously acquired “meanings” of the concepts. This consequently results in “changes” (i.e.,
weak restructuring) in the learner’s cognitive structure.
Furthermore, concept assimilation can also be accomplished within the whole cognitive
structure where a concept can be assimilated into other concepts. As stated above, concepts
serve as the basic components in the hierarchy of cognitive structure and they are not
detached from each other but are connected by means of relations, either in conjunction or
disjunction with each other. These relations can be extended, modified, and/or constructed
over time in the course of learning. For instance, in the course of learning, when the learner
finds the relations between a concept with other concepts in his/her cognitive structure, these
concepts are integrated (i.e., assimilated) into the cognitive structure based on the structural
schematic links with the cognitive structure, and consequently results in “changes” (i.e., weak
restructuring) in the learner’s cognitive structure. Similarly, once the learner finds the
relations between newly acquired concepts and his/her existing knowledge structure, the
concepts are progressively integrated (i.e., assimilated) into his/her cognitive structure.
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Consequently, concept assimilation increases the organization of knowledge (i.e.,
reorganization), which in turn results in the weak restructuring of cognitive structures but it
may not radically “change” the structural frame of knowledge because, radical changes
require many transformational steps leading to an increasing differentiation of cognitive
structures.
In sum, in the view of the present theoretical framework of cumulative learning,
concepts are not stored in segregated or disorganized form in the learner’s cognitive
structures but they are progressively subsumed under a corresponding category in accordance
with certain schematic rules throughout the course of learning. Therefore, it views that as
learning proceeds, the specific and general features of elements, functions, and relations of
concepts are continuously integrated (i.e., categorized) and then they are progressively
structured into categories, sub-schemas, and schemas. These cognitive structures
progressively develop over time through a schematization process. Therefore, the cognitive
structures become more complex. Accordingly, acquiring more complex concepts leads a
learner to construct more profound and complex cognitive structures in the course of learning,
and hence improves the knowledge and skills over time.
Concept categorization. With regard to the concept categorization, Rosch (1978)
defined it as classifying a number of objects that are considered equivalent and taxonomy as a
system used to relate categories to one another by means of a perceived similarity between
objects. Thus, he stated that “the greater the inclusiveness of a category within a taxonomy,
the higher the level of abstraction” (Rosch, 1978, p. 30). In other words, superordinate
categories make the categories one level more abstract. For example, furniture is
superordinate to a table (a basic level), while dinner table is subordinate to it. Rosch (1978)
thus identified the following two basic principles of categorization: 1) it involves a drive
toward “cognitive economy,” that is, it provides maximum information with the least
cognitive effort while retaining as many finite resources for finding similarities and
differences among stimuli as possible. This can be accomplished by appropriately mapping
categories to given attribute structures. 2) The perceived world is not an “unstructured total
set of equiprobable co-occurring attributes” (Rosch, 1978, p. 29). Thus, Rosch (1978)
claimed that perceived objects in the world are not initially categorized either at the most
abstract or at the most concrete level but that they must first be recognized as members of
their basic category (i.e., the most inclusive categories ― an average member of the category
― can be identified with). Thus, in order to identify them as members of their superordinate
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or subordinate category within the taxonomy, there has to be additional cognitive processing
(Rosch, 1978). For instance, for categories of concrete objects, prototypes develop through
the maximization of category resemblance and cue validity (i.e., similarity), which means that
“the validity of a given cue x as a predictor of a given category y increases” (Beach, 1964a,
1964b; Reed, 1972; as cited in Rosch & Lloyd, 1978, p. 30). Brooks (1978) stated that
participants store particular instances and use them to classify new cases with the old ones by
analogy when task or stimulus materials are changed (as cited in Rosch & Lloyd, 1978, p.
75).
The above stated arguments of categorization correspond to the proposed idea of
cumulative structuring processes, which is explained in section 2.2.5. That is, when new
pieces of information are perceived, a learner investigates what is known and what is
unknown by searching for relevant information/knowledge which can possibly be linked to
the perceived information and aggregates them in his/her cognitive structure. The learner then
tries to understand and interpret the new information by applying the cognitive structure
which contains the aggregated information/knowledge. The learner thus continuously
(re-)condenses the aggregated units of information/knowledge to form a concept and/or to
assimilate a concept into the cognitive structure based on the given learning goal. Therefore,
the concept now can contain resources for finding similarities and differences among stimuli,
which can be mapped to appropriate categories to given context. The learner may try to
aggregate additional unit(s) of information/knowledge by analogy until the learning goal in a
particular learning sequence is reached. This whole processes imply that the learner tries to
perceive and interpret the world in relation to the whole “structure” of the knowledge system
by consistently incorporating existing knowledge and new information. As a result, concrete
and specific information progressively becomes abstract as it reaches a higher level on the
hierarchy of knowledge.
Accordingly, cumulative learning in the presented theoretical framework views that
concept learning (concept formation and concept assimilation) occurs when a learner
associates new information/knowledge with existing knowledge and then integrates it by
assimilating, modifying, (re-)organizing, and (re-)structuring cognitive structures in a
meaningful way using analogy. This process of incorporating new information/knowledge
into the relevant part of existing knowledge by integrating aggregated units of packets of
information/knowledge in a more abstract way refers to schematization as previously
identified in the theoretical framework of the present study (see section 2.2.5).
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As a conclusion to section 2.2, “contemporary problems in cumulative learning,” the
theoretical concept of cumulative learning assumes that learners actively process and
interpret their environment in terms of their cognitive structures and that the components in
the cognitive structures are closely interrelated. In the course of learning, learners
progressively perceive, understand, and interpret a world by assimilating and accommodating
(i.e., accreting, tuning, (re-)organizing, and (re-)structuring) their cognitive structures in
cumulative patterns.
With regard to the cognitive processes, the theoretical concept of cumulative learning
assumes that learning can be broadly represented as the function of the interaction between
aggregation and abstraction: Learning is activated when a learner aggregates units of
information obtained from his/her environment and activates relevant parts of his/her existing
prior knowledge (i.e., aggregation). The learner then progressively integrates units of
information into existing knowledge structures (i.e., schematization) and constructs schemas
which consist of groups of more generalized concepts at different levels of abstraction. As
learning proceeds, these integrated assimilatory schemas become progressively complex and
profound that they can extend themselves to interpret the phenomena of the world, because
they provide an insightful framework for interpreting new information which can direct
assimilative and/or accommodative investigation in further learning sequences. As these
experiences accumulate in the course of learning, the learner’s knowledge and skills
progressively “change” over time, because cumulative learning assumes that any learned
capability at any stage of learning process affects the subsequent learning situation. Hence,
the learner’s cognitive structure becomes more enhanced and developed.
Correspondingly, the theoretical concept of cumulative learning assumes that the
mechanisms of learning that can result in the development of knowledge and skills are
cumulative and structural nature. It also views that there is no magic path in the “real
learning,” but only a slow and gradual development which requires a lot of time and consistent
effort of continual exposure to the topic like Norman (1980, p. 20) explicitly stated in his own
words (see section 2.1.8.1).
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2.3 Research Question
This section presents the research questions of the present study based on the
contemporary concept and theoretical framework of cumulative learning presented in the
previous section. Accordingly, the research questions for this study are not “assumed a
priori” (Bogdan & Biklen, 1998, p. 49) but are constructed on the basis of the theoretical
assumptions presented in this study as well as a preliminary investigation of an existing series
of interviews.
As stated in the previous section, the theoretical concept of cumulative learning in this
dissertation assumes the two mechanisms inherent to human learning – cumulative as well as
structural nature. Accordingly, it views that learning occurs when a learner associates new
information/knowledge with existing knowledge and then integrates it by assimilating,
modifying, (re-)organizing, and (re-)structuring cognitive structures in a meaningful way
using analogy. Consequently, it assumes that constructing well-organized as well as complex
cognitive structures in the course of learning is critical in cumulative learning.
As stated in the conclusion of the previous section, the present study views “real
learning” (Norman, 1980, p, 20) ― the learning that results in complex cognitive structures
― occurs progressively, cumulatively, and structurally, since constructing the complex
cognitive structures seems to involve many steps of modification lead by an increasing
differentiation as well as an interactive reconciliation of cognitive structures. Therefore the
present study assumes that learning is (1) a cumulative process wherein the learning in each
new sequence builds upon knowledge acquired in a previous sequence; and (2) a structuring
process in which less inclusive information/knowledge are subsumed under higher and more
inclusive ones.
Based on the assumptions stated above, the present study endeavors to determine the
learning mechanism inherent to human learning in relation to knowledge acquisition and
development: Is learning a cumulative process? Is learning a structuring process? Is the
learning in each new sequence built upon knowledge acquired in a previous sequence? These
questions form the following main research question of the present study:
1. How do learners acquire and develop knowledge and skills beyond those that are
currently available to them in the different sequences of learning?
1.1 Is learning a cumulative process wherein the learning in each new sequence builds
upon knowledge acquired in a previous sequence?
1.2 Is learning a structuring process in which less inclusive information/knowledge
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are subsumed under higher and more inclusive ones.
In an attempt to define the mechanism (or a class of phenomena) inherent to learning in a
scientifically adequate framework, the present study endeavors to investigate the cognitive
processes in relation to knowledge acquisition and development by compiling diverse
qualitative empirical results leading to the second research question:
2. What cognitive processes are inherent to learning in relation to knowledge acquisition
and development?
2.1 What are the steps of the cognitive processes involved in the course of learning?
2.2 What are the characteristics of each step of the cognitive processes?
In order to more clearly define the cognitive processes inherent to learning, the present study
also attempts to investigate the systematic use of learning strategies taken to facilitate
knowledge acquisition and development. This in turn forms the following third research
question:
3. What learning strategies are used in relation to knowledge acquisition and
development?
3.1What types of learning strategies are used among learners?
3.2 What are the characteristics of learning strategies used among learners?
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CHAPTER III
METHOD
The study presented here is a case study. Case studies are most often used in the field of
psychology in clinical research to describe the events and conditions of the participant(s), and
are specially used in educational psychology to describe a class of phenomena of the
participant(s) and/or conditions. Many qualitative studies are case studies and are presented
in order to illustrate theoretical aspects. Dreyfus and Dreyfus (1986) viewed case studies as
central to human learning. Therefore, the researcher of this study also assumed that this
method contributes to the cumulative development of understanding the phenomenological
aspects of the learning process because with in-depth qualitative analytic investigation,
human behavior can be meaningfully understood.
This study presents an in-depth analysis of publicly available archival interview
records, which were broadcasted on a TV program from August 2009 to June 2011 by Korea
Educational Broadcasting System (EBS), titled “The Royal Road to Learning.” The program
was to introduce effective and efficient learning processes and strategies in students preparing
for the college entrance examination in Korea, College Scholastic Ability Test (CSAT). The
data used in this study is a total of 49 cases involving 22 second and third-year high school
students, who were preparing to take the CSAT, and 27 first and second-year college and
university students who had taken the CSAT in Korea.
The study used the thematic content analysis method (Burnard, 1991) in which
principles of the coding scheme are generated from phenomena in the interviews. This method
is adapted from Glaser and Strauss’ (1967) “grounded theory” approach, the literature on
content analysis, and other sources concerned with the analysis of case studies (Creswell,
2005; Strauss, l987).
In order to analyze a class of phenomena in learning, the data from the broadcasted TV
program were transcribed. The transcription of the data is not from a full copy of the entire
conversation that took place during the interviews because not all parts of the interviews were
relevant with regard to the research questions for this study. For this reason, the researcher
extracted conversation and information (e.g., narration) from the interviews that was relevant
for the purposes of the present study following suggested “standard” guidelines (e.g.,
Creswell, 2005; Strauss, l987) to present a valid argumentation for findings.
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3.1 Data Collection
3.1.1 College Scholastic Ability Test (CSAT)
This section briefly introduces the education system in high school, college, and
university, and the CSAT7 in Korea. High schools (age 17 - 18 or 19) are divided into different
categories which include; public, private, vocational, and specialty high schools. Public and
private high schools are similar to the western education system. Specialty schools are highly
competitive and are divided into specific tracks that are geared towards a student’s career path
or areas (e.g., sciences and mathematics, foreign languages) they excel at. They offer similar
courses and programs to colleges and universities for they are implemented to support students
get accepted into colleges and universities. Vocational schools offer specialized fields (e.g.,
finance, technology, agriculture, etc), and are for usually students who do not plan on attending
college/university. In order to get accepted into a Korean college/university (age 19 - Adult)
generally students need to take the CSAT.
The CSAT is designed to measure the scholastic ability required for college education in
Korea. The test comprises 5 sections which include: Korean, mathematics, English, foreign
languages or Chinese classics, and a number of ‘electives’ in the sciences: physical and social
sciences. The test is administered only once a year, and hence, students are under immense
pressure to pass the test. They prepare for this test through intense memorization, studying, and
practicing solving problems during entire periods of high school.
The objectives of the CSAT are (a) to enhance fairness and objectivity of student
selection by measuring learning abilities and achievements required for college education; (b)
to allow students to select all or some of five tests as well as the subjects following the basic
principles of the seventh national curriculum, which focuses on the needs, interests, learning
abilities, and career goals of students; and (c) to test high-order thinking abilities together
with the students’ understanding of cross-disciplinary materials in different subjects (see
Table 3.1).
The number of items and the testing time is presented in Table 3.2. Scores range from 1
point to 3 points depending on the level of importance of the items. Multiple-choice items
have five choices.
7 The information was extracted from the website of the Korea Institute for Curriculum and Evaluation (KICE)
http://test.kice.re.kr/en/resources/abillityTest.jsp
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Test Subject
Korean Language -
Mathematics
(Select 1)
Math A Math I + Math II + + (Select 1 of 3 area-specific subjects: Calculus,
Probability & Statistics, & Discrete Mathematics)
Math B Math I
English To develop items covering cross-disciplinary materials. It aims at evaluation
based on thinking skills focused on cross-curricula issues ( items cover
cross-disciplinary materials)
Social
Studies/Sciences/
Vocational
Education
(Select 1)
Social
Studies
Up to 4 of 11 subjects can be selected: Ethics (Ethics and Thought +
Traditional Ethics), Korean History, Modern & Contemporary Korean
History, World History, Politics, Economics, Society & Culture, Law and
Society, Korean Geography, Economic Geography, and World Geography.
Sciences Up to 4 of 8 subjects can be selected: Physics I, Chemistry I, Biology I,
Earth Science I, Physics II, Chemistry II, Biology II, and Earth Science II.
Vocational
Education
- 1 subject can be selected out of 4 computer-related subjects: Agricultural
Information Management, Basic Information Technology, General
Computers, and Fishery and Shipping Information Processing.
- Up to 2 of 13 subjects can be selected: Understanding of Agriculture,
Techniques in Basic Agriculture, Introduction to Industry, Basic Drafting,
Commercial Economy, Principles of Accounting, Introduction to Fisheries,
General Marine Affairs, General Oceanography, Human Development,
Food and Nutrition, General Design, and Programming
Foreign Languages/
Chinese Characters and
Classics
1 subject can be selected from among German I, French I, Spanish I, Chinese
I, Japanese I, Russian I, Arabic I, and Chinese Characters and Classics.
Table 3.1 Tests and subjects of CSAT
Test Number of Item Testing Time(per item) Item Format
Korean Language 50 items Including 5
listening items
80 min (1.6 min) Multiple-choice
Mathematics
(Select 1)
Math A 30 items Math I 40%
Math II 40%
Electives 20%
100 min (3.3 min) Multiple-choice
(70%) & short
answer (30%)
Math B 30 items Math I 100 min (3.3 min) Multiple-choice
(70%) & short
answer (30%)
Foreign Language (English) 50 items Including 17
listening &
speaking items
70 min (1.4 min) Multiple-choice
Social
Studies/
Sciences/
Vocational
Education
(Select 1)
Social Studies 20 items
per subject
20 items x up
to 4 subjects
Up to 120 min: 30 min
per subject (1.5 min)
Multiple-choice
Sciences 20 items
per subject
20 items x up
to 4 subjects
Up to 120 min: 30 min
per subject (1.5 min)
Multiple-choice
Vocational
Education
20 items
per subject
20 items x up
to 3 subjects
Up to 90 min: 30 min
per subject (1.5 min)
Multiple-choice
Second Foreign Languages/
Chinese Characters & Classics
30 items
per subject
- 40 min (1.33 min) Multiple-choice
Table 3.2 Number of items and testing time of CSAT
3.1.2 Procedure
The general procedure followed in the study was to elicit data from students’ accounts
of their cognitive processes and learning strategies as they engaged in a variety of learning
tasks over time. In order to ensure that the data was appropriate for the purpose of the present
study, the researcher assessed the relative advantages and disadvantages of alternative
70
approaches (e.g., collecting new data) in terms of the suitability, target population, and
reliability.
As for the suitability, first, since the goals of the interviews are to determine what
learning processes and strategies are effective and efficient means of preparing for the CSAT,
in-depth investigation of these interviews were able to serve to identify the mechanism of
learning in problem solving. Therefore, they fulfilled the research purpose of investigating
the cognitive processes and learning strategies in problem solving. Second, the video of the
interviews presents in-depth information with conversation and images. Since conversation is
the common means that human beings use to learn about phenomena in the world, analysis of
conversation can be used for research purposes, as stated by Kvale (1996). In general, there
are several advantages to video analysis (Rothe, 2000, p. 92). Visual data presents strong
images that portray the key concepts or ideas. Thus, videotape can be reviewed repeatedly for
validation and reliability. There are also several advantages of interviews (Rothe, 2000, p. 98).
Interviews provide an opportunity for participants to express their thoughts about issues they
may not be adequately presented in other research methods (e.g., questionnaires, survey,
laboratory experiment, etc.) since they allow the interviewee to expand on ideas and make
clarifications (e.g., “for instance,” “what I mean is,” “in other words,” etc). Third, since the
EBS is a public media organization whose mission is mainly focused on education and
knowledge for Koreans, the credibility of the data is assured.
As for the target population, first, the study was particularly interested in students with
high achievement on their academic performance, because it is assumed that these students
use more elaborated learning processes and strategies. This is fulfilled as the interviewees’
school academic records, scores on the national mock (practice) test when preparing for the
CSAT, or scores on the actual CSAT were ranked within the top 10%. Second, since the
study intends to rule out age-related developmental effects in learning, the target population
of this study was students above the age of 15 situated in the context of intensive problem
solving. The grade levels of the interviewees ranged from high school students (45%) to
college/university students (55%).
As for the reliability, the method used in the interviews to collect the information seems
appropriate because the questions were appropriate for the goals of the interviews, were well
organized, and elicited reliable responses. Though the data was collected by means of
self-reporting interviews, the information is likely to be reliable as the method of interview
was consistent. The interview data was supervised to ensure that the participants’ responses
were consistent with their behavior as most of the interviews were recorded on-site and
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triangulated by way of interviews with their peers and teachers as well as experts’ opinions.
Furthermore, as the video included document reviews (learning materials such as textbooks,
reference books, workbooks, worksheets, note taking, etc.), it should be highly reliable.
Therefore, it does not seem necessary to conduct random checking of the information against
the records used by the persons who originally recorded the information. Thus, no biasing
effect is expected on the responses.
Possible disadvantage, however, is that there might not be a perfect fit between what the
researcher was trying to investigate and the purposes for which the data was collected. Also
the selected interviews had some constraints regarding the sources and interviewees as the
researcher was not able to personally contact the interviewees. The researcher thought it
would be beneficial to confirm interview transcripts and the coding scheme she developed as
it would enhance the internal validity of the study. Unfortunately, this did not materialize and
the researcher conducted the study amidst these constraints. Furthermore, explanations of
past experiences mentioned in the interviews might be to some extent “reconstructions” based
on the participant’s present circumstances. Thus, one-time interviews tend to be
time-specific.
3.1.3 Material
As stated above, this study is based on investigation and analysis of interviews
conducted with the top students of different high schools, colleges, and universities in Korea
on the process and strategies of learning. The series of interviews was originally produced
and broadcasted on a TV program by Korea Educational Broadcasting System (EBS). The
program is designed to investigate effective and efficient learning processes and strategies in
students preparing for the CSAT. The EBS is a public multi-platform media organization
whose mission is mainly focused on education and knowledge for Koreans. To gain
permission to use the material (interviews) for this study, the researcher contacted the EBS
and informed them regarding the purpose of the study and the handling of the material to
address the ethical issues. In order to ensure confidentiality, the present study conceals names
and any information which might possibly affect the privacy of the participants.
The video contains three data collection methods for gathering information on learning
processes and strategies used by the participants. First, the participants were asked to describe
the “particular strategies or method they used” in the process of learning. Second, teachers,
domain experts, peers, or family members of the participants were also interviewed parallel
to the participants’ interviews about learning processes and strategies used by the participants.
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The third approach was on-site observation for detecting the learning processes and strategies
used in classroom settings and/or at home. The majority of the interviews were conducted
both at the interviewee’s school and home.
The participants were asked to answer questions so that the interviews could be
recorded by video camera. The interviewees’ answers to a given question formulated by the
production team was then transcribed and coded by the researcher. While the structure of
these interviews varied (semi-structured to some extent) and the questions were open-ended,
with the intention of encouraging a full, meaningful answer using the interviewee’s own
knowledge and/or feelings. The following is a list of some interview questions that the
researcher compiled from the transcripts of the interviews:
What is your learning process and strategy when preparing for the CSAT?
Do you feel (or have you felt) competent about using your learning strategies?
How could the learning (strategies) be improved to be more consistent with the
day-to-day learning task you were performing?
What are the goals of the learning and studying you performed?
How were you able to improve your learning process?
How did you figure out particular learning strategies?
How is the effectiveness of learning being controlled (or managed)?
49 individual interviewees from this series of interviews were selected for their relevance to
the purpose of this study. Each interview explored learning processes and strategies
thoroughly during the average length of 25 minutes. The interviews dealt with learning all
school subjects in general. School subjects particularly applied are mathematics (N=16),
English (N=5), Korean language (N=5), Science (N=5), Social Studies (N=3). The interviews
were conducted with the participating students on an individual basis. Furthermore, five small
group experiments were also conducted with several students who volunteered while the
interviews were being produced.
3.1.4 Participants
The participants of this study were a total of 49 individual interviewees (the grade
levels ranged from second and third-year high school students (45%) to first and second-year
college/university students (55%) in Korea who participated in the broadcasting produced by
the EBS to investigate their learning strategies while preparing for college entrance exams
from the 2008 to 2011. Twenty-two participants were selected from the top students of
different high schools and the other 27 participants were current students of the
colleges/universities who had achieved good scores on the actual CSAT. Their school
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academic records, scores on the national mock (practice) test when preparing the CSAT, or
scores on the actual CSAT were ranked within the top 10% of the range (i.e., high achievers).
Five cases of the interviews included small experiments carried by production team
while interviewing participant ID number 3, 4, 5, 9, and 19 to see the effectiveness of the
participants’ learning strategies and/or cognitive processes. Table 3.3 presents the descriptive
data of the experiments.
Case Number of students Grade level Gender
3 6 High school (2nd
) Females
4 (1st Exp)
4 (2nd
Exp)
30 (approx.)8 High school (2
nd ) Males
5 High school (2nd
) Males
5 5 High school (1st ) 3 Males/2 Females
9 2 High school (1st ) Males
19 39 High school (1st or 2
nd )
9 Males
Table 3.3 Descriptive data of small experiments
Since all of the participants were at advanced levels of academic achievement in
different school subjects, they were identified as “high-achievers” for the purpose of defining
cognitive processes and learning strategies. “Average-achievers” were identified on the basis
of informal identifications revealed in the interviews as no definite level of academic
performance had been identified previously for these participants. Table 3.4 presents the
descriptive data of the participants and the school subjects particularly discussed in the
interviews.
Categories Subcategories Total
Grade level High school (2nd
grade) N=8 (16%)
High school (3rd
grade) N=14 (29%)
College/University (1st yr.) N=24 (49%)
College/University (2nd
yr.) N=3 (6%)
Gender Male N=23 (47%)
Female N=26 (53%)
Table 3.4 Descriptive data of participants
The data refers to a subset of the population which might be unrepresentative of the larger
population. Since the participants were chosen for the purpose of learning and understanding
effective (or at least successful) learning processes and strategies, the data does not necessarily
represent a “normal” population. Rather, it represents a specific population focused on the
8 “Approximately 30” were identified on the basis of informal identifications revealed in the video of the
interview as no definite number of students had been identified in the video.
9 The grade level was identified as 1
st grade on the basis of informal identifications revealed in the narration of
the interview while it was identified as 2nd
grade from the subtitles that had been displayed on the screen.
74
topic of defining effective or successful learning processes and strategies.
However, as the purpose of the study was to investigate the learning mechanism, cognitive
processes, and learning strategies in relation to knowledge acquisition and development
inherent to learning and the extent to which they can lead to effective learning, the
participants were selected because of their “knowledge” of the topic. Hence, the range of the
data generated may be assumed to be deeper and richer than that obtained from a random
sample.
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3.2 Data Analysis
A qualitative analysis of the data describing how participants approached and worked
on the different learning tasks when studying different school subjects was undertaken to
identify cognitive processes and learning strategies. The aim of the data analysis was to
produce a systematic investigation of the proposed research questions. Since the full
interviews contained various information other than participants’ learning processes and
strategies (e.g., school/university life, family stories, hobbies, etc), the researcher only
extracted conversation and information from the interviews that was relevant for the purposes
of the present study, and they were transcribed for each student. The criteria for extracting the
part of an interview for analysis included appropriateness for the purpose of this study, and
hence, parts of the interviews which were not relevant to the research questions were left out
in the transcript and analysis. The learning resources the participants used during learning
were also identified. This included the notes, textbooks, reference books, and workbooks
containing various learning tasks depending on the types of activities included in the
curriculum that they had been studying.
3.2.1 Stages of the Analysis
Analysis is identified on the basis of the participant and contains the following stages:
1) Preliminary exploratory analysis
The data was initially analyzed. Notes regarding the primary learning processes and
strategies talked about in each interview were made after the videos were viewed several
times. The purpose of these notes was to explore the data to obtain general ideas about the
data, to think about how it is organized, and to consider whether the study needed more
data or not. The researcher recorded ideas and principles that may be relevant to the
research questions during the initial phases of the analysis as she worked with the data.
2) Initial transcript
Only parts that were relevant to the presented research questions for each recording were
transcribed using an MS Word file. This initial transcript consisted only of verbatim
descriptions of each process and strategy mentioned by participants.
3) Generating initial coding scheme
The researcher read through the data intensively multiple times in order to generate sound
general themes and categories within the transcripts. Two broad themes were constructed
from the data analysis: cognitive processes (Theme 1) and learning strategies (Theme 2).
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A great variety of learning processes and strategies were described in the interviews.
Nevertheless, major categories that seemed to emerge had to do with aggregation and
abstraction (Theme 1); and cognitive, metacognitive, and social/affective strategies
(Theme 2). This can be conceived as the transition phase from the particular to the general
analysis (McCracken, 1988) in order to comprehend and conceptualize the overall
information presented in the data so that the codes can be meaningfully interconnected.
The entire coding process went through iterative validation between the interpretation and
the data as particular phenomena emerged. Nevertheless, as the literature states (e.g.,
McCracken, 1988) it is the researcher who identifies the superficial as well as deep
structures contained in the data and the researcher’s subjectivity (knowledge and
experiences) is likely to emerge throughout the coding and analysis processes. However,
the study followed suggested guidelines (Creswell, 2005; Strauss, l987) for presenting
valid argumentation for analysis and findings.
4) Constructing coding scheme
The transcript was then investigated and subsequently coded for the occurrence of
cognitive processes and learning strategies. The transcripts were coded by highlighting
remarks that revealed cognitive processes and learning strategies, transferring them to a
coding sheet, and writing down the categories and subcategories of the cognitive processes
and learning strategies as well as any comments or insights developed by the researcher as
a result of the detailed analysis of the transcripts. There were three stages in constructing
complete coding scheme:
a) In open coding, the researcher arranged similar phenomena to form large categories by
iteratively reading through the transcripts and checking the video. The categories were
freely constructed as many were necessary to draw out all potentially meaningful
aspects of the data.
b) Axial coding involved taking the constructs that emerge from open coding and
reassembling them in such a way that core categories are linked with their
subcategories, thus allowing more central concepts or core categories to be captured.
This process continued until all categories were constructed. These categories were
then analyzed for emerging phenomena (Creswell, 2005; Bogdan & Biklen, 1998). It
was important for the (sub-) categories to be constructed in a holistic perspective in
order to comprehend the participants’ overall learning processes. The subcategories
were grouped into broader super-categories to reduce the number of categories. For
example, the following categories were combined into a single category entitled
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‘Complex building/Classification’:
Defined characteristics of information to combine equivalent relevant attributes.
Analyzed the arrangements of elements in a set to associate separate parts in order to form a
new arrangement.
c) In selective coding, the final list of categories and subcategories were constructed
through deletion, combination, and modification. The transcripts were reinvestigated
alongside the final coding scheme to establish the degree to which the categories
covered all aspects of the interviews which were relevant for answering the research
questions. Adjustments were then made as necessary. Furthermore, in order to control
for researcher bias, the researcher discussed the list of categories with two colleagues
and then made necessary adjustments accordingly to finalize the coding scheme. Table
3.5 presents the list of the coding scheme. The subcategories branched off into more
specific ideas that emerged from the data. The classification of the coding scheme (see
section 3.2.3), and the descriptions of the coding schemes (see section 3.2.4 & 3.2.5)
are explained in detail in separate sections.
Theme Categories (number of subcategories)
Cognitive process Aggregation (7)
Abstraction (5)
Learning process Cognitive strategy (13)
Meta-cognitive strategy (7)
Social/Affective strategy (3)
Table 3.5 List of coding scheme
5) Coding
a) In individual coding, the transcript of each interview was investigated using the coding
scheme and then “coded” accordingly. The categories were identified on a computer
using a word processor. Each specific code consisted of an abbreviated form to
represent the specific ideas (e.g., Agg6a = Aggregation: schema completion)
b) In group coding, each coded section of the interviews was extracted from the transcript
and all items of each code were grouped together. Multiple electronic files of the
transcripts were used to ensure that the context of the coded sections was maintained.
Table 3.6 depicts a sample of actual transcription from an interview coded10
.
10
See section 4.1 and Appendix A for the complete coding used to analyze the data.
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Transcript of case 7 (Grade: HS3) Cognitive processes Learning strategies
1
2
3
4
5
6
7
8
9
10
11
12
N7: The participant takes one “special” day to study
science and to find out relations and connections
between each section of the subject. As each section
of the science subject is closely related to each other,
what one has learned in the previous section can be
applied to the other sections. S/he first examines the
table of contents of the textbook to determine the
importance of each section. And then, s/he simply
skims through the sections that s/he is already
familiar and that which does not require in-depth
knowledge and invests more time in studying the
sections where s/he feels incompetent.
[7.1], Agg1 identification
[7.3], Ab3 structural
[7.2], M6 problem
[7.4], M1 organization
[7.5], M2 planning
[7.6], M3 attention
Table 3.6 Sample coding of interview transcription
The researcher tried to ensure that everything transcribed from the interviews was written
down in context in order to avoid altering the meaning of what was said. However, since
the transcript did not contain the entire conversation that took place during the interviews,
this risk may still be present. This is one of the limitations of this study. Nevertheless,
once particular parts of the interviews had been extracted, the researcher tried to ensure
that the particular remarks or parts of conversations had a bearing on the context of the
whole interview.
6) Validate the coding scheme
The extracted parts were organized corresponding to the coding scheme, and then two
colleagues were asked to check the appropriateness of the coding scheme. Adjustments
were made as necessary to validate the findings.
7) Write- up
a) In open write-up, all of the extracted parts were put together for direct reference while
writing up the findings. The researcher started with the first category of the first theme
and then selected the various examples of data that had been grouped under that
category and wrote an interpretive commentary that linked the examples together. The
researcher then continued on to the next category and so on until the whole analysis
was written up. During the entire process, the researcher often referred back to the
original interview recordings and to the whole transcripts of the interviews to ensure
that the analysis delivered the original meanings and contexts as closely as possible.
b) In collective write-up, the findings were arranged according to the proposed research
questions along with the summary of transcription in the form of narratives and/or
actual conversation. Hence, the “result” section of this study can be characterized as a
comparison of the findings with research questions.
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3.2.2 Validity and Reliability
Many qualitative researchers have sought to adapt and supplement quality criteria
common to quantitative research to fit qualitative methods and data. According to Lincoln
and Guba (1989), the assessment of qualitative research draws on the criterion of
“trustworthiness,” which involves constructs of internal validity, external validity, and
reliability. Hence, verification of the data analysis concerns the generalizability (i.e., findings
can be generalized), the reliability (i.e., consistency of findings/results), and the validity of
findings (i.e., whether the study in fact investigates what was intended).
Validity can be viewed as the accuracy of the findings. Internal validity is understood as
“the extent to which it establishes how things really are and really work” or the authenticity
of the work. External validity refers to “the applicability to the larger population from which
the smaller sample was drawn” (Lincoln & Guba, 1989, p. 234), that is, to what populations
and context the effect of measurement variables can be generalized. In addition, reliability is
related to the consistency or stability of measurement, assuming that “every repetition of the
same, or equivalent, instruments to the same phenomena will yield similar measurements”
(Lincoln & Guba, 1989, p. 235).
Member checking. As a means of authenticating the data, this study could have adopted
“member checks” (Lincoln & Guba, 1989; Stake, 1995), in which the researcher asks the
participants to read and give their reactions to the researcher’s interpretations of data, about
which the participants have firsthand knowledge. Or another good way to check for validity
would have been to re-interview the participants and ask them to read through and confirm
the transcripts of their interviews. However, these methods were not possible due to the
constraints of the source. Nevertheless, in order to produce high-quality data that are credible,
accurate, and true to the phenomenon under study and to reduce the likelihood of
misinterpretation (Patton, 1990; Stake, 1995), the present study systematically reviewed and
compared the different aspects that emerged from the full set of data. Thus, the findings of
this study are coherent within the context of the collected data. Accordingly, this study sought
to strengthen the validity of these aspects.
Triangulation. The researcher indirectly established “triangulation” as a means to
enhance internal validity. Triangulation by method (Creswell, 2005) is established by
corroborating evidence from different methods of data collection shown in the interviews,
such as document reviews (e.g., participants’ note taking in their own notes, incorrect answer
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notes, textbooks, reference books, and workbooks) as well as the document reviews of the
summaries and/or key points of interviews uploaded to the EBS web site. “Triangulation by
theory” is established by using different theories, for example by selecting the most plausible
ones to explain the results (Bogdan & Biklen, 1998). This serves to establish the validity of
the findings in relation to previous findings as well as to extract core findings that are
consistent with the theoretical background. However, this study also could have employed a
direct triangulation of data sources and methods (Patton, 1990; Stake, 1995). For instance, the
researcher could have directly confirmed the interviews by conducting surveys to validate the
findings from each method. As the data contain some constraints with the sources and
interviewees, it was not possible to personally contact interviewees.
With regard to the reliability, it seems that the data corresponds to the theoretical
propositions, and hence the internal reliability of the study is assured. To ensure the internal
validity (i.e., control researcher bias) of this study, the researcher discussed the list of
categories and their operational descriptions with two colleagues: one from the field of
educational science that was fully familiar with the context of the study, and the second
colleague though not directly involved in the field of educational science but one who was
familiar with the process of category generation in qualitative analysis of case studies.
Necessary adjustments were made accordingly to finalize the coding scheme. This is another
limitation of this study, for the literature suggests that this should be done independently and
without seeing the researcher’s coding scheme in order to enhance the validity of the
constructed categories. However, since the coding scheme was first constructed by the
researcher based on the presented theoretical framework of cumulative learning, and there
were no previous studies for comparing and testing validity of the coding scheme, the
researcher believed that the validation of the list of categories should be grounded on the
initial one. And future studies should test the validity and reliability of the coding scheme.
The researcher asked one of the colleagues to read through the transcripts along with
the constructed coding scheme, and the colleague agreed in most cases with the original
coding analysis. Since the colleague was not fully familiar with the context of the study prior
to being asked to help validate the coding system, the question of whether or not the coding
was complete and accurate could not be fully accounted for.
With regard to the external validity, the limited group of participants makes it to some
degree difficult to make generalized statements beyond this group (“high-achievers).
Furthermore, considering that secondary education in Korea is primarily geared toward a
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college entrance examination (i.e., CSAT), one cannot rule out a cultural bias in the learning
process and strategies.
3.2.3 Classification of the Coding Scheme
This section presents the classification of the coding scheme used for the data analysis.
Two broad themes are constructed from data analysis: cognitive processes and learning
strategies. These themes are then further categorized and subcategorized. However, it should
be noted that the present study does not claim that this coding scheme explains all
phenomena in human learning in the direction of cumulative learning. Hence, the coding
scheme serves as a useful foundational approach for understanding the constructs the
researcher has attempted to analyze in this study.
Cognitive processes (Theme 1). Two categories were created based on the theoretical
argumentation in accordance with Selz (1913), Dörner (1976), and Lompscher (1972) to
illustrate the cognitive processes inherent to cumulative learning: (a) aggregation with 8
subcategories and (b) abstraction with 5 subcategories (see Table 3.7).
Categories Subcategories
Aggregation(7) Identification of attributes and features of a phenomenon
Serial array of similar phenomena
Complex building/ Classification
Decomposition of a complex
Complex reproduction
Complex completion
Schema completion
Schema instantiation
Abstraction(5) Comparison of phenomena with regard to (dis-)similarities
Generalization on the basis of similarities
Identification of structural intersections
Analogy building
Mental model
Table 3.7 Classification of categories of cognitive processes
Learning strategies (Theme 2). A list of 23 learning strategies (i.e., complex cognitive
skills) reported by students was constructed as (a) cognitive strategies (with 13
subcategories); (b) metacognitive strategies (with 7 subcategories) which were differentiated
again to show how a learning activity was planned, monitored, and evaluated; and (c)
social/affective strategies (with 3 subcategories), adapted from Chamot, Küpper, and
Impink-Hernandez’s (1988, pp. 17–19) studies on learning strategies in second language
acquisition (see Table 3.8).
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Cognitive strategy (13) Meta-cognitive strategy (7) Social/Affective strategy (3)
Organization:
Grouping
Note taking
Reorganization/Reconstruction
Planning:
Advance organization
Organizational planning
Attention
Self-management
Questioning for clarification
Cooperation
Self-reinforcement
Elaboration:
Inferencing
Elaboration
Transfer
Imagery
Keyword method
Summarizing
Monitoring:
Self-monitoring
Problem identification
Repetition:
Repetition
Auditory representation
Evaluation
Resourcing
Substitution
Table 3.8 Classification of categories of learning strategies
3.2.4 Descriptions of the Coding Scheme: Cognitive Process
This section presents the operational descriptions of the coding scheme with cognitive
processes – aggregation and abstraction – used for the data analysis. These categories are
referred to in symbolic terms in the coding scheme of this study as Agg=aggregation process,
Ab=abstraction process.
3.2.4.1 Aggregation
Aggregation in this study refers to the process of aggregating units (or pieces) of
information/knowledge by identifying and extracting them in terms of elements, relations,
and functions using analogies as learning proceeds. If the aggregated knowledge satisfies the
learning goal, then it is assimilated into the existing knowledge structure so that it can be
used in subsequent learning processes. In terms of the level of knowledge, in aggregation
process, the aggregated packets of information/knowledge are at the same level in the
hierarchy of cognitive structure as learners are yet to define the shared structural features of
the aggregated packets and cognitive structure (i.e., structural similarities) in this process.
Identification of attributes and features of a phenomenon (Agg1). This can be
accomplished by identifying attributes and features of information in a given context.
Attributes refer to a quality regarded as a characteristic or inherent part of something or
someone (e.g., a material object used in art to identify mythical figure). Features refer to a
distinctive attribute or aspect of something or someone (e.g., a part of the face, such as mouth
or eyes, which makes a significant contribution to its overall appearance). For example, to
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learn the concept of “string instrument,” a musical instrument that produces sound by means
of vibrating strings, the necessary precondition is the adequate identification of a set of
attributes and features to characterize the entities of this concept (e.g., size, shape, sound
producing mechanism, and playing technique). Hence, this is a fundamental exploratory
process in the course of learning.
Serial array of similar phenomena (Agg2). The identification of similar phenomena can
be accomplished by defining perceptual and surface similarities of (a) individuals and
elements, (b) underlying relations, and/or (c) functions of the aggregated information (Selz,
1913; as cited in Seel, 2003, p. 175). A learner compares the perceptual and surface
similarities of new information with his/her existing schemas to investigate whether the
problem situation can be subsumed under already known rules, principles, and laws and then
puts pieces of information together into a serial array to form larger chunks of information.
Such perceived phenomena are then encoded as cognitive operations referred to as
“internalized actions” (i.e., actions executed in thought) and associated with specific
conditions for execution (Seel, 2003). This “lawful activity” in turn promotes the learner’s
semantic knowledge concerning similar phenomena because s/he perceives them as a flowing
unified set rather than as an individual entity. For instance, in the process of learning the
concept of “string instrument,” a learner may arrange some similar phenomena (e.g., its size,
shape, sound producing mechanism, and playing technique) of the musical instrument that
produces sound by vibrating strings. Hence, recognizing similarity is a fundamental aspect of
mental processes as prior scientists have acknowledged the same (e.g., Aristotle’s principle of
association by resemblance).
Complex building/Classification (Agg3a). This can be accomplished by defining
characteristics of information to combine equivalent relevant attributes (Klix, 1971; as cited
in Seel, 2003, p. 176). By coordinating (i.e., classifying) multiple units of information and
their relational connections to existing schemas, the learner integrates the information into
hierarchically organized integrative cognitive structures. For instance, to learn the concept of
“string instrument,” the learner may arrange different examples of it, such as violin, harp,
guitar, and piano by identifying attributes. The learner then analyzes the arrangements of
elements in a set to associate separate parts in order to form a new arrangement (Jüngst,
1978). This idea is in line with “globalization,” a term coined by Meili (1944) involving the
synthesis or separation of data (as cited in Heller, 1976, p. 15). Here, the learner’s thinking
process focuses on forming concepts for the elements of the aggregated information and
begins to form new relationships. Some necessary abilities for this process are flexibility in
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thinking (Krutetskii, 1976) and reversibility, which enables the learner to relate all
possibilities to their consequences. The learner now advances from exploratory processes to
the beginnings of a comprehension of concepts, meanings, and relationships, and hence
identifies structural invariants across the series of information within the context of his/her
cognitive structure. As a result, internal complexity of the learner’s cognitive structure is
decreased.
Decomposition of a complex (Agg3b). This can be accomplished by defining, isolating,
and comparing characteristics of information with the goal of forming invariants for
quantities of attributes (Klix, 1971; as cited in Seel, 2003, p. 176). The features belonging to
each unit of different aggregated information should be separated in a meaningful way to
make them flexible and link them closely to each other within the context of a learner’s
cognitive structure. This ensures that they remain well integrated throughout the whole
schematization process in learning. The learner should be able to identify details embedded in
units of information and disassemble them into a set of concepts that have particular relations
to one another. Thus, the learner should be able to consider and represent simultaneously
several relationships and isolate a single aspect. This can be achieved by stimulating
corresponding information stored in one’s memory. With this process of differentiation of
multiple units of information (from more general to more specific), the information is more
distinctively integrated into the learner’s cognitive structure. This differentiation is derived
from the principle of equilibration, which is stated in section 2.1.6.
Complex reproduction (Agg4). This can be accomplished by identifying and extracting
information from multiple sources in terms of the following components: (a) individuals or
elements, which refers to comprehension of an object and understanding of its properties (e.g.,
“chair” or “a piece of furniture”); (b) relations, which refers to understanding of the attribute
relation. It requires the ability to isolate and compare attributes, find similarities and
differences between facts (i.e., identify basic operation types), and identify its characteristics
and actively construct relations between concepts; and (c) functions, which refers to
identifying the function (e.g., “to sit” or “to write”) of an object (Selz, 1913; as cited in Seel,
2003, p. 175).
More specifically, understanding relations refers to (re-)discovering or (re-)producing
information/knowledge through a learner’s own cognitive information processing activity.
For instance, in problem solving in mathematics a learner should be able to perceive the
relation between the elements of a problem (Krutetskii, 1976) and thus formulate provisional
rules and principles from given material (Thurstone, 1938). In this process, the ability to
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question his/her constructive point of view is critical. For instance, when a particular
procedure is applied to the same situation and always results in the same consequences, this
specific phenomenon (i.e., input-consequence relation; classically S-R relation) is encoded
and stored in his/her memory through self-modifying processes (i.e., assimilation and/or
accommodation). It can be facilitated through the use of a system of symbolization (Bruner &
Oslon, 1977) such as language or concrete imagination. Here, the learner arranges elements
in a visual structure (Michael, Zimmerman, & Guilford, 1950). This is accomplished by
internalizing the figurative, functional, and/or operative regularities and invariants identified
in the learning situations. This way, the learner is able to represent the learned content and
concepts with his/her own “words” and hence, s/he can better understand and organize the
information, which in turn is retained longer in his/her memory.
Complex completion (Agg5). A learner can accomplish this by associating previously
unrelated ideas to form a new unified idea. Units of simple information/knowledge are
grouped together in various ways to create complex knowledge. Here, the learner forms
(new) hypotheses and tries them out mentally or physically on the basis of the identified
associations and directs how given information should be processed (evaluation). While
doing this, the learner either extends the existing associations or creates new associations to
form cognitive “gestalts” via constructive associations. In general, the associated perceptual
images (either textual or visual) that have often followed one another in the previous learning
experience are more likely to follow one another in the future learning sequences. However,
sometimes the learner will consciously reject one of these so-called “prototypical scripts”
(Schank & Abelson, 1977) and direct his/her thinking differently upon recognizing that it
would conflict with the given situation. This is a critical point in this cognitive process
because the learner now begins to construct new associations which have not acquired in
previous learning. For instance, in the case of mathematical problem solving, the learner
might find new associations between elements in the information of the problem that are not
explicitly stated. That is, the learner begins to understand underlying conceptual connections.
This can be done through the formation of mathematical gestalts (Cameron, 1925).
Schema completion (Agg6a). A learner completes schemas by putting aggregated and
extracted information into a coherent structure (i.e., schema), a configuration of interrelated
features that define a concept, principle, or “laws.” Schemas are flexible integrations of
complex phenomena and actions. Existing mental representation can be conceived as a
hierarchical cognitive structure or as a system of semantic networks which can be constantly
elaborated, reorganized, and/or described in greater detail (Strittmatter, 1990). A schema can
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be completed if all relationships among all information to be integrated into the schema are
completely specified. A schema contains a general form (i.e., prototypical of a phenomenon)
that is constant but generalizable across particular occasions as well as slots that are filled
with information about a particular experience. Thus, a schema guides thinking processes and
makes a person capable of useful and meaningful inferences (i.e., schema anticipation; Selz
1913). The necessary precondition for this ability is the knowledge that a given object (e.g.,
eyes) is part of a larger complex (e.g., a face). This makes it possible for the learner to
complete a drawing of a face in which the eyes are missing. As stated in the theoretical
review, (re-)organization and (re-)structuring are at the center of schema completion. They
involve using prior knowledge to reach in-depth understanding through complex thinking
process. The creation of meaningful links between existing knowledge and new information
promotes the gradual acquisition of in-depth knowledge.
Schema instantiation (Agg6b). An abstract schema can be rendered less abstract during
the process of instantiation (i.e., supplying values for the variables in a schema). This
involves relating general modes of descriptions to concrete situations. That is, a learner
activates cognitive operations and associates them with specific conditions for execution. For
instance, the process of learning the concept like that of a “string instrument” involves a
process of generalization (e.g., string, sound, and vibration). By repeating different types of a
generalization (e.g., under-generalization or over-generalization) the learner can generate a
set of alternative general concepts, and may then use this particular schematic instance in a
subsequent learning situation.
The necessary condition for the execution of a particular operation is activation of all
possible conditions along with the attributes specifically connected to them by way of
concrete logical thought. This activation process includes direct as well as “indirect activation
of the attributes stored in defined traces of knowledge memory” (Seel, 2003, p. 49). By doing
this, the learner can determine and thus internalize the causal relations between action and
consequence. For instance, previously acquired concepts can be related back to the present
learning situation and interpreted in the learning content; and (newly) interpreted concepts
and/or principles can be applied to understand the concrete content. Accordingly, with
persistent practice and in-depth investigations in various problem solving situations, a learner
can determine which specific concepts from general abstractions can be applied in a
particular situation. In doing this, the learner reorganizes or reconstructs the contents of
his/her existing schemas by assimilating or accommodating them in accordance with the new
information. This allows the learner to build up ever more general concepts by acquiring
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various simple concepts in the course of particular experiences. These perceived results are
then stored for further use and hence influence activities in subsequent learning.
The above stated descriptions of the coding scheme of “aggregation” process are
summarized in the following table 3.9.
Code Subcategories Operational Description
Agg1 Identification of
attributes/features
of a phenomenon
It can be accomplished by identifying attributes and features of a
phenomenon of information: (a) attributes – a quality regarded as a
characteristic or inherent part of information; and (b) features – a
distinctive attribute or aspect of information.
Agg2 Serial array of
similar phenomena
It can be accomplished by defining perceptual and surface
similarities of (a) individuals/elements, (b) underlying relations,
and/or (c) functions of the aggregated information. A learner
compares the perceptual and surface similarities of given information
with his/her existing schemas and then puts the information into a
serial array. The learner forms and analyzes the arrangements of
elements in a set to associate parts to form a new arrangement or to
invert them.
Agg3a Complex building/
Classification
It can be accomplished by defining characteristics of information to
combine equivalent relevant attributes.
Agg3b
Decomposition
of a complex
It can be accomplished by defining characteristics of information to
isolate and compare these attributes with the goal of forming
invariants for quantities of attributes. The features belonging to each
unit of different aggregated information should be separated in a
meaningful way to make them flexible and link them closely to each
other. This ensures that they remain well integrated throughout the
schematization process in learning.
Agg4 Complex
reproduction
It can be accomplished by identifying and extracting information
from multiple sources in terms of the following components: (a)
individuals/elements, which requires comprehension of an object);
(b) relations, which requires the ability to isolate and compare
attributes, find the similarities and differences between facts,
understand the attribute relation, identify characteristics, and
construct relations between concepts; and (c) functions, which
requires identification of the function of an object.
Agg5 Complex
completion
It can be accomplished by associating previously unrelated ideas to
form a new unified idea. Units of simple information and knowledge
are grouped together in various ways to create complex knowledge.
Agg6a
Schema completion It can be accomplished by organizing aggregated and extracted
information into a coherent structure (i.e., a schema). The schema
can be completed if all relationships among all information and
knowledge to be integrated into a schema are completely specified.
Agg6b Schema
instantiation
An abstract schema is rendered less abstract during the process of
instantiation (i.e., supplying values for the variables in a schema).
Table 3.9 Coding list of Theme1: Aggregation (Category 1)
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3.2.4.2 Abstraction
Abstraction in this study refers to the process of structural mapping between entities of
information and knowledge and hence changes the description of it based on existing
knowledge. Accordingly, it reduces the amount of information in one’s cognitive structure in
a way that the information relevant to the learning goal is preserved, while the irrelevant
information is ignored. Learners abstract knowledge through thinking, analogical reasoning,
(re-)organizing, and (re-)structuring cognitive structures, and hence construct a higher level
of knowledge in the hierarchy of cognitive structure with abstraction process. The learners
perceive and process information at different levels of abstractions, and therefore, as
cognitive structures develop, the level of abstraction becomes higher.
Comparison of phenomena with regard to similarities and dissimilarities (Ab1). A
learner compares phenomena with regard to similarities and dissimilarities between the
superficial and underlying structure of information/knowledge by analogy. For instance, in
problem solving the learner compares and sorts phenomena on the basis of the results. The
content of the phenomena is compared and patterns are observed. The learner’s mental
comparison of the records (similarities vs. dissimilarities) leads to the recognition of patterns,
which are derived by abstracting the relationship between activity and effect. This abstracted
activity-effect relationship involves a coordination of conceptions (Piaget, 1977/2001) as
both of them are tangible representations of available conceptions. By abstracting problem
elements and meanings, the learner can identify similar situations with similar conceptual
structures and formulate a solution. Accomplishment of this presupposes some basis for
identifying what is significant and for extracting what is invariant across phenomena.
Throughout the course of learning, the learner gradually progresses from focusing on
superficial aspects of the phenomena to their underlying aspects. Here, the surface
identification may not be competent when a certain context requires complex knowledge (i.e.,
identifying deep structure). However, in the view of the presented theoretical framework of
cumulative learning, it is likely to be linked with relevant information and hence would
provide some basis for facilitating cumulative (re-)organization and (re-)structuring of the
cognitive construction (i.e., to help make the whole system of cognitive structure develop).
Generalization on the basis of similarities (Ab2). A learner can accomplish this by
extending the size of the description of information and knowledge and then transforming
descriptions along the set-superset in his/her cognitive structure. The learner extracts
commonalities from the superficial as well as the underlying structure of information and
knowledge by analogy and then generates a description that is larger than the given specific
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information using inferences (i.e., inductive, deductive, analogical) on the basis of similarities.
For instance, in concept learning, with generalization, a learner can correctly identify that a
particular entity should be categorized under a specific concept category without prior
learning. By repeating different types of generalization (e.g., over-generalization or
under-generalization), the learner can identify that the characteristic “sound is produced by
vibrating strings” is the important element of a learning situation (i.e., learning the concept of
“string instrument”) that should be abstracted. Thus, when these processes are repeated, the
units of information/knowledge that belong to the specific schema become increasingly
routinized and automated as the learner consistently associates the new information with
his/her existing knowledge through extensive practice.
As Jüngst (1978) stated, the learner abstracts information and knowledge by filtering
out invariant and irrelevant attributes and thus forms and analyzes abstract concepts through
the intensification of attributes and inversion. The ability to make accurate and broad
generalizations is critical to the process of learning because this extends the scope of
adaptability of acquired information and knowledge to further learning and other situations.
Such generalization can be accomplished for any object or situation that an individual
experiences and is likely to be applicable to a similar form and context. The learner can
generalize from one situation to another for s/he perceives that they are likely to be grouped
together when they share the same input-consequence mechanism.
Identification of structural intersections (Ab3). A learner accomplishes this by defining
structural relations (e.g., attribute subordinate-superordinate relation) between information/
knowledge that are formed through the intervention of cognitive operations that isolate and
compare attributes, find similarities, and form dimensions. In doing this, the learner is
actively indentifying characteristics of the information/knowledge and constructing structural
relations between them. It may either take the form of (a) sensory abstraction, which is based
on the selective isolation of attributes, or (b) operative abstraction, which is based on the
conjunctive association of independent attributes, which in turn leads to the formation of
concept structures (Klix, 1971; as cited in Seel, 2003, p. 176). For instance, learning
mathematics requires a learner to perceive formalized mathematical material (Krutetskii,
1976). That is, the learner should understand the definite structure of mathematical material.
A cognitive structure that is clear and well organized facilitates the learning and retention of
new information, while a cognitive structure that is confused and disorderly hampers learning
and retention, as Ausubel (1960) emphasizes. Accordingly, learning can be enhanced by
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elaborating on relevant information/knowledge of cognitive structure. In this way, the learner
comprehends and constructs complex structures.
Analogy building (Ab4). A learner can accomplish this by associating new information
with existing knowledge and then integrating it by assimilating and accommodating his/her
cognitive structure in a meaningful way using analogy (e.g., comparing one’s mental models
of the two structures by analogy ― existing knowledge and the present cognitive structure by
analogy). This allows the learner to create new connections between two analogues which
initially seem unrelated, because s/he progressively recognizes the similarities and
commonalities between the two situations. This in turn strengthens the learner’s cognitive
structures, and hence, helps the learner to store and retain information in their memory longer
and effectively. Through a continuous integration process, the learner constructs new
concepts, principles, and laws on the basis of basic structures. While doing this, the learner
breaks with old structures and anticipates (to some extent) new patterns. In other words, the
learner resolves internal conflicts by linking large parts of knowledge to specific sequences of
analogical arguments in a logically structured way, and hence the learner’s cognitive process
is enhanced through the specific situation.
For instance, solving a complex math problem requires basic mathematical knowledge
(e.g., mathematical terms, formulas) as well as higher-level knowledge requiring creativity
and logical thinking ability in order to fully understand mathematical concepts, principles,
and verification processes. Therefore, developing mathematical problem solving ability
requires a high-level of complex cognitive skills. The learner can acquire and develop
mathematical knowledge by using appropriate inquiry skills dialectically, such as convergent
(i.e., inductive, abductive reasoning, and/or deductive reasoning that verifies and confirms
whether an idea is appropriate or not), divergent (i.e., creating ideas, finding concepts, rules,
and principles), and analogical thinking. The learner seeks deep meanings and relationships
using analogy. Abstracted causal (i.e., IF-THEN) relationships of this kind promote the
development of a “new” conception. Consequently, the notion of anticipation, as emphasized
by Piaget (1971), is the key to the cognitive construction. Conceptions generated or learned
from the construction become the learner’s ability to anticipate the effect of a particular
activity without mentally or physically performing it.
Mental model (Ab5). A learner accomplishes this by creating mental models and/or
modifying pre-existing mental model(s). As the learner progressively develops his/her
knowledge and skills, a particular piece of knowledge which was at one time satisfactorily
anchored in his/her cognitive structure could be proved to be inadequate with newly gained
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information. When existing knowledge does not fit the learning situation (or a situation for
solving a problem), one experiences disequilibrium (i.e., the balance between one’s
knowledge and reality is broken). Upon encountering cognitive dissonance of this kind, the
learner examines the possible causes and hence starts to investigate the relationships between
his/her existing knowledge and the current learning situation (or that required to solve a given
problem). In other words, when new information/knowledge contradicts existing knowledge,
new strategies must be developed to resolve the resulting cognitive conflict by deleting
and/or modifying components in one’s cognitive structure (accommodation). The conflicting
information can also be assimilated into the learner’s existing knowledge structure if it can be
meaningfully disassembled. Then the learner reorganizes his/her cognitive structure
correspondingly.
A learner thus actively creates mental models (i.e., radical alternatives). Mental models
can be modified through abstraction (compressing or removing certain attributes) or
concretization – the addition of attributes (Selz, 1913; as cited in Seel, 2003, p. 175). In this
process, the learner clearly understands the intent of the problem and hence predicts and
constructs mental models that can be applied to solve it. Since various ideas can be
formulated to solve a problem, many models can also be generated for each problem. The
learner explores several induced mental models by generalizing the content and formalizing
and refining their representation. The learner then verifies these models to find out whether
they are appropriate to solve the given problem and critically analyzes and assesses them with
respect to properties of his/her existing knowledge. In doing this, the learner clarifies
concepts and thus (re-)organizes/ (re-)structures the learned content accordingly.
The above stated descriptions of the coding scheme of “abstraction” process are
summarized in the following table 3.10.
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Code Subcategories Operational Description
Ab1 Comparison of
phenomena with
regard to
similarities and
dissimilarities
It can be accomplished by comparing phenomena with regard to
(dis-)similarities between the superficial and underlying structure of
information/knowledge by analogy: (a) abstraction through a
filtering out of invariant and irrelevant attributes; and (b) formation
and analysis of abstract concepts through the intensification of
attributes and inversion.
Ab2 Generalization on
the basis of
similarities
It can be accomplished by extending the size of the description of
information/knowledge and transforming descriptions along the
set-superset in one’s cognitive structure. It can be accomplished
because the learner can generate a larger description than the given
specific information/knowledge through inferences (inductive,
deductive, and analogical) on the basis of similarities.
Ab3 Identification of
structural
intersections
It can be accomplished by defining relations within/between
concepts (e.g., attribute relation, subordinate-superordinate relation)
that are formed through the intervention of cognitive operations such
as isolate and compare attributes, find similarities, and form
dimensions. In doing this, the learner actively constructs structural
relations between concepts. It may either take the form of (a) sensory
abstraction, which is based on the selective isolation of attributes, or
(b) operative abstraction, which is based on the conjunctive
association of independent attributes, which in turn leads to the
formation of concept structures.
Ab4 Analogy building It can be accomplished by associating new information with existing
knowledge and then integrating it by assimilating and
accommodating one’s cognitive structure in a meaningful way using
analogy (e.g., comparing one’s mental models of the two structures
– existing knowledge and the present cognitive structure by
analogy). In doing this, the learner can create new connections
between two analogues which initially seem unrelated, because s/he
recognize the similarities and commonalities between the two
situations.
Ab5
Mental model It is accomplished by creating mental models and/or modifying
pre-existing mental model(s). When the learner identifies that new
information is a special case of the existing knowledge, s/he then
creates an instant mental model that links the new information to the
relevant parts of existing knowledge so that an appropriate link can
be made to it by analogy. The modification of the mental model can
be accomplished by abstraction (i.e., compressing or removing
certain attributes) or by concretization (i.e., adding attributes).
Table 3.10 Coding list of Theme 1: Abstraction (Category 2)
3.2.5 Descriptions of the Coding Scheme: Learning Strategy
Learning strategies refer to the systematic use of cognitive and metacognitive elements
in specific tasks (Ellis, Lenz, & Sabornie, 1987a; 1987b) or steps taken to facilitate the
acquisition, storage, retrieval, and use of information (Ehrman & Oxford, 1989). The goal of
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strategy use is to “affect the learner’s motivational or affective state, or the way in which the
learner selects, acquires, organizes, or integrates new knowledge” (Weinstein & Mayer 1986,
p. 315). Studies argue that learning strategies have significant impact on academic
performance (Bos & Anders, 1990; Swanson, 1993; Swanson & Alexander, 1997). In other
words, a deficiency of organizational and learning strategies demanded by a specific task
causes academic difficulties, a passive approach to school tasks (Torgesen, 1982), or a failure
to apply a learning strategy one has acquired to a new learning situation (Chan & Cole, 1986).
Therefore, a learner’s cognition and motivation is directly affected by learning behavior
(Mckeachie, Pintrich, Lin, & Smith, 1991). Studies have proposed various taxonomies for the
classification of learning strategies (e.g., Dansereau, 1985; Pressley, 1986; Weinstein &
Mayor, 1986). These strategies may include inferencing, deduction, imagery, summarization,
and transfer a focus on particular aspects of new information, analysis and monitoring of
information processing during its acquisition, the organization or elaboration of new
information in the encoding process, the evaluation of learning when it is completed, and
checking to confirm that the learning will be successful.
The following tables present the operational descriptions of the coding scheme used in
data analysis with learning strategies: cognitive strategies (Table 3.11), metacognitive
strategies (Table 3.12), and social/affective strategies (Table 3.13). These categories are
referred to in symbolic terms in the coding scheme of this study as C=cognitive strategy,
M=metacognitive strategy, and SA=social/affective strategy.
1) Cognitive strategies refer to the learning strategies used for manipulating incoming
information in ways that enhance learning. They may be grouped11
together as follows:
organization (grouping, note taking, reorganization/reconstruction); elaboration (inferencing,
elaboration, transfer, imagery, keyword method, summarizing), repetition (repetition,
auditory/visual representation,), resourcing, and substitution.
2) Metacognitive strategies refer to the learning strategies used for planning,
monitoring, and self-evaluation concerning learning processes, comprehension, and activities.
They may be grouped together as follows: planning (advance organization, organizational
planning, attention, self-management); monitoring (self-monitoring, problem identification);
and evaluation (self-evaluation).
3) Social/affective strategies refer to the learning strategies involving interaction with
11
This grouping is presented in Table 4.2. The cognitive strategies may be constrained to specific types of tasks
in the learning activity.
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others (i.e., asking questions for clarification, cooperative learning) to accomplish common
learning goals; and providing personal motivation by arranging rewards for oneself when a
learning activity has been successfully completed (self-reinforcement).
Code Subcategories Operational Description
Organization C1 Grouping Ordering, classifying, or labeling information (words,
terminology, concepts, and/or principles) on the basis of
their attributes or meaning; recalling information on the
basis of previous groupings.
C2 Note taking
Writing down key words or concepts in (abbreviated)
verbal, graphic, or numerical form while learning
(listening, reading, or writing) to assist performance of a
learning task.
C3 Reorganization/
Reconstruction
Constructing a meaningful or larger cognitive structure
by combining elements of information and knowledge in
a new way.
Elaboration C4 Inferencing
Using available information/knowledge to infer the
meanings or usage of unfamiliar/new information
associated with a (learning) task, to predict outcomes, or
to fill in missing information. Consciously applying
learned or self-constructed rules to understand
information, and thus constructing rules on the basis of
analysis of learned information/knowledge.
C5 Elaboration
Relating new information to prior knowledge; relating
different parts of new information to each other; and
making meaningful associations with the new
information and existing knowledge.
C6 Transfer Using previous knowledge and/or skills to assist
comprehension or production of a (learning) task. Using
the previous knowledge as a basis for understanding
and/or generating new information/ knowledge.
C7 Imagery Using visual images (mental or actual) to understand,
represent, or remember new information.
C8 Keyword
method
Remembering new information by generating easily
recalled images or words of some relationship with other
information/knowledge and the new information.
C9 Summarizing Making a mental, oral, or written summary of
information presented/learned in a (learning) task.
Repetition C10 Repetition Repeating learned contents through practice by writing or
speaking them.
C11 Auditory/visual
representation
Playing in back of one’s mind (i.e., mentally rehearsing)
the auditory/visual representation to understand or
remember information (words, phrases, or longer
sequences).
C12 Resourcing Using reference materials such as reference books,
workbooks, dictionaries, and/or internet sources in
addition to given learning materials (e.g., school
textbooks, class handouts).
C13 Substitution Selecting alternative approaches or revised plans to
accomplish a learning task.
Table 3.11 Coding list of Theme 2: Cognitive strategies (Category 1)
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Code/ Sub-categories Operational Description
Planning
M1 Advance
organization
Previewing the main ideas, concepts, or principles of the
material to be learned, often by skimming the text for the
organizing principle.
M2 Organization
al planning
Proposing strategies for handling an upcoming learning task;
generating a plan for the parts, sequence, main ideas, or
functions to be used in handling a (learning) task.
M3 Attention This can be differentiated in the following way: (a) directed
attention: deciding in advance to attend in general to a learning
task and to ignore irrelevant distracters; maintaining attention
during learning; and (b) selective attention: deciding in advance
to attend to specific aspects of information, often by scanning
for key words, concepts, and/or principles.
M4 Self-
management
Understanding the conditions that help one successfully
accomplish a learning task and arranging them for to account for
these conditions; controlling one’s learning performance to
maximize use of prior knowledge.
Monitoring
M5 Self-
monitoring
Checking, verifying, and/or correcting one’s comprehension or
performance in the course of learning and/or checking the
accuracy and/or appropriateness of one’s learning strategy/
process while it is taking place. This can be differentiated in the
following way: (a) comprehension monitoring: checking,
verifying, and/or correcting one’s understanding; (b)
performance monitoring: checking, verifying, or correcting
one’s learning performance (e.g., problem solving); (c) strategy
monitoring: tracking how well a strategy is working; (d) plan
monitoring: tracking how well a plan is working; and (e)
double-check monitoring: tracking across a (learning) task and
acts previously undertaken or possibilities considered.
M6 Problem
identification
Explicitly identifying the central point needing resolution in a
(learning) task or identifying an aspect of the task that hinders its
successful completion in the course of task.
Evaluation
M7 Self-
evaluation
Checking the outcomes of one’s own learning performance
against an internal measure of completeness and accuracy (e.g.,
a learning goal) after it has been completed; checking one’s
strategy use or ability to perform the task at hand. This can be
differentiated in the following way: (a) performance evaluation:
judging one’s ability to perform the task; and (b) strategy
evaluation: judging one’s strategy use when the task is
completed.
Table 3.12 Coding list of Theme 2: Metacognitive strategies (Category 2)
Code/ Subcategories Operational Description
SA1 Questioning for
clarification
Asking for explanation, verification, rephrasing, or examples about the
material from teachers or peers; asking for clarification or verification about a
(learning) task; posing questions to oneself.
SA2 Cooperation Working together with one or more peers to solve a problem, pool
information, check a (learning) task, model a learning activity, or get feedback
on performance.
SA3 Self-
reinforcement
Providing personal motivation by arranging rewards for oneself when a
learning activity has been successfully completed.
Table 3.13 Coding list of Theme 2: Social and affective strategies (Category 3)
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CHAPTER IV
RESULTS
This section provides findings from the analysis performed on the interview data.
During the interviews, quite a lot of what was being said appeared to be helpful as possible
answers to the research questions. However, there was the dilemma of what to extract and
what to leave out during the analysis of the interviews, because there was some discrepancy
between what the researcher was trying to investigate and the purposes for which the data had
been originally collected. That is, while the interviews focused on the specific learning
processes and strategies that are effective and efficient when preparing for the CSAT, the
present study focused on determining the learning mechanism, cognitive processes, and
learning strategies in relation to knowledge acquisition and development inherent to learning
and the extent to which they can lead to effective learning. Therefore, the researcher applied
some extent of mental “leap” for interpreting the phenomena presented in the interviews.
The analyses of segments of interviews with the 49 case studies are presented to
illustrate the scheme of the cognitive processes and learning strategies identified in the
interviews: six described in detail in the first section, and 43 summarized in brief in the
second section. Each of the complete interview transcripts of the six in-depth case studies
(case studies 1 to 6) is given in Table 4.1 to 4.6. These six cases were chosen for the in-depth
analysis from the 49 case studies for the following reasons: (1) They seem to contain more
relevant and detailed content in regard to the research questions of this study; and (2) they
represent learning different school subjects. The school subjects particularly applied in the
cases are mathematics (case 1 & 4), English (case 3), and Korean language (case 6). Case 2
and 5 represented learning all school subjects in general (case 2 & 5). Consequently, they
may show topic-specific learning phenomena between the different school subjects.
Furthermore, for the same reasons, they may also rule out possible topic-dependent
phenomena. Therefore, this in turn increases the reliability (i.e., consistency of
findings/results), and the validity of findings (i.e., whether the study in fact investigates what
was intended) as the present study intends to rule out topic-related effects while finding the
mechanism in learning. Analyzing the other 43 cases was to increase the generalizability of
the present study. Based on the analysis of the case studies, the third section presents overall
findings of the mechanisms of knowledge and skill development, cognitive processes,
learning strategies, and learning in problem solving.
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4.1 Case Studies 1 to 6
As the outset, the symbolic terms used in the transcripts and analysis of the data are as follows:
Participant’s ID is represented with “P” followed by ID number (P1 = the comment presented
in the interview with participant ID number 1).
“N” represents the narrated part of an interview by the narrator, which represents participant’s
comments about learning processes/strategies and the behavior recorded during interview. This
is represented with “N” followed by ID number (N1= the narration presented in the interview
with the case of participant ID number 1).
“T” represents teacher’s comment in the corresponding interview (T5=teacher’s comment
presented in the interview with the case of participant ID number 5).
“E” represents expert’s comment in the corresponding interview (E5=expert’s comment
presented in the interview with the case of participant ID number 5).
Grade level is presented as follows: HS2=High school 2nd
grade; HS3=High school 3rd
grade;
UF=University freshman; US=University sophomore; CF=College freshman; CS= College
sophomore.
4.1.1 Case 1
P1 (HS2) uses a unique strategy in learning and solving math problems, which is presented in
the following narration:
N1: The participant interprets and represents a mathematical problem by drawing shapes and
figures (i.e., using different squares, triangles, circles, lines, dots, graphs, & etc) and then
reorganizes it [the mathematical problem] like a story for s/he does not like memorizing
mathematical formulas.
After determining the problem, P1 represents the elements of the problem using visual
images to better understand the information in the problem statement ([1.1], C7 imagery),
and thus assists performance of the learning by drawing concepts in graphic form while
learning ([1.2], C2 note). While doing this, P1 reproduces the information in a visual
structure by internalizing the figurative, functional, and/or operative regularities and
invariants identified in the problem ([1.3], Agg4 reproduction). This can be accomplished by
identifying and extracting the information in terms of the following components: (a)
individuals/elements, which requires comprehension of an object; (b) relations, which
requires the ability to isolate and compare attributes, find similarities, identify characteristics,
construct relations between concepts, understand the attribute relation, and compare the
similarities and differences between facts; and (c) functions, which requires identification of
the function of an object. P1 then constructs a meaningful larger cognitive structure
(“reorganizes it like a story”) by combining elements of the information in a new way ([1.4],
C3 reorganization); and by relating and making meaningful associations of different parts of
the information to each other (code [1.5], C5 elaboration). While doing this, P1 uses
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available information/knowledge to infer the meanings and usage of the information with the
problem by applying learned or self-constructed rules (i.e., using visual images) to
understand the information by analogy ([1.6], C4 inferencing). P1’s unique learning
processes are clearly presented in the following snippets of comments:
P1: I solved math problems using two strategies, by the typical method (i.e., solving by
principles and formula) as well as by my own strategy (i.e., illustrating the problem with shapes
and figures), in order to compare them with regard to correctness and efficiency. In order to
compare the typical method [solving by principles and formula] with my own strategy [solving
by shapes and figures], I first collected the information about mathematical concepts, principles
and formulas from multiple conceptual books, and then reorganized them into a structure in my
conceptual note….Then I solved math problems using the two strategies [by the typical method
and by my own strategy]. The answers I got using the two strategies were different and most of
the time, the answers that I got using my own strategy were incorrect.
P1’s cognitive processes and learning strategies are clear from what s/he said. After
determining the goal of the learning task (i.e., compare two different strategies with regard to
correctness and efficiency), P1 first aggregates relevant units of information (i.e., concepts,
principles, formulas) from multiple resources by identifying attributes and features of a
phenomenon of information ([1.7], Agg1 identification; [1.8], C12 resourcing). P1 then
identifies and extracts the aggregated information in terms of their individuals/elements,
relations, and functions ([1.9], Agg4 reproduction); and associates the aggregated information
to each other in order to form a unified set by reorganizing them ([1.10], C3 reorganization)
in order to create a more unified complex knowledge ([1.11], Agg5 completion) by relating
and making meaningful associations of different parts of the information to each other using
analogy ([1.12], C4 inferencing; [1.13], C5 elaboration). And then P1 solves the problems
using two different mental models, one is constructed by the typical method and the other is
constructed by his/her own strategy ([1.14], Ab5 model). While doing this, P1 compares the
(dis-)similarities of the two mental models, which is presented in the following narration:
N1: The participant recognized that the condition under which his/her own strategy did not
work successfully was when s/he did not correctly understand the complex multiple relations
contained in the problem statement.
This illustrates that P1 identifies the attributes and features of the information contained in
the problems statement in an attempt to compare the phenomena of applying the two mental
models ([1.15], Agg1 identification); and then identifies the critical factor that affects the
results of problem solving by isolating and comparing the attributes contained in the problem
statement ([1.16], Agg3b decomposition).
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N1: The participant thought about the reasons why s/he could not understand the complex
multiple relations contained in the problem statement, and decided to use the strategy that s/he
used during elementary school, which was highly successful in solving math problems, in order
to find out some flaws in his/her current problem solving strategy.
This illustrates that P1 understands the conditions that help him/her more successfully
accomplish a learning task ([1.17], M4 management). P1 checks and verifies his/her
comprehension (i.e., comprehension monitoring) and performance (i.e., performance
monitoring) in problem solving in order to check the accuracy and appropriateness of the
cognitive process and learning strategy (i.e., strategy monitoring) in the course of learning
([1.18], M5 monitoring). P1 explicitly investigates the attributes/features of the condition
under which his/her problem solving strategy did not work successfully and identifies the
central point needing resolution in the further learning situations ([1.19], M6 problem); and
checks the outcomes of his/her own problem solving performance against an internal measure
of completeness and accuracy of the learning goal after it has been completed (i.e.,
performance evaluation), and thus checks his/her strategy use or ability to perform the task at
hand (i.e., strategy evaluation; [1.20], M7 evaluation). Upon figuring out the central point
needing resolution in the learning task (“s/he could not understand the complex multiple
relations contained in the problem statement”), P1 resources the basic-level conceptual book,
which is better fit into his/her particular learning style ([1.21], C13 substitution). With this
modified strategy, it appears to be a better support to his/her understanding of the complex
multiple relations involved in the problem statement. This in turn reflects that P1 actively
modifies his/her cognitive structure by identifying and learning the sub-components in the
structure that should be firmly corrected and completed in order them to be more successfully
incorporated into the whole cognitive structure. In doing this, P1 presumably constructs
relations between (units of) information (e.g., attribute relation, subordinate-superordinate
relation) and compares the similarities and differences between the information. This is
evidenced as follows:
N1: Therefore, s/he first read a basic-level (equivalent to elementary school level) conceptual
book, which is composed of drawings and stories. While reading the book, s/he could clearly
understand the relations between the concepts contained in the book, and thus, realized that
some of the math concepts (e.g., calculating distance, weight, and understanding numbers) are
closely associated with modern society.
This shows that P1 defines relations between the concepts. The plausible interpretations of
the underlying cognitive processes that resulted in this phenomenon would be as follows:
This can be accomplished through the intervention of cognitive operations; such as isolate
100
and compare attributes, find similarities, and form dimensions. In doing this, P1 actively
constructs structural relations between the concepts by operative abstraction, which is based
on the conjunctive association of independent attributes of the information ([1.22], Ab3
structural). Therefore, P1 realizes how the mathematical concepts are applied to the modern
society. As P1 compares the mathematical concepts and the phenomena in the modern society,
P1 generalizes the specific math concepts (e.g., calculating distance, weight, and
understanding numbers) through inferences on the basis of the similarities between the
mathematical concepts and the phenomena in the modern society ([1.23], Ab1 comparison;
[1.24], Ab2 generalization). P1 then reorganizes his/her cognitive structure in a meaningful
way using analogy by comparing his/her mental models of the two phenomena: mathematical
concepts and modern society. Therefore, P1 can create new connections between the two
phenomena which initially seem unrelated as s/he recognizes the similarities and
commonalities between the two ([1.25], Ab4 analogy).
P1’s intentions of the cognitive processes implemented in the course of leaning are
clearly presented in the following comment:
P1: ….I tried to identify the structural [emphasis added] similarities and differences between
the two strategies [solving by principles and formula vs. solving by shapes and figures], and
tried to figure out the flaws in my strategy, and then I modified my strategy correspondingly. I
continued to modify my strategy to fill the gap between the two strategies through deliberate
practice [emphasis added]….I was so happy that my own strategy could be used appropriately
to achieve a top rank score in a math test [in the end].
P1 actively compares the structural similarities and differences between the two problem
solving strategies: typical method and P1’s own strategy ([1.26], Ab3 structural). This can be
constructed through the intervention of cognitive operations (isolate and compare attributes,
find similarities, and form dimensions). In doing this, P1 actively compares his/her mental
models of the two problem solving strategies. P1 then modifies the mental model of his/her
own strategy by figuring out the flaws in it, and then reorganizes his/her cognitive structure
accordingly using analogy ([1.27], Ab4 analogy; [1.28], Ab5 model). As P1 continually
going through these modification, (re-)organization, and (re)structuring processes, P1
progressively completes problem solving schema ([1.29], Agg6b schema completion). It is
thus assumed that more advanced and stable problem solving schema can be constructed as
P1 progressively integrates extracted modifications into the schema while instantiating the
schema through deliberate practice ([1.30], Agg6b instantiation).
101
In sum, in the course of learning, (1) P1 first identified and set the goal of learning by
identifying the requirements of a learning task (in lines 8-9; 17-20; 25-26). To do this, P1 first
determined the factual knowledge (e.g., facts, concepts, principles, and rules) and procedural
knowledge (e.g., procedures for solving complex mathematics problems) that were required
to meet the learning goal (in lines 8-10; 21-22). (2) P1 then grouped these components of
information/knowledge together and reorganized them into a coherent structure (in lines
10-11), which presumably was accomplished by breaking them down on the basis of
constructive schematic principle using existing knowledge. By doing this, P1 acquired
concepts, principles, and procedures more complex than that s/he already had available.
Hence, new components of information and knowledge were systematically integrated into
P1’s existing knowledge structure in the course of learning (in lines 1-4; 22-24). (3)
Thereafter, these complexly reproduced components of information/knowledge were
progressively mapped onto P1’s cognitive structure (i.e., schematization; in lines 22-24) until
P1 constructed a whole structure which functions at a level that is satisfactory for completing
a learning task (in lines 25-27). Thereby, P1 could extend the range and complexity of
interconnected relationships that s/he was able to subsume under a hierarchical cognitive
structure (in lines 22-24). That is, the information/knowledge in the cognitive structure
interacted with each other and was appropriately subsumed under a relevant and more
inclusive conceptual structure. This whole learning process seemed to be accomplished by
systematic cumulative integration between simpler parts and complex whole cognitive
structures.
The analysis of case 1 seems to give evidence that P1 developed math problem solving
knowledge and skills by continuously (re-)organizing and (re-)structuring a cognitive
structure that is more advanced and complex than prior (or existing) ones by defining the
features, relations, and functions of the elements of the newly gained information. P1
continuously modified the cognitive structure based on the learned organizational principles
that can interconnect the newly gained information with existing schemas until P1 acquired
satisfactory schemas. Consequently, case 1 seems to support the assumptions of this study
that (1) learning is a cumulative process wherein the learning in each new sequence builds
upon knowledge acquired in a previous sequence; and (2) learning is a structuring process in
which less inclusive information/knowledge are subsumed under higher and more inclusive
ones. This in turn seems to answer the first research question in this study. The above stated
analysis is shown in the following table 4.1.
102
Table 4.1 Coding of interview transcription: Case 1
Transcript of case1(Grade: HS2) Cognitive processes Learning strategies
1
2
3
4
N1: ….The participant interprets and represents a mathematical problem by drawing shapes and figures (i.e.,
using different squares, triangles, circles, lines, dots, graphs, & etc) and then reorganizes it like a story for
s/he does not like memorizing mathematical formulas and then reorganizes it like a story for s/he does not like
memorizing mathematical formulas.
[1.2], Agg4 reproduction
[1.1], C7 imagery
[1.3], C2 note
[1.4], C3 reorganization
[1.5], C5 elaboration
[1.6], C4 inferencing
5
6
7
P1: I solved math problems using two strategies, by the typical method (i.e., solving by principles and
formula) as well as by my own strategy (i.e., illustrating the problem with shapes and figures), in order to
compare them with regard to correctness and efficiency.
8
9
10
11
12
13
P1: In order to compare the typical method [solving by principles and formula] with my own strategy
[solving by shapes and figures], I first collected the information about mathematical concepts, principles and
formulas from multiple conceptual books, and then reorganized them into a structure in my conceptual note
….Then I solved math problems using the two strategies [by the typical method and by my own strategy].
The answers I got using the two strategies were different and most of the time, the answers that I got using
my own strategy were incorrect.
[1.7], Agg1 identification
[1.9], Agg4 reproduction
[1.11], Agg5 completion
[1.14], Ab5 model
[1.8], C12 resourcing
[1.10], C3 reorganization
[1.12], C4 inferencing
[1.13], C5 elaboration
14
15
16
17
18
19
20
N1: The participant recognized that the condition under which his/her own strategy did not work
successfully was when s/he did not correctly understand the complex multiple relations contained in the
problem statement.
N1: The participant thought about the reasons why s/he could not understand the complex multiple relations
contained in the problem statement, and decided to use the strategy that s/he used during elementary school,
which was highly successful in solving math problems, in order to find out some flaws in his/her current
problem solving strategy.
[1.15], Agg1 identification
[1.16], Agg3b decomposition
[1.17], M4 management
[1.18], M5 monitoring
[1.19], M6 problem
[1.20], M7 evaluation
[1.21], C13 substitution
21
22
23
24
N1: Therefore, s/he first read a basic-level (equivalent to elementary school level) conceptual book, which is
composed of drawings and stories. While reading the book, s/he could clearly understand the relations
between the concepts contained in the book and thus, realized that some of the math concepts (e.g.,
calculating distance, weight, and understanding numbers) are closely associated with modern society.
[1.22], Ab3structural
[1.23], Ab1 comparison
[1.24], Ab2 generalization
[1.25], Ab4 analogy
25
26
27
28
29
30
P1: I tried to identify the structural [emphasis added] similarities and differences between the two
strategies, and tried to figure out the flaws in my strategy, and then I modified my strategy correspondingly.
I continued to modify my strategies to fill the gap between the two strategies through deliberate practice
[emphasis added]…. I was so happy that my own strategy could be used appropriately to achieve a top rank
score in a math test. Also, solving math problems using my own strategy has lots of advantages for
instance, it is not easily forgotten, and it is faster and fun than the typical strategy.
[1.26], Ab3 structural
[1.27], Ab4 analogy
[1.28] Ab5 model
[1.29], Agg6b schema
[1.30], Agg6b instantiation
103
4.1.2 Case 2
P2 (CF) learns mainly by “reading” texts, which is presented in the following narration:
N2: In order to fully attend to school lessons, s/he took fewer notes and concentrated on
listening to the class teacher’s lesson….S/he tried to find relations and/or interconnectivity
between school subjects instead of studying each subject in a separate manner to effectively
utilize the information [from different school subjects]….S/he focused on the interconnectivity
of the concepts from various topics of the school subjects, and hence, could keep the
concentration level high irrespective of the conditions [i.e., learning by reading] under which
he/she studied.
P2’s cognitive processes are clearly presented in the following snippets of comments:
P2: I tried to understand information by repeatedly reading whole parts of textbooks as well as
other reference books rather than studying particular sections of the books that were considered
“important.” I studied this way so I could grasp the information in the books as a whole
[interconnected one instead of as locally segregated pieces].
P2 first identifies attributes and features of a phenomenon of the information by reading
whole parts of the learning materials ([2.1], Agg1 identification) using various reference
materials in addition to school textbooks ([2.2], C12 resourcing).
P2: [preceding comment cont’d], Hence, while I was reading books, I tried to understand how the
whole parts of the books were connected with each other, and then I tried to mentally encode
them in an orderly fashion.
P2: I mainly studied by reading learning materials, which is a less efficient strategy for
concentrating during study when compared with learning by writing or speaking. So, in order to
supplement such inefficiency, I tried to associate the information from different subjects to each
other…and tried to link the current learning content with previously learned content….I wanted
to make my brain like a dictionary.
These comments provide evidence that P2’s learning is enhanced by elaborating relevant
information/knowledge of cognitive structure. This is accomplished in each part of learning is
meaningfully subsumed under and compiled into a comprehensive learning through
schematization in the course of learning.
While reading the materials, P2 identifies the characteristics of the information in terms
of its individuals or elements, relations, and/or functions ([2.3], Agg4 reproduction); and then
actively associates the information to each other to form to create a new unified complex
whole, which is presumably accomplished by organizing a unit of simple information in
various ways ([2.4], Agg5 completion). While organizing the information, P2 compares the
phenomena with regard to similarities and dissimilarities of superficial as well as the
underlying structure of information by analogy ([2.5], Ab1 comparison); and defines relations
between information through the intervention of cognitive operations, and hence, identifies
104
structural interconnectivity of it (e.g., attribute relation, subordinate-superordinate relation;
[2.6], Ab3 structural). It seems that P2 facilitates operative abstraction based on the
conjunctive associations of independent attributes, which in turn leads to the construction of a
cognitive structure: While doing this, the features belonging to each unit of the information
should be separated in a meaningful way to make it flexibly linked to each other. This
ensures that each of the information remain well integrated throughout the schematization
process in the course of learning. This in turn enables P2 to have a more lasting memory of the
learned content. P2 explicitly states that s/he tried to understand the learning content in terms
of its underlying structural interconnectivity with other information rather than just focusing
on the factual elements or surface features of learning content. This refers to (re-)discovering
or (re-)producing information/knowledge through P2’s own cognitive information processing
activity. While “reproducing” the information, the ability to question P2’s own constructive
point of view is critical for the following reason: When a particular information processing
procedure is applied to the same or similar learning situations and always results in the same
or similar consequences, this specific phenomenon (i.e., input-consequence relation) is
encoded and stored in P2’s memory through self-modifying processes (i.e., assimilation
and/or accommodation).
P2 classifies the information ([2.7], C1 grouping), and then constructs a meaningful
larger cognitive structure ( e.g., “encode them in an orderly fashion,” “I wanted to make my
brain like a dictionary”) by combining elements of the information in a new way ([2.8], C3
reorganization); and by relating and making meaningful associations of different parts of the
information to each other (“associate the information from different subjects to each other”;
[2.9], C5 elaboration). While doing this, P2 consciously applies learned or self-constructed
rules to understand information, and thus constructs rules on the basis of analysis of learned
information “to link the current learning content with previously learned content” by analogy
([2.10], C4 inferencing). Hence, P2 can operate the information as a whole interconnected one
instead of as locally segregated pieces. P2 understands the conditions that help him/her more
successfully accomplish a learning task ([2.11], M4 management).
P2: I tried to use my brain as a notebook. During the lesson, I tried to encode the whole parts of
the lesson including even the teacher’s gestures, tone of the voice, jokes, as well as information
like images in a movie. And then during the break time12
, I reviewed the lesson by retrieving the
12 A typical Korean school day includes six to seven 50-minute classes per day. There is a 10- minute break
between them and one hour lunch break.
105
images that I stored during the lesson as if I were watching a movie. In this way, I could
effectively review the whole lesson in a relatively short time as compared to reading materials.
P2 explicitly uses mental images to understand, represent, or remember new information
([2.12], C7 imagery); and repeats learned content through practice by mentally rehearsing
them ([2.13], C10 repetition). Through the use of a system of symbolization (i.e., visual
imagination) P2 arranges the elements of learned information in a visual structure. This can
be accomplished by internalizing the figurative, functional, and/or operative regularities and
invariants identified in the learning situations. This way, P2 is able to represent the learned
information with his/her own constructed “words.” This symbolization helps P2 to keep
stimulated content active in his/her memory and promote the storage of new content in
long-term memory. P2’s learning processes are also clearly presented in the following
snippets of comments:
P2: ….I put much of my effort in reviewing lessons. I set up a schedule to review lessons as
soon as possible. I reviewed one lesson at least three times in a month (i.e., daily, weekly, and
monthly): (1) I did the first review of the lessons of the day in the evening by reading the
textbooks. I tried to thoroughly understand the information by carefully checking if I had
missed any information of the lessons and tried to catch all the details. It took substantial time.
(2) Then I reviewed the whole lessons held during the week at the end of the week, and this did
not take that much time as I had already reviewed the information during the daily review. As
the concepts had already been reviewed, this time, I reviewed only important parts, trying to
understand the gist of the topics, and this was enough for understanding the whole parts of the
lesson. (3) Then I reviewed the whole lessons held during the month at the end of the month by
reading the subtitles and important parts of the sections, and this was enough for a good
grasping of the entire content of the materials. The reviews had speeded up as I had reviewed
lessons more and more. And thus, the concepts learned became more and more substantial over
time during the three years of high school days.
This implies that P2 learns from “reorganization” and “restructuring” of information rather
than merely stacking pieces of information in an isolated manner. While reviewing the
lessons of the day, P2 identifies attributes and features of a phenomenon of the information
from the learning material ([2.14], Agg1 identification). During weekly reviews, P2 only
reviews important parts of it trying to understand the gist of the information in the learning
material. This implies that P2 could define the characteristics of the information to isolate
and compare the attributes with the goal of extracting the gist of the information ([2.15],
Agg3b decomposition). P2 then progressively identifies and extracts the information from
the readings in terms of its individuals/elements, relations, and functions ([2.16], Agg4
reproduction). With this “reproduced” information, P2 associates (previously unrelated)
information to each other and forms a new unified idea. While doing this, units of
information are grouped together in various ways to create complex knowledge ([2.17],
106
Agg5 completion). P2 then organizes them into a schema, which is constructed by a
configuration of interrelated features that define different concepts ([2.18], Agg6a schema).
Throughout the multiple review processes, with the help of a schema, P2 can more and more
clearly identify all relationships between the information that can be integrated into the
schema. As P2 goes through multiple steps of conceptual modification, the schema becomes
progressively strengthened, and hence, P2 can develop knowledge and skills over time. In
terms of learning strategies, P2 clearly proposes strategies for handling learning tasks ([2.19],
M2 planning); attends to specific aspects of information by scanning for key information
during the 2nd
and 3rd
time of the reviews ([2.20], M3 attention); and repeats learned contents
by reviewing them multiple times ([2.21], C10 repetition).
While taking the interview, the production team carried out an experiment to see how
much information P2 could retrieve from a particular period of study. The result is presented
in the following narration:
N2: The result was that the participant concentrated for about 30 minutes without a break while
reading a book and described in detail substantial amount of the information in sentences. S/he
also drew an important illustration from the content, and thus added some other related
information that was not stated in the book.
While reading the book, P2 identifies the attributes and features of a phenomenon of
information ([2.22], Agg1 identification); and relates them to his/her existing knowledge
([2.23], Agg5 completion). In doing this, units of simple information are associated in various
ways to construct a meaningful cognitive structure by combining elements of information and
existing knowledge in a new way ([2.24], C3 reorganization; [2.25], C5 elaboration).
N2: During the entire high school days, s/he studied more than 200 workbooks. This enabled
him/her to clearly identify the missing information as well as misunderstandings, and to more
deeply understand the content.
P2: To check my level of understanding and knowledge, I solved a great number of problems
from multiple workbooks. Since the different publishers have their own unique styles in
forming questions, I thought that if I identify and then master all the different styles of the
questions, it would become a good strategy while preparing for CSAT….As a result; I was able
to easily solve problems while taking the CSAT.
The above snippets of comments illustrate that with persistent practice and in-depth
investigations in various problems solving, P2 instantiates the schema ([2.26], Agg6b
instantiation; [2.27], C10 repetition); and (re-)organizes and (re-)constructs the content of
his/her existing schemas by assimilating and/or accommodating them in accordance with the
new information ([2.28], C3 reorganization; [2.29], C5 elaboration). It is assumed that the
107
abstract schema is rendered less abstract during the process of practice by supplying values
for the variables in the schema.
P2 uses various reference materials in addition to given learning materials ([2.30], C12
resourcing). This allows P2 to build on more general concepts ([2.31], Ab2 generalization) by
acquiring various simple concepts during the course of learning experiences. And this made
P2 correctly determine the specific concepts that can be applied in a particular problem
solving situation. While doing this, P2 checks and verifies his/her comprehension and
performance in problem solving in order to check the accuracy and appropriateness of his/her
cognitive processes and learning strategies while they are taking place ([2.32], M5
monitoring). P2 thus checks the outcomes of his/her learning performance (i.e., performance
evaluation against an internal measure of completeness and accuracy of the learning goal
after it has been completed; [2.33], M7 evaluation).
In sum, in the course of learning, (1) P2 first determined the factual knowledge (e.g.,
facts, concepts, principles, and rules) in order to understand the learning content while
reading books (in lines 7-8; 24-28). (2) P2 then grouped the components of knowledge
together and reorganized them into a coherent structure (in lines 11-12). Hence, new
components of the information were systematically integrated into existing knowledge
structure in the course of learning. (3) Thereafter, these complexly reproduced components of
information were progressively mapped onto cognitive structure until P2 constructed a whole
comprehensive structure (i.e., schematization; in lines 11-12; 24-38). While doing this, P2
went through many steps of conceptual assimilation and modification and thereby, could
extend the range and complexity of interconnected relationships that s/he was able to
subsume under (or assimilate into) a hierarchical knowledge system (in lines 36-38). This
implies that P2 learned from “reorganizing” and “restructuring” of cognitive structure rather
than merely stacking pieces of information/knowledge in an isolated manner.
This whole learning process seemed to be accomplished by systematic cumulative
integration between simpler parts and complex whole cognitive structures. P2 developed
knowledge and strategies by continuously constructing a cognitive structure that is more
advanced and complex than prior (or existing) ones by defining the features, relations, and
functions of the elements of the newly gained information, and then constructing schemas
that interconnect these elements with existing schemas. Consequently, the analysis of case 2
seems to give evidence for the assumptions of this study that learning is a cumulative as well
as a structuring process, which in turn seems to answer the first research question in this
study. The above stated analysis is shown in the following table 4.2.
108
Table 4.2 Coding of interview transcription: Case 2
Transcript of case 2(Grade: CF) Cognitive processes Learning strategies
1
2
3
4
5
6
N2: ….In order to fully attend to school lessons, the participant took fewer notes and
concentrated on listening to the class teacher’s lesson …. S/he tried to find relations and/or
interconnectivity between school subjects instead of studying each subject in a separate manner
to effectively utilize the information ….S/he focused on the interconnectivity of the concepts
from various topics of the school subjects, and hence, could keep the concentration level high
irrespective of the conditions under which he/she studied.
7
8
9
10
P2: I tried to understand information by repeatedly reading whole parts of textbooks as well as
other reference books rather than studying particular sections of the books that were considered
“important.” I studied this way so I could grasp the information in the books as a whole
[interconnected one instead of as locally segregated pieces].
[2.1], Agg1 identification
[2.2], C12 resourcing
11
12
13
14
15
16
17
P2: Hence, while I was reading books, I tried to understand how the whole parts of the books
were connected with each other, and then I tried to mentally encode them in an orderly
fashion….I mainly studied by reading learning materials, which is a less efficient strategy for
concentrating during study when compared with writing or speaking. So, in order to
supplement such inefficiency, I tried to associate the information from different subjects to
each other … and tried to link the current learning content with previously learned content ….
I wanted to make my brain like a dictionary.
[2.3], Agg4 reproduction
[2.4], Agg5 completion
[2.5], Ab1 comparison
[2.6], Ab3 structural
[2.7], C1 grouping
[2.8], C3 reorganization
[2.9], C5 elaboration
[2.10], C4 inferencing
[2.11], M4 management
18
19
20
21
22
23
P2: I tried to use my brain as a notebook. During the lesson, I tried to encode the whole parts
of the lesson including even the teacher’s gestures, tone of the voice, jokes, as well as
information like images in a movie. And then during the break time, I reviewed the lesson by
retrieving the images that I stored during the lesson as if I were watching a movie. In this way,
I could effectively review the whole lesson in a relatively short time as compared to reading
materials.
[2.12], C7 imagery
[2.13], C10 repetition
109
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
P2: ….I put much of my effort in reviewing lessons. I set up a schedule to review lessons as
soon as possible. I reviewed one lesson at least three times in a month (i.e., daily, weekly, and
monthly): (1) I did the first review of the lessons of the day in the evening by reading the
textbooks. I tried to thoroughly understand the information by carefully checking if I had
missed any information of the lessons and tried to catch all the details. It took substantial time.
(2) Then I reviewed the whole lessons held during the week at the end of the week, and this
did not take that much time as I had already reviewed the information during the daily review.
As the concepts had already been reviewed, this time, I reviewed only important parts, trying
to understand the gist of the topics, and this was enough for understanding the whole parts of
the lesson.
(3) Then I reviewed the whole lessons held during the month at the end of the month by
reading the subtitles and important parts of the sections, and this was enough for a good
grasping of the entire content of the materials. The reviews had speeded up as I had reviewed
lessons more and more. And thus, the concepts learned became more and more substantial
over time during the three years of high school days.
[2.14], Agg1 identification
[2.15], Agg3b decomposition
[2.16], Agg4 reproduction
[2.17], Agg5 completion
[2.18], Agg6a schema
[2.19], M2 planning
[2.20], M3 attention
[2.21], C10 repetition
39
40
41
42
43
44
N2: The production team carried out an experiment to see how much information the
participant could retrieve from a particular period of study. The result was that s/he
concentrated for about 30 minutes without a break while reading a book and described in
detail detailed substantial amount of the information in sentences. S/he also drew an
important illustration from the content, and thus added some other related information that
was not stated in the book.
[2.22], Agg1 identification
[2.23], Agg5 completion
[2.24], C3 reorganization
[2.25], C5 elaboration
45
46
47
48
49
50
51
52
N2: During the entire high school days, s/he studied more than 200 workbooks. This enabled
him/her to clearly identify the missing information as well as misunderstandings, and to more
deeply understand the content.
P2: To check my level of understanding and knowledge, I solved a great number of problems
from multiple workbooks. Since the different publishers have their own unique styles in
forming questions, I thought that if I identify and then master all the different styles of the
questions, it would become a good strategy while preparing for CSAT….As a result; I was
able to easily solve problems while taking the CSAT.
[2.26], Agg6b instantiation
[2.31], Ab2 generalization
[2.27], C10 repetition
[2.28], C3 reorganization
[2.29], C5 elaboration
[2.30], C12 resourcing
[2.32], M5 monitoring
[2.33], M7 evaluation
110
4.1.3 Case 3
P3 (UF) uses a unique learning strategy, which is presented in the following narration:
N3: The participant memorized whole parts of the textbooks to thoroughly study all information
since all school test questions were asked based on the textbooks. ….Only when s/he fully
understood the content in the textbooks, it supported memorization and longer retention (simply
memorizing texts without understanding of the content is not useful in learning)….S/he
memorized the textbooks cumulatively and repetitively. For English, s/he first repeatedly read
aloud the main texts. While doing this, s/he tried to encode the order of the words in the
sentences as well as their structures. This made him/her become familiar with sentence
structure. After reading it ten times, s/he then self-tested [if she memorized all the sentences
correctly].
P3 memorizes entire texts from an English textbook by cumulatively repeating a chunk of
language (a sentence or phrase). This gives evidence that repetition is a necessary strategy for
memorization in language learning. However, P3 commented that simply memorizing the
content in the textbook without understanding it was not useful in learning: Only when P3
had fully understood the content, it supported the memorization and longer retention.
Evidence for this is found in the following comment:
P3: While I memorize the English sentences, I could naturally learn the grammars and
structures of them. I also memorized each sentence in a cumulative way. I first assigned
numbers in serial order for every sentence; then memorized the 1st sentence by repeating each
sentence about three times and then memorized the 2nd sentence; then I memorized the 1st and
2nd sentences together; then memorized the 3rd sentence; and then memorized the 1st, 2nd, and 3rd
sentences together; and continued this way sequentially until the end. Then I confirmed if I
memorized all the sentences correctly by getting tested with the help of others; and continually
reviewed them as often as possible….I had been using this strategy for six years and all the
information I had memorized had been accumulated and stored in my knowledge, and hence, I
was naturally able to learn grammar, use conjunctions, and use proper ordering of the words in
the English sentences.
In the process of memorization, P3 first identifies the attributes and features of a phenomenon
of the information ([3.1], Agg1 identification); and processes the information as larger
chunks (“assigned numbers in serial order for every sentence,” “encode the order of the words
in the sentences as well as their structures”; [3.2], Agg2 serial; [3.3], C2 grouping). The
perceived phenomena (the figurative, functional, and/or operative regularities) are then
internalized (“become familiar with sentence structure”; [3.4], Agg4 reproduction). Therefore,
P3 can construct a meaningful larger cognitive structure by combining elements of the
information in a new way ([3.5], C3 reorganization). This in turn promotes P3 to perceive the
information as a whole rather than in fragmented unrelated pieces. Hence, P3 comprehends
the structural relationships of the elements of the information (“I could naturally learn the
grammars and structures of them”; [3.6], Ab3 structural). In doing this P3 presumably
111
(un-)consciously coordinates units of the information and their relational connections, and
then integrates them into a hierarchically organized integrative cognitive structure.
In terms of learning strategies, P3 repeats the information through practice by writing
and speaking them ([3.7], C10 repetition) by playing in the back of his/her mind (i.e.,
mentally rehearsing) the auditory/visual representation in order to understand and remember
the information (words, phrases, or longer sequences; [3.8], C11 representation). In doing this,
P3 clearly plans strategies for memorizing the information and repeats it multiple times
accordingly ([3.9], M2 planning). P3 specifically attends to the key information during the 2nd
and 3rd
time of the reviews ([3.10], M3 attention). P3 checks and verifies understanding of the
information while memorizing it ([3.11], M5 monitoring), which is evidenced in the above
stated narrator’s comment (“only when s/he fully understood the content in the textbooks, it
supported memorization and longer retention”). P3 thus checks the outcomes of learning
performance against an internal measure of completeness and accuracy of the learning goal
after it has been completed (“if s/he memorized all the sentences correctly by getting tested
with the help of others”; [3.12], M7 evaluation).
N3: As more and more textbooks were memorized, his/her memory skill became more developed,
and thus the memorized content became retained longer in his/her memory. Eventually s/he was
able to memorize the textbooks almost perfectly. This enabled him/her to clearly understand and
identify test questions.
With persistent repetition, the time it took to complete the memorization was gradually
decreased and the memorized information was retained longer in P3’s memory. The plausible
interpretations of the cognitive processes that resulted in this phenomenon would be as
following: As P3 repeatedly “reproduces” the information during multiple reviews, P3
integrates the information into existing knowledge through modification, (re-)organization,
and (re)structuring, and hence, progressively completes a schema ([3.13], Agg6a schema).
Hence, the packs of memorized information operate as parts of the whole cognitive structure,
which are interconnected to each other within the structure instead of as locally segregated
pieces. Therefore, it is retained longer in P3’s memory. This reflects that persistent practice
results in an automatized structure (internal and external).
P3: I thought that I should do something else other than learning from lessons and books to get
a perfect test score….I memorized the whole parts of an English textbook to make sure I was
not missing any single piece of information in it…Until the 2nd
grade of junior school I studied
by solving multiple problems. But I recognized that the questions that I got wrong answers
were due to missing detailed information in textbooks. Since then I started to use this strategy
[memorize textbooks].
112
P3 explicitly identifies the point needing resolution in a learning task (“I recognized that the
questions that I got wrong answers were due to missing detailed information in textbooks”).
P3 understands the conditions that help P3 to successfully accomplish the learning task and
controls further learning strategies to maximize the efficiency of learning13
([3.14], M4
management; [3.15], M6 problem; [3.16], C13 substitution).
P3: For history, once I completed reading and understanding the whole texts in the textbook
[parts of targeting learning], I structured them under corresponding sub-headings; and then I
memorized the information by the sub-headings.
As P3 identifies the attributes and features of a phenomenon of the information given context
([3.17], Agg1 identification), P3 groups the information together to form larger chunks of
information (i.e., sub-headings; [3.18], Agg2 serial). This is presumably accomplished as P3
defines characteristics of the information to combine equivalent relevant attributes ([3.19],
Agg3a classification), and thus, defines characteristics of information to isolate and compare
these attributes with the goal of forming a coherent structure ([3.20], Agg3b decomposition).
P3: [preceding comment cont’d], then I compared the retrieved texts using the textbook to see if
they correctly matched with the book and if there was any missing information; and then I
modified them by adding some missing information and correcting mistakes.
P3 progressively develops a schema as P3 continually checks the completeness and
accurateness of the retrieved information ([3.21], Agg6a schema). P3 went through many
steps of conceptual modification, and hence could acquire complex forms of knowledge over
time. This implies that P3 learned from “reorganization” and “restructuring” of the
information rather than merely stacking pieces of information in an isolated manner.
P3: Completing such work took a series of steps and I tried to do it within two days of lessons
to maximize memorization….Eventually, I was able to mentally retrieve the book like a
photographic memory during the test. When I got a question that I could not answer right away,
I was able to look up the relevant part and was able to get the answer using this memory. It felt
like I was taking an open-book test.
Such photographic memory is presumably accomplished as P3 progressively develops more
complex and advanced schema by repeatedly (re-)organizing and (re-)structuring the
information into the schema ([3.22], Agg6a schema). While taking this interview, the
production team carried out an experiment to see the effectiveness of P3’s learning strategy
which is presented in the following narration:
13
Good school records make it easier to enter a good college/university in Korea as high school grade-point
average (GPA) is partially reflected for college/university admission along with CSAT score.
113
N3: Six high school students who volunteered and had never memorized the whole parts of the
textbooks [2nd
grade, females, “average achievers”], took the experiment to see if the
participant’s strategy was also applicable and effective with them. They were asked to study 9
pages of a particular section from a history textbook for two days, and then asked to represent
them in 30 minutes….The result showed that there was no single student who was able to
completely memorize the texts, but there was some improvement in learning for all of them.
They all mentioned that the strategy was useful in organizing and storing learned material in a
more structured way. All of them felt difficulty in summarizing the information even though
they understood the 9-pages-long information in a structured way. It seemed that summarizing
the learned content also demanded a specific strategy.
This reflects that acquiring systematically well-organized cognitive structure requires
persistent (re-)organization and (re-)structuring in order to lead to stable cognitive structure.
N3: [preceding narration cont’d] one of them mentioned that s/he used to not care about the
serial organization of the content in a textbook but upon trying out this strategy; s/he was able
to retrieve information in a sequential manner.
This reflects the effect of structural organization in learning process. That is, all pieces of the
factual knowledge P3 gains from learning sequences gradually transforms into procedural
knowledge as s/he maps factual knowledge to a schema. Each piece of declarative knowledge
is subsumed under the schematic links by individual pieces of knowledge that were initially
decomposed and becomes gradually subsumed under a coherent cognitive structure (i.e.,
schema). This enables the learner to form a larger integrated cognitive structure. The above
experiment supports that P3 (“high achiever”) has systematic structural organizational
approaches to learning tasks at their disposal than do “average achievers,” who seem not to
be concerned about the structural organization of information in learning sequences.
In sum, P3 developed knowledge and skills by continuously constructing a cognitive
structure that is more advanced and complex than ones (in lines 33-35). P3 went through
many steps of conceptual assimilation and modification and thereby, could extend the range
and complexity of interconnected relationships (in lines 18-19) that s/he was able to subsume
under interacting hierarchical knowledge system (“restructuring” of cognitive structure; in
lines and 36-38). The whole learning process seemed to be accomplished by systematic
cumulative integration between simpler parts and complex whole cognitive structures (in
lines 10-16; 36-38). Consequently, the analysis of case 3 seems to give evidence for the
assumptions of this study that learning is a cumulative as well as a structuring process, which
in turn seems to answer the first research question in this study. The above stated analysis is
shown in the following table 4.3.
114
Table 4.3 Coding of interview transcription: Case 3
Transcript of case 3 (Grade: UF) Cognitive processes Learning strategies
1
2
3
4
5
6
7
8
9
N3: The participant memorized whole parts of the textbooks to thoroughly study all information
since all school test questions were asked based on the textbooks.
N3: ….Only when s/he fully understood the content in the textbooks, it supported memorization
and longer retention (simply memorizing texts without understanding of the content is not useful
in learning)….S/he memorized the textbooks cumulatively and repetitively. For English, s/he
first repeatedly read aloud the main texts. While doing this, s/he tried to encode the order of the
words in the sentences as well as their structures. This made him/her become familiar with
sentence structure. After reading it ten times, s/he then self-tested [if she memorized all the
sentences correctly].
[3.1], Agg1 identification
[3.2], Agg2 serial
[3.4], Agg4 reproduction
[3.3], C2 grouping
[3.5], C3 reorganization
10
11
12
13
14
15
16
17
P3: While I memorize the English sentences, I could naturally learn the grammars and
structures of them. I also memorized each sentence in a cumulative way. I first assigned
numbers in serial order for every sentence; then memorized the 1st sentence by repeating each
sentence about three times and then memorized the 2nd
sentence; then s/he memorized the 1st
and 2nd
sentences together; then memorized the 3rd
sentence; and then memorized the 1st, 2
nd,
and 3rd
sentences together; and continued this way sequentially until the end. Then I confirmed
if I memorized all the sentences correctly by getting tested with the help of others; and
continually reviewed them as often as possible.
[3.6], Ab3 structural [3.7], C10 repetition
[3.8], C11 representation
[3.9], M2 planning
[3.10], M3 attention
[3.11], M5 monitoring
[3.12], M7 evaluation
18
19
20
21
N3: As more and more textbooks were memorized, his/her memory skill became more
developed, and thus the memorized content became retained longer in his/her memory.
Eventually s/he was able to memorize the textbooks almost perfectly. This enabled him/her to
clearly understand and identify test questions.
[3.13], Agg6a schema
22
23
24
25
26
N3: According to the Ebinghaus’ forgetfulness curve, one is able to memorize 70% of the
meaningless spellings after one day, hence, if one repeatedly exercises them the next day, it is
retained for a week; if one repeatedly exercises them again after a week has passed, then it is
retained for a month; and if one repeatedly exercises them again after a month has passed, then
it is retained for six months.
115
27
28
29
30
31
32
P3: I thought that I should do something else other than learning from lessons and books to get a
perfect test score….I memorized the whole parts of an English textbook to make sure I was not
missing any single piece of information in it….Until the 2nd
grade of junior school I studied by
solving multiple problems. But I recognized that the questions that I got wrong answers were
due to missing detailed information in textbooks. Since then I started to use this strategy
[memorize textbooks].
[3.14], M4 management
[3.15], M6 problem
[3.16], C13 substitution
33
34
35
P3: For history, once I completed reading and understanding the whole texts in the textbook
[parts of targeting learning], I structured them under corresponding sub-headings; and then I
memorized the information by the sub-headings.
[3.17], Agg1 identification
[3.18], Agg2 serial
[3.19], Agg3a classification
[3.20], Agg3b decomposition
36
37
38
P3: [preceding comment cont’d], then I compared the retrieved texts using the textbook to see if
they correctly matched with the book and if there was any missing information; and then I
modified them by adding some missing information and correcting mistakes.
[3.21], Agg6a schema
39
40
41
42
43
P3: Completing such work took a series of steps and I tried to do it within two days of lessons to
maximize memorization…. Eventually, I was able to mentally retrieve the book like a
photographic memory during the test. When I got a question that I could not answer right away,
I was able to look up the relevant part and was able to get the answer using this memory. It felt
like I was taking an open-book test.
[3.22], Agg6a schema
44
45
46
47
48
49
50
51
52
53
54
55
56
57
N3: Six high school students who volunteered and had never memorized the whole parts of the
textbooks [2nd
grade, females, “average achievers”], took the experiment to see if the
participant’s strategy was also applicable and effective with them. They were asked to study 9
pages of a particular section from a history textbook for two days, and then asked to represent
them in 30 minutes.
N3: The result showed that there was no single student who was able to completely memorize
the texts, but there was some improvement in learning for all of them. They all mentioned that
the strategy was useful in organizing and storing learned material in a more structured way. All
of them felt difficulty in summarizing the information even though they understood the 9-
pages-long information in a structured way. It seemed that summarizing the learned content also
demanded a specific strategy.
N3: One of them mentioned that s/he used to not care about the serial organization of the
content in a textbook but upon trying out this strategy; s/he was able to retrieve information in a
sequential manner.
116
4.1.4 Case 4
P4’s (HS2) learning processes are clearly presented in the following snippets of narration:
N4: ….S/he studied math by solving complicated math problems in many different ways. Many
students memorize necessary math formulas and apply them in problem solving, but s/he
attempted to prove the formulas and thus tried to construct his/her own particular ways to solve
the problems rather than simply memorizing them. By doing this, s/he was able to understand the
development and derivations of the formulas, which in turn improved his/her learning.
This illustrates that P4 generates math problem solving schemas by extracting general
solution strategies or principles from the various solution approaches ([4.1], Agg6b
instantiation). P4 thus acquires creative mathematical skills by constructing his/her own
problem solving mental models ([4.2], Ab5 model). To P4, problem solving is not merely to
practice the problems but to investigate and discover underlying concepts, rules, and
principles of the mathematical formulas. This seems allowed P4 to understand underlying
concepts, rules, principles, the relation between the elements of a problem; and thus allowed
p4 to formulate provisional rules and principles from given material. In this process, the
ability to question his/her constructive point of view is critical. When a particular problem
solving procedure is applied to the same problem solving situation and always results in the
same (or similar) consequences, this specific phenomenon (i.e., input-consequence relation)
is encoded and stored in P4’s memory through self-modifying processes (i.e., assimilation
and/or accommodation), which is presumably accomplished by internalizing the figurative,
functional, and/or operative regularities and invariants identified in the problem solving
situations. This way, P4 is able to continuously reconstruct the contents in his/her problem
solving schemas (“construct his/her own particular ways to solve the problems”).
N4: ….S/he consistently tried to solve complex math problems until s/he could clearly find the
mistakes and his/her incorrect knowledge.
It is assumed from this comment that, in the process of finding mistakes and incorrect
knowledge, P4 presumably (re-)produces newly learned (and hence, could find the mistakes
and incorrect knowledge) information/knowledge through P4’s own cognitive information
processing activity in terms of its individuals/elements, relations, and functions ([4.3], Ab4
reproduction); compares phenomena with regard to similarities and dissimilarities of the
information/knowledge between P4’s problem solving models or schemas (which result in
mistakes and incorrect knowledge) and newly learned models or schemas (which result in
correct knowledge), and extracts commonalities of the information/knowledge from the
117
surface as well as the underlying structure of the two models by analogy ([4.4], Ab1
comparison); and (re-)constructs/modifies mental model(s) ([4.5], Ab5 complex completion).
N4: Continuing this strategy, s/he was able to apply various procedures to different sets of
problems….This made him/her to acquire mathematical ability…and the number of problems
that s/he could solve in a given period of time gradually increased.
While applying various procedures to different sets of problems, P4 relates the information in
the given problem solving task to existing knowledge; relating different parts of the existing
knowledge to each other; and makes meaningful associations with the information in the
given problem solving task ([4.6], C5 elaboration). Here, P4 uses the existing knowledge as a
basis for understanding the problem ([4.7], C6 transfer). This comment seems to give
evidence that schema induction involves the generalization of an abstract schema from
multiple various procedural features ([4.8], Ab2 generalization).
N4: ….Until middle school, s/he practiced problems by referring to commentary books and
simply copying the procedure from the book, and looking up answers right away as soon as s/he
was not able to solve a certain problem. However, since high school, s/he did not refer the
commentary book until s/he solved a certain number of problems.
This shows that P4 selects alternative approaches to better accomplish the problem solving
tasks ([4.9], C13 substitution). P4’s intention of implementing such learning processes is
clear from the following comments:
P4: ….When I solve a math problem; I use several different problem solving strategies and then
compare them in order to find out the best strategy. [Therefore], I was able to clearly
understand how the formulas were developed, the underlying concepts as well as the
background information, and hence, was able to correctly handle complex and difficult
problems in actual tests, and this gave me a great sense of accomplishment.
In order to solve complex math problems, P4 actively identifies and interprets the underlying
concepts, and searches the best strategy (or principle) to apply by comparing the phenomena
with regard to similarities and dissimilarities between the superficial and underlying structure
of the different problem solving strategies by analogy ([4.10], Ab1 comparison). This is
because a complex problem hides the principle(s) that are needed for proper application
whereas a simple math problem usually provides the principle for solving the problem in the
statement of the question. In doing this, P4 associates given information in the problems with
his/her existing knowledge and then integrates them into cognitive structure in a meaningful
way ([4.11], Ab4 analogy); and then constructs new models by analogy ([4.12], Ab5 model).
Upon perceiving information in the problem solving task, P4 searches his/her
retrievable knowledge to identify some similar situations in the past in order to find solutions
118
from it; and then constructs the mental representations of the information in the problem and
progressively links the mental representations to one another until the problem solving mental
model fully “maps” into relevant phenomenon of his/her cognitive resources in order to
progressively adapt the mental representations of the information to the problem solving
situation (i.e., reorganization; [4.13], C3 reorganization).
While doing this, it is assumed that with the given information in the problem solving
task, P4 (a) activates existing schema(s) relevant to the task, (b) concretely and/or abstractly
generalizes the given information, (c) extracts commonalities and finding causal relationships
between the activated schema(s) and the given information, and then (d) (re-)organizes and
(re-)structures his/her cognitive structure accordingly. In doing so, P4 acquires general
problem-solving strategy (e.g., Polya’s four stages of general problem-solving strategy in
mathematics, which is mentioned in the section 2.2.3) that can be used as topic-independent
knowledge.
P4: When I am faced with difficult problems in further problem solving situations, I can
quickly determine the most appropriate solution process as I can apply the various ways that I
had practiced the current task.
This illustrates that upon acquiring well-organized schematic links of the information
processing for problem solving; P4 could more quickly activate and retrieve
situation-appropriate schemas in problem solving situations. It gives evidence that having
highly abstract cognitive structure through systematic organization of knowledge increases the
speed of searching and activating relevant schemas in future learning situations resulting in
efficient and effective learning.
P4: ….I first try to prove a formula and then apply it to solve a problem. When I can solve a
very complex problem, I feel Archimedes’ “Eureka moment” and I am getting more and more
motivated in solving more complex problems. Though it takes much time but it makes me more
confident with the answer.
This comment implicitly reflects that acquiring complex forms of knowledge, such as
complex math problem solving skills, which requires persistent (re-)organization and
(re-)structuring in order to lead to stable cognitive structure. P4’s intention of implementing
such learning processes is clear from what s/he said as below. P4 clearly understands the
conditions that help him/her successfully developing mathematical thinking ability ([4.14],
M4 management):
P4: I think solving a complex math problem that requires applying multiple principles in a
complicated way is much more useful for developing the mathematical thinking ability when
119
compared to solving a simple (i.e., less complex) problem that usually only requires knowing
basic concepts and applying a single principle or a simple calculation.
While taking the interview, the production team carried out two small experiments to see the
effectiveness of P4’s particular problem solving strategy:
N4: The production team carried out two experiments. (1) In the 1st experiment, a very complex
math problem was given to everyone in class [HS 2nd
grade, approximately 30 students, all
males]. After 20 minutes, the participant [P4] could start to solve the problem, and after 40
minutes, s/he could solve the problem while many of the others either gave up and/or could not
solve it.
N4: [preceding comment cont’d] s/he first set up a hypothesis and then proved it with the given
problem, and his/her solution steps were much longer and more complicated than others whose
strategies were much simpler.
This comment shows that P4 acquired general problem solving knowledge, such as the
Polya’s (1957) four stages of a general problem-solving strategy in mathematics: analysis,
planning, execution, and checking: This shows that P4 analyzed the problem statement,
planned (“set up hypothesis”), and executed (“proved it with the given problem”) it. It is
assumed that P4 could acquire these reasoning skills (general knowledge) such as inferences
([4.15], C4 inference) in the process of connecting and unifying schemas through continuous
structural mapping process in the course of learning.
As shown in the above comments, in the practice of problem solving, P4 deliberately
tried to find a related underlying principle and structural similarities that could possibly be
applied to solve the problem, and thus, P4’s math problem solving ability gradually
progressed from focusing on superficial aspects of the phenomena to their underlying aspects
(i.e., prove formula). This is because P4 was able to recognize the distinctions between the
two structures (i.e., deep structural underlying principles and superficial information
presented in the problem statement) in the course of learning and hence progressively
modified his/her problem solving processes/strategies in order to find deep structures in
learning tasks depending on his/her previous learning experiences. Consequently P4 better
performed in the actual tests shown in the 1st experiment, especially when faced with
complex problems.
N4: (2) In the 2nd
experiment, 5 students [HS 2nd
grade, males] were picked up among the
students who gave up solving the problem in the 1st experiment. This time, they were asked to
solve the same problem again and a hint [emphasis added; i.e., guideline for solving the
problem] was provided. This time they were seriously in solving the problem, and the result
was that all of them were able to solve the problem correctly.
The result of the 2nd
experiment shows that “average-achieving” students were able to
120
retrieve relevant information/knowledge in order to solve the problem with explicit cues;
otherwise, they failed to retrieve it. This reflects that average-achieving students were not
able to identify hidden deep structure but focused on surface structure of the given problem.
They thus had an incorrect or insufficient perception of the problem and failed to solve it
correctly. The five students commented after solving the problem as follows:
1st student: “The hint helped me to generate further sequences of problem solving”; 2
nd student:
“The hint led me to think the problem in a new idea”; 3rd
& 4th
students: I recognized that I can
solve a complex problem if I do not give it up.”
This seems to reflect the importance of persistent thinking in solving complex math problem.
Furthermore, an expert (a professor in math education) also emphasized the “power of
persistent thinking” in learning math. This seems to imply that acquiring complex forms of
knowledge, such as complex math problem solving skills, requires persistent
(re-)organization and (re-)structuring in order to acquire stable cognitive structure.
In sum, case 4 seems to give evidence that mental models can be modified through
abstraction (compressing or removing certain attributes; e.g., identifying math principles) or
concretization (the addition of attributes; e.g., setting up hypothesis and proving it). In this
process, P4 could more clearly understand the intent of the problem, and hence, P4 could
predict and construct mental models that could be applied to solve the given problem (in lines
1-5; 9-10; 13-15). P4 explored several induced problem solving mental models by
formalizing and refining the problem statement (i.e., defining underlying principles, setting
up hypothesis), and then verified these models to find out whether they were appropriate to
solve a given problem (proving hypothesis) and critically analyzed and assessed them with
respect to properties of his/her existing knowledge (in lines 13-15). In doing this, P4 could
clarify concepts and reorganize and restructure his/her cognitive structure accordingly (in
lines 9-11; 13-15).
In the course of learning, P4 developed knowledge and strategies by continuously
constructing a cognitive structure that is more advanced and complex than prior (or existing)
ones by defining the features, relations, and functions of the elements of the newly gained
information, and then constructed schemas that interconnect these elements with existing
schemas (in lines 9-11; 13-15). (1) P4 first determined the factual knowledge (e.g., facts,
concepts, principles, and rules) in order to understand the learning content while solving
math problems (in lines 2-3; 9-10). (2) P4 then reorganized the factual knowledge into a
schema (in lines 19-21). Hence, new components of information and knowledge were
121
systematically integrated into existing knowledge structure in the course of learning. (3)
Thereafter, these complexly reproduced components of knowledge were progressively
mapped onto cognitive structure (i.e., schematization) until P4 constructed a whole
comprehensive structure (in lines 13-21). While doing this, P4 went through many steps of
conceptual assimilation and modification and thereby, could extend the range and
complexity of interconnected relationships that s/he was able to subsume under (or
assimilate into) a hierarchical knowledge system (in lines 13-21). This case seems to provide
evidence that P4 learned from “reorganization” and “restructuring” of cognitive structure
rather than merely stacking pieces of information/knowledge in an isolated manner. This
whole learning process seems to be accomplished by systematic cumulative integration
between simpler parts and complex whole cognitive structures (in lines 2-5; 9-11; 14-16;
19-24), which in turn seems to give evidence for the assumptions of this study that learning
is a cumulative and a structuring process. The above stated analysis is shown in the
following table 4.4.
122
Table 4.4 Coding of interview transcription: Case 4
Transcript of case 4 (Grade: HS2) Cognitive processes Learning strategies
1
2
3
4
5
6
7
8
N4: The participant studied math by solving complicated math problems in many different ways. Many students
memorize necessary math formulas and apply them in problem solving, but s/he attempted to prove the formulas and
thus tried to construct his/her own particular ways to solve the problems rather than simply memorizing them. By doing
this, s/he was able to understand the development and derivations of the formulas, which in turn improved his/her
learning ….Until middle school, s/he practiced problems by referring to commentary books and simply copying the
procedure from the book, and looking up answers right away as soon as s/he was not able to solve a certain problem.
However, since high school, s/he did not refer the commentary book until s/he solved a certain number of problems.
[4.1], Agg6b instantiation
[4.2], Ab5 model
[4.9], C13 substitution
9
10
11
12
N4: ….S/he consistently tried to solve complex math problems until s/he could clearly find the mistakes and his/her
incorrect knowledge. Continuing this strategy, s/he was able to apply various procedures to different sets of problems,
and the number of problems that s/he could solve in a given period of time gradually increased. This made him/her to
acquire mathematical ability.
[4.3], Ab4 reproduction
[4.4], Ab1 comparison
[4.5], Ab5 complex comp.
[4.8], Ab2 generalization
[4.6], C5 elaboration
[4.7], C6 transfer
13
14
15
16
P4: When I solve a math problem, I use several different problem solving strategies and then compare them in order
to find out the best strategy. I was able to clearly understand how the formulas were developed, the underlying
concepts as well as the background information, and hence, was able to correctly handle complex and difficult
problems in actual tests, and this gave me a great sense of accomplishment.
[4.10], Ab1 comparison
[4.11], Ab4 analogy build
[4.12], Ab5 model
[4.13], C3
reorganization
17
18
19
20
21
22
23
24
P4: ….I first try to prove a formula and then apply it to solve a problem. When I can solve a very complex problem, I
feel Archimedes’ “Eureka moment” and I am getting more and more motivated in solving more complex problems.
Though it takes much time but it makes me more confident with the answer. When I am faced with difficult problems
in further problem solving situations, I can quickly determine the most appropriate solution process as I can apply the
various ways that I had practiced the current task. I think solving a complex math problem that requires applying
multiple principles in a complicated way is much more useful for developing the mathematical thinking ability when
compared to solving a simple (i.e., less complex) problem that usually only requires knowing basic concepts and
applying a single principle or a simple calculation.
[4.14], M4 self-
management
25
26
27
28
29
30
31
N4: The production team carried out two experiments. (1) In the 1st experiment, a very complex math problem was given to everyone in class [HS
2nd
grade, around 30 students, males]. After 20 minutes, the participant [P4] could start to solve the problem, and after 40 minutes, s/he could
solve the problem while many of the others either gave up and/or could not solve it. S/he first set up a hypothesis and then proved it with the
given problem, and his/her solution steps were much longer and more complicated than others whose strategies were much simpler.
N4: (2) In the 2nd
experiment, 5 students [2nd
grade, males] were picked up among the students who gave up solving the problem in the 1st
experiment. This time, they were asked to solve the same problem again and a hint [emphasis added; i.e., guideline for solving the problem] was
provided. This time they were seriously in solving the problem, and the result was that all of them were able to solve the problem correctly.
[4.15], C4 inference
123
4.1.5 Case 5
P5 (UF) uses “learning by teaching” method (i.e., teaching oneself and others). P5’s learning
processes are clearly presented in the following snippets of narration:
N5: ….The participant used “learning by teaching” method [teaching oneself (i.e.,
self-explaining) and others]. S/he first aggregated all detailed information from different
learning materials (e.g., class handouts, textbooks, and reference books) and used these for
creating a kind of “teaching note.”
P5 first aggregates relevant units of information/knowledge from multiple resources in
addition to given learning materials ([5.1], Agg1 identification; [5.2], C12 resourcing).
N5: For efficient note taking, s/he first wrote down the titles [headings & subheadings] and
learning objectives to set the direction of the study.
P5 proposes a strategy for handling learning tasks ([5.3], M2 planning); and attends to
specific aspects of information by scanning the key information in the learning material ([5.4],
M3 attention).
N5: [preceding comment cont’d] I used various types of symbols or abbreviated words that
could be substituted with some texts in the note (e.g., ▼= because); and added various
examples, descriptions, symbols, and pictures on it.
While constructing the note, P5 identifies attributes and features of a phenomenon of the
aggregated information ([5.5], Agg1 identification); defines characteristics of the information
to combine equivalent relevant attributes in order to classify the information under relevant
(sub-)headings ([5.6], Agg3a classification); and represents the elements of the information in
a visual structure (i.e., symbols, and pictures) by internalizing the figurative, functional,
and/or operative regularities ([5.7], Agg4 reproduction; [5.8], C7 imagery).
P5 classifies the information (words, terminology, sentences, concepts, and/or
principles) on the basis of their attributes or meaning ([5.9], C1 grouping); and writes down
key words or concepts in (abbreviated or more elaborated) verbal or graphic form while
constructing the note to assist learning ([5.10], C2 note). P5 uses various types of symbols or
abbreviated words that could be substituted with some texts in the note (e.g., ▼= because).
This way, P5 is able to represent the learned content and concepts with his/her own “words”
and hence, P5 can better understand and organize the information, which in turn is retained
longer in his/her memory.
N5: [preceding comment cont’d] While doing this, s/he tried to understand relations between the
information, and then, s/he represented it by speaking aloud using his/her own words.
124
This comment illustrates that P5 associates the aggregated information to each other to form a
unified set by reorganizing them ([5.11], C3 reorganization) to create a more unified complex
cognitive structure ([5.12], Agg5 completion) by relating and making meaningful
associations of different parts of the information to each other using analogy ([5.13], C4
inferencing; [5.14], C5 elaboration). Understanding relations refers to rediscovering and/or
reproducing information through one’s own cognitive information processing activity, which
is presumably accomplished by internalizing the figurative, functional, and/or operative
regularities and invariants of the information identified in the learning situations. This shows
that P5 attempted to actively process, (re-)organize, and (re-)structure the information. P5’s
learning processes are more clearly explained in the following comment:
P5: My learning process starts with note taking. I aggregate all information from the lesson as
well as related components (e.g., concepts, principles) that can be linked to the current learning
from my existing knowledge, and this is very helpful in understanding the current lesson.
This illustrates that P5 identifies attributes and features of a phenomenon of information
while aggregating units of information ([5.15], Agg1 identification) in order to find relevant
information from his/her existing knowledge ([5.16], Agg4 reproduction). While doing this,
P5 actively investigates relations between the given information and his/her existing
knowledge. P5’s intention of implementing the cognitive processes is clear from what s/he
said:
P5: [preceding comment cont’d] in note taking, it is important to meaningfully link one concept
to another rather than simply copying the information from the lesson into the note. I focus on
the connectivity [emphasis added] between the concepts and thus, I try to write them in the note
in a way that I can easily explain them to myself and other students later on.
This illustrates that P5 explicitly associates a unit of simple information/knowledge together
in various ways to create complex knowledge ([5.17], Agg5 completion); and defines and
constructs relations between the concepts (e.g. subordinate-superordinate relation, attribute
relation,) that are formed through the intervention of cognitive operations ([5.18], Ab3
structural). Evidence for this is also found in the following comments:
N5: Once a lesson is over, s/he reorganizes [emphasis added] the note taken from the lesson
during the break time. S/he first identifies the heading of the section and then organizes the
learned content under the identified heading in a structured way…in the learning process, s/he
first tries to understand the heading and the objectives of the section; then divides the heading
into sub-headings based on the similarity of content; and tries to find the interconnectivity
[emphasis added] between the sub-headings so that s/he can clearly understand the whole
structure [emphasis added] of the section and how each sub-section are related to each other
under the headings.
125
P5’s intention of implementing these cognitive processes (i.e., organizational structuring) is
clear from what s/he said:
P5: It is important to understand the heading of the section because it tells useful clues about
the relations with the content to be learned. I first try to define how the heading can be related
to the content and why it represents the content. It’s important to consider the connectivity
[emphasis added] between the heading and the content so that the content can be meaningfully
identified and linked with each other. Therefore, understanding the heading is the first step in
learning a particular section.
P5: ….I am not trying to memorize the content in the note by heart but I am trying to create one
full story which flows naturally by reorganizing [emphasis added] the information in it.
Therefore, with the note, I can easily figure out the structure of the section, and recognize how
each component is connected to each other within the structure. This becomes useful resource
for future learning sequences because I can easily find the structural flow of the information in
the note so that I can explain it structurally to others and to myself.
The above comments clearly show that P5 actively organizes information/knowledge in a
structural manner (i.e., schematization) during the entire course of learning processes. This
reflects that P5 went through many steps of conceptual modification and hence could acquire
complex forms of knowledge over time. P5 progressively develops a schema as P5
continually checks the completeness and accurateness of the retrieved information/knowledge
in the process of schematization ([5.19], Agg6a schema). This implies that P5 learned from
“reorganizing” and “restructuring” of information/knowledge rather than merely stacking
pieces of information/knowledge in an isolated manner. It shows evidence that
schematization involves (re-)organization and (re-)structuring of information/knowledge in
order to lead to a stable cognitive structure by comparing and/or contrasting sub-structures,
principles, concepts, and units of information.
More precisely, in the course of learning, (1) P5 identifies attributes and features of a
phenomenon of information ([5.20], Agg1 identification); defines characteristics of
information to combine equivalent relevant attributes ([5.21], Agg3a classification), and to
isolate and compare these attributes ([5.22], Agg3b decomposition); identifies information in
order to identify characteristics of it and then actively constructs relations between concepts
([5.23], Agg4 reproduction); associates the units of information together with existing
knowledge in various ways to create complex knowledge ([5.24], Agg5 completion);
organizes the associated information into a coherent cognitive structure ([5.25], Agg6a
schema); defines structural relations of the elements of P5’s cognitive structure (i.e.,
information, concepts, rules, principles) that are formed through the cognitive intervention
([5.26], Ab3 structural); and builds analogy by associating them with existing knowledge and
126
then integrating it by adding, modifying, reorganizing, and restructuring cognitive structure in
a meaningful way using analogy ([5.27], Ab4 analogy ).
P5 thus explicitly uses ([5.28], C11 representation) strategy to better understand and
remember information; and repeats learned content through practice by mentally rehearsing
them ([5.29], C10 repetition). These strategies presumably helped P5 to “reproduce” the
information in a symbol system (i.e., language) by internalizing the figurative, functional,
and/or operative regularities and invariants identified in the information; and thus to keep
stimulated content active in his/her short-term memory (STM) and promote the storage of
new content in long-term memory (LTM). This is suggested in the following comment:
P5: I try to understand the content by speaking it aloud several times until I can fully
understand all the information in the note. Then I rehearse it without referring to the note and
try to memorize it by considering the structure of the information.
P5’s learning processes of mathematics is clearly presented in the following narration:
N05: In solving math problems s/he first studied math concepts by fully explaining the concepts
to oneself by speaking aloud. S/he tried to describe the problem solving procedure and the
relevant concepts while doing this. This made him/her clearly understand the mathematical
concepts, and hence, s/he could more easily retrieve and apply the relevant concept while solving
problems in a test.
As previously stated, learning mathematics requires a learner to perceive formalized
mathematical material, which means that a learner should understand the definite structure of
any mathematical material. The necessary precondition for this ability is the adequate
identification of attributes and features of mathematical components in a given material.
Evidence for this is clearly found in this comment. Here, P5 identifies attributes and features
of a phenomenon of information ([5.30], Agg1 identification); and identifies information
from the problem solving task in terms of individuals/elements, relations, and functions of it
([5.31], Agg4 reproduction). P5 then organizes the reproduced information into a coherent
cognitive structure ([5.32], Agg6a schema) while P5 is describing the problem solving
procedure to herself. While doing this, P5 checks if s/he correctly understands the relevant
math concepts, rules, and principles ([5.33], Agg6b instantiation); and defines the relevant
part of his/her existing knowledge that can be structurally related to the information
contained in the problem statement ([5.34], Ab3 structural). This seems to show that learning
requires persistent (re-)organization and (re-)structuring leading to a comprehensive cognitive
structure. While P5 self-explains (elaborates) all the detailed explanations of the concepts
required in solving a particular problem, which was not included in a math commentary book,
127
P5 actively (re-)organizes and (re-)structures his/her cognitive structure. This is also
evidenced in the following comment:
N5: [preceding comment cont’d] while solving math problems by speaking aloud, the
participant could more carefully think about the underlying concepts and this would in turn
enrich his/ her mathematical thinking ability.
While self-explaining the study materials that go beyond the information presented in the
material, P5 deducts ([5.35], C4 inference) from existing knowledge by applying a general
principle to information in the current learning content, and thus, generalize them ([5.36],
Ab2 generalization). The analysis of the case 5 shows that these inferences helped P5 to
interconnect existing knowledge and new information in the learning content, often by way of
plausible analogical details that were not explicitly stated in the content. This inferencing
strategy seems to have a positive influence on the initial acquisition as well as the progressive
structuring of knowledge. The following comment seems to give evidence to this aspect. This
illustrates that P5 clearly understands the conditions that help him/her more successfully
accomplish a learning task ([5.37], M4 management):
P5: While I was trying to explain the math concepts and solution procedures in detail to myself,
I was able to clearly figure out the structures [emphasis added] of the problems, some missing
concepts and misunderstandings of my knowledge….In learning math, I think a good
understanding of basic concepts, principles, and formulas are an important pre-condition for
applying them in solving problems. I paid a lot of effort in learning concepts, and with this, I
was able to master all the detailed concepts and the procedures required in solving a math
particular problem. Consequently, though I did not practice many problems, [I think] I acquired
stable and flexible applicability of using the various concepts.
P5’s active schematization process is explicitly evidenced from the small experiment carried
out by the production team while interviewing P5. The experiment shows that P5 explicitly
tries to identify the schematic features of the given information. This gives evidence that P5
tried to reorganize and restructure the perceived information by associating them with his/her
existing knowledge. Therefore P5 was able to represent the learned content with his/her “own
words” and hence, s/he could better understand and organize the content ([5.38], Agg4
reproduction):
N5: The production team experimented with five volunteered students [HS 1st grade; 3 males/2
females; “average achievers”]: The five students as well as the participant took a history lesson
together. The participant [P5] was asked to use the same learning strategy that s/he used during
high school days. During the lesson, s/he diligently as well as continuously wrote down all
information taught in the lesson. During the 10 minutes of break time after taking the lesson,
some of the five students were discussing the information taught during the lesson with peers;
one of the five students was calmly reviewing the information; and s/he was also calmly
reviewing the information.
128
N5: All of the five students as well as the participant [P5] were asked to represent the previous
lesson to the rest of the class. Among all them, the participant [P5] explicitly attempted to
present the interconnectivity [emphasis added] between the information presented in the lesson
using his/her own words, while the others simply repeated or summarized the lesson.
This is also firmly evidenced in the comments provided by P5’s class teacher which is given
below:
N5: The teacher commented that compared to other students, s/he tried to deliver what s/he
understood from the lesson by using own words in a simple structured way while others were
simply trying to copy/repeat the lesson when they were asked some questions from the class.
The interview also contained an expert’s (a professor in the field of educational psychology)
comment (that the production team recorded) with regard to the “learning by teaching” or
“learning by self-explaining” strategies. The expert commented that such strategies help a
learner to elaborate and/or structuralize information, which in turn is retained longer in the
learner’s LTM, and is better retrieved from the LTM. This comment appears to be
comparable to the analysis identified above in the code [5.28] and [5.29].
Three of the five students’ commented after taking the experiment as below. This
seems to imply that one should by firmly identify the interconnectivity between the
information, and then organize them in a structured way as a whole so that the pieces of
information can be meaningfully incorporated into the whole cognitive structure:
1st student: “It helped me to restore the learned information/knowledge into my memory”; 2
nd
student: “I think it is a good learning strategy because I have to study the information in leaning
material more carefully (i.e., in detail) and structurally so that I can explain it to others”; 3rd
student: [While I was representing the previous lesson to the rest of the class], “I was
embarrassed in a moment. I understood the content but I felt difficult clearly explaining it to
others. I think it is because I did not fully understand the entire content as I only memorized
specific important parts of the content [i.e., learning material] as [locally segregated] pieces.
Therefore, I think understanding the whole flow of the information in the content [i.e., identify
structural connectivity of the information as a whole] is important as sis [P5] commented.
P5 actively posed questions to oneself and also asked for explanation, verification, or
examples about the material from a teacher or peer ([5.39], SA1 questioning):
N5: When s/he was not able to solve some problems, s/he then asked teachers for some help by
pointing out the specific concept that s/he could not understand when solving a particular
problem. This made the teachers could more efficiently correct his/her misunderstandings or
incomplete knowledge.
The math teacher’s comment also provides evidence that learning math requires structural
organization of information:
T5: One should first master basic concepts and then organizes them in a structured way. This is
critical in math problem solving because only with this, one can correctly identify/retrieve the
129
relevant concepts for solving a particular problem. If concepts are not well organized, one then
cannot correctly search/retrieve relevant information, and hence, cannot understand the
problem.
In sum, in the analysis of case 5, there seems to be evidence that P5 could develop knowledge
and strategies by continuously constructing a cognitive structure that is more advanced and
complex than prior (or existing) ones by defining the features, relations, and functions of the
elements of the newly gained information, and then constructing schemas that interconnect
these elements with existing schemas (in lines 7-29; 33-36; 44-49). Consequently, case 5
seems to give evidence for the assumptions of this study that learning is a cumulative and a
structuring process, which in turn answers the first research question in this study.
In terms of cognitive processes, (1) P5 first determined the factual knowledge (e.g.,
facts, concepts, principles, and rules) in order to understand the learning content while
reading learning materials (in lines 5-6; 9-11; 21-25; 33-34); and then (2) grouped these
components of knowledge together and reorganized them into a coherent cognitive structure
(in lines 5-8; 12-14). Hence, new components of information and knowledge were
systematically integrated into existing knowledge structure over time. (3) Thereafter, these
complexly reproduced components of knowledge were progressively mapped onto cognitive
structure until P5 constructed a whole comprehensive structure (i.e., schematization; in lines
21-29; 33-36). While doing this, P5 went through many steps of conceptual assimilation and
modification and thereby, could extend the range and complexity of interconnected
relationships that s/he was able to subsume under (or assimilate into) a hierarchical
knowledge system (in lines 12-14; 40-43). This case also gives evidence that P5 learned from
“reorganization”/“restructuring” of cognitive structure rather than mere stacking pieces of
information/knowledge. This whole learning process seems to be evidently accomplished by
systematic cumulative integration between simpler parts and complex whole cognitive
structures (in lines7-29; 33-36; 44-49). The above stated analysis is shown in the table 4.5.
130
Table 4.5 Coding of interview transcription: Case 5
Transcript of case 5 (Grade: UF) Cognitive processes Learning strategies
1
2
3
N5: The participant used “learning by teaching” method [teaching him/herself and others]. S/he first aggregated all
detailed information from different learning materials (e.g., class handouts, textbooks, and reference books) and used
these for creating a kind of “teaching note.”
[5.1], Agg1 identification [5.2], C12 resourcing
4 N5: For efficient note taking, s/he first wrote down the titles and learning objectives to set the direction of the study. [5.3], M2 planning
[5.4], M3 attention
5
6
N5: [preceding comment cont’d] I used various types of symbols or abbreviated words that could be substituted with
some texts in the note (e.g., ▼= because); and added various examples, descriptions, symbols, and pictures on it.
[5.5], Agg1 identification
[5.6], Agg3a classification
[5.7], Agg4 reproduction
[5.8], C7 imagery
[5.9], C1 grouping
[5.10], C2 note
7
8
N5: While doing this, s/he tried to understand relations between the information, and then, s/he represented it by
speaking aloud using his/her own words.
[5.12], Agg5 complex com [5.11], C3 reorganization
[5.13], C4 inferencing
[5.14], C5 elaboration
9
10
11
P5: My learning process starts with note taking. I aggregate all information from the lesson as well as related
components (e.g., concepts and principles) that can be linked to the current learning from my existing knowledge, and
this is very helpful in understanding the current lesson.
[5.15], Agg1 identification
[5.16], Agg4 reproduction
12
13
14
P5: In note taking, it is important to meaningfully link one concept to another rather than simply copying the
information from the lesson into the note. I focus on the connectivity between the concepts and thus, I try to write them
in the note in a way that I can easily explain them to myself and other students later on.
[5.17], Agg5 complex com.
[5.18], Ab3 structural
15
16
17
18
19
20
N5: Once a lesson is over, s/he reorganizes the note taken from the lesson during the break time. S/he first identifies
the heading of the section and then organizes the learned content under the identified heading in a structured way… in
the learning process, s/he first tries to understand the heading and the objectives of the section; then divides the
heading into sub-headings based on the similarity of content; and tries to find the interconnectivity between the
sub-headings so that s/he can clearly understand the whole structure of the section and how each sub-section are related
to each other under the headings.
21
22
23
24
25
26
27
28
29
P5: It is important to understand the heading of the section because it tells useful clues about the relations with the
content to be learned. I first try to define how the heading can be related to the content and why it represents the
content. It’s important to consider the connectivity between the heading and the content so that the content can be
meaningfully identified and linked with each other. Therefore, understanding the headings is the first step in learning a
particular section. I am not trying to memorize the content in the note by heart but I am trying to create one full story
which flows naturally by reorganizing the information in it. Therefore, with the note, I can easily figure out the
structure of the section, and recognize how each component is connected to each other within the structure. This
becomes useful resource for future learning sequences because I can easily find the structural flow of the information
in the note so that I can explain it structurally to others and to myself.
[5.20], Agg1 identification
[5.21], Agg3a classification
[5.22], Agg3b
decomposition
[5.23], Agg4 reproduction
[5.24], Agg5 complex comp
[5.25], Agg6a schema comp
[5.26], Ab3 structural
[5.27], Ab4 analogy
[5.19], Agg6a schema com.
131
30
31
32
P5: I try to understand the content by speaking it aloud several times until I can fully understand all the information in
the note. Then I rehearse it without referring to the note and try to memorize it by considering the structure of the
information.
[5.28], C11 representation
[5.29], C10 repetition
33
34
35
36
N5: In solving math problems s/he first studied math concepts by fully explaining the concepts to oneself by speaking
aloud. S/he tried to describe the problem solving procedure and the relevant concepts while doing this. This made
him/her clearly understand the mathematical concepts, and hence, s/he could more easily retrieve and apply the relevant
concept while solving problems in a test.
[5.30], Agg1 identification
[5.31], Agg4 reproduction
[5.32], Agg6a schema comp
[5.33], Agg6b instantiation
[5.34], Ab3 structural
37
38
39
40
N5: In general, a math commentary book does not include all the detailed explanations of the concepts required in
solving a particular problem but it rather indicates the procedure to solve the problem. However, while solving math
problems by speaking aloud, the participant could more carefully think about the underlying concepts, and this would
in turn enrich his/ her mathematical thinking ability.
[5.36], Ab2 generalization
[5.35], C4 inference
41
42
43
N5: ….When s/he was not able to solve some problems, s/he then asked teachers for some help by pointing out the
specific concept that s/he could not understand when solving a particular problem. This made the teachers could more
efficiently correct his/her misunderstandings or incomplete knowledge.
[5.39], SA1 questioning
44
45
46
47
48
49
P5: While I was trying to explain the math concepts and solution procedures in detail to myself, I was able to clearly
figure out the structures of the problems, some missing concepts and misunderstandings of my knowledge…In learning
math, I think a good understanding of basic concepts, principles, and formulas are an important pre-condition for
applying them in solving problems. I paid a lot of effort in learning concepts, and with this, I was able to master all the
detailed concepts and the procedures required in solving a math particular problem. Consequently, though I did not
practice many problems, [I think] I acquired stable and flexible applicability of using the various concepts.
[5.38], Agg4 reproduction
[5.37], M4 management
50
51
52
53
T05 (math teacher): One should first master basic concepts and then organizes them in a structured way. This is critical
in math problem solving because only with this, one can correctly identify/retrieve the relevant concepts for solving a
particular problem. If concepts are not well organized, one then cannot correctly search/retrieve relevant information,
and hence, cannot understand the problem.
54
55
56
57
58
59
60
61
62
N5: The production team experimented with five students [HS 1st grade; 3 males/2 females; “average achievers”]: The
five students as well as the participant took a history lesson together. The participant [P5] was asked to use the same
learning strategy that s/he used during high school days. During the lesson, s/he diligently wrote down all the information
taught in the lesson. During the 10 minutes of break time after taking the lesson, some of the five students were
discussing the information taught during the lesson with peers; one of the five students was calmly reviewing the
information; and s/he [P5] was also calmly reviewing the information. Then all of the five students as well as the
participant [P5] were asked to represent the previous lesson to the rest of the class. Among all them, the participant [P5]
explicitly attempted to present the interconnectivity [emphasis added] between the information presented in the lesson
using his/her own words, while the others simply repeated or summarized the lesson.
132
4.1.6 Case 6
It starts with the narration explaining P6’s (UF) initial learning strategy as follows:
N6: The participant used two textbooks for Korean language: One is for taking notes during
lessons and the other is for reorganizing the same after lessons. S/he carefully read all the details
in the textbook and highlighted important topics and then tried to paraphrase it. It was effective in
understanding and memorizing information.
This illustrates that after determining information ([6.1], Agg1 identification), P6 reproduces
the information ([6.2], Agg4 reproduction) while expressing the meaning of the information
using different words (“paraphrase it”), which presupposes the comprehension of the
elements, relations, and functions of the information. In doing this, P6 presumably constructs
a meaningful cognitive structure by combining pieces of information in a new way ([6.3], C3
reorganization). Another indication of this reorganization strategy is found in this interview
when P6 made “incorrect answer notes” after completing the test. Here P6 modifies learning
strategy to improve problem solving performance ([6.4], C13 substitution; [6.5], M7
evaluation) following the class teacher’s advice:
N6: When his/her mock test result of Korean language was not up to par, s/he thought it was
because of his/her inability to solve problems, and therefore tried to solve many different types
of problems … however, though s/he put tremendous effort into studying Korean language, the
outcome was still not successful. Therefore, s/he tried to find other ways. The class teacher
recommended that the participant create “incorrect answer notes.”
P6 first identifies and aggregates relevant information for all the problems that had been
answered incorrectly from multiple sources ([6.6], Agg1 identification; [6.7], C12 resources)
and then identifies missing concepts/knowledge and misunderstandings ([6.8], Agg4
reproduction), which is accomplished as P6 identifies the elements, relations, and functions
of the information. P6 then puts them into a note. This is evidenced in the following
narration:
N6: Following the class teacher’s suggestion, s/he started to create “incorrect answer notes.” At
the beginning, s/he was not able to figure out an effective way for creating the note (e.g., what
information should be put in the note). So, s/he simply cut the texts of the problems along with
answers from the commentary book and then pasted them in the note for all the problems that had
been answered incorrectly; and then tried to identify the missing concepts/knowledge and
misunderstanding when solving these problems by referring to reference books. S/he extracted
related information from multiple reference books and then put them in the note coherently; and
thereafter diligently reviewed this note.
P6 then progressively modifies the way of creating the incorrect answer notes in an attempt to
use the note more effectively for future learning and problem solving situations:
133
N6: Although the note became more and more complete, his/her test scores did not improve. So,
s/he created the note differently by finding out his/her own thoughts and modifying them as s/he
thought that just copying the answers from the commentary book was not effective in learning.
N6:S/he first analyzed the problems that had been answered incorrectly to find out where s/he
was incompetent to figure out exactly what the subject had to supplement; analyzed answer from
the multiple-choice questions to find out the reasons how and why each choice can or cannot be
the correct answer; and then systematically (re-)organized them in the note.
This implies that it is important to correctly identify the incomplete knowledge and
misunderstandings, and to continuously modify one’s cognitive structure through
(re-)organization and (re-)structuring (i.e., schematization) until one could solve problems
without conceptual errors while creating the note ([6.9], C3 reorganization). Here P6 actively
modifies his/her cognitive structure by identifying the sub-components (i.e., incomplete
knowledge and misunderstandings) in the cognitive structure that should be more firmly
corrected and completed in order them to be more successfully incorporated into the whole
cognitive structure. Otherwise, doing it would not be effective in learning.
While doing this, P6 was able to correctly figure out deficiencies in his/her own
knowledge and acquire the necessary knowledge to successfully solve the problems. P6
explicitly investigates the attributes/features of the condition under which his/her problem
solving strategy did not work successfully and identifies the central point needing resolution
in the further learning situations (e.g., reasoning skills, types of problems, etc.; [6.10], M6
problem). P6 clearly understands the conditions that help him/her more successfully
accomplish a learning task ([6.11], M4 management). P6 thus checks the outcomes of his/her
own learning performance against an internal measure of completeness and accuracy of the
learning goal after it has been completed and thus checks his/her strategy use or ability to
perform the task at hand ([6.12], M7 evaluation). More specifically, P6 does (a) performance
evaluation: judging his/her ability to perform the task; and (b) strategy evaluation: judging
his/her strategy use when the task is completed. Upon figuring out the central point needing
resolution in the learning task (lacks logical thinking), P6 modifies learning strategy to
accomplish a learning task ([6.13], C13 substitution). With this modified strategy, it seemed
to better support his/her understanding of the complex multiple relations between the problem
statement and the given answer choices. This illustrates that P6 actively constructs and
modifies his/her cognitive structure by identifying the sub-components in the cognitive
structure that should be more firmly corrected and completed in order them to be more
successfully incorporated into the whole cognitive structure.
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P6’s school teacher also indicated that taking notes on incorrect answers helps learners to find
missing knowledge and misunderstandings so that they are better prepared for further
learning (or studying), as provided in the following comments:
T6 (class teacher): When students make incorrect answers, most of the time, they simply think
that they simply made mistakes in solving problems. But in fact, it is because that they did not
correctly understand the concepts and principles required in solving the problems and/or they
incorrectly comprehended the texts of the problems….Most of the students report that creating
“incorrect answer notes” is difficult and time consuming but they also report that by doing it they
could clearly identify their misunderstandings and missing knowledge so that they can be better
prepared in their further learning.
This comment stresses the importance of identifying the sub-components in the cognitive
structure that should be firmly corrected and completed in order for them to be more
successfully incorporated into the whole cognitive structure.
T6: [preceding comment cont’d] In Korean language, one should first comprehend texts and
understand principles before solving problems. However, in reality, due to the insufficient time
for preparing the actual CSAT, most of the students are trying to solve problems when they lack
ability in fundamental knowledge (i.e., comprehending texts and understanding principles) and
hence, their knowledge and ability can hardly be improved….The problem is most of students
perceive that the Korean language is relatively easy to prepare for the test because it is their
mother language and hence familiar to them. Accordingly, they try to learn information only by
simply listening to lessons rather than learning them by actively constructing their own
understanding and interpretation. Hence, it seems that when they are solving problems, they
cannot identify appropriate solution steps by themselves.
The class teacher’s comment implies the importance of complex reproduction in the process
of aggregation. That is, a learner should actively process, (re-)organize, and (re-)structure
information by identifying it in terms of its elements, relations, and functions using his/her
own analogy rather than passively aggregating the information ([6.14], Agg4 reproduction;
[6.15], C3 reorganization; [6.16], C4 inferencing). In doing this, the learner can rediscover
and/or reproduce the information through one’s own cognitive information processing
activity.
P6 continued to modify the way in creating the incorrect answer notes by actively
identifying the flaw in his/her thinking processes as below:
N6: ….In spite of the participant’s consistent deliberate efforts, the test scores even worsened and
s/he could not figure out how to solve this obstacle. Going over the note however, s/he figured
out that most of the time, s/he struggled when choosing the correct answer between two choices
in multiple-choice questions, and always chose the wrong one. This led him/her to believe that
s/he had not studied the learning material in great detail and hence had chosen wrong answers
[which were indeed the case].
Going over the note, P6 checks and verifies his/her comprehension and performance in
135
problem solving in order to check the accuracy and appropriateness of the cognitive process
and learning strategy while it was taking place ([6.17], M5 monitoring): More specifically, P6
does (a) comprehension monitoring: checking and verifying understanding; and (b)
performance monitoring: checking and verifying learning performance (i.e., problem solving).
P6 explicitly investigates the attributes/features of the condition under which his/her problem
solving strategy did not work successfully and identifies the point needing resolution in the
further learning situations ([6.18], M6 problem); and checks the outcomes of learning
performance against an internal measure of completeness and accuracy of the learning goal
after it has been completed ([6.19], M7 evaluation).
N6: [preceding comment cont’d] from this s/he realized that s/he did not have sufficient ability
in extracting irrelevant attributes from the answer choices to a given problem, and hence,
created the note differently. S/he did not refer the commentary book hereafter and tried to
identify the related concepts and relations of each given choice of answers to the problem in the
texts using his/her own analogy. In doing this, s/he was able to reproduce the information from
the given text in the problem statement. For instance, from the given Korean language text (e.g.,
literary work), s/he associated “heat waves” or “baby cloud” with positive meaning; and “the
tendency of raining” or “there is no visitor” with negative meaning.
Here P6 reproduces the information from the given text by associating the information with
existing knowledge and then integrates it by reorganizing his/her cognitive structure in a
meaningful way using analogy. P6 associated “heat waves” and “baby cloud” with positive
meanings and “a chance of rain” and “there is no visitor” with negative meanings in a
literary text. This is derived from constructing a meaningful cognitive structure by relating
and making meaningful associations of different parts of the information to his/her existing
knowledge ([6.20], C5 elaboration).
In doing this, P6 first identifies information ([6.21], Agg1 identification); defines
characteristics of information to combine equivalent relevant attributes ([6.22], Agg3a
classification), and to isolate and compare these attributes ([6.23], Agg3b decomposition);
identifies and extracts information from multiple sources in terms of its individuals/elements,
relations, and functions ([6.24], Agg4 reproduction); and builds analogy by associating new
information with existing knowledge and then integrates it by reorganizing cognitive
structure in a meaningful way using analogy ([6.25], Ab4 analogy ).
P6 uses available information to infer the meanings and usage of the information with
the problem by applying self-constructed rules (using symbols) to understand the
information by analogy ([6.26], C4 inferencing). This illustrates that P6 links rather large
portions of knowledge to specific sequences of deductive arguments in a logically structured
136
way. Hence, P6’s cognitive process is enhanced through the specific situation. This allows
P6 to create new connections between two analogues which initially seem unrelated, because
P6 recognizes the similarities and commonalities between the two situations. P6’s existing
knowledge structure becomes involved in the solution of problem solving. Through this
continuous structural mapping process, P6 builds new concepts on the basis of basic
cognitive structures.
N6: [preceding comment cont’d] Furthermore, s/he identified them in the text using symbols.
For instance, ∆= negative meaning; ○= positive meaning; □=a paradoxical expression;
wave=thematic verse. With this, s/he was able to analyze the answer choices in terms of their
elements, underlying relations, and/or functions.
This illustrates that P6 understands relations (i.e., rediscovers and/or reproduces the
information) by his/her own analogy using a system of symbolization. After determining the
problem, P6 represents the elements of the problem using visual images to better understand
the information in the problem statement ([6.27], C7 imagery), and thus assists performance
of the learning by using the symbols while learning ([6.28], C2 note). While doing this, P6
reproduces the information in a visual structure by internalizing the figurative, functional,
and/or operative regularities and invariants identified in the text ([6.29], Agg4 reproduction).
N6: [preceding comment cont’d] Thus, while the participant was analyzing the note, s/he
recognized that s/he lacks logical thinking, and hence, started to write down his/her own mental
models of problem solving about how and why s/he drew the particular choice out of multiple
choices by analogy….Over time using this learning strategy, s/he was able to extract the incorrect
answers from the given choices using inferences from the given text. That is, s/he tried to
carefully investigate whether each of the given choices logically matched with the given text of a
problem in terms of elements, relations, and/or functions.
This illustrates that P6 associates the information in the given text with the answer choices
for the problem. Hence, a unit of simple information is grouped together in various ways to
create complex knowledge ([6.30], Agg5 completion). P6 compares phenomena with regard
to similarities and dissimilarities of the information between the given text and the answer
choices for the problem and then extracts commonalities from the surface as well as the
underlying structure of the information by analogy ([6.31], Ab1 comparison). P6 presumably
generates a larger description than the given information through inferences (i.e., inductive,
deductive, analogical) on the basis of superficial as well as structural (dis-)similarities and
hence transforms the descriptions of the information along the set-superset in P6’s cognitive
structure while doing this comparison ([6.32], Ab2 generalization). Hence, P6 presumably
could correctly identify and extract the given information in a way that the information
137
relevant to the problem solving goal is preserved, while the irrelevant information is ignored
(or distinguished). P6 identifies conceptual relations between the given text and the answer
choices of the problem and correctly abstracts them by filtering out invariant and
(ir-)relevant attributes from the given choices using analogy in order to avoid mistake
([6.33], Ab4 analogy); crates and/or modifies mental model(s) of problem solving ([6.34],
Ab5 model); and thus summarizes the information in the note ([6.35], C9 summarizing).
While writing down mental models of problem solving about how and why P6 drew the
particular choice out of multiple choices by analogy, P6 presumably generates a set of
attributes to characterize the entities for the analogical conclusion. Hence P6 may be able to
distinguish some (ir-)relevant aspects contained in the information in terms of these attributes.
By repeating this, P6 presumably progressively differentiate a set of different attributes
(elements, relations, and/or functions) that characterize each of the given answers as correct
or wrong in the given problem. When these processes are repeated in the course of learning,
the units of information/knowledge that belong to the schema that direct the processes
become increasingly routinized and automated as P6 consistently associates the given
information with the schema through extensive practice. Consequently, P6 seemed to be
progressively develop his/her problem solving performance from slow, conscious, and
difficult to more rapid, accurate, unconscious, and effortless automation, which is evidenced
as follow. The following statement gives evidence that the improvement of ability in solving
problems takes persistent time and efforts in analyzing and interpreting problems in the
course of learning:
N6: ….With a disciplined practice, creating the note did not take much time than before. Then
s/he practiced if s/he could use this strategy in actual test by timing each problem solving
process. With persistent practice, s/he could speed up so that the time taken to complete each
problem was reduced over time. Thus s/he became more and more confident in his/her strategy.
In sum, the analysis of case 6 seems to give evidence for the assumptions of this study that
learning is a cumulative and a structuring process, which in turn seems to answer the first
research question in this study. P6 developed knowledge and strategies by continuously
constructing a cognitive structure that is more advanced and complex than prior (or existing)
ones by defining the features, relations, and functions of the elements of the newly gained
information, and then constructing plausible organizational principles that interconnect these
elements with existing schemas until P6 acquired satisfactory schemas.
In the course of learning, (1) P6 first identified and set the goal of learning by
identifying the requirements of a particular learning task (in lines 5-10; 20-23; 44-47). To do
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this, P6 first determined the factual knowledge (e.g., facts, concepts, principles, and rules)
and procedural knowledge (e.g., procedures for solving problems) that are required to meet
the learning goal (in lines 15-16; 45-47; 55-60). (2) P6 then grouped these components of
knowledge together and reorganized them into a coherent structure (in lines 15-16; 51-60),
which presumably is accomplished by breaking them down on the basis of constructive
schematic principle using existing knowledge (in lines 44-50; 55-57). By doing this, P6
acquired concepts, principles, and procedures more complex than that s/he already had
available. Hence, new components of information/knowledge were systematically integrated
into P6’s existing knowledge structure in the course of learning, and thereafter, these
complexly reproduced components of knowledge were progressively mapped onto cognitive
structure (i.e., schematization) until P6 constructed a whole structure which functions at a
level that is satisfactory for completing a learning task (in lines 55-60). Thereby, P6 could
extend the range and complexity of interconnected relationships that s/he was able to
subsume under (or assimilate into) a hierarchical knowledge system (in lines 46-50).
This whole learning process seemed to be evidently accomplished by systematic
cumulative integration between simpler parts and complex whole cognitive structures. That
is, all of the information/knowledge the learners gained from learning sequences gradually
transformed into the cognitive structure as P6 mapped them to a schema. Each piece of
knowledge seemed to be properly subsumed under the schematic links by individual pieces
of knowledge that had been initially decomposed and were gradually subsumed under a
coherent cognitive structure (in lines 44-50; 55-60). This promoted fast activation of
associations and thus formed a larger integrated cognitive structure. This structure is then
preserved for further use (in lines 61-64). The above stated analysis is shown in the
following table 4.6.
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Table 4.6 Coding of interview transcription: Case 6
Transcript of case 6 (Grade: UF) Cognitive processes Learning strategies
1
2
3
4
N6: The participant used two textbooks for Korean language: One is for taking notes during lessons and the
other is for reorganizing the same after lessons. S/he carefully read all the details in the textbook and
highlighted important topics and then tried to paraphrase it. It was effective in understanding and
memorizing information.
[6.1], Agg1 identification
[6.2], Agg4 reproduction
[6.3], C3 reorganization
5
6
7
8
9
10
11
12
13
14
15
16
N6: When his/her mock test result of Korean language was not up to par, s/he thought it was because of
his/her inability to solve problems, and therefore tried to solve many different types of problems …
however, though s/he put tremendous effort into studying Korean language, the outcome was still not
successful. Therefore, s/he tried to find other ways. The class teacher recommended that the participant [P6]
to create incorrect answer notes.
N06: Following the class teacher’s suggestion, s/he started to create “incorrect answer notes.” At the
beginning, s/he was not able to figure out an effective way for creating the note. What information should be
put in the note for instance? So, s/he simply cut the texts of the problems along with answers from the
commentary book and then pasted them in the note for all the problems that had been answered incorrectly;
and then tried to identify the missing concepts/knowledge and misunderstanding when solving these
problems by referring to reference books. S/he extracted related information from multiple reference books
and then put them in the note coherently; and thereafter diligently reviewed this note.
[6.6], Agg1 identification
[6.8], Agg4 reproduction
[6.4], C13 substitution
[6.5], M7 evaluation
[6.7], C12 resources
17
18
19
20
21
22
23
N6: Although the note became more and more complete, his/her test scores did not improve. So, s/he created
the note differently by finding out his/her own thoughts and modifying them as s/he thought that just copying
the answers from the commentary book was not effective in learning.
N6: S/he first analyzed the problems that had been answered incorrectly to find out where s/he was
incompetent to figure out exactly what the s/he had to supplement; analyzed answer from the multiple-choice
questions to find out the reasons how and why each choice can or cannot be the correct answer; and then
systematically (re-)organized them in the note.
[6.9], C3 reorganization
[6.10], M6 problem
[6.11], M4 management
[6.12], M7 evaluation
[6.13], C13 substitution
24
25
26
27
28
29
T6 (class teacher): When students make incorrect answers, most of the time, they simply think that they
simply made mistakes in solving problems. But in fact, it is because that they did not correctly understand
the concepts and principles required in solving the problems and/or they incorrectly comprehended the texts
of the problems …. Most of the students report that creating “incorrect answer notes” is difficult and time
consuming but they also report that by doing it they could clearly identify their misunderstandings and
missing knowledge so that they can be better prepared in their further learning.
30
31
T6: [preceding comment cont’d] In Korean language, one should first comprehend texts and understand
principles before solving problems. However, in reality, due to the insufficient time for preparing the actual
[6.14], Agg4 reproduction [6.15], C3 reorganization
[6.16], C4 inferencing
140
32
33
34
35
36
37
38
CSAT, most of the students are trying to solve problems when they lack ability in fundamental knowledge
(i.e., comprehending texts and understanding principles) and hence, their knowledge and ability can hardly
be improved….The problem is most of students perceive that the Korean language is relatively easy to
prepare for the test because it is their mother language and hence familiar to them. Accordingly, they try to
learn information only by listening to lessons rather than learning them by actively constructing their own
understanding and interpretation. Hence, it seems that when they are solving problems, they cannot identify
appropriate solution steps by themselves.
39
40
41
42
43
N06: ….In spite of the participant’s consistent deliberate efforts, the test scores even worsened and s/he
could not figure out how to solve this obstacle. Going over the note however, s/he figured out that most of
the time, s/he struggled in choosing the correct answer between two choices in multiple-choice questions,
and always chose the wrong one. This led him/her to believe that s/he had not studied the learning material
in great detail and hence had chosen wrong answers [which were indeed the case].
[6.17], M5 monitoring
[6.18], M6 problem
[6.19], M7 evaluation
44
45
46
47
48
49
50
N6: [preceding comment cont’d] from this s/he realized that s/he did not have sufficient ability in extracting
irrelevant attributes from the answer choices to a given problem, and hence, created the note differently.
S/he did not refer the commentary book hereafter and tried to identify the related concepts and relations of
each given choice of answers to the problem in the texts using his/her own analogy. In doing this, s/he was
able to reproduce the information from the given text in the problem statement. For instance, from the given
Korean language text (e.g., literary work), s/he associated “heat waves” or “baby cloud” with positive
meaning; and “the tendency of raining” or “there is no visitor” with negative meaning.
[6.21], Agg1 identification
[6.22], Agg3a classification
[6.23], Agg3b
decomposition
[6.24], Agg4 reproduction
[6.25], Ab4 analogy
[6.20], C5 elaboration
[6.26], C4 inferencing
51
52
53
54
N6: [preceding conversation cont’d] Furthermore, s/he identified them in the text using symbols. For
instance, ∆= negative meaning; ○= positive meaning; □=a paradoxical expression; wave=thematic verse.
With this, s/he was able to analyze the answer choices in terms of their elements, underlying relations,
and/or functions.
[6.29], Agg4 reproduction [6.27], C7 imagery
[6.28], C2 note
55
56
57
58
59
60
N6: [preceding conversation cont’d] Thus, while the participant was analyzing the note, s/he recognized that
s/he lacks logical thinking, and hence, started to write down his/her own mental models of the problem about
how and why s/he drew the particular choice out of multiple choices by analogy….Over time using this
learning strategy, s/he was able to extract the incorrect answers from the given choices using inferences
from the given text. That is, s/he tried to carefully investigate whether each of the given choices logically
matched with the given text of a problem in terms of elements, relations, and/or functions.
[6.30], Agg5 completion
[6.31], Ab1 comparison
[6.32], Ab2 generalization
[6.33], Ab4 analogy
[6.34], Ab5 model
[6.35], C9 summarizing
61
62
63
64
N6: ….With a disciplined practice, creating the note did not take much time than before. Then s/he practiced
if s/he could use this strategy in actual test by timing each problem solving process. With persistent practice,
s/he could speed up so that the time taken to complete each problem was reduced over time. Thus s/he
became more and more confident in his/her strategy.
141
4.2 Case Studies 7 to 49 in Brief
In this section, 43 case studies (case 7 to 49) are presented in brief. Complete interview
transcripts of these studies are given in Appendix A.
Case 7 (Grade: HS3)
P7’s learning processes are presented in the following snippets of narration:
N7: The participant takes one “special” day to study science subject and to find out relations
and connections between each section of the subject. As each section of the science subject is
closely related to each other, what one has learned in the previous section can be applied to the
other sections. S/he first examines the table of contents of the textbook to determine the
importance of each section. And then, s/he simply skims through the sections that s/he is
already familiar and that which does not require in-depth knowledge and invests more time in
studying the sections where s/he feels incompetent.
P7 tries to correctly figure out deficiencies of his/her knowledge that s/he needs to meet in
learning by identifying attributes and features of a phenomenon of information ([7.1], Agg1
identification; [7.2], M6 problem); and by defining relations of each section of the subject
(e.g., attribute relation, subordinate-superordinate relation) that are presumably formed
through the intervention of cognitive operations (i.e., isolate and compare attributes, find
similarities, and form dimensions; [7.3], Ab3 structural). P7 previews the main ideas,
concepts, or principles of the material to be learned by skimming the text for the organizing
principle ([7.4], M1 organization); and thus proposes strategies for handling learning tasks
([7.5], M2 planning); attends to specific aspects of information by scanning the framework of
the learning material (i.e., TOC) as well as the sections where P7 feels incompetent during
the reviews ([7.6], M3 attention). This illustrates that P7 clearly understands the conditions
that help him/her more successfully accomplish a learning task ([7.7], M4 management).
N7: …and then s/he studies it across the whole sections. This way, s/he can more easily identify
the inter-relations [emphasis added] and this holds true within one particular subject of science
as well as in all other science subjects (e.g., physics, chemistry, biology, and earth science).
Since there are some concepts that are common to other [science] subjects, once s/he figures out
this association, s/he tries to fully reconfirm the relationship between them. This enables
him/her to study two science subjects effectively at one time.
By figuring out the interconnectivity between the sections, P7 constructs structural relations
between them. P7 thus identifies the characteristics of the information, and then actively
constructs relations between concepts ([7.8], Agg4 reproduction); associates the concepts to
form a new unified idea (“once s/he figures out this association, s/he tries to fully reconfirm
the relationship between them”; [7.9], Agg5 completion) and then (re-)organizes them into a
142
coherent structure ([7.10], Agg6a schema; [7.11], C3 reorganization). While identifying the
“inter-relations” between different science subjects, P7 relates different parts of the existing
knowledge to each other and then makes associations with the newly gained information in
the learning task ([7.12], C5 elaboration) using the existing knowledge as a basis for
understanding the new information ([7.13], C6 transfer).
N7: …and then finishes up his/her study by studying the workbook. In doing this, s/he was able
to intensively study the parts that s/he feels less confident in a structured way.
While studying the problems in the workbook, P7 applies the learned content into solving the
problem, and hence, the abstract schema that P7 constructed above is progressively rendered
less abstract during the process of practice by supplying values for the variables in a schema;
([7.14], Agg6b instantiation; [7.15], C10 repetition). With this, P2 structurally reorganizes and
restructures ([7.16], C3 reorganization) the content of his/her schemas by assimilating and/or
accommodating them in accordance with the new information ([7.17], C5 elaboration). Here
P7 uses the workbook to support learning ([7.18], C12 resourcing). This allows P7 to build up
more general concepts ([7.19], Ab2 generalization) by acquiring various simple concepts
during the course of learning experiences. This comment gives evidence that schema
induction involves the generalization of an abstract schema from multiple various procedural
features. While doing this, P7 checks his/her comprehension and performance in problem
solving in order to check the accuracy and appropriateness of the cognitive process while it is
taking place ([7.20], M5 monitoring). P7 thus checks the outcomes of his/her own learning
performance against an internal measure of completeness and accuracy of the learning goal
after it has been completed; [7.21], M7 evaluation).
Case 8 (Grade: UF)
P8 tries to correctly figure out deficiencies of own knowledge that P8 needs to meet in the
course of learning ([8.1], Agg1 identification); and defines relations between the particular
information to be learned with the other sections in the learning material (i.e., subordinate-
superordinate relation) by checking the TOC ([8.2], Ab3 structural), which is explained in the
following narrator’s comment:
N8: The participant makes preparation two minutes before lessons during break time by going
over the table of contents (TOC) of the upcoming lesson in order to check the relationship of the
section to the rest of the parts of the subject. Since the TOC contains the core information of the
section, checking it enables him/her to overview the entire frame and to identify the important
parts of it. Therefore, s/he was able to be more attentive in class.
143
P8 uses advance organization before beginning a lesson to construct a general preview of
his/her prior knowledge of the learning content to be learned, and thus generates some of the
information s/he anticipated would be included in the upcoming lesson ([8.3], M1
organization); and thus plans strategies for handling the upcoming learning task during the
lesson ([8.4], M2 planning). By doing this, P8 formulates a framework for the overall
learning process, and hence, during the lesson, P8 is able to attend more to the key
information that P8 identified during the previews ([8.5], M3 attention).
P8 thus compares his/her own idea to that of the class teacher by actively forming
questions before lesson and then finding answers during lesson ([8.6], Ab4 analogy). This
illustrates that P8 tries to correctly figure out deficiencies of his/her knowledge that s/he
needs to meet in learning ([8.7], M6 problem). Presumably, while doing this, P8 structurally
reorganizes and restructures ([8.8], C3 reorganization) the content of his/her existing schemas
by assimilating and/or accommodating them in accordance with the new information ([8.9],
C5 elaboration). When P8 encounters learning content that is inconsistent with his/her
expectations, P8 then poses questions to oneself and also asks for explanation and verification
from his/her teacher ([8.10], SA1 Questioning). P8 thus actively modifies his/her cognitive
structure accordingly ([8.11], Ab5 model). P5’s intention of implementing these cognitive
processes is clear from what s/he said:
N8: In every lesson, s/he tries to ask at least one question to the class teacher. Therefore, while
listening to lessons, s/he marks “Q” for the questions to ask to the class teacher in order to fill the
gap between his/her thoughts and the class teacher’s in terms of understanding/interpreting the
information in the learning material. When the marked question is resolved during the lesson,
s/he writes the answers right away. Actively finding out answers to the marked questions made
him/her more deeply comprehend the lesson.
P8 organizes extracted information into a structure immediately after lessons ([8.12], Agg6a
schema). By figuring out the interconnectivity between the concepts contained in the
different sections of the learning material, P8 constructs the structural relations among the
concepts. This enables P8 to (re-)organize and (re-)structure his/her cognitive structure:
N8: …. After the lessons, s/he immediately reviews the learned content with the note that s/he
took during the lesson for two minutes during break time. With this s/he could effectively
reorganize information, and hence, could remember it for a longer time.
This case illustrates that P8 continuously checks and verifies his/her comprehension to check
the accuracy and appropriateness of the cognitive process ([8.13], M5 monitoring); and
understands the conditions that help him/her successfully accomplish a learning task ([8.14],
M4 management).
144
Case 9 (Grade: HS2)
This case gives evidence for the importance of practice in problem solving. Most participants
of the interviews in the present study learned from practicing when solving various problems
with various practice books. Surprisingly, P9 repeatedly studied only one particular
mathematics workbook for a substantial period of time as stated in the following narration:
N9: The participant’s mathematics test score had been significantly increased in six months.
During this time s/he repeatedly studied only one particular math workbook ten times.
When P9 solved the problems while referring to the answer book, it was impossible to
determine whether s/he could solve the problems without referring to it or not. Therefore, P9
did not refer the answer book for the 1st and 2
nd time while solving the particular workbook:
P9: The 1st and 2nd time solving the workbook, I took away the answer book and never opened it
up. There were too many problems that I was not able to solve. I just passed them over ….It took
about two months for me to solve the workbook once, and there were more incorrect answers
than correct ones. However, having completed it all independently without referring to the
answer book, I was able to precisely determine my missing knowledge and misunderstandings,
and my level of problem solving ability.
As P9 became aware that missing knowledge impedes P9’s further improvement of learning,
P9 explicitly investigates the attributes/features of the condition under which his/her problem
solving strategy did not work successfully and identifies the information that P9 should have
corrected and completed for further learning situations ([9.1], M6 problem; [9.2], Agg1
identification). P9’s further learning processes are presented in the following snippets of
comments:
P9: When I solved it the 3rd time, I changed strategy. I now referred to the answer book for the
problems for which I did not know/understand the answer, and then tried to memorize the
solution procedure.
P9: By the time I solved the same workbook for the 6th time, I could automatically [emphasis
added] identify the solution steps and answers to solve the problems. And as the number of
problems solved in the workbook increased, the time it took for me to solve the problems and the
number of incorrect answers decreased.
P9: ….By the time I solved it for the 10th time I tried to solve the problems in various ways using
multiple approaches and tried to find out the most appropriate way to solve them.
These comments imply that by repeatedly practicing the problems along with the necessary
formulas and the solution steps, P9 progressively constructed a math problem solving schema,
and hence, was able to solve the problems in various ways by flexibly applying the various
knowledge (“multiple approaches”) contained in the schema ([9.3], Agg6b instantiation).
While attempting to solve the problems in several different directions, P9 presumably actively
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compares phenomena with regard to similarities and dissimilarities of information given in
the problem solving tasks and existing knowledge ([9.4], Ab1 comparison); and identifies
deep-structural similarities between the different problem solving models (“approaches”;
[9.5], Ab3 structural). While doing this, P9 compares problem solving mental models by
analogy ([9.6], Ab4 analogy); and creates and modifies the models accordingly to construct
the most appropriate one for the specific problem ([9.7], Ab5 model).
P9: As I had already acquired the basic concepts and principles, this improved my mathematical
skills.
This gives evidence that P9 could successfully construct the math problem solving schema
([9.8], Agg6a schema) from the various procedural steps by extracting certain general
solution strategy or principle in the process of repeatedly practicing the same workbook until
P9 constructed stable schema ([9.9], Ab2 generalization), and hence, P9 presumably could
construct more and more powerful cognitive links in his/her cognitive structure. And this may
have allowed P9 to use the newly acquired information/knowledge more efficiently and
creatively.
While solving the same workbook repeatedly, P9 continuously checks and verifies
his/her comprehension and performance in problem solving in order to check the accuracy
and appropriateness of the cognitive process and learning strategy while it is taking place
([9.10], M5 monitoring): More specifically, P9 does (a) comprehension monitoring: checking
and verifying understanding; (b) performance monitoring: checking and verifying learning
performance (i.e., problem solving); and (c) strategy monitoring: tracking how well a strategy
is working in solving all the different styles of the questions. P9 thus checks the outcomes of
own learning performance against an internal measure of completeness and accuracy of the
learning goal after it has been completed and thus checks his/her strategy use or ability to
perform the task at hand ([9.11], M7 evaluation). P9 clearly does (a) performance evaluation:
judging ability to perform the task; and (b) strategy evaluation: judging strategy use when the
task is completed. Upon figuring out the central point needing resolution in the learning task
(“I was able to precisely determine my missing knowledge and misunderstandings”), P9
progressively modifies learning strategies (i.e., solve problems by not referring answer book
→ refers to the answer book → memorizes the solution procedure → solves the problems in
various ways using multiple approaches; [9.12], C13 substitution). This clearly illustrates that
P9 actively modifies his/her cognitive structure by identifying the sub-components in the
cognitive structure that should be more firmly corrected and completed in order them to be
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more successfully incorporated into the whole cognitive structure. P9’s intention of
implementing these unique learning processes is clear from what s/he said:
N9: Seeing that the other students were studying various types of problems with multiple
workbooks, s/he became anxious and was not confident that if s/he was using a correct [i.e.,
efficient and effective] strategy to improve his/her problem solving skills in math. Nevertheless,
instead of solving the other workbooks like the other students do, s/he continued to study the
same workbook until s/he reached the 10th time as originally planned.
This illustrates that P9 clearly plans and implements the strategies for handling learning tasks
([9.13], M2 planning). P9 clearly understands the conditions that help him/her more
successfully accomplish a learning task ([9.14], M4 management). Studies (e.g., Schunk,
1990) propose that self-regulating learners tend to have positive beliefs in their capabilities in
a specific task situation. Furthermore, studies have also shown that positive self-efficacy
beliefs are more likely to promote the use of effective cognitive and regulatory strategies in a
systematic way (e.g., Neber & Schommer-Aikins, 2002). Evidence for this is found in the
following comment:
N9: …in a 2nd grade math practice test, s/he scored 100. This increased his/her confidence when
studying other school subjects, and s/he thus became a top-ranked student at the school.
While taking the interview, the production team carried out an experiment to see the
effectiveness of P9’s learning strategy:
N9: Assuming that repeated practice solving the same problems can have a positive effect, the
production team experimented with two volunteer students [HS 2nd
grade, males, “average
achievers”]. They were asked to repeatedly (10 times) practice one particular booklet of
practice material that the production team provided to them. The material contained various
questions related to the topic of “trigonometrical function.” They were asked to track each time
it took to complete the test as well as the number of wrong answers. And thus, they were asked
to clearly examine the reason why they got the wrong answers.
N9: [preceding comment cont’d] they were diligently worked on the material and at the 3rd
day
after the material was given, they were able to complete the entire questions in it [presumably
over 35 questions14
]. And at the 8th day after the material was given, they were able to practice
the material 10 times. Then they took a test about the topic they studied [post test] and strongly
positive effects were found [out of 9 questions, student 1 scored 1(pre) to 7(post) and student 2
scored 3(pre) to 6(post)].
N9: Student 1 commented that after practicing the material 4th times, s/he referenced answer
book and was able to clearly understand the solution steps and answers to solve the problems.
And as the number of problems solved in the given practice material increased, the time it took
for him/her to solve the problems and the number of incorrect answers decreased. Thus, s/he
commented that this particular experience increased his/her interest and confidence in studying
14
It was estimated on the basis of informal identifications revealed in the interviews as no definite numbers of
questions contained in the provided practicing material had been identified.
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math. After taking the post test, student 2 commented as follows: “After practicing the material
10 times, I felt easy in dealing with math problems [before this particular experience, s/he could
hardly solve math problems], and thus, I think it increased my practical ability and
understanding in math.”
Below is identified based on the informal screenshot revealed in the video:
Number of completing the material 1st 2nd
3rd
4th
… 7th
8th
9th
10th
Time took to complete the material 12h 9h 5h 3h … 3h10 min 2h30 min 2h 1h 15min
Number of wrong answers 35 20 15 11 … 15 13 7
It seems that in the process of repeatedly practicing the particular booklet of practice material,
the two students presumably could more firmly corrected and completed the relevant
sub-components contained in the problems by going through many steps of conceptual
modification, and hence, the sub-components could be more successfully incorporated into
the whole cognitive structure. Thereafter, positive effects were found. If so, this seems to give
evidence that the construction of stable schema (i.e., complex forms of stable knowledge)
involves many steps of conceptual modification by firmly mastering all necessary
sub-components in order for them to be more successfully incorporated into the whole
cognitive structure. Otherwise, merely practicing various features of multiple problems with
multiple workbooks without firmly mastering necessary sub-components would not be
effective in learning. Consequently, this reflects that in order to construct a stable schema,
which can be effectively used in further learning sequences, one should firmly master basic
sub-components so that they can be successfully integrated into a hierarchical cognitive
structures, which can be meaningfully used in the further learning sequences.
Case 10 (Grade: UF)
P10’s learning processes are presented in the following narration:
N10: The participant tried to find at least three different ways to solve each of the math problems.
S/he could even find out more than 30 different ways for unusual cases. By doing this, s/he was
able to improve his/her creative mathematical skills, and thus, was able to clearly identify the
intent of the examiner in a given test, and hence, could easily find out the most appropriate way to
solve a particular problem.
By solving a problem in multiple approaches, P10 instantiates (i.e., supplying values for the
variables in a schema) an abstract problem solving schema, and hence, the schema is
rendered less abstract as P10 relates the information/knowledge contained in his/her schema
to concrete problems solving situations ([10.1], Agg6b instantiation). While attempting to
solve a problem in several different approaches, P10 extracts the essential elements of the
information in the problem statement that should be reserved for solving the problem in the
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multiple approaches and thus compares phenomena with regard to the surface as well as the
underlying structural similarities and dissimilarities of the different approaches by analogy
([10.2], Ab1 comparison, [10.3], C4 inference); and thus identifies deep-structural relations
between the information in the problem and the different problem solving models ([10.4],
Ab3 structural). As P10 progressively identifies the commonalities that can be shared
between them, this presumably strengthens P10’s cognitive structure of math problem solving,
which helps P10 to understand the definite structure of mathematical material. While solving
a problem in multiple approaches, P10 continuously associates each of the different problem
solving models with the information in the problem to check the accuracy and appropriateness
of the models. Through these continuous integrating processes, P10 progressively
(re-)organizes and (re-)structures cognitive structure ([10.5], Ab4 analogy); and/or constructs
new model(s) of problem solving by analogy ([10.6], Ab5 model).
This illustrates that P10 could successfully construct the math problem solving schema
from the various procedural steps by extracting general solution strategies or principles from
them. This allowed P10 to understand underlying core concepts and rules, and hence, could
construct more powerful cognitive links in his/her cognitive structure ([10.7], C5 elaboration),
which in turn presumably allowed him/her to clearly understand underlying deep-structural
attributes of the information given in the problem statement (“was able to clearly identify the
intent of the examiner in a given test”). P20 thus understands the conditions that help him/her
more successfully improving his/her creative mathematical skills ([10.8], M4 management).
N10: ….S/he applied the same strategy in other subjects: For social studies, s/he tried to find a
new problem solving strategy in addition to using his/her existing knowledge. For instance, s/he
analyzed the given passage of the question by matching it with each of the given choices of
answers and then induced true and false of the each of the matches using inference.
This illustrates that P10 relates the information in the question with the information in the
each of the given choices of answers. While doing this, P6 first identifies the information
([10.9], Agg1 identification); defines characteristics of the information to combine
equivalent relevant attributes ([10.10], Agg3a classification), and to isolate and compare
these attributes ([10.11], Agg3b decomposition); identifies the information in terms of its
individuals/elements, relations, and functions ([10.12], Agg4 reproduction); and builds
analogy by associating the information in the question with the information in the each of the
given choices of answers ([10.13], Ab4 analogy ).
Here P10 clearly uses available information to infer the answer of the question based
on the logical deductions/inductions derived from matching the information in the question
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with the information in the each of the given choices of answers ([10.14], C4 inferencing).
P10 might have linked his/her existing knowledge to the specific instances of analogical
arguments in a logically structured way. Hence, P10’s cognitive process presumably is
enhanced through the specific situation.
Case 11 (Grade: HS2)
P11’s learning processes are presented in the following narration:
N11: Though the participant had solved lots of problems [persistently practiced problem solving],
his/her Korean language test score did not improve at all. After consulting with the teacher, s/he
realized that s/he had not studied learning materials in a systematic way, and hence, created a
symbol system [emphasis added] in order to organize them in a coherent structure.
P11 checks the outcomes of learning performance ([11.1], M7 evaluation; identifies the
condition under which his/her problem solving strategy did not work successfully and
identifies the central point needing resolution in the further learning situations ([11.2], M6
problem); and then selects alternative learning approaches ([11.3], C13 substitution). P11
understands the conditions that help him/her more successfully accomplish a learning task
([11.4], M4 management).
N11: While reading passages, s/he marked some words using specific symbols: For instance,
squares with key words of each paragraph; triangles and squares for contrasting words; waves for
examples, analogy, and quote. This was effective in understanding passages when solving
problems because by doing this, s/he could identify an important part of the passage at a glance.
This illustrates that P11 represents the elements of the information in a visual structure by
internalizing the functional and operative regularities identified in the learning situations
using a system of symbols ([11.5], Agg4 reproduction; [11.6], C7 imagery). By doing this,
P11 understands the schematic relations between the information, and hence more clearly
identifies key information. This refers P11 reproduces the information using inference by
applying learned or self-constructed rules (i.e., symbol system). In doing this, P11 classifies
the information (words, terminology, sentences, concepts) on the basis of its attributes or
meaning ([11.7], C1 grouping); marks down key words and/or concepts in graphic form to
assist learning ([11.8], C2 note). While using the symbols (e.g., squares, triangles) that could
be substituted with some texts in the note, P11 consciously applies the rules to understand
information, and thus constructs rules on the basis of analysis of learned information “to link
the current learning content with previously learned content” by analogy ([11.9], C4
inferencing). This way, P11 is able to represent the learned content and concepts with his/her
own “words” and hence, P11 can better understand and organize the information, which in
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turn is retained longer in his/her memory, which is evidenced in the following narration:
N11: Thus, s/he could remember the content of the passage longer and more precisely than before,
and hence, was able to solve problems correctly.
P11 identifies attributes and features of a phenomenon of the information from multiple
sources ([11.10], Agg1 identification; [11.11], C12 resourcing), and then organizes the
information in various ways, which is evidenced in the following narration:
N11: ….S/he organized the theories of Korean language and then summarized an important part
on a sheet of paper. For those words that s/he did not clearly know the meaning of, s/he
aggregated detailed information from multiple learning sources (e.g., textbooks, reference books,
workbooks) and then organized them on the paper.
N11: S/he thus organized literary works in various ways (e.g., the period of literary activities by
the authors).
Upon identifying the information, P11 puts the information into a serial array to form larger
chunks of information ([11.12], Agg2 serial; [11.13], C1 grouping); defines characteristics of
the information to combine equivalent relevant attributes, and to isolate and compare the
attributes ([11.14], Agg3a classification; [11.15], Agg3b decomposition); defines and/or
constructs relations between the information ([11.16], Agg4 reproduction); and organizes it
into a structure ([11.17], Agg6a schema; [11.18], C9 summarizing).
N11: [preceding comment cont’d] from this, s/he could identify the nature of a particular literary
work by associating the name of the author even if it was the work that s/he had never learned
before….With persistent practice, s/he could speed up so that the time taken to solve each
problem and the number of incorrect answers [of problem solving tasks] were reduced over time.
This comment illustrates that P11 associates previously segregated pieces of information to
each other to form a new large complex pack of information which consists of units of related
information ([11.19], Agg5 completion; [11.20], C5 elaboration). In doing this, P11 defines
the relations of the information in the process of cognitive operations (e.g., isolates and
compares attributes, finds similarities, and forms dimensions, which in turn leads P11 to form
corresponding schema ([11.21], Ab3 structural). P11 thus instantiates the schema with
persistent practice ([11.22], Agg6b instantiation).
Case 12 (HS2)
P12’s learning processes are presented in the following snippets of narration:
N12: The participant memorized English vocabulary cumulatively and repeatedly. In other words,
s/he memorized the words that s/he learned yesterday (1st set of words) along with the new words
learned today (2nd set of words); then, the following day, s/he memorized the words that s/he
learned from the previous days (1st & 2nd sets of words) along with the new words learned for the
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day (3rd set of words). S/he continued this way sequentially until s/he memorized all words in
English textbook, and continually tried to mentally rehearse them as many times as possible.
P12 repeats learned information through practice by mentally rehearsing them
(“constructively associating images and relations”), which thus evidenced as below ([12.1],
C10 repetition). This strategy helps P12 to keep stimulated content active in his/her memory
and promote the storage of new content in long-term memory.
N12: ….S/he thus grouped some related words together in various ways (e.g., categories, images,
functions, prefix, suffix); and then tried to naturally remember these words by constructively
associating images and relations. For instance, s/he tried to memorize and retrieve new words
by generating easily recalled images (textual and or figural) of a relationship between the word
in English and a homonym in Korean. Thus, with persistent repetition, the time it takes to
memorize the words was gradually decreased.
Upon identifying the attributes and features of a phenomenon of the information (i.e.,
categories, images, functions, prefix, and suffix of the English words) ([12.2], Agg1
identification), P12 puts them together into a serial array ([12.3], Agg2 serial); and defines
characteristics of them to combine equivalent relevant attributes ([12.4], Agg3a
classification), and to isolate and compare the attributes ([12.5], Agg3b decomposition). The
classified phenomena are then encoded as cognitive operations (i.e., internalizing the
figurative, functional, and/or operative regularities and invariants identified in the process of
memorization; [12.6], Agg4 reproduction). P12 thus identifies relationships of the series of
the words by figuring out their structural invariants (i.e., categories, images, functions, prefix,
and suffix of the English words) and groups them accordingly ([12.7], Ab3 structural; [12.8],
C1 Grouping). This presumably promotes P12 to perceive the grouped words as a flowing
unified set rather than as an individual piece. By coordinating units of the words and their
relational connections, the words are integrated into organized integrative cognitive structures.
P12 thus tries to memorize and retrieve new words by generating easily recalled images
(textual and or figural) of a relationship between the word in English and a homonym in
Korean ([12.9], C8 keyword; [12.10], C7 imagery).With persistent practice, P12 gains an
internally automatized structure. This case appears to be comparable with case 3.
Case 13 (Grade: UF)
P13’s learning processes are presented in the following narration:
N13: The participant had hard time understanding high school math lessons as s/he did not fully
understand basic math concepts. So s/he repeatedly studied one particular math workbook
regardless of whether the answers were right or wrong until s/he fully understood each problem in
the book. By the time s/he practiced the workbook five times, s/he was able to figure out exactly
what had to be corrected and learned more for solving math problems.
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This illustrates that P13 did performance evaluation ([13.1], M7 evaluation), and realizes the
incompetent point (lack of understanding of basic math concepts) which impedes further
improvement of learning ([13.2], M6 problem). P13 clearly understands the conditions that
help him/her more successfully accomplish a learning task ([13.3], M4 management).
Accordingly, P13 tires to identify the attributes and features of the information (i.e., basic
math concepts) that should be firmly corrected and completed in the learning situations by
repeatedly studying the same set of problems ([13.4], Agg1 identification; [13.5], C10
repetition). Presumably, in the process of repeatedly studying the same set of information,
P13 might have constructed math problem solving schema ([13.6], Agg6a schema) by
progressively identifying the sub-components (i.e., misunderstandings and incomplete
knowledge) in the cognitive structure (i.e., schema) that should be firmly mastered in order
them to be successfully incorporated into the cognitive structure. This seems evidenced in the
narrator’s (N13) comment that “s/he was able to figure out exactly what had to be corrected
and learned more for solving math problems.” Furthermore, P13 could instantiate the problem
solving schema by repeatedly solving the problems ([13.7], Agg6b instantiation). Hence, P13
was able to correctly figure out deficiencies in his/her own knowledge, and acquired the
necessary knowledge to successfully solve math problems. This case appears to be
comparable with case 9.
Case 14 (Grade: HS3)
P14’s learning processes are presented in the following snippets of narration:
N14: Even though basic math concepts are mastered, if one cannot correctly apply these concepts
when solving various types of problems, then they are useless. In order to correctly
understand/distinguish the different types of problems, s/he categorized and organized math
problems by type [emphasis added] and then studied/practiced similar type of problems together;
and then put them in a note.
This illustrates that with investigations of various “types” of the math problems, P14
reorganizes and restructures the content of existing schemas by assimilating and/or modifying
them in accordance with the newly gained information ([14.1], C1 grouping; [14.2], C3
reorganization). This presumably allows P14 to build up more general concepts and thus
correctly determine the specific procedural knowledge that can be applied in a particular
“type” of problem the problem. While grouping the problems into different categories, P14
identifies a phenomenon of information ([14.3], Agg1 identification); puts the information
into a serial way ([14.4], Agg2 serial); defines characteristics of information to combine,
isolate, and compare them ([14.5], Agg3a classification; [14.6], Agg3b decomposition); and
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identifies the elements of the information in different types by internalizing the identified
functional and operative regularities of the information ([14.7], Agg4 reproduction).
N14: Thereafter, s/he intensively studied the problems that s/he had answered incorrectly, and
tried to find different approaches in solving these problems. Repeating this process over and over
again, s/he was able to understand math concepts more deeply, and was also able to clearly
remember the reasons why s/he made incorrect answers. This in turn improved his/her
mathematical applicability while problem solving.
While practicing the problems by type, P14 might have compared/extracted some surface as
well as underlying structural commonalities (e.g., concepts, rules, principles, laws) of the
information that are grouped together in the each of the type by analogy ([14.8], Ab1
comparison); and might have associated the extracted information to form a unified idea for
each of the type ([14.9], Agg5 completion). In doing this, P14 associates the information with
his/her existing knowledge and then integrates them together into a new cognitive structure in
a meaningful way using analogy ([14.10], Ab4 analogy); and creates and/ or modifies mental
models of problem solving ([14.11], Ab5 model).
While attempting to solve a problem in several different approaches (“tried to find
different approaches in solving these problems”), P14 instantiates (i.e., supplying values for
the variables in a schema) an abstract problem solving schema, and hence, the schema is
rendered less abstract as P14 relates the information/knowledge contained in his/her schema
to concrete problems solving situations ([14.12], Agg6b instantiation). While doing this, P14
might have extracted the essential elements of the information in the problem statement that
should be reserved for solving the problem in the multiple approaches; might have compared
phenomena with regard to the surface as well as the underlying structural similarities and
dissimilarities of the plausible different approaches by analogy ([14.13], Ab1 comparison,
[14.14], C4 inference); and thus, P14 might have identified deep-structural relations between
the information in the problem and the plausible different problem solving models ([14.15],
Ab3 structural). As P14 progressively identifies the commonalities that can be shared
between them, this might have strengthened P14’s cognitive structure of math problem
solving, which might have lead P14 to clearly understand the definite structure of
mathematical material. P14 might have continuously associated each of the different problem
solving models with the information in the problem to check the accuracy and appropriateness
of the models. Presumably, through these continuous integrating processes, P14 progressively
(re-)organized and (re-)structured cognitive structure ([14.16], Ab4 analogy); and hence,
could successfully construct the math problem solving schema from the various procedural
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steps by extracting certain general solution strategy or principle from them ([14.17], Ab5
model). P14 thus did performance evaluation ([14.18], M7 evaluation), and identified the
incompetent point (“s/he intensively studied the problems that s/he had answered incorrectly”)
which impedes further improvement of learning ([14.19], M6 problem). P14 clearly
understands the conditions that help him/her more successfully accomplish a learning task
([14.20], M4 management).
Case 15 (Grade: HS3)
P15’s learning processes are presented in the following snippets of narration:
N15: As the participant had not yet fully acquired fundamental basic math concepts, s/he was not
able to solve even the basic level of high school math problems. Hence, to master the basic math
concepts, s/he thoroughly studied middle school math for three months: S/he first identified
learning content, studied concepts, and practiced to apply the concepts by solving various types of
problems. In the process of doing this, s/he realized that s/he was missing and misunderstanding
lots of core math concepts…and this improved his/her ability of concept application in problem
solving.
P15 did performance evaluation ([15.1], M7 evaluation), and identified the incompetent point
(“to master the basic math concepts”) which impedes further improvement of learning and
understands the conditions that help him/her to successfully accomplish a learning task and
control learning performance to maximize efficiency of learning ([15.2], M6 problem). While
doing this, P15 checks his/her comprehension of the learning material while it is taking place,
and hence realizes s/he realized that s/he was missing and misunderstanding lots of core math
concepts ([15.3], M5 monitoring). Upon figuring out the central point needing resolution in
the learning task, P15 changed learning approach accordingly and resourced the basic-level
conceptual book (“studied middle school math for three months”; [15.4], C13 substitution).
While studying the middle school math, P15 first identifies information ([15.5], Agg1
identification); and studies the sub-components (“concepts”) in the cognitive structure that
should be more firmly mastered in order them to be successfully incorporated into the whole
cognitive structure of math problem solving. While learning the concepts, P15 might have
gone through many steps of conceptual modification and hence might have acquired complex
forms of knowledge over time (“and this improved his/her ability of concept application in
problem solving”; [15.6], Agg6a schema). P15 thus instantiates the schema by applying
learned concepts into problem solving ([15.7], Agg6b instantiation). Presumably, P15
progressively “reorganizes” and “restructures” the schema, and hence, develops more
complex schema as P15 continually checks the completeness and accurateness of the learned
information in solving problems ([15.8], C3 reorganization).
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N15: ….Since the CSAT is to measure the problem solving skills as well as the ability to
understand the problem statements, s/he analyzed the previous examinations implemented by
KICE: S/he marked the concept(s) that the examiner asks in solving a problem with blue-pen; and
marked the parts that is easy to make mistakes with a red-pen and then put these into a note to avoid
making the same mistakes.
This illustrates that P15 clearly proposes strategies for handling learning tasks ([15.9], M2
planning); and attends to specific aspects of the information in the problem statement by
scanning for the information that P15 should pay more attention ([15.10], M3 attention).
While analyzing the previous examinations, P15 decomposes the information contained in the
problem statements in order to define the characteristics of it ([15.11], Agg3b
decomposition); classifies the information into different categories (the concept(s) that the
examiner asks in solving a problem, and the information that is easy to make mistakes ([15.12],
Agg3a classification; [15.13], C1 grouping); and then summarizes them in a note ([15.14], C9
summarizing). P15 clearly understands the conditions that help him/her more successfully
accomplish a learning task ([15.15], M4 management).
Case 16 (Grade: UF)
This case explicitly illustrates that schematization requires persistent (re-)organization and
(re-)structuring of cognitive structure in order to lead to a stable cognitive structure by
comparing and/or contrasting the sub-structures, concepts, and units of information contained
in the cognitive structure. This is presented in the following snippets of narration:
N16: The participant systematically [emphasis added] created notes for each school subject in
order to effectively study and organize vast amounts of learning materials. After organizing the
notes, s/he repeatedly wrote them down for the purpose of studying.
N16: For Korean language, s/he created “incorrect answers note” in order to reduce mistakes
while solving problems. For mathematics, s/he created two separate notes (one for organizing
concepts, and the other for organizing incorrect answers). In the concept note, s/he organized
important math concepts and formulas along with their derivations, and examples from multiple
conceptual books. As the textbook is core in learning history, s/he organized the history note
multiple times in various ways based on the content (e.g., by historic event, people, period, and so
on). For world history, s/he aggregated and extracted common facts and information from three
different textbooks. For geography, s/he focused on the interpretation of diagrams and maps, and
so, to get familiar with national maps, s/he reorganized it in the note.
In the process of creating the notes, P16 identifies attributes and features of a phenomenon of
the information in the learning material ([16.1], Agg1 identification); puts them into a serial
array to form larger chunks of information ([16.2], Agg2 serial); defines characteristics of
information to combine equivalent relevant attributes, and to isolate and compare these
attributes ([16.3], Agg3a classification; [16.4], Agg3b decomposition). P16 identifies the
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information from the readings in terms of its individuals/elements, relations, and functions
([16.5], Agg4 reproduction; [16.6], C3 reorganization). While doing this multiple times in a
variety of ways, P16 progressively specifies various relationships in the information and
associates (group) them in various ways to form a new unified idea (“s/he organized the history
note multiple times in various ways based on the content (e.g., by historic event, people, period,
and so on); [16.7], Agg5 completion; [16.8], C5 elaboration; [16.9], C1 grouping). P16
systematically (re-)organizes extracted information into a coherent structure, and hence
completes schema ([16.10], Agg6a schema; [16.11], C2 note; [16.12], C3 reorganization;
[16.13], C9 summarizing). P16 thus persistently instantiates schemas with the practice of
problem solving ([16.14], Agg6b instantiation), which is evidenced as follows:
N16: For economics, after learning basic economic concepts, s/he solved problems, and then
aggregated the problems wherein s/he made incorrect answers as s/he thought that the correct
interpretation of the problem and the application of learned concepts in solving problems were
more important than simply learning concepts.
P16 did performance evaluation ([16.15], M7 evaluation), and explicitly identifies conditions
that help him/her to successfully accomplish a learning task and control learning performance
to maximize efficiency of learning ([16.16], M6 problem); and clearly proposes strategies for
handling learning tasks ([16.17], M2 planning). P16 clearly understands the conditions that
help him/her more successfully accomplish a learning task ([16.18], M4 management).
This case illustrates that P16 actively modifies cognitive structure by identifying the
various types and relations of sub-components in the cognitive structure that should be more
firmly identified in order them to be more successfully incorporated into the whole cognitive
structure. This case clearly shows that P16 went through many steps of conceptual
modification and hence could acquire complex forms of knowledge over time. That is, all of
the information/knowledge P16 gains from learning sequences gradually transform into the
cognitive structure as P16 maps them to a complex schema. Each piece of declarative
knowledge is subsumed under the schematic links by individual pieces of knowledge that
were initially decomposed and then gradually subsumed under a coherent cognitive structure.
Case 17 (Grade: HS2)
P17 uses a unique learning strategy in learning and solving mathematics problems, which is
presented in the following narrator’s comment:
N17: The participant repeatedly studies the same math problem not by simply solving the
problem multiple times but by creating [emphasis added] new problems by transforming the
problem into various ways.
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The cognitive processes are clearly presented in the following two snippets of narration:
N17: The first step for doing this is identifying the problem statement by checking the
condition(s) and question(s) of it. By carefully investigating the problem, s/he identifies some
possible transforming conditions, and then creates a draft list of problems. Out of this list, s/he
then identifies the problems that could be and could not be transformed as a problem by referring
various concept books to ensure if each of the problems meets proper conditions to become “a
problem.” Then s/he discusses the items on the list with his/ her friends and modifies them
accordingly. If there are any problems that they cannot solve, then they get some help from their
teacher.
P17 constructs new math problems by constructing mental models. P17 then identifies
whether each of the models can be applied in solving each of the constructed problems. In
order to construct the problems, P17 transforms the original problem statement into various
ways ([17.1], C1 grouping), In doing this, P17 identifies and extracts the mathematical
objects (or elements) from the given problem statement and interprets them appropriately in
the given context ([17.2], Agg1 identification); defines characteristics of the information to
combine equivalent relevant attributes ([17.3], Agg3a classification), and to isolate and
compare the attributes with the goal of constructing a new problem ([17.4], Agg3b
decomposition); identifies and extracts available information from the original problem
statement in terms of its elements, relations, and functions ([17.5], Agg4 reproduction); and
associated the reproduced information into a new unified idea ([17.6], Agg5 completion).
P17 uses available information to infer the meanings and usage of the information for
constructing new problems by applying learned and/or self-constructed rules ([17.7], C4
inferencing).While associating the information, P17 constructs a meaningful cognitive
structure by combining elements of the information in various new ways ([17.8], C3
reorganization); and by relating and making meaningful associations of different parts of the
information to each other ([17.9], C5 elaboration).
Based on the packs of unified ideas, P17 constructs mental models by generalizing,
formalizing, and refining the various properties of the reproduced information ([17.10], Ab5
model); and thus verifies the models to find out whether they are context-appropriate or not
with respect to properties of his/her existing knowledge. P17 thereafter verifies the models
through discussion with peers ([17.11], SA2 cooperation). The mental models can be
modified through abstraction (compressing or removing certain attributes) or concretization
(the addition of attributes) as learning proceeds.
N17: Creating problems by oneself has never been easy. However, by analyzing, decomposing,
and reassembling the given (original) problem to create new problems, s/he could more deeply
understand the examiner’s intent as well as, more clearly find out his/her incompetent areas so
that s/he can improve those parts later on in the course of learning.
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This implies that P17 could progressively understand formalized mathematical material, and
hence, could successfully construct well-organized as well as complex problem solving
schemas ([17.12], Agg6b instantiation). Consequently, P17 could clearly find out the
sub-components (i.e., incomplete knowledge and misunderstandings) in his her cognitive
structure that should be more firmly mastered in order for them to be successfully
incorporated into the whole cognitive structure. Presumably, as P17 implemented various
structural changes while transforming the given problems into various ways, P17 could
extract useful commonalities among the different structural dimensions, and hence, could
find some general problem solving principles ([17.13], Ab3 structural; [17.14], Ab2
generalization). P17 clearly understands the conditions that help him/her more successfully
accomplish a learning task ([17.15], M4 management). This case seems to give evidence that
schema induction involves the generalization of an abstract schema from multiple procedural
changes.
Case 18 (Grade: HS3)
P18’s learning processes are presented in the following snippets of narration:
N18: The participant identified/extracted key words to study based on examination of previous
examinations, and then identified the related concepts that can be linked/associated to these key
words and then marked them in textbooks.
P18 first identified information in the learning material ([18.1], Agg1 identification); and
defines characteristics of the information to extract key information by isolating and
comparing the attributes of the information ([18.2], C8 keyword; [18.3], Agg3b
decomposition), and to link related information (“related concepts”) with the key information
([18.4], Agg3a classification; [18.5], C1 grouping). P18 identifies relations between the key
words and (aggregated) related concepts ([18.6], Agg4 reproduction); and then associates
them to create complex knowledge ([18.7], Agg5 completion). This helped P18 identify and
extract the key concepts that should be firmly mastered in order for them to be successfully
incorporated into the whole cognitive structure. This way, P18 could identify the gist of the
information in the learning material that should be firmly mastered during the course of
learning. This is evidenced in the following narrator’s comment:
N18: While doing this, s/he could clearly identify the concepts that more frequently showed up in
the tests, and hence, s/he could identify more or less important parts from the learning material
and focused more on studying the important parts.
The following comments illustrates that P18 organizes extracted information into a coherent
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structure ([18.8], Agg6a schema; [18.9], C3 Reorganization):
N18: S/he then transcribed the whole textbook by organizing these associated information.
While P18 was studying the learning material across sections, P18 defined structural relations
between the information within his/her whole cognitive structure ([18.10], Ab3 structural);
and thus identified and/or constructed meaningful relations between the information by
analogy ([18.11], Ab4 analogy). This helped P18 in applying different concepts when solving
a complex problem.
P18: With this, s/he was able to thoroughly study the learning material without missing any
single part of it.…S/he studied learning material of a particular subject across sections rather than
simply studying particular section separately. This was a big help in solving the CSAT type
problems which requires applying multiple concepts from across the sections when solving a
question.
This shows that P18 clearly understands the conditions that help him/her more successfully
accomplish a learning task (solving the CSAT type problems; [18.12], M4 management).
Case 19 (Grade: HS3)
P19 correctly figures out deficiencies in his/her knowledge that s/he needs to meet in learning
([19.1], M6 problem); plans strategies for handling learning tasks accordingly ([19.2], M2
planning); and therefore, selects alternative learning approaches ([19.3], C13 substitution):
N19: The participant thought that s/he fully mastered all the concepts in middle school math, but
later on, s/he realized that it was not true while s/he was studying high school math. Hence, s/he
started learning those basic concepts that s/he was vaguely aware of again until those concepts
became clear to him/her.
P19 formulated a framework for the learning process at the concrete as well as the abstract
level as follows:
N19: While doing this, s/he realized that some of the high school level’s math problems that s/he
solved in a complex way could be simply solved by applying basic concepts and principles learned
from middle school math. Hence, while studying the high school math, s/he intensively focused on
the sections and concepts that are closely associated with high school math.
The cognitive processes are clearly presented in the following snippets of narration:
N19: In doing this, s/he first checked the table of contents of middle school math and then
identified and extracted those parts that would need more attention and effort (i.e., more important
parts); and then, tried to figure out how the concepts in middle school math were related and
developed in high school math (i.e., how basic math concepts are developed into higher levels of
complex concepts/ principles).
P19 identifies attributes and features of a phenomenon of the information in the learning
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materials (middle/high school math; [19.4], Agg1 identification); puts the identified
information into a serial array ([19.5], Agg2 serial) to figure out how the properties of basic
middle school math has evolved into advanced high school math; and reproduces the
information in terms of its elements, relations, and functions to figure out how the properties of
basic middle school math can be applied in high school math ([19.6], Agg4 reproduction).
While doing the “reproduction,” P19 associates pieces of the reproduced information to each
other into a large unified idea ([19.7], Agg5 completion). In the process of figuring out how the
properties of basic middle school math has evolved into advanced high school math, P19
extracts commonalities from superficial as well as the underlying structure of information by
analogy ([19.8], Ab1 comparison); and defines structural relations of the information within
the cognitive structure (e.g., subordinate-superordinate relation; [19.9], Ab3 structural). As a
result of this, some previously vague mathematical concepts became clearer. With this, P19
deepened the understanding of mathematical concepts, and hence, could solve high school
math in a much simpler way using multiple approaches.
Presumably, implementing these processes, P19 could progressively identify how the
sub-components (i.e., information, concepts, principles) at lower levels are progressively
evolved and incorporated into those at higher levels in the cognitive structure (i.e., schema).
In other words, P19 might have progressively mapped the different levels of properties onto
the sub-components of cognitive structure. Thereby, P19 presumably could extend the range
and complexity of interconnected relationships that s/he was able to subsume under (or
assimilate into) a hierarchical cognitive structure. Therefore, P19 presumably could construct
well-organized math problem solving schema ([19.10], Agg6a schema).
N19: .…S/he thus organized the problems that s/he was not able to solve in a note and repeatedly
reviewed it. With this, s/he was able to clearly understand the principles in high school math.
This illustrates that P19 did performance evaluation ([19.11], M7 evaluation); realizes the
incompetent point which impedes further improvement of learning ([19.12], M6 problem);
and clearly understands the conditions that help him/her successfully accomplish a learning
task ([19.13], M4 management). Accordingly, P13 tries to identify the attributes and features
of the information that should be firmly mastered in the learning situations ([19.14], Agg1
identification); reorganizes the aggregated information in a note ([19.15], C3 reorganization);
and instantiates the schema by repeatedly practicing problems ([19.16], Agg6b instantiation;
([19.17], C10 repetition). This case seems to illustrate that the whole learning process was
accomplished by systematic cumulative integration between simpler parts and complex whole
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cognitive structures.
To examine the effects of P19’s learning strategy, the production team conducted small a
experiment with 39 high school students (2nd
grade) in a class that were divided in two groups,
to see the difference in their math problem solving processes between the groups. This is
explained in the following snippets of narration:
N19: One team [experimental group, 16 students] was given the learning material which
contained comprehensive information (e.g., term, definition, characteristic, concept, etc) related
to the entire 3yrs of middle school math about “figure” and the other team [control group, 23 students] was not provided with the material. After studying for one day, they worked on the 6
problems related to the 1st grade high school level of “figure” for 30 minutes, and were
interviewed to elicit responses regarding what they were thinking.
N19: Most of the students from the experimental group commented that there was a positive
effect. The class teacher assesses the test results of the two groups, and commented as below:
T19: One participant from the experimental group used more effective and efficient problem
solving processes in terms of its clarity and simplicity when compared to the one from the control
group though both of them got correct answers to the question. While the former clearly
understood the underlying concepts and their implications to apply in solving the question, the
latter solved the question by substituting complicated formula. Therefore, with regard to the time
efficiency, the former is better than the latter.
N19: ….One student from the experimental group commented that: “I recognized that a test
deals with all the accumulated information for each topic, and hence, I recognized that I should
basically know all the information from the middle school levels.” Another student commented
that: “I recognized that I forgot lots of information from the middle school math. And I see that
some of the high school levels of mathematical formulas are different from the middle school
level.” Another student commented that: “I see that there is some information that cannot be
applied in to the high school levels of math. And hence, I think I can ignore some parts of it.”
It shows that some of the students recognized how the properties of basic middle school math
has evolved into advanced high school, and thus, could identify underlying structural relations
and commonalities between them within their cognitive structures (e.g.,
subordinate-superordinate relation). Therefore, they could more clearly perceive formalized
mathematical material.
Case 20 (Grade: UF)
P20’s learning processes are presented in the following snippets of narration:
N20: After solving problems, s/he compared his/her thoughts with the explanations from the
answer (or commentary) book for all the problems irrespective of whether it was correct or wrong
in order to identify the missing knowledge, misunderstandings, and alternative solution processes.
This illustrates that P20 explicitly compares his/her problem solving procedures with the
procedures from the commentary book ([20.1], Ab1 comparison).
N20: S/he especially studied thoroughly the problems that had been answered incorrectly. First,
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s/he identified what part of his/ her explanation was incorrect and then identified the missing
concepts that s/he would need to learn more. Then s/he repeatedly wrote them down in order to
memorize them until s/he was able to correctly apply those concepts when solving problems.
P20 first tries to figure out deficiencies in his/her knowledge that s/he needs to meet in
learning ([20.2], Agg1 identification; [20.3], M6 problem); and then repeats the identified
information ([20.4], C10 repetition).
N20: S/he then did a comparative analysis of multiple answer books with regard to the
approaches and processes of problem solving, and realized that there were different approaches
for solving the problems between different answer books. In this case, s/he aggregated these
problems and grouped them together based on the types of the problems, and then compared the
different approaches and identified the similarities and the differences among them. S/he then
selected the most appropriate (best) and effective way among them.
P20 identifies and extracts the information from multiple sources (“different approaches for
solving the problems between different answer books”; [20.5], Agg4 reproduction; [20.6], C12
resource); classifies them based on the types of the problem ([20.7], Agg2 serial; [20.8], C1
grouping); and then extracts commonalities from superficial as well as the underlying
structure of the classified information by analogy ([20.9], Ab1 comparison).
N20: ….S/he thus identified and aggregated important concepts and the problems that had been
answered incorrectly in order to intensively study those parts. Then s/he created his/her own
answer book by combining his/her “incorrect answer notes” (a note of the wrong answers s/he
has gotten on a test) with comprehensive note (a note wherein s/he comprehensively organized
learned content).
P20 identifies attributes and features of a phenomenon of the information (“important
concepts and the problems that had been answered incorrectly”; [20.10], Agg1 identification);
and then organizes them into the “answer book” ([20.11], C3 reorganization; [20.12], C2 note).
P20 investigates the attributes/features of the condition under which his/her problem solving
strategy did not work successfully and identifies the central point needing resolution in the
further learning situations ([20.13], M6 problem). P20 clearly understands the conditions that
help him/her more successfully accomplish a learning task ([20.14], M4 management).
N20: S/he then practiced the problems by referring to his/her own answer book rather than simply
copying the solution processes and answers from the answer book. In doing this, s/he wrote down
all solution steps that were omitted in the answer book (e.g., steps for proving a formula or
calculation process), and thus, added different approaches for solving the same problem in the
note. These well-organized notes became a great reference material and s/he repeatedly reviewed
them.
While solving the problems by referring to the answer book, P20 identified different
approaches for solving the same problem and then added them in the note. This reflects that
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P20 identifies or constructs new problem solving models by analogy ([20.15], Ab5 model).
While attempting to solve the problems in several different directions, P20 presumably actively
compares phenomena with regard to similarities and dissimilarities of the information given
in the problem solving tasks and existing knowledge ([20.16], Ab1 comparison); and
identifies deep-structural similarities between the different problem solving models
(“approaches”; [20.17], Ab3 structural). In doing this, P20 progressively constructs new
connections as s/he recognizes the similarities and commonalities of the different solution
processes. Therefore, P20 progressively constructs complex math problem solving schema
([20.18], Agg6a schema), and hence, is able to solve the problems in various ways by flexibly
applying the various knowledge (“multiple approaches”) contained in the schema ([20.19],
Agg6b instantiation).
Case 21 (Grade: HS3)
P21’s learning processes are presented in the following snippets of narration:
N21: The participant went through five learning stages to perfectly memorize a textbook: (1)
S/he checked the table of contents (TOC) of the textbook to identify the structure and the
relations between each section; (2) thoroughly read them; (3) s/he read the textbook once again
and identified/marked core information and its relations between other supportive information ;
(4) s/he read the textbook once again by carefully investigating the information and relations
identified in Step 3; and then (5) checked the title and subtitles of the sections of the book by
rehearsing the studied information. With this, s/he was able to thoroughly study the entire parts of
the textbook…After going through the five steps, s/he tested it by filling a blank paper with the
content that s/he could remember in the form of mind map to check how correctly s/he could
memorize every detail of the content studied.
P21 previews the main ideas, concepts, and/or principles of the material to be learned by
checking the text for the TOC ([21.1], M1 organization). P21 clearly proposes strategies for
handling learning tasks ([21.2], M2 planning); and attends to key information and other
information relevant to it during the reviews ([21.3], M3 attention; [21.4], keyword method).
While doing this, P21 first identifies attributes and features of a phenomenon of the
information ([21.5], Agg1 identification); defines characteristics of the information (e.g., core
concepts) to combine equivalent relevant attributes, and to isolate and compare these
attributes ([21.6], Agg3a classification; [21.7], Agg3b decomposition); compares the
superficial as well as the underlying structure of the information using analogy (“to identify
the structure and the relations between each section”; [21.8], Ab1 comparison); and defines
structural relations (subordinate-superordinate relation) among the information within
cognitive structure ([21.9], Ab3 structural). While processing these processes, P21 associates
parts of the information to each other to form a whole interconnected unified idea ([21.10],
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Agg5 completion); and hence completes a schema (i.e., mind map; [21.11], Agg6a schema).
N21: While re-reading the textbook, s/he marked the part that could not be remembered using
different colored pens; and then read the textbook again focusing on the parts that s/he was not
able to remember while creating the mind map. This was repeated until s/he could perfectly
create the mind map, and hence, s/he was able to perfectly memorize the textbook.
As P21 became aware that missing knowledge impedes further improvement of learning, P21
explicitly investigates the incomplete knowledge (“the parts that s/he was not able to
remember while creating the mind map”; [21.12], M6 problem); and repeatedly organizes
extracted information into a coherent structure until s/he could perfectly organize them into the
mind map ([21.13], C10 repetition; [21.14], C9 summarizing). P21 clearly understands the
conditions that help him/her more successfully accomplish a learning task ([21.15], M4
management). P21’s further learning processes are presented in the following narration:
N21: During the examination period, s/he compared last semester’s tests with the textbooks to
identify the examiner’s intent when asking questions. By doing this s/he was able to find out
which part of the textbook should be more intensively studied and thus s/he was able to predict
questions that were likely to be asked in the test.
Here P21 identifies and extracts the relevant information, concepts, and principles that should
be mastered in order to solve the problems in the test ([21.16], Agg1 identification. While
doing this, P21 identifies (superficial as well as the underlying structural) relations between
the information in the textbook and the questions in the test and correctly abstracts them by
extracting relevant information from the textbook using analogy ([21.17], Ab1 comparison;
[21.18]; Ab3 structural, [21.19], Ab4 analogy).
Case 22 (Grade: UF)
P22’s learning processes are presented in the following snippets of narration:
N22: The participant studied by creating “incorrect answer notes.” Upon realizing that s/he had
made repeatedly the same mistakes even within a day (e.g., got incorrect answer during the day,
and made the same mistake in the evening study), s/he thought that s/he should modify the way she
was creating the incorrect notes. It took substantial time to create the incorrect note as s/he put all of
the problems that had been answered incorrectly. Therefore, s/he filtered the problems that had
been answered incorrectly and then extracted those problems that s/he got wrong more than two
times. This was because the problems that s/he got wrong only once were likely due to a simple
calculation mistake or due to some simple misunderstandings.
P22 explicitly investigates the attributes/features of the condition under which his/her
problem solving strategy did not work successfully and hence identifies the central point
needing resolution in the further learning situations (“with this, s/he was able to figure out the
sections s/he was incompetent”; [22.1], Agg1 identification; [22.2], M6 problem); and checks
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the outcomes of his/her own learning performance against an internal measure of
completeness and accuracy of the learning goal after it has been completed and thus checks
his/her strategy use or ability to perform the task at hand (i.e., performance evaluation,
strategy evaluation; [22.3], M7 evaluation). This illustrates that P22 actively modifies his/her
cognitive structure by identifying the sub-components in the cognitive structure that should
be more firmly corrected and completed in order them to be more successfully incorporated
into the whole cognitive structure.
N22: And then s/he created the incorrect answer notes using color papers instead of the typical
note: S/he separated each section of the textbook (e.g., blue section referred to “probability and
statistics”; orange section referred to “log”). With this, s/he was able to figure out the sections s/he
was incompetent, and hence, studied more intensively those weak areas.
This illustrates that P22 figures out deficiencies in his/her knowledge that s/he needs to meet
in learning by identifying the particular sections P22 was incompetent by classifying the
identified incompetent areas into different sections ([22.4], C1 grouping). Upon figuring out
the central point needing resolution in the learning task, P22 changes approach to accomplish
a learning task (“…and hence, studied more intensively those weak areas”; [22.5], C13
substitution. With this modified strategy, it seemed to be a better support to his/her
understanding of the missing concepts and misunderstandings. P22 clearly understands the
conditions that help him/her more successfully accomplish a learning task ([22.6], M4
management).
N22: ….S/he solved the problems in the note at least five times: after solving the problems (1st),
s/he modified the solution processes for the problems that had been answered incorrectly by
referring to the answer book (2nd
); s/he then solved them again (3rd
) and then modified the
solution processes one more time by referring to the answer book (4th); and solved them again
(5th)….With this strategy, s/he could reduce the time in creating the note and thus, the burden for
re-learning because it reduced the amount of material to be learned.
This illustrates that P22 actively modifies his/her cognitive structure by modifying and
constructing problem solving models by analogy ([22.7], Ab5 model). By repeatedly solving
the same problems in the note, P22 went through many steps of conceptual modification and
hence could acquire complex forms of knowledge over time ([22.8], C3 reorganization;
[22.9], C10 repetition). In doing this, P22 progressively creates new connections as s/he
recognizes the similarities and commonalities of the different solution processes ([22.10],
Ab4 analogy). Therefore, P1 progressively completes problem solving schema by continually
modifying mental models ([22.11], Agg6b schema completion). It is inferred that more
advanced and stable problem solving schema is constructed as P22 progressively integrates
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extracted modifications into the schema and then (re-)organizes and (re-)structures them. P22
thus instantiates the schemas through deliberate practice ([22.12], Agg6b instantiation).
Case 23 (Grade: HS3)
P23’s learning processes are presented in the following snippets of narration:
N23: At first, the participant observed and copied other students’ learning strategies. One of
them was creating a literature note. S/he exactly copied the friend’s strategy by organizing the
note using heading such as main subject, categories, and historical background. But later on, s/he
modified the strategy to make it more suited to his/her level of ability.
This shows that P23 modifies learning strategy to improve learning performance ([23.1], C13
substitution; [23.2], M7 evaluation).
N23: [preceding comment cont’d] during lessons, s/he took notes; and then added the
information learned from reference books and workbooks.
P23 takes notes during lesson, aggregates relevant information from multiple sources, and then
adds them in the note ([23.3], C2 note; [23.4], C12 resource; [23.5], C9 summarizing).
N23: ….s/he thus used symbols [emphasis added] for identifying the meaning of the words in the
literature text (e.g., poem) so that s/he could more clearly identify the meaning. Though creating
the notes took considerable time to complete, it was effective in learning.
After determining the information, P23 represents the elements of the problem using visual
images to better understand the information in the problem statement ([23.6], C7 imagery),
and thus assists performance of the learning by using symbols while learning ([23.7], C2
note). While doing this, P23 internalizes the figurative, functional, and/or operative
regularities using the system of symbolization, and reproduces the information by his/her
own analogy using a system of symbolization ([23.8], Agg4 reproduction). P23 clearly
understands the conditions that help him/her more successfully accomplish a learning task
([23.9], M4 management).
Case 24 (Grade: HS3)
P24s’ learning processes are presented in the following snippets of narration:
N24: ….Two students actively cooperated in their learning throughout their high school years.
For instance, they created a structural framework [emphasis added] while studying literature by
clearly identifying the relationship between the persons that showed up in the literature using
symbols [emphasis added].
P24s identified information and then actively constructed relations by arranging it in a visual
structure (system of symbolization), which was accomplished by internalizing the figurative,
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functional, and/or operative regularities and invariants identified in learning situations (i.e.,
reproduce information; [24.1], Agg4 reproduction; [24.2], C7 imagery); and structurally
organized the identified information in a note ([24.3], C2 note; [24.4], C3 reorganization).
These processes helped them to understand the structural relations of the information.
N24: ….The major events were organized in chronological order. By doing this, they could more
easily understand the literary works. Thus they could understand it more from a structural point
of view.
This illustrates that P24s actively (re-)constructed their cognitive structure while organizing
the information in a structured manner. While doing this, they firstly identified attributes and
features of a phenomenon of the information ([24.5], Agg1 identification); defined
characteristics of the information (e.g., core concepts) to combine equivalent relevant
attributes, and to isolate and compare these attributes ([24.6], Agg3a classification; [24.7],
Agg3b decomposition; [24.8], C1 grouping); extracted commonalities from the superficial as
well as from the underlying structure of the information using analogy ([24.9], Ab1
comparison); and defined structural relations (subordinate-superordinate relation) among the
information ([24.10], Ab3 structural); associated the information to each other in order to form
a whole unified idea ([24.11], Agg5 completion); and organized the extracted information into
a coherent structure, and hence completed a schema ([24.12], Agg6a schema). As they
recognized the information, they progressively created new connections using analogy
([24.13], Ab4 analogy).
N24: ….Initially, one of them tried to mimic and model the thought processes of the other,
whose academic achievement was far superior. As the learning proceeded, the lower-achieving
learner became better and better at internalizing these thought process and thus managed to
regulate his/her own learning process progressively over time. This led to a successful increase
in the lower-achieving learner’s academic abilities and achievement, thus reducing the learner’s
reliance on the other learner over time. In the end, the thought and learning processes of the two
learners shifted from a relationship of dependence to one of sharing. For instance, they
discussed clippings from the newspaper, shared their opinions, and gave feedback to each other.
This enlarged the scope of their thought and improved both of their logical thinking and writing
skills.
P24s actively cooperated in their learning throughout their high school years. Initially, one of
them tried to mimic and model the thought processes of the other, whose academic
achievement was far superior ([24.14], SA2 cooperation). This enlarged the scope of their
thought and improved both of their logical thinking and writing skills. They thus provided
personal motivation by arranging rewards for themselves when a learning activity has been
successfully completed ([24.15], SA3, reinforcement) which is evidenced in the following
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comment:
P24s: ….we sometimes rewarded ourselves to cheer us up when a targeted learning activity has
been successfully completed. For instance going shopping, take one day off, dining out and etc.
The data shows that P24s clearly understand the conditions that help them more successfully
accomplish a learning task, and arrange their learning processes to account for these
conditions ([20.16], M4 management).
Case 25 (Grade: HS3)
P25’s learning processes are presented in the following snippets of narration:
N25: The participant had some hard time in math but eventually ranked within 0.02% at the
national practice test. Previously, though s/he could easily solve simple problems, s/he had a hard
time when solving difficult levels of math problems. Hence, s/he reflected on his/her learning
strategies, and realized that s/he only practiced low and high level of math problems but not
practiced enough intermediate level. Upon realizing that, s/he differentiated the math problems into
5 different levels and studied them from the easiest level to the highest level at his/her pace.
P25 explicitly investigated the attributes and features of the condition under which his/her
learning strategy did not work successfully in problem solving situations and hence identified
the central point needing resolution in the further learning situations ([25.1], Agg1
identification; [25.2], M6 problem); and checked the outcomes of learning performance
against a learning goal after it has been completed, and thus checked his/her strategy use or
ability to perform the task at hand (i.e., performance as well as strategy evaluation; [25.3],
M7 evaluation). P25 clearly understands the conditions that help him/her more successfully
accomplish a learning task, and arranges learning processes to account for these conditions
([25.4], M4 management). While differentiating the math problems into different levels, P25
defines characteristics of the information to combine equivalent relevant attributes, and to
isolate and compare to these attributes ([25.5], Agg3a classification; [25.6], Agg3b
decomposition); and then groups the components of the information and reorganizes them
into a structure by their level of complexity ([25.7], C1 grouping; [25.8], Agg2 serial).
N25: ….S/he first reviewed the content learned in class by referring conceptual books.
This illustrates that P25 identifies the information learned in class referring to reference
books [25.9], Agg1 identification; ([25.10], C12 resourcing).
N25: S/he then practiced simple calculation problems which primarily can be solved simply using
proper concepts and then progressively moved to higher levels….S/he tried to integrate multiple
concepts learned from different sections.
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While practicing the 5 different levels of math problems, P25 compares the (dis-)similarities
between the different levels of information ([25.11], Ab1 comparison); and defines relations
between them (e.g., attribute relation, subordinate-superordinate relation; [25.12], Ab3
structural). While doing this, P25 progressively integrates the different levels of the
information, and then (re-)organizes and (re-)structures them in his/her cognitive structure in
a meaningful way using analogy ([25.13], Ab4 analogy ). Therefore, P25 progressively
recognizes how sub-components are structurally incorporated into the whole cognitive
structure (i.e., schema; [25.14], Agg5 completion). Presumably, as learning proceeds, the
integrated and assimilated schema becomes progressively complex and profound that it can
extend itself to interpret the phenomena of the world, because it provides an insightful
framework for interpreting new information which can direct further modifications (or
accommodation) in further learning sequences ([25.15], Agg6a schema). As these experiences
accumulate in the course of learning, P25’s cognitive structure becomes more enhanced and
developed, and hence, P25’s knowledge and skills progressively develops over time. This is
evidenced in the following comment:
N25: ….By doing this, s/he could determine how the basic concept has progressed into advanced
level.
P25’s further learning processes are clearly presented in the following narration:
N25: ….S/he collected all problems that had been answered incorrectly and identified related
topics/sections in the textbook. S/he then studied each corresponding section by going through
the entire 5 levels of problems and then solved those problems that had been answered incorrectly.
This is because once s/he completed the entire steps; s/he was able to figure out new approaches
to solve them. This made her more and more confident in math.
This gives evidence again that P25 explicitly investigates the attributes and features of the
condition under which his/her learning strategy did not work successfully in problem solving
situations and hence identifies the central point needing resolution in the further learning
situations ([25.16], Agg1 identification; [25.17], M6 problem); and checks the outcomes of
his/her own learning performance against an internal measure of completeness and accuracy
of the learning goal after it has been completed ([25.18], M7 evaluation). P25 plans strategies
for handling learning tasks and identifies the requirements of a particular learning task in
order to firmly master the sub-components in his/her cognitive structure (“studied each
corresponding section by going through the entire 5 levels of problems and then solved those
problems that had been answered incorrectly”; [25.19], M2 planning).
While doing this, P25 determines the factual knowledge (facts, concepts, principles,
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rules) as well as the procedural knowledge (procedures for solving more advanced levels of
math problems) at different levels. This presumably is accomplished by breaking them down
on the basis of constructive schematic principle using existing knowledge. By doing this, P25
acquires concepts, principles, and procedures more complex than that s/he already has
available. Hence, new components of information/knowledge are systematically integrated
into P25’s existing knowledge structure in the course of learning ([25.20], C3 reorganization;
[25.21] Agg5 completion). Thereafter, these complexly reproduced components of
knowledge are progressively mapped onto schemas (i.e., schematization) until P25 constructs
a whole structure which functions at a level that is satisfactory for completing a learning task
([25.22] Agg6a schema). Through this continuous structural mapping process, P25
(re-)discovers new problem solving approach by abstraction process ([25.23], Ab3 structural;
[25.24], Ab5 model). Thereby, P25 can extend the range and complexity of interconnected
relationships that s/he is able to subsume under interacting hierarchical knowledge system by
relating different levels of information to each other and then makes meaningful associations
with the information in the given learning task ([25.25], C5 elaboration; [25.26], C6 transfer).
The whole learning process seems to be accomplished by systematic cumulative integration
between simpler parts and complex whole cognitive structures.
Case 26 (Grade: UF)
P26’s learning processes are presented in the following narration:
N26: ….Before the lesson, s/he divided the note into three columns (preview, lesson, and
review), and then reviewed the table of content to be learned in upcoming lesson and checked the
relationship of the section (to be learned) within the whole parts of the textbook.
This shows that P26 develops a unique way of note taking ([26.1], C2 note) to reduce the
time to explore all sections in social studies. P26 clearly plans strategies for handling learning
tasks ([26.2], M2 planning).
N26: S/he thus predicted the content to be learned and put it in the preview column of the note.
By doing this, s/he was able to see the entire framework of the learning material, and thus was
able to identify those topics that would need more attention and effort (i.e., more important parts)
in the learning process in advance, and hence, s/he was able to be more attentive in class. During
the class, s/he took notes comparing it with the content of the preview; and during the test
period, studied by reviewing the content in the 1st and 2
nd columns.
P26 previews the main ideas, concepts, or principles of the material to be learned by checking
the text for the TOC ([26.3], M1 organization); and attends to key information and the entire
framework of the information to be learned during the reviews ([26.4], M3 attention; [26.5],
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C8 keyword). P26 clearly understands the conditions that help him/her more successfully
accomplish a learning task, and arranges the learning processes to account for these
conditions ([26.6], M4 management).
P26 first identifies attributes and features of a phenomenon of the information ([26.7],
Agg1 identification); defines characteristics of the information (e.g., core information) and
differentiates them accordingly; [26.8], Agg3a classification; [26.9], C1 grouping). Therefore,
P26 attends to the key information during the class. While defining the relationship between
the section to be learned and the whole sections in the learning material, P26 actively
constructs relations between the information ([26.10], Agg4 reproduction); defines structural
relations (subordinate-superordinate relation) between the information within the entire
framework of the learning content, associates the information to each other, and forms a large
unified idea ([26.11], Ab3 structural; [26.12], Agg5 completion). P26 then organizes the
extracted information into a note. This whole learning process seems to be evidently
accomplished by systematic cumulative integration between simpler parts and complex whole
cognitive structures. This case seems to suggest that organization strategy supported P26 in
constructing internal links of cognitive structure by relating relevant aspects of the newly
learned information to existing knowledge structure.
Case 27 (Grade: UF)
P27 proposes strategies for handling learning tasks by reorganizing the note multiple times
([27.1], M2 planning; [27.2], C10 repetition), which is evidenced in the following comment:
N27: The participant reorganizes and restructures [emphasis added] his/her note using three
steps, namely, an addition process, a division process, and a subtraction process.
The reorganization processes are clearly presented in the following comment:
N27: S/he first collects as much information as possible from multiple resources (class note
taking, reference books, workbooks, and the note takings of other students if necessary) and then
organizes them in his/her note in detail (s/he calls it as an addition process). In doing this, s/he
repeatedly studied the learning material and hence, was able to naturally memorize them.
P27 identifies attributes and features of a phenomenon of the information using multiple
resources, and then organizes them in a note ([27.3], Agg1 identification; [27.4], C12
resourcing; [27.5], C2 note).
N27: ….As there was insufficient time to study all of this information aggregated in the note, s/he
tried to systematically organize the information in order to reduce the time to review it later on.
Hence, s/he classified the content based on the titles and key words of each section (s/he calls it a
division process) and then extracted the core information out of it (s/he calls it a subtraction
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process).
P27 puts pieces of the information together into a serial array to form larger chunks of
information ([27.6], Agg1 identification; [27.7], Agg2 serial); defines characteristics of the
information to combine equivalent relevant attributes, and to isolate and compare these
attributes in order to extract the core information out of it ([27.8], Agg3a classification; [27.9],
Agg3b decomposition; [27.10], C1 grouping).
N27: ….By doing this, s/he was able to clearly identify the content within the whole framework
of the textbook and hence, was able to extract keywords that s/he should focus during study.
P27 identifies the aggregated information from multiple sources in terms of its
individuals/elements, relations, and functions ([27.11], Agg4 reproduction); and then
organizes the extracted information into a whole framework of his/her cognitive structure
([27.12], Agg6a schema) so that s/he can identify how each of the specific information is
related within the structure. While doing this, P27 defines structural relations
(subordinate-superordinate relation) among the information within the entire framework of the
learning content ([27.13], Ab3 structural); and thus continuously associates newly learned
information with existing knowledge using analogy ([27.14], Ab4 analogy; [27.15], C3
reorganization). This whole learning process seems to be accomplished by systematic
cumulative integration between simpler parts and complex cognitive structures.
P27: ….S/he organized four study plans: an hourly timetable, a daily study timetable, a weekly
schedule, and an action table for checking the progress of the study.
P27 proposes strategies for handling learning tasks ([27.16], M2 planning). It shows that P27
understands the conditions that help him/her more successfully accomplish a learning task,
and arranges the learning processes to account for these conditions ([27.17], M4
management).
Case 28 (Grade: US)
P28 studied the online lectures provided by EBS:
N28: S/he collected information from the online lectures and then put them in a note. Then s/he
reviewed the note as many times as possible (more than 20 times).
This illustrates that P28 aggregates information using online lectures ([28.1], Agg1
identification; [28.2], C12 resourcing); summarizes extracts the information in a note ([28.3],
C9 summarizing); and reviews them multiple times ([28.4], C10 repetition).
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Case 29 (Grade: UF)
P29’s learning processes are presented in the following narration:
N29: For social studies, the participant created “a concept note” by aggregating all of the
information from multiple learning sources. S/he first established the framework based on the
textbook and class handouts, and then wrote down the core content of each section in the note;
s/he then added additional information from reference books and workbooks. In this way, s/he
was able to effectively study a variety of materials.
P29 first identifies and aggregates all relevant information to a learning task ([29.1], Agg1
identification) using multiple resources ([29.2], C12 resourcing), and then constructs the
structural framework for the aggregated information, and organizes the information in a note
accordingly ([29.3], C2 note). While organizing the information into the structure, P29 puts
the information into a serial array to form larger chunks of it ([29.4], Agg2 serial; [29.5],
grouping); defines characteristics of the information to combine equivalent relevant attributes,
and to isolate and compare these attributes ([29.6], Agg3a classification; [29.7], Agg3b
decomposition). P28 then identifies and reproduces the information in terms of its
individuals/elements, relations, and functions ([29.8], Agg4 reproduction); and associates the
pieces of information to each other in order to integrate them into a larger unified framework
([29.9], Agg5 completion; [29.10], C5 elaboration; [29.11], C3 reorganization). P29 then
systematically organizes the associated information into a coherent structure, and hence
completes a schema ([29.12], Agg6a schema; [29.13], C2 note; [29.14], C9 summarizing).
N29: After aggregating and organizing the note, s/he extracted the most critical information from
among them by analyzing the (types of) questions that had appeared on the previous
examinations as s/he thought that the information that were repeatedly asked in the examinations
were essential information to learn. S/he tried to figure out the related sections for each question
and then intensively reviewed those sections.
This illustrated that P29 identifies core sub-components (“the most critical information”) in
the cognitive structure that should be firmly mastered in order that they are more successfully
incorporated into the whole cognitive structure so P29 can use the sub-components more
effectively during future learning and problem solving situations:
N29: S/he thus analyzed the given choice of answers from the multiple-choice questions to find
out the reasons how and why each choice can or cannot be the correct answer.
While analyzing the information in the given question with each of the given choice of
answers, P39 reproduces the information in terms of its individuals/elements, relations, and
functions ([29.15], Agg4 reproduction) using available information to infer the meanings and
usage of the information in the given context. While doing this, P29 applies existing
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knowledge to identify and/or create conceptual relations between the two information by
associating pieces of the separated information to each other in various ways, and thereby,
builds analogy ([29.16], Ab4 analogy; [29.17], C4 inferencing). P29 compares phenomena
with regard to similarities and dissimilarities of the information between the given text and
each of the answer choices of the problem, and then extracts commonalities from the surface
as well as the underlying structure of the information by analogy ([29.18], Ab1 comparison).
While doing this comparison, P29 presumably generalizes the given information through
inferences (i.e., inductive, deductive, analogical) on the basis of superficial as well as
structural (dis-)similarities of the information with the information/knowledge in his/her
existing knowledge, and hence may transform the attributes/features of the information along
the set-superset in P29’s cognitive structure ([29.19], Ab2 generalization). Hence, P6
presumably could correctly identify and extract the attributes/features of the given
information in a way that the information relevant to the problem solving goal is preserved,
while the irrelevant information is ignored (i.e., filtering out invariant and (ir-)relevant
attributes).
N29: For the problems that had been answered incorrectly, s/he tried to find out the reasons and
then summarized them in the corresponding sections of the note. By doing this, s/he was able to
identify the essential information as well as the missing information/knowledge that s/he should
learn further.
P29 identifies the sub-components (i.e., incomplete knowledge and misunderstandings) that
should be firmly corrected and completed in order them to be more successfully incorporated
into the whole cognitive structure ([29.20], Agg1 identification); and then summarizes them
in a note. This way, P29 progressively maps the sub-components onto a cognitive structure.
N29: If one does not learn and study the organized content in the note, the note is not useful in
learning and problem solving. Hence, for history, s/he reorganized the timeline of the historical
events in chronological order along with its causes and consequences on a B4-size paper, and
then added small incidents. S/he then checked if there was any missing information on the notes
in the process of learning.
In an attempt to use the aggregated/extracted information more effectively for future learning
and problem solving situations, P29 classifies the information into different groups ([29.21],
C3 reorganization; [29.22], Agg2 serial; [29.23], C1 grouping). P29 defines characteristics of
the information and then classifies ([29.24, Agg3a) and decomposes ([29.25], Agg3b) it. P29
then reproduces the information in terms of its individuals/elements, relations, and functions
([29.26], Agg4 reproduction); and organizes it into a structure ([29.27], C2 note; [29.28], C9
summarizing). In doing this, P29 continuously identifies missing information, and thereby,
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progressively completes a schema ([29.29], Agg6a schema).
In order to construct meaningful associations between different information that can be
used in later learning sequences, P29 reorganized them in a variety of ways. The organization
strategy seems to support P29 in constructing internal links by relating relevant aspects of the
newly learned information to existing knowledge structure. Thereafter, these complexly
reproduced components of information/knowledge were progressively mapped onto a
cognitive structure (i.e., schematization) until P29 constructed a whole structure which
functions at a level that is satisfactory for completing a learning task. Thereby, P29 could
extend the range and complexity of interconnected relationships that s/he was able to
subsume under interacting hierarchical knowledge system. This whole learning process was
also accomplished by systematic cumulative integration between simpler parts and complex
whole cognitive structures.
N29: For the content that was difficult to memorize, s/he did a lecture to him/herself like a
teacher. With this, s/he could efficiently memorize the content organized in the notes.
P29 internalizes learned content by rehearsing them, and hence the figurative, functional,
and/or operative regularities and invariants of this information are encoded as cognitive
operations ([29.30], C10 repetition). This helped P29 to keep stimulated content active in
his/her memory and promoted the storage of new content in long-term memory.
The analysis of the interview gives evidence that P29 explicitly checks the outcomes of
his/her own learning performance ([29.31], M7 evaluation); and identifies the point needing
resolution in the further learning situations ([29.32], M6 problem). P29 clearly understands
the conditions that help him/her more successfully accomplish a learning task ([29.33], M4
management).
Case 30 (Grade: UF)
P30’s learning processes are presented in the following snippets of narration:
N30: ….S/he designed a systematic way to do the literary analysis. For instance, for poem, s/he
analyzed the poem by four criteria (i.e., its subject matter, the state of the speaker, the meaning of
poetic words, and the atmosphere) referring to commentary books of literature. While reading the
passage(s) of problems, s/he analyzed and organized them by these four criteria.
P30 identifies attributes and features of a phenomenon of the information resourcing
commentary books of literature in addition to given learning materials ([30.1], Agg1
identification; [30.2], C12 resourcing); defines characteristics of the information to combine
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equivalent relevant attributes, and to isolate and compare these attributes ([30.3], Agg3a
classification; [30.4], Agg3b decomposition).
N30: S/he then carefully compared his/her own analysis to that of the commentary book and then
modified his/her thoughts (i.e., analysis) accordingly. With this, his/her analytic ability had
gradually increased.
N30: After solving the problems, s/he carefully compared his/her own analysis to that of the
commentary book; and then modified his/her thoughts. Consequently, his/her analytic ability had
gradually increased over time.
These comments show that P30 compares the two analysis (his/her own analysis to that of the
commentary book) with regard to similarities and dissimilarities ([30.5], Ab1 comparison);
and then modifies his/her cognitive structure by identifying the sub-components (i.e.,
incomplete knowledge and misunderstandings) in the cognitive structure that should be
corrected and completed in order them to be more successfully incorporated into the whole
cognitive structure. While doing this, P30 was able to correctly figure out deficiencies in
his/her own knowledge, and acquired the necessary knowledge to successfully solve the
problems. P30 explicitly investigates the condition under which his/her mental model for
solving problems did not work successfully ([30.6], M6 problem; [30.7], Ab5 model); and
hence builds analogy by associating newly gained information with existing knowledge in a
meaningful way using analogy ([30.8], Ab4 analogy). P30 thus systematically structures
aggregated information, which is clear from what s/he said:
N30: For fiction, s/he identified the structure by investigating people, events, and background.
P30 groups information ([30.9], Agg2 serial; [30.10], C1 grouping); defines characteristics
of the information to combine, isolate, and compare the attributes of the information
([30.11], Agg3a classification; [30.12], Agg3b decomposition). P30 then “reproduces” the
information by identifying its individuals/elements, relations, and functions ([30.13], Agg4
reproduction); and organizes it into a structure. It is inferred that as P30 continually defines
the information (its features, relations, functions), and then repeatedly (re-)organizes and
(re-)structures the information based on the different organizational principles that
interconnect the elements of the information, P30 could construct a cognitive structure that is
more advanced and complex than prior (or existing) ones ([30.14], Agg6a schema).
N30: As s/he thought that the important factors in the analysis of the novel are characters, events,
and background, s/he created a symbol that represented each of the factors and identified them in
the text accordingly. For instance, ∆= the clue of background; ○= important event; □= character;
wave=emotional state.
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This illustrates that P30 understands relations between the information by his/her own
analogy using a system of symbolization (i.e., rediscovers and/or reproduces the information).
After determining the information, P30 classifies it on the basis of its attributes or meaning
(i.e., words, terminology, sentences, concepts, and/or principles; [30.15], C1 grouping); and
then represents the elements of the information using visual images to better understand the
information in the problem statement ([30.16], C7 imagery; [30.17], C2 note; [30.18], C9
summarizing). This way, P30 was able to represent the learned content and concepts with
his/her own “words” and hence, P30 could better understand and organize the information,
which in turn is processed effectively in his/her working memory, and hence, the time spent
on problem solving was significantly reduced, which is evidenced in the following narration:
N30: [preceding comment cont’d] and then based on these symbols, s/he created a structural map.
This strategy was efficient because s/he did not have to reread the text while solving problems.
Furthermore, to quickly identify the meaning of the text of nonliterary writing, s/he used
diagrams to understand the meaning of text. Consequently, the time spent on problem solving
was significantly reduced.
P30 reproduces the information in a visual structure by internalizing the figurative, functional,
and/or operative regularities and invariants identified in the information ([30.19], Agg4
reproduction). While creating the structural map, P30 compares the information with regard
to similarities and dissimilarities of it ([30.20], Ab1 comparison); and defines relations
between concepts (e.g., attribute relation, subordinate-superordinate relation). In doing this,
P30 actively constructs relations between the information ([30.21], Ab3 structural). P30 thus
progressively builds analogy by associating newly gained information with existing
knowledge in a meaningful way ([30.22], Ab4 analogy). The date gives evidence that P30
checks the outcomes of his/her own learning performance ([30.23], M7 evaluation); and
identifies the point needing resolution in the further learning situations ([30.24], M6 problem).
P29 understands the conditions that help him/her more successfully accomplish a learning
task ([30.25], M4 management). It seems that the whole learning process was accomplished
by systematic cumulative integration between simpler parts and complex whole cognitive
structures.
Case 31 (Grade: UF)
P31’s learning processes are presented in the following snippets of narration:
N31: ….For science, the participant first studied concepts by reading conceptual books rather
than blindly memorizing them. At first, s/he read reference books focusing on the parts that the
school teacher pointed out as important concepts. S/he then underlined the content that s/he
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thought were important.
P31 identifies attributes and features of a phenomenon of the information resourcing other
sources in addition to given learning materials ([31.1], Agg1 identification; [31.2], C12
resourcing); and repeats it ([31.3], C10 repetition). P30 then identifies the condition under
which his/her learning strategy did not work successfully and identifies the point needing
resolution in further learning situations ([31.4], M6 problem); and then selects alternative
learning approaches accordingly ([31.5], C13 substitution). This modified strategy helps P30
to keep stimulated content active in his/her memory and promotes the storage of new content
in long-term memory. The following comments illustrate this point:
N31: By the time s/he read the conceptual book around 20 times, s/he realized that s/he
negligently read the book. Hence, s/he bought another conceptual book and read it again with a
fresh heart….Eventually s/he read each of the three different conceptual books more than 20
times. Therefore, s/he was able to fully master all the concepts in science.
This illustrates that P31 tried to master sub-components (e.g., information, concepts,
principles) in order for them to be firmly incorporated into the whole cognitive structure.
However, P31 figured out that this was not enough for successfully solving different types of
problems as described in the following narration:
N31: However, s/he still could not perfectly solve some problems. S/he reflected on his/her
learning and realized that s/he was incompetent in particular types of problems. Hence, s/he
grouped all the problems that had appeared in the previous mock tests to identify/distinguish the
types of problems and then grouped them based on the topics and sections of textbook. Then s/he
diligently practiced those identified problems by studying corresponding sections.
To identify the key point that needs resolution in future problem solving situations ([31.6],
M6 problem), P31 groups the information ([31.7], Agg2 serial; [31.8], C1 grouping); and
defines characteristics of the information to combine equivalent relevant attributes, and to
isolate and compare these attributes in order to distinguish the different types of problems
([31.9], Agg3a classification; [31.10], Agg3b decomposition). While relating each of the
identified different types of the problems with the corresponding topics and sections in the
textbook, P31 compares and extracts the surface as well as the underlying structural
commonalities between them ([31.11], Ab1 comparison; [31.12], Ab3 structural). While
doing this, P31 associates the attributes/features of the information in the problems with the
related information in the learning material in order to find relevant information (e.g.,
concepts, underlying principles) for solving the problems using analogy ([31.13], Ab4
analogy). Thereby, P31’s problem solving ability could be gradually developed from
focusing on superficial aspects of the information in the problems to its underlying aspects.
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Presumably, P31 could extract commonalities from among them, and hence, find a general
solution strategy or principle when solving the different types of problems ([31.14], Ab2
generalization).
N31: ….Thus, after taking the test by KICE, s/he discussed the problems with other students to
identify new types of problems, and then intensively studied those new types of problems. By
doing this, s/he was able to better understand concepts.
By discussion with peers, the problem solving process was verified and intensified ([31.15],
SA2 cooperation). The analysis of the interview gives evidence that P31 explicitly checks the
outcomes of his/her own learning performance ([31.16], M7 evaluation). P31 clearly
understands the conditions that help him/her more successfully accomplish a learning task
([31.17], M4 management).
Case 32 (Grade: UF)
P32’s learning processes are presented in the following snippets of narration:
N32: ….S/he used to study by solving problems in Korean language. However, realizing that this
did not help for preparing CSAT, s/he changed learning strategy… and finally scored 100 in CSAT
in Korean language.
P32 checks and verifies the accuracy and appropriateness of the learning strategy against the
learning goal after it has been completed ([32.1], M7 evaluation). Upon figuring out the point
needing resolution in further problem solving situations (“improve his/her analytic skills and
time management during the test”), P32 modifies learning strategy to accomplish a learning
task (“created incorrect answer notes”; [32.2], C13 substitution). With this modified strategy,
it seemed to better support his/her understanding of the complex multiple relations between
the problem statement and the given answer choices. P32 clearly plans strategies for handling
learning tasks ([32.3], M2 planning), which is explained in the following narration:
N32: ….In order to improve his/her analytic skills and time management during the test, s/he
created “incorrect answer notes” to more precisely determine his/her missing knowledge and
misunderstandings. S/he used three types of commentary footnotes while organizing the note: (1)
For the correct answer, s/he wrote down the reason why s/he picked the correct answer; (2) for
incorrect answer, s/he wrote down the reason why s/he chose the incorrect answer as well as the
reason why it could not be the correct answer; and (3) for the question s/he was not sure, s/he
wrote down the reason why s/he was struggling when choosing the correct answer as well as any
hidden pitfalls in the question. By doing this, s/he was able to clearly understand and identify
incomplete knowledge and misunderstandings, and hence, modified and completed them
accordingly. This in turn improved his/her problem solving skills and abilities.
As P32 became aware that missing knowledge impedes his/her analytic skills during the test,
P32 identifies incomplete knowledge and misunderstandings ([32.4], M6 problem; [32.5],
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Agg1 identification); and then summarizes them in a note ([32.6], C9 summarizing). As P32
identifies the information (i.e., incomplete knowledge and misunderstandings) that should be
firmly completed and corrected, P32 continuously modifies his/her cognitive structure
accordingly so it can be used to successfully solve problems ([32.7], C3 reorganization). It is
assumed that in the process of finding mistakes and incorrect knowledge, P32 compares the
newly learned information with his/her existing knowledge by analogy ([32.8], Ab1
comparison); integrates the newly learned information into his/her existing knowledge
through assimilation and accommodation ([32.9], Ab4 analogy); and either modifies his/her
mental model(s) of problem solving or constructs new model ([32.10], Ab5 model). Hence,
P32 presumably could extend the range and complexity of interconnected relationships that
s/he was able to subsume under hierarchical knowledge system.
N32: While solving problems, s/he identified the concepts that were not clear to him/her and
searched for their meanings in the dictionary. S/he then organized them in a separate note. By
doing this, s/he was able to master the concepts, and hence, was able to solve problems more
quickly and accurately.
P32 identifies and extracts information from useful source ([32.11], Agg1 identification;
[32.12], C12 resources); and then identifies missing concepts/knowledge and
misunderstandings by identifying the elements, relations, and functions of the information
([32.13], Agg4 reproduction). P32 then organizes the reproduced information into a structure,
and hence progressively completes a schema ([32.14], Agg6a schema).
N32: ….To manage the time while taking the test, s/he measured the time it took to solve three
problems at a time. And s/he marked “+” for over time, and “–” for less time. And thus s/he
identified the specific types of problems in the test pool which took an unusually long time to
solve, and then intensively practiced similar types of problems. With this strategy, s/he was able
to complement his/her weaknesses.
P32 instantiates the schema by solving problems ([32.15], Agg6b instantiation). The analysis
of the interview illustrates that upon figuring out the central point needing resolution in
further learning and problem solving situations (time management during the test), P32
clearly plans and applies strategies for handling for further test situations ([32.16], M7
evaluation; [32.17], M2 planning). P32 understands the conditions that help him/her more
successfully accomplish a learning task ([31.18], M4 management). This case seems to give
evidence that the whole learning process was accomplished by systematic cumulative
integration between simpler parts and complex whole cognitive structures.
Case 33 (Grade: UF)
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P33’s learning processes are presented in the following snippets of narration:
N33: ….Beginning in the 1st grade of middle school, the participant reorganized [emphasis
added] textbooks in a note. At the 1st time of reading, s/he tried to identify the structure of the
content in the textbook by continually reminding how unit(s) of information is related to each
other within the entire structure.
In order to reorganize the information in the textbook, P33 first identifies the whole structure
of the learning content, and defines structural relations between the information in it ([33.1],
Ab3 structural).
N33: At the 2nd time of reading, s/he tried to reorganize the information by focusing on the
interconnectivity [emphasis added] between the information, and this made him/her easier to
understand and remember the content. S/he then reorganized the learned content in the note…
For math, s/he organized basic concepts along with their derivations; and then solved practice
problems corresponding to each concept. While doing this, s/he tried to find out the most
appropriate solution processes and then organized them in the note along with their proof
processes.
N33: S/he then reorganized the learned content in the note… S/he then identified and numbered
the information for memorization, and then repeatedly wrote them down to memorize.
Upon identifying the information from the 1st reading ([33.2], Agg1 identification), P33
associates different parts of the information to each other in order to find the interconnectivity
between the information, which enables him/her to create complex knowledge ([33.3], Agg5
completion); and then organizes the complex knowledge into a note ([33.4], C2 note; [33.5],
C9 summarizing). While doing this, P33 “reproduces” the information in a structure by
internalizing the figurative, functional, and/or operative regularities identified in the
information ([33.6], Agg4 reproduction). By relating and making meaningful associations
(“interconnectivity”) of different parts of the information to each other ([33.7], C3
reorganization; [33.8], C5 elaboration), P33 constructs a meaningful larger cognitive
structure (“reorganizes”), and hence progressively completes a schema ([33.9], Agg6a
schema). P33 instantiates the schema through practice ([33.10], Agg6b instantiation); and
repeats it by writing them down to memorize ([33.11], C10 repetition). The analysis of the
interview gives evidence that P33 clearly understands the conditions that help him/her more
successfully accomplish a learning task ([33.12], M4 management).
Case 34 (Grade: UF)
P34’s learning processes are presented in the following snippets of narration:
N34: ….The participant aggregated all the mathematical concepts throughout the entire
curriculum of middle school by each section including definition, equations, and theorem in
order to clearly understand the basic math concepts. While doing this, s/he first aggregated the
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table of contents from textbooks of each grade level on a piece of paper and then checked for
similarities; and then reorganized them by grouping related sections together in order to create a
coherent structure by (sub-) headings (e.g., by equation, function, calculus).
In order to more clearly understand the basic math concepts, P34 first identifies the structural
framework of the information (content) in the entire curriculum of middle school in the
textbook ([34.1], Ab3 structural). While doing this, P34 defines (subordinate-superordinate)
structural relations between the information (e.g., concepts, principles) within the entire
framework of the learning content. Upon identifying the whole structural framework of the
information, P34 tries to clearly identify the sub-components that should be firmly mastered in
order for them to be successfully incorporated into the whole structure. Therefore, P34
identifies attributes and features of a phenomenon of the information from multiple sources
([34.2], Agg1 identification; [34.3], C12 resourcing); and then groups related sections together
in order to create a whole structure that contains all sub-components of the entire three years of
middle school math. P34 forms larger chunks of the information by several sub-headings for
the middle school math information (e.g., equation, function, calculus; [34.4], Agg2 serial;
[34.5], C1 grouping). This way, P34 attempted to figure out how each of the properties of
middle school math is interconnected with each other: That is, how the properties of lower
grades (1st and 2
nd grade) math are progressively evolved into advanced higher grade (3
rd grade)
math.
Accordingly, P34 defines characteristics of the information to combine equivalent
relevant attributes ([34.6], Agg3a classification); reproduces the information in a structure by
internalizing the figurative, functional, and/or operative regularities identified in the
information ([34.7], Agg4 reproduction); associates the information to each other in order to
form a larger unified idea, which is presumably accomplished by organizing a unit of simple
information in various ways ([34.8], Agg5 completion). P34 thus compares the similarities
and dissimilarities of superficial as well as the underlying structure of the information by
analogy ([34.9], Ab1 comparison); and defines the relationships of information by identifying
structural invariants across the series of information (“reorganized them by grouping related
sections together in order to create a coherent structure by (sub-) headings; [34.10], Ab3
structural). In doing this P3 progressively integrates units of information into a hierarchically
organized integrative cognitive structure (i.e., schema; [34.11], Agg6a schema).
N34: [preceding comment cont’d] s/he then intensively studied them following the organized
plan and then practiced problems referring to the note. With this strategy, s/he was able to
immediately retrieve the relevant concepts required for solving specific problems.
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P34 instantiates the schema with persistent practice ([34.12], Agg6b instantiation). This gives
evidence that the mastery of proficient problem solving requires practice of application. The
analysis of the interview gives evidence that P34 clearly understands the conditions that help
him/her more successfully accomplish a learning task ([34.13], M4 management). This case
seems to give clear evidence that the whole learning process was accomplished by systematic
cumulative integration between simpler parts and complex whole cognitive structures.
Case 35 (Grade: UF)
P35 used “mind mapping” strategy to organize information. While creating the map, P35 first
identifies the structural framework of the information and thus identifies relations between
each part of the information ([35.1], Ab3 structural), as explained in the following comments:
N35: ….S/he thought that elaboration and integration [emphasis added] concepts are important
when studying science. Hence, s/he first aggregated information by topics and created a mind
map (thus, s/he was able to identify the entire structure of the content to be learned).
N35: S/he then studied basic concepts. While doing this, s/he tried to find out the central concepts
and their key points: (1) S/he first identified basic concepts in the specific section. In order to
expand his/her analytic thinking ability, s/he linked related information to the basic concepts and
tried to identify how each concept/topic is related to each other.
Upon identifying the structural relations between the sections, P35 identifies the information
and tries to find out the key concepts of the learning material ([35.2], Agg1 identification;
[35.3], C8 keyword); adds and links related information into the core concepts by defining
characteristics of the information to combine relevant attributes ([35.4], Agg3a classification),
and to isolate and compare the attributes of it ([35.5], Agg3b decomposition; [35.6], C1
grouping); constructs relations between the information ([35.7], Agg4 reproduction);
associates the information various ways to figure out plausible interconnectivity between the
information ([35.8], Agg5 completion), and then organizes the reproduced information into a
structure ([35.9], C2 note; [35.10], C9 summarizing).
N35: (2) S/he then modified the mind map by updating this information and by adding more
information from reference books.
N35: (3) S/he then solved various problems through careful analysis. By doing this, s/he was able
to identify the relationship between sections, and this enabled him/her to successfully solve
problems that required applying integrated knowledge between different sections.
While modifying the mind map, P35 modifies the map by assimilation and/or
accommodation: compares newly learned information with existing knowledge analogy
([35.11], Ab1 comparison); and reorganizes the information based on the structural relations
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([35.12], Ab3 structural; [35.13], C3 reorganization). In doing this, P35 constructs a
meaningful cognitive structure by relating and making meaningful associations of different
parts of the information to each other ([35.14], C5 elaboration); integrates units of the
information into a hierarchically organized schema ([35.15], Agg6a schema); and instantiates
the schema with persistent practice ([35.16], Agg6b instantiation). P35 clearly understands
the conditions that help him/her more successfully accomplish a learning task ([35.17], M4
management).
Case 36 (Grade: CF)
P36’s learning processes are presented in the following snippets of narration:
N36: ….The participant aggregates information during lessons by note taking, and then
reorganizes the note by adding related additional information from textbooks, reference books,
and handouts; s/he then reviews this note until s/he completely understands and memorizes all
the content in the note. After that, s/he writes them down on a piece of paper to check how much
information s/he understood and was able to memorize.
P36 identifies information by taking notes ([36.1], Agg1 identification; [36.2], C2 note);
reorganizes the note by adding related information from multiple sources ([36.3], Agg4
reproduction; [36.4], C3 reorganization; [36.5], C12 resourcing); repeats the information
organized in the note by writing them down for memorization ([36.6], C10 repetition).
N36: During class lesson, s/he writes down her doubts and then tries to solve them by getting help
from class teachers. S/he also helps other students in their problem solving and this supported
his/her learning….S/he also repeatedly solved those problems that s/he answered incorrectly and
this enabled him/her to fully understand and master all the problems in the workbook.
P36 actively poses questions to him/herself while taking the lesson and also seeks for
explanation or verification about the questions to a teacher or peer ([36.7], SA1 questioning);
and works together with peers (sharing feedback on performance; [36.8], SA2 cooperation).
Case 37 (Grade: HS3)
P37’s learning processes are presented in the following snippets of narration:
N37: The participant carefully prepares the following day’s lessons the day before to thoroughly
preview the information to be learned beforehand….S/he summarizes and organizes concepts in
a note and then solves problems in order to check his/her level of understanding of the previewed
content….During the lesson, s/he tries to find solutions to the problems that s/he was not able to
solve while previewing.
N37: …s/he compares the information learned from the preview and the lesson, and then
identifies important parts of the content (i.e., the parts that s/he should focus special attention).
P37 previews the main ideas, concepts, or principles of the material to be learned ([37.1], M1
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organization). While previewing the information, P37 identifies attributes/features of a
phenomenon of the information ([37.2], Agg1 identification); summarizes the extracted
information in a note ([37.3], C9 summarizing; [37.4], C2 note); checks his/her level of
understanding of the previewed information ([37.5], M7 evaluation); actively poses questions
to oneself while previewing the information, ([37.6], SA1 questioning); and compares the
information from the preview and the lesson ([37.7], Ab1 comparison).
N37: S/he reviews learning content four times: (1) S/he reviews the content learned during the
lesson five minutes before the end of the lesson; (2) reviews it again during recess time; (3)
during self-study hours; and (4) during weekly reviews. By doing this, s/he reviews all the
information learned from lessons and identifies misunderstandings or incomplete knowledge,
and then organizes them in a note.
P37 clearly proposes strategies for handling learning tasks ([37.8], M2 planning); and repeats
learned contents by reviewing them multiple times ([37.9], C10 repetition). In doing this, P37
progressively completes schemas ([37.10], Agg6a schema).
N37: S/he then solves problems and compares the similarities and differences of the solutions to
the problems between him/herself and the school teacher….Once in a week, s/he collaborates
with other students: Each of them picks up three problems, and they put the collected problems
on a piece of paper. Then each student solves the problems and then they share and discuss their
solutions.
P37 instantiates the schema by solving problems ([37.11], Agg6b instantiation); and compares
his/her solution processes to the problems with the school teacher (superficial as well as
structural (dis-)similarities; [37.12], Ab1 comparison); and shares feedback on problem
solving with peers ([37.13], SA2 cooperation).
Case 38 (Grade: HS2)
P38’s learning processes are presented in the following snippets of narration:
N38: When the participant read the textbook, s/he easily forgot the content in the previous
sections of the book. Hence, s/he divided the content into several sections (e.g., paragraphs) (a)
when the topic changed; and (b) when s/he was not able to retrieve the previous content. S/he
then intensively studied each divided section and never moved to the next one until s/he fully
understood the current section.
N38: S/he repeatedly read the content in order to memorize: When s/he understood the
information in a particular section of the book, s/he then tried to rehearse its core information and
drew a schematic diagram [emphasis added] to understand the content better (checking).
P38 identifies and groups the information ([38.1], Agg1 identification; [38.2], C2 grouping);
repeats it in order to memorize ([38.3], C10 repetition); extracts the key information ([38.4],
C8 keyword). While drawing the diagram, P38 defines characteristics of the information to
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combine, isolates, and compares the attributes of it ([38.5], Agg3a classification; [38.6],
Agg3b decomposition); extracts commonalities from superficial as well as the underlying
structure of the information using analogy ([38.7], Ab1 comparison); defines (subordinate-
superordinate) structural relations between the information ([38.8], Ab3 structural); associates
the information to each other in order to form a new unified idea ([38.9], Agg5 completion);
and then organizes the extracted information into a structure (i.e., schematic diagram), and
hence, completes a schema ([38.10], Agg6a schema).
N38: ….S/he solved problems as much as possible using multiple resources (e.g., textbooks,
reference books, internet sites) and asked for help from teachers for the problems that s/he was
not able to solve.
P38 instantiates the schema through practice ([38.11], Agg6b instantiation) using multiple
resources ([38.12], C12 resourcing); and asks teachers for explanation and verification about
the problems that s/he was not able to solve from ([38.13], SA1 questioning). P38 clearly
understands the conditions that help him/her more successfully accomplish a learning task
([38.14], M4 management). This case seems to give clear evidence P38 progressively
developed knowledge by structurally organizing simpler parts of the information into
complex whole cognitive structures during the course of learning.
Case 39 (Grade: US)
P39’s learning processes are presented in the following snippets of narration:
N39: The participant studied middle school textbook to master basic level of English. S/he thus
memorized English words cumulatively and repeatedly: (1) 50 words from the previous day + 50
words for the day=100 words; (2) 100 words from the previous days + 50 words for the day = 150
words, and so on. S/he repeated this with a five-day cycle and then organized the words that s/he
still could not memorize in a note and reviewed them as often as possible.
This illustrates that P39 aggregates information ([39.1], Agg1 identification); internalizes the
information by reviewing it multiple times ([39.2], C10 repetition). P39 clearly proposes
strategies for handling learning tasks ([39.3], M2 planning); and organizes the information
(i.e., words) that s/he could not memorize in a note ([39.4], C2 note).
N39: S/he thus analyzed each part of the sentence in terms of its function (e.g., subject, object,
complement, modifier, etc.). By doing this, s/he was more easily able to understand a long and
complex sentence.
While analyzing each part of the sentence, P39 reproduces the information in terms of its
functions, which requires identification of the function of an object ([39.5], Agg4
reproduction). While doing this, P39 uses available information to infer the meanings and
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usage of the information by applying learned rules to understand the information using
analogy ([39.6], C4 inferencing). This helped P39 to keep stimulated content active in
his/her memory and promoted the storage of new content in long-term memory. This in turn
reflects that each piece of factual knowledge that P39 gained from learning sequences is
gradually subsumed under the relevant part of cognitive structures.
Case 40 (Grade: US)
P40’s learning processes are presented in the following snippets of narration:
N40: The participant divided the learning content in textbook by sections and then persistently
reviewed them. By doing this, s/he was able to structurally organize and integrate the
information.
N40: ….Thus, s/he did a lecture to him/herself: While doing this, s/he first wrote down the title
and heading of the content and then did a lecture to him/herself. S/he looked up the textbook for
the contents that s/he was not able to remember, s/he then added related information aggregated
from other sections, and then structurally organizes them in the note.
P40 identifies and groups the information ([40.1], Agg1 identification; [40.2], C2 grouping).
When P40 rehearses the learned information, P40 first defines the framework of it by
identifying the title and heading of the information, and then defines related information
([40.3], Ab3 structural; [40.4], C5 elaboration). P40 then identifies incomplete knowledge
and then actively modified his/her cognitive structure. This illustrates that P40 actively
modifies it by identifying the sub-components in the cognitive structure that should be more
firmly mastered and completed in order for them to be more successfully incorporated into
the whole cognitive structure. P40 (re-)organizes and (re-)structures the components of
existing schemas by assimilating and/or modifying them in accordance with the new
information ([40.5], C3 reorganization; [40.6], C2 note), and hence progressively completes a
schema ([40.7], Agg6a schema). This is presumably accomplished by grouping the
components of knowledge together on the basis of constructive schematic principle using
existing knowledge. By doing this, P40 acquires more complex concepts, principles, and
procedures than that s/he has already available. Hence, newly learned components of
information and knowledge are systematically integrated into P40’s existing knowledge
structure in the course of learning.
Case 41 (Grade: HS3)
P41’s learning processes are presented in the following snippets of narration:
N 41: Upon realizing that s/he was incompetent in solving problems which requires applying
multiple cross-sectional principles, s/he identified the types of each problem.
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P41 identifies the point needing resolution in problem solving situations and understands the
conditions that help him/her to successfully accomplish problem solving tasks and control
learning performance to maximize efficiency of learning ([41.1], M6 problem); checks the
outcomes of his/her problem solving performance ([41.2], M7 evaluation); and clearly
understands the conditions that help him/her more successfully accomplish a learning task
([41.3], M4 management).
N41: S/he first aggregated the problems that had been answered incorrectly and frequently asked
in previous tests and then grouped them together by related sections. S/he then persistently
practiced those problems….S/he also got some help from the teacher for the unsolvable problems
and tried to identify the reasons why s/he was not able to solve the problem.
This illustrates that P41 first aggregated the problems that had been answered incorrectly and
frequently asked in previous tests ([41.4], Agg1 identification). P41 then grouped these
aggregated problems together, and reorganized them into a structure by related sections
([41.5], C1 grouping; [41.6], Agg2 serial). This is accomplished by defining characteristics of
the information to combine, isolate, and compare the attributes of the information ([41.7],
Agg3a classification; [41.8], Agg3b decomposition). P41 thus actively posed questions to
him/herself and asked for explanation and verification about them from the teacher ([41.9],
SA1 questioning).
Case 42 (Grade: HS2)
P42’s learning processes are presented in the following snippets of narration:
N42: For math, s/he created three notes: (1) S/he first aggregated and organized the solution steps
of a problem in a practice note; (2) created an incorrect answer note for the problems that had
been answered incorrectly; and (3) identified incomplete information (i.e., missing information
and misunderstandings) from the two notes and then created a concept note.
P42 identifies the information (i.e., correct problem solution steps, the problems that had been
answered incorrectly ([42.1], Agg1 identification); and then organizes it in a note ([42.2], C2
note; [42.3], C9 summarizing). P42 understands the central point needing resolution in the
solution steps for future learning situations ([42.4], M6 problem); and clearly understands the
conditions that help him/her more successfully accomplish a learning task ([42.5], M4
management).
N42: ….For Korean language and English, s/he solved each problem three times and then
compared the solution processes, and then modified misunderstandings and incomplete
knowledge.
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P42 practices problem solving by solving them multiple times ([42.6], C10 repetition). By
repeatedly practicing the same problems, P42 actively compares the solution processes ([42.7],
Ab1 comparison). By doing this, P42 compares different problem solving models by analogy
([42.8], Ab4 analogy) and thus progressively modifies misunderstandings and incomplete
knowledge ([42.9], Ab5 model). By doing this, P42 acquires concepts, principles, and
procedures more complete than those s/he already has available. Therefore, P42 progressively
constructs more complete problem solving schemas ([42.10], Agg6a schema). This whole
learning process seems to be accomplished by systematic cumulative integration between
simpler parts and complex whole cognitive structure.
Case 43 (Grade: UF)
P43’s learning processes are presented in the following snippets of narration:
N43: ….The participant first carefully read textbooks to understand the information. While doing
this, s/he collected all questions and inquiries and tried to find solutions by using multiple sources
(reference books, teachers, internet, and so on). S/he then organized those collected information
into the textbook. In this way, s/he created his/her own special textbook [emphasis added]. S/he
then repeatedly read this textbook.
P43 first identifies attributes and features of a phenomenon of the information from textbooks
([43.1], Agg1 identification). While doing this, P43 poses questions to oneself and thus starts
the process of searching for the solutions to these questions from multiple sources ([43.2],
SA1 questioning; [43.3], C12 resourcing); and then adds the identified new information in the
textbook. While doing this, P43 comprehends individuals/elements of information, relations
between pieces of information, and their functions; combines elements of information in a
new way ([43.4], C3 reorganization); and makes meaningful associations of different parts of
the information to each other ([43.5], C5 elaboration). This way, the “special textbook” then
becomes a meaningful larger cognitive structure that P43 can use as a learning resource.
N43: ….After about 5 to 7 times of reading, s/he was able to identify the interconnectivity
between sections at a glance. Thus s/he was able to anticipate some problems that would be asked
in the upcoming test because s/he was able to identify more or less important information from
the textbook.
P43 repeats learned information by reviewing it multiple times ([43.6], C10 repetition); and
progressively comprehends the structural relationships of across the series of the information
(“identify the interconnectivity between sections”; [43.7], Ab3 structural). In doing this P43
coordinates units of information and their relational connections, and then integrates them in
hierarchically organized integrative cognitive structures.
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Case 44 (Grade: HS3)
P44’s learning processes are presented in the following narration:
N44: ….The participant first collected information from multiple sources (textbook, answer
book, multiple reference books) and then put that extracted information and knowledge into a
note with a coherent structure.
P44 identifies and extracts information from multiple sources ([44.1], Agg4 reproduction;
[44.2], C13 resourcing); and organizes the extracted information into a note ([44.3], C2 note;
[44.4], C9 summarizing).
Case 45 (Grade: UF)
P45’s learning processes are presented in the following narration:
N45: …After learning basic concepts, the participant tried to find out causal relationships
between the concepts, and then marked these relationships using arrows [emphasis added]. Once
these relationships were identified, s/he then wrote them down in a note in order to have a better
understanding of these relationships.
Upon mastering basic concepts, P45 progressively comprehends the relationships of the
information and identifies structural invariants across the series of information ([45.1], Ab3
structural). In doing this P45 associates units of information and their relational connections,
and then integrates them into cognitive structures ([45.2], Agg4 reproduction). This illustrates
that P45 explicitly associates a unit of simple information/knowledge together in various
ways to create complex knowledge. Therefore, P45 could construct a meaningful or larger
cognitive structure by combining elements of the information/knowledge in a new way
([45.3], C3 reorganization; [45.4], Agg5 completion). This in turn promotes P45 to perceive
the aggregated information as a whole rather than in fragmented bits and pieces. P45 then
organizes this identified/extracted information (i.e., relationships) into a note ([45.5], C2 note;
[45.6], C9 summarizing).
Case 46(Grade: HS3)
P46’s learning processes are presented in the following snippets of narration:
N46: ….The participant aggregated and extracted a certain amount of information from each
section of the math textbook and then grouped them together. S/he then divided them into a daily
study material.
This shows that P46 actively organizes information in a structural manner during the entire
course of learning processes (i.e., schematization). P46 first identifies attributes and features
of a phenomenon of the information ([46.1], Agg1 identification); defines characteristics of
the information to combine equivalent relevant attributes ([46.2], Agg3a classification). P46
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clearly proposes strategies for handling learning tasks ([46.3], M2 planning).
N46: After learning mathematical concepts, s/he practiced problems, and then analyzed the
solution strategy and concepts/principles applied in solving the problems. S/he organized them
into a note, and then persistently reviewed and studied them.
P46 identifies the relevant concepts, and principles that should be mastered in order to solve
the problems ([46.4], Agg1 identification). While practicing the problems, P46 tries to find
related concepts and underlying principles that are applied to solve the problems; and
organizes the extracted information into a note ([46.5], C2 note; [46.6], C9 summarizing).
N46: ….By doing this s/he was able to naturally learn the problems which require applying
multiple cross-sectional principles and concepts in a question.
This implies that P46 learned from “restructuring” of information rather than merely stacking
pieces of the information in an isolated manner. While analyzing the solution strategy and
concepts/principles applied in solving the problems, P46 identifies characteristics of the
information in a given problem statement ([46.7], Agg1 identification); associates the
information together in various ways to define related concepts and principles ([46.8], Agg4
reproduction); and builds analogy by associating given information in the problem with
existing knowledge ([46.9], Ab4 analogy). In doing this, P46 presumably modifies and
reorganizes his/her cognitive structure in a meaningful way using analogy ([46.10], C3
reorganization). Such perceived phenomena are then encoded as cognitive operations into
his/her cognitive structure. This in turn promotes P46 to perceive newly gained information
as a whole rather than in fragmented bits and pieces during the course of learning ([46.11],
Agg5 completion). Therefore, P46 could define principles and concepts required to
successfully solve a question by having a clear cross-sectional understanding.
Case 47 (Grade: HS3)
P47’s learning processes are presented in the following snippets of narration:
N47: S/he reorganized the history textbook by topics such as culture, politics, and economics….
S/he then aggregated additional information from various learning materials and then utilized
them in the textbook, and then persistently reviewed them.
P47 first identifies and aggregates information ([47.1], Agg1 identification); groups the
components of the information together and reorganizes them into a structure by topics
([47.2], C1 grouping; [47.3], Agg2 serial); aggregates more relevant units of information
from multiple resources ([47.4], C12 resourcing); and repeats them ([47.5], C10 repetition).
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Case 48 (Grade: UF)
P48’s learning processes are presented in the following snippets of narration:
N48: The participant first aggregated information from five different math textbooks and
carefully studied all the concepts. S/he then checked and verified them by solving problems, and
then organized the aggregated/extracted information into a note.
P48 aggregates information from five different math textbooks ([48.1], Agg1 identification;
[48.2], C12 resourcing). Upon studying all the aggregated concepts, P48 checks and verifies
learned concepts by solving various problems. In doing this, P48 confirms if s/he correctly
understands the relevant concepts and thus creates connections between the concepts and
given problems by extracting commonalities from superficial as well as the underlying
structure between them by analogy ([48.3], Ab1 comparison; [48.4], Ab4 analogy). P46 then
organizes the learned information into a note ([48.5], C2 note; [48.6], C9 summarizing).
Case 49 (Grade: UF)
P49’s learning processes are presented in the following snippets of narration:
N49: In order to improve problem solving ability in Korean language, the participant developed a
systematic problem-solving strategy. For instance, s/he grouped the text in the textbook together
by Genre (e.g., Nonliterary writing, Novel, Poem, and etc) and then analyzed them. ….For
social science, s/he tried to define the structural relationship of information (e.g., super-concept
and a sub-concept of information). By doing this, s/he was able to clearly understand the
structural relationship of the information in the textbook.
P49 clearly proposes strategies for handling learning materials ([49.1], M2 planning). P49
first identifies attributes and features of a phenomenon of the information ([49.2], Agg1
identification); groups components of the information together by Genre and reorganizes
them into a structure by topics ([49.3], C1 grouping; [49.4], Agg2 serial). This is
accomplished by defining characteristics of the information to combine equivalent relevant
attributes, and to isolate and compare these attributes ([49.5], Agg3a classification; [49.6],
Agg3b decomposition). P49 identifies the structural relations of the information in social
science study ([49.7], Ab3 structural).
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4.3 Overall Findings
This section presents the overall findings from the analysis of the 49 case studies. It
first presents the mechanisms of knowledge and skill development identified in the studies in
order to answer the first research question of the present study: It attempted to identify 1)
how the participants accumulate knowledge, and 2) how they (re-)structure their knowledge
in the course of learning. The next two sections present the overall findings of cognitive
processes and learning strategies in order to answer the second and third research questions of
the present study. It then presents the phenomena identified in problem solving situations.
4.3.1 Mechanisms of Knowledge and Skill Development
The present study questioned how learners develop knowledge and skills beyond that
which is currently available to them in the different sequences of learning. The overall
findings seem to give evidence that the participants develop knowledge and skills by
constructing a cognitive structure that is more advanced and complex than prior (or existing)
ones in the course of learning (e.g., case 3 in lines 33-35/case 5 in lines 7-29; 33-36; 44-49).
The results showed that the participants continuously constructed new cognitive structures as
they acquire new information and knowledge by constructing plausible organismic principles
that interconnect the new information with existing knowledge until they acquire satisfactory
schemas.
More precisely, as shown in most of the cases (especially cases 1 to 6), in the course of
learning, (1) the participants first identified and set the goal of learning by identifying the
requirements of a particular learning sequence. To do this, they first determined the
information ― the factual knowledge (e.g., facts, concepts, principles, & rules) and/or
procedural knowledge (e.g., procedures for solving a problem) ― that are required to meet
the learning goal (e.g., case 1 in lines 8-9; 17-20; 25-26/case 2 in lines 7-8; 24-28/case 4 in
lines 2-3; 9-10/case 5 in lines 5-6; 9-11; 21-25; 33-34/case 6 in lines 5-10; 20-23; 44-47). (2)
They defined the features, relations, and functions of the elements of the information and then
grouped these components of information together and/or broke them down on the basis of
schematic principle. Hence, the piece of information was reproduced as meaningful
sub-components (i.e., related pieces) that can be subsumed under a relevant and more
inclusive category of the cognitive structure, so that they can be flexibly assimilated,
modified, and restructured into the cognitive structure (e.g., case 1 in lines 10-11/case 2 in
lines 11-12/case 3 in lines 18-19/case 4 in lines 19-21/case 5 in lines 5-8; 12-14/case 6 in
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lines 15-16; 44-60). (3) Thereafter, these complexly “reproduced” components of information
and were progressively mapped (i.e., structural mapping) onto the cognitive structure until
they constructed a whole structure which functions at a level that is satisfactory for
completing a particular learning task (e.g., case 1 in lines 22-24/case 2 in lines 11-12;
24-38/case 4 in lines 13-21/case 5 in lines 21-29; 33-36). By doing this, new components of
information are systematically integrated into their existing knowledge structure in the course
of learning (e.g., case 1 in lines 25-27). Hence, the participants could extend the range and
complexity of interconnected relationships of the information that they were able to subsume
under (or assimilate into) their interacting hierarchical knowledge system (e.g., case 1 in lines
22-24/case 2 in lines 36-38/case 3 in lines and 36-38/case 4 in lines 9-11; 13-21/case 5 in
lines 12-14; 40-43/case 6 in lines 46-50; 55-60). With this, they acquired concepts, principles,
and procedural knowledge more complex than those they already had available. This whole
learning process was accomplished by systematic cumulative integration between simpler
parts and complex whole cognitive structures (e.g., case 3 in lines 10-16; 36-38/case 4 in
lines 2-5; 9-11; 14-16; 19-24/case 5 in lines 7-29; 33-36; 44-49/case 19, 25, 26, 27, 29, 31, 32,
40, 42).
Structural mapping. As stated previously, structural mapping in this study refers to
mapping the information and knowledge from one sub-component of the structure to another
one in the hierarchy of the whole cognitive structure. It is shown in the data that the
participants organized aggregated information and knowledge into categories, subsumed
these categories under relevant sub-schemas in their existing knowledge structures, and
merged the sub-schemas into superior schemas.
The structural mapping process in problem solving is shown as follows: in order to use
a practice problem A as a model for solving a given new problem B, the participants tried to
apply the underlying concepts and principles of A to solve the problem B (e.g., case 6 in lines
9-12/case 9 in lines 22-25/case 46). However, in some instances, they were not able to
identify the underlying structure but rather focused on surface features of the information
contained in the problem. As a result, they formed an incorrect or insufficient perception of
the problem and failed to solve problems correctly (e.g., case 4 in lines 29-31). Thus, they
were sometimes misled by the surface features of the information into using the wrong
correspondence (mapping), which in turn produced an incorrect solution to the problem (e.g.,
case 6 in lines 44-50). In other words, when they had not yet acquired sufficient
deep-structural knowledge, they were misled by the surface features of the information
contained in a problem that is not causally or logically related to their solution. In contrast,
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when they had already acquired deep-structural knowledge, they were not likely to be
affected by the surface features, and therefore they performed better in problem solving.
The participants typically prepared for the CSAT by solving practice problems. The
practice materials they used usually included descriptions of the concepts and the principles
for the underlying problems and their application when solving the problems. In general, it is
possible to solve the problems in the actual tests by applying the facts, concepts, and
principles in the textbooks. And usually, the problems are solvable through the application of
a single principle, but some of them required higher-order complex thinking skills which
require deep understanding and the application of multiple principles. In order to apply a
principle, its relevant parts need to be associated with the problem, and correct inferences
should be drawn about the problem and its solution based on the way in which it
corresponded to the principle. Once the participants had mapped the principle to the task,
they substituted related components of the information contained in the task (e.g., case 4 in
lines 1-5; 9-10; 13-15).
The overall findings give evidence that applying the principle is more involved than
simple substitution as Novick and Holyoak (1991) stated. When a problem is complex, the
participants appear to use a variety of strategies in deciding whether to refer to the principle,
and they usually refer back to it many times in order to verify whether the identified
principle(s) can be correctly applied (i.e., appropriateness) to the problem or not (e.g., case 4
in lines 9-12/case 9 in lines 28-29/case 10 in lines 1-6/ case 17). They related and compared
the information in the given problem to their existing knowledge on the basis of the
superficial as well as the structural (dis-)similarities, and generated a larger description than
the given information through inferences (i.e., inductive, deductive, analogical). Hence, they
transformed the descriptions of the information along the set-superset in their cognitive
structures while doing this comparison. This way, they could correctly identify and extract
the given information in a way that the components of the information relevant to the
problem solving goal are preserved, while the irrelevant parts of it was ignored (or
extinguished). Hence, they could correctly abstract the relevant components (attributes,
features) of the information by filtering out invariant and (ir-)relevant components using
analogy (e.g., case 6).
In the course of learning, the participants developed and improved their cognitive
structure by (a) adding and/or assimilating separate units or sets of units of information and
knowledge (i.e., categorization or grouping); and by (b) structuring (or mapping) units of the
information and knowledge by actively transforming and applying relevant rules and
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principles to these units (e.g., case 14, 16, 20, 29, 31, 32, 40). This corresponds to Piaget’s
(1968) notion of “structuralism.” By subsuming a unit of information/knowledge into another
one, the participants increased the complexity of the cognitive structure (e.g., case 1 in lines
22-24/case 2 in lines 36-38/case 3 in lines and 36-38/case 4 in lines 13-21/case 5 in lines
12-14; 40-43/case 6 in lines 46-50). This more complex cognitive structure in turn resulted in
the creation of new knowledge that included all or parts of the prior knowledge structure as a
substructure and generated new (i.e., novel) performances (e.g., case 1 in lines 1-4; 21-14/
case 5 in lines 44-49/case 6 in lines 44-50/case 7 in lines 1-7). This mechanism is used
repeatedly in the course of learning, leading to the gradual development and improvement of
cognitive structure over time (e.g., case 21 in lines 11-14/cases 9, 13, 14). In addition, this
“constructive schematic (or organismic) mechanism” increased the competence of knowledge
the participants had acquired (e.g., case 9 in lines 30-31/case 14 in lines 1-5). It seems that
this mechanism functions on the basis of rules and principles the participants had acquired as
well as, to some extent, “incidental” trial-and-error experiences in the learning sequences
which occurred as the participants’ cognitive structures continuously interact in various ways
with a changing mechanism (e.g., case 6).
The results also showed that the participants (high-achieving learners15
) in this study
are more actively trying to identify functions, relations, and the interconnectivity of each
component of information to complete complex schemas using analogies and/or inferences
while average-achieving learners seem to rely on acquiring factual knowledge as locally
segregated pieces rather than as a whole interconnected one (e.g., case 3 in lines 44-57; case
5 in lines 59-62).
The above stated findings seem to give evidence that entire cognitive processes in
learning interact with each other cumulatively and that learning in each sequence thus
depends on the previous one. That is, in any learning situation, learners go through a series of
sequential steps towards higher-level (i.e., more complex) knowledge on the basis of their
current level of understanding of the learning task. Each part of learning is meaningfully
compiled into a comprehensive learning through schematization during the course of learning.
Therefore, the participants’ cognitive structures are gradually developed and improved over
time. These findings provide clear evidence of contingent organizational structuring (i.e.,
schematization) in the course of learning, which is consistent with the proposed theoretical
15
As their school academic records, scores on the national mock (practice) test when preparing the CSAT, or
scores on the actual CSAT were ranked within the top 10% of the range (see section 3.1.4).
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assumption of cumulative learning in the present study.
4.3.2 Cognitive Processes
The present study identified the steps of cognitive processes involved in the different
sequences of learning in terms of knowledge acquisition and elaboration. The data suggests
that the cognitive processes can be separated into two broad categories ― aggregation and
abstraction of knowledge ― which will be explained in more detail in a later section. It was
shown that most of the participants tried to aggregate, extract, and condense information from
a series of learning tasks. This allowed them to build stronger semantic cognitive structure
than before with persistent practice (the extreme continuum of this would be
“automatization”). The participants’ school learning is divided into topics and sections of the
learning materials. They studied various degrees of new information of a topic and then
moved on to the next topic with a substantial amount of practice. In learning a particular topic,
they first tried to understand information by aggregating all relevant information from various
learning sources (e.g., textbooks, reference books, workbooks, note-taking, media, etc.) and
their existing knowledge. Thereafter, they organized the aggregated packets of information
and knowledge into coherent categories (i.e., categorization or structural mapping), subsumed
the categories into relevant sub-schemas (or local schemas) by abstraction, and finally
merged them into a superior schema by generalization.
4.3.2.1 Aggregation
In processes of aggregation, the participants tried to understand information of a
learning task by collecting all information that they deemed to be relevant to the current
learning situation from multiple sources of learning materials (e.g., textbooks, reference
books, and/or work books) and/or mediums (e.g. teachers or peers). Much of the work at the
beginning of this aggregation process concentrated on learning textbooks and reference books,
where core concepts and principles were highlighted and discussed along with example(s)
illustrating the application of these concepts and principles. By associating (i.e., classifying)
multiple units of information and their relational connections, the participants integrated the
information into hierarchically organized cognitive structures. As a result, internal
complexity of their cognitive structures was decreased (e.g., case 31 in lines 9-13)
Instances were shown that the participants complexly “reproduce” information (refer the
instances with code Agg4, reproduction) by internalizing the figurative, functional, and/or
operative regularities and invariants identified in the learning situations (e.g., case 5 in lines
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5-6/case 30 in lines 12-15). This way, they were able to represent the learned content (e.g.,
concepts, rules, principles) with their own “words” and hence, they could better understand
and organize the information, which in turn was retained longer in their memory. This was
particularly evident in the participants, who attempted to actively (re-)organize and transform
that information that had been aggregated rather than simply repeating the information
without attempting to filter or actively process the information (e.g., case 5 in lines 59-62).
The data shows that most of the participants actively associated (previously unrelated)
ideas to form a new unified idea by grouping units of information/knowledge together in
various ways to create more complex knowledge (refer the instances with code Agg5,
completion), which in turn supported the participants understanding of underlying conceptual
connections (e.g., case 29).
The learning materials were sometimes insufficient for solving highly complex
problems as they did not always present the factual information and/or procedural application
of the information required to solve specific problems. In these instances, the participants
tried to find the missing information (factual, relational, structural, or implicational) on their
own by actively and creatively interconnecting their existing knowledge and the given
learning (or problem solving) situation by analogy. Thus, when a simple aggregation of
existing knowledge was not sufficient for successful task completion, abstraction (i.e., higher
levels of cognitive structure) was used.
In general, there appear to be two kinds of aggregation: deliberate and spontaneous.
Deliberate aggregation occurred when the current learning represented an earlier learning that
the participants perceived to be relevant for completing the learning task and therefore find
interconnectivity between them (e.g., case 2 in lines 4-6/case 5 in lines 15-20; 59-62/case 33
in lines 4-6/case 43 in lines 5-7). Spontaneous aggregation occurred when the learning
situation did not present a relationship between earlier learning situations and current learning,
and therefore does not construct any meaningful relations (or interconnectivity) between
them (e.g., case 3 in lines 3-5/case 4 in lines 5-8/case 5 in lines 12-14/case 29 in lines 16-20).
Deliberate aggregation seems much more successful than spontaneous aggregation as more
participants could define and aggregate a relevant principle when they found and/or
constructed interconnectivity between the aggregated information and earlier learning
situations (i.e., existing knowledge). In the practice of problem solving, most of the time the
participants deliberately tried to find a related underlying principle and structural similarities
that could possibly be applied to solve the problem. Superficial inducement seemed less
prevalent among the participants and thus, their learning gradually progressed from focusing
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on superficial aspects of the phenomena to their underlying aspects of the information (e.g.,
case 4 in lines 13-16/case 5 in lines 37-43; 51-54/case 9 in lines 22-23). This is because they
were able to recognize the distinctions between the two (i.e., superficial and deep underlying
structural) features of the information with regard to cognitive structuring, and hence
progressively modified their learning processes/strategies in order to find deep structural
features of the information in learning tasks depending on their previous learning experiences
(e.g., case 24 in lines 1-3/case 30 in line 11/case 33). Consequently they performed better in
actual tests, especially when faced with complex problems (e.g., case 4 in lines 25-31).
Furthermore, the data shows evidence for practice effect from the most of the
participants. With persistent practice and in-depth investigations in various problem solving
situations, the participants could determine which specific concepts (from general
abstractions) can be applied in a particular situation. In doing this, they (re-)organized and
(re-)structured, or (re-)constructed the sub-components of their existing schemas by
assimilating or modifying them in accordance with the new information. This allowed the
participants to build up more general concepts, which are then stored for further use and
hence influence activities in subsequent learning (e.g., cases 4, 6, 9, 11, 13, 15, 16, 25, 31, 32,
33, 34, 35, 38, 41, 42, 46).
4.3.2.2 Abstraction
The participants abstracted information and knowledge by filtering out invariant and
irrelevant attributes, and thus formed more abstract as well as general concepts through the
intensification of the attributes of information and knowledge (e.g., case 4 in lines 9-12/case
6 in lines 55-60). The overall findings seem to give evidence that a cognitive structure that is
clear and well organized facilitated the learning and retention of new information, while a
cognitive structure that is confused and disorderly hampered learning and retention, as
Ausubel (1960) emphasized. Accordingly, learning appears to be enhanced by elaborating on
relevant information/knowledge of cognitive structure. Instances were shown in multiple
cases (e.g., cases 1, 4, 5, 9, 10, 13, 14, 15, 16, 17, 19, 25, 33, 46).
The data shows that analogy building (refer the instances with code Ab4 analogy)
allows the participants to create new connections between two analogues which initially seem
unrelated, because they recognize the similarities and commonalities between the two
situations (e.g., case 6 in lines 44-50). This further strengthens their cognitive structures, and
hence, helps them to store and retain information in their memory longer (e.g., case 2 in lines
15-23/case 11 in lines 6-9; 20-21). Through continuous structural mapping processes they
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strengthened the knowledge (e.g., concepts, principles) they already have available, and thus
constructed new concepts and principles. While doing this, they seemed to break with old
structures and constructed (to some extent) new structures (e.g., cases 1, 5, 9, 13, 15, 19, 25,
33, 34, 35, 45). When new information contradicts with their existing knowledge, they
developed new strategies to resolve the resulting cognitive conflict by deleting and/or
modifying components in their cognitive structure (accommodation) in a logically structured
way (e.g., case 2, 5, 7, 18, 33, 35, 43). Hence, their cognitive process seemed to be enlarged.
For instance, solving a complex math problem requires basic mathematical knowledge
(e.g., mathematical terms, formulas) as well as higher-level knowledge requiring creativity
and logical thinking ability in order to fully understand mathematical concepts, principles,
and verification processes. Therefore, developing mathematical problem solving ability
requires a high-level of complex cognitive skills. The data shows that the participants tried to
understand the definite structure of mathematical material. They sought deep meanings and
relationships of the information in the material using analogy. They acquired and developed
mathematical knowledge by using appropriate inquiry skills dialectically, such as convergent
(i.e., inductive, abductive reasoning, and/or deductive reasoning that verifies and confirms
whether an idea is appropriate or not), divergent (i.e., creating ideas, finding concepts, rules,
and principles), and analogical thinking (e.g., case 1 in lines 21-24/case 4 in lines 13-16/case
5 in lines 44-49). Abstracted causal (i.e., IF-THEN) relationships of this kind seemed to
promote the development of a new conception (e.g., case 17 in lines 3-13). The data shows
that the conceptions constructed or learned progressively became the participants’ ability to
anticipate the effect of a particular activity without explicitly mentally or physically
performing it (e.g., cases 1, 5, 9, 25). Consequently, the notion of anticipation, as emphasized
by Piaget (1971), appears to be the key to the cognitive construction.
As previously stated, when existing knowledge does not fit the situation (e.g., a
situation for solving a problem), one experiences disequilibrium (i.e., the balance between
one’s knowledge and reality is broken). Upon encountering cognitive dissonance of this kind,
the participants examined the possible causes and hence started to investigate the
relationships between their existing knowledge and the current learning situation (or that
required to solve a given problem; e.g., case 6 in lines 5-9; 39-50; 55-60). The conflicting
information is assimilated into the participants’ existing knowledge structures if it can be
meaningfully disassembled. Otherwise, it made the participants to modify the
sub-components in their cognitive structures (accommodation). Then they (re-)organized and
(re-)structured their cognitive structures correspondingly (e.g., case 20 in lines 4-21/ case 2 in
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lines 36-38/case 4 in lines 9-12). Thus, they actively created mental models corresponding to
the requirements of specific learning tasks (i.e., radical alternatives; e.g., case 4 in lines
13-16/case 10 in lines 1-10 /case 17 in lines 3-13).
The aforementioned findings of cognitive processes seem to demonstrate that learners
who engage in learning continuously aggregate and abstract knowledge in order to construct
complex cognitive structures (i.e., schemas), thus gradually improving their knowledge in the
course of learning. However, the analysis of the data appears to indicate that the boundaries
between the identified cognitive process steps are neither as clear nor as sequential as the
coding schemes imply. Hence, the boundaries should be considered more as broad tendencies
that the researcher suggests for the sake of understanding the sequences of the cognitive
processes in cumulative learning.
4.3.2.3 Generalization
As stated in section 2.2.4.3, abstraction process is assumed to be closely related to
generalization process, and thus abstraction and generalization is assumed to be often
occurring simultaneously in learning situation. The data seems to show multiple instances of
the generalization, though the present study did not apply the separate coding category for it.
The investigation for the evidence of the generalization is rather incorporated under the
subcategories of “abstraction” process (refer the instances with code Ab2, generalization on
the basis of similarities). The data seems to give evidence that the entire abstraction process
stated above is not loosely detached as a series of unrelated pieces but is closely linked as “a
system” which progressively connects all parts of it. And thus abstraction and generalization
seems to occur simultaneously in learning situation as assumed to be. Therefore, this section
describes some instances found for the generalization.
As stated in section 2.2.4.3, generalization under the theoretical assumption of
cumulative learning in the present study refers to the process of modifying a learner’s
understanding of a principle in such a way that the surface features of information do not
affect its mapping or its application to the problem. Generalization is assumed to cause the
learner to associate various pieces of information to each other on the basis of structural
features instead of just surface features of them, and to be misled less by surface features
during structural mapping. Consequently, it is assumed that generalization allows the learner
to apply the principle acquired from learning situations (e.g., practice problems) to more
learning situations and problems. In other words, as repeating different types of a
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generalization (e.g., over or under-generalization), the learner can progressively differentiate
a set of different attributes that characterize the concept of particular phenomenon. When
these processes are repeated, the units of information/knowledge that belong to the
corresponding schema become increasingly routinized as the learner consistently associates
the new information with his/her existing knowledge through extensive practice.
Consequently, the learner progressively develops his/her performance from slow, conscious,
and difficult to more rapid, accurate, unconscious, and effortless one.
The analysis of the present study seems to suggest that the ability to make accurate and
broad generalizations is critical to the process of learning because this extends the scope of
adaptability of acquired information and knowledge to further learning and/or problem
solving situations (e.g., case 17 in lines 3-9). Such generalization seems to be accomplished
for any object or situation that an individual experiences and is likely to be applicable to a
similar form and context. The participants could generalize from one situation to another
when they perceived that they are likely to be grouped together when they share the same
input-consequence mechanism.
The present study revealed a number of factors influencing the generalization. First, the
data suggests that it is easier to generalize familiar instances than unfamiliar instances
because relevant existing schemas provide a general framework in learning and/or problem
solving situations (e.g., case 14 in lines 1-5/case 31 in lines 9-13/case 41 in lines 1-5): (1)
P14 felt more competent in solving (similar) types of problems that s/he had practiced; (2) as
P31 realized that s/he was incompetent in particular types of problems, P31 grouped all the
problems that had appeared in the previous mock tests to identify and distinguish the types of
problems and then grouped them based on the topics and sections of textbook. Then P31
practiced those identified problems by studying corresponding sections; (3) upon realizing that
P41 was incompetent in solving problems which requires applying multiple cross-sectional
principles, P41 identified the types of each problem, and then grouped them based on the
related section in the textbook. P41 then practiced those different types of problems in an
attempt to get familiar with such problems.
The participants felt incompetent to solve unfamiliar types of problems but felt
competent to solve ones that were to some extent similar to those they had practiced earlier.
Thus, in solving a complex problem, merely solving a substantial amount of problems was
not effective to solve a complex problem since this demands the application of generalized
multiple principles rather than a simple copying of solutions from previous problem solving
experiences. Upon recognizing this in the process of learning, the participants tried to identify
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underlying concepts and principles contained in problems and combine them in different
ways for practical use to improve their problem solving abilities and skills (e.g., case 4 in
lines 9-16/case 17 in lines 3-8).
Second, in the process of problem solving (or forming problems in case 17), the
participants actively searched their existing schemas to find the relevant principle for
successfully solving the problems (e.g., case 4 in lines 19-21). However, simply activating
relevant principles was not enough for successful problem solving (or forming problems)
because they had to determine its generality before applying it to the given problem (e.g.,
case 17 in lines 4-8). To create problems, P17 was able to identify the structural and
superficial similarities and commonalities between the problems during the practice.
Consequently, “generalization” may be regarded as a critical aspect in learning.
As indicated in the analysis, it seems possible that participants engage in superficial
reasoning (i.e., through superficial structure) while problem solving without deliberate
attention and practice using deeper reasoning (e.g., case 6 in lines 5-19). This was seen in
cases wherein they stored information they learned from practice material in their memory
without attempting to engage in deeper reasoning, and this information was then easily
forgotten. In the case, when they did activate the information it was often applied simply on
the basis of superficial similarities. When the participants recognized this aspect, they
progressively tried to identify deep structural features of the information for correct mapping
(e.g., case 6 in lines 20-23; 40-60). In order to do this, they had to generalize the examples
and principles they had learned from the practice material. For instance, they used multiple
practice problems and tried to find the similarities and commonalities between them and thus,
their mapping was often based on structural similarities rather than superficial similarities
(case 14 in lines 1-5/case 31 in lines 9-13/case 41 in lines 1-5).
Third, the data suggests that studying various practice problems is effective for
generalization. Studying various practice problems was particularly effective for the
participants, who might have already acquired sufficient principles and knowledge to solve
the problems. What made this practice useful to them seems to have been the strategies that
they used with the practice problems. While they were solving various exercise problems,
most of the time they tried to identify similarities and commonalities between the problems
(e.g., case 14 in lines 1-5/case 31 in lines 9-13/case 41 in lines 1-5). In contrast, the method
of simply copying explanations of their underlying principles from reference material without
drawing deep inferences seemed to produce less generalization (e.g., case 6 in lines 17-19).
Lastly, it seems that the generalization of learning experiences requires deliberate
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attention, time, and effort in cases in which it is necessary to acquire complex principles
generalization (e.g., case 4 in lines 17-24). The concepts and principles acquired in learning
generally required deliberate attention and practice before they become general or automatic
(in the ultimate continuum of generalization). The participants went through many steps of
modification in the course of differentiating their cognitive structures and generalizing
concepts and principles (e.g., case 1 in lines 10-11/case 2 in lines 11-12/case 3 in lines
18-19/case 4 in lines 13-21/case 5 in lines 5-8; 12-14/case 6 in lines 15-16; 44-60).
Consequently, it seems that generalization occurs slowly and is difficult to attain. After
solving the problems, most of the participants deliberately reflected on their solutions in order
to figure out the underlying fundamental concepts and principles that could be meaningfully
applied in future problem solving situations (e.g., case 4 in lines 13-24/case 9 in lines
28-29/case 17 in lines 4-8). Thus, they tried to modify the solutions to the practice problems
in order to highlight the application of each principle. Consequently, when they had already
acquired well-structured operational schemas through practice, their problem solving abilities
were gradually improved (e.g., case 9 in lines 30-31). However, when the participants studied
with multiple types of practice problems without deliberate abstraction and generalization, a
constrained type of transfer seemed to occur, and hence, it was not effective in improving
problem solving ability. For instance, in a test, P6 had difficulty while solving different types
of problems than when s/he practiced in the process of learning. This was because P6 merely
copied the solutions to the practice problems without drawing inferences from the solution to
the practice problems. Hence, generalization had not occurred. Accordingly, it seems that the
degree of generalization depends on a learner’s approach to learning.
4.3.3 Learning Strategies
The study identified the learning strategies that are used in the different sequences of
learning in relation to knowledge acquisition and development. The study was generally
satisfactory in identifying learning strategies by means of an investigation of interviews with
the participants. Since the interviews were based on “self-reports” from the participants to
given open-ended questions, some of them were more productive than others. Overall, it
seems that the categories and subcategories emerging from the data analysis are generally
consistent and of value for the field of cognitive and educational science.
Since the participants described how their learning process and strategies developed in
the course of minimum one to maximum six years during their secondary school days, the
researcher was able to identify a variety of different strategies they had tried and used as well
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as changes in their strategy use over time. Notable strategies used in high frequencies were
previewing and outlining contents to-be-learned, determining relevant information by
searching existing knowledge, taking notes, studying various learning materials in addition to
the textbooks to acquire deeper knowledge, and applying various strategies to practice
solving problems. The variety of identified strategies is integrated into the existing strategy
classifications of previous research (e.g., Chamot, Küpper, & Impink-Hernandez, 1988;
O’Malley & Chamot, 1990; Weinstein & Mayor, 1986). The data emerged in three broad
categories, namely: (a) cognitive strategies related to individual learning tasks which
occurred with sufficient regularity; (b) metacognitive strategies dealing with planning,
monitoring, and self-evaluation concerning learning processes, comprehension, and activities;
and (c) social and affective strategies involving interaction with others to accomplish
common learning goals. The distinction between cognitive and metacognitive strategies was
not clear-cut as the literature indicated (e.g., Brown, Bransford, Ferrara, & Campione, 1983).
Nevertheless, the classification scheme based on a division of learning strategies into three
categories appeared to be useful for describing the strategies derived from interviews.
As shown in the analysis of the case studies (section 4.1) and the frequencies of
learning strategies (see Table 4.7 & 4.8 in the following sections), the participants used the
following strategies particularly often during the course of learning: (1) cognitive strategy:
reorganization, note taking, grouping, repetition, resourcing, summarizing, and elaboration;
(2) metacognitive strategy: self-management, problem identification, self-evaluation, and
organizational planning; and (3) social and affective strategies: questioning for clarification,
and cooperation.
Most of the participants used various learning strategies in a systematic way (1) to
deeply understand learning contents; and (2) to systematically regulate their learning
processes. The study found evidence for goal directness, and contingent organizational
planning and utilization (refer to the instances with code “M2 planning” in the transcript; e.g.,
case 1 in lines 8-9; 17-20; 25-26/case 6 in lines 5-10; 20-23; 44-47/case 25 in lines 1-6). The
participants proposed strategies based on the goal for handling an upcoming learning task and
thus generated a plan for the parts, sequences, or functions to be used in handling the learning
task. They applied various types of strategies to the learning tasks in order to understand the
learning content in terms of its underlying structural interconnectivity with other information
and knowledge rather than just focusing on the factual elements or surface features of the
learning content. They first identified and perceived the learning task, set goals, and then
activated their existing knowledge to integrate new information into their existing knowledge
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structure (i.e., either assimilate or accommodate it into the appropriate schema; e.g., case 25
in lines 14-17/case 29 in lines 16-20/case 30 in lines 5-10; 16-20/case 31 in lines 14-16/case
32 in lines 6-13). Thus, they were clearly aware of their learning processes and performances
and actively monitored and managed their own learning by flexibly employing various types
of learning strategies in the course of learning situations (refer to instances with code “C13
substitution” in the transcript; e.g., case 1 in lines 17-20/case 3 in lines 27-32/case 32 in lines
1-3). Thus, successful learning processes and core cognitive strategies (e.g., linking of new
information to existing knowledge, elaboration and (re-)organization of information and
knowledge) were continuously applied (i.e., transferred) to future learning situations in the
manner of procedural knowledge.
Furthermore, when the participants focused on the “learning,” they tend to use various
cognitive strategies to enhance their understanding and retention of information, and to
acquire high-level complex knowledge. On the other hand, when they focused on “problem
solving,” they tend to use more metacognitive strategies (e.g., code M4 management; M7
evaluation) to apply information and knowledge they had learned more effectively. In other
words, the participants’ learning processes and strategies appear to be flexible, and changing
depending on the consequences of present learning performance. Consequently, there were
some changes in the ways by which the participants used cognitive, metacognitive, and
social/affective strategies in the course of learning, which depended on their active problem
identification, self-monitoring, and self-evaluation of their understanding of the learning tasks
and the effectiveness of strategies they had used previously (e.g., case 6 in lines 39-60).
P6 actively checked and verified his/her comprehension and performance in problem
solving in order to check the accuracy and appropriateness of the cognitive processes and
learning strategies while it was taking place (M5 monitoring); explicitly investigated the
attributes/features of the condition under which his/her problem solving strategies did not
work successfully and identified the point needing resolution in the further learning situations
(M6 problem); and checked the outcomes of learning performance against an internal
measure of completeness and accuracy of the learning goal after it had been completed (M7
evaluation). Accordingly, the ways in which P6 used cognitive strategies progressively
changed from one particular learning sequence to the next over time (case 6 in lines 12-16;
44-47) and as learning proceeded, P6 became progressively better at internalizing the thought
process in problem solving and thus managed to efficiently regulate learning processes.
Strategies whose use seemed to be increased over time included cognitive strategies
such as elaborating on existing knowledge (as explained above), deducing general rules and
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principles (e.g., case 6 in lines 45-47/case 9 in lines 28-31), as well as metacognitive
strategies such as planning, monitoring, and evaluating learning processes as well as
performance (e.g., case 15 in lines 7-11/case 16 in lines 13-16). Therefore, it appears that the
participants used learning strategies to affect “the way in which the learner selects, acquires,
organizes, or integrates new knowledge” (Weinstein & Mayer, 1986, p. 315).
It was evident in the interviews that most of the participants were highly
“self-regulated” (Zimmerman, 1989) learners (e.g., case 1 in lines 14-24/case 2 in lines
24-38/case 3 in lines 11-17/case 27 in lines 14-15/case 39 in lines 1-8). They clearly
perceived the requirements of specific learning tasks and identified appropriate strategies for
specific learning situations. Thus, they seemed to be well aware of their ability to adapt them
(e.g., case 1 in lines 14-20). Consequently, they actively regulated their own learning over
time and thus actively used various cognitive and metacognitive strategies.
Most of the participants used systematic strategies (e.g., case 3 in lines 44-57/case 5 in
lines 59-62). The data supports that they have a large variety of systematic approaches to
learning tasks and problem solving strategies at their disposal (e.g., case 3 in lines 55-57). For
instance, in problem solving, to correctly identify a problem, the participants actively
identified the central aspect that needed to be resolved in further learning situations (refer to
the instances with code “M6 problem” in the transcript; e.g., case 7 in lines 1-7/case 9 in lines
11-18/case 11 in lines 1-5). After completing the problem solving, they explicitly reviewed
their own performance in relation to their abilities, strategy use, their concept level, and so on
(refer to the instances with code “M7 evaluation” in the transcript; e.g., case 3 in lines
44-57/case 22 in lines 1-8). The small group experiment with P3 showed that even the type of
note taking used by the participant entailed an active manipulation of ideas.
It seems evident from the present study that awareness of schematization is one of the
key factors for developing knowledge (e.g., case 1 in lines 8-11/case 5 in lines 15-32/case 11
in lines 10-19/case 21 in lines 1-14/case 24 in lines 1-5/case 29 in lines 6-20/ case 30 in lines
16-20/case 34 in lines 1-6/cases 7, 17, 33, 35, 43). It appear to be schematization requires
persistent (re-)organization and (re-)structuring in order to lead to more and more stable
cognitive structure by comparing and/or contrasting sub-components (units of information,
concepts, principles, & etc.) in the structures. The participants acquired new information and
elaborate it on the basis of previously acquired information to different extents. They often
used their existing knowledge effectively and connected it with new information in the course
of learning processes. They seem to know how to effectively (re-)organize and store
information they have learned into their cognitive structures, and thus, they actively
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(re-)organized and (re-)structured their cognitive structures using a variety of strategies
depending on the demands of the learning tasks until they constructed satisfactory cognitive
structures that correspond to their learning goals. This is then stored in their LTM in the form
of a constructive interacting cognitive structure (i.e., schema) for further use. This recursive
process seemed to cause their knowledge and skills to be progressively extended,
structuralized, and become more condensed. It is shown in many ways.
For instance, P21 and P35 used “mind mapping” strategy. They structurally organized the
aggregated information into the map by defining the structural relations between the
information, and thus continually modified the map in the course of learning. Furthermore, the
note taking strategies of the participants were well-elaborated in terms of structural
organization (e.g., case 5 in lines 9-29/case 26 in lines 3-11/case 27 in lines 1-13). They
efficiently aggregated and extracted relevant information in learning tasks by linking new
information with their existing knowledge. They explicitly tried to find out how each
component of the information can be meaningfully interconnected to the entire content of the
learning material. Focusing on the interconnectivity between the information, they were
continually able to integrate parts of learning into overall learning. Furthermore, this is likely
due to the positive effect of cumulative learning in the procedural knowledge that they
applied to the task. That is, there may be some “general” procedural knowledge that they
acquired in the course of learning, which in turn became (to some degree) automatic with
persistent practice. This highly routinized knowledge seemed to free up attention for more
complex tasks which demand more mental power (or mental energy) for complex
understanding, such as complex math problem solving.
In sum, the participants consistently monitored and evaluated their strategies to make
sure whether they are appropriate and effective for performing a particular learning task.
Their approach to tasks was purposeful, and they monitored their comprehension and
performance for overall meaningfulness and effectiveness rather than focusing on learning
individual components of information. Thus, they persistently met difficulties and obstacles
by actively attempting to use various learning strategies. It seems obvious that learners who
engage in such active regulative involvement in their own learning actively try to use various
types of learning strategies (e.g., Dweck, 1986), which may greatly affect their development
of knowledge acquisition. The results show that all participants use a great variety of learning
strategies that direct their thoughts and behaviors throughout the learning process in order to
better understand, learn, store, and retain acquired information and knowledge. The strategies
seem to be deliberately planned, used, monitored, and evaluated in early stages of their use.
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Over time, they seem to become progressively internalized and automatized at the end of the
continuum. Thus, it seems that substantial time and effort are required to internalize new
strategies for mastering complex cognitive tasks. This is consistent with O’Malley and
Chamot’s (1990) view that it is difficult to apply an unfamiliar strategy to a demanding task
because the learner will then have to deal with two different complex tasks at the same time
as the cognitive strategy itself is a complex skill. In the following sections, the kinds of
learning strategies used in the participants’ learning are identified under the assumption that
cognitive process can be better investigated by identifying complex cognitive skills learners
use to perceive and process their learning experiences.
4.3.3.1 Cognitive Strategies
Cognitive strategies manipulate incoming information in ways that enhance learning.
The strategies that were used and well identified in the analysis are given in the table 4.7.
Core cognitive learning strategies that were used in different ways in combination with other
strategies seemed to be reorganization/reconstruction and elaboration: the former often
combined with note taking, grouping, and summarizing while the latter often combined with
inferencing and transfer. This suggests that these strategies may be more closely related with
cumulative learning when compared to other strategies.
Cognitive strategies Nr. of cases Nr. of instances
Reorganization/reconstruction 28 cases 36 times
Note taking 25 cases 29 times
Grouping 23 cases 25 times
Repetition 22 cases 24 times
Resourcing 21 cases 22 times
Summarizing 18 cases 19 times
Elaboration 17 cases 21 times
Substitution 13 cases 14 times
Inferencing 11 cases 15 times
Imagery 9 cases 9 times
Table 4.7 Frequencies of cognitive strategies
The data shows that in order to construct associations that can be meaningfully used in
later learning sequences, the participants use various learning strategies. For instance, since
language learning presupposes a great deal of memorization, the participants used various
cognitive-memory strategies in order to facilitate and enhance their memory processes. For
instance, they tried to memorize English words by grouping (classification), transforming
(i.e., making changes to their form), and rehearsing them (mentally reciting words). They
thus tried to find the meaning of words/sentences by dividing them into parts (decomposition)
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that they understood, and associated and synthesized new information into existing
knowledge (association) to facilitate and enhance their memory processes. This suggests that
organization and elaboration strategies seem particularly important to learning.
Grouping (C1) strategy (e.g., case 11 in lines 14-15/case 25 in lines 14-17) was found
to be related to the three cognitive processes of aggregation: The participants compared the
perceptual, surface, structural similarities between information and grouped it in various
ways, which presupposes the combination as well as the decomposition of the information
(refer to the instances with code Agg3a classification; Agg3b decomposition). Decomposition
was shown as the features belonging to each unit of different aggregated information needed
to be separated in a meaningful way to make them flexible and to link them closely to each
other so that they remain well integrated throughout the schematization process. This appears
to clearly reflect the organizational principle of cumulative learning.
Note taking (C2) strategies of the participants were well-elaborated in terms of
structural organization (e.g., case 5 in lines 9-29/case 26 in lines 3-11/case 27 in lines 1-13).
Most of the participants efficiently aggregated and extracted information by linking new
information with their existing knowledge, and tried to figure out the structural
interconnectivity between the information. Therefore, they were continually able to integrate
parts of learning into overall learning.
Two indications of reorganization/reconstruction (C3) strategies were found in the
interviews: Many participants mentioned that they made (a) comprehensive notes (b) and/or
“incorrect answer notes” after solving problems. It seems that reorganizing learned content
helps them to correctly identify incomplete knowledge and misunderstandings, and this
seemed to promote future learning in a meaningful way (e.g., case 6).
Indications of using inferencing (C4) strategy (e.g., induction, deduction, and/or
analogy) was shown in the process of learning when aggregated knowledge was refined and a
new (form of) knowledge was generated/constructed through interaction between new
information and existing knowledge. In problem solving, the strategies that encouraged the
participants to infer correct action seems to promote learning more effectively than those that
involved simply copying a solution without making any inferences (e.g., case 6 lines 10-16).
The indications of using inferencing strategy were clearly found with P5. S/he actively made
inference about the learning materials that goes beyond the information presented in the
material, and hence seemed to learn more. Inference strategy seems to have a positive
influence on the initial acquisition of knowledge as well as the progressive structuring of
information/knowledge. This finding corresponds to previous studies that propose that
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self-explanation improves learning in a variety of task domains (e.g., Brown & Kane, 1988;
Recker & Pirolli, 1995).
Learning seems to be based on the identification of both incomplete and incorrect
knowledge and their modification. The data indicates that learning strategies involving efforts
focusing on what one did wrong are useful for the participants (i.e., high-achievers). This
might be because they already have acquired sufficient factual domain knowledge. Hence,
focusing on finding and correcting errors might be more helpful than searching for more of
missing knowledge in the learning materials. However, missing knowledge seems to impede
average learners’ further improvement in learning: They often got the problem wrong when
they derived pieces of missing knowledge and thus learned incorrect knowledge. Hence, for
average learners focusing on searching for more of missing knowledge in the learning
materials might be more helpful than finding and correcting errors.
It was shown that the elaboration (C5) strategy is related to complex reproduction
(Agg4) and complex completion (Agg5) processes. While the participants identified and
extracted information from multiple sources in terms of the various components (elements,
relations, and functions) of the information, they often tried to make associations between
different parts of the information to each other (e.g., case 29 in lines 1-5). They thus tried to
associate different parts of the information in various ways to construct more complex
knowledge (e.g., case 33 in lines 4-10). The elaboration strategy thus encompasses the transfer
(C6) strategy (e.g., case 4 in lines 9-12/case 12 in lines 9-11/case 25 in lines 14-17).
Imagery (C7) strategy often co-occurred with symbolization (e.g., case 1 in lines 1-4/
case 2 in lines 18-23/case 5 in lines 5-6/case 6 in lines 51-54/case 11 in lines 6-9/case 22 in
lines 7-9/case 24 in lines 1-3/case 30 in lines 12-15). P2 explicitly mentally rehearsed
(recited) and reviewed learned materials that could be physically or mentally pictured. This
helped P2 keep stimulated content active in his/her memory, and thus promoted the storage of
new content in LTM. The participants used various types of symbols or abbreviated words
that could be substituted with some texts (e.g., ▼= because). This way, they were able to
represent the learned information with their own “words” and hence, they could better
understand (or internalize) and organize the information, which in turn was retained longer in
their memory. This reflects that they attempted to actively process, (re-)organize, and
transform the information by internalizing the figurative, functional, and/or operative
regularities and invariants identified in the learning situations.
Keyword method (C8) was shown in cases (e.g., case 12 in lines 7-12/case 18 in lines
1-3/case 21 in lines 1-10; case 26 in lines 6-11). The participants tried to identify the key
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information in learning material and thus more attention was directed to the key information
correspondingly. They tended to expand the range and the complexity of the packets of
aggregated units of information/knowledge based on the identified key information.
Summarizing (C9) strategy (e.g., case 29 in lines 1-5) was shown to be related to
complex building/classification (Agg3a) process as it presupposes defining characteristics of
information to combine equivalent relevant attributes. The participants often summarized
information by taking notes. Many of them took notes and then studied the information in the
notes in further learning situations, and often referred back to the notes when they were in the
midst of solving problems. The participants who studied the learning materials by
summarizing, and/or reorganizing the learned information on their own seemed to learn more
efficiently than those who merely repeated the information in the learning materials without
attempting to “reproduce” it with their own “words” using deeper inferences (i.e.,
internalization, cognitive operation).
Evidence for repetition (C10) strategy was found in most of the interviews. The
participants cumulatively repeated a chunk of information until they constructed a whole
structure which functions at a level that is satisfactory for completing a learning task. This
seems to give evidence that repetition is a necessary strategy in learning. However, many
participants commented that memorizing information without deep understanding of the same
was not useful in learning in the long run. Only when one has fully understood the
information, it supported memorization and longer retention (e.g., case 3 in lines 3-5).
Auditory representation (C11) was explicitly shown in case 5 (in lines 30-32). P5
explicitly used this strategy as s/he tried to understand the learning content by speaking it
aloud multiple times to better understand and remember information.
Most of the participants used various reference materials, such as reference books,
workbooks, dictionaries, encyclopedias, and/or internet sources in addition to the textbooks
(resourcing strategy, C12). They (e.g., case 5 in lines 1-3/case 27 in lines 3-6/ case 29 in lines
1-5) aggregated relevant units of information/knowledge from multiple resources (e.g., class
handouts, textbooks, and reference books) in addition to given learning materials. Thus, P28
aggregated information from online lectures provided by EBS.
Substitution (C13) strategy interacts with metacognitive strategies as it is derived from
actively monitoring one’s learning process. Most of the participants (e.g., case 1 in lines
17-20/case 3 in lines 27-32/case 32 in lines 1-4) were clearly aware of their learning
processes and performance, and thus actively monitored and managed their own learning by
flexibly employing various types of learning strategies in the course of learning situations.
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Upon figuring out the central point needing resolution in further learning situations they
actively reflected and modified their learning strategies correspondingly in order to improve
their learning performance.
4.3.3.2 Metacognitive Strategies
The results of the present study are consistent with previous studies (e.g., Brown et al.,
1983; O’Malley & Chamot, 1990) in that they indicate that metacognitive strategies involve
planning, monitoring, and evaluating the outcomes of learning. The self-management,
problem identification, self-evaluation, and organizational planning appear to be the
dominant one as shown in Table 4.8. As previously stated, most of time, the metacognitive
strategies occurred in combination with cognitive strategies.
Metacognitive strategies Nr. of cases Nr. of instances
Self-management 36 cases 36 times
Problem identification 22 cases 28 times
Self-evaluation 21 cases 25 times
Organizational planning 18 cases 20 times
Self-monitoring 8 cases 8 times
Attention 8 cases 8 times
Advance organization 5 cases 5 times
Table 4.8 Frequencies of metacognitive strategies
The participants who effectively used the metacognitive strategies to support their learning
appear to have understood the concept of “learning as a systematic
organization/construction.” They seemed to have a high level of schematic awareness. They
prepared for a learning task by first making a general review of their knowledge of the topic
to be learned and then generating some of the information that they expected to learn in the
learning material (Advance organization, M1). They (e.g., case 37 in lines 1-4; case 7 in lines
1-7) carefully previewed the main ideas, concepts, or principles of the material to be learned.
This strategy often co-occurred with organizational planning (M2). This suggests that the two
strategies are closely related in learning situations. Prepared for a learning task, the
participants then carefully and systematically planned their learning goals (M2). The
participants who used organizational planning with their learning task often developed a
general plan of study, formulated a framework for the overall learning process at the concrete
as well as the abstract level, and then planned individual goals that would result in the
completion of particular learning tasks.
As the learning proceeded, most of the participants revised some of the initial goals
and/or actively and flexibly managed their learning process, illustrating self-management
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(M4). They clearly understood the conditions that help them more successfully accomplish a
learning task (e.g., case 1 in lines 14-12/case 2 in lines 14-15/case 3 in lines 27-32/case 6 in
lines 17-23).
In terms of attention (M3), most of the participants used directed attention to focus on a
task while they were actively engaged in it. Selective attention was also used as a regulatory
strategy: They attended to specific aspects of information in learning material by scanning for
key information (e.g., case 7 in line 1-7); the sections where they feel incompetent during the
reviews of learning content (e.g., case 8 in lines 1-5); and the specific aspects of the
information (e.g., structural and/or interconnectivity between the information).
Self-monitoring (M5) and self-evaluating (M7) were used intensively for problem
solving tasks. In order to correctly identify a problem in their learning processes (M6,
problem identification), the participants monitored their understanding of presented
information as well as the effectiveness of their selected strategy for learning and problem
solving (e.g., case 1 in lines 14-20/case 6 in lines 39-43).In problem solving situations, they
monitored problem statements, concepts, principles, and types of the problems. Thus, they
persistently checked both problem solving tasks and their previous learning to determine
whether their performance goals were realized or not. This corresponds to previous studies on
self-monitoring effect (e.g., Chi et al., 1989). For instance, P5 frequently uttered assessments
of his/her understanding during learning, and noted negative self-assessments concerning a
failure to understand something (e.g., “That does not make sense. I don’t see how it works”)
as well as positive self-assessments (e.g., “That’s right. This is correct approach”). This
helped P5 to monitor his/her understanding of the learning content more accurately. This
indicates that accurate self-monitoring and self-evaluating influence learning.
4.3.3.3 Social and Affective Strategies
Social and affective strategies were also identified. Questioning for clarification (SA1)
occurred when the participants addressed questions to themselves as they worked through a
learning activity. For instance, P37 (in lines 16-18) actively posed questions to him/herself
and also asked for explanation and verification from his/her peers. This is also evidenced in
case 8 (in lines 6-11). When P8 encountered learning content that was inconsistent with
his/her expectations, P8 then posed questions to him/herself and also asked for explanation
and verification from his/her teacher.
Cooperation (SA2) was also identified along with cognitive and metacognitive
strategies (case 17 in lines 7-9). According to Vygotsky (1978), learning proceeds from the
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social level to the personal level. That is, learners mimic the thought process of their teacher,
peers, or other sources (e.g., reference books) in the early stages of learning until they can
solve problems on their own and reduce their reliance on others. Evidence for this was found
in the interview with P24s, in which two students actively cooperated in their learning
throughout their high school years. Initially, one of them tried to mimic and model the
thought processes of the other, whose academic achievement was far superior. As the
learning proceeded, the lower-achieving learner became better and better at internalizing
these thought process and thus managed to regulate his/her own learning process
progressively over time. This led to a successful increase in the lower-achieving learner’s
academic abilities and achievement, thus reducing the learner’s reliance on the other learner
over time. In the end, the thought and learning processes of the two students shifted from a
relationship of dependence to one of sharing. Self reinforcement (SA3) was also identified in
providing personal motivation by arranging rewards for him/herself when a learning activity
has been successfully completed (e.g., case 24 in lines 15-16).
4.3.4 Learning in Problem Solving
This section presents the overall findings in problem solving in relation to the three
research questions of the present study. The data clearly shows the importance of practice in
problem solving. Most participants learned from practicing when solving various problems
with various practice books. However, P9 repeatedly (10 times) studied only one particular
mathematics workbook for 6 months. In the period of the time, P9 comprehensively focused
on studying math as P9 commented that s/he only thought about math for a whole day. As the
number of problems solved in the workbook increased, the time it took for him/her to solve the
problems and the number of incorrect answers decreased: By the time P9 solved the same
workbook for the 6th
time (in lines 24-27); P9 could automatically identify the solution steps to
solve the problems. And by the time P9 solved it for the 10th
time (in lines 28-29); P9 could
solve the problems in various ways using multiple approaches. As P9 had already acquired the
basic concepts/principles while reviewing the workbook for 9 times, this improved his/her
mathematical skills (in lines 30-31). The learning dependent progression 1st
to 10
th seems to
indicate that the flexible schema instantiation presupposes a stable schema completion. P9
could successfully instantiate the math problem solving schema with the various procedural
steps because P9 had constructed stable and powerful cognitive links in his/her cognitive
structure in the course of the practice. This in turn reflects that the problem solving practice
becomes efficient when the practice results meaningful “changes” in a learner’s cognitive
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structure, allowing him/her to construct a stable schema which can be flexibly applied to
future learning situations.
Most of the participants deliberately practiced problem solving, especially with complex
level of problems, in order to increase their speed and accuracy for solving though they could
solve the problems without errors (e.g., case 9 in lines 11-31/case 32 in lines 18-22). This had
to do with the specific context of the actual CSAT test, where time limitations are strictly
applied (and therefore test anxiety is expected) and there is little time to retrieve relevant
knowledge. The time needed to complete a task decreased in proportion to the amount of
times the participants had practiced it as they gradually acquired procedural knowledge when
solving problems from practice. In general, there was also a corresponding increase in
accuracy of the problem solving except in few instances. For instance, when the participant
does not correctly recognize and identify his/her misunderstandings and/or incomplete
knowledge for a given problem solving task, the practice effect was not found (e.g., case 6 in
lines 10-19).
The participants seem to have acquired useful procedural knowledge through
cumulative schematization during practice. That is, all of the information (i.e., factual
knowledge) they gained from learning sequences gradually transformed into procedural
knowledge as they mapped factual knowledge to their schema. Therefore, each piece of the
learned information that was initially decomposed and unrelated to each other was gradually
subsumed under relevant parts of a cognitive structure. Hence, the participants could
meaningfully associate the information to each other, and form a larger integrated cognitive
structure. Hence, the factual knowledge progressively transformed to procedural knowledge.
This complexly completed structure was then preserved for further use. This enabled them to
solve the problems of the same (or similar) type by applying fewer but more complex chunks
of information/knowledge which contain more developed sub-skills than before.
Practice seems to speed up when the participants implemented certain groups of
interconnected sub-components of information/knowledge (i.e., sub-skills) as a whole (i.e.,
not segregated) into the learning and problem solving situation. Accordingly, with a
substantial amount of practice their understanding of the information in learning tasks and
problem solving skills became faster and needed less effort as this reduced the amount of
attention they needed to induce and activate appropriate procedures to understand
information, and to solve a particular problem. Hence, in a strictly time-constrained context
like CSAT, learners who have practiced a lot with clear sufficient understanding of the
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learning content were able to complete tasks more successfully.
Furthermore, when the participants practiced for the current learning task they seemed
to be naturally affected by their earlier training for a different task. Generally, prior practice
made a difference in how long it took for them to learn the current learning task. They stated
that prior practice generally decreased the time it took for them to learn the current learning
task and to solve problems (e. g., case 6 in lines 62-64/case 11 in lines 17-19).
The frequency of references to learning resources also declined with practice in
problem solving (e.g., case 6 in lines 44-47/case 9 in lines 19-31). When the practice
materials and the new learning tasks shared some features, the participants were able to
identify similarities and commonalities between them and this made the learning process
easier for them (e.g., case 9 in lines 24-31). This is due to the fact that the mastering of basic
skills facilitates the learning of practical skills, which contains the basic skills as
sub-components. Therefore, it seems that the effect of practice can be enhanced by a learner
figuring out the similarities and commonalities between the practice task and the current task.
Thus, practice became optimally efficient when it causes meaningful “changes” in the
participants’ cognitive structures: when such changes allow the participants to construct
stable schemas and to successfully apply the schemas in further learning situations.
The data shows that once the participants had completed the schematization processes,
they practiced relevant problems (e.g., case 7 in lines 14-15/case 15 in lines 3-6). As they
studied the problems, they often referred back to the learning materials (textbooks, reference
books, notes) or problems they had already solved. In this phase, as the participants (e.g.,
case 9 in lines 15-18/case 25 in lines 14-17) had already acquired relevant knowledge for
solving the problems, they could correct some misunderstandings and identify missing
knowledge (e.g., case 20 in lines 1-3/case 32 in lines 4-13/case 37 in lines 11-13). When this
occurred, they tried to fill the gap by aggregating missing parts of knowledge and information,
and continuously modified incorrect knowledge by schematization. They thus continuously
tried to increase their experiential knowledge by practicing problems (e.g., case 20 in lines
17-21/case 32 in lines 14-17/case 37 in lines 14-15). This enabled them to increase their
problem solving capabilities in terms of speed, accuracy, and strategies.
Previous research has proposed that in many task domains experts seem to mentally
plan solutions to problems that novices can only solve concretely. For instance, Koedinger
and Anderson (1990) found that expert geometers plan solutions to geometry proofs using
abstract, diagrammatic schemas in working memory. This seems to some extent comparable
to the findings of the present study concerning high and average achievers. For instance, the
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present study found that the participants (i.e., high achievers) show holistic and thus concrete
approaches in planning, monitoring, and evaluating their learning process and in their usage
of strategies. Thus, their experiences and skills acquired in each sequence of learning is
systematically organized and hence meaningfully stored in their cognitive structure in a more
abstract form. It then is positively activated in a subsequent learning situation (i.e.,
transformed). In a similar way, the present study found that average-achievers are less
competitive in organizing schematic cognitive structure, which in turn results in a lack of
correct understanding in the process of figuring out incomplete knowledge and/or correcting
misunderstandings. They also commented that they could not apply their existing knowledge
and solve the problems within the time limits of the actual test. This reflects the fact that their
schemas had not yet been sufficiently (or stably) developed in an abstracted form but
consisted of pieces of information/knowledge stored in segregated or disorganized form.
Consequently they needed more mental energy to activate relevant knowledge.
In sum, the phenomena identified in the course of problem solving from the case
studies seem to give evidence that (1) problem solving is a cumulative process wherein the
practice in each new sequence builds upon knowledge acquired in a previous sequence; and
(2) problem solving is a structuring process in which less inclusive sub-components (i.e.,
information, concepts, principles) that consist a problem solving schema are progressively
incorporated into the higher and more inclusive ones to complete more complex schemas.
This in turn seems to answer the first research question in the present study.
In the course of problem solving, the participants progressively fine-tuned their
cognitive structures more sophisticatedly by aggregating, abstracting, and generalizing
information/knowledge. In a give problem, the participants identified information contained
in the problem statement; investigated what is known and what is unknown by searching for
relevant information/knowledge in their existing knowledge which can possibly be linked to
the identified information, and aggregated them in their cognitive structure; tried to interpret
the information in the problem by applying their cognitive structure which contains the
aggregated information/knowledge; abstracted and generalized the aggregated units of
information/knowledge in order to apply them in problem solving. This whole processes
implies that they perceived and interpreted the problem in relation to the whole “structure” of
their knowledge system by consistently incorporating their existing knowledge and the
information in the problem. As a result, concrete and specific information in the problem
statement was progressively mapped into their cognitive structures through assimilation and
accommodation. This gives evidence that they went through schematization in the course of
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problem solving, which in turn supports the assumption of the present study (see section
2.2.5).
As a conclusion to section 4.3, the participants actively processed the information/
knowledge in the course of learning in terms of their cognitive structures. That is, they
progressively perceived, understood, and interpreted newly gained information by
assimilating and accommodating (i.e., accreting, tuning, (re-)organizing, and (re-)structuring)
their cognitive structures in cumulative patterns. It seems to give evidence that supports the
assumptions of the cumulative learning proposed in the present study that the mechanisms of
learning that can result in the development of knowledge and skills are cumulative and
structural nature.
With regard to the cognitive processes, the participants continuously aggregated and
abstracted the information/knowledge in the course of learning until they constructed and
completed satisfactory schemas for completing a specific learning task. In the given learning
task, they activated and aggregated units of information/knowledge obtained from their
environment that is relevant to the current learning task (i.e., aggregation). They then
progressively integrated the aggregated information/knowledge into their existing knowledge
structures (i.e., schematization), and constructed schemas which consist of groups of more
generalized sub-components at different levels of abstraction. As learning proceeds, these
schemas became progressively complex and profound that provided an insightful framework
for interpreting new information. As these experiences accumulated in the course of learning,
the participants’ knowledge and skills are progressively “changed” (or developed) over time.
With regard to the learning strategies, the participants used various learning strategies
in different ways in order to manage and support their learning processes. The core cognitive
strategies that are used in combination with other strategies seemed to be
reorganization/reconstruction and elaboration. Thus, the participants effectively used various
metacognitive strategies in order to systematically (re-)organize and (re-)structure, or
(re)construct their cognitive structures in the course of learning. This also gives evidence to
support the assumptions of cumulative learning proposed in the present study: the cumulative
as well as structural nature inherent to learning. It thus gives evidence that the “real learning”
can be accomplished in the course of slow and gradual development of knowledge and skills
which requires a lot of time and consistent effort of continual exposure to the topic.
Correspondingly, the theoretical assumptions of cumulative learning seem to be fulfilled.
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CHAPTER V
DISCUSSION
This chapter concludes the present study. At the outset, it presents the limitations of the
study followed by the theoretical as well as practical implications of the study. The third
section presents some recommendations for future studies. Lastly, it presents the conclusions
of the study in relation to the proposed research questions.
5.1 Limitations of the Study
This study has several limitations that reduce the generalizability of the results. Firstly,
as the present study used a selected series of existing interviews which was not originally
designed for the purpose of the study, there was not a perfect fit between what the researcher
was trying to investigate and the purposes for which the data was collected. Therefore, it was
difficult to conduct an investigation of “detailed” cognitive processes involved in learning.
As a result, often the distinction between aggregation and abstraction processes was to some
extent less distinctive in the interviews than the study would wish. The researcher
experienced difficulties when precisely classifying a particular process as to the definite
instance of when and where it occurred. This suggests that the distinction between them may
be less distinctive or that aggregation and abstraction processes often occur simultaneously in
combination. However, this is not a completely unexpected consequence, as the present study
used existing interview data which was not originally designed for an investigation of
“detailed” cognitive processes involved in cumulative learning. Therefore, while the data
supports the assumptions of cumulative learning in general (i.e., cumulative as well as
structural nature in learning), this study is to some extent not complete in its description of
the details of the cognitive processes inherent to cumulative learning. This is mainly because
the participants’ explanation of their learning processes and strategies in the interviews were
limited, and their subsequent learning could not be tracked through a follow up research.
Hence, the findings of the present study should rather be considered as proximal phenomena
which the study suggests for the purpose of understanding the cognitive processes in
cumulative learning. Nevertheless, this study still suggests making a distinction between the
two broad cognitive processes as it is useful for describing how a learner integrates all of the
different parts of learning into a whole coherent structure.
Secondly, the purposeful sampling of participants makes it to some degree difficult to
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make generalized statements beyond this group (“high-achievers”). Furthermore, although
this study analyzed 5 cases from small experiments that compared differences in the
cognitive processes and learning strategies of different groups, the experiments were not
initially designed for the purpose of a systematic comparison of different levels of learning
ability. Therefore, more cases involving learning processes with different levels of academic
achievement during secondary school should be investigated.
Thirdly, although this study was able to define the cognitive processes and learning
strategies of the participants from transcripts of video-taped interviews with acceptable
reliability, the participants’ reports might have been affected by the data collection procedure
employed. Thus, as the interviews had constraints with regard to the sources and interviewees,
the study was not able to confirm interview transcripts and the coding scheme the researcher
developed and coded. Therefore, it decreases the internal validity of the study. Furthermore,
as the present study is not an experimental study, there is no sufficient scientific experimental
comparison in regard to the effectiveness of learning processes and strategies beyond that of
self-explanation from interviewees. Hence, it reduces the generalizability of the results.
However, this shortcoming is complemented to some degree as this study included several
informal experiments that compared differences in the cognitive processes and learning
strategies of different groups as well as some supplementary comments from experts and
teachers which verifies some of the participants’ comments contained in the interviews.
Fourthly, the comments by the interviewees concerning their learning procedures could
have been misinterpreted by the researcher. The transcription of data was not an exact copy
of the entire conversation that took place during the interviews. The researcher extracted
conversation and information from the interviews that was relevant for the purposes of the
present study. Therefore, there may be some degree of misunderstandings, misinterpretations,
and inaccuracy that might have been “reconstructed” on the basis of the researcher’s specific
interpretations.
Lastly, considering that secondary education in Korea is primarily geared toward a
college entrance examination (i.e., CSAT), the data may reflect a cultural trait peculiar to
Korea rather than learning and problem solving in general. Students are encouraged more to
learn technical skills and knowledge to improve their scores on achievement tests than to
improve their domain-specific knowledge (e.g., linguistic ability, mathematical ability). This
might inhibit the students from developing deep understanding and productive and creative
thinking abilities. Hence, it will be necessary to investigate this issue further in other
countries of the world.
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5.2 Implications
This section presents the implications of the present study. As the present study aimed
to provide a detailed theoretical account of how cumulative learning functions in human
learning process, it starts by describing the theoretical implications of the present study in
relation to the presented theoretical foundation described in chapter II. The findings also
appear to have several practical implications for those who want to apply the principles of
cumulative learning in instruction.
5.2.1 Theoretical Implications
This section provides the theoretical implications of the present study in relation to the
two fundamental mechanisms inherent to human learning that can result in the development
of knowledge and skills ― the cumulative as well as the structural nature ― which is
described in chapter II. This section also provides the implications of the proposed cognitive
process classification scheme based on the distinction between aggregation and abstraction
for knowledge acquisition.
5.2.1.1 Cumulative Nature in Learning
Based on the reviews of Gagné’s cumulative learning model and its implication in
section 2.1.1, various arguments that assert cumulative nature in learning in section 2.1.2,
cumulative learning in the field of machine learning in section 2.1.7, and the learning
mechanism as an accumulation of knowledge in section 2.1.8.1, the present study proposed
the cumulative nature inherent to human learning.
Firstly, the present study greatly supports the notion of cumulative learning, propagated
by Gagné (1970, 1977, 1979, 2005), that learning is cumulative by nature. Gagnés argument
that learning is cumulative and that human intellectual development consists in building up
increasingly complex structures seems to be justified in the present study. From the data
analysis there is evidence that the learning/problem-solving capabilities consists in building
up increasingly complex structures of relevant intellectual knowledge and skills. The study
found evidence that the learning of higher-level knowledge and skills, such as higher-order
rules and principles depends upon the prior mastery of subordinate concepts and skills. When
a learner practiced for the current learning task, s/he seemed to be naturally affected by
his/her earlier learning and/or practice (i.e., training) for a different task. The present study
found evidence in that prior learning and/or practice generally made a difference in how long
it took for the learner to learn the current learning task. It also found evidence in that prior
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learning and/or practice generally decreased the time it took the learner to learn the current
learning task and to solve problems. In other words, when the learning/practice materials and
the new learning/problem solving tasks shared some features, the learner was able to identify
superficially as well as by structural similarities and commonalities between them and this
made the learning/problem solving process easier for the learner. Furthermore, as learning
proceeds, the learner’s cognitive ability gradually progressed from focusing on superficial
aspects of the phenomena to their underlying aspects of the learning/problem solving tasks.
This made the learner to construct more complex cognitive structures than prior ones. This
also provided evidence in that the mastering of basic components of cognitive structures
facilitates the learning of more complex ones, which contain the basic components of
cognitive structures as sub-components. Therefore, it seems that the effect of learning can be
enhanced by the similarities and commonalities between the prior learning and the current
learning task. Consequently, the present study found that learning becomes efficient when it
causes meaningful “changes” in the learner’s cognitive structure, allowing him/her to
construct highly stable schemas which can be applied to future learning situations.
However, while the present study corresponds to Gagné’s view that all learning is a
function of prior learning; it did not find clear evidence for his belief that the external
hierarchical sequencing of learning experiences can ensure internal (mental) organization.
While the present study found evidence in that learning lies within the individual’s internal
organization, it did not find evidence that supported external organization can also ensure the
internal organization. Rather, the present study appears to concur with Piaget (1976) on this
point, who argued that learning lies within the individual and that external organization
cannot ensure internal organization. The present study seems to show evidence in that mere
external organization of sub-components may not ensure internal organization unless a
learner actively organizes his/her internal cognitive structure in the course of learning.
Consequently, the present study can be used to present the idea that learning sequences
should encourage the internal organization of a learner’s cognitive structures along with
ensuring optimal external organization of sets of subordinate components that should be
mastered for specific learning task.
Secondly, the present study greatly asserts the “contemporary” concept of cumulative
learning that deals with the gradual development of knowledge and skills over time. It
assumed that a learner actively processes and interprets his/her environment in terms of
his/her cognitive structures and that the components in the cognitive structure are closely
interrelated. The study found evidence for this in that during the course of learning, a learner
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progressively perceives, understands, and interprets a world by continuously assimilating and
accommodating his/her cognitive structures in cumulative patterns. Therefore, the present
study asserts the proposed assumption that cumulative learning is the precondition for the
development of knowledge and skills over time, and hence concurs with the proposed
theoretical argumentation for cumulative learning in this theoretical framework —
cumulative learning reconsolidates the information/knowledge a learner has obtained through
various experiences and allows it to be reproduced and exploited for further learning
situations due to its cumulative effect. This appears to provide plausible implications for the
theoretical grounds that any learned capability at any stage of learning process affects the
subsequent learning situation. It may thus be used as the theoretical ground for finding the
mechanisms in human learning that emphasizes learning sequences are not segregated but
closely interconnected in the course of learning.
Third, the present study also assumed that the cumulative integration may have “a
certain vitality of its own” (Seel, 2003, p. 250) in the cognitive structuring process as a whole.
This assumption is fulfilled as the present study found evidence that strengthening the power
of the schematic links by firmly organizing and structuring learned content seemed to
ultimately increase the speed and the power of the whole schematization process in learning,
and hence resulted in an increasing learning power. Though it initially seemed to take longer
time for constructing well-organized and stable cognitive sub-structures, it ultimately
appeared to generate a certain degree of acceleration force as learning progressed. This aspect
may also be used as the theoretical ground for encouraging a learner to construct
well-organized and stable sub-structures at the initial stage in the course of learning.
5.2.1.2 Structural Nature in Learning
Based on the reviews of Ausubel’s subsumption theory in section 2.1.3, schema and
schema theory in section 2.1.4, Kant’s theory of schematism in section 2.1.5, Piaget’s process
of equilibration in section 2.1.6, and the learning mechanism as a change in cognitive
structure in section 2.1.8.2, the present study proposed the structural nature inherent to human
learning.
Firstly, the present study supports Ausubel’s theory of subsumption which proposed
that new information can be connected into relevant existing knowledge structure through
subsumption process. The subsumption theory corresponds to a great extent to the proposed
idea of cumulative learning. The present study found clear evidence in that the
information/knowledge a learner gains from learning sequences gradually transforms into
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more complex knowledge as s/he structurally maps the information/knowledge to a schema.
This promotes activation of associations and thus forms a larger integrated cognitive structure.
This structural mechanism enables the learner to understand and learn the similar and/or
related information by applying fewer but more complex chunks of knowledge which
contains more developed groups of interconnected sub-components of knowledge than before.
Consequently, the present study may provide theoretical grounds for explaining how new
information is learned in relation to existing knowledge structure.
Secondly, based on the reviews of the organizational nature of human learning proposed
in Kant’s theory of schematism, the present study assumed that any component of the
learner’s knowledge will have its own organizational property. The present study also
assumed the structural nature in learning, namely schematization, the process of organizing
information/knowledge into the relevant part of the hierarchical cognitive structure by
integrating aggregated information/knowledge in a more abstract way. Kant’s notion of
schematism seems justified in the present study as there is clear evidence for schematization
in learning. The study found that as learning proceeds a learner progressively integrates
pieces of information into sub-components (i.e., categories, concepts, principles) and the
relevant super-components (i.e., schemas) of the learner’s cognitive structures. These
integrated assimilatory schemas became progressively complex and profound in the course of
learning through the schematization process.
Thirdly, the present study reviewed the structural nature of human learning proposed in
Piaget’s equilibration process. The present study seems to correspond to Piaget’s (1976)
epistemology and conception of learning, which viewed humans as applying their existing
schemas to their environment through the process of equilibration. Piaget viewed that
equilibration occurs when people are in a state of equilibrium, and then they experience a
cognitive conflict as they realize the flaws in their thinking. This then induces people to adopt
a more plausible idea that resolves the cognitive conflict and hence attain a more stable state
of equilibrium. Piaget’s notions correspond to the conception of cumulative learning for it
views that new information is cumulatively integrated into existing knowledge structure by
continually assimilating, modifying, (re-)organizing, and (re-)structuring it in a more abstract
way through schematization in order to understand (i.e., resolve cognitive dissonance) the
new information.
Fourthly, the present study reviewed the concept of schema and schema theory, which
assumes that the learner instantiates existing schemas and constructs new ones by relating
new information to old schemas by analogy, in order to define the function of schema in
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cumulative learning. Based on the notions presented in the schema theory, the present study
proposed that learning process depends on the available information, existing knowledge, and
schematization process. The study found clear evidence that the cumulative integration of these
three parts gradually constitutes learning. And the knowledge constructed in this learning
process can be stored and made available for further learning processes. That is, in the course
of learning, a learner aggregates units of information obtained from his/her environment (i.e.,
external sources) and activates relevant parts of his/her existing prior knowledge (i.e., internal
sources). Thereafter, the learner progressively integrates units of information into existing
knowledge structures and reorganizes it (i.e., schematization), and hence, a packet of
aggregated units of information/knowledge (concrete and specific) is progressively
condensed into an abstract cognitive structure (i.e., schema). Therefore, the present study
seems to support the proposed assumption that new learning depends upon activating prior
knowledge, integrating newly gained information/knowledge into a pre-existing knowledge
structure, and then (re-)organizing and (re-)structuring the cognitive structure through the
cumulative interaction between prior knowledge and new information. Such interaction and
integration also promote the activation of other related areas of knowledge within the domain
because a learner seems to be disposed to associate various parts of cognitive sub-structures
(e.g., concepts, categories, sub-schemas) in his/her cognitive super-structure (schema), hence
expanding the schema and learning more.
The data also suggests that once a learner had completed schematization processes of
specific learning materials, s/he then practiced relevant problems as the learner was confident
that s/he has acquired some relevant knowledge for solving problems. As the learner studied
the problems, s/he often referred back to the learning materials (textbooks, reference books,
notes) or problems s/he had already solved. While solving the problems, s/he also corrected
some misunderstandings and rediscovered missing knowledge. When this occurred, the
learner tried to modify cognitive structures by aggregating missing parts of
information/knowledge. And thus, the learner gradually fine-tuned his/her cognitive structure
through (re-)organization and (re-)structuring until s/he could solve problems without
conceptual errors. Consequently, the learner “fine-tuned” his/her cognitive structure more
sophisticatedly by aggregating, abstracting, and generalizing their knowledge through the
schematization process. This continuous schematization made the learner continually link and
categorize each part of the information/knowledge in relation to the whole structure of the
cognitive system by consistently incorporating existing knowledge and new information.
The data shows that schema induction in learning involves the generalization of an
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abstract schema by extracting general concepts and/or principles by repeatedly reviewing the
information in learning material until a learner constructs a stable schema. The schema
induction in problem solving also involves the generalization of an abstract schema from
multiple various procedural features by extracting general concepts and/or principle and
solution strategy by repeatedly practicing problem solving until s/he constructs a more
advanced and stable problem solving schema. Therefore the learner could construct more and
more powerful cognitive links in his/her cognitive structure. And this allowed the learner to use
the newly acquired information/knowledge more efficiently and creatively. As the learner
correctly identifies and understands misunderstandings and incomplete knowledge in a given
learning task, s/he continuously aggregated relevant information/knowledge, and then
modified, reorganized, and restructured his/her cognitive structure through cumulative
schematization in the course of learning.
Previous research has proposed that in many task domains experts seem to mentally
plan solutions to problems that novices can only solve concretely. For instance, Koedinger
and Anderson (1990) found that expert geometers plan solutions to geometry proofs using
abstract, diagrammatic schemas in working memory. This seems to some extent comparable
to the findings of the present study concerning high and average achievers. For instance, the
present study found that high achievers show holistic and thus concrete approaches in
planning, monitoring, and evaluating their learning process and in their usage of strategies.
Thus, the experiences and skills acquired in each sequence of learning is systematically
organized and hence meaningfully stored in their cognitive structure in a more abstract form.
It is then positively activated in a subsequent learning situation (i.e., transformed). In a
similar way, the present study found that average-achievers are less competitive in organizing
schematic cognitive structure, which in turn results in a lack of correct understanding in the
process of figuring out incomplete knowledge and/or correcting misunderstandings.
5.2.1.3 Cognitive Processes in Cumulative Learning
One of the primary reasons for conducting this study was to determine whether a
cognitive process classification scheme based on the distinction between aggregation and
abstraction would be useful for knowledge acquisition. The study revealed plausible patterns
of cognitive processes in the interviewed students ― aggregation and abstraction.
The framework and functional explanations proposed in the present study appears to
have plausible implications for the theoretical analysis of the cognitive process in cumulative
learning. The combination of the cognitive processes between the aggregation of existing
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knowledge and the abstraction that deals with higher levels of cognitive structure should be
used throughout the entire schematization process for meaningful cumulative learning. That
is, in the course of learning, a learner actively aggregates relevant units of information or
knowledge. This aggregated knowledge would be refined and/or a new (form of) knowledge
is generated and/or constructed through interaction between new information and existing
knowledge by way of abstraction using inferences. It then becomes (re-)organized and
(re-)structured by schematization as the learner continually links and categorizes each part of
the information/knowledge in relation to the whole structure of the knowledge system by
consistently aggregating and abstracting existing knowledge and new information. These two
processes in combination allow the learner to continuously integrate newly gained
information and knowledge into the existing knowledge structure over time. By doing this,
any part of previously learned knowledge can continually affect the content, relations, and
structures of knowledge as it continually constructs a structured whole system. Thereby, the
learner gradually develops and improves his/her knowledge and skills over time. In this view,
more meaningfully structured knowledge can be better reserved for future use. The present
study shows evidence that most general knowledge is preserved at the end of the continuum of
the abstraction process in the cumulative learning processes. And hence, the learner could be
independent from superficial features and created abstract concept and this becomes the basis
for high order cognition in cognitive structure. This can be used as the theoretical ground in that
the learning sequences should encourage a learner to firmly master all necessary concrete and
specific information, and to progressively abstract and generalize it.
In sum, the theoretical concept and the framework of cumulative learning which was
developed in this dissertation may be used for further empirical research in human learning: It
may further our understanding of the ontological mechanism of human learning and pave the
way for more sophisticated explanations of the same in future studies. Understanding and
knowing precisely what happens in the course of cognitive processes in learning is critical for
complex problem solving in a complicated world. Educational research which deals with
changing cognitive processes and/or structures that possibly reflect the functional mechanism
and the development of the cognitive system (Butterfield, Siladi, & Belmont, 1980) would
give us a meaningful direction for further advancements in the various fields of learning
known under the umbrella term of “cognition.”
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5.2.2 Practical Implications
This section provides the practical implications of the present study in relation to
instruction. Firstly, since the present study proposes that meaningful comprehensive learning
may be accomplished through the cumulative systematic integration of parts of the cognitive
structure into the whole, it seems important to emphasize the role of systematic integration of
information/knowledge gained in the learning process. Due to the constructive schematic
nature of cognitive structures, particular pieces of information and knowledge that are not
sufficient in the present context (i.e., do not fully meet the requirement of the given learning
task) may later become sufficient through the further process of integration between prior
knowledge structure and new information. In other words, any piece of knowledge can be
utilized in further sequences of (formal, informal, conscious, and unconscious) learning
experiences in various ways. Accordingly, learners should be encouraged to systematically
integrate information/knowledge gained in the learning process and it would be beneficial by
not “wasting” any piece of “potential” source in learning.
Particularly, this study found that self-regulated schematization is likely to produce
more meaningful learning processes. In this regard, a self-regulated schematization of
abilities and attitudes should be encouraged in order for the learners to actively construct their
cognitive structure by inducing and activating various learning strategies. This can be
promoted by encouraging learners to link each part or component of knowledge into a
complex comprehensive knowledge system rather than simply collecting series of
sub-components of information and knowledge in the learning environment. Therefore,
instruction should emphasize schematization throughout instruction, the structure of the
learning content and an active analytical investigation of this content within the context of
learners’ existing knowledge by employing deduction, induction, analogical reasoning, and
inferencing strategies.
Accordingly, cumulative learning can be supported by instruction that deliberately
emphasizes the important structural connection in the learning process. This can happen in
many different ways. For example, students can be encouraged to create their own personal
structure of knowledge in any learning situation. Instruction focused on creating structural
connections in the course of the learning process would enable students to consistently
review new information and existing knowledge, thereby enhancing the learning of new
concepts, principles, and ideas. Since much school learning deals with either new information
and/or the restructuring of existing information, the concept of systematic cumulative
integration should have substantial pedagogical value in learning with respect to its efficiency.
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Thus, when a learner is not well-trained for schematization, providing a cognitive structure
that the learner can model after would be beneficial to the learner’s schematic knowledge
integration. In this regard, before the instruction, presenting a broad cognitive structure
related to the task the learners have to achieve might be helpful to the learners’ performance.
In addition, during and/or at the end of instruction, learners should have chances to reflect on
their learning processes with respect to their systematic integration of learned information. In
other words, if learners have some chance to be able to check and modify their cognitive
structures by comparing them with a provided cognitive structure before, during, and at the
end of the instruction, it would facilitate and enhance the learners’ schematization process.
Secondly, the data shows that cumulative systematic integration of all the accumulated
information/knowledge in the course of learning requires substantial time and effort (several
months to years) to complete. Consequently, whether this schematization is always effective
and beneficial or not in the learning situation when there is a time constraints is hard to
ascertain. However, if learning situations encourage learners to reproduce information in a
‘quick and easy’ manner, this will lead them to focus on simple memorization and shallow
information processing which will lead to low-level knowledge and skills. In this situation, it
would be hard to expect them to facilitate constructive reflection of their critical thinking and
integration of their knowledge as these cognitive activities require sufficient time and effort.
Furthermore, as the nature of assessment influences the approaches to learning which
learners adopt, if they perceive simple reproduction of information will successfully work in
the problem-solving situations (e.g., CSAT, school assessment), they will be naturally
‘encouraged’ to focus on practicing to solve problems with low-level of knowledge. In this
case, they will hardly perceive the benefit of constructive schematization in their learning
processes which is needed for meaningful cumulative learning to occur in the course of
learning. For instance, considering that secondary education in Korea is primarily geared
toward a college entrance examination (i.e., CSAT), one cannot rule out the fact that the
nature of school learning under this situation is not driven by the form and nature of
assessment. Therefore, school assessment should test higher order skills and be created in a
way to assess if the learners understand the information and its complex structural relations
more completely. This way, school assessment will encourage the learners to organize their
knowledge and structure them in a complex way so that it is made increasingly usable in
other areas beyond the current learning situation throughout their entire school learning years.
Therefore, instruction should also encourage them to acquire the underlying principles
of different concepts in their learning that allows subsequent transfer in association with other
231
information during the course of learning situations and consequently expands their
knowledge in a complex way. This allows the learners to deeply process information, and
elaborate on their learning.
Third, the identified steps of cognitive processes in the present study may be used as a
model for instruction and/or the assessment of learning status. For instance, in the course of
instruction, learners must be encouraged to systematically integrate new components of
information and knowledge into their existing complex cognitive structures. This can extend
the range and complexity of interconnected relationships and they are able to subsume under
the interacting knowledge system. For doing this, the learners should be encouraged to (1)
first identify and set the goal of learning by clearly identifying the requirements of a
particular learning task. To do this, they should be encouraged to determine and aggregate the
information (factual and procedural) that is required to meet the specific learning goal. (2)
And then the learners should be encouraged to activate relevant pre-existing schemas in order
for them to construct a (new) schematic principle that meets the requirements of the current
learning task. (3) Afterwards, the learners should be encouraged to group the identified
components of information together and reorganize them into a coherent structure. This can
be accomplished by breaking them down on the basis of constructive schematic principle
using relevant existing knowledge. (4) As a fourth step, the learners should be encouraged to
progressively map the complexly reproduced components of information onto cognitive
structures until they construct a whole structure which functions at a level that is satisfactory
for completing the current learning task. By doing this, the learners are acquiring concepts,
principles, and procedures which are more complex than those they already have available.
These processes maybe used in the assessment of learning status for assessing the learners’
knowledge in terms of (1) simple basic components of knowledge, (2) identifying
interconnectivity between the basic components of knowledge, and (3) identifying structural
relations between the components of knowledge in a whole structure which functions at a
higher level for completing a learning task.
Lastly, the study identified the kinds of learning strategies and the cognitive processes
that facilitate these learning strategies. Different cognitive processes and learning strategies
are used depending on the types of tasks participants are learning. The core learning
strategies that appear to be closely related in schematic organizational learning are 1) the
cognitive strategies of reorganization/reconstruction, repetition, and elaboration; 2) the
metacognitive strategies of self-management, problem identification, self-evaluation, and
organizational planning; and 3) the social and affective strategies like questioning for
232
clarification, and cooperation. This suggests that (re-)organization, repetition, and elaboration
strategies should be encouraged in the course of learning. According to Mayer (1984), (a)
organization strategies support learners in constructing internal links, which can be
established by relating relevant aspects of the new material to each other by identifying main
ideas, central concepts and their interrelations, and structuring contents; while (b) elaboration
strategies support them in constructing external links, which involves relating the new
material to the learner’s prior knowledge. Thus, in most cases the participants used
combinations of various cognitive and metacognitive strategies. This suggests that these
learning strategies should be encouraged to be used flexibly in the course of learning to
promote cumulative schematic organizational learning.
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5.3 Future Research
The findings of this qualitative study suggest several directions for future research.
Firstly, future studies should include the collection of quantitative data. The theoretical
assumption of cumulative learning posits that the learning process depends on the available
information, existing knowledge, and the schematization process. Although the assumption is
proved in this study with qualitative in-depth interviews of learning processes by showing
that the learners’ knowledge and skills improved over time by cumulative learning, no
sophisticated laboratory experimental data is provided. For example, it is unclear which
cognitive process is involved in which specific learning sequence and how the schematization
process evolved in different sequences of learning. Investigation in learning dependent
progression (e.g., Ifenthaler & Seel, 2005; Seel, Ifenthaler, & Pirnay-Dummer, 2009) of
specific schema in multiple points of learning would help to better understand the effects of
cumulative learning and would provide us with more refined directions for future research.
Therefore, future study is needed to examine the effect of cumulative learning on the
development of knowledge and skills with enough time and laboratory control and
treatments.
A plausible next step of the present study would be to design the study for an
investigation of “detailed” cognitive processes involved in learning for the acquisition of
larger pieces of knowledge, such as a whole semester’s of mathematics learning or other
subjects. This would involve observing the changes of cognitive processes for a period of
time sufficient for detecting “distinctive” changes in the usage of individual pieces of
knowledge. Time series studies which detect periodic (weekly, bi-weekly, or monthly)
learning dependent progression (e.g., test score, problem solving assessment) along with the
changes of the cognitive processes may suit the purpose of the study better so that the
participants’ explanation of their learning processes and strategies are adequately detected
and their subsequent learning could be tracked thoroughly. This way, the “future study”
would be able to investigate some meaningful relations between leaning performance (e.g.,
test score, problem solving assessment) and the changes of the cognitive processes.
Secondly, as mentioned earlier, validity was not adequately tested because the coding
scheme was constructed by the researcher and there were no previous studies for testing the
validity of the coding scheme. In addition the reports and comments by the interviewees
concerning their learning procedures could have been misinterpreted in many ways by
different researchers. Therefore, future studies should further test the validity and reliability
234
of the coding scheme. The validity of using particular criteria (i.e., aggregation and
abstraction) to identify cognitive processes in learning should be investigated through further
studies.
In addition, further research is needed to formalize these assumptions more precisely or
in a different way. These issues should be empirically validated through studying small parts
of learning sequences within a large body of data on general learning sequences. It would be
recommendable to implement a typical laboratory experiment on a problem solving task and
conduct a traditional quantitative analysis of tasks. Since the view of cognitive processes in
cumulative learning proposed in this study is one possible way in which these processes can
be used with contextual relevance, the researcher does not believe that cognitive processes
can only be defined on the basis of the two general categories proposed in this study. Other
studies might propose a different category of cognitive processes as well as an operational
definition. Much more work is needed on the formalization of the proposed cognitive
processes, and on a further clarification of their interrelationships. The researcher expects that
identifying the cognitive steps involved in cumulative learning may further our understanding
of the ontological mechanism of human learning and pave the way for more sophisticated
explanations of the same in future studies.
Thirdly, future research also needs to address in detail empirical support for the role of
schematization in cumulative learning, its representation, and its role in guiding learning
processes. For instance, Gagné (1985) argues that the cognitive strategic capabilities evolve
over time as byproducts of a multitude of problem solving situations and that they thus
cannot be trained in a direct manner. This seems a meaningful topic for future research in
relation to the current study.
Fourthly, as the present study did not find evidence that corresponded to Gagné’s belief
that the external hierarchical sequencing of learning experiences can ensure internal (mental)
organization, future research may investigate if a specific external organization can ensure
individual learner’s internal organization.
Fifthly, research topics may also include a systematic analysis of existing learning
mechanisms using concepts of cumulative learning in order to describe them in terms of
knowledge accumulation and transfers. The development of an effective method for
measuring the amount of knowledge accumulated and the amount of knowledge
meaningfully transferred (or integrated) into existing knowledge structures should also be
further investigated.
235
Sixthly, as this study did not focus on the effects of individual differences, it is still
necessary to consider individual differences such as the learners’ cognitive levels, and their
prior learning experiences. For example, low or average achievers might not get as much
benefit from schematization as high achievers. As previously stated, this is the case because
they do not have factual knowledge which is sophisticated enough to reflect essential
properties and phenomena of new learning tasks, and their meaning is not yet precise as they
still include features which need to be more precisely developed through a process of
abstraction. Therefore, further research is needed to examine how individual differences
affect acquiring knowledge in cumulative learning.
Seventhly, it seems that other psychological factors (e.g., motivation, feelings, emotion,
etc) affected learning over time in this study. However, because this study focused on the
cognitive processes involved in cumulative learning, information that was not relevant to the
purpose of the study was not deeply investigated. Nevertheless, these factors should be
investigated in a future study.
Lastly, the present study assumes that the process of cumulative learning is transferable
between domains as well as within a single domain. Therefore, it might be possible to use the
cumulative effects of learning to enhance domain-independent procedural knowledge. The
researcher suspects that interdisciplinary interests will likely increase in the future judging
from the current academic paradigm, which includes multi-dimensional interdisciplinary
fields like neurotechnology, educational psychology, etc. Therefore, this issue seems to be of
significant importance for understanding learning in general and hence it should be explored
more extensively in future research.
236
5.4 Conclusions
This section represents the conclusions of the present study in relation to the proposed
research questions stated in section 2.3. It starts by briefly representing the purpose of the
present study. As the outset, it should be noted again that the present study aimed to
investigate the mechanisms inherent to “real learning” (Norman, 1980, p, 20) − the learning
that results in complex cognitive structures – as the present study concurs with Norman’s
notion “real learning” which takes months or years to accomplish in the field of educational
science. The present study thus assumed that the real learning occurs progressively,
cumulatively, and structurally. The notions underlying the assumption are the two
mechanisms inherent to human learning that are stated in the conclusion of the literature
review of cumulative learning in section 2.1 — the cumulative as well as structural nature of
learning. Taking the two mechanisms into account, the present study assumed that the real
learning is only accomplished progressively as it viewed that constructing complex cognitive
structures is the precondition of the real learning is The present study assumed that
constructing complex cognitive structures involves many steps of modification led by an
increasing differentiation as well as an interactive reconciliation of cognitive structures.
As stated in the conclusion of the literature review of cumulative learning in section 2.1,
human learning is implicitly assumed to be cumulative in nature, but there is no
comprehensive theoretical foundation for this assumption. In the field of human learning, the
term cumulative learning has not been explicitly used since Gagné (1968), whereas it has
been used and studied in recent times in the field of machine learning, where the concept of
cumulative learning plays an important role (e.g., Pfeffer, 2000). The early approaches in the
fields of cognitive and educational psychology do not offer sufficiently profound reflection
on cumulative learning, the present study tried to bridge the gap of more than twenty years
between the early approaches and the present. Therefore, the present study endeavored to
define the “contemporary” theoretical framework of cumulative learning in section 2.2.
Based on the theoretical framework presented in section 2.2, the present study assumed
that learning is (1) a cumulative process wherein the learning in each new sequence builds
upon knowledge acquired in a previous sequence; and (2) a structuring process in which less
inclusive information/knowledge are subsumed under higher and more inclusive ones. Under
the above stated assumptions, the present study endeavored to determine the learning
mechanism inherent to learning in relation to knowledge acquisition and development. The
conclusions of each research questions are described in the following paragraphs.
237
Mechanisms of learning. The present study questioned how learners acquire and
develop knowledge and skills beyond those that are currently available to them in the
different sequences of learning. Firstly, it questioned whether learning is a cumulative
process wherein the learning in each new sequence builds upon knowledge acquired in a
previous sequence or not. The analysis of this study supports the general assumption that
learning consists of an accumulation and integration of a large number of small pieces of
knowledge (facts, ideas, rules, principles, procedures, technical details, heuristics, etc.) and
that these pieces of knowledge are (re-)organized, (re-)structured, and (re-)constructed in the
course of learning sequences. This provides evidence for the assumption that learning (i.e.,
the acquisition of knowledge and skills) is recursive in nature, as a learner continuously
modifies his/her cognitive structure by adapting new information to fit the existing
knowledge structure throughout the course of learning. The findings support the assumption
that the learning process is supported by the learners’ prior knowledge and that each sequence
of learning influences further learning. Therefore, learners acquire and elaborate knowledge
beyond those that are currently available to them in the different sequences of learning.
Secondly, the present study questioned whether learning is a structuring process in
which less inclusive information/knowledge are subsumed under higher and more inclusive
ones or not. The findings indicate that the cognitive processes inherent to cumulative learning
are constructive, meaning that the process of cumulative learning can be conceived as a
continuous build up of increasingly complex, interacting structures of knowledge. That is, in
any learning situation, learners go through a series of sequential steps towards higher-level
(i.e., more complex) knowledge on the basis of their current level of understanding of domain
knowledge. Each part of learning is meaningfully compiled into a comprehensive learning
through schematization during the course of learning. These cognitive structures interact with
each other and become increasingly inclusive, abstract, and complex, thereby promoting a
gradual development of knowledge and skills over time in the course of learning. These are
clear evidence of contingent organizational structuring (i.e., schematization) in the course of
learning, which is consistent with the proposed theoretical assumption of cumulative learning.
This supports the notion that knowledge and cognitive strategies are actively constructed by
the learner (Wittrock, 1974), which in turn confirms Piaget’s notion of a constructivist
account of development and intelligence. The schematization process is accomplished by
aggregation, abstraction, and generalization using analogies and inferences.
Thus, the study shows that a schematization process is a deliberate process that can be
accomplished when learners are able to identify incomplete or incorrect information/
238
knowledge and correct it through aggregation and abstraction. A super-schema, which
contains structural invariants of sub schemas, can be constructed through progressive
connections between sub-schemas. Thus, an organismic mechanism of this kind by itself
helps a learner to focus his/her attention on meaningful attributes that create plausible links
between elements for the complex interactions that will eventually be successfully
internalized. Learners can accomplish this by continuously utilizing information/knowledge
and thoughts in a coherent way when attempting to connect parts into a whole. Accordingly,
learning process is recursive in which existing schemas (i.e., the ones a learner already
possesses) are subordinated to the new cognitive structure (i.e., replacing a simpler structure
with a more complex one). In sum, the present study asserts the two mechanisms inherent to
human learning, namely, cumulative as well as structural nature.
Cognitive processes in cumulative learning. In an attempt to define the mechanisms
(or phenomena) inherent to learning in a scientifically adequate framework, the present study
questioned what cognitive processes are inherent to learning in relation to knowledge
acquisition and development. Firstly, the present study questioned what the steps are in the
cognitive processes involved in the course of learning, and it revealed plausible patterns of
cognitive processes in the interviews — aggregation and abstraction. The present study
provided a comprehensive investigation of the cognitive processes inherent to cumulative
learning through the analysis of diverse learning processes and strategies. Such a view aims
at facilitating the development of a theoretically grounded description of human learning
systems and providing a general understanding of the roles and the conditions for applying
the cumulative aspect of learning. The present study investigated the cognitive processes
revealed by students of different school subjects in Korea (e.g., Mathematics, English,
Korean Language, Science, and Social Studies) and gives good evidence that knowledge and
skills gradually develop over time in the course of learning through aggregation and
abstraction: In the course of learning, a learner actively aggregates relevant units of
information. This aggregated information is progressively refined and/or a new form of
knowledge is generated and/or constructed through interaction between new information and
existing knowledge by way of abstraction. It then becomes (re-)organized and (re-)structured
by schematization as the learner continually links and categorizes each part of the
information/knowledge in relation to the whole structure of the knowledge system by
consistently aggregating and abstracting existing knowledge and new information. These two
processes in combination allow the learner to continuously integrate newly gained
239
information into the existing knowledge structure over time. Consequently, concrete and
specific knowledge progressively becomes more abstract and general.
Secondly, the present study questioned what made up the characteristics of each step of
the cognitive processes. Aggregation deals with the units of packets of knowledge acquired
that are at the same level in the structure, while abstraction deals with knowledge that are
acquired at a higher level in the hierarchy of cognitive structure. In the process of aggregation,
given learning task, a learner (1) identifies attributes and features of a phenomenon of the
information from the learning task, (2) aggregates relevant units of information/knowledge
from existing knowledge by identifying their elements, relations and functions to the
information in the learning task, (3) identifies and extracts perceptual and surface similarities
between the aggregated information/knowledge and existing schemas, and (4) progressively
schematizes the extracted information/knowledge into a coherent cognitive structure (e.g.,
category, sub-schemas, schema). In the process of abstraction, the learner (1) extracts
commonalities from the underlying structure as well as the superficial features of the
aggregated information/knowledge, (2) defines structural schematic links of the
information/knowledge by integrating it into existing schemas through assimilation and by
modifying the existing schemas. (3) When the existing schemas cannot be matched to process
new information, the learner then constructs new mental models by analogy and modifies
schemas. These newly constructed schemas can then be applied to further learning situations.
The emergence of complex forms of cognitive structures through the course of learning
seems to allow for a slow, enduring, and gradual change in one’s cognitive structure in terms
of complexity and condenseness. While the complexity may be accomplished by the process
of aggregation, the condenseness (or compactness) may be generated by the process of
abstraction. One factor found to influence cognitive processes and strategy use was the
degree of understanding the students interviewed had in a particular domain. Another factor
influencing cognitive processes was the learning task itself. The specific demands of the
learning task seemed to influence the cognitive processes and strategies employed. For
instance, in solving complex mathematics problems the participants used more sophisticated
analogical thinking and inferencing processes (i.e., analogy building process).
Learning strategies in cumulative learning. In order to more clearly define the
cognitive processes inherent to learning, the present study questioned what learning strategies
are used in relation to knowledge acquisition and development. Firstly, the present study
questioned what types of learning strategies are used among the participants. The study
240
revealed the three categories of learning strategies: cognitive strategies, metacognitive
strategies, and social/affective strategies.
Secondly, the present study questioned what the characteristics of learning strategies
used among learners are. Cognitive strategies manipulate incoming information in ways that
enhance learning. They may be grouped together as follows: organization (grouping, note
taking, reorganization/reconstruction); elaboration (inferencing, elaboration, transfer, imagery,
keyword method, summarizing), repetition (repetition, auditory representation,), resourcing,
and substitution. Metacognitive strategies systematically organize learning processes in ways
that plan, monitor, and evaluate learning procedures and performance. They may be grouped
together as follows: planning (advance organization, organizational planning, attention, and
self-management); monitoring (self-monitoring, and problem identification); and evaluation.
Most of time, the metacognitive strategies occurred in combination with cognitive strategies.
Social and affective strategies provided motivation to more successfully accomplish learning
tasks. In general, the findings supported that “self-regulated” (Zimmerman, 1989) learners
actively regulate their own learning over time and thus actively use various cognitive and
metacognitive strategies as advocated by Bandura (1986) in his social-cognitive theory of
learning. This also corresponds to studies which have shown that self-regulated learners are
likely to pay more attention to learning processes, understanding, and mastery of tasks (e.g.,
Meece, Blumenfeld, & Hoyle, 1988; Pintrich & DeGroot, 1990).
In conclusion, the present study provides evidence for the assumptions of this study that
learning is a cumulative as well as a structuring process. The history of the investigation of
human thinking goes back to the first distinction between the soul and the body. Though
psychologists and scientists have tried out diverse approaches to understand and reveal what
the inherent mechanism of the mind might be in relation to human thinking since the early
20th
century, this still seems a matter whose nature we will not be likely to clearly describe as
Kant stated more than two centuries ago. Consequently, proving the cognitive processes in
human learning in any definite manner might be an impossible study to complete. Indeed it
might be true that such study might be only accomplished in the way of disproving a
hypothesis but not proving a hypothesis as Popper (1965) suggests. Nevertheless, despite the
shortcomings of the present study, the researcher hopes these findings will draw attention to
aspects of cumulative learning which are likely to be proven as important mechanisms of
learning.
241
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NOTES
1. I thank the Korea Educational Broadcasting System (EBS) for allowing me to use the interview
data for this study. The dissemination of the interview data is regulated as follows: “No
restrictions when used for scientific/statistical purposes, but any publication require written
permission from EBS.”
2. Preparation of this paper was supported in part by Wissenschaftliche Gesellschaft in Freiburg im
Breisgau.
3. Part of this study was published in the Encyclopedia of the Sciences of Learning by Springer
(2011).
4. Die Seiten 270 - 271 (Lebenslauf) enthalten persönliche Daten. Sie sind deshalb nicht Bestandteil
der Online-Veröffentlichung.
248
Appendix A: Coding of interview transcription (Case 7 to 49)
Transcript of case 7 (Grade: HS3) Cognitive processes Learning strategies
1
2
3
4
5
6
7
N7: ….The participant takes one “special” day to study science subject and to find out relations and
connections between each section of the science subject. As each section of the science subject is
closely related to each other, what one has learned in the previous section can be applied to the
other sections. S/he first examines the table of contents of the textbook to determine the importance
of each section. And then, s/he simply skims through the sections that s/he is already familiar and
that which does not require in-depth knowledge and invests more time in studying the sections
where s/he feels incompetent.
[7.1], Agg1 identification
[7.3], Ab3 structural
[7.2], M6 problem
[7.4], M1 organization
[7.5], M2 planning
[7.6], M3 attention
[7.7], M4 management
8
9
10
11
12
13
N7: …and then s/he studies it across the whole sections. This way, s/he can more easily identify the
inter -relations and this holds true within one particular subject of science as well as in t all other
science subjects (e.g., physics, chemistry, biology, and earth science). Since there are some concepts
that are common to other subjects, once s/he figures out this association, s/he tries to fully
reconfirm the relationship between them. This enables him/her to study two science subjects
effectively at one time.
[7.8], Agg4 reproduction
[7.9], Agg5 completion
[7.10], Agg6a schema
[7.11], C3 reorganization
[7.12], C5 elaboration
[7.13], C6 transfer
14
15
N7: …and then finishes up his/her study by studying the workbook. In doing this, s/he was able to
intensively study the parts that s/he feels less confident in a structured way.
[7.14], Agg6b instantiation
[7.19], Ab2 generalization
[7.15], C10 repetition
[7.16], C3 reorganization
[7.17], C5 elaboration
[7.18], C12 resourcing
[7.20], M5 monitoring
[7.21], M7 evaluation
Transcript of case 8 ( Grade: UF) Cognitive processes Learning strategies
1
2
3
4
5
N8: ….The participant makes preparation two minutes before lessons during break time by going
over the table of contents (TOC) of the upcoming lesson in order to check the relationship of the
section to the rest of the parts of the subject. Since the TOC contains the core information of the
section, checking it enables him/her to overview the entire frame and to identify the important parts
of it. Therefore, s/he was able to be more attentive in class.
[8.1], Agg1 identification
[8.2], Ab3 structural
[8.3], M1 organization
[8.4], M2 planning
[8.5], M3 attention
6
7
8
9
10
N8: In every lesson, s/he tries to ask at least one question to the class teacher. Therefore, while
listening to lessons, s/he marks “Q” for the questions to ask to the class teacher in order to fill the gap
between his/her thoughts and the class teacher’s in terms of understanding/interpreting the
information in the learning material. When the marked question is resolved during the lesson, s/he
writes the answers right away. Actively finding out answers to the marked questions made him/her
[8.6], Ab4 analogy
[8.11], Ab5 model
[8.7], M6 problem
[8.8], C3 reorganization
[8.9], C5 elaboration
[8.7], SA1 Questioning
249
11 more deeply comprehend the lesson.
12
13
14
N8: ….After the lessons, s/he immediately reviews the learned content with the note that s/he took
during the lesson for two minutes during break time. With this s/he could effectively reorganize
information, and hence, could remember it for a longer time.
[8.12], Agg6a schema [8.13], M5 monitoring
[8.14], M4 management
Transcript of case 9 (Grade: HS2) Cognitive processes Learning strategies
1
2
N9: ….The participant’s mathematics test score had been significantly increased in six months.
During this time s/he repeatedly studied only one particular math workbook ten times.
3
4
5
6
7
N9: Seeing that the other students were studying various types of problems with multiple
workbooks, s/he became anxious and was not confident that if s/he was using a correct [i.e., efficient
and effective] strategy to improve his/her problem solving skills in math. Nevertheless, instead of
solving the other workbooks like the other students, s/he continued to study the same workbook until
s/he reviewed it the 10th
time as originally planned.
[9.13], M2 planning
[9.14], M4 management
8
9
10
N9: ….And finally, in a second-grade math practice test, S09 scored 100. This increased his/her
confidence when studying other school subjects, and s/he thus became a top-ranked student at the
school.
11
12
13
14
15
16
17
18
N9: The 1st and 2
nd time solving the workbook, s/he took away the answer book and never opened it
up. This was because when s/he solved the problems while referring to the answer book, it was
impossible to determine whether s/he could solve the problems without referring to it or not, and
thus, there were too many problems that s/he was not able to solve. So, s/he just passed them over
….It took two months for him/her to solve the workbook once, and there were more incorrect
answers than correct ones. However, having completed it all independently without referring to the
answer book, s/he was able to precisely determine his/her missing knowledge and
misunderstandings, and his/her level of problem solving ability.
[9.2], Agg1 identification [9.1], M6 problem
19
20
21
P9: When I solved it the 3rd
time, I changed strategy. I now referred to the answer book for the
problems for which I did not know/understand the answer, and then tried to memorize the solution
procedure.
22
23
24
25
26
27
N9: By repeatedly practicing the problems along with the necessary formulas and the solution steps,
s/he naturally came to understand the underlying principles of solving these math problems.
N9: By the time s/he solved the same workbook for the 6th
time, s/he could automatically [emphasis
added] identify the solution steps and answers to solve the problems. And as the number of problems
solved in the workbook increased, the time it took for him/her to solve the problems and the number
of incorrect answers decreased.
[9.3],Agg6b instantiation
[9.4], Ab1 comparison
[9.5], Ab3 structural
[9.6], Ab4 analogy
[9.7], Ab5 model
[9.10], M5 monitoring
[9.11], M7 evaluation
[9.12], C13 substitution
250
28
29
P9: ….By the time I solved it for the 10th
time I tried to solve the problems in various ways using
multiple approaches and tried to find out the most appropriate way from them.
30
31
P9: As I had already acquired the basic concepts and principles, this improved my mathematical
skills.
[9.8], Agg6a schema
[9.9], Ab2 generalization
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
N9: Assuming that repeated practice solving the same problems can have a positive effect, the
production team experimented with two volunteer students [HS 2nd
grade, males, “average
achievers”]. They were asked to repeatedly (10 times) practice one particular booklet of practice
material that the production team provided to them. The material contained various questions
related to the topic of “trigonometrical function.” They were asked to track each time it took to
complete the test as well as the number of wrong answers. And thus, they were asked to clearly
examine the reason why they got the wrong answers.
N9: [preceding comment cont’d] they were diligently worked on the material and at the 3rd
day
after the material was given, they were able to complete the entire questions in it [presumably over
35 questions]. And at the 8th
day after the material was given, they were able to practice the
material 10 times. Then they took a test about the topic they studied [post test] and strongly
positive effects were found [out of 9 questions, student 1 scored 1(pre) to 7(post) and stduent2
scored 3(pre) to 6(post)].
N9: Student 1 commented that after practicing the material 4th
times, s/he referenced answer book
and was able to clearly understand the solution steps and answers to solve the problems. And as the
number of problems solved in the given practice material increased, the time it took for him/her to
solve the problems and the number of incorrect answers decreased. Thus, s/he commented that this
particular experience increased his/her interest and confidence in studying math….After taking the
post test, student 2 commented as follows: “After practicing the material 10 times, I felt easy in
dealing with math problems [before this particular experience, s/he could hardly solve math
problems], and thus, I think it increased my practical ability and understanding in math.
Transcript of case 10 (Grade: UF) Cognitive processes Learning strategies
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N10: ….The participant tried to find at least three different ways to solve each of the math problems.
S/he could even find out more than 30 different ways for unusual cases. By doing this, s/he was able
to improve his/her creative mathematical skills; and thus, was able to clearly identify the intent of the
examiner in a given test and hence, could easily find out the most
appropriate way to solve a particular problem. This increased his/her confidence in solving any
type of math problem, and hence, s/he finally got a perfect score at the CSAT math.
[10.1], Agg6b instantiation
[10.2], Ab1 comparison
[10.4], Ab3 structural
[10.5], Ab4 analogy
[10.6],Ab5 model
[10.3], C4 inference
[10.7], C5 elaboration
[10.8], M4 management
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N10: ….S/he applied the same strategy in other subjects: For social studies, s/he tried to find a new
problem solving strategy in addition to using his/her existing knowledge. For instance, s/he analyzed
the given passage of the question by matching it with each of the given choices of answers and then
induced true and false of the each of the matches using inference.
[10.9], Agg1 identification
[10.10], Agg3a classification
[10.11], Agg3b decomposition
[10.12], Agg4 reproduction
[10.13], Ab4 analogy
[10.14], C4 inferencing
Transcript of case 11 (Grade: HS2) Cognitive processes Learning strategies
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N11: ….Though the participant had solved lots of problems, his/her Korean language test score did
not improve at all. After consulting with the teacher, s/he realized that s/he had not studied learning
materials in a systematic way, and hence, created a symbol system [emphasis added] in order to
organize them in a coherent structure. Thus, s/he could remember the content of the passage longer
and more precisely than before, and hence, was able to solve problems correctly.
[11.1], M7 evaluation
[11.2], M6 problem
[11.3], C13 substitution
[11.4], M4 management
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N11: While reading passages, s/he marked some words using specific symbols: For instance, squares
with key words of each paragraph; triangles and squares for contrasting words; waves for examples,
analogy, and quote. This was effective in understanding passages when solving problems because by
doing this, s/he could identify an important part of the passage at a glance.
[11.5], Agg4 reproduction [11.6], C7 imagery
[11.7], C1 grouping
[11.8], C2 note
[11.9], C4 inferencing
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N11: ….S/he organized the theories of Korean language and then summarized an important part on a
sheet of paper. For those words that s/he did not clearly know the meaning of, s/he aggregated
detailed information from multiple learning sources (e.g., textbooks, reference books, workbooks)
and then organized them on the paper.
[11.10], Agg1 identification [11.11], C12 resourcing
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N11: S/he thus organized literary works in various ways (e.g., the period of literary activities by the
authors).
[11.12], Agg2 serial
[11.14], Agg3a classification
[11.15], Agg3b decomposition
[11.16], Agg4 reproduction
[11.17], Agg6a schema
[11.13], C1 grouping
[11.18], C9 summarizing
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N11: [preceding comment cont’d] from this, s/he could identify the nature of a particular literary
work by associating the name of the author even if it was the work that s/he had never learned
before....With persistent practice, s/he could speed up so that the time taken to complete each
problem and the number of incorrect answers were reduced over time.
[11.19], Agg5 completion
[11.21], Ab3 structural
[11.22], Agg6b instantiation
[11.20], C5 elaboration
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N11: Thus, s/he could remember the content of the passage longer and more precisely than before,
and hence, was able to solve problems correctly.
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Transcript of case 12 (Grade: HS2) Cognitive processes Learning strategies
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N12: ….The participant memorized English vocabulary cumulatively and repeatedly. In other
words, s/he memorized the words that s/he learned yesterday (1st set of words) along with the new
words learned today (2nd
set of words); then, the following day, s/he memorized the words that s/he
learned from the previous days (1st & 2
nd sets of words) along with the new words learned for the day
(3rd
set of words). S/he continued this way sequentially until s/he memorized all words in English
textbook, and continually tried to mentally rehearse them as many times as possible.
[12.1], C10 repetition
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N12: ….S/he thus grouped some related words together in various ways (e.g., categories, images,
functions, prefix, suffix); and then tried to naturally remember these words by constructively
associating images and relations. For instance, s/he tried to memorize and retrieve new words by
generating easily recalled images (textual and or figural) of a relationship between the word in
English and a homonym in Korean. Thus, with persistent repetition, the time it takes to memorize
the words was gradually decreased.
[12.2], Agg1 identification
[12.3], Agg2 serial
[12.4], Agg3a classification
[12.5], Agg3b decomposition
[12.6], Agg4 reproduction
[12.7], Ab3 structural
[12.7], C1 grouping
[12.9], C8 keyword
[12.10], C7 imagery
Transcript of case 13 (Grade: UF) Cognitive processes Learning strategies
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N13: ….S/he had hard time understanding high school math lessons as s/he did not fully understand
basic math concepts. So s/he repeatedly studied one particular math workbook regardless of whether
the answers were right or wrong until s/he fully understood each problem in the book. By the time the
subject practiced the workbook five times, s/he was able to figure out exactly what had to be
corrected and learned more for solving math problems.
[13.4], Agg1 identification
[13.6], Agg6a schema
[13.7], Agg6b instantiation
[13.1], M7 evaluation
[13.2], M6 problem
[13.3], M4 management
[13.5], C10 repetition
Transcript of case 14 (Grade: HS3) Cognitive processes Learning strategies
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N14:....Even though basic math concepts are mastered, if one cannot correctly apply these concepts
when solving various types of problems, then they are useless. In order to correctly understand/
distinguish the different types of problems, s/he categorized and organized math problems by type
and then studied/practiced similar type of problems together; and then put them in a note.
[14.3], Agg1 identification
[14.4], Agg2 serial
[14.5], Agg3a classification
[14.6], Agg3b decomposition
[14.7], Agg4 reproduction
[14.1], C1 grouping
[14.2], C3 reorganization
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N14: Thereafter, s/he intensively studied the problems that s/he had answered incorrectly, and tried
to find different approaches in solving these problems. Repeating this process over and over again,
s/he was able to understand math concepts more deeply, and was also able to clearly remember the
reasons why s/he made incorrect answers. This in turn improved his/her mathematical applicability
while problem solving.
[14.8], Ab1 comparison
[14.9], Agg5 completion
[14.10], Ab4 analogy
[14.11], Ab5 model
[14.12], Agg6b instantiation
[14.13], Ab1 comparison
[14.14], C4 inference
[14.18], M7 evaluation
[14.19], M6 problem
[14.20], M4 management
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[14.14], Ab3 structural
[14.16], Ab4 analogy
[14.17], Ab5 model
Transcript of case 15 (Grade: HS3) Cognitive processes Learning strategies
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N15: ….As the participant had not yet fully acquired fundamental basic math concepts; s/he was not
able to solve even the basic level of high school math problems. Hence, to master the basic math
concepts, s/he thoroughly studied middle school math for three months: S/he first identified learning
content; studied concepts; practiced to apply the concepts by solving various types of problems.
While doing this, s/he realized that s/he was missing and misunderstanding lots of core math
concepts…and this improved his/her ability of concept application in problem solving.
[15.6], Agg1 identification
[15.7], Agg6a schema
[15.8], Agg6b instantiation
[15.1], M7 evaluation
[15.2], M6 problem
[15.3], C1 grouping
[15.4], M5 monitoring
[15.5], C13 substitution
[15.9], C3 reorganization
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N15: ….Since the CSAT is to measure the problem solving skills as well as the ability to understand
the problem statements, s/he analyzed the previous examinations implemented by KICE: S/he
marked the concept(s) that the examiner asks in solving a problem with blue-pen; and marked the
parts that is easy to make mistakes with a red-pen and then put these into a note to avoid making the
same mistakes.
[15.12], Agg3b decomposition
[15.13], Agg3a classification
[15.10], M2 planning
[15.11], M3 attention
[15.14], C9 summarizing
[15.15], M4 management
Transcript of case 16 (Grade: UF) Cognitive processes Learning strategies
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N16: ….The participant systematically [emphasis added] created notes for each school subject in
order to effectively study and organize vast amounts of learning materials. After organizing the
notes, s/he repeatedly wrote them down for the purpose of studying.
N16: For Korean language, s/he created “incorrect answers note” in order to reduce mistakes while
solving problems. For mathematics, s/he created two separate notes (one for organizing concepts and
the other for organizing incorrect answers). In the concept note, s/he organized important math
concepts and formulas along with their derivations, and examples from multiple conceptual books.
As the textbook is core in learning history, s/he organized the history note multiple times in various
ways based on the content (e.g., by historic event, people, period, and so on). For world history, s/he
aggregated and extracted common facts and information from three different textbooks. For
geography, s/he focused on the interpretation of diagrams and maps, and so, to get familiar with
national maps, s/he reorganized it in the note.
[16.1], Agg1 identification
[16.2], Agg2 serial
[16.3], Agg3a classification
[16.4], Agg3b decomposition
[16.5], Agg4 reproduction
[16.7], Agg5 completion
[16.10], Agg6a schema
[16.6], C3 reorganization
[16.8], C5 elaboration
[16.9], C1 grouping
[16.11], C2 note
[16.12], C3 reorganization
[16.13], C9 summarizing
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N16: For economics, after learning basic economic concepts, s/he solved problems, and then
aggregated the problems wherein s/he made incorrect answers as s/he thought that the correct
interpretation of the problem and the application of learned concepts in solving problems were more
important than simply learning concepts.
[16.14], Agg6b instantiation [16.15], M7 evaluation
[16.16], M6 problem
[16.17], M2 planning
[16.18], M4 management
Transcript of case 17 (Grade: HS2) Cognitive processes Learning strategies
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N17: ….S/he repeatedly studies the same math problem not by simply solving the problem multiple
times but by creating new problems by transforming the problem into various ways.
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N17: The first step for doing this is identifying the problem statement by checking the condition(s) and
question(s) of it. By carefully investigating the problem, s/he identifies some possible transforming
conditions, and then creates a draft list of problems. Out of this list, s/he then identifies the problems
that could be and could not be transformed as a problem by referring various concept books to ensure
if each of the problems meets proper conditions to become a “problem.” Then s/he discusses the items
on the list with his/ her friends and modifies them accordingly. If there are any problems that they
cannot solve, then they get some help from their teacher.
[17.2], Agg1 identification
[17.3], Agg3a classification
[17.4], Agg3b decomposition
[17.5], Agg4 reproduction
[17. 6], Agg5 completion
[17.10], Ab5 model
[17.1], C1 grouping
[17.7], C4 inferencing
[17.8], C3 reorganization
[17.9], C5 elaboration
[17.11], SA2 cooperation
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N17: Creating problems by oneself has never been easy. However, by analyzing, decomposing, and
reassembling the given (original) problem to create new problems, s/he could more deeply understand
the examiner’s intent as well as, more clearly find out his/her incompetent areas so that s/he can
improve those parts later on in the course of learning.
[17.12], Agg6b instantiation
[17.13], Ab3 structural
[17.14], Ab2 generalization
[17.15], M4 management
Transcript of case 18 (Grade: HS3) Cognitive processes Learning strategies
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N18: ….The participant identified/extracted key words to study based on examination of previous
examinations, and then identified the related concepts that can be linked/associated to these key words
and then marked them in textbooks.
[18.1], Agg1 identification
[18.3], Agg3a classification
[18.4], Agg4 reproduction
[18.6], Agg5 completion
[18.7], Agg3b decomposition
[18.2], C8 keyword
[18.5], C1 grouping
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N18: While doing this, s/he could clearly identify the concepts that more frequently showed up in the
tests, and hence, s/he could identify more or less important parts from the learning material and focused
more on studying the important parts.
7 N18: S/he then transcribed the whole textbook by organizing these associated information. [18.8], Agg6a schema [18.9], C3 Reorganization
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P18: With this, s/he was able to thoroughly study the learning material without missing any single part
of it….S/he studied learning material of a particular subject across sections rather than simply studying
particular section separately. This was a big help in solving the CSAT type problems which requires
[18.10], Ab3 structural
[18.11], Ab4 analogy
[18.12], M4 management
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11 applying multiple concepts from across the sections when solving a question.
Transcript of case 19 (Grade: HS3) Cognitive processes Learning strategies
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N19: ….The participant thought that s/he fully mastered all the concepts in middle school math, but
later on, s/he realized that it was not true while s/he was studying high school math. Hence, s/he started
learning those basic concepts that s/he was vaguely aware of again until those concepts became clear
to him/her.
[19.1], M6 problem.
[19.2], M2 planning
[19.3], C13 substitution
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N19: While doing this, s/he realized that some of the high school level’s math problems that s/he
solved in a complex way could be simply solved by applying basic concepts and principles learned
from middle school math. S/he intensively focused on the sections and concepts that are closely
associated with high school math.
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N19: In doing this, s/he first checked the table of contents of middle school math and then identified
and extracted those parts that would need more attention and effort (i.e., more important parts); and
then, tried to figure out how the concepts in middle school math were related and developed in high
school math (i.e., how basic math concepts are developed into higher levels of complex concepts/
principles
[19.4], Agg1 identification
[19.5], Agg2 serial
[19.6], Agg4 reproduction
[19.7], Agg5 completion
[19.8], Ab1 comparison
[19.9], Ab3 structural
[19.10], Agg6b schema
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N19: ….S/he thus organized the problems that s/he was not able to solve in the note and repeatedly
reviewed it. With this, s/he was able to clearly understand the principles in high school math.
[19.14], Agg1 identification
[19.16], Agg6b instantiation
[19.11], M7 evaluation
[19.12], M6 problem
[19.13], M4 management
[19.15], C3 reorganization
[19.17], C10 repetition
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N19: One team [experimental group, 16 students] was given the learning material which contained
comprehensive information (e.g., term, definition, characteristic, concept, etc) related to the entire
3yrs of middle school math about “figure” and the other team [control group, 23 students] was not
provided with the material. After studying for one day, they worked on the 6 problems related to the
1st grade high school level of “figure” for 30 minutes, and were interviewed to elicit responses
regarding what they were thinking.
N19: Most of the students from the experimental group commented that there was a positive effect.
The class teacher assesses the test results of the two groups, and commented as below:
T19: One participant from the experimental group used more effective problem solving processes in
terms of its clarity and simplicity when compared to the one from the control group though both of
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them got correct answers to the question. While the former clearly understood the underlying concepts
and their implications to apply in solving the question, the latter solved the question by substituting
complicated formula. Therefore, with regard to the time efficiency, the former is better than the latter.
N19: ….One student from the experimental group commented that: “I recognized that a test deals
with all the accumulated information for each topic, and hence, I recognized that I should basically
know all the information from the middle school levels.” Another student commented that: “I
recognized that I forgot lots of information from the middle school math. And I see that some of the
high school levels of mathematical formulas are different from the middle school level.” Another
student commented that: “I see that there is some information that cannot be applied in to the high
school levels of math. And hence, I think I can ignore some parts of it.”
Transcript of case 20 (Grade: UF) Cognitive processes Learning strategies
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N20: ….After solving problems, s/he compared his/her thoughts with the explanations from the answer
(or commentary) book for all the problems irrespective of whether it was correct or wrong in order to
identify the missing knowledge, misunderstandings, and alternative solution processes.
[20.1], Ab1 comparison
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N20: S/he especially studied thoroughly the problems that had been answered incorrectly. First, s/he
identified what part of his/her explanation was incorrect and then identified the missing concepts that
s/he would need to learn more. Then s/he repeatedly wrote them down in order to memorize them until
s/he was able to correctly apply those concepts when solving problems.
[20.2], Agg1 identification [20.3], M6 problem
[20.4], C10 repetition
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N20: S/he then did a comparative analysis of multiple answer books with regard to the approaches and
processes of problem solving, and realized that there were different approaches for solving the
problems between different answer books. In this case, s/he aggregated these problems and grouped
them together based on the types of the problems, and then compared the different approaches and
identified the similarities and the differences among them. S/he then selected the most appropriate
(best) and effective way among them.
[20.5], Agg4 reproduction
[20.7], Agg2 serial
[20.9], Ab1 comparison
[20.6], C12 resource
[20.8], C1 grouping
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N20: ….S/he thus identified and aggregated important concepts and the problems that had been
answered incorrectly in order to intensively study those parts. Then s/he created his/her own answer
book by combining his/her “incorrect answer notes” (a note of the wrong answers s/he has gotten on a
test) with comprehensive note (a note wherein s/he comprehensively organized learned content).
[20.10], Agg1 identification [20.11], C3 reorganization
[20.12], C2 note
[20.13], M6 problem
[20.14], M4 management
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N20: S/he then practiced problems by referring to his/her own answer book rather than simply copying
the solution processes and answers from the answer book. In doing this, s/he wrote down all solution
steps that were omitted in the answer book (e.g., steps for proving a formula or calculation process),
[20.15], Ab5 model
[20.16], Ab1 comparison
[20.17], Ab3 structural
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and thus, added different approaches for solving the same problem in the note. These well-organized
notes became a great reference material and s/he repeatedly reviewed them.
[20.18], Agg6a schema
[20.19], Agg6b instantiation
Transcript of case 21(Grade: HS3) Cognitive processes Learning strategies
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N21: ….The participant went through five learning stages to perfectly memorize a textbook: (1)
S/he checked the table of contents (TOC) of the textbook to identify the structure and the relations
between each section; (2) thoroughly read them; (3) s/he read the textbook once again and
identified/marked core information and its relations between other supportive information ; (4) s/he
read the textbook once again by carefully investigating the information and relations identified in
Step 3; and then (5) checked the title and subtitles of the sections of the book by rehearsing the
studied information. With this, s/he was able to thoroughly study the entire parts of the
textbook….After going through the five steps, s/he tested it by filling a blank paper with the content
that s/he could remember in the form of mind map to check how correctly s/he could memorize
every detail of the content studied.
[21.5], Agg1 identification
[21.6], Agg3a classification
[21.7], Agg3b decomposition
[21.8], Ab1 comparison
[21.9], Ab3 structural
[21.10], Agg5 completion
[21.11], Agg6a schema
[21.1], M1 organization
[21.2], M2 planning
[21.3], M3 attention
[21.4], C8 keyword
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N21: While re-reading the textbook, s/he marked the part that could not be remembered using different
colored pens; and then read the textbook again focusing on the parts that s/he was not able to remember
while creating the mind map. This was repeated until s/he could perfectly create the mind map, and
hence, s/he was able to perfectly memorize the textbook.
[21.2], M6 problem
[21.13], C10 repetition
[21.14], C9 summarizing
[21.15], M4 management
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N21: During the examination period, s/he compared last semester’s tests with the textbooks to identify
the examiner’s intent when asking questions. By doing this s/he was able to find out which part of the
textbook should be more intensively studied and thus s/he was able to predict questions that were
likely to be asked in the test.
[21.16], Agg1 identification
[21.17], Ab1 comparison
[21.18], Ab3 structural
[21.19], Ab4 analogy
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N21: ….When his/her math scores dropped, s/he challenged him/herself by solving difficult math
problems and studied concept books, but all were unsuccessful. Therefore, s/he memorized the math
textbook and other supplemental materials… S/he thus repeatedly practiced all problems that had been
answered incorrectly.
Transcript of case 22 (Grade: UF) Cognitive processes Learning strategies
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N22: The participant studied by creating “incorrect answer notes.” Upon realizing that s/he had made
repeatedly the same mistakes even within a day (e.g., got incorrect answer during the day, and made
the same mistake in the evening study), s/he thought that s/he should modify the way she was creating
the incorrect notes. It took substantial time to create the incorrect note as s/he put all of the problems
that had been answered incorrectly. Therefore, s/he filtered the problems that had been
[22.1], Agg1 identification [22.2], M6 problem
[22.3], M7 evaluation
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answered incorrectly and then extracted those problems that s/he got wrong more than two times. This
was because the problems that s/he got wrong only once were likely due to a simple calculation
mistake or due to some simple misunderstandings.
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N22: And then s/he created the incorrect answer notes using color papers instead of the typical note:
S/he separated each section of the textbook (e.g., blue section referred to probability and statistics;
orange section referred to log). With this, s/he was able to figure out the sections s/he was incompetent,
and hence, studied more intensively those weak areas.
[22.4], Ab3 structural
[22.4], C1 grouping
[22.5], C13 substitution
[22.6], M4 management
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N22: ….S/he solved the problems in the note at least five times: after solving the problems (1st), s/he
modified the solution processes for the problems that had been answered incorrectly by referring to the
answer book (2nd
); s/he then solved them again (3rd
) and then modified the solution processes one more
time by referring to the answer book (4th); and solved them again (5th)… With this strategy, s/he
could reduce the time in creating the note and thus, the burden for re-learning because it reduced the
amount of material to be learned.
[22.7], Ab5 model
[22.10], Ab4 analogy
[22.11], Agg6b schema
[22.12], Agg6b instantiation
[22.8], C3 reorganization
[22.9], C10 repetition
Transcript of case 23 (Grade: HS3) Cognitive processes Learning strategies
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N23: ….At first, the participant observed and copied other students’ learning strategies. One of them
was creating a literature note. S/he exactly copied the friend’s strategy by organizing the note using
heading such as main subject, categories, and historical background. But later on, s/he modified the
strategy to make it more suited to his/her level of ability.
[23.1], C13 substitution
[23.2], M7 evaluation
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N23: [preceding comment cont’d] During lessons, s/he took notes; and then added the information
learned from reference books and workbooks.
[23.3], C2 note
[23.4], C12 resource
[23.5], C9 summarizing
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N23: ….S/he thus used symbols for identifying the meaning of the words in the literature text (e.g.,
poem) so that s/he could more clearly identify the meaning. Though creating the notes took
considerable time to complete, it was effective in learning.
[23.8], Agg4 reproduction [23.6], C7 imagery
[23.7], C2 note
[23.9], M4 management
Transcript of case 24 (Grade: HS3) Cognitive processes Learning strategies
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N24: ….Two students actively cooperated in their learning throughout their high school years. For
instance, they created a structural framework [emphasis added] while studying literature by clearly
identifying the relationship between the persons that showed up in the literature using symbols….
[24.1], Agg4 reproduction [24.2], C7 imagery
[24.3], C2 note
[24.4], C3 reorganization
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N24: The major events were organized in chronological order. By doing this, they could more easily
understand the literary works. Thus they could understand it more from a structural point of view.
[24.5], Agg1 identification
[24.6], Agg3a classification
[24.7], Agg3b decomposition
[24.8], C1 grouping
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[24.9], Ab1 comparison
[24.10], Ab3 structural
[24.11], Agg5 completion
[24.12], Agg6a schema
[24.13], Ab4 analogy
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N24: Initially, one of them tried to mimic and model the thought processes of the other, whose
academic achievement was far superior. As the learning proceeded, the lower-achieving learner
became better and better at internalizing these thought process and thus managed to regulate his/her
own learning process progressively over time. This led to a successful increase in the
lower-achieving learner’s academic abilities and achievement, thus reducing the learner’s reliance on
the other learner over time. In the end, the thought and learning processes of the two learners shifted
from a relationship of dependence to one of sharing. For instance, they discussed clippings from the
newspaper, shared their opinions, and gave feedback to each other. This enlarged the scope of their
thought and improved both of their logical thinking and writing skills.
[24.14], SA2 cooperation
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P24s: ….We sometimes rewarded ourselves to cheer us up when a targeted learning activity has been
successfully completed. For instance going shopping, take one day off, dining out and etc.
[24.15], SA3, self reinforce
[24.16], M4 self-manage.
Transcript of case 25 (Grade: HS3) Cognitive processes Learning strategies
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N25: ….The participant had some hard time in math but eventually ranked within 0.02% at the
national practice test. Previously, though s/he could easily solve simple problems, s/he had a hard time
when solving difficult levels of math problems. Hence, s/he reflected on his/her learning strategies,
and realized that s/he only practiced low and high level of math problems but not practiced enough
intermediate level. Upon realizing that, s/he differentiated the math problems into 5 different levels
and studied them from the easiest level to the highest level at his/her pace.
[25.1], Agg1 identification
[25.5], Agg3a classification
[25.6], Agg3b decomposition
[25.8], Agg2 serial
[25.2], M6 problem
[25.3], M7 evaluation
[25.4], M4 management
[25.7], C1 grouping
7 N25: ….S/he first reviewed the content learned in class by referring conceptual books. [25.9], Agg1 identification [25.10], C12 resourcing
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N25: S/he then practiced simple calculation problems which primarily can be solved using proper
concepts and then progressively moved to higher levels…s/he tried to integrate multiple concepts
learned from different sections. By doing this, s/he could determine how the basic concept has
progressed into advanced level.
[25.11], Ab1 comparison
[25.12], Ab3 structural
[25.13], Ab4 analogy
[25.14], Agg5 completion
[25.15] Agg6a schema
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N25: ….S/he collected all problems that had been answered incorrectly and identified related
topics/sections in the textbook.
[25.16], Agg1 identification
[25.21] Agg5 completion
[25.17], M6 problem
[25.18], M7 evaluation
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N25: S/he then studied each corresponding section by going through the entire 5 levels of problems
and then solved those problems that had been answered incorrectly. This is because once s/he
completed the entire steps; s/he was able to figure out new approaches to solve them. This made her
more and more confident in math.
[25.22] Agg6a schema
[25.23], Ab3 structural
[25.24], Ab5 model
[25.19], M2 planning
[25.20], C3 reorganization
[25.25], C5 elaboration
[25.26], C6 transfer
Transcript of case 26 (Grade: UF) Cognitive processes Learning strategies
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N26: ….The participant developed a unique way of note taking to reduce the time to explore all
sections in social studies.
N26: Before the lesson, s/he divided the note into three columns (preview, lesson, and review), and
then reviewed the table of content to be learned in upcoming lesson and checked the relationship of the
section (to be learned) within the whole parts of the textbook.
[26.1], C2 note
[26.2], M2 planning
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N26: S/he thus predicted the content to be learned and put it in the preview column of the note. By
doing this, s/he was able to see the entire framework of the learning material, and thus was able to
identify those topics that would need more attention and effort (i.e., more important parts) in the
learning process in advance, and hence, s/he was able to be more attentive in class. During the class,
s/he took notes comparing it with the content of the preview; and during the test period, studied by
reviewing the content in the 1st and 2
nd columns.
[26.7], Agg1 identification
[26.8], Agg3a classification
[26.10], Agg4 reproduction
[26.11], Ab3 structural
[26.12], Agg5 completion
[26.3], M1 organization
[26.4], M3 attention
[26.5], C8 keyword
[26.6], M4 management
[26.9], C1 grouping
Transcript of case 27 (Grade: UF) Cognitive processes Learning strategies
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N27: ….The participant reorganizes and reconstructs his/her note using three steps, namely, an
addition process, a division process, and a subtraction process.
[27.1], M2 planning
[27.2], C10 repetition
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N27: S/he first collects as much information as possible from multiple resources (e.g., class note
taking, reference books, workbooks, and the note takings of other students if necessary) and then
organizes them in his/her note in detail (s/he calls it as an addition process). In doing this, s/he
repeatedly studied the learning material and hence, was able to naturally memorize them.
[27.3], Agg1 identification [27.4], C12 resourcing
[27.5], C2 note
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N27: As there was insufficient time to study all of this information aggregated in the note, s/he tried to
systematically organize the information in order to reduce the time to review it later on. Hence, s/he
classified the content based on the titles and key words of each section (s/he calls it a division process)
and then extracted the core information out of it (s/he calls it a subtraction process).
[27.6], Agg1 identification
[27.7], Agg2 serial
[27.8], Agg3a classification
[27.9], Agg3b decomposition
[27.10], C1 grouping
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N27: ….By doing this, s/he was able to clearly identify the content within the whole framework of the
textbook and hence, was able to extract keywords that s/he should focus during study.
[27.11], Agg4 reproduction
[27.12], Ab3 structural
[27.13], Agg6a schema
[27.14], Ab4 analogy
[27.15], C3 reorganization
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P27: ….S/he organized four study plans: an hourly timetable, a daily study timetable, a weekly
schedule, and an action table for checking the progress of the study.
[27.16], M2 planning
[27.17], M4 management
Transcript of case 28 (Grade: US) Cognitive processes Learning strategies
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N28: ….The participant studied the online lectures provided by Korea Educational Broadcasting
System (EBS). S/he collected information from the online lectures and then put them in a note. Then
s/he reviewed the note as many times as possible (more than 20 times).
[28.1], Agg1 identification
[28.2], C12 resourcing
[28.3], C9 summarizing
[28.4], C10 repetition
Transcript of case 29 (Grade: UF) Cognitive processes Learning strategies
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N29: ….For social studies, the participant created “a concept note” by aggregating all of the
information from multiple learning sources. S/he first established the framework based on the textbook
and class handouts, and then wrote down the core content of each section in the note; s/he then added
additional information from reference books and workbooks. In this way, s/he was able to effectively
study a variety of materials.
[29.1], Agg1 identification
[29.4], Agg2 serial
[29.5], grouping
[29.6], Agg3a classification
[29.7], Agg3b decomposition
[29.8], Agg4 reproduction
[29.9], Agg5 completion
[29.12], Agg6a schema
[29.2], C12 resourcing
[29.3], C2 note
[29.10], C5 elaboration
[29.11], C3 reorganization
[29.13], C2 note
[29.14], C9 summarizing
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N29: After aggregating and organizing the note, s/he extracted the most critical information from
among them by analyzing the (types of) questions that had appeared on the previous examinations as
s/he thought that the information that were repeatedly asked in the examinations were essential
information to learn. S/he tried to figure out the related sections for each question and then intensively
reviewed those sections.
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N29: S/he thus analyzed the given choice of answers from the multiple-choice questions to find out
the reasons how and why each choice can or cannot be the correct answer.
[29.15], Agg4 reproduction
[29.16], Ab4 analogy
[29.18], Ab1 comparison
[29.19], Ab2 generalization
[29.17], C4 inferencing
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N29: For the problems that had been answered incorrectly, s/he tried to find out the reasons and then
summarized them in the corresponding sections of the note. By doing this, s/he was able to identify the
essential information as well as the missing information/knowledge that s/he should learn further.
[29.20], Agg1 identification
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N29: If one does not learn and study the organized content in the note, the note is not useful in learning
and problem solving. Hence, for history, s/he reorganized the timeline of the historical events in
chronological order along with its causes and consequences on a B4-size paper, and then added small
incidents. S/he then checked if there was any missing information on the notes in the process of
[29.22], Agg2 serial
[29.24], Agg3a classification
[29.25], Agg3b decomposition
[29.26], Agg4 reproduction
[29.21], C3 reorganization
[29.23], C1 grouping
[29.27], C2 note
[29.28], C9 summarizing
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20 learning. [29.29], Agg6a schema
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N29: For the content that was difficult to memorize, s/he did a lecture to him/herself like a teacher.
With this, s/he could efficiently memorize the content organized in the notes.
[29.30], C10 repetition
[29.31], M7 evaluation
[29.32], M6 problem
[29.33], M4 management
Transcript of case 30 (Grade: UF) Cognitive processes Learning strategies
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N30: ….The participant designed a systematic way to do the literary analysis. For instance, for poem,
s/he analyzed the poem by four criteria (i.e., its subject matter, the state of the speaker, the meaning of
poetic words, and the atmosphere) referring to commentary books of literature. While reading the
passage(s) of problems, s/he analyzed and organized them by these four criteria.
[30.1], Agg1 identification
[30.3], Agg3a classification
[30.4], Agg3b decomposition
[30.2], C12 resourcing
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N30: S/he then carefully compared his/her own analysis to that of the commentary book and then
modified his/her thoughts (i.e., analysis) accordingly. With this, his/her analytic ability had gradually
increased.
N30: After solving the problems, s/he carefully compared his/her own analysis to that of the
commentary book; and then modified his/her thoughts. Consequently, his/her analytic ability had
gradually increased over time.
[30.5], Ab1 comparison
[30.7], Ab5 model
[30.8], Ab4 analogy
[30.6], M6 problem
11 N30: For fiction, s/he identified the structure by investigating people, events, and background.
[30.9], Agg2 serial
[30.11], Agg3a classification
[30.12], Agg3b decomposition
[30.13], Agg4 reproduction
[30.14], Agg6a schema
[30.10], C1 grouping
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N30: As s/he thought that the important factors in the analysis of the novel are characters, events, and
background, s/he created a symbol that represented each of the factors and identified them in the text
accordingly. For instance, ∆= the clue of background; ○= important event; □= character;
wave=emotional state.
[30.15], C1 grouping
[30.16], C7 imagery
[30.17], C2 note
[30.18], C9 summarizing
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N30: [preceding comment cont’d] And then based on these symbols, s/he created a structural map.
This strategy was efficient because s/he did not have to reread the text while solving problems.
Furthermore, to quickly identify the meaning of the text of nonliterary writing, s/he used diagrams to
understand the meaning of text. Consequently, the time spent on problem solving was significantly
reduced.
[30.19], Agg4 reproduction
[30.20], Ab1 comparison
[30.21], Ab3 structural
[30.22], Ab4 analogy
[30.23], M7 evaluation
[30.24], M6 problem
[30.25], M4 management
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Transcript of case 31(Grade: UF) Cognitive processes Learning strategies
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N31: ….For science, the participant first studied concepts by reading conceptual books rather than
blindly memorizing them. At first, s/he read reference books focusing on the parts that the school
teacher pointed out as important concepts. S/he then underlined the content that s/he thought were
important.
[31.1], Agg1 identification [31.2], C12 resourcing
[31.3], C10 repetition
[31.4], M6 problem
[31.5], C13 substitution
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N31: By the time s/he read the conceptual book around 20 times, s/he realized that s/he negligently
read the book. Hence, s/he bought another conceptual book and read it again with a fresh
heart….Eventually s/he read each of the three different conceptual books more than 20 times.
Therefore, s/he was able to fully master all the concepts in science.
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N31: However, s/he still could not perfectly solve some problems. S/he reflected on his/her learning
and realized that s/he was incompetent in particular types of problems. Hence, s/he grouped all the
problems that had appeared in the previous mock tests to identify/distinguish the types of problems
and then grouped them based on the topics and sections of textbook. Then s/he diligently practiced
those identified problems by studying corresponding sections.
[31.7], Agg2 serial
[31.9], Agg3a classification
[31.10], Agg3b decomposition
[31.11], Ab1 comparison
[31.12], Ab3 structural
[31.13], Ab4 analogy
[31.14], Ab2 generalization
[31.6], M6 problem
[31.8], C1 grouping
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N31: ….Thus, after taking the test by KICE, s/he discussed the problems with other students to
identify new types of problems, and then intensively studied those new types of problems. By doing
this, s/he was able to better understand concepts.
[31.15], SA2 cooperation
[31.16], M7 evaluation
[31.17], M4 management
Transcript Interviewee ID: P32 / Grade: UF Cognitive processes Learning strategies
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N32: The participant used to study by solving problems in Korean language. However, realizing that
this did not help for preparing CSAT, s/he changed learning strategy… and finally scored 100 in
CSAT in Korean language.
[32.1], M7 evaluation
[32.2], C13 substitution
[32.3], M2 planning
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N32: ….In order to improve his/her analytic skills and time management during the test, s/he created
“incorrect answer notes” to more precisely determine his/her missing knowledge and
misunderstandings. S/he used three types of commentary footnotes while organizing the note: (1) For
the correct answer, s/he wrote down the reason why s/he picked the correct answer; (2) for incorrect
answer, s/he wrote down the reason why s/he chose the incorrect answer as well as the reason why it
could not be the correct answer; and (3) for the question s/he was not sure, s/he wrote down the reason
why s/he was struggling when choosing the correct answer as well as any hidden pitfalls in the
question. By doing this, s/he was able to clearly understand and identify incomplete knowledge and
misunderstandings, and hence, modified and completed them accordingly. This in turn improved
[32.5], Agg1 identification
[32.8], Ab1 comparison
[32.9], Ab4 analogy
[32.10], Ab5 complex comp
[32.4], M6 problem
[32.6], C9 summarizing
[32.7], C3 reorganization
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13 his/her problem solving skills and abilities.
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N32: While solving problems, s/he identified the concepts that were not clear to him/her and searched
for their meanings in the dictionary. S/he then organized them in a separate note. By doing this, s/he
was able to master the concepts, and hence, was able to solve problems more quickly and accurately.
[32.11], Agg1 identification
[32.13], Agg4 reproduction
[32.14], Agg6a schema
[32.12], C12 resources
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N32: ….To manage the time while taking the test, s/he measured the time it took to solve three
problems at a time. And s/he marked “+” for over time, and “–” for less time. And thus s/he identified
the specific types of problems in the test pool which took an unusually long time to solve, and then
intensively practiced similar types of problems. With this strategy, s/he was able to complement
his/her weaknesses.
[32.15], Agg6b instantiation [32.16], M7 evaluation
[32.17], M2 planning
[32.18], M4 management
Transcript of case 33 (Grade: UF) Cognitive processes Learning strategies
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N33: ….Beginning in the 1st grade of middle school, the participant reorganized textbooks in a note.
At the 1st time of reading, s/he tried to identify the structure of the content in the textbook by
continually reminding how unit(s) of information is related to each other within the entire structure.
[33.1], Ab3 structural
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N33: At the 2nd
time of reading, s/he tried to reorganize the information by focusing on the
interconnectivity between the information, and this made him/her easier to understand and remember
the content. S/he then reorganized the learned content in the note… S/he then identified and numbered
the information for memorization, and then repeatedly wrote them down to memorize… For math,
s/he organized basic concepts along with their derivations; and then solved practice problems
corresponding to each concept. While doing this, s/he tried to find out the most appropriate solution
processes and then organized them in the note along with their proof processes.
[33.2], Agg1 identification
[33.3], Agg5 completion
[33.6], Agg4 reproduction
[33.9], Agg6a schema
[33.10], Agg6b instantiation
[33.4], C2 note
[33.5], C9 summarizing
[33.7], C3 reorganization
[33.8], C5 elaboration
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N33: S/he then reorganized the learned content in the note ….S/he then identified and numbered the
information for memorization, and then repeatedly wrote them down to memorize.
[33.11], C10 repetition
[33.12], M4 management
Transcript of case 34 (Grade: UF) Cognitive processes Learning strategies
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N34: ….The participant aggregated all the mathematical concepts throughout the entire curriculum of
middle school by each section including definition, equations, and theorem in order to clearly
understand the basic math concepts. While doing this, s/he first aggregated the table of contents from
textbooks of each grade level on a piece of paper and then checked for similarities; and then
reorganized them by grouping related sections together in order to create a coherent structure by (sub-)
headings (e.g., by equation, function, calculus).
[34.1], Ab3 structural
[34.2], Agg1 identification
[34.4], Agg2 serial
[34.6], Agg3a classification
[34.7], Agg4 reproduction
[34.8], Agg5 completion
[34.9], Ab1 comparison
[34.3], C12 resourcing
[34.5], C1 grouping
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[34.10], Ab3 structural
[34.11], Agg6a schema
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N34: [preceding comment cont’d] S/he then intensively studied them following the organized plan and
then practiced problems referring to the note. With this strategy, s/he was able to immediately retrieve
the relevant concepts required for solving specific problems.
[34.12], Agg6b instantiation [34.13], M4 management
Transcript of case 35 (Grade: UF) Cognitive processes Learning strategies
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N35: ….The participant thought that elaboration and integration concepts are important when studying
science. Hence, s/he first aggregated information by topics and created a mind map (thus, s/he was able
to identify the entire structure of the content to be learned).
[35.1], Ab3 structural
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N35: S/he then studied basic concepts. While doing this, s/he tried to find out the central concepts and
their key points: (1) S/he first identified basic concepts in the specific section. In order to expand
his/her analytic thinking ability, s/he linked related information to the basic concepts and tried to
identify how each concept/topic is related to each other.
[35.2], Agg1 identification
[35.4], Agg3a classification
[35.5], Agg3b decomposition
[35.7], Agg4 reproduction
[35.8], Agg5 completion
[35.3], C8 keyword
[35.6], C1 grouping
[35.9], C2 note
[35.10], C9 summarizing
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N35: (2) S/he then modified the mind map by updating this information and by adding more
information from reference books.
[35.11], Ab1 comparison
[35.12], Ab3 structural
[35.15], Agg6a schema
[35.13], C3 reorganization
[35.14], C5 elaboration
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N35: (3) S/he then solved various problems through careful analysis. By doing this, s/he was able to
identify the relationship between sections, and this enabled him/her to successfully solve problems that
required applying integrated knowledge between different sections.
[35.16], Agg6b instantiation [35.17], M4 management
Transcript of case 36 (Grade: CF) Cognitive processes Learning strategies
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N36: ….The participant aggregates information during lessons by note taking, and then reorganizes
the note by adding related additional information from textbooks, reference books, and handouts;
s/he then reviews this note until s/he completely understands and memorizes all the content in the
note. After that, s/he writes them down on a piece of paper to check how much information s/he
understood and was able to memorize.
[36.1], Agg1 identification
[36.3], Agg4 reproduction
[36.2], C2 note
[36.4], C3 reorganization
[36.5], C12 resourcing
[36.6], C10 repetition
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N36: During class lesson, s/he writes down her doubts and then tries to solve them by getting help
from class teachers. S/he also helps other students in their problem solving and this supported his/her
learning.
[36.7], SA1 questioning
[36.8], SA2 cooperation
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N36: S/he also repeatedly solved those problems that s/he answered incorrectly and this enabled
him/her to fully understand and master all the problems in the workbook.
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Transcript of case 37(Grade: HS3) Cognitive processes Learning strategies
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N37: ….The participant carefully prepares the following day’s lessons the day before to thoroughly
preview the information to be learned beforehand.
[37.1], M1 organization
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N37: S/he summarizes and organizes concepts in a note and then solves problems in order to check
his/her level of understanding of the previewed content.
[37.2], Agg1 identification [37.3], C9 summarizing
[37.4], C2 note
[37.5], M7 evaluation
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N37: During the lesson, s/he tries to find solutions to the problems that s/he was not able to solve
while previewing.
[37.6], SA1 questioning
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N37: From the notes, s/he compares the information learned from the preview and the lesson, and
then identifies important parts of the content (i.e., the parts that s/he should focus special attention).
[37.7], Ab1 comparison
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N37: S/he reviews learning content four times: (1) S/he reviews the content learned during the lesson
five minutes before the end of the lesson; (2) reviews it again during recess time; (3) during self-study
hours; and (4) during weekly reviews. By doing this, s/he reviews all the information learned from
lessons and identifies misunderstandings or incomplete knowledge, and then organizes them in a
note.
[37.10], Agg6a schema [37.8], M2 planning
[37.9], C10 repetition
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N37: S/he then solves problems and compares the similarities and differences of the solutions to the
problems between him/herself and the school teacher.
[37.11], Agg6b instantiation
[37.12], Ab1 comparison
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P37: ….Once in a week, s/he collaborates with other students: Each of them picks up three problems,
and they put the collected problems on a piece of paper. Then each student solves the problems and
then they share and discuss their solutions.
[37.13], SA2 cooperation
Transcript of case 38 (Grade: HS2) Cognitive processes Learning strategies
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N38: ….When the participant read the textbook; s/he easily forgot the content in the previous sections of
the book. Hence, s/he divided the content into several sections (e.g., paragraphs) (a) when the topic
changed; and (b) when s/he was not able to retrieve the previous content. S/he then intensively studied
each divided section and never moved to the next one until s/he fully understood the current section.
[38.1], Agg1 identification
[38.3], C2 grouping
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N38: S/he repeatedly read the content in order to memorize: When s/he understood the information in a
particular section of the book, s/he then tried to rehearse its core information and drew a schematic
diagram [emphasis added] to understand the content better (checking).
[38.5], Agg3a classification
[38.6], Agg3b decomposition
[38.7], Ab1 comparison
[38.8], Ab3 structural
[38.9], Agg5 completion
[38.10], Agg6a schema
[38.3], C10 repetition
[38.4], C8 keyword
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N38: S/he solved problems as much as possible using multiple resources (textbooks, reference books,
internet sites) and asked for help from teachers for the problems that s/he was not able to solve.
[38.11], Agg6b instantiation [38.12], C12 resourcing
[38.13], SA1 questioning
[38.14], M4 management
Transcript of case 39 (Grade: US) Cognitive processes Learning strategies
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N39: ….The participant studied middle school textbook to master basic level of English. S/he thus
memorized English words cumulatively and repeatedly: (1) 50 words from the previous day + 50 words
for the day=100 words; (2) 100 words from the previous days + 50 words for the day = 150 words, and so
on. S/he repeated this with a five-day cycle and then organized the words that s/he still could not
memorize in a note and reviewed them as often as possible.
[39.1], Agg1 identification
[39.2], C10 repetition
[39.3], M2 planning
[39.4], C2 note
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N39: S/he thus analyzed each part of the sentence in terms of its function (e.g., subject, object,
complement, modifier, etc.). By doing this, s/he was more easily able to understand a long and complex
sentence.
[39.5], Agg4 reproduction [39.6], C4 inferencing
Transcript of case 40 (Grade: US) Cognitive processes Learning strategies
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N40: ….The participant divided the learning content in textbook by sections and then persistently
reviewed them. By doing this, s/he was able to structurally organize and integrate the information.
[40.1], Agg1 identification
[40.3], C2 grouping
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N40: Thus, s/he did a lecture to him/herself: While doing this, s/he first wrote down the title and heading
of the content and then did a lecture to him/herself. S/he looked up the textbook for the contents that s/he
was not able to remember, s/he then added related information aggregated from other sections, and then
structurally organizes them in the note.
[40.3], Ab3 structural
[40.8], Ab5 model
[40.7], Agg6a schema
[40.4], C5 elaboration
[40.5], C3 reorganization
[40.6], C2 note
Transcript of case 41 (Grade: HS3) Cognitive processes Learning strategies
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N 41: ….Upon realizing that s/he was incompetent in solving problems which requires applying multiple
cross-sectional principles; s/he identified the types of each problem.
N41: S/he first aggregated the problems that had been answered incorrectly and frequently asked in
previous tests and then grouped them together by related sections. S/he then persistently practiced those
problems.
[41.4], Agg1 identification
[41.6], Agg2 serial
[41.7], Agg3a classification
[41.8], Agg3b decomposition
[41.1], M6 problem
[41.2], M7 evaluation
[41.3], M4 management
[41.5], C1 grouping
Transcript of case 42 (Grade: HS2) Cognitive processes Learning strategies
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N42: ….For math, the participant created three notes: (1) S/he first aggregated and organized the solution
steps of a problem in a practice note; (2) created an incorrect answer note for the problems that had been
answered incorrectly; and (3) identified incomplete information (i.e., missing information and
[42.1], Agg1 identification
[42.2], C2 note
[42.3], C9 summarizing
[42.4], M6 problem
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4 misunderstandings) from the two notes and then created a concept note. [42.5], M4 management
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N42: ….For Korean language and English, s/he solved each problem three times and then compared the
solution processes, and then modified misunderstandings and incomplete knowledge.
[42.7], Ab1 comparison
[42.8], Ab4 analogy
[42.9], Ab5 model
[42.10], Agg6a schema
[42.6], C10 repetition
Transcript of case 43 (Grade: UF) Cognitive processes Learning strategies
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N43: ….The participant first carefully read textbooks to understand the information. While doing this,
s/he collected all questions and inquiries and tried to find solutions by using multiple sources (reference
books, teachers, internet, and so on). S/he then organized those collected information into the textbook. In
this way, s/he created his/her own special textbook. S/he then repeatedly read this textbook.
[43.1], Agg1 identification [43.2], SA1 questioning
[43.3], C12 resourcing
[43.4], C3 reorganization
[43.5], C5 elaboration
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N43: ….After about 5 to 7 times of reading, s/he was able to identify the interconnectivity between
sections at a glance. Thus s/he was able to anticipate some problems that would be asked in the upcoming
test because s/he was able to identify more or less important information from the textbook.
[43.7], Ab3 structural
[43.6], C10 repetition
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N43: After reading 8 to 10 times, s/he tried to encode all information as perfectly as possible. So, when
s/he was taking a test, s/he felt that she was taking an open-book test as s/he could retrieve most
information from the book.
Transcript of case 44 (Grade: HS3) Cognitive processes Learning strategies
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N44: ….The participant first collected information from multiple sources (textbook, answer book,
multiple reference books) and then put that extracted information and knowledge into a note with a
coherent structure.
[44.1], Agg4 reproduction [44.2], C13 resourcing
[44.2], C2 note
[44.3], C9 summarizing
Transcript of case 45 (Grade: UF) Cognitive processes Learning strategies
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N45: ….After learning basic concepts, the participant tried to find out causal relationships between the
concepts, and then marked these relationships using arrows. Once these relationships were identified, s/he
then wrote them down in a note in order to have a better understanding of these relationships.
[45.1], Ab3 structural
[45.2], Agg4 reproduction
[45.4], Agg5 completion
[45.3], C3 reorganization
[45.5], C2 note
[45.6], C9 summarizing
Transcript of case 46 (Grade: HS3) Cognitive processes Learning strategies
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N46: ….Instead of reading the textbook repeatedly, the participant aggregated and extracted a certain
amount of information from each section of the math textbook and then grouped them together. S/he then
divided them into a daily study material.
[46.1], Agg1 identification
[46.2], Agg3a classification
[46.4], Agg1 identification
[46.3], M2 planning
[46.5], C2 note
[46.6], C9 summarizing
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N46: After learning mathematical concepts, s/he practiced problems, and then analyzed the solution
strategy and concepts/principles applied in solving the problems. S/he organized them into a note, and
then persistently reviewed and studied them.
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N46: ….By doing this s/he was able to naturally learn the problems which require applying multiple
cross-sectional principles and concepts in a question.
[46.7], Agg1 identification
[46.8], Agg4 reproduction
[46.9], Ab4 analogy
[46.11], Agg5 completion
[46.10], C3 reorganization
Transcript of case 47 (Grade: HS3) Cognitive processes Learning strategies
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N47: ….The participant reorganized the history textbook by topics such as culture, politics, and
economics.
[47.1], Agg1 identification
[47.2], C1 grouping
[47.3], Agg2 serial
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N47: S/he then aggregated additional information from various learning materials and then utilized them
in the textbook, and then persistently reviewed them.
[47.4], C12 resourcing
[47.5], C10 repetition
Transcript of case 48 (Grade: UF) Cognitive processes Learning strategies
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N48: ….The participant first aggregated information from five different math textbooks and carefully
studied all the concepts. S/he then checked and verified them by solving problems, and then organized the
aggregated/extracted information into a note.
[48.1], Agg1 identification
[48.3], Ab1 comparison
[48.4], Ab4 analogy
[48.2], C12 resourcing
[48.5], C2 note
[48.6], C9 summarizing
Transcript of case 49 (Grade: UF) Cognitive processes Learning strategies
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N49: ….In order to improve problem solving ability in Korean language, the participant developed a
systematic problem-solving strategy.
[49.1], M2 planning
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N49: For instance, s/he grouped the text in the textbook together by Genre (e.g., Nonliterary writing,
Novel, Poem, and etc) and then analyzed them.
N49: ….For social science, s/he tried to define the structural relationship of information (e.g.,
super-concept and a sub-concept of information). By doing this, s/he was able to clearly understand the
structural relationship of the information in the textbook.
[49.2], Agg1 identification
[49.5], Agg3a classification
[49.6], Agg3b decomposition
[49.7], Ab3 structural
[49.3], C1 grouping
[49.4], Agg2 serial
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