Cumulative Learning and Schematization in Problem Solving

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

Soli Deo Gloria

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

v

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 4.7 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.

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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

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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%


(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


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



Multiple-choice

Vocational

Education

20 items

per subject

20 items x up

to 3 subjects



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.

72

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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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).


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.



[1.11], Agg5 completion

[1.14], Ab5 model

[1.8], C12 resourcing




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.


[1.16], 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.


[1.27], 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


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].



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.





[2.7], C1 grouping





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.





[2.18], Agg6a schema

[2.19], M2 planning

[2.20], M3 attention


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.





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.









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


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].


[3.2], Agg2 serial


[3.3], C2 grouping


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




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.


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].


[3.15], M6 problem


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.


[3.18], Agg2 serial

[3.19], Agg3a classification


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.


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.


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



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.


[4.2], Ab5 model


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


[4.5], Ab5 complex comp.



[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.


[4.11], 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,

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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



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.




[5.8], 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



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.



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.


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.



[5.22], Agg3b

decomposition


[5.24], Agg5 complex comp

[5.25], Agg6a schema comp


[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


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.



[5.32], Agg6a schema comp



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.



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.



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.

134

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

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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


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

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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


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


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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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.




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.





[6.7], 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.


[6.10], M6 problem




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


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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].


[6.18], M6 problem


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.



[6.23], Agg3b

decomposition


[6.25], Ab4 analogy



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.




[6.33], 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.

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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


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

145

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

146

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)


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)


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)


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.



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

154

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





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).

155

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

156

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


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.


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.

157

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.



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).


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)


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


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).



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)


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


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).



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




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).



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)


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)


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


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)


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)


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



Case 32 (Grade: UF)


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)

181


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)


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

182

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)


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).



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).



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


([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)


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)


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.



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



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).



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


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)


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.

190



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)


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)


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

191

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.



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)


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)


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


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

194

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,

195

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

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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

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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

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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.

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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)


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.



[7.2], M6 problem


[7.5], M2 planning

[7.6], M3 attention


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.






[7.13], 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.









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.




[8.4], 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



[8.7], 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



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


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



[9.6], Ab4 analogy

[9.7], Ab5 model




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


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.


1

2

3

4

5

6

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.




[10.5], Ab4 analogy

[10.6],Ab5 model




251

7

8

9

10

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.





[10.13], Ab4 analogy



1

2

3

4

5

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.


[11.2], M6 problem



6

7

8

9

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.


[11.7], C1 grouping

[11.8], C2 note


10

11

12

13

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.


14

15

N11: S/he thus organized literary works in various ways (e.g., the period of literary activities by the

authors).

[11.12], Agg2 serial





[11.13], C1 grouping


16

17

18

19

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.





20

21

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.

252


1

2

3

4

5

6

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.


7

8

9

10

11

12

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.


[12.3], Agg2 serial





[12.7], C1 grouping

[12.9], C8 keyword

[12.10], C7 imagery


1

2

3

4

5

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.





[13.2], M6 problem




1

2

3

4

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.


[14.4], Agg2 serial




[14.1], C1 grouping


5

6

7

8

9

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.




[14.11], Ab5 model





[14.19], M6 problem


253



[14.17], Ab5 model


1

2

3

4

5

6

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.





[15.2], M6 problem

[15.3], C1 grouping




7

8

9

10

11

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.



[15.10], M2 planning





1

2

3

4

5

6

7

8

9

10

11

12

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.


[16.2], Agg2 serial








[16.9], C1 grouping

[16.11], C2 note



254

13

14

15

16

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




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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. 6], Agg5 completion

[17.10], Ab5 model

[17.1], C1 grouping




[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.






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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.






[18.2], C8 keyword

[18.5], C1 grouping

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6

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




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11 applying multiple concepts from across the sections when solving a question.


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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


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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


[19.5], Agg2 serial






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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.




[19.12], M6 problem




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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.”


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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.


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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.



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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.


[20.7], Agg2 serial


[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


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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



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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.



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.









[21.2], M2 planning


[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




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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.





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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.


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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



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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], C1 grouping



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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







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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.



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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


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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.


[23.7], C2 note



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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….


[24.3], C2 note


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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.




[24.8], C1 grouping

259






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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.


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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.


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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.




[25.8], Agg2 serial

[25.2], M6 problem



[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.





[25.15] 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.


[25.21] Agg5 completion

[25.17], M6 problem


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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


[25.24], Ab5 model




[25.26], C6 transfer


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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.








[26.5], C8 keyword


[26.9], C1 grouping


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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


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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.


[27.5], 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).


[27.7], Agg2 serial




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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.






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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.



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).






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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.


[29.4], Agg2 serial

[29.5], grouping







[29.3], C2 note



[29.13], C2 note


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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.






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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.


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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






[29.27], C2 note


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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.



[29.32], M6 problem



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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.





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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.


[30.7], 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






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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.


[30.16], C7 imagery

[30.17], C2 note


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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.






[30.24], M6 problem


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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.



[31.4], M6 problem


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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







[31.6], 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.




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.



[32.3], 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



[32.9], Ab4 analogy

[32.10], Ab5 complex comp

[32.4], M6 problem



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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.




[32.12], 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




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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.


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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.






[33.4], C2 note




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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.




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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).



[34.4], Agg2 serial






[34.5], C1 grouping

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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


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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).


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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.






[35.3], C8 keyword

[35.6], C1 grouping

[35.9], C2 note


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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.






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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.



[36.2], C2 note




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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.



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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.


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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


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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.


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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).


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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


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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.



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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.



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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.


[38.3], 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).








[38.4], 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




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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.



[39.3], 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


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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.


[40.3], 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.


[40.8], Ab5 model




[40.6], C2 note


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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.


[41.6], Agg2 serial



[41.1], M6 problem



[41.5], C1 grouping


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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


[42.2], C2 note


[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.


[42.8], Ab4 analogy

[42.9], Ab5 model




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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




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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.



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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.


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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



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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.





[45.5], C2 note



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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.




[46.3], M2 planning

[46.5], C2 note


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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.



[46.9], Ab4 analogy




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N47: ….The participant reorganized the history textbook by topics such as culture, politics, and

economics.


[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.




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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.



[48.4], Ab4 analogy


[48.5], C2 note



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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.





[49.3], C1 grouping

[49.4], Agg2 serial

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Cumulative Learning and Schematization in Problem Solving

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Transcript of Cumulative Learning and Schematization in Problem Solving