Investigating changes in prospective teachers’ views of a ‘good teacher’ while engaging in...

Investigating changes in prospective teachers’ viewsof a ‘good teacher’ while engaging in computerizedproject-based learning

Ilana Lavy Æ Atara Shriki

Published online: 6 February 2008� Springer Science+Business Media B.V. 2008

Abstract Prospective teachers’ (PT) beliefs and views regarding teaching and learning

affect the way they teach after they have graduated. Often, these views are a consequence

of their past experience as pupils themselves. In recent years there has been a great effort

toward changing the teaching and learning of mathematics, which raises the need to

generate a change in views and beliefs. The present study aims at exploring the effect of

learning via computerized project-based learning (CPBL) on the change in views of third

year prospective mathematics teachers regarding the image of the ‘good teacher.’ To assess

the change in views of our pre-service teachers regarding the image of the ‘good teacher,’

an integrated tool was employed: two open questionnaires, written portfolios, and tran-

scripts of class discussions. In this paper we focus mainly on results obtained from the

questionnaires, supported by evidence from the portfolios, and show that CPBL is an

effective approach for supporting the change in views of our pre-service mathematics

teachers regarding the image of the ‘good teacher.’

Keywords Good teacher � Prospective teachers � Beliefs � Computerized project-based

learning � Inquiry-based learning � Affective � Cognitive and didactical aspects

Introduction

Becoming a mathematics teacher is a complex process that involves the development of

knowledge, skills, beliefs, and self-awareness with regard to the ability to engage in

teaching mathematics. The beliefs and knowledge that prospective teachers (PT) bring to

teacher education programs affect how and what they learn (Calderhead, 1996). One of

these beliefs concerns the image of the good (ideal) teacher.

I. Lavy (&)Computer Science and Information Systems, Emek Yezreel Academic College, Emek Yezreel MobilePost Office, 19300 Emek Yezreel, Israele-mail: [email protected]

A. ShrikiMathematics Education, Oranim, Academic College of Education, 36006 Tivon, Israele-mail: [email protected]

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J Math Teacher Educ (2008) 11:259–284DOI 10.1007/s10857-008-9073-0

The issue concerning the image of the ‘ideal teacher’ has occupied philosophers and

researchers since the need for teaching was raised, starting with Socrates and Plato who

served as models for good teaching, and continuing with philosophical-PT-perceived

descriptions of the desired attributes of the ‘ideal teacher.’ The characteristics and nature

of that image have undergone meaningful changes over the years, reflecting the social needs

and cultural perceptions of the period that these philosophers and researchers lived in.

Reichel and Arnon (2005) summarized the various prototypical ‘educational images’ of

the ‘good teacher’ throughout the years. Among them Socrates and Martin Buber perceived

the ideal teacher as one who is in constant dialogue with his students and consequently

advances their knowledge. Plato believed that the ideal teacher should be a philosopher

who helps his students to develop their thinking. Max Adler referred to social aspects of the

‘ideal teacher,’ namely—the teacher as a socializing pedagogue, whose main concern is to

help students to become a part of society. Feuerstein believed in mediated learning

experience, that is—the teacher as a mediator. Seymour Papert believes that a teacher

should encourage critical thinking, Donald Schon talked about a reflective teacher, Piaget

and Vygotsky believed in a constructivist teacher who emphasizes the processes occurring

during active learning rather than the products.

In the past people tended to believe that anyone who knows a certain subject matter

fairly well can teach it to high school students. Today it is recognized that to become a

good teacher, one should hold various skills and not just the knowledge of the subject

matter (Noddings, 1992). Shulman (1987, 1993, 2000), for example, referred to knowledge

of learners and their characteristics, knowledge of educational contexts, and knowledge of

educational products. These types of knowledge develop as teachers gain more experience

and confidence.

Teachers act and develop professionally according to the way they perceive the image

of the ‘good teacher’ (Wilson, Cooney, & Stinson, 2005). Hence we believe that one of our

responsibilities as teacher educators is to facilitate the construction of that image. For that

matter education instructors in pre-service programs should model good mathematics

teaching by posing worthwhile mathematical tasks; engaging teachers in mathematical

discourse; enhancing mathematical discourse through the use of a variety of tools,

including computers; creating learning environments that support and encourage mathe-

matical reasoning and teachers’ dispositions and abilities to do mathematics; expecting and

encouraging teachers to take intellectual risks in practicing mathematics and to work

independently and collaboratively (NCTM, 2000).

In the present study we examined changes in our PT perception of the image of the

‘good teacher’ while engaging in activities which were developed according to the

aforementioned NCTM’s recommendations. We believe such a change can imply the need

for processes such as becoming aware of the complexity of teaching even before the PT

actually practices it in class.

In what is to be followed we introduce the contextual framework of the present study

and present a brief relevant PT’ perceived background.

Contextual framework

In recent years, mathematics teacher educators have been calling to maintain a reform in

mathematics education (e.g. NCTM, 1989, 2000). These calls have influenced the devel-

opment of a vision regarding the image of the new generation of teachers, and suggested

various guidelines and ideas for qualifying them. In designing teacher training programs,

260 I. Lavy, A. Shriki

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teacher educators should consider the fact that PTs begin their math methods course with

interconnected ideas about mathematics, about teaching and learning mathematics, and

about schools (Ball, 1988). These ideas originate in their past experience as pupils, and will

eventually shape and affect their practice as mathematics teachers in the future (Ernest,

1989; Malinen, 2000; Skott, 2001). Part of these views is concerned with the image of the

‘good teacher’ relating to aspects concerning the teacher’s personality, and her professional

knowledge.

To adjust the PTs’ existing views to recommendations of the aforementioned reform,

there is a need to produce a change in their views. Most teacher educators agree that

teachers should engage their students in problem-solving activities and mathematical

writing, discuss pedagogical ideas (Farmer, Gerretson, & Lassak, 2003), and guide

mathematical explorations to encourage their student to construct mathematical knowledge

(NCTM, 2000). Liljedahl (2005) found that engaging PTs in problem-solving activities

within a non-traditional setting resulted in changing their beliefs regarding the meaning of

mathematical doing and teaching mathematics. It is the teacher educators’ responsibility to

support this process of change, by exposing the PT to innovative learning environments, in

which such a process can take place.

Inspired by the calls for implementing teaching innovations in general, and focusing on

problem posing and problem solving in particular, we applied a project-based learning

(PBL) approach in a didactical course that focuses on theories and didactical methods

implemented in teaching middle school/high school geometry and algebra.

Based on our experience, we find the PBL approach is one that provides such activities

as well as gaining insights regarding its importance to the teaching and learning processes

(Lavy & Shriki, 2003).

In the present study we aim at exploring the influence of learning via computerized

project-based learning (CPBL) on the change in views of third year prospective mathe-

matics teachers regarding the image of the ‘good teacher.’ To achieve this aim, an

integrated assessment tool was employed: two open questionnaires, transcripts of class-

room discussions, and written portfolios (Lavy & Shriki, 2004). The reason for choosing

the image of the ‘good teacher’ as one of our focal points of interest stems from the fact

that the characteristics which underlie this image represent various dimensions, such as: the

teacher’s content knowledge, general pedagogical knowledge, curriculum knowledge,

pedagogical content knowledge, knowledge of learners, knowledge of educational con-

texts, knowledge of educational ends (Shulman, 1987), as well as the teacher’s personality

(Reichel & Arnon, 2005).

In this paper we focus mainly on results obtained from the questionnaires, supported by

evidence from the students’ portfolios.

Theoretical background

In what follows we present a brief theoretical background regarding beliefs, image of the

‘good teacher,’ project-based learning and portfolio as an evaluative tool.

Beliefs

The research literature concerning teachers’ beliefs focuses on their nature and role, the

factors that influence their formation, and on questions such as: how, why, and when they

Investigating changes in prospective teachers’ views of a ‘good teacher’ 261

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should be changed. Over the past 20 years there has been a substantial amount of research

on teachers’ beliefs because ‘‘what teachers believe is a significant determiner of what gets

taught, how it gets taught, and what gets learned in the classroom’’ (Wilson & Cooney,

2002, p. 128). Most research about mathematics teachers’ beliefs has focused mainly about

their beliefs regarding mathematics, the teaching of mathematics, and the learning of

mathematics.

Belief is a ‘‘conviction of the truth of some statement or the reality of some being or

phenomenon especially when based on examination of the grounds for accepting it as true

or real’’ (Merriam-Webster’s third new international dictionary, 1993).

According to Hart (2002), beliefs are ‘‘part of our subjective knowledge with a

strong affective component. This is different from knowledge which can, at some level,

be socially agreed upon as true or false’’ (p. 162). Richardson (1996) recognizes

teachers’ beliefs as their own theories. These theories consist of sets of interrelated

conceptual frameworks. Beliefs can be visualized as personal appraisals, judgments, and

views that comprise a subjective knowledge about self and the environment. As such,

they involve cognitive as well as affective components. Beliefs are organized around an

object or a specific situation, and influence one to respond in a certain manner. Beliefs

are contextual, formed as a consequence of experience, and emerge during action. Since

beliefs are subjective, they do not need formal justification (Philippou & Christou,

2002).

Life experiences are a major contributor to the formation of beliefs (Richardson, 1996).

In turn, teachers’ activities and actions are primarily dependent on their beliefs rather than

on what they know or are really capable of achieving (Philippou & Christou, 2002). Belief

about the nature of mathematics knowledge and learning mathematics determine the way

one views involvement with the subject, as well as the choice of teaching strategies and

learning activities. Thus ‘‘it seems unlikely that fundamental changes in teaching could

occur without fundamental changes in accompanying beliefs’’ (Wilson & Cooney, 2002,

p. 142). Obviously, the reform in mathematics education demands that teachers possess

beliefs about mathematics, learning, and teaching that are different from those who are in

line with traditional school mathematics (Battista, 1994). That is to say that teacher edu-

cators who wish to help their PTs to internalize some of the benefits which are associated

with teaching in line with the reform, should at the same time change their beliefs with

regard to teaching and learning mathematics.

Changing one’s beliefs is not easy (Chapman, 2002). It occurs only under specific

conditions in which the individual is faced with new information and experiences that

‘disagree’ with established beliefs. Therefore ‘‘because many teachers’ beliefs and prac-

tices are deeply tied to school mathematics traditions, the success of current mathematics

education initiatives depends on our identifications of viable ways to encourage and enable

teachers to make significant shifts in their beliefs’’ (Lloyd, 2002, p. 150).

Image of a ‘good teacher’

Reichel and Arnon (2005) summarized studies on students’ perceptions regarding the good

teacher. They distinguished between those who found that professional knowledge is the

prominent attribute of the ‘good teacher’ and those that found the teacher’s personality is

the prominent one. Research which found professional knowledge to be the most promi-

nent characteristic of the good teacher, mentioned attributes such as: knows the subject

matter well, explains patiently and clearly, teaches in an interesting manner, and instructs


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the students how to solve problems. Research which found personal qualities to be the most

prominent characteristic of the good teacher, denoted attributes such as: respects his

students, understands the students’ problems, is a good listener, is warm, caring, believes in

his students’ ability, and has a sense of humor. It should be mentioned that those char-

acteristics are general and not specific to a certain discipline. Reichel and Arnon (ibid) also

analyzed 55 questionnaires of PTs in their last year of study, and 34 questionnaires of

experienced teachers who were completing their educational studies to get a bachelor’s

degree. They were all asked to refer to the image of the ideal teacher. The researchers

found that the PTs’ responses could be classified into two main categories: the teacher’s

personality and the professional knowledge of the teacher. 98% of the PTs referred to the

personality of the teacher and 76% related to the teacher’s professional knowledge. The

most prominent characteristic was ‘‘empathetic and attentive teacher’’ (91% of the PTs

referred to that characteristic); far behind were characteristics like ‘‘attitude to the disci-

pline’’ (57%), ‘‘general personal characteristics’’ (52%), ‘‘a leader’’ (44%), and ‘‘didactical

knowledge in the field of teaching methods’’ (42%). The teacher–student responses were

rather similar, and were also classified into the same two main categories, with a greater

emphasis on the category of the teacher’s personality. Again, the most prominent char-

acteristic was ‘‘empathetic and attentive teacher’’ (94% of the teachers and students

referred to that characteristic). Two main differences between the two groups were found:

while 91% of the teachers and students mentioned the characteristic of ‘‘professional

knowledge’’ only 67% of the PTs did so; 68% of the teachers referred to ‘‘having

disciplinary knowledge’’ while only 28% of the PTs mentioned it.

Wilson et al. (2005) examined the perspectives of nine high school teachers as regards

‘good mathematics teaching,’ using three interviews. They found that all their interviewees

‘‘emphasized the importance of prerequisite teacher knowledge, promoting mathematical

understanding, engaging students, and effectively managing the classroom environment’’

(p. 91). As to the prerequisite knowledge, they reported that teachers should have

knowledge of mathematics and their students. Some of their interviewees believe that what

constitutes good teaching is making connections in mathematics, helping students to

visualize mathematics, assessing students’ understanding, and simultaneously considering

various aspects concerned with class management.

Project-based learning

PBL is a teaching-and-learning strategy that involves students in complex activities, and

enables them to engage in exploring important and meaningful questions through a process

of investigation and collaboration (Krajcik, Czerniak, & Berger, 1999). The PBL approach

allows students to pose problems, ask questions, make predictions and decisions, design

investigations, collect and analyze data, use technology, share ideas, build their own

knowledge by active learning, and so on. The approach is based on the idea that students

should work relatively autonomously over a long period of time and conclude their work

with products or presentations (Jones, Rasmussen, & Moffitt, 1997; Thomas, Mergendoller,

& Michaelson, 1999). The PBL approach has various advantages (e.g. Krajcik et al., 1998):

it develops a sense of personal contribution to the process of learning; increases motivation;

raises self-satisfaction; helps in developing long-term learning skills and a deep, integrated

understanding of content and process; increases the ability to share ideas; promotes

responsibility and independent learning; provides answers to different learning needs;

develops the ability to collect and present data; and so forth.


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In our study, the PTs used dynamic geometry computer software, which served as a

fruitful environment for posing problems, making experiments, observing stability/insta-

bility of phenomena, stating and verifying or refuting conjectures easily and quickly

(Marrades & Gutierrez, 2000). Since the computerized environment was an integral part of

the PBL approach, we termed it as CPBL (Lavy & Shriki, 2003).

Students may also encounter several difficulties while learning through the PBL

approach. Among them: inability to generate meaningful questions trouble with managing

complexity and time problems with processing data and developing a logical argument to

support claims (Krajcik et al., 1998). As a consequence it is recommended to incorporate

some ‘‘scaffolds’’ within the PBL process to help the students overcome their difficulties.

By implementing the CPBL approach we intended to help PTs learn and discuss

the concepts of the discipline, either educational or mathematical, and enable them to

experience the meaning of ‘innovative learning approach.’

Portfolio as an evaluative tool

To explore the PTs’ change in beliefs regarding the image of the ‘good teacher,’ we looked

for tools that would provide us with various types of data, so that we could validate our

findings. We believe that at least one of the tools should enable the PTs to reflect on the

various processes they are about to undergo, since personal development occurs while

reflecting on activities (Cooney & Krainer, 1996). Moreover, teachers’ ability to be

reflective is linked to teacher change (Wilson & Cooney, 2002). Through reflection stu-

dents may become aware of the viewpoints that underlie their mathematical performances

in terms of what it means to solve problems and to reason (NCTM, 2000), as well as

developing new ways of making sense of what it means to engage in and teach mathe-

matics (Simon, 1994). Farmer et al. (2003) found that opportunities for discussion,

journaling, and reflective writing centered on mathematical ideas and issues of pedagogy,

allowed teachers to construct mathematical and professional meanings for themselves.

Following that, we chose the portfolio to be one of the sources of information. A Portfolio

is a record of one’s process of learning. It is a purposeful collection of examples of work

collected over a period of time. A Portfolio includes what one has learned, how one thinks,

how one creates and analyzes things, and so forth. As such, it enables the evaluation of the

learner’s progress and performance (Arter & Spandel, 1991). In using a portfolio as an

assessment tool, the focus is on the learner’s successes rather than his failures. Thus a

portfolio has the potential to motivate students and to advance their ability to reflect on the

processes they are going through and to carry out self-evaluation. Various studies on the

use of portfolios indicate that they can usefully serve purposes of assessment of profes-

sional competence and development (Campbell, Cignetti, Melenyzer, Nettles, & Wyman,

1997).

In our larger study we used three evaluative tools: a portfolio, two open questionnaires,

and transcripts of class discussions. In the present study we present partial data from the

first two tools.

The study

In this section we present relevant information regarding the study participants, the course

in which the study took place and the means of data collection and analysis.


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The study participants

25 PTs (8 males and 17 females) from an academic college, in their third year of studying

toward a B.A. degree in mathematics education, participated in the course. The students

represented all talent levels. The students studied toward being teachers of mathematics

and computer science, or teachers of mathematics and physics in high school. 40% of the

students are Israeli born 20% of them are immigrants from the former Soviet Union, and

40% are Arab-Israeli citizens. It should be mentioned that all students from the former

Soviet Union have been in Israel for more than 10 years, and one cannot recognize major

differences between the three student groups regarding either their social interactions with

other classmates or their learning ambitions and abilities. In the context of this paper we

will not discuss differences between the three groups with regards to change in their views

regarding the image of the ‘good teacher.’

The course

The course, in which the research was carried out, is a two-semester course and is part of

the students’ teacher training in mathematics education. This course is the first mathe-

matical method course the PTs attend and takes place in the third year of their studies (out

of four). In the first semester, learning and teaching methods are demonstrated via

geometrical topics, and in the second semester via algebraic topics. The research was

carried out in the first semester.

In their first year, the PTs studied basic mathematics courses such as calculus, algebra, and

geometry. In their second year the PTs participated in a general methods’ course in which

they learned topics such as: how to design a lesson, how to open and summarize a lesson, how

to ask questions and guide a class discussion, and so forth. This course was general and did

not relate to any specific discipline, and the PTs who participated in the course came from

various disciplines. In the third year the only didactics course that was taught is the afore-

mentioned course, and all the other courses that were taught in this year were advanced

mathematics courses. The course in which the research took place was focused on didactical

methods implemented in teaching middle school geometry and algebra.

In the fourth year the PTs have to participate in an additional methods course which

focuses on didactical methods implemented in teaching and learning high-school Calculus

and Analysis. In addition, they have to take a two-semester course which focuses on the

integration of computer software in mathematics classes.

The design of the discussed mathematical methods’ course, in which this research took

place, was based on the guidelines provided by the Israeli Ministry of Education as regards

the goals of teaching mathematics in middle and high school. These goals deal with issues

that concern the development of students’ understandings regarding mathematical pro-

cesses and students’ mathematical–logical thinking. In addition students should be

provided with opportunities to raise assumptions, arrive at conclusions and generalizations,

and so forth. These goals do not include any reference to teacher education.

The course was designed according to the rationale, principles, and recommendations of

the NCTM Standards (1989, 2000). In the framework of the course the PTs are engaged in

worthwhile mathematical tasks followed by discussions, in which they discuss pedagogical

and mathematical issues, as well as examine their beliefs about teaching and learning

mathematics. The tasks are deliberately designed to help the PTs rethink their conceptions

of what mathematics is and how it is learned.


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The learning environment

As previously mentioned, the course focuses on didactical methods implemented in

teaching middle school. In designing the learning environment we took into consideration

the fact that the context in which teaching occurs, affects what is being taught and learned

in the classroom (Wilson & Cooney, 2002). We also took into account that ‘‘Perhaps the

greatest obstacle for teachers is a lack of personal familiarity with mathematical problem-

solving and sense-making—processes that most have never experienced themselves, as

students or teachers’’ (Lloyed, 2002, p. 149). Hence we chose CPBL to be one of the main

methods of learning in that semester.

The PTs could choose to work individually or in pairs. All the students chose to work in

pairs and there was one group of three students. The project was geometry-oriented, and

the students used dynamic geometrical software in the various stages of the project. The

project included the following phases:

1. Solving a given geometrical problem that served as a starting point for the project;

2. Using the ‘what if not?’ strategy (Brown & Walter, 1993), to create a range of new

problem situations on the basis of the given problem;

3. Choosing one of the new problem situations and posing as many relevant questions as

possible;

4. Concentrating on one of the posed questions and looking for suitable strategies for

solving it;

5. Raising assumptions and verifying/refuting them;

6. Generalizing the findings and drawing conclusions;

7. Repeating stages 3–6, up to the point at which the student decided that the project had

been exhausted.

The PTs were exposed to three modes of interactions: (i) Team work; (ii) Whole class;

and (iii) Student–teacher interaction.

The team work interaction included the cooperative work of two PTs to accomplish the

project and the preparations for presenting the work during the class sessions. The whole

class interaction included presentations of the projects and monitoring a classroom dis-

cussion afterward. Throughout the discussions the PTs talked about their difficulties and

asked for their classmates’ advice. The classroom discussions that followed the presen-

tations provided the PTs with a supportive environment while working on the project and

actually served as ‘‘scaffolds’’ (Krajcik et al., 1998). During the class discussions we

helped the presenters to focus on their major ideas, monitor the discussion, and understand

the essence of their classmates’ suggestions and questions. The class discussions also

provided opportunities for considering instructional and pedagogical issues like—how

students build their knowledge, how to base instruction on previous knowledge, how to

uproot misconceptions, in what ways the learning environment influences and supports (or

prohibits) learning, what skills students should possess, and so forth.

All of the PTs played a double role in these presentations and discussions: they were

presenters on one hand, and an active audience on the other. All class sessions were

videotaped, and the transcripts were used as an evaluative tool. The student–teacher

interaction included a continuous exchange of e-mails. The PTs were asked to handle a

portfolio on a regular basis. As mentioned, the portfolio served as one of the evaluative

tools for assessing the change in views of the PTs regarding their image of the ‘good

teacher.’ Each PT was asked to send us a copy of her portfolio by e-mail at least once a

week. Each e-mail was answered immediately to maintain motivation. Our feedback was


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not judgmental. We mainly asked for further clarifications and elaborations and suggested

fresh viewpoints.

Data collection and analysis

Three main sources of data informed the study: The PT portfolios, our observations during

the presentations and the class discussions that followed them, and two questionnaires

(before and after the implementation of the CPBL approach).

The portfolio

When planning the research we had a dilemma that concerned the items that should be

included in the portfolio. We wondered whether we should provide the PTs with a detailed

list of instructions regarding what they ought to reflect on or merely give them the

autonomy to refer to anything they wished. Since it was our first experience with portfolio

as an evaluative tool we chose to provide the PT the autonomy of writing, hoping to learn

about their main focal points of interest. We only provided some general instructions, and

asked the PT to describe aspects that concerned their ways of thinking and working, their

progress, difficulties, experiments, conjectures and doubts, and to reflect on them. In

addition, they were asked to refer to the various interactions they were involved in and to

the way those interactions influenced their own work.

The questionnaires

To examine the change in the PTs’ beliefs regarding teaching, the nature of mathematics,

and the practice of mathematics we asked the PTs to refer to a three-question question-

naire. This questionnaire was given before and after the implementation of the CPBL

approach. The questions were: What is a good mathematics teacher? How do you perceive

mathematics? How do you perceive the work of the mathematician?

The PTs were asked to write their names on the questionnaires; there were no time and

space limitations.

In Table 1 we describe the three main sources of data and the focal points of each one as

a source of information.

In taking an interpretive stance toward research, our goal in analyzing the data was to

understand and assess the PTs’ learning and growth both from their own perspectives (as

was expressed by their reflective writing in the portfolio and from the questionnaires) and

from our own perspective (through the analysis of the class session transcripts). Each

source had its unique contribution to the assessment of the PTs’ change in views regarding

the image of the ‘good teacher,’ and actually these sources mutually stimulated each other.

The integrated tool enabled us to examine each PT’s current views as well as the route that

led him/her to that state.

Analysis was ongoing and continually informed the data-gathering process. When

the data-collection phase was completed, we followed the process of analytic induction

(Goetz & LeCompte, 1984), reviewing the entire corpus of data to identify themes and

patterns and generate initial assertions regarding the focal points of interest mentioned in

Table 1.


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Since the results are broad and comprehensive, within the scope of this paper we limit

ourselves to focusing on the PTs’ change in views regarding the image of the ‘good

teacher’ as came to fruition in the students’ responses to the question: ‘‘what is a good

mathematics teacher?’’, which was one of the questions included in the questionnaires

(before and after the implementation of the CPBL approach). In addition, we examined the

students’ portfolios looking for additional support of the observed change.

To examine the PTs’ change in views of the ‘good teacher’ we followed three

consecutive phases:

Firstly, we scanned all the students’ utterances from the questionnaires throughout the

lenses that concerned their perception of the image of a ‘good teacher.’

When scanning the students’ utterances, we noticed that they could be classified

according to three main aspects: affective, cognitive, and didactical, as shall be explained

broadly in the results section.

Secondly, we analyzed the content of each utterance and concluded that the utterances

concerning the didactical aspect can be further classified into ‘operatively oriented’

(an utterance which implies or includes practical operations) and ‘declarative’ (an utterance

which is expressed as a general saying without recommendations for implementations). It

should be stressed that the students’ utterances were translated from Hebrew and the

distinction between affective, cognitive, and didactical as well as declarative and opera-

tively oriented utterances might appear slightly different in English. To validate our

classification into the three aspects (affective, cognitive, and didactical) and the distinction

between declarative and operatively oriented utterances, we gave the whole corpus of

utterances to 10 experienced mathematics teachers and asked them to classify the utterances

according to the three aspects and then into declarative and operatively oriented ones. In a

majority of the cases the teachers’ classifications resembled ours. There were a negligible

number of disagreements. In these cases we discussed the disagreements with the validating

teachers until we reached an agreement.

Thirdly, we counted the number of utterances that relate to each of these aspects and

their classifications before and after the implementation of the CPBL approach. The reason

which underlies this counting was to learn about the PTs’ priorities. We assume that an

utterance with higher frequency is probably considered to be more important or relevant

for them.

Table 1 The research data sources

Data source Focus

The PTs’ portfolios Development of mathematics knowledge, changes intheir perceptions as regards characteristics of learning/teaching, issues they chose to focus on

The PTs’ written responses to the three openquestions (before and after the implementationof the CPBL approach)

How they perceive mathematics and the work of themathematician. How they perceive the image of thegood mathematics teacher

Transcripts of videotapes of the presentations andthe class discussions that followed thema

The ways they chose to present their difficulties to theirclassmates, the types of conflicts they shared withtheir classmates and the way they reacted to theirclassmates’ suggestions, the structure they chose fortheir presentation, the methods used to lead a classdiscussion

a Analysis of the transcripts is not included in the present paper


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Fourthly, for each of the aforementioned aspects, we looked for relevant evidence in the

students’ portfolios to support our conclusions.

Finally, from the above analyzed data and their integration we could gain a sense of the

PTs’ perceived image of the ‘good teacher.’ We use the term ‘perceived image’ since it

reflects the students’ said beliefs regarding various aspects that concern the image of the

‘good teacher,’ and does not necessarily reflects their practical abilities and skills

(Philippou & Christou, 2002; Wilson & Cooney, 2002).

Results

The PTs’ utterances as expressed in the study came from a handful of the study subjects.

As was previously mentioned, the students’ utterances were classified according to three

main aspects: affective, cognitive, and didactical. By ‘affective’ we mean all the utterances

relating to various emotions accompanying the teacher’s work: Emotions concerning the

teacher’s personality (e.g. patient) and emotions concerning the interactions between a

teacher and her students (e.g. reducing mathematics anxiety). By ‘cognitive’ we mean all

the utterances referring to the knowledge about the subject matter (what Shulman (1987)

termed as ‘content knowledge’) as well as mathematical skills and reasoning abilities of the

‘good teacher.’ By ‘didactical’ we mean all the utterances referring to knowledge about

teaching and knowledge about class management (what Shulman (1987) termed as general

pedagogical knowledge, pedagogical content knowledge, knowledge of learners, and

knowledge of educational ends).

Examination of the PTs’ utterances referring to the didactical aspect revealed a further

classification. We noticed that there were two types of utterances which we termed as

‘declarative’ and ‘operatively oriented.’ A ‘declarative’ (D) utterance is an utterance expressed

as a general saying without recommendations for implementations; an ‘operatively’ oriented’

(OO) utterance is an utterance implying or including practical operations.

For example, an utterance such as ‘‘knows how to explain well’’ was categorized as

‘didactical’, since it refers to knowledge about teaching. However, since there is no

indication of any specific method for its implementation, it was classified as ‘declarative.’

For additional examples see Tables 5 and 6.

In the following section we describe the two images—the perceived image of the ‘good

teacher’ before the implementation of the CPBL approach and afterward, and explain how

we used the comparison between them for characterizing the change in perception of the

image of the ‘good teacher.’ Then we will refer to the contribution of the CPBL approach

to this change.

The PTs’ perceived image of the ‘good teacher’

In this section we describe the PTs’ perceived image of the ‘good teacher,’ according to

each of the three mentioned aspects (affective, cognitive, and didactical). Then we describe

an ‘integrated image,’ comprising all three aspects.

To compare the perceived image of the ‘good teacher’ before and after the imple-

mentation of the CPBL approach, we relate to the amount as well as the content of

utterances uttered regarding each aspect, as was obtained from the answers to the question:

what is a ‘good mathematics teacher’? We believe the amount (number) of a certain type of

an utterance reflects on the importance the PTs attribute to it.


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Table 2 demonstrates the number of the PTs’ responses according to the three aspects:

affective, cognitive, and didactical. In addition, the didactical aspect appears in further

classification: declarative and operatively oriented, before and after the implementation of

the CPBL approach.

From Table 2 one can notice that the total number (before and after) of cognitive

utterances is rather low (100) relative to affective (206) and didactical utterances (300). In

addition, Table 2 shows that utterances relating to the affective aspect were reduced to half

after the implementation of the CPBL approach (from 138 to 68), while utterances relating

to the cognitive aspect doubled their number (from 33 to 67), and utterances relating to the

didactical aspect more than tripled their number (from 66 to 234). Before the implemen-

tation of the CPBL approach and according to their frequency (138), affective attributes

were of major importance in designing the students’ perception of the image of the ‘good

teacher.’ This result is consistent with previous studies (e.g. Kobobi, 1992; Fridman &

Corngold, 1993) which found that the most important attribute of the ‘good teacher,’ as

perceived by students, is her personality. It is also consistent with Reichel and Arnon

(2005) who found that PTs’ who were asked to refer to the image of the ideal teacher

mainly referred to the personality of the teacher.

After experiencing the CPBL approach and according to their frequency (234), didac-

tical attributes gained major importance in designing the PTs’ perception of the image of

the ‘good teacher’. The results are consistent with previous studies (Milgram, 1988;

Goldberg, 1994) which found that the most important attribute of the ‘good teacher,’ as

perceived by students, is her knowledge. It should be noted that in these studies the term

‘teachers’ knowledge’ refers to what we termed cognitive as well as didactical knowledge

without distinguishing between them.

The most apparent change is in the number of didactical utterances under the category

‘operatively oriented’ (from 28 to 134).

With regard to the content of the students’ utterances, we separate our discussion

according to the three previously mentioned aspects. The following sections will discuss

each aspect broadly.

The PTs’ perceived image of the ‘good teacher’—affective aspect

In Table 3 we present the most frequent1 utterances that relate to the affective aspect of the

image of the ‘good teacher’ before the implementation of the CPBL approach and

afterward. The percentage in the following Tables (3–5) represents the ratio between the

number of PTs who explicitly wrote the specific utterances and the number of the study

Table 2 The number of utter-ances according [to] the threeaspects

Aspect Before After Total

Affective 138 68 206

Cognitive 33 67 100

Didactical:

Declarative 38 100 138

Operatively oriented 28 134 162

Total 66 234 300

1 Utterances with frequency less than 5 were not included.


123

subjects. For example, the utterance: ‘‘be a human being’’ was uttered by 18 students out of

25 (72%).

Before

The students’ utterances that relate to affective aspects can be sorted out into three cate-

gories which mutually affect each other: the ‘good teacher’s’ personality, the ‘good

teacher’s attitudes toward her student and the ‘good teacher’s attitudes toward her

profession.

Regarding the teacher’s personality, for the majority of the students (around 70%) it was

very important that the ‘good teacher’ have certain personal qualities such as patience and

humaneness. These two qualities might influence her attitude toward her students.

Observation of Table 3 shows that although the PTs pointed only two personal qualities of

the ‘good teacher,’ the high percentage can imply their importance.

As to the attitude toward her students, the PTs related to the importance of changing

students’ attitude toward mathematics. This change can be caused either by reducing thefear of mathematics or by creating positive atmosphere in the mathematics classroom. As

to the means for reducing fear of mathematics, the ‘good teacher’ should encouragestudents even when they fail and express warmth toward her students. For the creation of a

positive atmosphere in the mathematics classroom, the ‘good teacher’ should be sensitiveto the students’ needs and inspire the students to become interested in mathematics evenafter school hours.

With relation to the ‘good teacher’s’ attitude toward her profession the PTs indicated

the attribute loves to teach.

Table 3 The PTs’ utterances and frequencies concerning affective aspects before experiencing the CPBLapproach and afterward

Frequency (%) The PTs’ utterances

18 (72%)

17 (68%)

16 (64%)

15 (60%)

15 (60%)

13 (52%)

12 (48%)

10 (40%)

9 (36%)

7 (28%)

6 (24%)

1. Is a human being

2. Is patient

3. Changes students’ attitude toward math

4. Reduces fears of math

5. Encourages students even when they fail

6. Is sensitive to the students’ needs

7. Creates a positive atmosphere in the classroom

8. Expresses warmth toward her students

9. Loves to teach

10. Is pleased with her students’ success

11. Inspires the students to become interestedin mathematics even after school hours

Before Affective

21 (84%)

20 (80%)

16 (64%)

11 (44%)

1. Provides the impression that mathematics is not boring

2. Enables the students to experience the beauty of mathematics

3. Is pleased with her students’ success

4. Loves to teach

After


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From the PTs’ utterances it can be inferred that the PTs perceived the image of the

‘good teacher’ as a ‘motherly’ type of personality. Before the implementation of the CPBL

approach and concerning affective aspects, the ‘good teacher’ was perceived by the PTs as

a protective figure who is sensitive to her students’ needs, especially with regard to neg-

ative emotions and attitudes such as mathematics anxiety. The PTs’ affective utterances

imply what status mathematics, as a discipline, has among pupils—many PTs reported they

experienced anxiety during their mathematics studies both in school and college. Conse-

quently, they believe the mathematics teacher has an important role in supporting her

students to reduce their anxiety. These observations were supported by some of the PTs’

narratives in their portfolios. For example,

During my school studies I had two types of math teachers—those who were

empathetic and caring and those who were mainly concerned with teaching the

subject matter. I preferred the first type. It was much more enjoyable to learn math in

a supportive environment. However, in the class of the second type of teacher, I often

felt insecure … I was afraid to fail, because I could not anticipate the teacher’s

reaction.

After

Analysis of the PTs’ affective utterances after the implementation of the CPBL approach

shows that they still relate to the good teacher’s attitude toward her students and toward

her profession. It should be mentioned that there is no reference to specific attributes

relating to the personality of the ‘good teacher.’ With regard to the ‘good teacher’s’

attitudes toward her students, the majority of the PTs thought it important that she providethe impression that mathematics is not boring and that she enable the students toexperience the beauty of mathematics and be pleased with her students’ success. With

relation to the good teacher’s attitude toward her profession, they indicated that she should

love to teach, which is quite similar in its percentage to the identical utterance made before

working on the project.

From the examination of the utterances relating to affective aspects that were expressed

after experiencing the CPBL approach, it can be concluded that there was a shift in the

PTs’ perception of the ‘good mathematics teacher’ from a ‘motherly’ character whose

main concern is to help students to overcome negative feelings such as mathematical

anxiety, toward a figure that encourages the students to see the beauty of mathematics.

Namely, a shift can be observed from a protective image, which induces confidence,

toward an educational image that encourages the students to see the positive sides of

mathematics. Nevertheless the ‘good teacher’ still has to convey the impression that

mathematics is not boring.

The utterances that were expressed only after working on the project: Providesthe impression that mathematics is not boring and enables the students to experience thebeauty of mathematics, reflect the impact of experiencing the CPBL approach on the

perceived image of the ‘good teacher’ with relation to affective aspects. Participation in

the project exposed the students to the nature of inquiry-based learning in which they were

engaged with non-routine work—different from what they were used to. Consequently,

they had a chance to develop new perspectives regarding the ‘spirit’ of mathematics,

especially gaining a viewpoint which enables them to see how beautiful mathematics can

be. As was expressed in some of the PTs’ portfolios:


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Work on the project enabled me to experience success. I believe that students who

experience success will get over math anxiety and develop a positive attitude toward

math’’.

The PTs’ perceived image of the ‘good teacher’—cognitive aspect

Table 4 presents the most frequent utterances relating to the cognitive aspect of the ‘good

teacher’s’ image before the implementation of the CPBL approach and afterward. As was

previously mentioned, all the utterances that related to issues of reasoning, mathematical

abilities, and knowledge of the subject matter of the ‘good teacher’ were classified as

cognitive aspects.

Before

The PTs’ utterances that were classified as ‘cognitive’ relate to the cognitive abilities of the

‘good teacher’ and the mathematical ‘behavior’ she has to exhibit in her class. With regard

to her cognitive abilities in mathematics, the majority of the PTs had in mind an image of a

person who is able to solve every mathematical problem. It should be noted that this

utterance implicitly refers to knowledge of the subject matter, as was frequently expressed

in the PTs’ portfolios: ‘‘Obviously one who chooses to be a mathematics teacher must knowmathematics well enough to teach it … It is a necessary but not sufficient condition forbeing a good mathematics teacher’’. As to the mathematical ‘behavior’ she has to dem-

onstrate in class, they perceived the ‘good teacher’ as a person that learns from hermathematical mistakes and avoids repeating them. These utterances imply a perception

according to which math lessons should include issues and problems which are well

prepared down to the smallest details; there is no room for ‘surprises.’ That is, all possible

questions that might be raised in class are known to the teacher in advance, and she has all

the answers ready for them. In case she makes mistakes—she has the mathematical skills

to learn from them and avoids repeating them in the future as was reflected in some of PTs’

portfolios before they had to present their project in class, in which they had to function as

teachers:

…The part of the project I was mostly afraid of was the presentation … I knew I had

to be very well prepared. That is, I had to predict the possible questions of the

Table 4 The PTs’ utterances and frequencies concerning cognitive aspects before experiencing the CPBLapproach and afterward

Frequency (%) The PTs’ utterances

21 (84%)

12 (48%)

1. Can solve every mathematical problem

2. Learns from her mathematical mistakes and avoids repeating them

Before Cognitive

20 (80%)

18 (72%)

16 (64%)

13 (52%)

1. Tries to discover new mathematics regularities

2. Does not have to know the answer to all the questions

3. Is curious about new mathematical ideas

4. Teaches and learns at the same time

After


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students and have answers ready for them. However, I had no idea what questions

they might ask. I was afraid of being in a situation in which I would not be able to

provide clear answers…

After

The PTs’ utterances after experiencing the CPBL approach relate also to both the cognitive

abilities of the ‘good teacher’ and to the mathematical ‘behavior’ she has to exhibit in

class. With regard to her cognitive abilities, the PTs related to attributes they encountered

during the work on the project. Namely, there was a shift from a person who organizes her

lessons down to the smallest detail to avoid ‘surprises’—toward an open-minded person,

who is curious and not afraid of new experiences. That is, the good teacher should try todiscover new mathematical regularities.

Another shift is from a person who is able to solve every mathematical problem toward

a person who does not have to know the answer to all the questions. With regard to the

mathematical ‘behavior’ she has to exhibit in class, the PTs said that the ‘good teacher’

should be a person who learns and teaches at the same time. The utterances relating to

cognitive aspects described after experiencing the CPBL are consistent with the course of

learning via that approach. It can be concluded that the PTs’ perception of the ‘good

teacher’ regarding the required knowledge of the subject matter was influenced by the new

learning/teaching approach. While the PTs worked on their projects they often asked for

our advice. Consequently, they had the opportunity to observe us, as their teachers, dealing

with problems which we had not previously encountered. In such a learning/teaching

approach there are situations in which the teacher teaches and learns at the same time, and

does not necessarily know instantly the answers to all questions raised. Actually, the

teacher is no longer the source of knowledge; rather she is a learning instructor.

As was said in part of the PTs’ portfolios:

one of the main things I learned from the project is that it is essential for teachers to

be open-minded…teachers should be aware of the fact that learning is a dynamic

process and one cannot be expected to know all the possible answers.

Observing you [the researchers] during the class presentations, I noticed that when

you face situations for which you had no ready answers for us, you seemed to be

challenged and not embarrassed as I thought you might be. You encouraged us to

find the solutions together. It made me realize that teachers can teach and learn at the

same time.

The PTs’ perceived image of the ‘good teacher’—didactical aspect

Table 5 presents the most frequent utterances that relate to the didactical aspect of the

image of the ‘good teacher’ before the implementation of the CPBL approach and after-

ward. With regard to this aspect, it can be seen that the total number of didactical

utterances, either declarative or operatively oriented after the implementation of the CPBL

approach, more than tripled their number. The same is also true for the number of distinct

utterances (5 before vs. 14 after). The most apparent change was in the amount of

declarative type of utterances.


123

Before

The PTs’ utterances—either declarative or operatively oriented regarding didactical

perspectives before the implementation of the CPBL approach—related to the teacher as a

mediator between her class and the subject matter and general means for achieving edu-

cational goals. As to the former, the PTs indicated various ways in which the ‘good

teacher’ should mediate between her students and the subject matter. The PTs pointed out a

general purpose: Teaches in an interesting manner and various skills for fulfilling this

purpose such as knows to explain well, Tries various teaching methods to find the optimalone, Uses many examples and games, and adjusts the learning materials to the students’level. These utterances concerning the didactical skills which refer to the ‘good teacher’s

functioning before experiencing the CPBL approach are general in their essence. Namely,

they do not mention any specific learning/teaching method; they do not specify any kind of

games and examples. The PTs suggested that the teacher should try various teachingmethods to find the optimal one but do not really indicate any specific method. In addition

they believe that the ‘good teacher’ should know to explain well, but they do not mention

how or what exactly they mean by that. From the PTs’ utterances we can conclude that at

this stage their didactical knowledge is rather basic and superficial, as could be expected

considering the fact they had just begun their first methods course.

Table 5 The PTs’ utterances and frequencies concerning didactical aspects before experiencing the CPBLapproach and afterward

Frequency(%)

The PTs’ utterances

22 (88%)

16 (64%)

1. Knows how to explain well

2. Teaches in an interesting manner

D Before Didactical

11 (44%)

9 (36%)

8 (32%)

1. Tries various teaching methods to find the optimal one

2. Uses many examples and games

3. Adjusts the learning materials to the students’ level

OO

21 (84%)

19 (76%)

17 (68%)

17 (68%)

15 (60%)

11 (44%)

1. Encourages her students to ask questions

2. Develops creative thinking

3. Explains each topic clearly and simply

4. Improves mathematical thinking

5. Teaches in an engaging manner

6. Manages to understand the students’ difficulties

D After

21 (84%)

20 (80%)

19 (76%)

17 (68%)

16 (64%)

14 (64%)

14 (64%)

13 (52%)

1. Frequently uses class discussions

2. Provides her students the opportunity to explore mathematics anddoes not give them the solutions right away

3. Motivates her students to experience the process that amathematician goes through while looking for mathematic regularity

4. Inspires the students to look for a deep understanding and not justsuccess

5. Enables all students to take an active part in the mathematicslessons

6. Uses many examples in class for clarification

7. Lets the students think first and then establishes the subject matter

8. Integrates interesting activities in his teaching

OO


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The following utterances: Tries various teaching methods in order to find the optimalone, Uses many examples and games, adjusts the learning materials to the students’ levelrefer to general means for achieving educational goals, without any indication of the goals

themselves. Analysis of the didactical utterances—either declarative or operatively

oriented before the implementation of the CPBL approach—reveals that before the

implementation of the CPBL the ‘good teacher’ is perceived by the PTs as a kind of ‘ideal

person.’ This person knows how to explain well and teaches in an interesting manner.

However, specific educational goals are not mentioned. The image of the ‘good teacher’

appears to be one who operates in a ‘vacuum.’ She knows how to teach, but does not act

according to any educational goals.

After

Observation of the PTs’ utterances, either declarative or operatively oriented, after they

experienced the CPBL approach, reveals a different picture. Starting with five distinct

utterances (either declarative or operatively oriented) before the implementation of the

CPBL, they reached fourteen distinct utterances afterward.

Analysis of the didactical utterances after experiencing the CPBL approach reveals that

they related to three main categories: the teacher as a mediator between her class and the

subject matter, educational goals and means for achieving them, and teacher–student

interaction. As to the category: the teacher as a mediator between her class and the subject

matter, there were utterances such as: Explains each topic clearly and simply, teaches in anengaging manner. With regard to the category: educational goals and means for achieving

them, there were utterances such as: Develops creative thinking, Improves the mathe-matical thinking, which relate to educational goals while the operatively oriented

utterances indicate means for achieving educational goals. As to the category: the teacher–

student interaction, there were utterances such as: Manages to understand the students’difficulties, Encourages her students to ask questions.

Table 6 demonstrates the distribution of the PTs’ utterances after the implementation of

the CPBL approach according to three categories: the teacher’s mediation, the impact

teaching has on students, and the teacher–student interaction.

The indication of ‘D’ points to a declarative utterance, and its index points to the

utterance’s ordered number as appears in Table 5. Similarly, ‘OO’ designates an operative

oriented utterance.

Under the category: the impact teaching has on students, the PTs’ declarative utter-

ances refer to educational goals such as the development of creative thinking and the

improvement of mathematical thinking. Means for achieving these educational goals

appear under the same category as operatively oriented utterances (OO1–4, 7). Obser-

vation of OO1–4, 7 as well as D2, 4 reveals that most of them are influenced by the

constituents of the CPBL approach. Namely, utterances OO2 and OO7 and D1 refer to

inquiry-based activity, and utterance OO1 refers to the use of class discussions as was

done during their participation in the CPBL approach. This was reflected in some of the

PTs’ portfolios:

One of the most beneficial parts of the project was the class discussion. As a future

teacher I believe that class discussion has to be an integral part of every math lesson.

As a pupil I don’t remember class discussions in which we were able to discuss our

mathematical ideas…I found the discussions very helpful, since together we


123

Tab

le6

Th

ep

rosp

ecti

ve

teac

her

s’d

idac

tica

lu

tter

ance

saf

ter

the

imp

lem

enta

tio

no

fth

eC

PB

Lap

pro

ach

The

teac

her

’sm

edia

tion

Educa

tional

goal

san

dm

eans

for

achie

vin

gth

emT

he

teac

her

–st

uden

tin

tera

ctio

n

Dec

lara

tive

D3.

Expla

ins

each

topic

clea

rly

and

sim

ply

(68

%)

D5.T

each

esin

anen

gag

ing

man

ner

(60

%)

D2

.D

evel

ops

crea

tive

thin

kin

g(7

6%

)

D4

.Im

pro

ves

mat

hem

atic

alth

ink

ing

(68

%)

D1

.E

nco

ura

ges

her

stud

ents

toas

kq

ues

tio

ns

(84

%)

D6.

Man

ages

tounder

stan

dth

est

uden

ts’

dif

ficu

ltie

s(4

4%

)

Op

erat

ivel

yo

rien

ted

OO

6.

Use

sm

any

exam

ple

sin

clas

sfo

rcl

arifi

cati

on

(64

%)

OO

8.

Inte

gra

tes

inte

rest

ing

acti

vit

ies

inher

teac

hin

g(5

2%

)

OO

1.

Fre

quen

tly

use

scl

ass

dis

cuss

ion

s(8

4%

)

OO

2.

Pro

vid

esh

erst

ud

ents

the

op

po

rtu

nit

yto

exp

lore

mat

hem

atic

san

dd

oes

no

tg

ive

them

the

solu

tio

ns

rig

ht

away

(80

%)

OO

3.

Mo

tivat

esh

erst

ud

ents

toex

per

ien

ceth

ep

roce

ssth

ata

mat

hem

atic

ian

go

esth

roug

hw

hil

elo

ok

ing

for

mat

hem

atic

reg

ula

rity

(76

%)

OO

4.

Insp

ires

the

stu

den

tsto

loo

kfo

ra

dee

pu

nd

erst

and

ing

and

no

tju

stsu

cces

s(6

8%

)

OO

7.

Let

sth

est

ud

ents

thin

kfi

rst

and

then

esta

bli

shes

the

sub

ject

mat

ter

(64

%)

OO

5.

En

able

sal

lst

ud

ents

tota

ke

anac

tiv

ep

art

inth

em

ath

emat

ics

less

on

s(6

4%

)


123

succeeded in proving things and learning a lot. It was nice to see how everyone

contributed a little bit and together we learned a great deal.

After the implementation of the CPBL the ‘good teacher’ has both educational goals and

the means for achieving educational goals. The main educational goals concern developing

students’ thinking, and directing them while looking for mathematical regularities. The

means include: providing the opportunity to explore and pose questions, and enabling the

students to discuss their problems and questions:

The first time I felt the meaning of ‘doing mathematics’ was while I worked on the

project. The fact that I could pose any problem and ask any question without

knowing whether they were valid or solvable developed my mathematical knowledge

and thinking. I believe inquiry activities are a good way to work with students, since

it encourages them to explore and discover, and consequently develop their skills.

The integrated PTs’ perceived image of the ‘good teacher’

Integrating the three aspects: affective, cognitive, and didactical, it is possible to draw a

complete image of the ‘good teacher,’ as was perceived by the PTs before and after their

experience in learning via CPBL.

Before

The ‘good teacher’ has a ‘motherly’ character, whose main concern is to reduce the fear of

mathematics and encourage the students especially when they fail. Namely, the teacher has

to confront negative feelings and attitudes students bring with them to the mathematics

classroom. This teacher is well prepared for her lessons. Her lessons are taught in a

traditional manner, in which all the questions are well organized and solved in advance.

The teacher is the only source of knowledge, which means that she has to know the

answers to all questions. When she is mistaken, she has to learn from the mistakes and

avoid repeating them.

As to didactical perspectives, the ‘good teacher’ knows how to explain well and teaches

in an interesting manner. She tries various teaching methods to find the optimal one and

adjusts the learning materials to the students’ level by means of using many examples and

games. However, the educational goals of this teacher were not explicitly expressed by the

PTs. From the utterances, it could only be inferred that the main educational goal is to

enhance mathematical understanding.

After

The main role of the ‘good teacher’ is no longer taking care of negative attitudes toward

mathematics, instead she takes the position of a professional advocator. That is, she

exposes her students to the positive aspects of mathematics. She is in a continual state of

learning. She does not have to know the answer to all questions raised in class, since there

are situations in which new directions of inquiry are raised. Instead, she encourages the

students to investigate, ask questions, look for understanding, and see the beauty of


123

mathematics. She has explicit educational goals, such as developing creative thinking,

mathematical thinking and understanding. The means aimed at achieving these goals are

explicitly expressed: use interesting inquiry activities and class discussions.

Thus, it can be concluded that the above-perceived images of the ‘good teacher’ before

and after experiencing the CPBL approach indicate a change in the PTs’ views of the ‘good

teacher’ in all three aspects.

Discussion

In this section we discuss the nature of change in viewing the image of the ‘good teacher’

by our PTs as was described in the results section and refer to the effect of the CPBL

approach on the observed change. The change in views has implications for the PTs’

change in beliefs regarding teaching and learning. Considering the PTs’ beliefs is signif-

icant, since what they believe is an important determiner of what and how the subject

matter is taught (Wilson & Cooney, 2002).

The nature of change in viewing the image of the ‘good teacher’

Examination of Tables 3–6 reveals that the change in the PTs’ views comes to fruition in

the modification in nature and scope of the utterances. Moreover, Tables 5 and 6, which

relate to the didactical aspect, reveal that there was a shift from declarative to operatively

oriented utterances. After experiencing the CPBL approach, the utterances became more

comprehensive on one hand and more detailed on the other. These utterances included a

larger range of attributes a ‘good teacher’ should have than there were at the beginning of

the semester.

The PTs’ utterances regarding the affective aspect before the engagement in the CPBL

are consistent with Reichel’s and Arnon’s (2005) findings according to which personality

characteristics of the ‘good teacher’ were of the main concern in their PTs’ views.

However, observation of Table 3 shows that the number of utterances after experiencing

the CPBL approach was reduced to half of the utterances before. A possible explanation for

this shift was found in the PTs’ portfolios from which we realized that they believe one can

see and enjoy the beauty of mathematics only when one overcomes negative feelings (for

example—math anxiety), toward mathematics. Participation in the CPBL made them

realize that they could cope with activities they had never done before such as uncovering

mathematical regularities. As a result, the anxiety was replaced by feelings such as curi-

osity and appreciation of the beauty of mathematics.

Observation of the PTs’ utterances can tell about their priorities—what is more

important and what is less. As for the affective aspect, they did not mention attributes such

as charisma, a figure to imitate, morality, a person who has the ability to impose discipline,

and so forth. Examining the perspectives they did not point out one can learn about the way

the PTs, as a mirror of society, perceive the image of teachers in recent years. The

perceived image of the ‘good teacher’ with regard to the affective aspect reflects a

supportive character rather than an authoritative one.

The number of utterances relating to the cognitive aspect doubled after the imple-

mentation of the CPBL approach and shifted from a person who has to know the answer to

all questions raised in class to a person who is curious about discovering new mathematical

ideas and regularities. This change in the PTs’ views impies a teacher image which is


123

consistent with the recommendations of the NCTM standards (2000), according to which

teachers should be able to lead mathematical explorations and allow their own students to

construct mathematics knowledge. Inquiry-based lessons invite situations in which teacher

might not have the answer to all questions raised in class. Thus only if the teacher does not

feel threatened by being unable to answer the students’ questions right away, will she be

able to consider teaching in such an innovative setting. With regard to the cognitive aspect,

the PTs did not refer to aspects such as developing the ability to reflect on personal modes

of thinking or on processes involved in problem solving. It should be noted that although

during the whole period of working on the project, we emphasized the importance of

reflecting on the various modes of thought, the PTs did not relate explicitly to this issue.

Reflecting on processes was a new experience for the students, and it appears that one

semester is too short a period for internalizing the importance and benefits of this activity.

The most evident change occurred regarding the didactical aspect. Analysis of the

utterances in Table 5 reveals an interesting phenomenon. Most of the utterances before the

implementation of the CPBL approach involve the teacher’s centrality. This can be con-

cluded from the fact that in seven out of nine distinct utterances the word ‘student’ is not

mentioned. Namely, the PTs merely considered issues that related to the teacher’s role,

even though this teacher was described as a ‘vague’ figure, with undefined educational

goals.

However, seven out of fourteen utterances after the implementation of the CPBL

approach (D1, D6, OO2–5, OO7) included the word ‘students.’ The PTs began to inter-

nalize the fact that teaching is not simply a teacher-centered activity but is affected by the

presence of students. That is, teachers should adjust their teaching according to ‘rever-

beration’ coming from the class (for example, OO5–7 in Table 5). The description of such

a teacher specifies educational goals (such as D2, D4) and a detailed list of means aimed at

achieving them (such as OO1–3). Yet, the PTs did not discuss any issue concerning

curriculum knowledge, knowledge of learners, knowledge of educational contexts, and

knowledge of educational ends (Shulman, 1987). With regard to the didactical aspect the

PTs did not relate to perspectives from the teacher’s point of view such as the establish-

ment of a ‘didactic contract’ (Balacheff, 1990) between teacher and students which include

agreeable behaviors in the mathematics classroom, or to socio-mathematical norms such

as: what counts as mathematically efficient, mathematically sophisticated, mathematically

elegant, acceptable mathematical explanation and justification (Yackel & Cobb, 1996).

Although life experiences are a major contributor to the formation of beliefs

(Richardson, 1996), it is interesting to note that the PTs’ 12 years as learners of mathe-

matics in school provided them with images of what it means to teach mathematics rather

than learn it. The ability to refer to the above-disregarded issues necessitates the consid-

eration of both the student and teacher perspectives. This finding demonstrates the fact that

the PTs have not yet developed a wide perspective as regards constituents of the teacher’s

work.

Nevertheless, the changing nature of statements made by the participants indicated that

they had gained insights into the complexity of teaching even before beginning their work

inside high school classrooms.

In addition, referring to Reichel’s and Arnon’s (2005) findings regarding the differences

between pre-service and in-services perspectives, it can be said that our PTs underwent a

professional development process. Namely, at the beginning, the PTs perceived the cate-

gory of personal attributes as the most important aspect of the ‘good teacher.’ By the end,

they perceived the image of the ‘good teacher’ to be more like the in-service teachers in

Reichel’s and Arnon’s (ibid) study. It might be said that professional knowledge became


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more meaningful than personal attributes of the ‘good teacher.’ This finding is also

consistent with the perceptions of the teachers described in Wilson et al. (2005).

The effect of the CPBL approach

Changing one’s beliefs is not easy (Chapman, 2002). It occurs only under specific con-

ditions in which the individual is faced with new information and experiences (e.g. the

CPBL approach) that differ from the existing beliefs. Teachers’ beliefs about the way in

which students engage in mathematical activity and learn are critical factors in their

willingness and ability to design and implement inquiry-based instruction (Lloyd, 2002). A

teacher who perceives mathematical knowledge as a human empiricist creation will invest

more time and resources in helping students to make conjectures, test hypotheses, and

provide reasons through their individual involvement (Philippou & Christou, 2002).

In applying the CPBL approach we intended to engage our PTs in an innovative

teaching and learning environment which would enable them to explore important and

meaningful questions through a process of investigation and collaboration (Krajcik et al.,

1999). We believe such an experience would provide them with a sense of personal

contribution to the process of learning, increase their motivation, increase their ability to

share ideas, develop their ability to collect and present data, and so forth (Krajcik et al.,

1998). In addition, we hoped that the PTs’ experience with this approach would improve

their mathematical knowledge, and expose them to the benefits of this method as a means

of teaching and learning mathematics which would consequently change their beliefs.

Analysis of the data obtained from the questionnaires and the written portfolios reveals

that the CPBL approach had a meaningful effect on the change in views of the PTs with

respect to the three aspects—affective, cognitive, and didactical.

Regarding the cognitive aspect, the change that was expressed by shifting from a

teacher who does not have to know the answers for all questions raised in class, to one who

is curious about discovering new mathematical ideas and regularities, can be attributed to

participation in the CPBL approach. During the work on the project, the nature of the

learning environment invited situations in which questions from the class were addressed

to us, and the PTs had the opportunity to observe the way we coped with and handled these

situations. We conveyed the message that such situations are challenging for the teachers

as well as for the students, and consequently motivated further learning.

As for the didactical aspect, observing Table 5 demonstrates a list of utterances refer-

ring to constituents of the CPBL approach, as was described in the theoretical background

section. For example: Encourages her students to ask questions, provides her students the

opportunity to explore mathematics, and does not give them the solutions right away,

develops creative thinking, integrates interesting activities in his teaching, frequently uses

class discussions, and so forth.

Regarding the affective aspect, we noticed a shift from a ‘motherly’ type of personality

whose main concern is to minimize situations in which mathematical anxiety might be

raised, toward a personality who is responsible for exposing the students to the beauty of

mathematics. At first glance, this change might appear negative since it is valuable for

teachers to provide their students with moral support. However, the fact that the teacher’s

protective personality dominated the PTs’ image as regards the ‘good teacher’ before they

began the course, implies that the PTs took the pupils’ point of view. Reducing the

importance of being a caring teacher in favor of conveying the impression that


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mathematics is beautiful and interesting implies that the PTs referred to the situation

merely from a teacher’s point of view.

Final remarks

1. In our study we aimed at exploring the change in views of our PTs regarding the image

of the ‘good teacher’ as a consequence of experiencing the CPBL approach. Since

teaching experience and the courses that are learned are the factors that were found to

be the most meaningful for professional development (Latterell & Wohlhuter, 2004),

teacher educators should consider critical issues that relate to the kind of courses that

should be included in teacher education programs and their progression, as well as the

kind of experiences that the PTs should experience in parallel. We believe that the fact

that in such a relatively short time several changes occurred simultaneously as a

consequence of experiencing the CPBL approach, indicates that the integration of this

approach into teacher training programs is beneficial.

2. Beliefs are supposed to impel action, while experiences and reflection on action can

modify beliefs. Namely, there is a mutual interaction and influence between one’s

system of beliefs and one’s behavior (Philippou & Christou, 2002). Understanding the

PTs’ mathematical beliefs and the conditions under which they might be changed is

critical to teacher educators (Szydlik, Szydlik, & Benson, 2003). Pre-service programs

should consider the structure of beliefs the PTs bring to teacher education and

Provide experiences that help them to overcome common myths and misconceptions

about mathematics, its teaching and learning… Change can occur only when students

engage in personal explorations and are involved in powerful experiences in math-

ematical thinking and conceptual understanding that motivate a new perspective on

students’ views towards learning. This subsequently leads to modified classroom

practice, though a change in beliefs does not necessarily translate into changes in

practice (Philippou & Christou, 2002, p. 216).

Although there exist relationships between beliefs and practice, they may be

inconsistent (Wilson & Cooney, 2002). Thus, additional research is needed to

examine the relationship between the two images: the PTs’ perceived image before

and after the implementation of the CPBL approach and their actual practice as

future school teachers.

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