Addressing preservice student teachers negative beliefs and anxieties about mathematics

1.3.5 Preservice mathematics education courses Whilst some studies suggests that teacher education programs can assist in changing the attitudes and mathematical self-concepts of preser

Addressing preservice student teachers’ negative beliefs and anxieties about mathematics

Ms Sirkka-Liisa [Lisa] Marjatta Uusimaki

B.A., Bed (Secondary)

Centre for Mathematics, Science and Technology

Queensland University of Technology

April 2004

A 72 credit point thesis presented in fulfilment of the requirements of

the Master of Education (Research) ED12

DECLARATION

I, Sirkka-Liisa (Lisa) Marjatta Uusimaki, hereby declare that, to the best of my knowledge and belief, the work in this dissertation contain no material previously published or written by another person nor material which, to substantial extent, has been accepted for the award of any other degree or diploma at any institute of higher education, except where due reference is made

Signature………

Date………

ACKNOWLEDGMENTS

I wish to express my sincere gratitude to my principal supervisor Dr Rod Nason, Senior Lecturer in Mathematics Education, Queensland University of Technology for his brilliant ideas, excellent support and guidance throughout this study His assistance in the structuring and editing of this thesis has been greatly appreciated

I would like to also thank my associate supervisor Dr Gillian Kidman, Lecturer in Science Education, Queensland University of Technology for her outstanding contribution to this study that included advice and support in the analysis of the quantitative and the qualitative data, and in the formatting of the thesis to meet American Psychological Association (APA) guidelines I am truly grateful to Gillian for her encouragement and time she so freely gave

I would like to also thank and acknowledge Mr Andy Yeh for his assistance in the programming of the Online Anxiety Survey

Special thanks also to Mr Paul Shield who helped with the quantitative analysis of the Online Anxiety Survey data

Sincere thanks to the Director of the Centre of Mathematics, Science and Technology Education, Professor Campbell McRobbie for his kindness and support that he so generously offered throughout this study

Finally, this thesis is dedicated to my son Marcus Uusimaki whose unconditional love and support inspired me to research issues of quality in education and to give my all

in this study

ABSTRACT

More than half of Australian primary teachers have negative feelings about mathematics (Carroll, 1998) This research study investigates whether it is possible to change negative beliefs and anxieties about mathematics in preservice student teachers so that they can perceive mathematics as a subject that is creative and where discourse is possible (Ernest, 1991) In this study, sixteen maths-anxious preservice primary education student teachers were engaged in computer-mediated collaborative open-ended mathematical activities and discourse Prior to, and after their mathematical activity, the students participated in a short thirty-second Online Anxiety Survey based on ideas by Ainley and Hidi (2002) and Boekaerts (2002), to ascertain changes to their beliefs about the various mathematical activities The analysis of this data facilitated the identification of key episodes that led to the changes in beliefs The findings from this study provide teacher educators with a better understanding of what changes need to occur in pre-service mathematics education programs, so as to improve perceptions about mathematics in maths-anxious pre-service education students and subsequently primary mathematics teachers

TABLE OF CONTENTS

DECLARATION ……… i

ACKNOWLEDGMENT ……… ii

ABSTRACT ……… iii

Chapter 1 1.1 Introduction ……… 1

1.2 Background of study ……… 1

1.3 Overview of literature ……… 2

1.3.1 Maths-anxiety ……… 3

1.3.2 Teacher beliefs ……… 3

1.3.3 Overcoming maths-anxiety ……… 4

1.3.4 Assessment of maths-anxiety ……… 4

1.3.5 Pre-service mathematics education courses ……… 5

1.4 Significance of the study ……… 5

1.5 Chapter overview ……… 6

1.6 Summary ……… 6

Chapter 2 2.1 Introduction ……… 8

2.2 Maths-anxiety ……… 8

2.3 Consequences of maths-anxiety ……… 12

2.4 Teacher beliefs about mathematics ……… 13

2.5 Prior school experiences and the origins and the development of negative maths-beliefs ……… 15

2.6 Overcoming maths-anxiety in pre-service teachers 16 2.6.1 Beliefs ……… 17

2.6.2 Conceptual understanding of mathematics ……… 18

2.6.3 Subject matter knowledge and pedagogical knowledge ………… 19

2.7 Assessment of maths-anxiety ……… 21

2.8 Pre-service mathematics education courses ……… 23

2.8.1 Constructivist and social constructivist theories ……… 24

2.8.2 Collaboration ……… 25

2.9 Communities of learning and computer supported collaborative learning 27 2.10 Summary ……… 29

2.11 Theoretical framework for the study ……… 30

Chapter 3

3.1 Introduction ……… 32

3.2 Research methodology ……… 32

3.3 Participants ……… 33

3.4 Collection of data ……… 34

3.4.1 Semi-structured pre-enactment and post-enactment interviews … 34 3.4.2 Online Anxiety Survey ……… 34

3.4.3 Knowledge Forum notes ……… 35

3.4.4 Written reflections ……… 35

3.5 Procedure ……… 35

3.5.1 Phase 1: Identification of origins of maths-anxiety ……… 35

3.5.2 Phase 2: Enactment of intervention program ……… 36

3.5.3 Phase 3: Summative evaluation ……… 43

3.6 Data analysis ……… 43

3.6.1 Analysis of qualitative data ……… 43

3.6.2 Analysis of Online Anxiety Survey quantitative data ……… 44

3.7 Summary ……… 45

Chapter 4 4.1 Introduction ……… 46

4.2 Results from interview data ……… 46

4.2.1 Pre-interview results ……… 46

4.2.2 Comparison of pre- and post-interview results ……… 57

4.3 Results from reflection documents ……… 63

4.4 Online Anxiety Survey results ……… 65

4.4.1 Introduction ……… 65

4.4.2 Overall analysis of the Online Anxiety Survey results ………… 66

4.4.3 Session 1: Number sense activity ……… 68

4.4.4 Session 2: Space and measurement activity ……… 70

4.4.5 Session 3: Number and shape activity ……… 73

4.4.6 Session 4: Division operation activity ……… 75

4.5 Computer-mediated support tools ……… 77

4.6 Summary ……… 80

Chapter 5 5.1 Introduction ……… 82

5.2 Overview of study ……… 82

5.3 Overview of results ……… 83

5.4 Limitations ……… 87

5.5 Implications ……… 88

5.6 Summary and recommendations ……… 89

References ……… 91

Appendix 1: Phone interview questions ……… 103

Appendix 2: Pre-enactment Interview ……… 104

Appendix 3:Post-enactment interview …… ……… 105

Appendix 4: Online Anxiety Survey ……… 106

LIST OF TABLES Chapter 3

Table 3.1 The four mathematical activities ……… 37

Chapter 4 Table 4.1 The nature of mathematics ……… 48

Table 4.2 Reasons for teaching mathematics ……… 48

Table 4.3 Teacher knowledge and qualities ……… 49

Table 4.4 Maths-confidence ……… 51

Table 4.5 The origins of maths-anxiety ……… 52

Table 4.6 Situations causing maths-anxiety ……… 54

Table 4.7 Types of mathematics causing maths-anxiety ……… 55

Table 4.8 Perceptions of how to overcome maths-anxiety ……… 55

Table 4.9 Perceptions on how to reduce maths-anxiety in future students … 56 Table 4.10 The nature of mathematics ……… 59

Table 4.11 The relevance of mathematics ……… 59

Table 4.12 Teacher knowledge ……… 60

Table 4.13 Maths-confidence ……… 61

Table 4.14 Pairwise comparison: Overall results ……… 66

Table 4.15 Pairwise comparison: Session one results ……… 68

Table 4.16 Pairwise comparison: Session two results ……… 70

Table 4.17 Pairwise comparison: Session three results ……… 73

Table 4.18 Pairwise comparison: Session four results ……… 76

Table 4.19 Perceptions of computer-mediated software ……… 78

LIST OF FIGURES Chapter 2

Figure 2.1 The process of solving maths problems ……… 11

Figure 2.2 The theoretical framework ……… 30

Chapter 3 Figure 3.1 Intervention Program ……… 33

Figure 3.2 Online Anxiety Survey ……… 38

Figure 3.3 MipPad model and tabular representation ……… 39

Figure 3.4 MipPad model, language and symbol representation ……… 40

Figure 3.5 Shape and measurement activity ……… 41

Figure 3.6 Number and shape activity ……… 42

Figure 3.7 Division operation activity ……… 43

Chapter 4 Figure 4.1 Box plots overall positive feelings……… 67

Figure 4.2 Box plots overall negative feelings…….……… 68

Figure 4.3 Number sense activity (positive feelings responses)……… 69

Figure 4.4 Number sense activity (negative feeling responses)……… 69

Figure 4.5 Space and measurement activity (positive feeling responses)…… 71

Figure 4.6 Space and measurement activity (negative feeling responses)…… 72

Figure 4.7 Number and shape activity (positive feelings responses)………… 74

Figure 4.8 Number and shape activity (negative feelings responses)………… 75

Figure 4.9 Division operation activity (positive feelings responses)………… 76

Figure 4.10 Division operation activity (negative feelings responses)………… 77

CHAPTER 1 INTRODUCTION

1.1 Introduction

The purpose for this research study was to investigate whether supporting sixteen self-identified maths-anxious preservice student teachers within a supportive environment provided by a Computer-Supported Collaborative Learning (CSCL) community would reduce their negative beliefs and high levels of anxiety about mathematics

1.2 Background of study

A considerable proportion of students entering primary teacher education programs have been found to have negative feelings towards mathematics (Cohen & Green, 2002; Levine, 1996) These negative feelings about mathematics often manifest in a phenomenon known as maths-anxiety (Ingleton & O’Regan, 1998; Martinez & Martinez, 1996; Tobias, 1993)

Maths-anxiety can be described as a learned emotional response to, for example, participating in a mathematics class, listening to a lecture, working through problems, and /or discussing mathematics (Le Moyne College, 1999) People who experience maths-anxiety can suffer from, all or a combination of the following: feelings of panic, tension, helplessness, fear, shame, nervousness and loss of ability to concentrate (Trujillo, & Hadfield, 1999) Maths-anxiety surfaces most dramatically when the subject either perceives him or herself to be under evaluation (Tooke & Lindstrom, 1998; Wood, 1988)

A review of the literature clearly suggests that teachers’ beliefs have great influence on their students’ attitudes and beliefs about mathematics Hence, of concern is the persistent argument found in the research literature for the transference

of maths-anxiety from teacher to students (Brett, Woodruff, & Nason 2002; Cornell, 1999; Ingleton & O’Regan, 1998; Martinez & Martinez, 1996; McCormick, 1993; Norwood, 1994; Sovchik, 1996) and the difficulty in bringing to an end its continuity

The need for preservice teacher education mathematics courses to address the

towards mathematics has long been recognised in the research literature Many mathematics education courses have attempted to reduce maths-anxiety by focusing

on methodology and mathematical content as well as on learners’ conceptual understanding of mathematics (Couch-Kuchey, 2003; Levine 1996; Tooke & Lindstrom, 1998) Others have focused on having the preservice teacher education students re-construct their mathematical knowledge within the context of constructivist frameworks However, most preservice teacher education mathematics education courses have at best reported limited success only in ameliorating preservice teachers’ negative beliefs and high levels of anxiety towards mathematics

Therefore, in this study, a three-phase Intervention Model was developed and implemented to assist preservice student teachers to overcome not only their negative beliefs about mathematics but also their high level of anxieties about mathematics The first phase of this model, the identification phase, involved both the identification

of the maths-anxious preservice students and the semi-structured interviews The interviews questions focused on issues, such as, the origins and causes of negative beliefs about mathematics, preservice student teachers’ perceptions about the nature

of mathematics, what they believe characterize effective mathematics teaching and their ideas on how to overcome maths-anxiety The second phase, the intervention phase, involved the enactment of the intervention program This included the participants working in groups in non-intimidating workshop situations, learning novel mathematical activities with the help of innovative computer-mediated software and taking part in an Online Anxiety Survey The third phase, the evaluation phase, the collection and analysis of data from interviews, an Online Anxiety Survey, and written reflections about the preservice student teachers’ experiences in the project that in turn, when analysed were used to ascertain and explicate changes in students’ negative beliefs and anxieties

1.3 Overview of the literature

The conceptual framework to inform this study was derived from an analysis and synthesis of the research literature from the following fields:

1 Maths-anxiety

2 Teacher beliefs about mathematics

3 Overcoming maths-anxiety in preservice teachers

4 Assessment of maths-anxiety

5 Preservice mathematics education courses

To provide an advance organiser for the detailed review of the research literature that follows in Chapter 2, a brief overview of each of these areas is now presented

1.3.1 Maths-anxiety

In order to understand maths-anxiety and the development of maths-anxiety, Martinez and Martinez (1996) emphasised the importance of understanding the interactions between the cognitive and the affective processes of solving mathematical problems The development of confidence in contrast to maths-anxiety is dependent

on positive factors from the affective domain such as supportive environments, empathy and patience Positive factors from the cognitive domains of the problem-solving process involve the development of conceptual understanding of mathematics, and mathematics relevance to real life According to Martinez and Martinez (1996, p 6), when negative factors dominate the mathematics problem-solving process, “the by-product will be anxiety”

1.3.2 Teacher beliefs

The research literature shows that teachers’ beliefs about mathematics have a powerful impact on their practice of teaching Schoenfeld (1985) suggests that how one approaches mathematics and mathematical tasks greatly depends upon one’s beliefs about how one approaches a problem, which techniques will be used or avoided, how long and how hard one will work on it

It is suggested that teachers with negative beliefs about mathematics influence

a learned-helplessness response from students, whereas the students of teachers with positive beliefs about mathematics enjoy successful mathematical experiences that results in their seeing mathematics as a discourse worthwhile of study (Karp, 1991) Thus, what goes on in the mathematics classroom is directly related to the beliefs teachers hold about mathematics Hence, teacher beliefs play a major role in their students’ achievement and in their formation of beliefs and attitudes towards mathematics (Cooney, 1994; Emenaker, 1996; Kloosterman, Raymond, & Emenaker,

1.3.3 Overcoming maths-anxiety

An awareness of the learned negative belief[s] and affect[s] and then the ability to monitor these emotions are necessary components to overcome and control maths-anxiety (Martinez & Martinez, 1996; Pintrich, 2000) To overcome maths-anxiety, Martinez and Martinez (1996) state that “as with any negative behaviour, effecting change must begin with admitting that there is in fact a problem” (p 12) Hence, the realization and the acceptance of negative feelings are essential in the quest to overcome maths-anxiety Thus, becoming maths-confident in contrast to maths-anxious requires direct conscious action (Martinez & Martinez, 1996) To reflect and to think about one’s thinking is referred to as meta-cognition Martinez and Martinez (1996) argue, the meta-cognitive approach challenges anxieties through: (a) the analysis of thought processes about mathematics, (b) the translation of anxieties about mathematics into thoughts; and then (c) the analysis of these thoughts over an extended period of time

To overcome maths-anxiety, it is also necessary to recognize particular anxiety causing mathematics (Martinez & Martinez, 1996) For example, a person who says that he or she ‘hates’ mathematics may find on further reflection, that he or she ‘hates’ specific types of mathematics For many prospective teachers learning mathematics has meant only learning its procedures and may have, in fact, been rewarded with high grades in mathematics for their fluency in using procedures (Tucker, Fay, Schifter & Sowder, 2001)

Also, for learning to be most effective it is crucial that the learning environment is safe, supportive, enjoyable, collaborative, challenging as well as empowering Doerr and Tripp (1999) argue that conducive to learning are learning environments that provide opportunities to express ideas ask questions, make reasoned guesses and work with technology while engaging in problem situations that elicit the development of a deep understanding of mathematics and significant mathematical models

1.3.4 Assessment of maths-anxiety

A number of researchers (e.g., Ainley & Hidi, 2002; Hickey, 1997; Jarvela & Niemivirta, 1999; Pintrich, 2000) support the need for the development of methodologies and measures that access the dynamics of students’ subjective experiences or reactions whilst they are engaged in a learning activity Ainley and

Hidi suggest that such methodologies and measures provide a new perspective from which to consider the relation between what the person brings to the learning task and what is generated by the task itself

To monitor emotions, a self-reporting instrument known as an On-line Motivation Questionnaire (OMQ) that is administered before and after the specific learning tasks has been found to be successful amongst primary and secondary students in determining whether a learning situation is “an annoyer” or “a satisfier” (Boekaerts, 2002) The development of the On-line Motivation Questionnaire was guided by the theoretical model of adaptive learning (Boekaerts, 1992, 1996) This theory according to Boekaerts (2002) predicts students’ appraisals (motivational beliefs) of a learning situation and explains more variation in their learning intention, emotional state, and effort than domain-specific measures

1.3.5 Preservice mathematics education courses

Whilst some studies suggests that teacher education programs can assist in changing the attitudes and mathematical self-concepts of preservice and in-service primary school teachers to more positive ones (Bobis, & Cusworth, 1997; Philippou

& Christou, 1997), other studies imply that teachers maintain their negativity toward mathematics and mathematics teaching after they begin to teach (Cockroft, 1982; Freeman & Smith, 1997; Kanes & Nisbet, 1994; Pateman, 1989) To reverse this negativity about mathematics, Carroll (1998) suggested the re-examination of teacher education programs She felt that there must be more focus on the development of

“confidence in the ideas of the teachers who must be encouraged to analyse and critically evaluate their current knowledge, beliefs and attitudes and modify [these] to include new ideas” (p.8)

1.4 Significance of the study

This project has both practical and theoretical outcomes for preservice mathematics education and for research into computer supported collaborative learning (CSCL)

In terms of practical outcomes, this study seeks to improve the quality of teaching and learning in primary school mathematics by providing maths-anxious preservice teachers with the means to combat their negative feelings about mathematics through: a) the development of an understanding and awareness of their

learned negative feelings about mathematics, b) the development of repertoires of mathematical content and pedagogical knowledge using CSCL that will allow for the development of confidence in mathematics, and, c) identification of self and identity (Brett, 2002).

In terms of theoretical outcomes, this study will extend Boekaerts (2002) model of adaptive learning theory from its present context with primary and secondary school students to contexts with self-identified maths-anxious preservice student teachers It will also advance the body of theoretical knowledge within the field of CSCL especially with respect to its application within the field of teacher education of maths-anxious preservice student teachers

Chapter 1 provides information on the background of the research The significance of the study is examined and an overview of relevant literature is presented Chapter 2 reviews the relevant literature and provides a foundation for the study pertaining to what constitutes maths-anxiety, its origins and causes, consequences of maths-anxiety on the individual, the student as well as the impact negative beliefs about mathematics has on students’ numeracy outcomes Chapter 3 outlines the exploratory mixed-method design that was used in the study including the data collection and analysis A description of the data collection is given and a description of participants as well as the criteria used in selecting these participants In Chapter 4 the findings from the research study are presented Finally, Chapter 5 presents the discussion of the results, a summary and conclusion in regards to the relevant literature as well as the implications and limitations of the study for teacher preservice courses

1.6 Summary

The aim of this research study was to investigate whether supporting sixteen self-identified maths-anxious preservice student teachers (a) to develop mathematical reasoning, (b) to reflect on their learning, (c) to challenge and then to modify negative beliefs and attitudes about mathematics provided by a CSCL community would reduce their negative beliefs and high levels of anxiety about mathematics It is argued that enhancing the preservice student teachers’ repertoires of mathematical subject matter knowledge will lead to, reductions in their negative beliefs and anxieties about mathematics and to enhancement of their sense of identity as future

primary mathematics teachers as well as valued members within their learning community Most importantly, the broader implications of the study relate to the positive impact that these preservice student teachers will have on their future student numeracy outcomes

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction

“Maths-anxiety is not just a simple nervous reaction, nor is it a harmless myth: it is a debilitating affliction that restricts math performances among both children and adults worldwide” (Martinez & Martinez, 1996, p 9)

More than half of Australian primary teachers have negative feelings about mathematics (Carroll, 1998) Research suggests that it is a teacher’s personal school experiences that influence the developments of negative feelings about mathematics (Brown, McNamara, Hanley & Jones, 1999; McLeod, 1994; Nicol, Gooya & Martin, 2002; Trujillo, & Hadfield, 1999; Williams, 1988) As a consequence of these personal school experiences a considerable proportion of students entering primary teacher education programs have been found to have negative feelings towards mathematics (Carroll, 1998; Cohen & Green, 2002; Ingleton & O’Regan 1998; Lacefield, 1996; Levine, 1996; Philippou & Christou, 1997) Negativity about mathematics often manifests in what has long been identified as maths-anxiety (Barnes 1984; Bessant, 1995: Blum-Anderson, 1994; Cemen, 1987; Fairbanks, 1992; Hadfield, Martin & Wooden 1992; Ingleton & O’Regan, 1998; Martinez & Martinez, 1996; McCormick, 1993; Norwood, 1994; Richardson & Suinn, 1972; Tobias, 1993; 1978)

An early definition of maths-anxiety suggests that it is “… feelings of tension and anxiety that interfere with the manipulation of numbers and the solving of mathematical problems in a wide variety of ordinary life and academic situations” (Richardson & Suinn, 1972, p 551) According to Cemen (1987), maths-anxiety can

be described as a state of anxiety which occurs in response to situations involving mathematics which is perceived as threatening to self-esteem (Trujillo & Hadfield, 1999) Such feelings of anxiety can lead to panic, tension, helplessness, fear, distress, shame, inability to cope, sweaty palms, nervousness, stomach and breathing

difficulties and loss of ability to concentrate (Trujillo & Hadfield, 1999) Research studies have found that maths-anxiety is related to test anxiety which means that it surfaces most dramatically when the subject either perceives him or herself to be under evaluation (Ikegulu, 1998; Tooke & Lindstrom, 1998; Wood, 1988) Althoughearly research suggests that the term maths-anxiety was rather an expression of general anxiety and not a distinct phenomenon (Olson & Gillingham, 1980), more recent research into maths-anxiety has recognized it not only to be more complex than general anxiety but also more common than earlier suggested (Ingleton & O’Regan, 1998) It is because of its complexity that there is not a universal agreement as to what constitutes maths-anxiety

The origins of maths-anxiety and negative beliefs about mathematics can be categorised into three areas: (a) environmental, (b) intellectual and (c) personality factors (Hadfield & McNeil, 1994; Trujillo & Hadfield, 1999):

1 Environmental factors included negative experiences in the classroom,

parental pressure, insensitive teachers, mathematics being taught in a traditional manner as rigid sets of rules, and non-participatory classrooms (Trujillo & Hadfield, 1999; Stuart, 2000)

2 Intellectual factors including teaching being mismatched with learning styles,

student attitude and lack of persistence, self-doubt, lack of confidence in mathematical ability and lack of perceived usefulness of mathematics (Trujillo

& Hadfield, 1999)

3 Personality factors included reluctance to ask questions due to shyness, low

self-esteem and, for females, viewing mathematics as a male domain (Levine, 1996; Trujillo & Hadfield, 1999)

From this it can then be seen that the origins of maths-anxiety are as diverse as are the individuals experiencing maths-anxiety For some, people maths-anxiety is related to poor teaching, or humiliation and/ or belittlement whilst others may have learnt maths-anxiety from the maths-anxious teachers, parents, siblings or peers, or who may link their anxiety to numbers or to some operations generally (Martinez & Martinez, 1996; Stuart, 2000) Thus, to understand maths-anxiety, it must be recognized for its complexity Maths-anxiety is not a discrete condition but rather it is

a “construct with multiple causes and multiple effects interacting in a tangle that

defies simple diagnosis and simplistic remedies” (Martinez & Martinez, 1996, p.2) A definition by Smith and Smith (1998) takes into consideration this intricacy by encompassing both the affective and the cognitive domain of learning Smith and Smith state that maths-anxiety is a feeling of intense frustration or helplessness about one’s ability to do mathematics Maths-anxiety can be described as a learned emotional response to participating in a mathematics class, listening to a lecture, working through problems, and /or discussing mathematics to name but a few examples (Hembree, 1990; Le Moyne College, 1999) This definition stipulates that maths-anxiety is not exclusively a product of the affective domain but also of the cognitive domain of learning

According to Martinez and Martinez (1996), the cognitive domain of learning can be described as the logical component of learning For instance, logical thought processes, information storage, and retrieval, aptitude for learning mathematics, mathematics learning readiness and teaching strategies all belong to the cognitive domain Martinez and Martinez state that “the cognitive domain affects maths-anxiety when there are gaps in knowledge, when information is incorrectly learnt, and when the learning readiness and teaching strategies are mismatched” (pp 5-6)

The affective domain of learning is the emotional component of learning (Martinez & Martinez, 1996) This is the province of beliefs, attitudes and emotions about learning mathematics, of memories of past failures and successes, of influences from maths-anxious or maths-confident adults, of responses to specific learning environment and teaching styles (Gellert, 2001; Martinez & Martinez, 1996; Pehkonen & Pietila, 2003) The affective domain provides a context for learning (Martinez & Martinez, 1996) and if the affective domain provides a positive context, students can be motivated to learn, whatever their mathematical aptitude However,

“if the affective domain provides a negative context, even students with superior math-learning ability may develop maths-anxiety” argue Martinez and Martinez (1996, p 6)

Figure 2.1 demonstrates the interactions between the cognitive and affective processes of solving mathematical problems The figure shows a number of factors involved in the mathematical problem-solving process For example, the development

of confidence in contrast to anxiety is dependent on positive elements from the affective (e.g., supportive environment, empathy, patience) and/or the cognitive

domains of the problem-solving process (e.g., development of conceptual understanding of mathematics, relevance to real life, challenging) If however, negative elements dominate the mathematical problem-solving process, “the by-product will be anxiety” (Martinez & Martinez, 1996, p 2) Hence the development

of confidence in mathematics is a critical emotion in the process of learning(Ingelton

& O’Regan, 1998)

Figure 2.1 The Process of Solving Mathematical Problems (Source: Martinez &

Martinez, 1996, p 2)

Confidence is defined according to Barbalet (1998, p 86) as “an emotion with

a subjective component of feelings, a physiological component of arousal and a motor component of expressive gesture” Confidence functions in opposition to shame, shyness and modesty, which are described as emotion of self-attention or “thinking what others think of us” (Barbalet, 1998, p 86) Ingleton and O’Regan (1998) suggest that confidence has its origins in particular experiences of social relationships, such as

“where a person receives acceptance and recognition in contrast to the onset of anxiety and shame where a person is denied this acceptance or recognition” (p.3)

2.3 Consequences of maths-anxiety

Some of the consequences that result from being maths-anxious as opposed to maths-confident include:

This figure is not available online

Please consult the hardcopy thesis available from the QUT Library

1 The fear to perform tasks that are mathematically related to real life incidents, such as sharing or dividing a restaurant bill amongst diners or developing a household budget

2 Avoidance of mathematics classes

3 The belief that it is acceptable to fail/dislike mathematics

4 Feelings of physical illness, faintness, fear or panic

5 An inability to perform in a test or test-like situations

6 Participation in tutorial sessions that provide little success (McCulloch Vinson, Haynes, Sloan, & Gresham 1997)

Some commonly held beliefs associated with maths-anxiety and mathematics avoidance identified by Kogelman and Warren (1978) still hold true today Specifically some of these are:

1 Inherited mathematical ability or some people have a mathematical mind and some don’t

2 Mathematics requires logic not intuition

3 You must always know how you got the answer

4 There is one best way to do a mathematical problem

5 Men are better at mathematics than women

6 It is always important to get the answer exactly right

7 Mathematicians solve problems quickly in their heads

8 Mathematics is not creative

9 It is bad to count on your fingers (Sam, 1999) The implication of such negative beliefs and negative school mathematics experiences on many primary teacher education students has resulted in the continuity

of the maths-anxiety phenomenon Of concern is the persistent argument found in the research literature for the transference of maths-anxiety from teacher to students (Brett, et al., 2002; Cornell, 1999; Ingleton & O’Regan, 1998; Martinez & Martinez, 1996; McCormick, 1993; Norwood, 1994; Sovchik, 1996) and the difficulty in bringing to an end its continuity

2.4 Teacher beliefs about mathematics

A review of the literature indicates that teachers’ beliefs have much influence

on their students’ attitudes and beliefs about mathematics “A belief is the acceptance

of the truth or actuality of anything without certain proof “, according to McGriff Hare (1999, p 42) Beliefs are one’s subjective knowledge including whatever one considers as true knowledge, without the lack of convincing evidence to support these beliefs (Pehkonen, 2001) Since beliefs are cognitive in nature and developed over a relatively long period of time they seldom change dramatically without significant intervention (Lappan, et al., 1988; McLeod, 1992) Schoenfeld (1985) suggests that how one approaches mathematics and mathematical tasks greatly depends upon one’s beliefs about how one has to approach a problem, which techniques will be used or avoided, and how long and how hard one will one work on the mathematical task

Research findings suggest that beliefs about the nature of mathematics affect teachers’ conception of how mathematics should be presented (Ernest, 1988, 1991, 2000; Hersh, 1986) According to Hersh (1986, p.13):

One’s conception of what mathematics is affects one’s conception of

how it should be presented One’s manner of presenting it is an

indication of what one believes to be most essential in it…The issue

then it is not, what is the best way to teach? But, what is mathematics

really about?

Indeed, it is because the two domains of teacher belief and knowledge are intertwined and difficult to separate that makes them particularly of concern to teacher education programs where this bottleneck should be addressed simultaneously

A number of other studies have shown that teachers’ beliefs about mathematics have a powerful impact on the practice of teaching (Charalambos, Philippou & Kyriakides, 2002; Ernest, 1988, 2000; Golafshani, 2002; Putnam, Heaton, Prawat, & Remillard, 1992; Teo, 1997) McLeod (1992) states that, "the role

of beliefs is central in the development of attitudinal and emotional responses to mathematics" (p 579)

Drawing on the philosophy of mathematics, Ernest (1991) distinguishes two dominant epistemological perspectives of mathematics, namely the absolutist and the fallibilist beliefs about the nature of mathematical knowledge Absolutists believe that mathematics consists of a set of absolute and unquestionable truths that is certain and exact This is where mathematical knowledge is believed to be objective, value free and culture free In contrast, fallibilists believe that “mathematical truth is fallible and corrigible, and can never be regarded as beyond revision and correction” (Ernest,

1991, p.18) In other words, mathematics can be seen as the outcome of social processes and where mathematics knowledge is understood as fallible and always open to revision Rule-based ways of teaching are often associated with teachers with absolutist beliefs about the nature of mathematics (Ernest, 2000) Some research findings propose that maths-anxiety is often associated with teaching methods that are conventional (absolutist and rule-bound) (Sloan, Daane & Giesen, 2002) It has been noted that rule-based methods of instruction are commonly employed by primary teachers who possess high levels of anxiety and negative attitudes toward mathematics (Karp, 1991; Sloan et al., 2002) and thus the cycle is perpetuated

Other researchers have provided other classification systems to describe the different philosophies of teaching mathematics and their implications (Wiersma & Weinstein, 2001) For example, Lerman (1990) identified the dualistic1 and relativist2ways teachers depict when teaching mathematics Teachers teaching from a dualistic standpoint teach mathematics as a set of absolute and unquestionable truths Teaching from the relativist standpoint on the other hand is where mathematics is taught as a

“dynamic, problem-driven and continually expanding field of human creation and invention, in which patterns are generated and then distilled into knowledge” (Ernest,

1996, p 808) A review of the literature indicates that maths-anxiety is more likely to emerge in classrooms where teachers employ absolutist/dualistic or content-focused with emphasis on performance modes of teaching (Ernest, 2000)

A review of the research literature indicates that feelings of maths-anxiety in preservice teachers are often associated with negative beliefs about mathematics and the teaching of mathematics (Brett, et al., 2002; Cohen & Green, 2002; Karp, 1991; Middleton & Spanias, 1999) It is suggested that teachers with negative beliefs about mathematics influence a learned helplessness response from students this is a form of

a response where students seem to have lost the capacity to be accountable for their own behavior and performance, because of repeated unfavorable past performances (McInerney & McInerney, 1998) In contrast, the students of teachers with positive beliefs about mathematics enjoy successful mathematical experiences that result in their seeing mathematics as a discourse worthwhile of study (Karp, 1991) Thus, what goes on in the mathematics classroom is directly related to the beliefs teachers hold

about mathematics Hence, teacher beliefs play a major role in their students’ achievement and in their formation of beliefs and attitudes towards mathematics (Cooney, 1994; Emenaker, 1996; Kloosterman, et al., 1993; Roulet, 2000; Schofield, 1981)

2.5 Prior school experiences and the origins and the development of

negative maths-beliefs

As already discussed (see Section 2.2.), the developments of negative beliefs about mathematics can be and, in many cases, are influenced by siblings and fellow peers (Stuart, 2000) Negative beliefs about mathematics also have their origins in prior school experiences such as the experience of being a mathematics student, the influence of prior teachers and of teacher preparation programs (Borko, et al., 1992; Brown & Borko, 1992), as well as prior teaching experience (Raymond, 1997) For example, many negative beliefs held by teachers can be traced back to the frustration and failure in learning mathematics caused by unsympathetic teachers who incorrectly assumed that computational processes were simple and self-explanatory (Cornell, 1999) In their study Martinez and Martinez (1996) found that sixty percent of student teachers tested using a Math Anxiety Self Quiz claimed to be highly maths-anxious, thirty percent claimed to be moderately maths-anxious and many attributed their anxiety to hostile teaching strategies These hostile teaching strategies (Martinez

& Martinez, 1996, p 34) included:

1 Verbally abusing students for errors – being called math-dumb, bonehead, knucklehead, and pea brain

2 Punishing behaviour and deficiencies with math exercises

3 Exposing students to public ridicule by assigning board problems and badgering the un-prepared

4 Isolating the learners – “Keep your eyes on the board There will be no talking, no exchanging of notes or papers and no questions for anyone but the teacher”

5 Ram-rodding information – “Listen up I’m going to say this one time and one time only”

6 Input/Output teaching – Without interacting with students, teacher inputs information to them through lectures and study assignments; students output information to teacher by doing homework and taking tests

The consequences of these sorts of math-hostile teaching strategies ultimately impact negatively on student behaviour as well as in their attitude towards mathematics both in primary and secondary school and later in their deliberate avoidance of careers that require extensive mathematical knowledge (Martinez & Martinez, 1996; Tucker, et al., 2001)

There is an assumption amongst teachers that by controlling or hiding one’s maths-anxiety behind well-planned and well-explained mathematics lessons that students will not come to “learn the anxiety” of his or her teacher (Martinez & Martinez, 1996, p.10) However, students and particularly young children do learn the anxiety as they pick up on the covert signals displayed by the teacher, in other words they tend to see the strain behind the smile or hear tension in a voice (Martinez & Martinez, 1996) Thus, for a positive and successful teaching and learning experience

to occur “what the teacher says about math and what the teacher feels about math must match” (Martinez & Martinez, 1996 p.11)

2.6 Overcoming maths-anxiety in preservice teachers

“Tell me mathematics and I forget, show me mathematics and I may remember… involve me and I will understand If I understand mathematics, I will be less likely to have maths-anxiety And if I become a teacher of mathematics I can thus begin a cycle that will produce a generation of less likely maths-anxious students for the generation to come” (Williams, 1988, p.101)

Because of its complex nature encompassing both the affective and cognitive domains of learning, interventions focusing on both elements are needed to overcome maths-anxiety

2.6.1 Beliefs

To overcome negative beliefs and anxiety about mathematics requires a fundamental shift in a person’s system of beliefs and conceptions about the nature of mathematics and mental models of teaching and learning mathematics (Levine, 1996) Seligman (1991) believes that it is possible to successfully overcome maths-anxiety

through motivation and desire that in many cases make up for the lack of mathematical talent in people For instance, Wieschenberg (1994) claims that mature-age students returning to tertiary education, after years of absence, and without a proper mathematics background, can learn to enjoy mathematics because of their strong desire to learn mathematics coupled with their enthusiastic and committed approach to the teaching of mathematics Martinez and Martinez (1996) agree that for adult learners re-learning basic mathematical concepts is particularly gratifying because, in general, adult learners tend to over-learn This results in positive effects, where mathematics is seen as less intimidating whilst simultaneously building the learners’ maths-confidence To do this successfully, Ernest (2000) suggests that both encouragement and a genuine interest in the learners’ work by the educator/facilitator

is needed, in contrast to the public criticism and humiliation and/or belittling of students which have been shown to have negative effects that remain with the learner

Martinez and Martinez (1996) state that “as with any negative behaviour, effecting change must begin with admitting that there is in fact a problem” (p.12) Hence, the realization and the acceptance of negative feelings are essential in the quest to overcome maths-anxiety Thus, to become maths-confident in contrast to maths-anxious requires direct conscious action (Martinez & Martinez, 1996) To reflect and to think about one’s thinking is referred to as meta-cognition Martinez and Martinez (1996) argued, the meta-cognitive approach challenges anxieties through: (a) the analysis of thought processes about mathematics, (b) the translation of anxieties about mathematics into thoughts; and then (c) the analysis of these thoughts over an extended period of time Literally, the approach calls for becoming immersed

in mathematics and in the process of mathematical learning This is supported by Raymond (1997) who suggested that “early and continued reflection about mathematical beliefs and practices, beginning in teacher preparation may be the key

to improving the qualities of mathematics instruction and minimizing inconsistencies between belief and practice” (p 574) Indeed, reflection that involves thinking and acting on those aspects of learning and teaching mathematics that frustrate, confuse or perplex (Bryan, Abell, & Anderson, 1996), can help the maths-anxious preservice student teacher to untangle the web of deeply held negative beliefs and anxieties about mathematics In doing so, the relearning of mathematics and the discovery of the causes or the origins to one’s learnt negative beliefs and anxieties can become the

occasion and the process for a positive conceptual change towards mathematics learning and teaching (Bryan, et al 1996; Cosgrove & Osborne, 1985; Ferrari, 2000; Schon, 1983, 1987) The process of conceptual change requires the student to: (a) make explicit his/ her ideas about the mathematical concept (b) explores the concept, (c) clarify his / her view of the concept (d) consider others’ points of view (e) recognize discrepancies among views and resolve the discrepancies and (f) apply refined explanation to solve a new problem, which means to refine ideas and re-evaluate solution (Cosgrove & Osborne, 1985) Indeed, this occasion and process of a conceptual change can be the initial step toward the development of an appreciation

of mathematics as a system of human thought (Ball, 2001) that is both threatening and non-intimidating

non-It is recognized that particular kinds of mathematics cause feelings of anxiety (Martinez & Martinez, 1996) A person who says that he or she hates mathematics may find on further reflection, that he or she hates specific types of mathematics For instance, there may be a strong dislike for algebra whilst mental computation activities are seen as fun and challenging For many prospective teachers, learning mathematics has meant only learning its procedures and, in fact, may have been rewarded with high grades in mathematics for their fluency in using procedures (Tucker, et al., 2001) While procedural fluency is necessary, it is not an adequate foundation for teaching mathematics whereas an orientation towards making sense of mathematics must be considered fundamental both to learning and to teaching mathematics (Tucker et al., 2001) Making sense of mathematics includes a conceptual understanding of what mathematics is about

2.6.2 Conceptual understanding of mathematics

A conceptual understanding of mathematics means to be engaged in multiple mathematical processes and to understand how various mathematical concepts are related to one another in a useful and meaningful way (Lesh, 1985; Lesh & Doerr, 2002) Without a conceptual knowledge of mathematics, McCulloch Vinson et al., (1997) argue that mathematical power is diminished and leads to an increase in maths-anxiety

To understand the benefits of learning mathematics in an open-ended manner that promotes a conceptual understanding of mathematics in contrast to learning mathematics in a traditional manner, Boaler (2002, p 114) contrasted students who

were taught mathematics in traditional classroom settings (where they were taught to watch and faithfully reproduce procedures and to follow different textbook cues), with students taught mathematics through open, group-based projects Findings suggested that students who were taught mathematics in the traditional manner could indeed perform well in similarly reproduced situations However, difficulties were found to occur in situations that required mathematics to be used in open, applied or discussion-based situations In contrast students who had been taught through open-ended projects were not only able to use mathematics in different situations but

“outperformed the other students in a range of assessment, including the national assessment” (Boaler, 2002, p 114)

2.6.3 Subject matter knowledge (SMK) and pedagogical content knowledge (PCK)

Whilst there is evidence that subject matter knowledge is a predictor of mathematical learning and teaching effectiveness, Darling-Hammond, Wise and Klein (1999) caution that beyond a certain level “additional content knowledge seems to matter less to enhance effectiveness than knowledge of teaching and learning” (p 199) Thus, in addition to a conceptual understanding of mathematics, teachers also need to know pedagogy For teachers who are maths-anxious, an understanding of pedagogical knowledge is particularly important, especially since poor teaching strategies or methods is what were in many situations, the cause of their maths-anxiety Hence, it is not enough to have a conceptual knowledge of mathematics to be able to teach it effectively (Richardson, 1999) Ball and Cohen (1999) note “in order

to connect to students with content in effective ways, teachers need a repertoire of ways to engage learners effectively and the capacity to adapt and shift modes in response to students” (p 9) Pedagogical content knowledge (PCK) a term originally developed by Shulman (1987) and his colleagues, “is a unique kind of knowledge that intertwines content with aspect of teaching and learning” (Ball, Lubienski & Mewborn, 2001, p 448) and is referred to as a way of knowing the subject matter that allows it to be taught (Richardson, 1999, p 284) PCK involves: (a) knowing the subject matter, (b) knowing how students learn, (c) being aware of students’ preconceptions that may get in the way of learning, and (d) knowing various representations of mathematical knowledge in the form of metaphors or examples that makes sense

Research suggests that teachers in the western culture have a somewhat inferior mathematical content knowledge and pedagogical knowledge base of mathematics to their counterparts in countries such as China In her often quoted and well-known study, Ma (1999) contrasted the mathematics content knowledge and PCK of American elementary school teachers with their counterparts in China She found that the knowledge of the American teachers studied was relatively instrumental, unconnected and devoid of conceptual grounding On the other hand the Chinese teachers, with fewer years of formal education and inferior mathematical qualifications, had acquired a strong conceptual grounding in mathematics which Ma calls “profound understanding of fundamental mathematics that influenced the ways

in which they worked with children” (p.13) Her findings suggest that: (a) formal qualifications in mathematics are not reliable indicators of effective mathematics teaching in primary years, (b) there is no evidence to suggest that teachers’ mathematics subject matter knowledge develops as a consequence of teaching

In their study, Prestage and Perks (2000) found that whilst preservice teachers could do mathematics they did not necessarily hold multiple and fluid conceptions of the mathematics that underlie teacher knowledge or knowledge needed to plan for others to come to learn mathematics Hence, “the difficulty in addressing primary preservice teachers’ weak syntactic knowledge in the training years is a cause for considerable concern; indeed, there are no grounds for supposing that the issue is tackled at any later stage” argues Ma (1999, p.18) The initial transition from school learner to school teacher, if it is to be successful, must often involve a considerable degree of ‘unlearning’ (i.e discarding of mathematical ‘baggage’) In terms of both subject misconceptions and attitude problems, lack of attention to this potential impediment is thought to “help to account for why teacher education is often such a weak intervention – why teachers, in spite of course and workshops, are most likely to teach math just as they were taught” (Ball, 1988, p.40)

In addition to a teacher’s subject matter knowledge, pedagogical knowledge and academic ability, other important characteristics of teacher effectiveness include personal factors such as enthusiasm, flexibility, perseverance, confidence (Darling-Hammond, 2000; Good & Brophy, 1995) Another important feature is for the teacher

to appreciate the subject matter he or she is teaching, and to have an understanding of the nature of the subject matter, as well as an awareness of his or her attitudes towards

it (Cockroft Report, 1982) As evidence suggests it is the combination of all of these factors that ensures teacher effectiveness

Moreover, for learning to be most effective, the learning environment needs to

be safe, supportive, enjoyable, collaborative, challenging and empowering The aim then, is to create a learning environment where peer tutoring and collaborative learning is highly valued and where students have opportunities to both engage in and reflect on the discourse as they share and build their knowledge (Bobis & Aldridge, 2002) Doerr and Tripp (1999) argue that learning environments that provide opportunities where it is safe to express ideas, ask questions, make reasoned guesses

as well as work with technology while engaging in problem situations elicit the development of not only significant mathematical models but more importantly a deep mathematical understanding

2.7 Assessment of maths-anxiety

An awareness of the learned negative belief[s] and affect[s] and then the ability to monitor these emotions are necessary components to overcome and control maths-anxiety (Martinez & Martinez, 1996; Pintrich, 2000) A number of researchers (e.g., Ainley & Hidi, 2002; Hickey, 1997; Jarvela & Niemivirta, 1999; Pintrich, 2000) support the need for the development of methodologies and measures that access the dynamics of students’ subjective experiences or reactions whilst they are engaged in a learning activity Ainley and Hidi suggest that such methodologies and measures provide a new perspective from which to consider the relation between what the person brings to the learning task and what is generated by the task itself

In motivational research, constructs are often discussed in terms of their status

as either trait or state and one way of conceptualizing the relation between trait and state, is to see these in terms of levels of specificity (Ainley & Hidi, 2002) Trait refers to “an individual’s relatively enduring predisposition to attend to certain objects, stimuli and events and to engage in certain activities” while state refer to

“attention or concentration that is directed to the object or (situation) experience (Ainley & Hidi, 2002, p 44) Some researchers (e.g., Ainley & Hidi, 2002; Hidi & Berndorff, 1998; Renninger, 2000; Schiefele, 1996) believe that both individual and situational interest influence learning Thus, the importance for the development of measures that can reflect the dynamics of students’ experience as they are engaged

with a learning task since this will allow further identification of relationships between individual and situational factors (Ainley & Hidi, 2002)

In their study measuring interest and learning, Ainley and Hidi (2002, p.43) explored attention at the micro level of students’ subjective experiences, monitoring and recording what students were doing as they proceeded through a learning task They argue that the approach offers “an insight into changes in motivation that might occur as students make choices and navigate their way through a learning task” (p.43) They identified three critical issues mainly: (a) the relationship between person and situation (b) the identification of the specific processes through, which interest influence learning and achievement, and, (c) the relationship between specific motivational construct insights The results of their study showed that topic interest influenced students’ affective responses, which in turn influenced the degree that students persisted with the task and that was related to the outcome or the scores on the test at the end of each task (Ainley & Hidi, 2002)

To monitor emotions, a self-reporting instrument known as an On-line Motivation Questionnaire (OMQ) that is administered before and after the specific learning tasks has been found to be successful amongst primary and secondary students in determining whether a learning situation is “an annoyer” or “a satisfier” (Boekaerts, 2002) The development of the On-line Motivation Questionnaire was guided by the theoretical model of adaptive learning (Boekaerts, 1992, 1996) This theory according to Boekaerts predicts students’ appraisals (motivational beliefs) of a learning situation and explains more variation in their learning intention, emotional state, and effort than domain-specific measures

Boekaerts argues that there are certain cues in a learning situation that students may interpret as threatening or challenging She claimed that cues related to excitement and challenge feelings of autonomy, competence led to optimistic appraisal of a learning situation, (i.e that is a “satisfier”) whilst, cues that are related

to threat, loss, harm, boredom lead to pessimistic appraisal (i.e an “annoyer”) Of course, in addition to a learning situation that is seen by a student as either favourable

or unfavourable, Boekaerts (2002) cautions that, ”the same or similar environmental cues can be seen as dissimilar on different occasions by different students or the same students on different occasions” (p.81) She gives the example of how two students experiencing maths-anxiety differ in the links they have established between this

particular domain-specific belief and their appraisal of actual math problems For instance, “the first student may focus on cues that inform him that he cannot master the task or cannot complete it without help” that is, there is a negative link between maths-anxiety and subjective competence While, “the second student may focus on his feelings of displeasure, finding the task boring and irrelevant” That is, there is a negative link between math anxiety and the task attraction and its perceived relevance (Boekaerts, 2002, p 82) The On-Line Motivation Questionnaire reliably captures students’ cognitions and feelings in relation to specific learning tasks, and thus effectively opens new ways of studying and understanding motivation in the classroom (Boekaerts, 2002)

2.8 Preservice mathematics education courses

Whilst some studies suggest that teacher-education programs can assist in changing the attitudes and mathematical self-concepts of preservice and in-service primary school teachers to more positive ones (Bobis, & Cusworth, 1997; Philippou

& Christou, 1997), other studies imply that teachers’ maintain their negativity toward mathematics and mathematics teaching after they begin to teach (Cockroft, 1982; Freeman & Smith, 1997; Kanes & Nisbet, 1994; Pateman, 1989) To reverse this negativity about mathematics, Carroll (1998) suggested the re-examination of teacher education programs She felt that there must be more focus on the development of

“confidence in the ideas of the teachers who must be encouraged to analyse and critically evaluate their current knowledge, beliefs and attitudes and modify [these] to include new ideas (p.8).”

Cobb and Bauersfeld (1995) suggest that to improve the mathematical knowledge bases, alter beliefs and improve attitudes of practicing teachers as well as

to improve teacher development in mathematics education requires three factors: (a) teacher’s own motivation to change his/her practice, (b) access to model-eliciting activities that teachers’ try out in their own classrooms and (c) encouragement of regular and ongoing collaboration with other teachers to discuss classroom experiences (Hjalmarson, 2003; Siemon, 2001) Research has also noted (e.g., Tucker, et al., 2001) that to empower preservice student teachers with weak mathematics backgrounds, teacher education programs must take into account

preservice student teachers’ prior classroom experiences in which their ideas for

solving problems are elicited and taken seriously, and their sound reasoning affirmed,

as well as their mistakes challenged in ways that help them make sense of their errors Tucker et al (2001) further note that for teachers who are able to cultivate good problem-solving skills among their students, they must, themselves, be problem solvers, aware that confusion and frustration are not signals to stop thinking, and confident that with persistence they can work through to new insight That is, they will have learnt to notice patterns and think about whether and why these patterns hold, posing their own questions and knowing what sorts of answers make sense

2.8.1 Constructivist and social constructivist theories

To address these factors, most teacher-education programs have adopted methods based on principles of constructivist and social constructivist theories in their mathematics education subjects (e.g., Borko et al., 1990; Martin, 1994; Peck & Connell, 1991; Wilcox, Schram, Lappan, & Lanier 1991) The application of principles of constructivist and social construction theories has resulted in the establishment of learning environments where: (a) students construct their own knowledge from personal experiences rather than passively accepting information from the outside world (Brown, Collins & Duguid, 1989; Collins, Brown & Newman, 1989; Collins & Green, 1992; Resnick, 1987), (b) the creation of learning communities where students engage in discourse about important ideas (Putnam & Borko, 2000) and (c) students use reflection as a means of reconceptualising knowledge and beliefs (Beattie, 1997) However, some of the concerns with constructivist theories of learning, communities of learners and authentic tasks are that neither the beginning nor experienced teacher completely understand “what these ideas mean, what it might mean to draw on them in practice and the complications they raise for teaching and learning” (Lampert & Ball, 1999, p.39)

Mathematics courses that have been able to reduce maths-anxiety have tended

to focus not only on methodology and mathematics content but also on the learners’ conceptual understanding of mathematics (Levine, 1996) In a study by Couch-Kuchey (2003) for example, using a constructivist approach to alleviate learning anxiety amongst 41 early childhood preservice teachers showed the effectiveness of a mathematics methodology course in reducing levels of anxiety A significant reduction of maths-anxiety was noted, particularly in the practicum that was due to the various mathematical activities that had been introduced in the mathematics methodology course that were then carried out in the practicum (Couch-Kuchey,

2003) Nonetheless, the study did not report on whether there had been any changes in the preservice teachers’ perceived negativities about mathematics Similarly in a study

by Tooke and Lindstrom (1998) the effectiveness of a mathematics methods course showed a reduction in maths-anxiety amongst preservice primary teachers whilst there was no mention of any changes to preservice teachers’ negative beliefs about mathematics

It is common to note in most current preservice mathematics education courses, students are required to apply constructivist frameworks and to actively, and reflectively, construct (or reconstruct) knowledge Yet, according to Lampert (1988), most preservice teachers find this type of experience daunting because they have based their own learning on the assumption that their lecturers knew the truth and all that they needed to do was write it down, memorise it, and reproduce it on a test to prove they knew it Teachers faced with too much unresolved uncertainty during their preservice education programs may therefore find the experience disabling (Floden & Buchmann, 1993) Because most programs based on constructivist principles seem to have done little to resolve this issue of uncertainty many of these programs have reported only limited long-term success in improving preservice teachers’ repertoires

of mathematical subject-matter knowledge and pedagogical-content knowledge (Brett

et al., 2002)

2.8.2 Collaboration

The benefits of collaboration (i.e., the use of group work) in mathematical learning have been well documented (Johnson & Johnson, 1986; Kimber, 1996; Watson & Chick, 1997) For example, research suggests that cooperative and collaborative learning bring positive results such as deeper understanding of content, increased overall achievement in grades, improved self-esteem and higher motivation

to remain on task Collaborative and cooperative learning also helps students become actively and constructively involved in content, taking ownership of their own learning that leads to their development as critical thinkers, resolving group conflicts and improving teamwork skills Most importantly, cooperative learning techniques facilitate the student’s ability to solve problems and to integrate and apply knowledge and skills, the very art of learning (Koschmann, Kelson, Feltovich, & Barrows, 1996) Moreover, evidence suggests that cooperative teams achieve higher levels of thought and retain information longer than students who work individually especially in

subjects such as science and mathematics (Slavin, 1989; Totten, Sill, Digby & Russ, 1991) Benefits applicable particularly for maths-anxious students include: (a) opportunities for verbalising concerns in a safe environment, (b) allowing students to resolve conflicts that result in better understanding, (c) development of a diversity of problem solving techniques and (d) the promotion of responsibility (Watson & Chick, 1997) However, Stacey (1992) noted that some collaborative problem solving situations have shown the tendency among groups to choose ideas and approaches that are easily accessible, but not necessarily appropriate or correct, hence showing that a collaborative environment need not lead to successful conceptual development (Watson & Chick, 1997) While this may be the case Watson and Chick (1997) claim that collaboration or the use of group work in mathematics discourse is both encouraged and required by some curriculum documents (see, Australian Education Council [AEC], 1991; National Council of Teachers of Mathematics [NCTM], 1989, 2000) Co-operative methods that emphasize group goals and individual accountability significantly improve student achievement as well as have a positive effect on cross-ethnic relations and student attitudes towards school (Slavin, 1995)

According to Nason and Woodruff (2003, p 348) collaborative discursive component enhances and ensures the authenticity of classroom mathematical activities and enrich students understanding “of mathematical concepts and processes…” as well as “…their understanding about the nature and the discourse of mathematics” Moreover, a collaborative discursive component “enables teachers to individualize instruction and to accommodate students’ needs, interests, and abilities” (Lindquist, 1989; Watson, & Chick 2001)

2.9 Communities of learning and Computer Supported Collaborative

of knowledge-building (Brett, et al., 2002) Knowledge Forum is a single communal multimedia database into which students may enter various kinds of text or graphic notes and has been identified as potentially democratising contexts which allow for multiple “voices” to have space and opportunity to contribute and define the discourses (Brett et al., 2002; DiMauro & Jacobs, 1995; Sproull & Kiesler, 1991) Knowledge Forum allows for the provision of multiple perspectives that can shift the learner’s focus from the details of the task to the big picture, from isolated elements in

a situation to interacting relationships, or from particular events to generalized relationships (Brett et al., 2002)

Findings like this suggest that a CSCL environment could provide a particularly useful support for mathematically-anxious preservice teachers because the users themselves could define the function and disposition of the mathematical inquiry conference in order to meet their needs It is the view of Brett et al (2002) that CSCL environments have the potential to provide support mechanisms for making the uncertainty associated with the application of socio-constructivist principles during preservice teacher education less threatening

In knowledge-building communities (Scardamalia & Bereiter, 1996) students are engaged in the production of what Nason and Woodruff (2003) and Bereiter (2002) refer to as conceptual artefacts (e.g., ideas, models, principles, relationships, theories, interpretations etc) These artefacts can be discussed, tested, compared, and hypothetically modified (Nason & Woodruff, 2003)

A review of the research literature (e.g., Bereiter, 2002; Brett et al., 2002), however, indicates that most common “school math problems” do not provide contexts that facilitate knowledge-building activity that leads to the construction of mathematical conceptual artefacts According to Lesh (2000), in almost all “textbook” mathematical problems, students are required to search for an appropriate tool (e.g., operation, strategy) to get from the givens to the goals, and the product that students are asked to produce is a definitive response to a question or a situation that has been interpreted by someone else Most “textbook” mathematical problems thus require the students to produce “an answer” and not a complex conceptual artefact such as that generally required by most authentic mathematical problems found in the worlds outside of schools and higher education institutions Most textbook mathematical problems also do not require multiple cycles of designing, testing and refining that

occurs during the production of complex conceptual artefacts Most textbook mathematics problems therefore do not elicit the collaboration between people that most authentic mathematical problems outside of the educational institutes elicit (Nason & Woodruff, 2004) Another factor that limits the potential of most textbook mathematical problems is the nature of the answer produced by these types of problems Unlike complex conceptual artefacts that provide stimuli for ongoing discourse and other knowledge-building activity, the answers generated from textbook mathematical problems do not provide students much worth discussing

Nason and Woodruff (2003), however, have found that knowledge-building activity within CSCL environments can be greatly facilitated by having the participants engage in the investigations of novel mathematical tasks such as open-ended mathematical investigations (Becker, 2000; Morse & Davenport, 2000; Ogolla, 2003), number sense activities (McIntosh, 1995), and model-eliciting activities (Lesh

& Doerr, 2002) that: (a) generate conceptual artefacts (Bereiter, 2002) that participants can engage in discourse about, and (b) allow for multiple approaches and solutions

The types of conceptual artefacts that can be generated from these types of novel, open-ended mathematical activities can include a pattern, a procedure, a strategy, a method, a plan or a toolkit Nason and Woodruff (2003) argued that engagement in novel, open-ended mathematical activities provides rich contexts for mathematical knowledge-building discourse and thus facilitate the establishment and maintenance of online mathematics knowledge-building communities Furthermore, most of these types of activities are inherently motivating and maintain student engagement because of the associated experiences of success and value placed on the activities Findings like this suggest that engaging maths-anxious preservice teachers

in the exploration of novel, open-ended mathematical activities within a CSCL environment could provide a particularly useful support for maths-anxious preservice teachers, because the users themselves could define the function and disposition of the math inquiry conference in order to meet their needs

2.10 Summary

The literature reviewed in this chapter set out to investigate the causes and origins of maths-anxiety, its consequences, the impact it has on learning and teaching mathematics, and suggested ways of overcoming maths-anxiety It was found that

whilst studies suggest that teacher education programs can assist in changing the negative attitudes and mathematical self concepts of preservice and in-service primary school teachers to more positive ones (Bobis, & Cusworth, 1997; Philippou & Christou, 1997), other studies argue that teachers maintain their negativity toward mathematics and mathematics teaching after they begin to teach (Cockroft, 1982; Freeman & Smith, 1997; Kanes & Nisbet, 1994; Pateman, 1989) Indeed, there exists

a gap in the literature that does not address how to adequately overcome commencing preservice education students’ negative beliefs and anxieties about mathematics

To address the negative beliefs about mathematics, a number of studies suggested the re-examination of teacher education programs (e.g Ball, 2001; Carroll, 1998) For example, Carroll (1998) recommended that there must be a focus on the development of “confidence in the ideas of the teachers who must be encouraged to analyse and critically evaluate their current knowledge, beliefs and attitudes and modify [these] to include new ideas” (p 8)

Findings from the literature also suggested that a computer supported collaborative learning (CSCL) environment could provide support for maths-anxious preservice teachers It is the view of Brett et al (2002) that CSCL environments have the potential to provide support mechanisms for making the uncertainty associated with the application of socio-constructivist principles during preservice teacher education less threatening and resolvable

2.11 Theoretical framework for the study

The review of the research literature clearly indicates that if significant changes are to occur in negative beliefs about mathematics held by a significant proportion of preservice teacher education students, then first and foremost, the development of safe and non-threatening learning environments are crucial to ensure that maths-anxious pre-service student teachers can feel safe to explore and communicate about mathematics in a supportive group environment and to explore and relearn basic mathematical concepts (Bobis & Aldridge, 2002; Doerr & Tripp, 1999) (see Section 2.6.2) This is reflected in Component 1 of the theoretical framework presented in Figure 2.2 that was developed to inform the design and implementation of the study

Reducing Maths-anxiety and negative beliefs about Mathematics.

Safe and intimidating learning environment (e.g

non-Bobis & Aldrige 2002;

Doerr & Tripp, 1999)

(e.g Brett et al., 2002)

4.

Community of Learners

(e.g Brett et al., 2002;

Seashore, Krus e &

Bryk, 1995)

5.

On-Line Anxiety Survey

(Boekaerts, 2002)

Figure 2.2 The theoretical framework

Furthermore, the literature review also indicates that maths-anxious preservice student teachers need the opportunity to engage in practical inquiry and reflection about mathematics and mathematics teaching (Borko, Michalec, Timmons, & Siddle, 1997; McGowen & Davis, 2001; Stipek, Givvin, Salmon, & MacGyvers, 2001) (see Sections, 2.2, and 2.6) as can be afforded by engagement in novel mathematical tasks that allows for multiple approaches (e.g., model-eliciting problem solving activities, open-ended mathematical investigations, etc.) within the context of CSCL environments (see Sections, 2.8 and 2.9) This is reflected in Component 3 of the theoretical framework Component 2 relates to the types of mathematical activities that can be adopted to facilitate engagement in meaning-making mathematical activity

Component 3 of the theoretical framework relates to the benefits of CSCL environments As was noted in the literature review, CSCL environments may provide a particularly useful support for maths-anxious preservice teachers because the users themselves define the function and disposition of the math inquiry conference in order to meet their needs

Component 4 of the theoretical framework, Community of Learners, was based on research into the development of a community of learners (e.g., Brett et al., 2002; Watson & Chick, 2001) that was reviewed in Sections 2.8 and 2.9

Also, the literature review indicates that it is crucial to assist maths-anxious preservice students to become aware of their learned negative beliefs and emotions about learning mathematics, and that self-monitoring these emotions allows for them

to overcome and control maths-anxiety (Martinez & Martinez, 1996; Pintrich, 2000) (see Section 2.7) This is reflected in Component 5 of the theoretical framework This component of the framework which was manifested in the development of the thirty second Online-Anxiety Survey used in this research project to enable an awareness of participants emotional state was based on many aspects of Boekaerts (2002) ideas and research into students’ motivational experiences (see Section2.7)