addressing preservice student teachers’ negative beliefs...
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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
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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…………………………………….
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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.
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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.
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TABLE OF CONTENTS DECLARATION ……………………………………………………………….. iACKNOWLEDGMENT ……………………………………………………… iiABSTRACT …………………………………………………………………….. iii Chapter 1
1.1 Introduction ……………………………………………………………… 11.2 Background of study …………………………………………………….. 11.3 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 ………………………………………………… 51.5 Chapter overview ………………………………………………………... 61.6 Summary ………………………………………………………………… 6
Chapter 2
2.1 Introduction ……………………………………………………………… 82.2 Maths-anxiety ……………………………………………………………. 82.3 Consequences of maths-anxiety …………………………………………. 122.4 Teacher beliefs about mathematics ……………………………………… 132.5 Prior school experiences and the origins and the development of negative
maths-beliefs …………………………………………………………….. 152.6 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 …………………………………………… 212.8 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 272.10 Summary ………………………………………………............................. 292.11 Theoretical framework for the study ……………………………………... 30
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Chapter 3
3.1 Introduction ……………………………………………………………… 323.2 Research methodology …………………………………………………... 323.3 Participants ………………………………………………………………. 333.4 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 ……………………………………………………………… 464.2 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 ……………………………………… 634.4 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 ………………………………………... 774.6 Summary …………………………………………………………………. 80
Chapter 5
5.1 Introduction ………………………………………………………………. 825.2 Overview of study ………………………………………………………... 825.3 Overview of results ………………………………………………………. 835.4 Limitations ……………………………………………………………….. 87
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5.5 Implications ………………………………………………………………. 885.6 Summary and recommendations …………………………………………. 89
References ……………………………………………………………………… 91 Appendix 1: Phone interview questions ………………………………………… 103Appendix 2: Pre-enactment Interview ………………………………………….. 104Appendix 3:Post-enactment interview ……..………………………………….. 105Appendix 4: Online Anxiety Survey ……………………………………………. 106
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LIST OF TABLES Chapter 3
Table 3.1 The four mathematical activities …………………………………. 37 Chapter 4
Table 4.1 The nature of mathematics ………………………………………. 48Table 4.2 Reasons for teaching mathematics ……………………………….. 48Table 4.3 Teacher knowledge and qualities ……………………………….... 49Table 4.4 Maths-confidence ……………………………………………….... 51Table 4.5 The origins of maths-anxiety …………………………………….. 52Table 4.6 Situations causing maths-anxiety ………………………………... 54Table 4.7 Types of mathematics causing maths-anxiety ………………….... 55Table 4.8 Perceptions of how to overcome maths-anxiety …………………. 55Table 4.9 Perceptions on how to reduce maths-anxiety in future students …. 56Table 4.10 The nature of mathematics ……………………………………….. 59Table 4.11 The relevance of mathematics ………………………………….... 59Table 4.12 Teacher knowledge ……………………………………………..... 60Table 4.13 Maths-confidence ………………………………………………… 61Table 4.14 Pairwise comparison: Overall results …………………………….. 66Table 4.15 Pairwise comparison: Session one results ………………………... 68Table 4.16 Pairwise comparison: Session two results ……………………….. 70Table 4.17 Pairwise comparison: Session three results ……………………… 73Table 4.18 Pairwise comparison: Session four results ……………………….. 76Table 4.19 Perceptions of computer-mediated software ……………………... 78
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LIST OF FIGURES
Chapter 2 Figure 2.1 The process of solving maths problems ………………................ 11Figure 2.2. The theoretical framework ………………………………………. 30 Chapter 3 Figure 3.1 Intervention Program …………………………………………….. 33Figure 3.2 Online Anxiety Survey …………………………........................... 38Figure 3.3 MipPad model and tabular representation ……………………….. 39Figure 3.4 MipPad model, language and symbol representation ……………. 40Figure 3.5 Shape and measurement activity …………………………………. 41Figure 3.6 Number and shape activity ……………………………………….. 42Figure 3.7 Division operation activity ……………………………………….. 43 Chapter 4 Figure 4.1 Box plots overall positive feelings………………………………… 67Figure 4.2 Box plots overall negative feelings…….………………………….. 68Figure 4.3 Number sense activity (positive feelings responses)………………. 69Figure 4.4 Number sense activity (negative feeling responses)……………….. 69Figure 4.5 Space and measurement activity (positive feeling responses)……... 71Figure 4.6 Space and measurement activity (negative feeling responses)…….. 72Figure 4.7 Number and shape activity (positive feelings responses)………….. 74Figure 4.8 Number and shape activity (negative feelings responses)…………. 75Figure 4.9 Division operation activity (positive feelings responses)………….. 76Figure 4.10 Division operation activity (negative feelings responses)…………. 77
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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
related issues of negative beliefs about mathematics and high levels of anxiety
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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:
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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,
1993; Roulet, 2000; Schofield, 1981).
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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
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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
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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.
1.5 Chapter overview 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
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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.
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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).
2.2 Maths-anxiety 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
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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). Although
early 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
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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
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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:
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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
13
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,
14
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 relativist2
ways 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
1 This is very similar to Ernest’s (1991) classification of absolutist beliefs 2 This is very similar to Ernest’s (1991) classification of fallabilist beliefs
15
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”.
16
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
17
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
18
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 non-
threatening and non-intimidating.
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
19
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.
20
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
21
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
22
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
23
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,
24
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,
25
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
26
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 Learning (CSCL) Communities of learning have been used effectively as means to promote
reflective educational practices and researchers are calling for these environments to
be used as means to advance the status of the profession (Brett et al. 2002; Seashore,
Kruse, & Bryk, 1995). Brown and Campione (1994) defined a learning community as
a group of individuals who engage in discourse for the purpose of advancing the
knowledge of a collective--to participate in what Scardamalia and Bereiter (1995) call
knowledge-building. Computer supported collaborative learning (CSCL)
environments such as Knowledge Forum® have been successful in providing this sort
27
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
28
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
29
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.
30
Reducing Maths-anxiety and negative beliefs about Mathematics.
1.Safe and non-
intimidating learning environment (e.g.
Bobis & Aldrige 2002; Doerr & Tripp, 1999)
2.Engagment in novel
open-ended mathematical
activities(e.g. Becker, 2000).
3.Computer Supported
Collaborative Learning (CSCL)
(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.
31
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).
32
CHAPTER 3
RESEARCH DESIGN AND METHODOLOGY
3.1 Introduction The purpose of this research study was to investigate whether supporting
sixteen self-identified maths-anxious preservice student teachers within a supportive
environment provided by a CSCL community would reduce their negative beliefs and
high levels of anxiety about mathematics. In this chapter, the research design and
methodologies utilized in the study are presented.
3.2 Research Methodology An exploratory mixed method design was used in this study where the process
began with gathering qualitative data to explore phenomenon, followed by the
collection of quantitative data to elaborate on relationships found in the qualitative
data (Creswell, 2002).
The strength of this design is in its emphasis on the qualitative aspect of the
study. The qualitative aspect of the study utilizes what Charmaz (1990) refers to as a
constructivist approach where there is a need to understand and to explore in this
instance, maths-anxiety in preservice student teachers. A grounded theory design was
used to inform the analysis of the qualitative aspects of the study as it provided a
means for developing theory “grounded” in the participant’s views rather than using
existing theory and also it provided for the modification of existing theories. Creswell
(2002) defines grounded theory as “a systematic, qualitative procedure used to
generate a theory that explains, at a broad conceptual level, a process, an action, or
interaction about a substantive topic” (p. 439). Grounded theory approach is both
rigorous and systematic and offers a macro-picture rather than a micro-analysis of
educational situations (Creswell, 2002). This approach allows for the generation of an
understanding of a process related to a substantive topic such as math-anxiety a
process, that in grounded theory research is “a sequence of actions and interactions
among people and events pertaining to a topic” (Creswell, 2002, p. 448).
33
The methodology used in this study incorporated three stages, namely:
Phase 1: Identification of participants and pre-interviews.
Phase 2: Enactment of intervention program.
Phase 3: Summative evaluation of the intervention program.
The following sections: 3.5.1, 3.5.2 and 3.5.3 will fully describe each phase of
the study.
Figure 3.1 provides a diagrammatic representation of the study’s three phases.
4.
5.
Pre- Interview2.
Workshops1.
Online Anxiety Survey
2.
Maths Activities (KF and M ipPad)
3.
Reflections1.
Post-Interview2.
Self Identification of M aths Anxiety
1.
Phase 3 Evaluat ion
Phase 1 Identif ication Phase 2 Intervent ion
Cycl ical components of
the model
Non-cyclical components of
the model
Key
Figure 3.1. Intervention Program
3.3 Participants Initially, a cohort of approximately 300 third-year preservice primary teacher
education students (254 female and 46 male) enrolled in a mathematics education
curriculum unit at a major metropolitan university in Eastern Australia were
approached to take part in the study via an email message. The email indicated that
the researchers were seeking self-identified maths-anxious preservice students who
were willing to take part in the research project. Within a period of a few days, over
45 students replied to this email and volunteered to participate in this project. From
this group, sixteen (n=16) students were purposively sampled (Creswell, 2002) to take
part in the study: fifteen female preservice education students and one male preservice
education student. All but three of the sample of preservice education students, were
mature-aged students. The criteria for the selection included: a) Degree of maths
anxiety, b) Motivation to participate in regular one-hour workshops, and 3) Access to
the internet. This was ascertained via the means of a brief semi-structured telephone
34
interview (Appendix 1). This formed the initial phase of the Identification phase of
the three-phased Intervention Model (see Figure 3.1).
3.4 Collection of data There were four instruments used in the collection of research data:
1. Semi-structured pre- and post-interviews.
2. Online Anxiety Survey
3. Knowledge Forum Notes
4. Written reflections.
The development and theoretical underpinnings each of these instruments and
their purpose are discussed in this section.
3.4.1 Semi-structured Pre-enactment and Post-enactment Interviews.
The Pre-enactment Interview helped to identify the causes of preservice
students’ maths-anxiety and particular mathematics that were anxiety-causing (see
Appendix 2). The analysis of data was used to inform the selection of appropriate
mathematical learning activities in later stages of the study. The post-enactment
interviews (see Appendix 3) that took place at the end of the study provided
information regarding any changes that may or may not have occurred to the
participants’ maths-anxiety. The collection of individual data was necessary as it
ensured that reliable information was gained and it provided the opportunity for
participants to clarify any concerns (Creswell, 2002).
3.4.2 The Online Anxiety Survey
The purpose for the Online Anxiety Survey (Uusimaki, Yeh, & Nason, 2003)
was to: (a) have the participants recognize and accept their feelings about the
introduced mathematical activity, both before commencing and at the completion of
the mathematical activity, and (b) measure trends in the participants’ negative or
positive emotions about mathematics as their learning process was unfolding. Self-
reporting instruments such as rating scales are commonly used to measure peoples’
attitudes or reactions to various stimuli and as a visual analogue a scale “is a reliable,
valid and sensitive self-report measure” (Gift, 1989, p. 288). Particularly, the scale is
helpful in that it avoids the problems of language, so the interpretations to
descriptions are avoided (Gift, 1989). The on-line anxiety scale (see Appendix 4) that
35
was developed for the study was based on ideas by Ainley and Hidi (2000) and
Boekaerts (2002).
3.4.3 Knowledge Forum notes
Knowledge Forum notes that included as attachments the participants’
individual and group mathematical models plus their comments about other
participants’ models were collected and viewed on a regular basis by the researcher.
Knowledge Forum has automatic tracking programs which provide data about
patterns of browsing and commenting. This enabled the researcher also to assess
changes in the participants’ patterns of collaboration and discourse.
3.4.4 Written reflections
The written personal reflections focused on what the participants learnt about
their own feelings in undertaking the project and mathematical activities. Participants
were encouraged to document issues that their group encountered in the process of
group work (Hare & O’Neill, 2000; Walker, 1985). This allowed also for the
development of meta-cognition as the participants were able to reflect upon their
group thinking as well as their own, make note of their group difficulties such as, not
working as a team and logistical problems preventing regular meeting times, as well
as progressively commenting on how the group dealt with any difficulties.
Participants were also given the opportunity to articulate these personal reflections in
their post-test interviews.
3.5 Procedure The procedures for each of the three phases of this study will be described
under this section together with the specific details about each of the four
mathematical activities are provided in Section 3.5.2.
3.5.1 Phase 1: Identification of origins of maths-anxiety In order to ascertain the causes and negative feelings about mathematics held
by the participants, individual thirty minute semi-structured interviews were
conducted. Prior to the submission of questions, verbal permission was sought from
each participant to tape-record the interviews for subsequent transcription. In order to
protect the confidentiality of the participants, each participant was immediately given
an anonymous pseudonym.
36
The design of questions was informed by the research literature. Questions 1
and 2 that were based on the philosophy of mathematics were informed by Ernest
(2000) (see Section 2.4). Questions 3 and 4 related to teacher knowledge and teacher
qualities (e.g., Ball & Cohen, 1999: Richardson, 1999) (see Section 2.6.3). Question 5
related to maths-confidence (e.g. Barbalet, 1998; Ingleton & O’Regan, 1998;
Martinez & Martinex 1996) (see Section 2.2) and Question 6 to computer confidence
(Brett et al, 2002) (see Section 2.9). Questions 9, and 10 related to maths-anxiety
(e.g., Martinez & Martinez, 1996; Smith & Smith, 1998) (see Section 2.2). Questions
7 and 8 related to the formation of beliefs and attitudes towards mathematics (e.g.,
Cornell, 1999; Emenaker, 1996; Kloosterman, Raymond & Emenaker, 1993) (see
Section 2.6.1). Questions 11 and 12 related to means of overcoming math-anxiety
(Brett et al., 2002; Carroll, 1998; Raymond, 1997) (see Section 2.6). The focus
questions for both the pre-enactment interview and the post-enactment interviews are
presented in Appendices 1 and 2.
3.5.2 Phase 2: Enactment of Intervention Program
There were four mathematics activities in this phase:
Activity 1: Number sense activity
Activity 2: Space and Measurement activity
Activity 3: Number and shape activity
Activity 4: Division operation activity
Each of the mathematical activities chosen for this phase were carefully and
deliberately selected based on participants’ interview responses with respect to the
causes of their maths-anxiety and the types of mathematical learning experiences they
had had as students of mathematics. For example, in Activity 2, the Space and
measurement activity was selected because in the interviews, most of the participants
indicated anxiety about spatial knowledge and algebra. This Space and measurement
mathematical activity involved the use of spatial diagrams in the open-ended
investigation of the relationship between perimeter and area of rectangles towards the
development of general rules that could be expressed in the form of natural language,
arithmetic equations and/or algebraic rules.
Based on the review of the research literature, it was expected that the
mathematical activities chosen would reduce participants’ high levels of anxiety
whilst also providing the participants with opportunities to simultaneously enhance
37
their repertoires of knowledge about mathematics. The various mathematical activities
chosen for the study are presented in Table 3.1
Table 3.1 The four mathematical activities.
Syllabus Strand
Mathematical Activity
Number Operations -
Mental Computation
What are the best way(s) of working out problems such as 68 + 49 in your head?
Space and Measurement
Farmer Browns best sheep paddock fronts the river and he has 100 metres of fencing. He needs help to find out the largest rectangular area he can enclose using the 100 metres of fencing.
Algebra In ancient times, people discovered that numbers have shapes. For example, they discovered that all odd numbers had the shape of an L or a gnoman (the L-shaped part of a sundial) * * * For example: 3 * * and 5 * * *
• Using MipPad, see if you can generate a rule to work out the sum of the first 5 odd numbers.
• Then try to develop a rule for the sum of the first 10 odd numbers.
• Then try to develop a general rule.
Number Operations
a) How can 3 X 19 be generated from 3 X 20 and 3 X 15 b) How can you model the following two notions about division: a. Partitioning (4 X ? = 24) b. Quotitioning (? X 4 = 24)
Collins, Brown, and Newman (1989) suggest that “ideal” learning
environments direct learners’ cognitive activity toward goals that are concerned with
gaining knowledge, what Bereiter and Scardamalia (1989) call personal knowledge
building goals. Hence, to ensure a conducive learning environment, workshop
situations were deliberately designed to portray a safe and supportive environment to
help participants feel secure to take risks, and feel supported by each other.
In the first workshop participants were allocated into small groups of three
with whom they collaborated as a team throughout the study. Participants were then
introduced to a 30 seconds pre and post Online Anxiety Survey (Uusimaki, Yeh, &
38
Nason, 2003) to measure their subjective experiences prior to and after partaking in
each of the various mathematical activities introduced in the workshop situations (See
Appendix 4 for example of both pre- and post- Online Anxiety Survey).
Figure 3.2. Online Anxiety Survey.
Figure 3.2 shows an example of the pre-session Online Anxiety Survey and
the choices of the affective responses used in the study. It takes approximately 30
seconds to complete this specifically designed Likert style online anxiety activity. To
record the emotional response triggered by the mathematical activity the participant
was required to slide a bar horizontally along a scale to indicate his or her feeling both
prior to and at the completion of the mathematical activity. Once the participant had
completed the activity, the program then records a numerical value (unknown to the
participant) that was stored as a percentage.
In order to assist participants with the development of their mathematical
models, they were introduced to the functions of the computer mediated software
programs Knowledge Forum and Mathematics Ideas and Process Pad (MipPad) (Yeh
& Nason, 2003). The purpose for using Knowledge Forum was that it provides an
effective platform for facilitating learning that is centred on ideas and deeper levels of
understanding (Brett, et al. 2002). Also, a safe environment where participants’
interpretations are revealed and shared as afforded by Knowledge Forum provide
participants with a sense of ownership over what and how they interpret and make
39
sense of their own learning. However, more importantly participants come to
appreciate how other participants interpret and make sense of the various
mathematical activities. After the completion of the mathematical model, participants
were required to (a) post the model on the Knowledge Forum (b) make comments on
other participants/groups’ models, and (c) revise their model.
In Activities 2, 3 and 4 participants were asked to generate mathematical
models with the help of MipPad (Yeh & Nason, 2003), a computer mediated
comprehension tool that allows for the viewing of each participant’s learning process
and development of their mathematical model.
MipPad provides various icons the user can select and use as stamps in a paste
like fashion and the resulting diagram or model can then be explained via the
inclusion of text. Comprehension modelling tools (Woodruff & Nason, 2003a; 2003b)
such as MipPad also enables users to use different types of mathematical
representations simultaneously. For example, in Figure 3.3, both the array model and
tabular representations are utilised to facilitate the process of solving a problem.
Figure 3.3. MipPad model and tabular representation.
In Figure 3.4 the array model, natural language and mathematical symbol
representations are utilised. Following the completion of their math modelling with
MipPad, users can then use the animation feature of MipPad to “replay” their model-
building procedure. That is, they can use the “reverse” function of MipPad to return to
the first steps in the process of generating the solution to the mathematical activity
40
and then use the “play” function of MipPad to view in sequence the process that led to
the completion of the activity.
Figure 3.4. MipPad model, language and symbol representation
3.5.2.1 Activity 1: Number sense activity
As in every other activity, there were four components within this activity:
1. Introduction of mathematical activities, with whole-group discussions.
2. A pre-activity Online Anxiety Survey.
3. Computer mediated collaborative knowledge-building.
4. A post-activity Online Anxiety Survey
The rationale beginning with a relatively easy mathematical activity based on
number sense ensured the successful completion of the task by all participants. The
activity also established the recognition on the participants’ part that there are many
different possible ways of reaching a solution. The rationale for selecting a mental
computation activity such as, ‘working out 68 + 49 in the head’ was based on the
notion that it leads to better number sense, flexibility working with numbers as well as
it allows for the recognition of diversity in arriving at the same answer (McIntosh,
1995). Whilst mental computation incorporates mental arithmetic it predominantly
focuses on the thinking processes adopted in the strategy rather than the product.
Hence mental computation is a personal process (McIntosh, 1995; Tucker, et al.,
2001).
41
3.5.2.2. Activity 2: Space and Measurement mathematical activity
In this mathematical activity, the participants were presented with the
following activity and diagram (Figure 3.5):
Farmer Brown has a field on the banks of the river. He has 100 metres
of fencing to enclose three sides of a rectangular grazing area. What
would be the dimensions of the rectangle with the largest possible area
that he could enclose with the 100 metres of fencing? What if he had
50 metres of fencing? 200 metres? 1000 metres?
Figure 3.5. Space and Measurement activity
The rationale for selecting this mathematical activity was that it was a problem
that could be solved using a variety of problem-solving strategies such as trial-and-
error, make a simpler problem, act it out, make a table and look for a pattern. Also, it
was a mathematical problem that enabled many different levels of “correct” answers.
For example, a participant could succeed at generating a particular numerical answer,
a verbal rule, a numerical rule or a generalised algebraic rule. The Farmer Brown
mathematical activity also provided a context for subtly inducting the participants into
authentic engagement with measurement concepts (perimeter and area) and algebra,
two of the mathematical domains that had been identified in the pre-interviews as
being anxiety-causing. Also, the research literature indicates that these two domains
of mathematical knowledge are amongst the least understood by preservice student
teachers (see Baturo & Nason, 1996; Nason & Verhey, 1988; Simon & Blume, 1994).
River
50 metres (Length)
25 metres (Breadth) 25 metres 1250 square metres (Area)
42
3.5.2.3. Activity 3: Number and shape activity
In this mathematical activity, the participants were presented with the
following problem (Figure 3.6):
In ancient times, people discovered that numbers have shapes. For example, they discovered that all odd numbers had the shape of an L or a gnoman (the L-shaped part of a sundial).
* * *
For example: 3 * * and 5 * * * Using MipPad, see if you can generate a rule to work out the
sum of the first 5 odd numbers. Then try to develop a rule for the sum of the first 10 odd
numbers. Then try to develop a general rule.
Figure 3.6. Number and shape activity.
The rationale for this activity was to integrate number sense notions with the
study of shape whilst at the same time extending the participants’ experiences with
algebra beyond what had been done in the previous mathematical activity. It was also
chosen to provide the participants with a challenging activity that would require them
to create rather than just apply existing mathematical knowledge.
3.5.2.4. Activity 4: Division operation activity
Division is probably the least understood mathematical operation amongst
teachers (Ball, Lubienski & Mewborn, 2001). Much of the confusion associated with
the division operation can be traced back to its dual personalities of partitive (sharing)
division and quotitive (continued equal subtraction) division. Research on teachers’
knowledge of division for example, has revealed that teachers use predominantly a
partitive or sharing conception of division (Ball, 1990a, 1990b; Graeber, Tirosh &
Glover, 1989; Simon, 1993). This has resulted in many teachers not being able to
reason through problems involving division by zero, division by fractions and
decimals, or dividing a smaller number by a larger number. To understand the
difference between partitive and quotitive division participants were given examples
and explanations on both. In partitive division, the total and the number of shares are
43
given and the number of each share is the unknown quantity that has to be generated
or computed.
To help the participants gain deeper insights into the division operation, they
were presented with the following activity (Figure 3.7):
How can you model the following two notions about division:
a. Partitioning (4 x ? = 24)
b. Quotitioning (? x 4 = 24)
Figure 3.7. Division operation activity
Using MipPad, the participants were required to generate models to explain
both aspects of division.
3.5.3 Phase 3: Summative evaluation
At the end of the study, all participants were required to produce a written
reflection about their experiences in the project. These were then analysed in order to
identify potential relationships between perception of higher mathematical
competence and lower levels of anxiety.
Following the written reflections, semi-structured interviews were conducted
by the researcher to further investigate any changes that may or may not have
occurred. These interviews also allowed for the verbalising of participants written
experiences in the project. These thirty minute interviews were tape-recorded and
subsequently transcribed.
3.6 Data analysis Both qualitative and quantitative methods of data analysis were utilized in this
study. Qualitative data from the Pre- and Post-enactment Interviews, observations,
Knowledge Forum shared data base and the written reflections were analysed utilizing
a grounded theory approach. Quantitative data from the Online Anxiety Survey was
analysed using multivariate analysis of variance (MANOVA) with repeated measures
and graphical analysis.
3.6.1 Analysis of qualitative data
Incorporating a grounded theory approach, data analysis was conducted in an
ongoing hermeneutic cycle (Guba & Lincoln, 1989). Collected data in the form of
observations, Knowledge Forum notes and Online Anxiety Surveys were reviewed
44
weekly to identify trends to inform further data collection (Strauss & Corbin, 1998).
As trends were identified these were then compared with existing theory.
3.6.1.1 Analysis of Pre- and Post-enactment Interview data
The analysis of this date proceeded in this way:
1. Familiarization and organisation of interview transcripts prior to formal
analysis.
2. Transcripts of interviews were analysed, and data reduced by determining
the frequencies for the major variables in the proposed study. Thus a
content analysis was conducted (Burns, 2000).
3. Identification of major themes and subsequent categories under each
theme. Analysing techniques were based on a grounded theory design
(Creswell, 2002) that involved constantly comparing data with emerging
theories.
The categorical or nominal data that was extracted from interviews required a
method of standardising the frequency distribution of responses that would allow
comparisons. Proportions were converted into percentages to allow for these
comparisons, a comparison of the number of cases in a given category with a total
size of the distribution (Levin, 1977)
3.6.1.2 Analysis of written reflections
Each reflection paper was scrutinised to allow for comparisons, contrasts and
insights to be made and demonstrated. This was followed by coding, that was
conducted to determine themes, issues, topics, concepts and propositions (Burns,
2000). A content analysis was carried out (Burns, 2000) with the frequencies of the
major variables determined to permit the analysis and comparison of meanings.
3.6.1.3 Analysis of Knowledge Forum Notes
Knowledge Forum notes were analysed and compared to assess changes in the
participants’ patterns of collaboration and discourse. In particular this analysis
focused both on the models participants produced and the notes, the thinking behind
these notes and the processes used and the mathematics learned during the workshop
situation. This data was analysed after each of the implementations of the CSCL
workshop environment.
45
3.6.2 Analysis of Online Anxiety Survey quantitative data
Prior to, and after their model building activity, the students participated in the
Online Anxiety Survey, to ascertain changes to their perceptions about the various
mathematical activities. The analysis of this data facilitated the identification of key
episodes that led to the changes in perceptions. A graphical analysis in the form of a
bar graph was created for each participant, and for each of the workshop activities.
This increased the readability of the survey findings (Levin, 1977). A comparison of
pre- and post percentages were made in relation to each of the six affective/feeling
responses via a graphical analysis in the form of a line graph. That is, the on-line
anxiety scale recorded an interval level of measurement (Levin, 1977) as the
respondents indicated a measure “which yield equal intervals between points on the
scale” (Levin, 1977, p.6). Parametric tests were used to analyse the data with the
statistical data analysis used in this study first tested for overall significant differences
between pre-intervention and post-intervention scores. This analysis was based on a
multivariate analysis of variance (MANOVA) with repeated measures. The
independent variable was the pre-and post- activity intervention whilst the six feeling
responses (comfortable, fine, confident, worry, nervousness and frustrated) were the
dependent variables or response categories. Pillai’s Trace test was used due to small
sample size (n=16). Further analysis was used by means of paired sample T-test to
determine whether there was significant change within each of the feeling responses.
Box plots were used to further elaborate on findings and to give an overall
visual record of participants’ pre- and post-activity anxiety measure.
3.7 Summary
In this chapter, a rationale for the use of the exploratory mixed design was
presented. The chapter also outlined the selection and criteria for the choice of the
sixteen participants. The rationale for the three phase Intervention Program was given
and the computer-mediated tools MipPad and Knowledge Forum introduced. The use
of multiple data from pre- and post-interviews, written reflections Knowledge Forum
notes and the data from the Online Anxiety Survey ensured that the research findings
were consistent with the data collected.
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CHAPTER 4
RESULTS
4.1 Introduction
The results of this research study will be presented in four sections. The first
section focuses on the results derived from the analysis of the qualitative data from
the pre- and post- enactment interviews. The second section focuses on the results
from the analysis of qualitative data from the reflection documents produced by each
of the participants. The third section of the chapter focuses on the results derived from
the analysis of quantitative data from the Online Anxiety Survey. In this section, the
results derived from the analysis of quantitative data will be complemented by
qualitative data derived from the post- enactment interviews. The following section
focuses on the participants’ perceptions with regards to the information and
communication technology tools. The chapter concludes with a summary of the
findings derived from the analysis of the qualitative and quantitative data.
4.2 Results from interview data In this section, the results from the analysis of data from the pre-enactment interview
will first be presented. Following this, the results from the analysis of the post-
enactment interview data will be presented and compared with the outcomes derived
from the analysis of pre-enactment interview data.
4.2.1 Pre-interview results
Based on the literature review presented in Chapter 2, the following
information was needed to inform the planning of the intervention program that aimed
to facilitate preservice teachers’ overcoming their negative beliefs and anxieties about
mathematics:
1. Preservice teacher students’ perception about the nature of mathematics.
2. Reasons for teaching mathematics.
3. Teacher knowledge for teaching mathematics.
4. Teacher qualities for teaching mathematics.
5. Maths-confidence.
6. Origins of negative beliefs and anxieties towards mathematics.
7. Situations and types of mathematics causing maths-anxiety.
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8. Perceptions about ways to overcome negative beliefs and anxieties about
mathematics.
9. Ways these preservice teachers would assist their students to
prevent/overcome maths-anxiety.
4.2.1.1 Preservice teachers’ perception about the nature of mathematics
When asked to describe what they thought mathematics is, all of the
participants gave responses that indicated that they had multi-dimensional
conceptions’ about the nature of mathematics. For example, Linda thought
mathematics is:
I suppose mathematics is usually related to numbers and measurements and data and those sorts of things, to help you live in our world. That’s a bad answer but a hard question.
This response indicated that her conception of mathematics focused on numeration
and measurement and that it had a utilitarian purpose.
Rose also felt that mathematics focused on numbers and that it had a utilitarian
purpose. However, she also felt that mathematics was something she did not like
doing. She explained mathematics as:
Numbers! A headache! Mathematics to me is basically a subject and a subject that I’m pretty much scared of and avoid at all costs. Yet, it’s crucial to every day life and I understand that, but I still don’t like doing it at all.
Marge’s conception also focused on numbers and the utilitarian aspects of
mathematics. However, she also felt that mathematics was about problem solving too.
She indicated that:
Before I came to Uni to me it [mathematics] was numbers and working things out but after having done the first two mathematics units I’ve realized that you can relate it to real world things, that it’s more problem solving about what you’re suppose… what you need to work out or whatever
The analysis found certain commonalities in the participants’ responses to the
focus question, What is mathematics? These commonalities are listed in Table 4.1. As
is indicated in this table, the most common responses related to mathematics being
hard (25%), problem-solving (24%), real world and numbers (both 22%).
Table 4.1
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The nature of mathematics
Category of response Percentages %
Hard 25
Problem solving 24
Real world 22
Numbers 22
Other 13
Procedures 12
Different concepts 10
4.2.1.2 Reasons for teaching mathematics
All of the participants’ responses to this question focused on the utilitarian
aspect of mathematics. Sixty-nine percent of the participants indicated that
mathematics should be taught because it was relevant for real world activities such as
shopping, building fences and going to the bank. Thirty one percent of participants
believed that mathematical skills should be taught because it prepared students for
their futures.
Table 4.2
Reasons for teaching mathematics
Category of response Percentages %
Relevant for real world activities 69
Life skills to prepare for future 31
These two categories of responses were included in Sally’s response. She
stated that:
Kids have to….children need life skills and to be an active participant in our life they need certain skills, reading, writing is one of them and maths is another, they need to know where numbers fit into their life they have to know how to use numbers and how to work with numbers ‘cause numbers form a part of our every day life.
Similarly Rose noted:
I think because our every day lives involve it [mathematics] and because in every single way every profession that anybody could ever do, any career, will involve some form of mathematics whether it be engineering where it’s big algebraic type equations or work in a supermarket
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4.2.1.3 Teacher knowledge and qualities
In terms of what the participants felt constituted teacher knowledge, the results
indicated that pedagogical knowledge was perceived to be more important than
specific subject matter knowledge. Fifty-seven percent of the participants identified
the ability to teach mathematics as being important when teaching mathematics.
Forty-three percent of the participants’ in turn thought that conceptual knowledge was
important.
Table 4.3
Teacher knowledge and qualities
Category of response Percentages %
What knowledge is needed to teach mathematics?
Pedagogical knowledge 57
Conceptual knowledge 43
What are the important teacher qualities? Personal qualities 65
Professional qualities 35
Many of the participants felt that teachers needed both pedagogical content
knowledge (PCK) and subject-matter knowledge (SMK). Ann, for example, stated
that teachers:
…need to know how to teach it and also understand what they’re teaching and I guess they need to value it. They’ve got to find it important.
When discussing SMK knowledge, many of the participants focussed on knowledge
of mathematical concepts and processes. Some extended this to knowledge of
mathematical principles. Tiffany for example noted that:
They [the teacher] need to have a good basic understanding of all the mathematical principles, especially of what they are teaching.
Some of the participants, however, extended the notion of SMK knowledge to include
positive dispositions to mathematics3. Carla, for example, indicated that it is not
enough to have a good basic understanding of mathematics rather teachers also need
to have:
3 Ball (2001) indicated that dispositions towards mathematics is a very important dimension of subject-matter knowledge
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A love for mathematics and to be confident with interacting with the relationship of mathematics, and then knowing how to impart that. For example, I’ve only done well in mathematics at uni. I was completely dumb and hopeless in primary and secondary school but with my mathematics lecturer [at uni] I felt very comfortable and very at ease that is until I went out on prac…where all the same old labels that I had and prejudices and everything came up and I just froze. If I’m going to be able to teach mathematics and have children enjoying it and loving it, I have to be consistent otherwise I’m just going to pass on those underground, you know, attitudes that were passed onto me.
Sixty-five percent of the participants thought that the personal qualities
teachers’ needed in order to teach mathematics well included patience, understanding,
enthusiasm and empathy. For example, Linda suggested the teacher needed to have:
A lot of compassion and a lot of patience and a good understanding of the children in her class and how they think and where they’re coming from. They also need to have the ability to see things from many different perspectives and to be able to think outside of the way that she sees things.
Susan explained that a personal quality she felt a teacher needed to have is:
Confidence, definitely, I guess not a love for the subject but you know an interest in the subject and just an overall comfortableness with the subject so that the lack of confidence or the resentment towards the subject isn’t actually passed onto the kids.
Thirty-five percent of the participants thought that the professional qualities a
teacher needed to teach mathematics well included being flexible and knowledgeable.
Petra felt teachers needed to have:
Knowledge… well, you need a good understanding of mathematics, you need to be well equipped to teach it, you have to understand it and you have to bring different ways of teaching because there’s so many different learners. I am such a hands on learner and kids love you know, learning by doing instead of just rote learning which is the way I was taught.
Ally thought that: …a teacher has to have a lot of real world knowledge of mathematics how it can be applied and not just knowledge of what has to be taught in the classroom. As a teacher it is about being able to help develop skills that they [kids] can take outside the classroom and remember and use on a practical basis.
It was interesting to note that when defining the professional quality of being
knowledgeable, Petra and Ally focused on both SMK and PCK. Ally’s definition of
SMK also was very interesting because it went beyond understanding of content
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knowledge to include knowledge about what Shulman (1987) and Ball (2001) refer to
as knowledge about mathematics in culture and society.
4.2.1.4 Maths-confidence
When the participants were asked about their maths-confidence, most
participants indicated that they did not feel very confident about mathematics. The
results in regards to the participants’ maths-confidence are presented in Table 4.4.
Table 4.4.
Maths-confidence
Category of responses Percentages %
How confident are you about your mathematical skills?
Not at all 19
Not very 50
Semi-confident 31
Quite confident 0
Confident 0
Many of the participants’ responses to this question revealed the reasons for
their lack of confidence. Some indicated that the rote-learnt nature of their repertoire
of mathematical knowledge was the main reason for feeling not confident. This was
typified by Ron’s response:
…I just sort of passed mathematics at school. I had to rote-learn. I had to learn formulas and those sorts of concepts and I didn’t really understand them. For tests I just had to learn what we had to do but I never actually understood what we did.
Other participants indicated that their reasons for feeling not confident were related to
concerns about teaching maths. For example, Karen felt her confidence in maths was:
Basically okay, my main concern would be teaching the higher grades I think. With the younger grades, I think I would be O.K. but when it comes to teaching the higher grades, that’s where I’m anxious and not very confident
4.2.1.5 The origins of maths-anxiety
The analysis of data (see Table 4.5) revealed that 66% of the participants
perceived that their negative beliefs and anxiety towards maths emerged in primary
school.
Table 4.5
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The origins of maths-anxiety
Category of responses Percentages %
When did you learn to dislike mathematics?
Primary school 66
Secondary school 22
Tertiary education 11
Why did you learn to dislike mathematics?
Teacher 72
Can’t remember 28
Nineteen percent of the respondents identified negative experiences as early as
Year 1 and Year 2. Tina recalls the time in Year 1 as a time when:
I used to make lots of mistakes and I was always frightened… I vividly remember ...... getting into huge trouble because I couldn’t fit a puzzle together. I vividly remember that. Just absolutely getting caned by this teacher.
Donna remembers in Year 4 how:
We had a whole heap of sums that we had to do and I worked through them and I got all of them wrong and then I was made to stand up and I was just belittled and so from then on I believed I couldn’t do it (maths).
One of the similarities between Tina and Donna’s recollections seems to lie with their
teachers. These two respondents were among the 56% of participants’ who
specifically identified their primary school teachers as the source for their learnt
dislikes and fears of mathematics. It is not only teacher attitudes that appeared to
cause problems, but also teacher actions – or in the case of Linda, teacher inactions.
Linda remembers that in primary school, and specifically in Year 5, she learnt to
dislike mathematics:
When I was in grade 5 and we started doing division and I was away the very first day they introduced division and I came back the next day and I had no clues what everyone else in the class seemed to know really well. And my teacher never took the time to actually sit down and go through it with me so I was trying to play catch up and I feel like I’ve been playing catch up ever since, just because I’ve missed, not just with division though, that’s followed me the whole way through. Everyone else seems to understand this. I don’t. I’ll try and figure it out myself.
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The analysis revealed that 22% of the participants identified secondary school
(compared to 66% identifying primary) as a time when they learnt to dislike
mathematics. For example, Kim felt that mathematics became hard in Year 11:
Up until then I was doing quite well and then I had a change of teacher and I just lost it and it was never explained properly, it was just a lot of writing on the board that didn’t make sense and in the end I just sort of backed away from it.
Like the 66% of participants who identified primary school experiences as
where their negative beliefs and anxieties towards mathematics originated, these
participants specifically identified secondary school teachers as the major contributing
factor for their learnt dislike of mathematics. Petra’s comment about one of her
secondary school mathematics teachers exemplified the type of comments made by
these four participants about some of their secondary mathematics teachers.
I had a teacher called Mr O, a bit of a Hitler looking fellow but I just have visions of him throwing dusters at students you know to get their attentions and he just never explained anything… just wrote it on the board and then you just copied it and then you just had to really go home and try and work it out so I was pretty stressed about that cause I kept thinking you need to talk about it, you need to go through it together and ask whether you understand it.
Eleven percent of the participants identified tertiary education as a time when
their dislike of and anxiety towards mathematics emerged. For example, Diane said:
I had no problem with it (mathematics) through primary school and high school and I had a great teacher in high school. I think that maybe what helped me to overcome it but I think, once I hit uni, it was kind of a whole frightening sort of thing… having to relearn things that you hadn’t done since you were in primary school and I think, it frightened me. I think it was just a bad experience from there. I’ve been trying really hard to get beyond that by choosing it as an elective to get over it but I’m still frightened by it. It’s ridiculous.
4.2.1.6 Situations causing maths-anxiety
As can be seen in Table 4.6, the participants felt most anxious about
mathematics when they had to communicate their mathematical knowledge in some
way (48%), for example, in test situations or verbal explanations.
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Table 4.6
Situations causing maths-anxiety
Category of responses Percentages %
What causes your maths-anxiety?
Communicating math knowledge 48
Practicum situations 33
‘When I can’t do it’ 20 Also, causing a lot of anxiety was the teaching of mathematics in practicum situations
(33%) due to insecure feelings of making mistakes or not being able to solve
mathematical problems correctly. For example, Petra explained:
Oh probably when I know I have to teach it because I sort of really don’t have I mean I can look at the syllabus and know what I’ve got to teach but I’ve got to know it myself to teach it so I suppose I feel anxious thinking that I’ve got to go out into the classroom and teach it when I don’t feel very equipped or confident with it, actually putting it into practice.
Likewise, Rose explains that her most anxious moments are:
When I’m being called on to answer questions… and I don’t know the right language and I try to answer the question as best I can but you don’t really get your meaning across because you don’t understand the language and you don’t know what language to use. Testing…like I said when kids ask me questions. In the prac experience, there were two teachers the other prac teacher took the [mathematics] class but the students were asking questions, so I still had to handle that part of it. Just when somebody tests my knowledge… It does and it makes me feel as if I don’t know what I am talking about.
The use of mathematical language, insecure feelings about making mistakes and not
being able to solve problems correctly when in teaching situations appeared to be in
particular problematic for these preservice teachers.
4.2.1.7 Types of mathematics causing most maths-anxiety
Two strands from the Queensland Studies Authority (2003) syllabus caused
most anxiety amongst the participants these were: algebra and patterns (33%) and
space (31%). Number operations especially division, was also a concern (21%).
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Table.4.7
Types of mathematics causing maths-anxiety
Category of responses Percentages %
What mathematical concepts cause your maths- anxiety? Number 22
Patterns & Algebra 33
Measurement 11
Chance and Data 3
Space 31
The anxiety caused by these strands was well exemplified by Ann’s response:
Long division!!! Couldn’t ever do that. Dividing. Can’t do that. Time tables. You know how they used to learn the times tables. I still can’t do them because they sing that song. One, ones are one and all that and I never had a very good memory so I could never learn them. I’m making myself sound really bad… And with addition and subtraction, I still use my fingers to count up things… I used to do it under my desk so the teacher couldn’t see ‘cos you’re supposed to know just what 6 plus 6 is without counting it on your fingers sort of thing.
4.2.1.8 Overcoming maths-anxiety
Table 4.8 indicate that 46% of the participants wanted to have more math
education in order to overcome their maths-anxiety Thirty-one percent of the
participants believed that a supportive learning environment would alleviate their
maths-anxiety whilst 23% felt that the teacher was important in overcoming maths-
anxiety.
Table.4.8
Perceptions of how to overcome maths-anxiety
Category of responses Percentages %
What can help you to overcome your maths-anxiety?
More math education 46
Supportive learning environment 31
Being an effective teacher 23
To overcome their negative beliefs and anxieties about mathematics, the
participants felt they would benefit from further mathematical education in the form
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of revising the basics. Also, of importance was a supportive learning environment
involving group work that was seen to help the participants in facing his/her
mathematical fears. Ron suggested the following to overcome his maths-anxiety,
Well, learning strategies of how to teach it effectively and how to understand the concepts. I really sort of have to go back to basics I suppose and sort of re-teach myself concepts of mathematics, for example, the way I did it at school and it makes much more sense now. It was just a mechanical operation at school. You just carried the one and it didn’t sort of mean anything, now there’s sort of, you can see the logic behind the way they do it now, which is totally different, so I have to re-learn that myself.
Carla in turn took a more philosophical approach by suggesting:
I think with anything, by facing it and saying this is what I’m anxious about. Often in my own personal life experience, anxiety comes from fear and fear is just the unknown and I just think this will be a wonderful opportunity to embrace the unknown, and at least, maybe walk away with, you know, and attitude of joy. I can do this. I think if I can stand in front of a classroom with that attitude, sure my kids are going to catch that.
4.2.1.9 Perceptions on how to reduce maths-anxiety in their future students
The final interview question related to how the participants as teachers could
help reduce maths-anxiety in their future students. As can be seen in Table 4.9 the
overwhelming response was for them as teachers to have a positive disposition (64%)
in relation to all mathematical concerns including mindfulness and recognition of
negativity towards mathematics in the students, and the creation of a supportive
classroom environment. Thirty-six percent of participants felt that scaffolding
mathematical ideas and concepts together with hands on activities were ways to
reduce maths-anxiety in their future students.
Table 4.9
Perceptions on how to reduce maths-anxiety in future students
Category of responses Percentages %
How would you help your future students to overcome maths-anxiety?
Positive reinforcement 64
Scaffolding / hands on 36
Ally explains the importance of what the majority of participants referred to as
positive reinforcement:
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It’s important to build a child’s self esteem. If they’re having problems with their self-esteem then you need to encourage them and I know it can be hard within a classroom to have one-on-one with children but you need too and to explain it in different ways…use smaller numbers or take it back to the basics help them to get a good feeling about themselves so that they will actually enjoy what they’re doing rather than struggling and just decide they don’t like it
Diane in turn suggested that to help students overcome maths-anxiety, teachers need to:
…step them through it. Always allow the children time to ask questions. If they don’t understand, they’re not to be made like out as if they’re stupid or silly in any way. They’ve really got to be able to, if they’ve got a problem, put their hand up immediately and say, I don’t understand. There’s no use having a child sitting there blankly and not noticing it, because the child will never learn. That’ll cause anxiousness throughout the rest of their lives when it comes to mathematics. You’ve just got to allow them to ask questions, even if it’s a silly question, treat it seriously.
The need for the teacher to be approachable and for the classroom environment to be
non-threatening was also emphasised by Jill’s response who thought that it is
important to:
… really let them know that if they do have a problem that they can come to you and that you’re not going to get mad at them. You should make it clear to the whole class that it’s not embarrassing if you have to repeat things for certain people cause’ I know when I was at school I hated asking once and twice and three times cause you know other people would look at you as though you’re an idiot. I think you need to let them know that they can come to you for help and not to be embarrassed about it and not everybody gets mathematics, some people get it, some people don’t and I don’t get it
4.2.2 Comparison of pre- and post-enactment interview results
The results from the post-enactment interview questions saw some interesting
changes to preservice teacher students’ conceptions about the following four issues:
(a) the nature of mathematics, (b) the relevance of mathematics, (c) teacher
knowledge and qualities, and (d) maths confidence.
4.2.2.1 Nature of mathematics When asked to describe what mathematics is, most of the participants
confirmed responses they had given earlier (See section 4.2.1.1). That is, they
indicated that they had multi-dimensional conceptions about the nature of
mathematics and that it had a utilitarian purpose. There was a consensus amongst the
participants that mathematics is about problem solving, numbers, procedures, it
related to real life and involved thinking. Belinda’s answer exemplified the typical
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response by the majority of participants who identified mathematics as being problem
solving:
Mathematics is a lot of problem solving it’s not just to do with numbers any more, so much, it is more to do with problems.
As well as problem solving, doing, thinking and asking questions were also identified
as being important aspects of mathematics. For example, Petra stated:
Mathematics is not a spectator sport. It is a subject which is learned by doing and thinking about problems. Asking questions is an important part of the learning process.
Furthermore, Sally noted a relationship between numbers and patterns when she
explained:
It is the relationship between numbers and patterns… I see now that it is more to do with looking for patterns and relationships between those patterns that involve numbers or that involve groups or object that children can count and put into groups and things like that.
Interestingly, for some of the participants mathematics ceased being as frightening as
it had been previously. For example, Tina felt that:
Mathematics is a wonderful thing that involves lots of problem solving, decision making and I’ve discovered making sense of things, basically having an understanding of what the problems are and how to solve them. I don’t think it’s as frightening as it was because you’re not alone… it’s still very daunting but now I know there are avenues for help.
As can be seen in Table 4.10 the analysis reveals a shift in the participants’
dispositions toward the nature of mathematics. For example, 62.5% of the responses
in contrast to the pre-enactment response score of 24% suggested that mathematics is
about problem solving. Also, of significance is that a new category was identified
with 25% of the responses suggesting that mathematics is about thinking strategies.
The analysis further showed that 31% of responses indicated that mathematics was
about procedures suggesting a shift from the 12.5% of pre-test responses. There was
a decrease from 25% to 12.5% of responses indicating a positive shift in beliefs about
mathematics being hard. Interestingly there was an increase in responses believing
mathematics is about numbers from 22% to 37.5%.
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Table 4.10
Nature of mathematics
Category of responses. Percentages %
Pre-interview
Post-interview
What is mathematics? Problem solving 24 63 Thinking strategies - 25 Real world 22 25 Numbers 22 38 Different concepts 10 19 Procedures 13 31 Hard 25 13 Other 13 6 4.2.2.2. The relevance of mathematics.
The analysis as can be seen in Table 4.11 noted an increase in responses to the
relevance of teaching mathematics for real world activities from 69% to 88%.
Table 4.11
The relevance of mathematics
Category of responses. Percentages %
Pre-interview
Post-interview
Why teach mathematics? Relevant for real world activities 69 88 Life Skills to prepare for future 31 94
Interestingly the results indicate a significant increase from 31% to 94% of
responses believing teaching mathematics relates to life skills as it prepares students
for their futures [e.g., employment, shopping, budgeting]. Rose explained,
It’s really, really important. It’s a fundamental part of life. It’s involved with so many occupations and just daily life... It’s a way of connecting with students who aren’t necessarily literate students or who are good at different things.
Sally elaborated on this and suggested that,
So much of what we do at a sub-conscious level involves mathematics especially in our younger years and then obviously as we get older what we do at a conscious level involves mathematics and I think you need mathematics to function in society even at the basic level of going to the shop and buying some milk. There are other areas for kids to get enjoyment out of
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our younger years and then obviously as we get older what we do at a conscious level involves mathematics and I think you need mathematics to function in society even at the basic level of going to the shop and buying some milk. There’s other areas for kids to get enjoyment out of and also to learn and possibly to look for a future career in that doesn’t involve numbers but is still patterns and shapes and things like that… it gives them a lot more scope.
4.2.2.3 Teacher knowledge and qualities.
The post-interview responses to the question what knowledge is needed to
teach mathematics saw little change to the pedagogical knowledge percentage from
the pre-enactment interview. Fifty-six percent of the participants in the post-
enactment interview (as opposed to 57% in the pre-enactment interview) felt that
having good repertoires of pedagogical content knowledge was important for teachers
(See Table 4.12).
However, as can be seen there was a significant increase in participants’
perception about the importance of teachers having good repertoires of mathematical
conceptual knowledge.
Table 4.12
Teacher knowledge
Category of responses. Percentages %
Pre-interview
Post-interview
What knowledge is needed to teach mathematics? Pedagogical knowledge 57 56 Conceptual knowledge 43 88 What are the important teacher qualities?
Personal qualities 65 94 Professional qualities 35 75
This increase suggests that the participants had begun to understand that the
ways they will teach mathematics will be quite different from the ways they
themselves, learnt as students and that the success of these different ways of teaching
mathematics is heavily dependent on teachers having deep conceptual understandings
of the mathematical content being taught. For example Rose explains:
We need content knowledge. But, teachers need to be students as well, they need to be able to learn along side student to be able to create new knowledge and expect many different ways of doing things
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This point about teachers needing deep understanding of the mathematics content
being taught also was exemplified by Petra’s response:
The teacher needs a lot of knowledge, about all the different strands of mathematics. You need to know about algebra, you need to know about chance and data you need to know about problem-solving and problem-posing - all those things
In addition to content knowledge, the analysis revealed an increase in
responses in regards to the importance of personal qualities of the teacher from 65%
to that of 94% (see Table 4.12). Participants identified qualities such as flexibility,
creativity and adaptability, understanding, patience and listening skills as important.
Petra explained personal teacher qualities she felt to be important:
You need to be understanding, you need to listen. Listening is a really important thing, I think. And also making it [mathematics] relevant to people’ everyday lives, making it meaningful, I think.
Responses regarding the professional qualities of the teacher saw a significant
increase from 35% to 75%. Carla explains,
A professional quality that I would like to see in a teacher is that she knows about mathematics, she knows about the discipline of mathematics that she or he keeps up to date with all the new literature that’s coming out, can attend conferences, can collaborate with her colleagues or his colleagues and just is on top of the academic [mathematical] concepts of what’s happening in schools
4.2.2.4 Maths-confidence
There were significant changes in the participants’ maths-confidence as can be
noted in Table 4.13.
Table 4.13
Maths-confidence
Category of responses. Percentages %
Pre-interview
Post-interview
How confident are you about your own math–skills?
Not at all 19 0 Not very 50 19 Semi- confident 31 44 Quite confident 0 31 Confident 0 6
The decrease in not being maths-confident according to most participants does
not necessarily mean that they feel maths-confident content wise rather that they feel
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they are not alone and that there are others who like themselves do not understand
mathematics. More importantly, they have come to appreciate that there are many
different ways of learning mathematics. For example Marge explains:
I couldn’t say that content wise I am very confident yet. What it’s helped me with is seeing how many different ways people go about doing things… that was the best thing for me
Belinda in turn said:
I’m not overly confident with my mathematics but I’m not as scared to give it a go… I will sit down and look at it for hours where as before I would have just said okay I don’t understand that … give up.. I just saw when we were doing our math groups there’s just more people that feel the same way as well so it’s not like I’m a dummy and I think just group work and having someone to talk about it helped.
Rose further elaborated on this confidence by noting:
I think my skills still have something to be desired, but I think in learning as a group of people that have similar issues and frustration and worries, it’s good to see that you’re not alone and that you’re not the only one… I feel a little more confident in asking people questions and letting people know when I don’t understand… I feel confident in that even if I have a wrong answer it’s not going to mean the end of the world or that people are going to think less of me. It’s good to collaborate with a lot of different people.
There was an increase in participants’ feeling semi-confident from 31% to 44%. Rose
explained her feelings of semi-confidence by stating:
Although I still find the subject area difficult and challenging I understand its place and worth and have developed new conceptions about what mathematics is and how it should be taught, especially number sense. I value the importance of discussion in mathematics, flexibility in teaching methods, incorporation of group work, developing and maintaining motivation and the use of technology in creating interest and understanding in this key learning area.
Belinda a highly maths-anxious participant explains how she challenged her maths-
avoidant behaviour:
I have always hated long division and still never learnt how to do it, but I went out and had someone teach me how to do it, that is a big achievement for me to overcome the first thing that turned me right off math. The praise for my accomplishments encouraged me to try and do better in my weakened areas, such as long division areas. I have learned that no matter how hard it looks, give it a go even if you can’t do it at least you know you have tried it not just given up at first glance. Also, there is always going to be more than one way of working out a problem.
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Feelings of being quite confident to feeling confident about mathematics were
two new categories identified. Participants feeling quite confident indicated that they
no longer felt afraid about the different processes involved in mathematical problem
solving and that it is acceptable to make mistakes. Participants also recognized the
benefits of group work.
Petra explained her feelings of confidence,
Oh I feel a lot more confident even though I was very nervous about the program. I’m a lot more confident because I understand it [mathematics] a lot more – I’ve learnt with this program about how important it is not actually about getting the answer but it is the process – how you are doing it. I enjoyed the group work we all learn differently, it is quite amazing – we all have different processing skills
Carla felt,
100% more confident because what I’ve learnt in this project is that to embrace the freedom of making a mistake, means that it’s a stepping stone to finding a solution where as prior to coming into this research program I felt shame I felt intimidated because a mistake meant a mark less and that classified me basically as a dummy. Today after being involve in this program I realised that mistakes are to be celebrated and it’s like being able to say to the kids “okay that way didn’t work, let’s try something else.
However, the participants feeling quite confident had not as yet extended their
levels of confidence to feeling confident about teaching mathematics with
understanding to students in a classroom. Ron as the only male in the research study
was the only participant who felt confident in teaching mathematics for
understanding. He stated that:
I think I could now teach children these concepts and have them understand the concepts, which is the most important thing…
4.3 Results from reflection documents Participants’ written reflections revealed some interesting thoughts that were
consistent with but also complemented their post-enactment interview responses.
There was, for example, an agreement amongst all participants that “visual
representational and concrete models are essential” for effective mathematics
teaching. This is because as Susan noted:
…they connect students to the real world, to concepts and components that are existent in the world and society that they live in. They are related to their real lives and are not isolated from these.
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The participants felt that the kind of mathematical activities that students
would enjoy in class included, “activities that are useful in the real world not just
those that arrive at the correct answer” [Kim]. “Finding Nemo is a current movie that
children love… and would understand/relate to” was suggested by Ann and her group
as an idea to develop their mathematical model for the quotitioning activity. Indeed,
all participants thought that the use of stories is not only a popular teaching strategy
but also an effective way to teach mathematics. Sally explained:
using a story to develop a problem, rather than just providing a numerical problem is a better approach to teach children mathematics. By using a familiar story with children’s language, children will be more likely to be able to develop a clear picture of the concept that is being taught.
According to most of the participants the most surprising aspect of the project
related to how people all learn in different ways. Linda, in particular felt that:
this was perhaps the most influential learning that I gained.
Donna came to realize that interpretation is a very individual thing, which
became evident to her:
when I viewed the work of other participants in the group and the activities they designed to demonstrate how they would approach the given tasks and also the feedback and responses that were given to mine and other participants’ models. I’m quite sure when I am in the classroom that I will experience this day in, day out, with my students and the work they produce.
All participants felt that the use of language especially mathematical language
was very important, Ann felt that she:
“..learnt to be careful with the kind of language used with children…”
Donna believed that:
when explaining concepts to students, the correct use of language can be the difference between them understanding the concept being taught or not. It is in using children’s language that will help students’ build their mathematical understanding, make connections and make sense of mathematics.
Group work and the development of a community of learners were strongly
supported by the participants. Kim in particular felt that feedback from the group
contributed to her development of confidence. She explained:
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Comments from peers are imperative in the learning curve and contribute to confidence in developing models to make sense and support reflection. This creates an awareness to provide students with activities within groups to support language and communication skills
Tiffany also felt that the use of group work was very effective in finding a solution.
As she explained:
By constructing our own models first we used the best of our ideas put together in a comprehensible package working with peers also helped to see the problem with new eyes so that we could see different ways problems could be improved and other strategies used.
Marge felt strongly that her:
… mathematical learning developed not only as a student but also as a developing teacher within and with the support of a community of learners.
This was also supported by Ron who felt that:
A community approach to problem solving or learning whether it is on Knowledge Forum or between children in a classroom is an excellent way to learn, sharing ideas, seeing a problem from a different perspective really helped me to develop my model
He further stated:
…if students learn in a supportive community environment they are bound to gain a greater mathematical understanding. I feel this is what I lacked when I was in primary school.
4.4 Online Anxiety Survey results 4.4.1 Introduction
The Online Anxiety Survey measured three positive feeling responses as
defined by the participants: (a) comfortable (a sense of personal comfort), (b)
confident (a sense of I can do this activity), and (c) fine (I feel good about this
activity), and three negative feeling responses as defined by the participants: (a)
nervous (physical feelings such as for example, a nervous stomach), (b) worried (a
sense of fear for activity), and (c) frustrated (a sense of anger and hopelessness
towards the situation). The four novel open-ended mathematics activities that were
chosen to address participants’ maths-anxiety in each work shop session were:
1. Number sense activity.
2. Space and measurement mathematical activity.
3. Number and shape mathematical activity.
4. Division operation activity.
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As outlined in Section 3.4.2, the Online Anxiety Survey was administered
before the commencement and after the completion of each of the four mathematics
novel open-ended mathematical activities. The following section will present the
overall results for the Online Anxiety Survey for the four sessions, followed by the
individual session results.
4.4.2 Overall analysis of the Online Anxiety Survey results
The statistical data analysis used in this study initially tested for overall
significant differences’ between pre-intervention and post-intervention scores of the
Online Anxiety Survey. As outlined in Section 4.6.3, the analysis was utilised a
multivariate analysis of variance (MANOVA) with repeated measures. The
independent variable identified as the pre- and post-activity intervention, whilst the
dependent variables were the six feeling responses (comfortable, fine, confident,
worry, nervousness and frustrated). Pillai’s Trace test was used due to small sample
size (n = 16) of the study and was found to be statistically significant (Pillai’s Trace =
.647, F(6,56) = 17.01, p = .000). Further analysis with the use of Pairwise
comparison was used to determine the significance of the mean differences between
the pre- and post-intervention score, as shown in Table 4.14
Table. 4.14
Pairwise comparison: Overall results
Feeling Mean difference Pre and Post
activity
Std. Error Sig. (a)
Comfortable 23.097(*) 2.849 .000 Confident 19.032(*) 2.893 .000 Fine 20.935(*) 2.405 .000 Nervous -27.661(*) 3.364 .000 Worried -18.774(*) 3.828 .000 Frustrated -6.694 4.064 .105 Based on estimated marginal means * The mean difference is significant at the .05 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments)
The pairwise comparison test compared the difference between each pair of
means that is, the comparisons were between subject t tests. Comparisons were made
on the individual means using the standard errors of each mean (Becker, 1999). As
can be seen, the results show that there were significant mean differences in the
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positive feeling responses comfortable, fine, and confidence as well as in the negative
feeling responses nervousness and worried. However, no significant mean differences
were found in the feeling response of frustration.
Box plots were used to further elaborate on findings and to give an overall
visual record of participants’ pre- and post anxiety distribution. The box represents
the inter quartile range which contains the middle fifty percent of values in the
sample, that is from the twenty-fifth percentile to the seventy-fifth percentile. The line
across the box indicates the median. The whiskers extend respectively from the
twenty-fifth and seventy-fifth percentiles to the lowest and highest scores. An outlier
is shown as an open circle either above or below the upper or lower whisker and
represent data outside the regular data distribution. A positive skew is determined by
the mean being higher than the median and that the upper whisker is longer than the
lower whisker (Lane, 2000).
The overall results of the project show the impact of the four mathematics
activities on the participants’ feelings. Figure 4.1 represents the three positive feelings
– comfortable, confident and fine. As can be seen from Figure 4.1, there has been a
positive shift in the feeling responses comfortable and confident for most participants.
However, the degree the bottom whiskers have not shifted in these two feeling
responses indicate at least one participant not feeling comfortable nor confident with
the activity. The degree to which the bottom whiskers move up in the positive feeling
response “fine” suggests that all participants enjoyed the mathematical learning
experiences.
626262626262N =
post-f inepre-f ine
post-confidentpre-confident
post-comfortablepre-comfortable
120
100
80
60
40
20
0
-20
Figure 4.1. Box Plots Overall positive feelings
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The overall impact of the intervention on participants’ negative feelings
showed a decrease in the negative feelings, worried and nervousness; this can be
noted by the degree the top whiskers move down in these negative feeling responses.
Only a slight decrease was noted in the feeling of frustration.
626262626262N =
post-frustratedpre-frustrated
post-w orrypre-w orry
post-nervouspre-nervous
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100
80
60
40
20
0
-20
22
Figure 4.2. Box plots Overall negative feelings 4.4.3 Session 1: Number sense activity.
Table 4.15 represents the results from the number sense activity, and shows a
positive increase in the levels of participants’ feelings of comfort, fine and in
confidence at the completion of this activity. The results also showed a decrease in
the negative feelings of nervousness and worry whilst the feeling of frustration
remained stable.
Table 4.15
Pairwise comparison: Session one results
Measure Mean difference Pre and Post
activity
Std. Error Sig. (a)
Comfortable 27.438(*) 4.447 .000 Confident 21.125(*) 4.082 .000 Fine 19.688(*) 5.105 .002 Nervous -35.313(*) 6.548 .000 Worry -16.375(*) 6.572 .025 Frustrated -4.125(*) 9.996 .686 Based on estimated marginal means * The mean difference is significant at the .05 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments)
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Figure 4.3 (see below) clearly indicates that the part of the intervention based
around the Number Sense activity had a positive impact on the participants, with the
bottom whiskers moving up in each of the positive feeling responses. This is
indicative of a positive learning experience for all participants.
161616161616N =
post f inepre f ine
post confidentpre confident
post comfortablepre comfortable
120
100
80
60
40
20
0
-20
13
13
Figure 4.3. Number Sense activity (positive feeling responses)
In the box plot see below (Figure 4.4) the impact of this part of the
intervention on the majority of participants’ negative feelings suggests a decrease in
nervousness and worry. Interestingly, the top whisker for the negative feeling
response frustration saw a slight increase.
161616161616N =
post frustratedpre frustrated
post w orriedpre w orried
post nervouspre nervous
120
100
80
60
40
20
0
-20
5
Figure 4.4. Number Sense activity (negative feeling responses)
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In order to identify why a slight increase in frustration occurred, recourse was
made to post-activity interview data. The analysis of this data indicated that the types
of frustration they felt after completing the activity were different to those they had
felt prior to the activity. Before the activity, their frustration was related to confusion
and doubts about how to get started. In contrast, the frustration felt after the activity
was related to problems with the computer and also to uncertainty about the
correctness of their solutions to the mathematics activity. This is well exemplified by
Susan. To describe the complexity of her feelings relating to this mental computation
activity, Susan explained both her feelings of frustration and why she did not feel
particularly nervous:
I think what happened in that (activity) was that I started to do the problem, and I was quite nervous and overall I was just… but, I had in my previous mathematics experience had experiences with mental computation so I had sort of some strategies that I could apply. But then when I got into it and I started applying it, it wasn’t making sense. And, so I got frustrated. And, then my computer wouldn’t work so that added to the frustration. Then at the end I was more relaxed that it was all over, but at the same time, I was frustrated because I knew the strategies but I just didn’t know if they were right – the ones I was applying – or if they… you know, so it was like all these mixed emotions. But, because I’d had experience with it (mental computation) previously, I guess I wasn’t as nervous.
4.4.4 Session 2: Space and measurement activity,
Table 4.16 showed that the pair comparison test results from the part of the
intervention based around the space and measurement activity.
Table. 4.16
Pairwise comparison: Session two results
Measure Mean difference Pre and Post
activity
Std. Error Sig. (a)
Comfortable 15.938(*) 6.369 .024 Confident 11.938(*) 4.596 .020 Fine 18.875(*) 3.970 .000 Nervous -17.563(*) 5.632 .007 Worry -6.313 8.123 .449 Frustrated -2.125 6.418 .745 Based on estimated marginal means * The mean difference is significant at the .05 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments)
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The results demonstrate an increase in the positive feelings of comfortable,
fine, and confidence, whilst a slight decrease was noted in the negative feeling of
nervousness. No notable decrease was noted in worry or frustration
The box plot (Figure 4.5) allows for a visual display of the distribution of data.
It is interesting to note the encouraging impact this part of the intervention had on
participants’ positive feelings. As can be seen, the degree the bottom whiskers have
moved up in the positive feelings comfortable, confident and fine suggest that all
participants had a positive learning experience
161616161616N =
post f inepre f ine
post confidentpre confident
post comfortablepre comfortable
120
100
80
60
40
20
0
Figure 4.5. Space and measurement activity (positive feeling responses)
As can be seen in the box plot below (Figure 4.6) the impact this part of the
intervention had on the participants’ negative feelings is interesting. While all the
participants had recorded having positive feelings about this activity, there has been a
decrease in participants’ nervousness, but with a slight increase is evident in the
negative feeling worry whilst the negative feeling frustration remain stable.
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161616161616N =
post frustratedpre frustrated
post w orriedpre w orried
post nervouspre nervous
120
100
80
60
40
20
0
-20
6
Figure 4.6. Space and measurement activity (negative feeling responses) To understand these response results particularly with respect to non-significant
findings for worry and frustration, recourse again was made to the post-activity
interview data. Recourse to post-activity interview data from the outlier participants
indicated that the non-significant changes to worry and frustration could be attributed
to the face-to-face and on-line knowledge-building discourse associated with this
particular mathematics problem. Susan who was more worried and frustrated after the
activity described her experience:
I remember! I’ll be honest. During the actual exercise I wanted to rip it apart. I did not understand a thing. It was like my brain froze. I remember I actually started crying. It was horrible. But then I let a few days go by and then one night I said “That’s it. I’m going to take the time out and just spent time on it seriously”. Like, I must have spent an hour and just did all these calculations and then because I had taken the time and patience to work through it systematically then I guess I found and answer that I felt comfortable with.
The other two outlier participants were Tiffany and Linda. Tiffany explained:
Farmer Brown was a really hard one for me. I think I was feeling fine because I knew that I could probably ask somebody… there was support there to find an answer if I got really stuck, whereas if I hadn’t had that support, I wouldn’t have been fine. I think I was comfortable because I had some mathematical knowledge and I knew that I could find an answer if I asked for help. The frustration comes into not knowing what to do and what’s expected and things like that. That really makes me grit my teeth when I don’t know how to do something…It’s like making a cake: you can have the ingredients there; you can see the cake at the end; but you don’t know how to get through the process of getting that cake made. Worry and nervousness comes back into it too, worried that you’ll say the wrong answer or do
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the wrong thing. And probably nervousness comes into you’re not quite sure of what you’re doing. It comes into the frustrated part too; what are you going to do if you can’t find an answer. How are you going to explain it to your peers’.
Linda also felt frustration and believed that the mathematics activity chosen was not
particularly relevant to her. She explained how she:
…was starting to get a little frustrated with completing the tasks and finding out exactly what we needed to do – what was required of us. I didn’t find this problem to be a very authentic task for me to do. While I was completing it, I was thinking “quite frankly, I don’t care about Farmer Brown and his sheep. I don’t care. I’m not a farmer. I don’t’ have sheep. And I sort of felt a bit frustrated at doing that when it had no relevance to me and what I was doing.
4.4.5 Session 3: Number and shape activity
Table 4.17 demonstrate the results from the part of the intervention based
around the number and shape mathematics activity. It shows an increase in the
positive feelings comfortable, fine, and confidence and a decrease in both nervousness
and worry whilst there is little change to feelings of frustration.
Table. 4.17
Pairwise comparison: Session three results
Measure Mean difference Pre and Post
activity
Std. Error Sig. (a)
Comfortable 18.313(*) 5.553 .005 Confident 17.063(*) 6.492 .019 Fine 17.375(*) 5.426 .006 Nervous -24.750(*) 6.998 .003 Worry -26.188(*) 7.534 .003 Frustrated -.313 8.795 .972 Based on estimated marginal means * The mean difference is significant at the .05 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments)
There was a positive impact on the majority of participants’ feelings. The
degree the bottom whiskers moved up in the positive feeling ‘fine’ suggests that all
participants felt fine about this activity.
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161616161616N =
post f inepre f ine
post confidentpre confident
post comfortablepre comfortable
120
100
80
60
40
20
0
-20
Figure 4.7. Number and shape activity (positive feeling responses) As can be seen in Figure 4.7, the bottom whiskers remain consistently at ‘0’ for the
feelings comfortable and confident for the number and shape activity. To understand
this response result recourse was made to the post-activity interview data. Recourse
to post-activity interview data from the identified participants indicated that the non-
significant changes to comfort and confidence could be attributed to the particular
mathematics problem. Karen explained her experience:
I did not actually like the activity. I had a bit of trouble with that one. I actually went onto the internet and did a bit of research for it – on nominal L shaped numbers and things like that. And that was the one I did actually go and see the lecturer about. And he said “You’re making it harder than what it actually is”. So…I was finding it quite hard and then when I finished it, I still didn’t feel confident with it and took it to the lecturer and talked to him about it. And he said, “You’ve got it there. You’re worrying about nothing”.
The impact of the number and shape mathematics activity on participants’
negative feelings (see Figure 4.8), showed a decrease in nervousness, worry and
frustration in all participants’.
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161616161616N =
post frustratedpre frustrated
post w orriedpre w orried
post nervouspre nervous
120
100
80
60
40
20
0
-20
Figure 4.8. Number and shape activity (negative feelings)
To understand these positive shifts to all three negative feeling responses
recourse to post-interview data was made. Linda’s experience typified the positive
response by participants regarding this mathematics activity. She explained:
I was really surprised when completing the assignment that I actually figured out what to do, because I read the thing and thought: “No way am I going to be able to figure out any pattern to this. This makes no sense to me”. And I did it and thought “Hang on, I can see the pattern there” – because I’m usually the last person to see the pattern in anything. So that is why I was nervous: I was being asked to find a pattern in something.
Petra had also a positive experience with the mathematics activity but
experienced frustration with the computer mediated soft ware, she explained:
I was good at that. I was fine with the activity. That was great. I did all the flowers. I was happy with that. I was frustrated not about the actual activity but about MipPad. I was actually trying to get the work on to MipPad. Unless you’re really used to it – which I am now – you get frustrated when, if you lose things.
4.4.6 Session 4: Division operation activity.
Table 4.18 demonstrated that the results from this division operation
mathematics activity showed an increase in the positive feelings comfortable, fine,
and confidence. The results also show a significant decrease in nervousness, worry
and frustration.
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Table. 4.18
Pairwise comparison: Session four results
Measure Mean difference Pre and Post
activity
Std. Error Sig. (a)
Comfortable 27.750(*) 6.653 .001
Confident 25.188(*) 6.869 .002
Fine 26.125(*) 4.572 .000
Nervous -29.875(*) 7.054 .001
Worry -25.250(*) 7.007 .003
Frustrated -20.813(*) 4.975 .001
Based on estimated marginal means * The mean difference is significant at the .05 level. a Adjustment for multiple comparisons: Least Significant Difference (equivalent to no adjustments)
As the box plot in Figure 4.9 show, the intervention based around this
mathematics activity had a positive impact on the participants’ feelings. This can be
clearly noted with the degree the bottom whiskers have moved up in the positive
feeling responses comfortable, confident and fine.
161616161616N =
post finepre f ine
post confidentpre confident
post comfortablepre comfortable
120
100
80
60
40
20
0
-20
4
Figure 4.9. Division operation activity (positive feeling responses)
The box plot in Figure 4.10 show the impact this part of the intervention had
on participants’ negative feelings. And, as can be seen there was a significant
decrease in all three negative feelings clearly indicating that all participants had a
positive learning experience.
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161616161616N =
post frustratedpre frustrated
post w orriedpre w orried
post nervouspre nervous
120
100
80
60
40
20
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-20
Figure 4.10. Division operation activity (negative feelings)
Recourse was made to post-interview data to understand this data. The analysis of this
data indicated that the majority of participants enjoyed this mathematics activity. Jill
however, identified herself as someone who experienced some confusion with the
division maths activity she explained her feelings:
Because I wasn’t sure which was which? And I’d it written down? – the lecturer had said partitioning was sharing so I wrote that down, and I saw other people but they’re going “No, no it’s the other one”. I should have taken the lecturers word for it, because he was right. But I thought I’ve done it wrong again. And then I started to feel bad again. And then I went through all my notes again I had all these hand-written notes…I had Partitioning is sharing and I thought “I WAS RIGHT” So I felt a bit better after that. The other one was the array model. And I knew there were a set model and I thought, “I don’t remember what the array model was” I didn’t feel comfortable. Luckily I found the array model and used that and I felt fine after doing that. Once I’d worked out which was which – like the array model I thought, that’s how I can use it here” And once I’d worked out the division – which was which – I was fine then. I thought “that’s how I can do this one.
4.5 Computer mediated support tools Table 4.19 show the results in regards to participants perceptions about
computer mediated support tools, Maths Ideas and Process Pad (MipPad) and
Knowledge Forum that was used in the study.
Table 4.19
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Perceptions of computer mediated software
Category of responses Percentages %
Perceptions of MipPad It was good 62 It was problematic 30 It was useful 8 Perceptions of Knowledge Forum It was good 61 It was problematic 23 It was useful 16
Results show that although 62% of responses thought that the comprehension
software program MipPad was good. However, 30% thought the program to be
problematic.
The animation function found in MipPad was identified not only as very
popular amongst all participants but was perceived as an important part in seeing how
learning takes place. For example Sally explained:
By using MipPad I was able to see how animation can be used to help children understand mathematical concepts in a creative, fun and enjoyable way…The animation function allows you to see the processes that students would undertake to solve a real world problem…Prior to this research program, I would not have considered using a program such as MipPad to teach mathematics.
Tiffany likewise felt that:
…models that used animation to explain the procedures, and talked through the steps of solving the activity were more effective in promoting learning. Acting out the problem and showing how the problem could be solves was a superior way to show the working of the problem than viewing an ordinary document that could not be interacted with.
Donna explained:
MipPad was good. I think that helps with the thinking process as well because you’re sitting there trying to visualise what you’re going to do… the animation, you can rewind it back and even see how other people have what steps they’ve taken to get there and the results of whatever it is they are doing
Carla felt the “bomb”4 was a crucial component of MipPad. She explained:
4 The “bomb” is a tool that enables the user to destroy everything they have created in a MipPad window.
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The bomb [in MipPad] takes all the shame out of making mistakes. It is much more innovative than red crosses slashed across students’ work. I believe it would be very constructive to sit with my students, look at their work using the animation tool, scaffold where and how they may need to try different strategies, they could then save any work they want to and using the bomb tool to start again… it is crucial for students’ to be taught how to celebrate and embrace mathematical mistakes because mistakes are simply stepping stones to bigger and better learning outcomes.
In the early stages of the study, Carla indicated feelings of great shame about
making mistakes. That she found the “bomb” a useful tool for alleviating her fears for
making mistakes when learning mathematics was a most unexpected but very
satisfying finding.
Most of the 30% of the participants who indicated that MipPad was
problematic tended to focus on issues such as problems in initially learning how to
use the software. Kim, for example, felt that MipPad initially was:
Really frustrating to use… but then after using it a few times I became a bit more confident with it and now I think it’s good. I think it would be really useful for children,, the kids would really enjoy the interactive part and being able to build up a problem themselves… the other thing I thought about MipPad that was really beneficial for me and I think would be for children was the animation because until I sort of understood what that was I thought oh you can just develop a particular problem for children without actually realising how you present it to them. By having it that way you had to suddenly think well I can’t do it that way I’ve got to dot this first, then that then that… So that was a really good learning curve for me
Other issues of concern to the 30% of participants who found MipPad
problematic included the participants not being able to down load the program into
their home computers and not being able to edit copy or paste pictures.
With respect to Knowledge Forum, 61% of the participants thought it was
good whereas 23% of responses found it problematic. Some of the positive elements
identified included a non-threatening environment, and participants being able to give
and receive feedback. A typical response for example was noted by Rose:
I really enjoyed it, Knowledge Forum was excellent in starting discussions with people, reading people’s notes getting the feedback…sometimes it’s better to get feedback that isn’t face to face so it’s not personal. You have time to think about it, like you’re not
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being confronted about it. You can then take the feedback and run with it.
Also, identified were helpful factors such as collaboration and being able to work on
Knowledge Forum when time permitted. For example Donna thought Knowledge
Forum was:
Great, I think it was really good in as much as that we weren’t able to get together time wise. So from different areas we were able to collaborate. However, the only criticism I can make is that there were areas of Knowledge Forum that I had to learn about and I think I would like to have had a little bit more grounding in how to use the program
Knowledge Forum was also identified to help build student maths-confidence and to
reflect on their learning, Kim explains:
Using the Maths Forum (Knowledge Forum), helped to scaffold and support learning. Having a praise contribution was a positive input, which build confidence and supported reflection. I have learnt reflection is a valuable tool in gaining mathematical understanding. By reflecting on questioning and design I often saw things from another perspective and a greater understanding was developed.
The sense of a community of learners was important to Ron who felt that:
Knowledge Forum community provided me with feedback and support and showed me that there are many other interesting and innovative ways to explain mathematical notions about division. I think if students learn in a supportive community environment they are bound to gain a greater understanding. I feel this is what I lacked when I was in primary school.
While the comments mainly portray positive comments about Knowledge Forum
there were situations which caused frustration amongst participants’ particularly
involving network/server difficulties. Jill explanation typifies participants’ frustration
with Knowledge Forum:
It was good when it worked… I found it rather frustrating because I couldn’t get into it and look at other people’s notes.
Petra explains:
I got a little frustrated with Knowledge Forum just because it was down a fair bit of the time. But in general it’s been great. It’s been great to communicate with other students and see different models and everything up on the screen and just the way people have done things – the way they’ve been thinking. It’s been pretty good.
4.6 Summary The research study found that there was a significant reduction in the
participants’ levels of maths-anxiety. Significant increases were recorded in the
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participants’ levels of positive feelings associated with maths-anxiety (comfort,
confidence and feeling fine) whilst significant reductions were recorded in negative
feelings associated with maths-anxiety (nervousness and worry). No significant
changes were recorded in the participants’ levels of frustration. However, an analysis
of post-interview data indicated that qualitative changes occurred to the participants’
feelings of frustration during the course of engagement in the mathematical problem
solving activities. Whereas frustration at the beginning of an activity was related to
concerns about being able to start the process of problem solving, by the end of the
activities, feelings of frustration tended to be related to frustration with the computers
(e.g. the server being down, or not being able to download the software program on
the home computer) or frustration about the relevance/worth of a specific
mathematics activity.
The study also found that the participants’ beliefs about mathematics had
changed during the course of the research study. For the majority of participants, a
movement occurred from an absolutist to a fallibilist view about the nature and
discourse of mathematics. The results also indicate a positive shift in participants’
beliefs about the importance and relevance of mathematics as it relates to culture and
society. The findings further indicated a substantial shift in the participants’
perceptions about the importance of teachers having sound repertoires of
mathematical conceptual knowledge whilst also emphasising the importance of
personal qualities such as, flexibility, adaptability, listening skills and patience.
The findings clearly suggest that the decrease in participants’ maths-anxiety
and changes to the participants’ beliefs about mathematics can be attributed to a
number of factors such as, the continuous support from their group members via the
computer-mediated Knowledge Forum community, and the support they received
from the researcher and facilitator within the non-intimidating workshop
environments. Another important factor that played a crucial role in the reduction of
maths-anxiety and positive changes to participants’ beliefs that emerged during the
course of the post-interviews was the time allowed to explore and engage in
asynchronous computer-supported collaborative discourse. Also associated with this,
the results suggest an increase in maths-confidence that can be attributed to the
participants not feeling alone and that there were others’ who like themselves, did not
understand mathematics.
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CHAPTER 5
DISCUSSION AND CONCLUSION
5.1 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 CSCL community would reduce their negative beliefs and
high levels of anxiety about mathematics. To assist these sixteen maths-anxious
preservice student teachers, a three-phase Intervention Program was developed and
implemented in this study.
In this chapter, an overview of the study is presented (see Section 5.2) and the
major findings of the study are reviewed in relation to relevant literature and previous
research (see Section 5.3). Limitations of the study are also discussed (see Section
5.4) and the implications for further research and teaching are considered (see Section
5.5.). Finally, recommendations to inform future research and practice in the field of
addressing preservice student teachers’ negative beliefs and anxieties about
mathematics are presented (see Section 5.6).
5.2 Overview of study The study began by identifying the origins of the preservice teachers’ negative
beliefs about and anxieties towards mathematics. This required the development of a
set of interview questions that were based on findings from the analysis and synthesis
of the research literature. The information that was derived from the interview
questions was used to assist in the selection and the development of learning activities
utilised in the second phase of the Intervention Program to address the preservice
teacher education students’ negative beliefs about and anxieties towards mathematics.
In the second phase of the intervention program, the participants were initially
inducted into the face-to-face workshop and on-line computer-supported collaborative
learning community aspects of the program. Following their induction into the
intervention program, the participants then collaboratively engaged in a sequence of
four open-ended mathematical activities. During the course of these learning
activities, the participants engaged in face-to-face workshops and on-line
collaborative knowledge-building activity mediated by Knowledge Forum. In the final
83
three activities, the mathematical knowledge-building was facilitated by the use of the
comprehension modelling tool, MipPad. The participants used MipPad to develop
their mathematical models. These MipPad artefacts were then posted as attachments
to Knowledge Forum notes for comment and discussion with other members of the
learning community. At the beginning and end of each learning activity, the
participants were required to record their feelings about the mathematical learning
activity on the Online Anxiety Survey.
In the final phase of the program, written reflections were collected from the
participants. They also were administered a post-enactment interview. This data plus
that derived from the pre-enactment interviews and the Online Anxiety Survey were
then analysed to investigate changes to the participants negative beliefs and levels of
anxiety about mathematics and the factors that influenced these changes.
5.3 Overview of results Most of the findings from this study regarding the causes of negative beliefs
and anxieties about mathematics were consistent with the findings reported in the
research literature. (e.g., Brown, et al., 1999; Carroll, 1998; Cornell, 1999; Ernest,
1996; Nicol, et al., 2002; Trujillo, & Hadfield, 1999). For example, this study found
that the origins of maths-anxiety in most of these participants could be attributed to
prior school experiences (cf., Levine, 1996; Martinez & Martinez, 1996). Whilst the
literature suggests that negativity toward mathematics originates predominantly in
secondary school (e.g., Brown, et al., 1999; Nicol, et al., 2002), data from this study
suggests that negative experiences of the participants in this study most commonly
originated in the early and middle primary school. The perceived reasons for these
negative experiences were attributed to the teacher, particularly to primary school
teachers (72%) rather than to specific mathematical content or to social factors such
as family and peers.
Situations which caused most anxiety for the participants included
communicating one’s mathematical knowledge, whether in a test situation or in the
teaching of mathematics such as that required on practicum. This is consistent with
findings in the literature that suggests that maths-anxiety surfaces most dramatically
when the subject is seen to be under evaluation (e.g., Tooke & Lindstrom, 1998).
Specific mathematical concepts, such as algebra, followed by space and number
84
sense, caused most concern amongst the participants. This finding too is consistent
with previous research with respect to maths-anxiety (e.g., Ball, 2001).
The research study found that overall there was a significant reduction in the
participants’ levels of maths-anxiety. The findings from the Online Anxiety Survey in
regards to each of the four mathematics activities saw significant increases in most of
the participants’ levels of positive feelings (i.e., comfort, confidence and feeling fine).
Significant reductions were also, noted in the negative feelings (i.e., nervousness and
worry). Interestingly, the findings suggested no significant changes in participants’
levels of frustration except for the last learning activity.
An analysis of post-enactment interview data indicated that qualitative
changes occurred to the participants’ feelings of frustration during the course of
engagement in the mathematical problem solving activities. Whereas frustration at the
beginning of an activity was related to concerns about the mathematical activity and
being able to start the process of problem solving, by the end of the activities, feelings
of frustration tended to be related to frustration with the computers (e.g. the server
being down, or not being able to download the software program on the home
computer) or frustration about the relevance/worth of a specific mathematical activity.
During the last activity (the division activity), there was a significant decrease in
participants’ frustration. This decrease in frustration could be attributed to
participants’ feelings of confidence and enthusiasm toward using the innovative
computer comprehension tool MipPad (because of its animation factor that allowed
for viewing the learning process), as well as to Knowledge Forum that provided the
non-intimidating and safe on-line environment where participants’ felt safe to explore
and share their mathematical ideas and models.
In line with the research literature (e.g., Brett et al. 2002), the results from this
study also indicated that computer supported mathematical knowledge building
communities can facilitate positive change to participants’ mathematical content
knowledge, concepts about the nature and discourse of mathematics and conception of
learning mathematics.
The insights into participants’ subjective experiences as they navigated their
way through the various mathematics activities suggested that their interest in
overcoming their fear of mathematics and fear of failure in the task influenced their
85
affective responses but not intended effort (c.f., Ainley & Hidi, 2002; Boekaerts,
2002). For instance, by the time the participants were engaged in the final
mathematics activity, although they still experienced feelings of anxiety about
mathematics, this was no longer manifested by avoidance behaviour. For example,
Belinda now felt comfortable enough to confront her phobia towards long division;
she independently sought help outside the community to acquire knowledge necessary
for her to be able to teach division with deep understanding. Further, these insights
were helpful to the participants in their quest for effecting change (to overcome their
negative beliefs and anxieties about mathematics), for they allowed reflection and
awareness of their emotional state (Boekaerts, 2002; Martinez & Martinez, 1996).
It was found that the overall decrease in participants’ negative beliefs and
levels of maths-anxiety can be attributed to the following factors:
1. Computer support from other group members within the computer-
mediated Knowledge Forum learning community.
2. Non-intimidating workshop environments.
3. Asynchronous computer-supported collaborative discourse.
Findings from the study clearly suggested that the continuous support from
other participants within the learning community provided via the means of
Knowledge Forum notes and comments together with the on-going support they
received in the non-intimidating workshop environments from the researcher and
facilitator during the various mathematics activities facilitated the decrease in maths-
anxiety and negative beliefs about mathematics. Participants came to value small-
group work, individual effort and the power of the community of the group as a whole
in resolving what to accept as valid in their growing repertoire of mathematical
knowledge. Also, the extra time the participants’ were allowed to explore key
mathematical concepts and processes when engaged in asynchronous computer-
supported collaborative discourse played a crucial role not only in the decrease of
maths-anxiety but also in the development of their understanding and conceptual
knowledge about the specific mathematics that were subsumed within the
mathematics problems (Brett et al., 2002). Hence, findings from this study clearly
support and also in line with other research (c.f. Brett et al., 2002; Scardamalia &
Bereiter, 1995) that participating in a CSCL environment increased the depth of
86
participants learning as well as it fostered interactivity among the participants that in
turn led to the development of their community.
The increase in maths-confidence also could be attributed to the participants
not feeling isolated or alone. That is, there were others who like themselves, did not
understand mathematics. This finding is in line with the research literature about the
importance of the development of teaching and learning communities (Ball, 2001;
Boaler, 2002; Lampert & Ball, 1999; Ma, 1999) and in the development of identities
as mathematics teachers (Brett, et al. 2002).
The study also found that the participants’ beliefs about mathematics had
changed during the course of the research study in a number of different ways. First,
for the majority of participants, a movement occurred from an absolutist to a fallibilist
view about the nature and discourse of mathematics (Ernest, 1996). According to Ball
(1988) and Ernest (2000), change in viewpoints about the nature and discourse of
mathematics is a necessary but not complete condition required for changes from
negative beliefs and maths-anxiety in preservice primary school teachers. The
findings from this study tend to confirm this notion.
Second, changes also were noted in the participants’ conceptions about the
rationale for teaching and learning mathematics in the primary school. Participants by
the end of the study saw mathematics as being relevant not only for real world
activities but crucial in preparing their prospective students for their futures. They
also went beyond the utilitarian reasons for teaching and learning mathematics and
had begun to appreciate the importance of exploring the structural aspects of
mathematics that underlie deep understanding. This finding was consistent with the
findings from many previous studies (e.g. Brett, et al. 2002; Cornell, 1999; Ingleton &
O’Regan, 1998; Martinez & Martinez, 1996; McCormick, 1993; Norwood, 1994;
Sovchik, 1996).
Third, the majority of participants’ also came to recognize mathematics being
problem-solving and about mathematical thinking strategies whilst simultaneously
seeing mathematics as challenging and at times hard. This was closely associated to
the changes that occurred to their beliefs and levels of maths-anxiety.
Fourth, the results indicated a significant increase placed by the participants on
the importance of both pedagogical content knowledge and subject matter knowledge
87
in order to teach mathematics well. Especially since the main motivational factor for
participating in this research project was participants’ fear for inadequate mathematics
teaching. The personal qualities of the teacher such as patience, understanding,
empathy, enthusiasm and confidence were seen as vital to the development of student
learning (e.g., Ball & Cohen, 1999; Ball, 2001; Richardson, 1999; Shulman, 1987).
5.4 Limitations. This study was limited by the short period of time allowed for this research
study not only in regards to the time allowed for the learning and teaching of
mathematics but also in learning how to use the computer-mediated tools MipPad and
Knowledge Forum. During the course of the study, the participants often commented
that they felt that they needed a longer period of time in order to become familiar and
expert in using the computer-mediated tools. In future studies, participants should be
given a significant period of time to become familiar with the computer tools.
Another comment which the participants made was that they would have
preferred to have been able to engage in the project for a longer period of time
including times when they are engaged in practicums. They felt that this would have
enhanced their experiences by enabling them more time to reflect on what they had
done and also on the implications of the new learning for their future roles as teachers
of mathematics. They also suggested that having primary school students engaged in
the on-line mathematical knowledge-building activities are worthy of consideration
for future research studies in this field.
There was concern amongst some participants in regards to maintaining a
positive attitude towards mathematics once they began teaching. Indeed, the
overwhelming request was for follow-up and on-going support. Other concerns the
participants felt had the potential to impact on their teaching mathematics related to
the school culture, time constraints and crowded classes.
The Online Anxiety Survey was effective in being able to record the
emotional state of the participants before and after each of the four mathematics
activities. However, one of the difficulties of the Online Anxiety Survey was that
there were no “behavioural” anchors to the Likert scale. That is the scale did not focus
on different aspect of comfortableness, nervousness, confidence, worry, frustration
and feeling fine. Even so, and whilst the thirty-second Online Anxiety Survey had no
88
psychometric properties it has face validity. Nevertheless, the results from the online
anxiety activity must be treated with some caution. In future studies, it would be
worthwhile to consider the different aspects of comfortableness, nervousness,
confidence, worry, frustration and feeling fine especially during post-activity
interviews and reflections.
In a study of this kind it is important to note that no attempt has been made to
generalise the findings to the full cohort of students. It is up to the reader if they feel
their context is similar to that described in this study to consider the applicability of
these findings to their context. However, the findings from this study suggest that it
could be transferred to a large cohort of students by partitioning the full cohort into
learning communities of approximately 15-20 students where each learning
community would engage in face-to-face and on-line knowledge-building
collaboration similar to those experienced by the participants in this study. Thus, in
some ways, the findings from this study could provide a blue-print for reforms to
preservice teacher education.
5.5 Implications. Many of these findings from this research study have clear implications
preservice teacher education programs. First, the findings that many of the
participants’ maths-anxiety was teacher-caused indicates the need for preservice
mathematics educators to ensure that mathematics education workshops be conducted
in warm, non-intimidating and supportive learning environments where they are able
to: (a) freely explore and communicate about mathematics in a supportive group
environment (b) explore and relearn basic mathematical concepts, and (c) apply their
re-learnt knowledge in real-life and authentic situations.
Second, the findings also clearly indicate that preservice teachers with
negative beliefs and high levels of maths-anxiety need more time to reflect and to
discuss mathematical ideas. Furthermore, the findings from this study indicate that
this time to reflect and discuss ideas should be able to occur both synchronously and
asynchronously. Thus, there is a need for face-to-face workshops where preservice
teachers can engage in synchronous discourse and on-line discourse such as that
provided by the Knowledge Forum environment where preservice teachers can
engage in asynchronous discourse. Also, the overwhelming request by the participants
89
was to be supported both emotionally as well as with practical matters such as lesson
plans, whilst on their practicums.
Third, although the findings indicated that engagement in mathematics
activities can do much to reduce negative beliefs and high levels of anxiety about
mathematics, it is important that the mathematics activities selected reflect on and
address the preservice teachers’ specific concerns about mathematics. One size clearly
does not fit all.
Fourth, the findings clearly indicate that to effectively address negative
beliefs and anxieties about mathematics the focus and goal of preservice student
teacher education courses should focus on both teacher learning and student learning.
Finally, as evidenced by the latent themes in the participants’ responses, it is
clear that isolation and evaluation anxieties will not be allayed via merely arming
preservice teachers with content knowledge. This could act to further problematise
the individual and dismiss the fundamental importance of the individual feeling part
of an emerging mathematics community in which they perceive themselves to be
supported.
5.6 Summary and recommendations The aim for 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 was
argued that enhancing these preservice student teachers’ repertoires of mathematical
subject knowledge, would facilitate reductions in these preservice student teachers
negative beliefs and anxieties about mathematics and also enhance their sense of
identity as future primary mathematics teachers and as valued members within their
learning community. Indeed, most of the findings from this study clearly suggest that
participating in a CSCL environment not only increased the depth of participants’
mathematical learning but it also provided a safe forum where participants’ could
share their ideas without a fear of being ridiculed. The cooperation amongst
participants that led to the development of their learning community resulted in a
decrease in maths-anxiety. A positive change to beliefs about mathematics was
evident amongst some of the participants with findings indicating a development of
90
new ways of thinking about mathematics as a discourse worthwhile rather than
something to be avoided at all cost. The positive changes towards mathematics these
participants have experienced have the potential for a positive impact on student
numeracy outcomes as well as on student attitudinal and emotional responses about
mathematics.
Clearly, there is need for further research and development of innovative
interventions that focuses both on the affective and the cognitive domains of learning
mathematics, to effect permanent change to negative beliefs about mathematics in
maths-anxious pre-service student teachers.
An analysis to the existence of maths-anxiety amongst mature-aged female
preservice student teachers would be of interest, especially since many mature-aged
women are returning to tertiary study to train as teachers.
Future studies into addressing and understanding the general acceptance and
existence of negative beliefs and misconceptions about mathematics in our society
should focus on how this relates to our culture of learning and teaching mathematics.
91
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Appendix 1
Self-identification phone interview:
1. Have you access to Internet and a computer at your home?
2. Why are you interested in this research project?
3. Can you avail yourself to attend workshops regularly?
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Appendix 2
“Mathematics is a central element in human history, society and culture” (Ernest, 2000, p.8)
Pre-service student pre-enactment interview:
1. What is mathematics?
2. Why teach mathematics?
3. What “knowledge” do you think a teacher need to teach mathematics?
4. What other qualities do you think a teacher need to teach mathematics?
5. How confident are you about your own math skills?
6. How confident are you about using computers?
7. Why do you like/dislike mathematics?
8. When/ how do you think you learnt to like/dislike mathematics?
9. When do you feel most anxious about mathematics?
10. Are there particular kinds of mathematics that makes you feel anxious?
11. What do you think can help you overcome your feelings of math anxiety?
12. What could a teacher do to help students to overcome their negative
feelings about mathematics?
13. Was there anything else you want to tell me that you think is important?
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Appendix 3
“Mathematics is a central element in human history, society and culture” (Ernest, 2000, p.8)
Pre-service students’ post- enactment interview:
3. What is mathematics?
4. Why teach mathematics?
3. What “knowledge” do you think a teacher need to teach mathematics?
4. What other qualities do you think a teacher need to teach mathematics?
5. How confident are you about your own math skills now after your participation in this research project? 6. How did you find using MipPad? 7. How did you find using Knowledge Forum?
8. Was there anything else you want to tell me that you think is important?
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Appendix 4
Computer based anxiety scale
Pre-Session Locate how you feel right now just before commencing the math activity session Uncomfortable ComfortableNot nervous Nervous Not fine Fine Not worried Worried Not confident Confident Not frustrated Frustrated
Post-session
Locate how you feel right now just after completing the math activity session Uncomfortable ComfortableNot nervous Nervous Not fine Fine Not worried Worried Not confident Confident Not frustrated Frustrated