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Mathematical attitudes and beliefs : reviewing the parent-child relationship Stout, Melissa C. 2000

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MATHEMATICAL ATTITUDES AND BELIEFS: RE-VIEWING THE PARENT-CHILD RELATIONSHIP by MELLISA C. STOUTE Dip. Ed. The University of British Columbia, 1998 B.A. The University of Western Ontario, 1994 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in THE FACULTY OF GRADUATE STUDIES Department of Curriculum Studies We accept this thesis as conforming tty the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 2000 © Mellisa C. Stoute (2000) In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada DE-6 (2/88) ABSTRACT The original puipose of this study was to investigate the relationship of parents' own math beliefs and attitudes with their children's math beliefs and attitudes. An extensive review of the literature on the affective domain resulted in the decision to use a novel genre of inquiiy and representation. Because a few previous studies, all quantitative, had found a positive but weak relationship between the beliefs and attitudes of parents and children, it was decided to see whether a change in methods and methodology would affect or elucidate past results. This change involved the use of a case study research design, as well as variations of Ginsburg's (1981) 'clinical interview' for data collection to produce Lawrence-Lightfoot's (1983) 'portraiture' for data analysis and reporting. This current investigation showed that using these new approaches supported the results of past positivist studies: that is, parents' own math beliefs and attitudes play some role in the way in which their children feel and react to mathematics. More importantly, the changed approaches to methods and methodology also enhanced the appreciation for the complexity of the parent-child relationship by giving a humanistic touch to past statistical results, and suggested further avenues for exploring the whole area of children's mathematical beliefs and attitudes. T A B L E OF CONTENTS A B S T R A C T . . . . ii T A B L E OF CONTENTS iii LIST OF TABLES. . . . . . .. v A C K N O W L E D G E M E N T S vi C H A P T E R ONE: INTRODUCTION 1 Overview of methodology and methods .' 6 Summary .... 7 C H A P T E R T W O : L I T E R A T U R E REVIEW * .9 Background: The affective domain '.. 9 Mathematical beliefs 14 Attitudes toward mathematics.. 19 Influences on mathematical beliefs and attitudes 25 C H A P T E R T H R E E : M E T H O D S & M E T H O D O L O G Y 32 The Constructivist Lens 3 2 The Study •. ... 36 Interviewing in theory 38 The Questions ; 41 Interviewing in practice : 43 Analyzing and reporting the data 44 C H A P T E R FOUR: PORTRAITURE 49 The Richards Family 49 The Hill Family...: 58 The Forrester Family 65 The Lebrun Family 72 C H A P T E R FIVE: ANALYSIS 80 The Richards Family 81 The Hill Family \. .! ...83 The Forrester Family 85 The Lebrun. Family 87 iv CHAPTER SIX: DISCUSSION 90 What about math beliefs and attitudes? 90 Re-viewing the methodology and methods 96 CHAPTER SEVEN: REFLECTIONS 102 My journey ... 102 What have we learnt here? : 103 What can we take from this? 105 REFERENCES 108 APPENDIX A: Administrator Informed Consent Form 116 APPENDIX B: Student Informed Consent Form 118 APPENDIX C: Parent Informed Consent Form 120 APPENDIX D: Guiding Questions... 122 V LIST OF TABLES -• ' Table .1: Definitions of Attitude 20 vi ACKNOWLEDGEMENTS I must statt by giving credit to God. He walked with me through it all and carried me when the road seemed toughest. I would like to thank my thesis advisor, Dr. Susan Pirie who believed in me sometimes more than I believed in myself. You were my eyes and my ears, and my guide through the maze of research. I look forward to working with you again. My sincere thanks to the other committee members, Dr. Ann Anderson and Dr. Linda Fan-Darling for being so understanding and accommodating. Your words of wisdom went a long way into moulding the works in progress. To my mom and dad, and my sister, Meiyl - the ones that reminded me to pursue my dreams. (Thank you, thank you, thank you)1"'"'0'1. Your love, support and prayers provided a safe haven for working so many miles away from home. To the rest of my family and to my friends here and at home - thanks for being my personal cheering squad. A special thanks goes to the four families that took the time to let me into their busy schedules. Without them, I would have no stoiy to tell. Thanks also to eveiyone else who walked into my life day after day, year after year. There were so many who saw me through to this point. Last but not least, a heartfelt thanks to my fiance Sherwin who gave me the space to take this journey thousands of miles away. I owe you one, honey! Now it is time to start a whole other adventure, but this time with you. CHAPTER ONE: INTRODUCTION THE BIRTH OF M Y QUESTION This research project saw its beginnings in my landlady's kitchen. I was helping her daughter with some math homework, when she came in to check on our progress. As was the norm at the house, our conversation swung to the difficulty of mathematics and its usefulness to mankind. It was during this conversation that my landlady remarked: "Don't worry Julia... You're a chip off the old block. I was never really good at this stuff." My heart sank to hear these words, and by the look on the daughter's face, I could i see that she too felt a bit of discouragement. My landlady is an accomplished violinist, and her daughter seems to be following in her footsteps. Because of this, she believes that a child's abilities are gene-related and that her daughter may have inherited her musical talent as well as her lack of mathematical ability. This got me thinking. From my observations (and according to her previous grades), the girl at the centre of our discussion, Julia, was mathematically competent for an average Grade 10 student. However, she was currently slipping in her math and experiencing some math anxiety. I wondered if the mother's negative remarks were affecting the daughter on a subconscious level. Do parents pass on their math beliefs and attitudes to their children? The exploration began. As fate would have it, within a few months of this episode, I had a conversation with another parent who believed that her child had inherited her lack of mathematical 2 ability. Because this child and I had had conversations about his low math self-concept, hearing what his mother had to say made me question whether the boy's belief about himself had anything to do with his mother's belief that one's mathematical skills was predetermined 'in the genes'. I was curious as to parents' ability to influence their children's attitudes and beliefs, and I chose to explore the issues in this study. Commonsense suggests that our contemporary technologically advanced world would make mathematics easier to understand for students today, than two decades ago. Surprisingly, despite the implementation of computers and graphing calculators in the classroom, there is substantial literature that suggests a continuing high incidence of math anxiety (Sherman & Christian, 1999). So many students are slipping through the cracks or dropping math at the first opportunity. Why? Although Herold (1979) wrote about reasons why children fail at math back in the 1970s, one of his suggestions still seems applicable to concerns raised by modem researchers like Sherman and Christian. Perhaps, our students may be under too little pressure. Unfortunately, in our culture, many people take pride in "never having been able to understand math" (Herold, 1979, p. 53). We are ashamed to say we cannot read, but we are not ashamed to say we cannot do math (Kloosterman & Cougan, 1994; McLeod, 1992; Sherman & Christian, 1999). If parents feel this way and let their children know it, then the message is clear: "Don't bother trying!" Like an urban legend, there is a general belief amongst the populace that a person needs some sort of 'special brain' to do mathematics. However, Herold points out that the truth of the matter is that any person with normal learning ability can handle the 3 mathematics offered through to high school. It is not until one reaches the more advanced levels that a 'gift for mathematics' is required. With the increasing emphasis on the importance of mathematics in the curriculum came a growing interest in the development of affective variables such as attitudes and beliefs. The reader should note that the majority of the research on this aspect of the affective domain was carried out in the 1960s and 1970s, and these works continue to be central to any study done today. The interest in students' attitudes and beliefs was and still is partially due to the generally increasing poor view students hold of math, as well as to the assumption that there must be a strong causal relationship between attitudes and achievement in mathematics (Kulm, 1980). Over the years, the evidence for this causal relationship between attitude and achievement has never been demonstrated as strong (see Cobb, 1986; Garofalo, 1989a; Kloosterman,-1991), but the correlation, although low, was consistently positive, especially as children progressed from their late elementary years to their middle school years (McLeod, 1992; Ma & Kishor, 1997). Many of the people who have conducted research in this area believe that attitudes are in fact important, although in a more complex way than an attitude "test" alone can tell (Brown & McEntire, 1984; McLeod, 1992). Yet recent reports (see Valian, 1998) continue to refer to large-scale quantitative studies, which examine the roots of students' attitudes without, I believe, appreciating the intricacy of the issues surrounding the affective domain. For instance, is it possible for a mother to talk about her own negative math experiences with her daughter, and have that same daughter turn around and echo the opposing attitudes of the father? Will a child not reflect the attitudes of the same-sex 4 parent? These are just some of the questions that may arise out of an investigation such as this; because, I attempted to expose the complexity of the influences on students' attitudes and beliefs by going beyond a simple testing approach. One of the loudest criticisms by those reviewing research done in the 1960s and 1970s on attitudes and beliefs is that there has been too much reliance on correlational methods and on indirect measures of behaviour, such as questionnaires and attitude scales (Aiken, 1970; McLeod, 1987a, 1987b, 1992, 1994; Kloosterman & Cougan, 1994). If researchers are to make progress in building theory and gathering relevant data about the role of the affective domain in the learning and teaching of mathematics, they need to provide data on a wide range of issues (McLeod, 1987c). Some of these issues can.be analysed at a certain level through the use of quantitative techniques, but qualitative data can also add substantially to the completeness of our understanding of these issues. Most of the research to date on students' mathematical attitudes and beliefs has followed the traditional paradigm of quantitative research, and this approach has produced some very valuable information. In. recent years however, research in mathematics education on the whole has made successful use of a variety of qualitative as well as quantitative techniques (McLeod, 1992). Perhaps it is time for these techniques to be introduced to the study of influences on students', mathematical attitudes and beliefs. There have been growing changes in how we view the mathematics learner that are influencing the way we approach research issues (McLeod, 1987a, 1987b; Anderson, 1997). In the past, mathematics education was viewed as the teacher pouring a set of facts into the minds of students, (also known as the empty-vessel principle). However, • 5 today mathematics educators recognise that students are actively engaged in constructing their own knowledge of mathematics, rather than just absorbing it. This type of thinking is the basis for constructivism. Since the late 1980s, McLeod (1987a, 1992) has been a proponent for a "constructivist approach to research" on affect (1987a, p. 133), but constructivism by itself cannot provide the framework my study needs. It does not cater for the social aspect involved in the development of one's attitudes and beliefs. In order to recognize the role of other contributors to children's construction of their math attitudes and beliefs, I heeded the calls for a constructivist approach, but in fact used a social constructivist lens for how I would view the learner. This principle will be discussed later in Chapter Three. According to Aiken (1970) and McLeod (1992), there are at least three techniques available for measuring attitudes and beliefs: (1) observational methods; (2) interviews; and (3) self-report methods like questionnaires and attitude scales. As previously stated self-reporting paper-and-pencil instruments are to date the method of choice; but qualitative approaches like interviews and observational methods can provide us with a rich set of data on students' responses to mathematics and themselves. Having located two quantitative studies that looked at my exact research interests -that is, the effect of parents' mathematical beliefs and attitudes on their children's mathematical beliefs and attitudes, I decided to pursue the same question, but from a qualitative standpoint. In light of the calls for new research approaches in this particular area, the puipose of this study was to explore the research issue of parents' and children's mathematical beliefs and attitudes, by using the specific qualitative method of case study. A case "is a 6 special something to be studied, a student, a classroom, a committee.. ./perhaps, but not a problem, ... or a theme. .. .It is often useful to organise our study around an issue or several of them" (Stake, 1995, p. 133). The two main issues that I was interested in were, whether there still exists a relationship between what children and parents say or believe about mathematics, and about themselves as past learners of mathematics and whether a change in methodology would support past quantitative results and better inform us about students' math attitudes and beliefs. Overview of methodology and methods The Study: A belief in social constructivism helps me to understand how the parent as well as the child plays a role in the construction of a child's mathematical beliefs and r attitudes. In keeping with the tenets of this philosophy, qualitative research methods were sought. As a result, I chose to use a case study approach to address the subject of the influence of parents' math attitudes and beliefs on their children's math attitudes and beliefs. According to Yin (1994), this research method was appropriate for my investigation because it fit his three main criteria for choosing a case study design. First, the investigation would pose a "why" question (Why use qualitative methods in research on the role of parents on students' math beliefs and attitudes?). Second, I as the investigator had little control over the events taking place. Last, I was investigating a "contemporary phenomenon" (Yin, 1994, p. 1) - attitudes and beliefs. The main source of evidence was derived from interview questions that were designed along the lines of the clinical interview method as is described by Ginsburg (1981). This technique is focused, yet flexible enough to allow the interviewer to follow 7 up on responses as well as to allow those responses to take the lead in order to capture what lies behind the immediate appearances of the meaning of a statement (Posner & Gertzog, 1982). Finally the data were analysed in order to create "portraits" (Lawrence-Lightfoot, 1983) of each family interviewed with a view to revealing the mathematical attitudes and beliefs of the students and parents. Selection of Students: Since late elementary and early junior high school years are considered to be important times in the development of attitudes toward and beliefs on mathematics (Wilhelm & Brooks, 1980; McLeod, 1992), I looked to students in Grade 7 to form a putposeful sample. These students were selected by their teacher on the basis of theh aptitude in that math class, their math self-concept (on a particular occasion), their having two parents, as well as their ability to communicate freely and openly with Others. Nature of the study: Being explanatory and exploratory in nature, and using only four family units, there was no attempt to make statistical generalizations from the results of the investigation. However, because of the rich, thick description provided, it is hoped that the portraits can resonate with the reader's own experience and thus, illuminate the phenomenon of parents' and children's math beliefs and attitudes. Summary It seems safe to state that the research into the development of students' math beliefs and attitudes is in need of a jump-start. This is evidenced by the diminishing emergence of new studies in this area. Because the correlations between the influence of 8 parents' beliefs and attitudes and their children's beliefs and attitudes have consistently been quite low, little progress has been made in building theory and gathering relevant data about the role of parents in students' math beliefs and attitudes (McLeod, 1992, 1994). If we support McLeod's findings, then it becomes necessary for researchers to use qualitative approaches that can add to "the completeness of our understanding of these issues" (McLeod, 1992, p. 588). I hope that this study can be the start of a rejuvenation process. It can make a significant contribution to the present body of knowledge since: (1) it looks at one aspect of parental influence - the role of parents' own attitudes toward and beliefs about mathematics, (2) it uses a social constructivist lens for viewing the math learner, and (3) it uses qualitative techniques such as clinical interviewing and portraiture as a way of getting at the 'voice' of the participants, and reporting their beliefs about and attitudes toward math. \ 9 CHAPTER TWO: LITERATURE REVIEW JOURNEYING THROUGH THE M A Z E Background: The affective domain Most mathematics educators would agree that learning mathematics is a cognitive endeavour. Nevertheless, we cannot deny that in the cognitive fields of mathematics, affect, does play an important role in students' decisions about how they approach mathematics and how much mathematics they will need in the future (Ma, 1999; McLeod, 1992; Reyes, 1984; Schoenfeld, 1989). When I talk about my own math teaching, I would be just as likely to mention students' passion for or loathing of mathematics as to report on their cognitive achievements. Sometimes I think I spend even more time discussing the former than the latter with colleagues. As a society on the whole, we tend to have more affective than cognitive discussions about mathematics, as comments about liking or disliking math are more common than the mathematical content actually learnt. These informal observations appear to support the view that affect plays a significant role in mathematics learning and instruction. In a philosophical look at education, Scheffler (1991) states that "Education, that is to say, the development of mind and attitudes in the young- is split into two grotesque parts: unfeeling knowledge and mindless arousal" (p. 3). Even though affect and cognition sometimes appear to be worlds apart, my purpose here is not to reduce cognition to emotion or emotion to cognition. Instead, like Scheffler, I hold that "cognition cannot be cleanly sundered from [the affective domain] and assigned to -10* science. All these spheres of life involve both fact and feeling; they relate to sense as well as sensibility" (p. 3). I think Maturana (1991) best describes the intertwining influence of the cognitive and affective worlds when he says: We human beings operate and exist as an intersection of our conditions of observers (in conversation) and living systems, and as such we are multidimensional beings, actual body-nodes of a dynamic intercrossing network of discourses and emotions that continuously move us from one domain of actions to another in a continuous flow of many changing conversations (p. 43). The entangled relationship between affect and mathematical learning and teaching provides us with major reasons for studying affective factors in mathematics education (Reyes, 1984). For example, Reyes and later Hart (1989) both find that possessing a positive attitude toward math is an important educational outcome, regardless of the level of achievement. They, along with others, (see Daskalogianni & Simpson, 1999; Garofalo, 1989a; Kloosterman, 1995; and Ma & Kishor, 1997) are convinced that positive attitudes will improve the ability of students to leam math. This statement is also supported by the National Research Council's 1989 report on the future of mathematics education, which puts considerable emphasis on the need to change the public's beliefs and attitudes about mathematics. It may be obvious that affective issues are a central concern of students, teachers, and even parents - but what about researchers? Despite this external motivation, I discovered that research on affective issues in mathematics education continues to reside on the "periphery of the fteld"(McLeod, 1992, p. 575). According to Schoenfeld (1992), in early research, there was a "sharply delineated distinction between the cognitive and 11 affective domains" (p. 358) that originated from Bloom's (1956) Taxonomy of Educational Objectives. Constructs like mathematics anxiety, for. example, were said to clearly reside in the affective domain because they were centred on how individuals felt about mathematics. On the other hand, concepts such as mathematics achievement resided within the cognitive domain since they solely dealt with one's subject-matter knowledge. From all the research to date, it would seem that as more research was done, our understanding of the cognitive and affective domains became enlightened, as we began to appreciate' the overlapping nature of affect and cognition. As one reads the literature, one cannot help but become aware of problems with the definitions used for what is typically called the affective domain. From the discussions about the affective domain, it appears that psychologists, and mathematics educators interested in research on beliefs and attitudes toward mathematics have difficulty communicating clearly with one another owing to, in part, the lack of common usage of terms.. Depending on the author one reads, it would seem that the word affect has several meanings. For example, in Reyes (1984), she says "affective refers to students' feelings about mathematics, aspects of the classroom, or about themselves as learners of mathematics" (p. 558). Along the same lines, Leder's (1987) report on attitudes toward mathematics uses affect as a general term to represent all the feelings that seem to be related to mathematics learning and teaching. Mandler (1989) however, recognized that there was a dilemma surrounding the meaning of the word affect and offers the following interesting metaphoric interpretation: 12 Unfoitunately, the term has meant many things to many people, acquiring interpretations that range from "hot" to "cold". At the hot end, affect is used coextensively with the word emotion, implying an intensity dimension; at the cold end, it is often used without passion, referring to preferences, likes and dislikes, and choices. I interpret the use that it has acquired in the problem-solving field to be hot, rather than cold... (p. 3 - 4). As a result, Mandler uses the words affect and emotions interchangeably at times in his work. Given all these different viewpoints, I would have to say that the common thread amongst them is that the affective domain refers to a "wide range of feelings and moods that are'generally regarded as something different from pure cognition" (McLeod, 1987b, p. 170). For the puipose of my research, I found that McLeod's (1987b, 1992) reviews of research on the affective domain offered the clearest explanation. Based on the research done in the past, he believed that there exist at least three facets of the affective experience of mathematics students that are worthy of further study. First, students hold certain beliefs about mathematics and about themselves that play an important role in the development of their affective responses to mathematical situations. Second, since inteiTuptions and blockages are unavoidable in the leaning of mathematics, students will experience both positive and negative emotions as they leant mathematics; these emotions are likely to be more noticeable when tasks are new. Third, students will develop positive or negative attitudes toward mathematics (or parts of the mathematics cuniculum) as they encounter the same or similar mathematical situations repeatedly. In addition, beliefs, attitudes and emotions describe a wide range of affective responses to mathematics, with vaiying stability, thus while beliefs and attitudes are generally stable, 13 emotions may change rapidly. For example, a student who believes that math is rule-based is likely to express the same attitude the next day (and even 10 years later). However, a student who gets frustrated while working on a non-routine problem may express strong positive emotions, like the 'Aha experience', just a few minutes later when the problem is solved. According to McLeod (1992), beliefs, in general, are largely cognitive rather than affective in nature. Even with respect to the sub-levels of math, beliefs, beliefs about mathematics often involve very little affect, and are frequently based as much on cognitive responses as on feelings or affective responses. On the other hand, beliefs about oneself may have more of an affective component (see McLeod, 1987b, 1992). According to Leder (1987), attitude toward mathematics is a construct used to refer to feelings about math that are relatively consistent. Such feelings might include how much a student likes math, or how confident he/she feels about doing math. McLeod (1987b) brings out an important point when he states that attitudes sometimes have a belief component, but "they are distinguished from beliefs by the feelings that accompany the beliefs" (p. 172). Mandler (1989) describes emotion as an affective reaction that is more intense than beliefs or attitudes. Emotions usually involve some physiological arousal like tense muscles and rapid heartbeat, as well as some redirection of the person's attention. Typical emotions would include joy, anxiety, frustration, and surprise. Of the three affective variables, the emotional reactions of students have received the least attention with respect to research because of the "lack of a theoretical framework within which to interpret the role of emotions in the learning of mathematics" (McLeod, 1992, p. 583). Because of this lack of theoretical framework, I tried to focus only on the mathematical beliefs and attitudes which parents and students hold, since the nature of these two constructs better lend themselves to being studied. However, as you the reader will discover from the portraits, it is difficult to ignore the emotions people bring with them when talking about math. Before reviewing the literature on beliefs and attitudes, I would like to prepare you the reader for the dilemma faced in defining the constructs of belief and attitude. In a book titled A Question of Values, Hunter Lewis states: In the world of human thought... the most fruitful concepts are those to which it is impossible to attach a well-defined meaning (cited in -Pajares, 1992, p. 308). In contrast, Pajares himself (1992) argues that "articulate conversation must demand not only clarity of thought and expression but also preciseness of word choice and meaning" (p. 309). Unfortunately, educational psychology does not bestow such precision upon its constructs, and instead, the issue of definitions in this area of the affective domain closely resembles Lewis' description. Thus, finding the meaning of beliefs and attitudes in mathematics education becomes at best a "game of player's . choice" (Pajares, 1992, p.309). Mathematical beliefs According to Garofalo (1989a, 1989b), with support from McDonough (1998), research'in mathematics education has consistently shown that our children's success or failure in problem-solving depends on much more than their knowledge of the needed mathematical content. One factor that can influence the direction and outcome of their 15 performance is the beliefs that they hold about mathematics. However, in spite of the arguments that people's beliefs are important influences on the way in which they conceptualise tasks and learn from experience, the implicit interest and fascination that educators and researchers alike have in beliefs have not become explicit (Pajares, 1992). In fact, studies aimed at understanding the beliefs of students have been scarce during the 1990s. Why? Although Rokeach (1968) defined beliefs as "any simple proposition, conscious or unconscious, inferred from what a person says or does, capable of being preceded by the phrase, 'I believe that..."' (p. 113), perhaps as a global construct, belief (like affect and attitude) does not lend itself to be easily defined. Depending on the researcher, beliefs have been referred to as -attitudes, values, judgements, perceptions, conceptions, dispositions, and internal mental processes, to name a few (for a list of more aliases see Pajares, 1992). Dewey (1933) described belief as "something beyond itself by which its value is tested; it makes an assertion about some matter of fact or some principle or law" (p. 6). Abelson (1979) defined beliefs in terms of people manipulating knowledge for a specific puipose or under a necessaiy condition. Brown and Cooney (1982) explained that beliefs are dispositions to action and major deteiminants of behaviour. Even though these dispositions are time and context specific, they are qualities that have significant implications for research and measurement. Sigel (1985) characterised beliefs as "mental constructions of experience -often condensed and integrated into schemata or concepts" (p. 351) that have enough truth to guide behaviour. In Schoenfeld's (1992) discussion, beliefs were interpreted as an individual's 16 understandings and feelings that shaped the ways that that individual conceptualised and engaged in .mathematical behaviour. Beliefs can be descriptive (It is time for math class to begin), evaluative (I do not enjoy doing math), or prescriptive (I need to do my homework, or my teacher will have a fit), but elements of each are present in most beliefs (Pajares, 1992). Rokeach (1968) claimed that all beliefs are comprised of three integral parts. A cognitive component that represents knowledge, an affective component capable of arousing emotion and a behavioural component activated when action is required. Other key features about the nature of beliefs are as follows: they are formed early and tend to self-perpetuate, persisting even against contradictions caused by reason, time, schooling, or experience (Abelson, 1979; Rokeach, 1968). Second, individuals tend to develop a belief system i t that houses all their beliefs acquired through the process of cultural transmission (Abelson, 1979; Brown & Cooney, 1982), Third, they can colour not only what an individual recalls but also how he/she recalls it, and if necessary completely distorting the recalled event in order to sustain the belief (Pajares, 1992). Finally, there is the self-fulfilling prophecy by which beliefs influence perceptions that influence behaviours that are consistent with, and that reinforce the original beliefs (Pajares, 1992). However, Rokeach cautioned that understanding beliefs requires "making inferences about individuals' underlying states, which may be fraught with difficulty because individuals are often unable or unwilling, for many reasons, to accurately represent their beliefs" (Pajares, 1992, p.314). It is for this reason that those who review 17 research on affective issues contend that beliefs cannot be directly observed or measured, but must be inferred from what people say, intend, or do (Pajares, 1992; McLeod, 1992). As McLeod indicates, students' beliefs can also be categorized in teims of the object of the belief; thus, there is research referring to beliefs about mathematics, beliefs about self, beliefs about mathematics teaching, and beliefs about the contexts in which mathematics education occurs. McLeod (1992), Garofalo (1989a, 1989b), and Schoenfeld (1985) are among the many authors known for writing of the importance of considering beliefs about mathematics as a discipline. For example, Garofalo states that many secondary students believe, "almost all mathematical problems can be solved by the direct application of the facts, rules, foimulas, and procedures shown by the teacher or given in the textbook" (1989a, p. 502). This notion can lead to the conclusion that "mathematical thinking consists of being able to learn, remember, and apply facts, rules, and procedures" (1989a, p. 503). This kind of thinking has significant implications for a researcher like Peter Kloosterman who studies the effect of affective variables on motivation. He states that "from a motivational perspective, students who believe that mathematics is rules and procedures are motivated to try to memorise those rules and procedures. They are not interested in'trying to grasp underlying concepts or connections between mathematical topics" (Kloosterman, 1995, p. 135). Other beliefs about the nature of mathematics that students typically seem to hold include: (1) 'math problems have one and only one right answer', (2) 'math is a solitary activity, done by students in isolation', (3) 'students who have understood the math they have studied will be able to 18 solve any assigned problem in five minutes or less', and (4) 'the math learned in school has little or nothing to do with the real world' (see Kloosterman, 1995; McLeod, 1992). Although student beliefs about mathematics have received the most attention in research on problem solving, studies of beliefs about oneself as a learner of mathematics have also played a central role in research on affect (McLeod, 1994). According to Eisenberg (1991), one's belief about oneself or the perceptions of one's own ability is an extremely important factor in the mathematics that one does, as a low self-opinion about one's ability can be devastating. For example, the myth that some people lack mathematical minds is an important part of self-concept (Kloosterman, 1995). As Kloosterman further explains, this type of thinking does not affect motivation for students who think they are good in math. However, it can be devastating for students who have the impression that they just "weren't cut out" (p. 13.8) to do math. It should be noted that what we know about beliefs about the math self is as a result of research on constructs such as self-concept, confidence, and causal attributions (McLeod, 1992). So far this discussion has concentrated on students' beliefs about mathematics and about themselves as learners. But as mentioned earlier, there is a corresponding set of beliefs that students hold about mathematics teaching (e.g. 'teaching is telling') and about the social context of instruction (e.g. 'learning is competitive') that is also important to the study of affect in mathematics education (McLeod, 1992). With respect to what is known about beliefs about mathematics teaching, the majority of the research is centred on teachers' beliefs about mathematics and mathematics teaching (Kloosterman, 1995; McLeod, 1992; Pajares, 1992); thus we have little information on students' beliefs about \ 19 mathematics instruction. The same is true for students' beliefs about the social context. Even though McLeod (1992) states that "recent research on mathematics learning has given increased attention to the social context of instruction and more generally to cultural issues in mathematics education" (p. 581), the overall picture of research on affect shows that what has been done in this area is quite small in comparison to the work done on other types of beliefs. , Two of the goals set out by the.National Council of Teachers of Mathematics ate: (1) valuing mathematics, and (2) becoming confident in one's ability to do mathematics (NCTM, 1989). These can be explored by considering students' beliefs about math, and beliefs about themselves as doers of math. These two categories of beliefs are important not only because "they influence how one thinks about, approaches, and follows through on mathematical tasks but also because they influence how one studies mathematics and how and when one attends to mathematics instruction" (Garofalo, 1989a, p. 502). As a result of the N C T M stated goals, and the fact that I wanted to maintain consistency in some areas with the previous two studies that researched my question, I focused solely on beliefs about math and beliefs about oneself as a math learner, rather than address some of the other potential research areas mentioned above. Attitudes toward mathematics As is the case with the belief construct, research on attitudes has a relatively long history, but it too has been troubled by the lack of clear and precise definitions: see (Aiken, 1970, 1976; Leder, 1987; Reyes, 1980; Kulm, 1980; Ma & Kishor, 1997) for detailed reviews and analyses. Just as there were several definitions for the word belief, 20 so too there are numerous ways of defining the word attitude. Often the term attitude is referred to in the literature as beliefs (Neale, 1969) or feelings (Daskalogianni & i Simpson, 1999), and there is even the tendency to avoid explicit definition and instead settle for operational definitions "implied by items of instruments measuring attitude" (Kulm, 1980, p. 356). Furthemiore, Leder (1987) states that "many researchers seem to select for their definition a measurement procedure that is convenient for their puipose of their study. Those concerned solely with measurement typically defined attitude as unidimensional, while those concerned with building theory have tended to use a broad multistmctural definition" (p. 261). Leder even cites researchers Fishbein & Ajzen (1975) who once identified more than 500 different methods of measuring and in turn defining attitudes. Nevertheless, as can be seen from the sample definitions of attitude summarized below, there is much overlap amongst available definitions for attitude and between these definitions and definitions of beliefs discussed earlier. Table 1: Definitions of Attitude Author Year Definition v Allport 1935 A mental and neural state of readiness, organized through experience, exerting a directive or dynamic influence upon the individual's response to all objects and situations with which it is related. Rokeach 1968 An organization of several beliefs focused on a specific object or situation predisposing one to respond in some preferential manner. 21 Aiken Ajzen McLeod Daskalogianni & Simpson 1970 A learned predisposition or tendency on the part of an individual to respond positively or negatively to some object, situation, concept, or person. 1988 A disposition to respond favourably or unfavourably to an object, person, institution or event. 1992 The affective responses that involve positive or negative feelings of moderate intensity and reasonable stability. 1999 The amalgam of the emotional experiences of a topic and the beliefs about the nature of the topic which leads to a predisposition to respond with similar emotions and similar expectations in similar experiential settings. With so many viewpoints to consider, it begs the question of what we really mean by attitude. For this study, I used McLeod's definition because it seemed to capture the meaning of other definitions, but in a concise manner. Thus, liking or disliking calculus and being bored by factorisation are seen as 'attitudes' by this researcher. However, it should be noted that mathematics is not a unidimensional factor; there are many different kinds of mathematics, and with them can come a variety of feelings about each type of mathematics (McLeod, 1992). Because of the exploratory nature of this project, I examined attitudes toward mathematics in a general sense, since there is little evidence of elementary-aged children identifying the various areas of mathematics (McDonough, 1998). 22 According to McLeod (1992), attitudes toward mathematics appear to develop in two different ways. First, "attitudes may result from the automatizing of a repeated emotional reaction to mathematics" (p. 581). For example, if a child has repeated negative experiences with say problem solving, the emotional impact usually tends to lessen in intensity over time. Eventually the emotional reaction to solving problems will become more unconscious, there will be less physiological arousal, and the response (dislike of problem solving) will become a stable one that can probably be measured as an attitude, through a questionnaire or by observation. A second source of attitudes is "the assignment of an already existing attitude to a new but related task" (McLeod, 1992, p. 581). By this McLeod means that a student who has a negative (or positive) attitude toward say geometric proofs may attach that same attitude to proofs in algebra. There exists a further hypothesis concerning the bringing together of beliefs and attitudes. In their analysis of the formation and effects of attitudes towards mathematics, Katrina Daskalogianni and Adrian Simpson (1999) found that the source of an individual's attitude towards math is the product of his/her beliefs about math and of their previous experiences with it. It is their view that when a student is faced with a math problem, he/she first tries to associate it with previous math problems encountered in the past. It is at this point that an attitude towards math, or towards that broad category of the math question, is activated. If the mathematical situation is novel, the student first tries to associate it with previous ones and with the approaches used and at the same time, their beliefs about what math is about and how it should be approached come to the surface and can predispose them to act mathematically in a particular way. If the mathematical 23 situation is already known, the student is then predisposed to directly approach it in the way that is consistent with their previous experiential settings and the corresponding beliefs about it (p. 11). From these researchers' viewpoint, there seems to be an inter-play between attitudes and beliefs. Since the 1960s, mathematics educators have been troubled by the growing dislike for formal mathematical teaching (Neale, 1969; McLeod, 1992). As a result, mathematicians and mathematics educators have been placing quite an emphasis on discovery modes of learning and on mathematical recreations, partly because such activities are thought to give a truer picture of mathematics, but more importantly, to promote positive attitudes toward mathematics (Neale, 1969). Neale explains how as teachers in the classroom, we should be aware that allowing students to enjoy the discovery of the "orderliness of mathematical relationships" (p. 631) has the potential of illuminating the basic structure of mathematics as a subject, as well as turning reluctant students into eager mathematical explorers. Implicit in such an observation is a belief that (1) certain attitudes toward mathematics are thought to be important objectives of instruction (NCTM, 1989), and (2) a positive attitude toward math is thought to play a crucial role in causing students to learn math. However, when researchers carried out quantitative studies that tested the effect of attitude on mathematical achievement, the results consistently showed low correlations. In keeping with the scientific approach, such results should be dismissed because they lacked the statistical significance needed to support a causal relationship between attitudes and mathematical performance. In spite of such evidence, the following claim still seems to be a widespread belief that mathematics educators use today to support their reasons for promoting positive attitudes in students:-Favourable attitudes toward school subjects on the whole maximize the possibility that a student will willingly learn more about that subject, remember what he/she has learned, and use what he/she has learned (Mager, cited in Neale, 1969, p. 633). We need therefore to investigate this whole area further to see whether such statements can really be substantiated. Notwithstanding the difficulties involved in researching attitudes toward math, there have been periods of substantial progress in our knowledge of math attitudes (McLeod, 1987a, 1992). From the 1960s until the present day, the research has showed some success in identifying important patterns of student responses to mathematics, especially in the area of gender-related differences (Fennema & Sherman, 1976; McLeod, i 1994). At the same time however, other researchers were dissatisfied with the results, noting that the theoretical background for the studies was lacking (Leder, 1987; McLeod, 1987a), that the results sometimes ran contrary to expectations (see Ma, 1999), and that "complicated statistical analyses of questionable questionnaire data were not necessarily reflecting accurately what students were thinking and feeling" (McLeod, 1994, p.640). According to those who have been reviewing research on math attitudes (see Neale, 1969; Kulm, 1980; Leder, 1987; McLeod, 1987b, 1992, 1994), the problem was finding adequate explanations of the relationship between attitudes and achievement when the results consistently showed low statistical correlations between the two variables. They all suggest that a possible solution.may lie in the approach to research in this area. 25 Instead of using the quantitative methods favoured in the past, they recommend the use of methods that help to capture the complexity of math attitudes. My study heeds their advice, but more of this will be discussed later. Now, it is time to turn our attention to an area of the research that has been given little attention in the past decade (in comparison to the work done on affect and achievement) -the factors that contribute to the construction of students' math attitudes toward and beliefs about themselves in terms of their mathematical ability and interest (McLeod, 1992). Influences on mathematical beliefs and attitudes According to McLeod (1992), many major evaluation studies have provided us with useful background information on attitudes and beliefs about self. National assessment reports from the United States asked children in grades 3, 7, and 11 if they were good at doing mathematics; and the percentage of students who responded positively dropped from 65% in grade 3 to 53% in grade 11. This indicates what happens to levels of confidence as a student progresses through school (McLeod, 1992, p. 580). Those reports also found that the percentage of students who say they enjoy mathematics declined from 60% in grade 3 to 50% in grade 11. So we have some evidence of what happens to students' mathematics attitudes as they advance through school. But why are so many of our students not confident about their mathematical ability? Why does a poor attitude toward the discipline plague learners at different levels of schooling? According to the research on the affective domain, children's attitudes and beliefs may be influenced by several factors, which in turn may affect their mathematical 26 achievement. The following is my own simple diagrammatic representation of this relationship, based on past research: Teachers & School Environment Parents & Home Environment Society E.g. peers, media Students' Beliefs/ Attitudes Math Achievement To date, three main influential factors have been identified - teachers, parents, and the society at large (see Eccles & Jacobs, 1986; McLeod, 1992; and Bishop, 1993). These researchers all agree that answers may lie in the social context provided by the school and the home of students, as well as society at large. According to Bishop (1993), societies establish educational institutions for intentional reasons - and formal mathematics education is directly shaped and influenced by those institutions in different ways. Additionally, he explains that a society also encompasses individuals, groups and institutions, which do not have any formal or intentional responsibility for mathematics learning. Nevertheless they may "frame expectations and beliefs, foster certain values 27 and abilities, and offer opportunities and images, which will undoubtedly affect the ways in which mathematics is viewed, understood and ultimately learnt by individual learners" (Bishop, 1993, p. 3). There are many individuals in our society who could potentially influence a young person, but one of the principal groups who seem to have the greatest potential for affecting his/her formal mathematics education is the adult members of the family (Bishop, 1993; Wilhelm & Brooks, 1980;Parson, Adler, & Kaczala, 1982; Wigfield, 1983; Okagaki & Sternberg, 1993). According to these researchers, it is parents' perception, memory, and image of how school was for them, which affect what they say and believe about mathematics today. For many of today's adults, the image of mathematics is not only a veiy traditional one, but also it tends to be a mainly negative one as well (based on their own experiences of math in school). It also seems that the negative experience is assumed to be so widespread that to claim mathematical ignorance and inadequacy is socially acceptable, however unpleasant it may be personally. The collective parental memory of school mathematics can in fact be largely negative, and so it is possible for this memory to be easily transmitted as a negative image to the next generation, thereby influencing not only the mathematical expectations of the children, but their motivations for studying mathematics and their predisposition for continuing, or not, to study the subject (Aiken, 1970; Yee & Eccles, 1988; Bishop, 1993; Garofalo, 1989a, 1989b). Thus the messages which would be received by our students are that mathematics is not a very important subject, that it is a difficult subject, and that only certain people in society will be able to achieve well in it. Of course, 28 positive messages extolling the joys of mathematics may also be passed from parent to child; unfortunately, those stories seldom make it to the mainstream. In the current study, I question whether parents still have an overall negative image of their math experiences in school, and whether there is a largely negative image echoed in their children's math experiences today. Aiken (1970) cites three ways in which parents can affect children's beliefs and attitudes: (1) by parental expectations (beliefs) of child's achievement, (2) by parental encouragement, and (3) by parents' own attitudes. From the articles reviewed for my study, the majority of them focused on either the impact of parental expectations, or parental support (see Gottfried, Fleming, & Gottfried, 1994; Holloway, 1986; Parsons, Adler, & Kaczala, 1982; Raymond & Benbow, 1986; Wang, Wildman & Calhoun, 1996; Wigfield, 1983); and this is also reflective of the research focus of many investigators of affective issues in mathematics education. One of the most fascinating findings of these studies is that, parents' beliefs about children's ability and the children's own beliefs appeal" to be related in small, yet significant and meaningful ways. For instance, parents who believed their children were not able in math were found to have children who were low in then math ability perceptions. Unfortunately, it appears that veiy few researchers are concerned with the impact of parents' own attitudes and beliefs. Since only two of the articles (Wilhelm & Brooks, 1980; Andre, Whigham, Hendrickson & Chambers, 1997) I reviewed did in fact look into the impact of parents' attitudes and beliefs, I decided to focus my attention in this direction. 29 The first study by Wilhelm & Brooks (1980) looked into the existence of a relationship (by grade and by sex) between student attitudes to math and parental attitudes to math, using the Sandman's Mathematics Attitude Inventory. Significant relationships were found between Parents' Anxiety, Self-concept, Enjoyment, and Perceived Value of Mathematics and pupil attitudes on the same scales. The second study by Andre 'et al (1997) also had the same research purpose as the first; however, they added questions that asked the parents their perception of their child's competence, and their expectation of their child's performance. These researchers also found notable correlations between parental and students' attitudes; however, their results are quite centred around gender differences. Both sets of results tell the same tale: yes there is a relationship, but quantitatively it is veiy small. Thus it appears that more work is needed to give credibility to this area. According to McLeod (1987a, 1992, 1994) what seems to be missing from both studies is the view that students actively construct their own beliefs and attitudes. In order to appreciate that viewpoint, as well as see the role parents play in the formation of these complex constructs, attitudes and beliefs need to be measured by methods other than paper-and-pencil tests. Do children see themselves as affected by what their parents believe about math? We can think of parental influences as being essentially extrinsic, because they originate from someone outside the learner. However, the extent of their influence will depend on how internalised they become, and thus, the internalisation process is key to understanding motivation for learners (Bishop, 1993). Bishop goes on to say that the internalisation of beliefs and expectations, as with any 30 aspect of motivation, will be dependent on the significance to the learner of the people mediating those beliefs and expectations. From this review of the literature, it is clear that investigating students' math beliefs and attitudes adds to our understanding of the possibility of "breadth, complexity, and subtlety" (McDonough, 1998, p. 269) of these affective variables. Although the research on attitudes and beliefs has produced useful data for the mathematical community, there are many who feel that a new approach to the affective domain could yield substantially more progress, especially in the development of better theoretical frameworks, and in making connections between research on affect and contemporary theories of learning (see Kulm, 1980; McLeod, 1987b, 1992, 1994; Reyes, 1980). This new approach would need to take into account the view that "learners are actively engaged in constructing their knowledge of mathematics, rather than just absorbing it" (McLeod, 1987a). Referred to as a constructivist view, McLeod further states that this way of regarding the student learner is already impacting upon paradigms for research on cognitive issues in mathematics learning and teaching. Thus, it is time for constructivism to influence how mathematics educators approach research on children's attitudes toward and beliefs about mathematics. Instead of carrying out a statistical investigation similar to the two studies that examined the role of parents' own attitudes and beliefs (Wilhelm & Brooks, 1980 and Andre et al, 1997), I adopted a different methodological approach in order to see if its usage would allow for a deeper understanding of the research on students' math attitudes 31 and beliefs. In the following chapter, I offer a detailed look into how applying qualitative methods concurs with the constructivist orientation suggested in past research. f 32 CHAPTER THREE: METHODS & METHODOLOGY LAYING THE GROUND RULES The Constmctivist Lens For at least the last three decades, those reviewing research have been calling for a different way in which to approach the research on the affective domain. Abelson (1976) notes that research on attitudes [and the same could be said for beliefs] is confusing and contradictory, and suggests that "the present state of .. .theory is frankly in a mess" (p. 40). Mandler (1984), also supported by Ma & Kishor (1997) observe that research in this area is generally not cumulative, and that researchers have been preoccupied with measurement issues, while neglecting the development of theory. In mathematics education, many of the reviewers of the research on affect have asked for greater emphasis on theoiy development to guide research on attitudes and beliefs, especially as numerous authors have noted the relatively weak relationship between affect and achievement (Kulm, 1980; Leder, 1987; Ma & Kishor, 1997; McLeod & Adams, 1989). In an attempt to expand the conceptions of the affective domain in mathematics education, McLeod (1987a, 1987b, 1987c, 1994) has been quite outspoken in demanding a constructivist lens for re-viewing research issues in this area. But what does a constructivist approach to researching attitudes and beliefs look like? In establishing a constructivist position for studying the development of attitudes, McLeod (1987a) referred to the works of Mandler (1984) and Skemp (1979) and discovered that there are "barriers which children face as they learn mathematics" (p. 33 137). These intemiptions or blockages either prevent a schema from reaching completion (Mandler, 1984) or keep a student from reaching a goal (Skemp, 1979). The student's interpretation of that barrier, however, will*depend on his or her knowledge, beliefs, and previous experience. According to McLeod (1987a, 1987c, 1989), the work done by Mandler and Skemp provide useful information on how barriers can produce negative as well as positive feelings about math. On the other hand, these studies use von Glaserfeld's view of constructivism that "treats the cognizing subject as the organizer of his or her own experience and the constructor of his or her own reality" (Kilpatrick, 1987, p. 10). It is as though our children are "informationally closed" (Kilpatrick, 1987, p. 9), self-contained systems. They ignore the fact that students are often influenced by external factors as they construct their beliefs and attitudes. As a result, we need to add another dimension unto our constructivist lens - the social aspect. Acknowledging the social aspect of learning mathematics is not a trivial problem, because the social domain includes linguistic factors, cultural-factors, interpersonal interactions (such as peer interactions), as well as teaching and the role of the teacher (Bauersfeld, 1996; Cobb, 1994; Ernest, 1994; Lerman, 1992). However, if we acknowledge the existence of an informal mathematics education system (Bishop, 1993) by which children learn math through unintentional measures, then we must also acknowledge that parents are involved in that informal process. As a result, the social domain as described above should be extended to include specifically the parent-child interactions, and the role of the parent. Still, a fundamental problem the psychology of mathematics education faces is: "how to reconcile the private mathematical knowledge, skills, learning, and conceptual development of the individual with the social nature of [formal and informal] school mathematics, .. .its context, [and its] influences" (Ernest, 1994, p. 62)? One approach Ernest and other authors who agree with his line of thinking offer to solve this dilemma is to propose a social constructivist theory of learning mathematics (Bauersfeld, 1996; Cobb, 1994; Ernest, 1994; Rogoff, 1990). It should be noted here that social constructivism comes out of considerations of cognition, but can be broadened to explorations of the affective domain. On the surface this, theory recognises that "both social processes and individual sense making have central and essential parts to play in the learning of mathematics" (Ernest, 1994, p. 63); but, the precise nature of this perspective is far more complex to describe. The following is a brief description that tries to do justice to the depth of such a construct. Social constructivism is used to refer to widely divergent positions, which share the notion that the social domain impacts on the development of individuals in some formative way, and that individuals construct (or adapt) their meanings in response to their experiences in social contexts (Bauersfeld, 1996; Ernest, 1994; Rogoff, 1990). Such a description is vague enough to accommodate a range of positions -from the slightly socialized version of radical constructivism based on a Piagetian theoiy of mind to more socio-cultural and sociological perspectives based on a Vygotskian linguistic approach. Researchers who have attempted to develop a form of social constructivism based on Piagetian constructivist theoiy usually start from a radical constructivist position and add on social aspects of classroom interaction to it (Cobb, 1994; Ernest, 1994). This kind • 35 of thinking is referred to as a "compleraentarist" (Ernest, 1994, p. 66) version of social constructivism because it prioritises the individual aspects of knowledge construction, but acknowledges the importance of social interaction as secondary (Cobb, Wood, & Yackel, 1992; Cobb, 1994; Ernest, 1994). However, if we are to acknowledge the separation of the social and individual domain that a complementarist approach assumes, then how do we compensate for the role of other factors like language and culture? Ernest (1994) questions the Piagetian version of social constructivism when he asks: "If these are ontologically disparate realms, how can transfer from one to the other take place" (p. 68)? If we are to use this framework to examine the beliefs and attitudes of students, then we would have to ask how such a framework could explain the 'transfer' of attitudes and beliefs from parent to child? Lerman (1992, 1994) proposed that some of the difficulties linked with such a form of social constructivism might be overcome by replacing the Piagetian theory of mind with the Vygotskian theory of thought and language. Lerman further argued that any form of social constructivism that maintained a radical constructivist account of individual learning of mathematics failed to adequately account for language and the social dimension. Commonsense dictates that the interactions between parent and child are full of language; thus, a Vygotskian form of social constructivism is more appropriate for this study. Vygotskian social constructivism distinguishes itself from the Piagetian form by taking an integrated rather than complementary approach to constructivism. According to Ernest (1994), this other approach views the individual and the social domains as "indissolubly interconnected, with human subjects formed through their interactions with 36 each other (as well as by their internal processes) in social contexts" (p.69). Here, the mind is viewed as social and conversational (language aspect) because of the following assumptions: (1) "individual thinking of any complexity originates with and is formed by internalized conversation, (2) all subsequent individual thinking is structured and nurtured by this origin, and (3) some mental functioning is collective" (Ernest, 1994, p.69). Even though it would be difficult to single out only one of these assumptions, I was intrigued by the first one about internalisation because, given the interactive nature of the parent-child relationship, it helps us to understand how the math beliefs and attitudes of parents can be passed on to their children. Just as the Vygotskian version of social constructi vism has been used by researchers to show the crucial role of the teacher in student learning (Cobb, 1994; Ernest, 1994), I feel that it can also be used to enlighten the research on the role of parents in student learning. Bauersfeld (1996) says it best when he states: We may have to understand learning... from a perspective that takes into account many more forces and influences than the traditional understanding does. (p. 10) The study Qualitative research methods were adopted to explore the mathematical attitudes and beliefs of students and parents involved in the study. Given the research problem and the explanatory and exploratory nature of the study, a case study seemed appropriate as a research design. The aim of a case study can be to describe and explain a given phenomenon that is of interest to the researcher. Yin (1994) defines a case study as "an empirical inquity that investigates a contemporary phenomenon within its real-life 37 context, especially when the boundaries between phenomenon and context are not clearly evident" (p. 13). This definition is appropriate for this study because there are several inteitwining factors that can influence a student's math attitudes and beliefs; however, the heart of this study focused only on the influence of parents' math attitudes and beliefs. By delimiting the object of the study, I have also met one of Merriam's (1998) key requirements for doing case study research, that is, a case study must possess a sense of boundedness. Finally, my decision to choose a qualitative case study stemmed from the fact that this design is chosen when researchers are interested in "insight, discovery, and interpretation rather than hypothesis testing" (Merriam, 1998, p. 28). With a framework in place and the research design decided, I moved my attention to selecting an appropriate grade level to sample. In a review of an investigation of the genesis of negative attitudes toward mathematics, McLeod (1987b) put forth the suggestion that seventh grade is an important point in the development of attitude, a finding that agrees with other.research in this area. As a result, I focused my attention on getting seventh graders to participate in my study. Once I received permission from the principal of the "Young Buds" Elementary School in Vancouver, BC (see Appendix A), I approached one of the seventh grade Math teachers and asked for the participation of four students. Why only four? Merriam (1998) states that the "crucial factor is not the number of respondents but the potential of each person to contribute to the development of insight and understanding of the phenomenon" (p. 83). With this in mind, when selecting students to participate, the teacher was to choose students who were articulate in expressing their views about math 38 beliefs and attitudes. More importantly, I wanted children based on two criteria - (1) their perception of themselves as math learners, and (2) their mathematical ability. Based on these criteria, the following matrix was produced as a guide for the teacher: Student who is good in math, and thinks he/she is good in math Student who is poor in math, but thinks he/she is good in math Student who is good in math, but thinks he/she is poor in math Student who is poor in math, and thinks he/she is poor in math Ascertaining a child's math self-perception is problematic. I decided to base this on a single judgement made by the student, and reported to me by the teacher. I was told that the teacher asked the class to rest their heads (so that they could not see each other) and raise their hands when they agreed with the statement she made. One example of such a statement is "I think I am good in math". Once the students raised their hands, she then judged them on their math ability and their ability to express themselves. Finally four students were chosen, and they were each given consent forms for themselves and their parents (see Appendices B & C). It should be noted that all the students participating in this project were girls; since gender was not a focus of this study, a balanced sample was not sought. Interviewing in theoiy [A]ny fundamentally new approach to a scientific problem inevitably leads to new methods of investigation and analysis. Vygotsky cited in Ginsburg, 1997, p. 1. 39 In keeping with McLeod's (1987a, 1992) suggestions for constructivist researchers on affect to use individual observation, clinical interviews, and teaching experiments, I decided to use the clinical interview as my medium for data collection. One of the major advantages of an interview as opposed to a survey or questionnaire is that when questions fail to produce detailed answers (as with paper-and-pencil tests),-the researcher can probe further (Kloosterman, 1997). The clinical interview (Ginsburg, 1981) is a technique first developed by Jean Piaget "to explore the richness of children's thought, to capture its fundamental activities, and to establish the child's cognitive competence" (p. 4). Simply put, the clinical interview is an "exchange between two or more people in which the interviewer seeks to elicit information from the interviewee about how the latter thinks and leams" (Long & Ben-Hur, 1991). This technique gives a researcher an insight into the many aspects of a child's mind by using a semi-structured, open-ended type of questioning that allows a child's responses to direct the flow of questions (Ginsburg, 1981, 1997; Long & Ben-Hur, 1991), In keeping with the clinical interview's procedure, I, the interviewer, started out with some common questions (see Appendix D), and developed other insitu questions based on the interviewee's answers. However, this was done in a context of trying not to direct the students' responses (Ginsburg, 1981) or to elicit what Piaget referred to as suggested convictions (Posner & Gertzog, 1982). Long & Ben-Hur (1991) also stress how crucial it is for interviewers to remain non-judgmental in their acceptance of responses and at the same time show respect for what the interviewee has to offer. Once in the interviewer seat, I found that being so neutral was sometimes easier said than done! 40 With this type of interview, there is a real danger of misinteipreting or biasing participants' responses. To counter this, I followed Piaget's recommendation to make counter-suggestions by rephrasing questions or asking new questions in the interviewee's language (Long & Ben-Hur, 1991; Posner & Gertzog, 1982). Ginsburg (1997) further supports this action by suggesting the need for the interviewer to occasionally give what he/she assumes are incorrect interpretations of what the interviewee means so as to ensure that the interviewee is not responding in a manner that he/she assumes the interviewer expects. There were times throughout my interview sessions when I resorted to this as well. Even though the clinical interview allows freedom in questioning, a certain degree of uniformity must be maintained if one wishes to obtain comparable results. As open-ended as it might be, Ginsburg (1981) suggests that some consistency be accomplished by having an original 'task' (or in my case, an original focus) in mind. In addition, I used the same set of initial questions with each participant so as to maintain the spirit of constancy. During my sessions with the parents and the children, I allowed them to explore their feelings, but always keeping them on the topic of math beliefs and attitudes. Now it should be noted that in Ginsburg's (1997) review of the clinical interview, he emphasised the need for deliberate nonstandardisation. This may seem to conflict with my using the same set of questions throughout the sessions, but my intent was to bring a sense of coherence, and not standardisation in the hope of adding validity to the data collection process. 41 Finally, the goal of this study was to discover and explore how using qualitative methods enlightened past research on the role of parents' beliefs and attitudes on students' beliefs and attitudes. Thus, the use of the clinical interview was appropriate since it allowed the researcher to see how a person "thinks, sees the world, [and] constructs personal reality" (Ginsburg, 1997). Ginsburg also states that "the clinical interview can provide a kind of 'thick description' of the mind (to borrow from Geertz, 1973)... -more than many standard procedures"(p. ix). The veiy nature of paper-and-pencil instruments such as questionnaires precludes exploration and does not allow for the immediate pursuit of interesting phenomena (Ginsburg, 1981, 1997). The Questions As discussed in the literature review, it is often difficult to analyse research concerning attitudes and beliefs because of the way in which they are defined. For some psychologists, attitude and belief are different constructs. For example, Hart (1989) cites Rajecki as saying: "beliefs about an object is included as one of three components of attitude; the other two components are an affective or evaluative component and intended behaviour toward the object. Other psychologists define attitude and belief as the same psychological construct, thus removing the evaluative and behavioural components from attitude" (p.41). However, I am in agreement with the viewpoint that there is an intimate relationship between the two constructs (Daskalogianni & Simpson, 1999). In order to foimulate my list of common questions found in Appendix D, I used sub-constructs found in quantitative tests developed for examining mathematical attitudes and beliefs (see Aiken, 1974; Brown & McEntire, 1984; Fennema & Sherman, 1976; Kloosterman & 42 Stage, 1992; Sandman, 1980). This was done so that the questions would encourage talk on what a person believed about math and his/herself as a math learner, and in turn how those beliefs affected their attitude toward the subject. The following is a listing of these sub-constructs: Math as a subject: A person's views about the nature of mathematics. Enjoyment of math: The pleasure a person derives from doing mathematical activities. Value/usefulness of math: A person's views regarding the value of mathematical knowledge. Confidence in learning math: A person's perception of his or her ability to learn and perform well in math situations. Motivation in math: A person's desire to do work in mathematics beyond what is required. Math as a male domain: A person's views about mathematics being male-oriented. Causal attributions of math achievement: The reason a person uses to attribute their success or failure at mathematical tasks. ' To better prepare myself for the questioning stage of the interview, I decided to run a trial session with my roommate. In spite of the fact that she was neither a parent nor a seventh-grader, she was still a person with math experiences; thus, I was able to receive feedback on the quality of the questions, and on the timing of the interview session. However, even though I made provisions for this quality-check, other concerns about the applicability of questions still arose during the actual interview. 43 Interviewing in practice Once the four consent forms were received, each family was contacted in order to arrange an interview session with the researcher. These sessions were conducted so that each family member was interviewed alone and at a time convenient to him or her. All interviews were informal, in keeping with the clinical interview method and it was intended that they last approximately twenty-five to thirty minutes. In actuality, the interviews lasted from thirty minutes to an hour. Each interview session was audio-taped since this format is considered less distracting and intrusive, and more accurate than note taking. However, I also wrote post-field notes immediately after each interview session so as to.capture some of the overall experience and impressions (including for example, participants' body movements and facial expressions, and brief descriptions of what was happening around us). At the start of each session, I began by explaining my purpose of collecting a database of responses regarding people's mathematical beliefs and attitudes. In an attempt to create an atmosphere of trust and respect, I assured each person that there was no right or wrong answer because their views were their own. This was followed by a series of personal questions about issues like educational background, career goals, hobbies, family life, etc., so as to put the interviewee at ease and establish rapport. Even though people often have no problems proclaiming their lack of mathematical competence, I found that in this one-on-one setting, some of the adults seemed hesitant to talk about their math self to someone they considered a "brains in math". In order to create a sense of openness, I then found myself having to downplay my mathematical abilities, as well as to re-assure the participants that it was not my intent to judge then-mathematical beliefs or attitudes. Analysing and reporting the data In order to minimise biases I may have formed and to maintain uniformity throughout the data collection, all the tapes were listened to at the veiy end of the data collection. I decided against transcribing the entire tapes, and instead used "thin description" (Geertz, 1973, p. 9-10) to report the happenings of each interview, independent of intentions or circumstance. These descriptions of events summarised what each person had to say with respect to his/herself as a math learner and about math and were referenced to time codes on the tapes for ease of accessing the data which remained the actual responses on the tapes. Stake (1995) asserts that "to assist the reader... case researchers need to provide opportunity for vicarious experiences" (p. 86) and he continues that "the write-up can be organised in any way that contributes to the reader's understanding"(p. 124). Van Maanen (1988) suggests the report can offer impressionist tales that are "personalised accounts of fleeting moments of fieldwork case in dramatic form" (p. 7). With my obligation to the reader in mind, having completed the summaries, I chose to write "portraits" (Lawrence-Lightfoot & Davis, 1997) of each family with the child as the centrepiece. I constantly referred to the tapes and field notes for the vocal nuances and bodily messages. This relatively new method of inquiry and documentation was designed to "capture the richness, complexity, and dimensionality of human experience in social and cultural context, conveying the perspective of the people who are 45 negotiating those experiences" (Lawrence-Lightfoot & Davis, 1997, p. 3). According to Lawrence-Lightfoot (1983), portraits are shaped through dialogue between the portraitist/researcher and the subject, and can only be told through recollection because they seem to "evolve as much out of intuition, autobiography, and serendipity as they do from purposeful intention" (Lawrence-Lightfoot & Davis, 1997, p.3). The portraits of the families are not intended to be literal 'snapshots' because the translation of images has been probed, layered, and interpreted by the portraitist, in other words the portraits bear the hallmark of the artist. They express my perspective as the researcher/artist, since they were shaped by the evolving relationship between the subject and myself. Even though I only had one interview session with each subject, the data collected are still viable, as portraits have the ability to express a "haunting paradox, of a moment in time and of timelessness" (Lawrence-Lightfoot & Davis, 1997, p.4). They are vital documents of who the subject is and who the subject may become, even if they no longer look like the subject in his/her eyes. Portraiture distills an essence of the subject through the researcher/artist's integration of the data, and Eisner (1981) makes the point that validity in the arts is "the product of the persuasiveness of personal vision... what one seeks is illumination and penetration" (p.6). Although Stake (1995) declares that "no amount of caring for the case will assure its worth", he continues that this worth lies in the "interactive communication, first between a single researcher with the case, later with the reader" (p. 136). Even though I am interested in getting at the 'voice' of the parents and students involved in this study, I am not purporting to give them 'voice'. Allowing individuals to 46 articulate their experiences in their 'own voice' requires the researcher to check and re-check with the individuals to ensure that their story is being accurately reflected. Even then one cannot be sure all personal bias is ruled out. I did not want to represent a distinguishing of their voice from my own. By using portraiture to report what transpired between the participants and myself, I know whose voice I am speaking from - my own: just as the artist who paints a portrait does. According to Lawrence-Lightfoot (1983) and Lawrence-Lightfoot & Davis (1997), a portraiture is a piece of text that comes as close as possible to painting with words. It conveys the authority, wisdom, and perspective of the subjects, and allows the researcher to reveal a perspective of the subject that he or she might not have considered before. Portraits can also be used as documents of inquiry and intervention that would hopefully lead toward new understandings and insights, as well as instigating change. Like others who use portraiture, my purpose here is not to offer a complete and full representation of the families, but rather a slice of reality that would "transform our vision of the whole" (Lawrence-Lightfoot & Davis, 1997, p. 5). It is my hope that my choice of perspective would allow you the reader to experience the research on attitudes and beliefs differently - that is to meet, through me, the people behind the pencil-and-paper questionnaires. If I am to create a portrait with words, then I am going to need paint to bring those words to life. According to Lawrence-Lightfoot & Davis (1997), a portraiture must link imagination and interpretation, and the artist/researcher can only accomplish that with "thick description" (Geertz, 1973). Clifford Geertz (1973) describes thick description as 47 "the researcher's constructions of other people's constructions of what they are up to" that is not only creative and interpretive, but indeed fictional as well (p. 9). "The line between the mode of representation and substantive content is as undrawable in cultural analysis as it is in painting" (Geertz, 1973, p. 15). As a result, he sees the researcher's imagination as a quintessential part of cultural and (I would also add) social depiction. This is evidenced by his statement: "it is not against a body of uninterpreted data... that we must measure the cogency of our explications, but against the power of scientific imagination to bring us in touch with the lives of strangers" (p. 16). However, one cannot forget that this is research and Lawrence-Lightfoot & Davis (1997) cites Geertz's warning against the use of interpretation, imagination, and creativity as central elements of a study. "These 'humanistic' dimensions must always be in close communion with rigorous and systematic attention to details of social reality and human experience. Behaviour, interaction, encounter, and gesture must be attended to with exactness, and retained, 'because it is through the flow of behaviour - or more precisely of social action -that cultural forms find articulation'" (Geertz cited in Lawrence-Lightfoot & Davis, 1997, p. 8). In heeding their warning, I have used my field notes and summaries, carefully checked against the original audio-tapes and third person confirmation so as to add an element of rigor to the investigation through a form of triangulation. Finally, my choice of method for reporting the data is appropriate because it allows me to be consistent with my intent of taking on a new approach to the research on the role of parents on students' math attitudes and beliefs. The analytic rigor of past positivist approaches often gave us a perspective that was somewhat distant, discerning, 48 and skeptical. It is no wonder that there has been a lack of interest in this area of the affective domain since the 1980s. However, "with its focus on narrative, .. .portraiture intends to address wider, more eclectic audiences" (Lawrence-Lightfoot & Davis, 1997, p. 10). Furthermore, Lawrence-Lightfoot & Davis (1997) claim that portraiture allows the researcher to speak in a language that is neither coded nor exclusive, as well as to produce texts that will seduce readers into thinking more deeply about issues that are of interest to the researcher. As you enter the portrait gallery in the next chapter to view the four families, I hope to "deepen the conversation" (Geertz, 1973, p. 29) on the role of parents in their children's mathematical attitudes and beliefs, in a way that has not been done before. 49 CHAPTER FOUR: PORTRAITURE THE PORTRAIT G ALLERY The Richards Family As soon as you come through the main door, you get the feeling of a house that is alive. There seems to be something going on in eveiy room. Mrs. Richards is in the kitchen cooking. Mr. Richards is in the basement watching Sunday sports on the television. Vee is upstairs chatting with a sibling in a raised voice, perhaps to be heard over some of the other noises in the house. Even though the interview was scheduled a week in advance, it was as though my anival was unexpected. I felt like an intruder. Both parents and Vee went into the kitchen to decide who would interrupt their plans first. From what I overheard, Vee got little say in the discussion and she was told to go first. Unlike the other homes, this family did not seem too concerned about securing a quiet place for the interview sessions, and so Vee and I settled into the living room next to the kitchen, with the hum of the blender in the background. Sitting on the edge of a couch across from me, Vee's slumped posture suggested that she was a bit disappointed about having to go first. Even though she answered my questions about her personal life, her tone of voice was hesitant; and she sounded as though she would have preferred to be doing something else. Outside of school, dancing occupies a lot of her time as she indulges herself in ballet, modern dance, jazz, and tap. As she slipped back comfortably into the couch, you could see that simply talking about 50 her favourite pastime was relaxing for her. Her rate of speech picked up, and she seemed in a better mood to talk about beliefs and attitudes about or toward math. Vee was selected to be part of the study because her teacher saw her as a strong math student, and because she also rated herself as "good" in an informal poll carried out in her grade-seven math class. However, on the day of the interview, Vee appeared to be quite modest about her mathematical abilities. When asked to rate her ability as a math student, Vee shrugged her shoulders, looked to the floor and said, "Okay, I guess... I would rate myself as 8.5 out of 10... Stepping into a class, I won't know anything, but I usually get it after one lesson." Vee seemed aware of her mathematical competency because she made reference to being one of the few in class who would always have the right answers; and yet, she was reluctant to speak highly of her talent. Just in rating her mathematical ability with a number seemed problematic, as she struggled to find a number that was high in ranking but would still seem modest in nature. In an unenthusiastic tone, she went on to state: "Math is more BORING than other subjects. [Capitalisation used to stress the intonation in her voice.] In terms of liking it,. I would give it a 7 out of 10. I like geometry and not fractions, but for no particular reason." Becauseof the way she stressed the word "boring", I asked Vee to explain what the word meant to her. She responded by stating how she felt that her math class was not much of a "challenge" for her. Her motivation for working hard at math was the same for all her classes - "I like getting stuff done." By her own admission, there was nothing special about math, as Vee is an all-round student that excels in all her classes, and she did not believe that it takes special talent to do well in math. She attributed her success to 51 "listening a lot to the teacher" as well as to "working with things in [her] head and spending time with them." As if to suggest that the road to mathematical success was easy for anyone, Vee sighed and said, "You just need to understand eveiything, as math is about understanding... once you understand it, it comes to you." When asked how she knew that she was competent in math, Vee said "Well from my test results, and because I usually know the answers to the questions that the teacher asks in class and I know that I have it right. I'm usually the one with my hand up most times." From what was said with respect to her views about her math self, there is no doubt that Vee is quite self-confident. Nonetheless, when I probed her about her reluctant tone as she discussed having her hand up in class, she claimed that she would prefer not to have to be the one with all the answers in her math class, since doing so made her uncomfortable. As we switched to talking about her views on math as a subject, .Vee's posture remained the same - she was laid-back and slightly slouching on the couch. Her tone of voice was still one of reluctance. Vee also appeared to be slightly embarrassed by all the commotion around us. She kept looking toward the kitchen and rolling her eyes every time the noises got too loud. When asked to comment on the usefulness of math as a subject, Vee had mixed feelings. In the early part of the session she stated only some parts were applicable to the real world, for nothing more than eveiyday uses like grocery calculations. However, later on she expressed a "need" to know math. Even though Vee has no particular career goals in mind, she knows that she would like to attend university. "I would need math for the future." Because she wants to be successful in life, she said that she needed to work hard in math now, so that she could have good grades for next 52 year. Vee expressed an awareness of the importance of her math grades for a smooth progression through her schooling. In addition to understanding the subject, Vee believes memorisation of steps plays an important role in learning. This may seem a bit contradictory in light of her earlier comment on understanding ("You need to understand eveiything in math 'cause math is about understanding... once you understand it, it comes to you"); however, I believe she was referring to the need to have a good grasp of basic algorithmic procedures (for instance, "2 + 2 is 4"). I also got the feeling that Vee is a child that ponders over things. Not only did she seem to be reflecting on her words as she spoke, but she also stated "You should take whatever time is needed to complete a problem." Also, earlier in the conversation, Vee commented on needing to spend time working through problems in her head. For Vee, math is not a subject that requires speed. Vee also did not believe that math could be discovered on one's own and she seemed surprise that such a possibility existed. As far as she was concerned, math had to be taught to its learners. Wondering if she was even familiar with discoveiy learning, I explained what the terms meant. Yet, Vee maintained that she had never experienced a class that used discoveiy learning. When my chat with Vee came to an end, she excused herself so that she could get her mother. Mrs. Richards had been caught by surprise as she had expected to have more time to finish her preparations in.the kitchen. Why she simply did not switch with her husband might have been a mystery; but then again, "you know how men get with their sports!" To ease Mrs. Richards into talking about her math beliefs and attitudes, we began the interview talking about her. As she sat comfortably back into the chair on my left with her legs crossed, her arms were as much a part of the conversation as her lips. Besides being a "proud mother of four", Mrs. Richards is a secondary French teacher with over twenty years of experience. She has a love of dance, and is thrilled that her daughter Vee shares in this love as well. The excitement alone in Mrs. Richards' voice tells that dance class is a special mother-daughter time for her and Vee. Next, she reflected fondly upon her times in Australia where the family lived for 15 years prior to moving here to Vancouver. "Vee was the only one bom there, did she tell you?" After almost 5 minutes talking about life 'down under', I explained the purpose of the interview session. Like most of the adults interviewed, Mrs. Richards wondered about which of her experiences she should use to talk about math. Should she talk about her i . experience with math while as a student or now as an adult? Having free choice, she chose to talk about her times in high school. Right off the bat, Mrs. Richards claimed to have never liked math because it did not come easily to her. "I only liked it when I got the right answer." As she reflected on her math experiences in school, Mrs. Richards' entire demeanour changed. The woman who was gesturing at the beginning of the conversation was not the same woman before me. Her voice was lower and her arms were now veiy close to her body. She looked tight. In her own words, "Math was very painful for.me... Liking a subject is hard to separate from finding it easy." 1 could not help but be curious about what could have happened to cause such pain; but Mrs. Richards closed up. There was a lot of sudden silence, which was in stark contrast to amount of excitement there was previously in the conversation. Even though this was the type of math experience I was interested in, I 54 could see that Mrs. Richards was very uncomfortable; thus, I decided not to piy. From what I gathered, success in math required a lot of hard work-on Mrs. Richards' part -more than subjects like English or French which came more "naturally" to her. She "resented the pressure placed on [her] to stay on top of [her] math grades, [since] it was one of those subjects in which a high score would boost one's chances for scholarships." The more she spoke, the more I sensed that fear might also have played a role in her attitude toward math. "This one subject could have made or broken [her] chances of going on to university and a better life... I had to do welHn it!" It seemed as though she had no choice. Despite this feeling toward math, Mrs. Richards claimed that her math grades were usually amongst her highest, and that because of this, she could hide her true feelings and portray herself as someone with a love of math - an image that she says she assumes her family has of her. In spite of her negative feelings, Mrs. Richards was still able to find math useful for "training the brain to think intellectually". In discussing the nature of .math she said, "Math is quite abstract and can be painful on the brain because of the extreme level at which the brain must work." Again, a reference to the word "pain". While describing herself sometime during the interview, Mrs. Richards chuckled and said: "I was a minimalist child!" When asked to clarify this statement, she explained that as a child, she only did the "bare minimum"; thus, doing math required her to push herself more than she wanted to be pushed. Remarking on the applicability of math, she said: "I only use the simple math like adding and subtracting in my job." She looked off to one side as if to reflect upon her thoughts, then she sighed saying: "Sort of makes you 55 question all the hard work you had to put in back then... what was it all for?" Finally, Mrs. Richards does not believe it takes special talent to do well in math... "I did well, and I have no special talent." Instead she asserted that practice and hard work were vital for learning math! "You need to practice beyond what you think is enough!" Suddenly Mrs. Richards was distracted by one of her other daughters who was leaving for a job interview. I had had sufficient information for my research, plus I felt that I was now responsible for Mrs. Richards' somewhat sombre mood. She had become less animated toward the end of the interview and, the change in behaviour occurred right after her mentioning her painful experiences. I wrapped up our conversation and waited for Mr. Richards to appear for his interview. The minute he walked into the room, the atmosphere felt formal. We shook hands, and he sat in the same chair as his wife did earlier. As with his daughter and his wife, I began with some personal questions to 'break the ice'. "Tell me a little about yourself..." "What do want to know?" "What do you do for a living?" "I'm self-employed." "Doing what?" "Let's say... the beverage making industry." [Brief silence] "How long did you say this would take?" "Twenty-five to thirty minutes, sir" I sensed that Mr. Richards would have preferred not to talk about his personal life. His answers were quite short, and I wondered if he would react the same way when 56 talking about math attitudes and beliefs. Even though he did not say the exact words, his mannerisms seemed to suggest that "we skip the pleasantries and get down to the bottom line." I knew I was taking him away from the game on the television, so I got down to my questions about math. He crossed his feet and I crossed mine. With respect to liking math as a subject, Mr. Richards rated it as a 7 out of 10. Slowly shrugging his shoulders, he said: "Math was like any of the other subjects... I felt the same toward all my classes." He recalled liking topics like geometry and trigonometry, which he referred to as the "discipline parts of math." Even though he considered himself an "average student with average" grades, Mr. Richards was "very serious-minded as an adolescent". However, he did mention that he only pushed himself enough to get "decent grades". Mr. Richards' memories of high school could be divided into two segments. During the first part, he was a somewhat "distracting child" in that he "goofed around in class"; however, once he figured out the importance of scoring top marks, he knew that it would take hard work to achieve his goals. Mr. Richards was still sitting upright, but now he was tapping his crossed leg. We continued discussing his feelings about math as a subject. "Math is the science of numbers. I learnt it because it was compulsory." He also stated that math had no applicability in the real world, but as soon as he made that statement he also said, "Well I guess it serves some usefulness since I do figures at work and 1 have to deal with percentages and probabilities." .1 am not sure if an expression on my face signalled a sense of dissatisfaction with his first response, but Mr. Richards' latter statement appealed to have been made to get a positive reaction out of me. He kept staring at me 57 intensely, as if watching my every move. He believes that understanding everything is unnecessary for doing well in math, and instead finds that memorisation is an important factor. "You need to know the sequential stuff like BOMDAS so that they become second nature." Later on he also added, "One needs to be able to think logically to leam math." As far as Mr. Richards was concerned, you either "have what it takes to do math or you do not.. .and if you do not, then you should work hard to achieve success." Throughout the interview, Mr. Richards' answers were often abrupt. Unless probed, he offered little explanations to his usually one-lined responses. He looked at the clock across the room and I looked at my watch. My time was up.. I thanked him for his time and began to pack my things away. As I was leaving, Mr. Richards inquired about my research and my homeland. Suddenly the atmosphere in the room became less tense, as his tone of voice was now more relaxed. I was surprised by the change in his behaviour, because I would have thought that he would want to resume watching the game on television. Instead, we spent considerable time talking about the commonalties between his former Australia and my Trinidad. 58 The Hill Family I had overestimated the time it would take to get to their house, and as a result I arrived almost fifteen minutes prior to my scheduled time. I hesitated, but since it was cold and dark outside, I decided to approach the house. I knocked on the door twice, but there was no answer. I knocked again.. .no answer. I wondered if they were home. Ire-checked the address. I knocked again. My knuckles were getting sore. I was beginning to feel a bit silly standing out there. As I looked around to.see if anyone was watching me, I noticed a man with a child on his shoulders coining from across the sheet and toward the house. After a brief exchange, I learnt that the man was Mr. Hill. He welcomed me and we proceeded inside the house. As I stood .in the foyer, the little girl who was on the father's shoulders excitedly went to get the rest of the family. Mrs. Hill greeted me and pointed out that I was early. She and one daughter were in the process of finishing their dinner, and apparently Mr. Hill and the other child had left them for a while. Now, everyone was. returning to the dinner table to resume the family meal. I apologised and sat in a little area across from where they ate. As I browsed through a newspaper I could hear what seemed like a typical family conversation. Each person was discussing his or her day's activities. When it came time for the interviews, I joined Mrs. Hill in the living room. While I got my materials ready, she was hying to persuade her younger daughter to leave us alone for a few minutes. The child could not understand why I wanted to talk to eveiyone else in her family and not her. "Do me, do me", she pleaded. I interjected and told Mrs. Hill that I would certainly interview this daughter too. After a few minutes of 5? asking her what she liked about her math class, she was satisfied and left the room with the same burst of energy with which she entered. Sitting on the floor across from me, Mrs. Hill began to talk about herself and her . family. The atmosphere was quite relaxed and informal. Her life revolved around running her own business as well as being a wife and a mother of two girls. As we began to discuss how she viewed herself as a math leant er, Mrs. Hill pointed out that it was difficult for her to separate "liking math" from "understanding math". "If I had to rate my liking of math, it would be a 5 out of 10... It's not that I didn't like math, I just didn't understand it. I had no one to tell me." The passion in her voice suggested that even though Mrs. Hill was an "average" student, she had to "struggle to maintain that level, especially during the later years". "1 wanted to graduate with the rest of my class. I didn't want to have to stay behind in grade 12, so 1 had to work really, really hard." There were several occasions throughout the interview, where Mrs. Hill stressed the lack of help available to her. There was a sense of helplessness in her voice. "I got frustrated after tiying... there was no one there for me." "My teacher wasn't even there for me -imagine, she just told me to aim at getting the minimum passing grade so that I could graduate!" "I really wanted to do better, but no one would help me". "I understand the basics, but if you show me [more advanced concept's], I'll usually get it." Finally shaking her head in a woeful manner, she said: "If I had the kind of help Jay [her daughter] has, who knows what I could have done!?" All Mrs. Hill's verbal and non-verbal expressions convey a woman who feels she was dealt a bad hand. 60 As far as Mrs. Hill is concerned, there is a dichotomous nature to mathematics. "You either understand it or you don't. It either clicks or it doesn't." If you are a person for whom math does.not come naturally, for whom math does not "click", then "you have to be prepared to put in a lot of hard work". "You also need the right attitude - the wanting to leant, open to someone explaining it to you." For a brief moment, I thought 'that Mrs. Hill was talking about someone else rather than herself. The increased speed in her voice made it sound as though she was trying to convince me of something. Later, I found out that she was referring to her daughter, Jay, because she revealed how she thought Jay had the wrong approach to the math help she was receiving. "There was a time when Jay would get so frustrated whenever someone tried to explain stuff and she couldn't get it... she had no patience whatsoever." Mrs. Hill strongly believes that all math must be shown to the learner, because, "how else would you leam?" She also does not feel that full understanding of everything is necessary since "there's a lot of stuff Grant [the husband] does that I couldn't, but 1 still aced math!" Mrs. Hill refers to her husband as a "brains" and is aware that she is not as competent as he is; however, she is "still proud of what [she] did accomplish because [she] did it against all odds". We were running a bit over the allotted time and the younger daughter 'desperately' needed her mother's attention. I ended my session with Mrs. Hill and waited to see who would be my next participant. A few minutes later, Jay nervously walked into the room and sat on the couch on my right. I assured her that she had nothing to be nervous about, since I was there to listen to whatever she chose to divulge 61 -about her feelings toward math. "Okay..." she said and leaned back into the seat with her anus outstretched and crossed in front of her. This twelve-year old has a strong love of sports, as her extracurricular activities include swimming, baseball, basketball, volleyball, and track. Whatever the season, Jay is involved in one or more sporting events. Her career aspirations are currently set on Hollywood, as she would like to become an actress. Excitedly she says: "I just love Drama class!" However, she does point out that this dream may change. Jay was selected for this study because she rated herself as 'good' in the informal class poll, and her teacher did not agree with her assessment. Surprisingly on the day of our interview, Jay rated her math ability as "average". "In my class, there are a couple of smart kids and I'm not with them... I don't get zero on every test, so I must be average." It is only of late that Jay has been able to realise some success in math. She attributes the majority of mis turn-around to the tutor her parents hired. The smile on her face while talking about times spent with the tutor suggested that Jay is grateful for the intervention and enjoys having "someone to explain math to [her]". She talked about how confident she felt in her math and how she now has a chance to please her father. "Getting good grades is good because I don't [usually] do well, so when I get a good mark, I know my dad will be happy and I will be happy." Comments such as "My dad says that [math] is one of the most important subjects you will need when you grow older," and "My dad is a math brain and he wants me to become like him" were often heard throughout the interview. Veiy seldom did I hear about Jay's own thoughts and feelings. Whenever I asked Jay if she shared the 62 sentiments of her father, her eyebrows would narrow as she said "Yeah..." Perhaps she does believe in what her father has told her, and so his words are. her words. I tried my best to keep a neutral facial expression while she spoke; because I did not want to insinuate that there was something wrong with her having similar thoughts as her father. Jay seems veiy close to her father and there was never any hint of resentment of his influence, on her part. She was simply pleased as punch to finally be able to perform at a level that met with her father's expectations. We finished our session and she went off to . find her father. When Mr. Hill walked into the room, I sensed a definite presence about him. He appeared to be a no-nonsense type of man. I noticed that the younger daughter did not come in and disturb us as she had done during the previous two interviews. I began the session as I had done with all the other participants -asking a series of personal questions to ease any tensions. "Tell me a bit about yourself?" -"What do you want to know?" -"Well, are you from back east like your wife?" "No." "Where were you born?" "B.C." "Where in B.C.?" "Northern B.C." "Where in northern B.C.?" "Why do you want to know?" 63 "I don't!" [Pause] "You know ... never mind." My whole demeanour was changing before my eyes. I did not want my mood to affect my ability to carry out an effective interview, so I decided to stop with the personal questions. Mr. Hill continued: "You know what, I don't see why my personal information is any of your business.. .These days, there are too many people with access to our private lives..." There ensued a whole 5-minute conspiracy speech. Mr. Hill was convinced that my research could have some kind of power over his control of his privacy, even though I told him that pseudonyms were used to protect the participants. Not wanting to provoke him any further, I tried to drop the matter, Mr. Hill sat back into the chair looking slightly ruffled, and 1 retreated into my seat. This was an interesting session, because whenever I would pose a question to Mr. Hill, he would pull apart its meaning. For instance, he said: "What do you mean by 'how • much I like math'? It's not a question of liking math - it's a matter of knowing what you need and what you use." For reasons even Mr. Hill said he could not fully explain, he pursued a Bachelor's degree in the field of Mathematics. He reasoned that he had to have liked math more than the other subjects, since "that was the discipline in which [he] received [his] degree. "I wanted a degree and math was the path of least resistance." It was hard to pin-point Mr. Hill's attitude toward math, but from what T could gather, it seemed that he neither had positive nor negative feelings about math. He was successful at it, so he pursued it. Although he was not amongst the top achievers in his university classes, Mr. Hill is still very confident in his mathematical capabilities and he has the "degree to prove it". 64 In an assertive tone he said: "Math is not difficult, it's just good common sense. It's logical. If you understand A, you'll understand B, which will lead you to C." Throughout the interview, he took great pains to stress a need for full mathematical understanding. "You need to understand everything. It's like building blocks - if you miss One, it's over." "You need to grasp every level so the house of cards doesn't fall." He felt quite passionate about ensuring that children maintained full understanding of mathematical concepts, because he saw what the lack of understanding did to his daughter, Jay. He admitted to struggling with the fact that his daughter needed his help and he was incapable of effectively helping her. "It was frustrating to see Jay struggle with things that seemed so simple to me." "I often lost my patience with her." "It's only been recently that we discovered that she was missing a lot of the fundamentals." His opening up about losing his patience allowed a real humanistic side of Mr. Hill to shine through the tough shell he portrayed. In the final stages of the interview, Mr. Hill spent some time blaming the educational system for the failures of our children. He explained: "You can't miss a concept because it will catch up to you later. You have to understand it eveiy day in class. Teachers don't realise that kids miss out on stuff. They just go ahead teaching and it's the children who suffer." When asked how he managed to be successful in the same educational system, he answered: "1 was a logical thinker. I was quicker at grasping concepts." On that note, I felt that I had gathered sufficient information. I thanked him for his time and his insights, and went in search for the rest of the family to bid my farewells. 65 The Forrester Family On arriving at the house, 1 found the front door slightly ajar. There was no doorbell, so I held the door and knocked for a while shouting "hello". After this went on for some time, I decided to step in and take a look.. Just in case. I smelt something wonderful seemingly coming from the kitchen. I did not venture too much into the home since I found it peculiar that there was no sign of life. I called out for Mrs. Forrester a few more times, but no response. 1 stepped back into the porch area, as I felt uncomfortable standing in the house by myself. Suddenly, there was the sound of movement. I called out to Mrs. Forrester again. This time she came to the door. She inquired about how long .1 was waiting and apologised for not being there when I arrived. She had to run down to the basement and knew that I would be along soon, so she had left the door open for me. I. felt welcomed and a sense of being 'at home' - as opposed to being in a stranger's house. The smell of chicken casserole can do that to you. Along with taking care of a family of five, Mrs. Forrester also works on a shift system as a medical assistant at the local hospital. On that day, like most days, she was preparing dinner for the family to ensure that eveiyone else would have something to eat on returning home. "If I don't do this, I'm not sure what or if they'll eat." Wanting to keep an eye on the cooking, we settled at a small table in the kitchen and continued to talk about her interest in the arts and in nature but more to the point, her thoughts on math. When Mrs. Forrester looks back on her experience in school, she recalls two distinct periods, "in elementary school, I LOVED math especially when we would get 66 problem-solving sheets to work oh outside. I had so much fun with that. Then when we got to a certain grade, we had to show our working, and I COULDN'T." [Capitalisation used to stress the intonation in her voice.] In her earlier years, Mrs. Forrester had "a genuine love for math", but because she had problems presenting algorithms in the format that was expected by the teacher, she got frustrated with the math of later years. "When I tried to show my working it would be a mass of confusion with tonnes of paper." Even though she pointed out that in today's world, she is able to reason things out without the pressure of having to conform to algorithmic thinking, she still said that • her love of math went from an 8 to a 3. "Is it still a 3 today?" "Yes!" [There was no hesitation.] Hints of disgust in her tone suggested that having to conform to what the teacher required was a big issue for Mrs. Forrester, especially when that conformity challenged her freedom of expression. "I loved Literature the best 'cause I could explore and express myself." There were several occasions during the interview where she would put stress on the joys of being in her English class as opposed to her math class. Once her marks began to slip, Mrs. Forrester became a bit "withdrawn". She stopped speaking out in math class, and she also did not discuss her difficulties with her parents. With pride in her voice, she talked about how "smart" her mother was with shaip mental abilities. Not wanting to disappoint her mother, she "hid". However, Mrs. Forrester seemed very comfortable talking about her experience with math with me, now many years later. She knew she did not do well toward the end of her high school years 67 and she also knew the reason why that was. There was neither shame nor pride in her talking about the experience -just honest admissions. Because she strongly holds on to the negative math experiences, she views math as "difficult, strict, and rigid"; and yet she feels that it is something eveiyone can do. "It boils down to understanding, and everyone can understand!" In spite of her negative experiences with the math, Mrs. Forrester also had positive comments to make. For instance, she had said: "students should work in groups, because math is a sharing environment.. .it's all about getting there," and "I think math can be discovered - that's the beauty of it!" Her description of why math should not be time-driven was like that of an artist, because she referred to the need to be able to spend as much time with the work, as was needed. Even though she still claims to dislike the subject, she claims not to have been "devastated" by the experiences. She was still able to "attain marks respectable enough to allow [her] to maintain [her] friendship with [her] honour-roll friends", and in her words, she claims that this was important to her. The change in nature of the subject from elementary to secondary, "challenged" Mrs. Forrester in a way that she said was unexpected. She found if hard to rebel against her loss of expression, because she knew that "having high scores was the name of the game". "[She] resented that!" However, she laments, "marks were a sign of high achievement and I wanted that." Once she was in high school, she could "no longer live the carefree life, of primary school days". For a person who loves the expressiveness and freedom the arts and the outdoors offer, Mrs. 'Forrester claims that the sudden "restrictiveness" and "discipline" required of the math classes in secondary school were "difficult to come to terms with". 6 8 Later that same day, I returned to the home to interview Mr. Forrester and his daughter Sherry. Since Mr. Forrester had arrived home first, I started the interview session with him. He gave instructions to his older daughter about intercepting incoming calls, and then closed the surrounding blinds to insure our privacy. On first impression, he seemed like a veiy down-to-earth type of guy and his body language seemed to support this impression throughout our conversation. He looked quite relaxed in his chair, with neither his arms nor his legs crossed. Like his wife, he enjoyed a lot of outdoor activities like tennis, swimming, and biking; and he involved himself in these activities whenever he was not working at his job. He seemed quite comfortable talking about himself and even admitted to dropping out of university after one year. As Mr. Forrester sat in a relaxed manner across the table, he began to describe his great love of math. "Oh, my liking of math was definitely a 9 out 10!" "It came first before all my other subjects because it was easier than ttying to comprehend French, Literature or Chemistry." Mr. Forrester finds himself to be mathematically competent now, and when he was in high school; and his grades supported his knowledge of his ability. Nevertheless, "[he] gives credit where credit is due", and as such he-spoke at great length about his gratitude to his teachers. "I also had good teachers at every level, and plus I was interested in math." He enjoyed his math experiences throughout school and even though he worked hard at it to keep his overall average in high standing, the enthusiasm in his voice also suggested that he was motivated from within. "If math was not compulsory, I would still take it". 69 The positive sentiments could still be heard as Mr. Forrester talked about the nature of math itself. "Math is very precise which is good because you can see the end result as opposed to Literature, where there's too much interpretation." Because he prefers "things to be sfraightforward", he "disliked" subjects like Literature because "they weren't black and white like math". Even though he feels math has "lost some of it applicability", his belief seems linked to his own personal experience. "Math WAS applicable, but now it has changed". He gave this example: "they have metric now and I learned the imperial system!" However, Mr. Forrester still pointed out the importance of math, "especially in the business world". "You need to know about profit and loss". Finally when questioned about his feeling about math as a male-oriented subject, he pondered for a while and carefully said, "Perhaps when I was in school, but I don't feel that is true now." He explained that his statement is based on the fact that there were more boys than girls in his math classes. In these times, he believes there to be an "even distribution". Although it felt as if we could have chatted all night long, Mr. Forrester was mindful of the fact that Sherry still had her homework to attend to. Once I had gathered sufficient information,.! wrapped up my interview with him and waited as he tracked down his daughter. I was surprised by Sherry's upbeat nature as she was described (by her teacher) as a girl so "doubt[ihg] of her mathematical abilities" that I assumed she lacked confidence in all aspects of her life. However, her energy lifted the room the moment she entered and sat at the table. She expressed how excited she was to be "part of a study", and started to ask me about my studies on campus. Once I was able to steer the 70 conversation back to talking about her, 1 learnt that in addition to school, Sherry spends about 16 hours weekly taking gymnastics. Outside of those two main events, she goes ice-skating and swimming with friends. Although she appears to have a demanding schedule, Sherry proudly claims: "Yeah, but I still have a social life!" "I just loooove ICQ!" ICQ is a computer programme that allows people to chat instantaneously online; and through this medium, Sherry seems to enjoy the best of both the physical and virtual worlds. This high-spirited 12 year-old already knows that she wants to be a litigation lawyer, because of her love for "constructive arguing". Just meeting Sherry leads one to feel that there is nothing she cannot do. Sherry's family was chosen to participate because it was determined that Sherry' was a student who was good at math, but thought she was not. Going into the interview with this knowledge, I became quite confused because the girl in front of me did not present herself as lacking in confidence in her math abilities. "I think I'm okay now". I was intrigued by one of her earlier statements on whether or not she liked math. "I like different things at different times." As 1 tried to figure out if I was receiving mixed messages, I came to realise that Sherry feels differently about herself depending on the situation. Perhaps, she was caught on an off day when her teacher conducted the informal class poll. In fact, she emphatically said, "I like math because it's challenging, yet straightforward." Overall, I would say that Sherry is very confident in her math abilities, although there are times when she had her doubts. "I was good before, but now I'm not sure... The entire class recently failed a test which I was sure I did well in." Even though Sherry maturely says that grades don't always reflect how smart a person is, 71 she still relies heavily on them to gauge her performance. Thus, it is not surprising that her opinions about her math self changes as her math grades fluctuate. When asked about the first word that came to mind upon hearing 'math', Sherry fondly said with a smile: "Adding.. .it was the first thing I learnt in school." As a subject, math is precise, involves "figuring things out", and requires full understanding. Believing that math should not be time-restricted, Sherry crossed her arms and strongly said: "I tend to work on things for a while, so 1 guess... It should take as long as it takes to do all the steps". It sounded as though she was trying to justify her actions. According to Sherry, one does not need any special talent to learn math, but instead, "they need to understand [it] and a good teacher." When it comes to the role of the teacher, she has no trouble holding that person accountable. Without a second thought she said, "If you make a mistake, it's probably.the teacher's fault." It was only as an afterthought that she also suggested that blame could be found in a person's carelessness. Her side comments about her math teachers seemed to suggest that Sherry often found fault with those who taught. Because of a fleeting remark she made, I felt there was definitely some teacher conflict that occurred in her past. Suddenly, 1 remembered my promise to Mr. Forrester. Her dad had made the point of my leaving enough time for Sherry to attend to her homework; and, Sherry and I had been talking for a little over an hour. Sherry had wanted to keep on talking about her feelings about math, but she also admitted that part of her enthusiasm to prolong the interview was attributed to her not wanting to do any schoolwork. ' 72 The Lebrun Family Twas pleasantly impressed when I first met Mrs. Lebrun because she seemed so well dressed for such an early hour in the day. It felt more like a Sunday evening than a Monday morning. She had an air of elegance about her so much so that suddenly I felt a need to be more formal. Mrs. Lebrun escorted me to the living room, and offered me something to drink. I declined. I felt too nervous to be handling beverages while sitting on her furniture. As a native Of France, Mrs. Lebrun has a somewhat distinctive accent. She made me aware of the difficulties people sometimes have in understanding her, and as such, she insisted that I feel free to ask my questions as many times as necessaiy. "I'm not pretentious at all. If you don't understand, you have to ask." As we talked about her views on herself as a math learner, Mrs. Lebrun was always up-front with her responses and she seemed very sincere. "I would say that I was okay in math.. .1 succeeded at it, but preferred Literature and Social Studies." Her memories of her math experiences are indeed pleasant and positive, and there was a lot of laughter and smiling during our conversation. Wanting to succeed at all her subjects, Mrs. Lebrun worked diligently to maintain being in the top third of her math class. She was the model student who "listened and tried to participate when necessaiy." She was always confident in her abilities, and only got frustrated when "things got too technical." "I don't like it when things are technical 'cause I don't understand it!" Although Mrs. Lebrun stopped pursuing math courses after high school, she is still veiy proud of her accomplishments. She believes that math does not involve special talent because "the average person can do basic math." She also sees high school as involving both "the 73 teaching of basic and some advanced math"; and since she successfully completed high school, she claims to have done her duty. From her own admission, there is no doubt that Mrs. Lebrun could have gone on to pursue more advanced math; but "it was a question of INTEREST." She believes that one should pursue the things that one is interested in. Even though Mrs. Lebrun made a point of not being able to talk "fairly" about math since it was not part of her present life, she still praised its benefits. "You need math to survive... math develops the intellect." When I asked her about her concern for being "fair", Mrs. Lebrun explained that she believed that "only a person who worked with math could say positive things about it". She feared that she would not be able to say a lot of positive things, as she did not work with math. In her opinion, nothing more that the basics are necessary in today's world; however, she approvingly said: "I would support someone wanting to do advanced math, even though 1 may think it unnecessary." At one point in the interview, she appeared to be uncomfortable with my use of the adjective "bright" to describe students. "1 don't like words like 'bright' and 'intelligent' because a man walking down the street could be intelligent without the math!" Her feelings on this matter were "close to [her] heart given the difficulties that [her] daughter, Shelby, had experienced with math". Mrs. Lebrun is very sensitive when it comes to her daughter, and when she talked about her, you could feel the change in the atmosphere. Instead of being light and jovial, things now seemed slightly tense. It was clear that Shelby's well being meant a lot to her. This was evidenced by the fact that a great portion of my interview with Mrs. Lebrun was spent discussing Shelby. Having only recently moved to Vancouver, the 74 Lebruns had been quite concerned about how Shelby would manage the transition to a western Canadian educational system. They made it their top priority to pick the best school as well as meet with all Shelby's teachers. Due to differences in the way that math is structured here (as compared to Fiance), Shelby could no longer maintain her math grades at a satisfactory level. Her parents became quite concerned and approached her teachers for advice. The result was that Shelby was placed in a remedial math class in addition to her regular class. I learnt that these two classes operate from different outlines, which often leave Shelby "in a state of confusion". The Lebruns have invested in a tutor to help Shelby make sense of her math world. In addition, Mrs Lebrun also stated that they "even buy math games, puzzles,, and worksheets to take on their vacations", in order to reinforce their commitment and support. An interesting moment also occurred when Mrs. Lebrun was discussing her thoughts on math teachers. In an apprehensive voice she said, " I find math teachers are so..." This statement was accompanied by her outlining the shape of a square with her hands. "You see with art, there is more than one alternative... more freedom for interpretation." Realising that 1 was.a math teacher, Mrs. Lebrun was cautious in what she was saying. "I'm not saying this about you..." She was trying not to hurt my feelings. This was the first time that .1 saw her wanting to hold back her thoughts. Because I took some time to respond, Mrs. Lebrun quickly said: "I don't want you to think I'm negative!" She wanted to ensure that her present thoughts did not affect the positive image she had painted earlier in the conversation. Once I assured her that she could speak freely without offending me, she relaxed and continued talking. There were . 7 5 a couple of phone calls that came to the house while we spoke, and after an hour into the interview, they were becoming more frequent. Mrs. Lebrun was gracious enough to let the machine answer the calls, but I could tell that she wanted to get back to her schedule. We wrapped up our conversation and confirmed the times for the sessions with her husband and her daughter. Before meeting Shelby, everything that I knew about her mathematically was obtained from her regular math teacher, as well as her parents. She had been selected to participate in this study because she was poor at math and thought herself to be poor as well. While arranging the interview session with the family over the phone, her parents insisted that I interview them first so that they could see the type of questions I was asking, before I could be allowed to interview Shelby. They warned that now was a sensitive time for Shelby, and that they wanted to ensure that I did nothing to aggravate the situation. Knowing how Shelby's teacher felt, as well as how protective her parents were of her made me quite nervous. 1 felt as though 1 had to handle her with gloves -1 had to watch what 1 said and what I did. Although I looked forward to talking with her, I was also apprehensive about our meeting. As I was waiting for Shelby, both her parents came into the room with her and they introduced her to me. They ensured her that I was "okay" and that she had nothing to be nervous about. 1 think their presence was having an opposite effect - Shelby and 1 both looked uneasy. Once they left us alone, a timid Shelby took a seat on the settee to my right. Shelby was quite soft-spoken and her braces made it doubly hard to hear everything she said. Not wanting to call attention to this, I.chose to move into the 76 conversation more, rather than constantly asking her to repeat herself. For the intents and purposes of this study, whenever those around Shelby spoke about her, they tended to define her by her problems with math. However, now that I have met Shelby, I see that she is also a twelve year-old with a love of the arts - for example, potteiy, painting, and drawing. She is not sure what career she might pursue at this moment, but it would either involve translating or working as a chef in a restaurant. To her credit, she is already bilingual. Despite her weaknesses in math, Shelby still likes it as a subject. In fact she rated it with a " 9 " . What she does not like, is the fact that she has to take two math classes that are as opposite as night and day. "It is hard to say how I really feel, because [the two classes] do-different things which confuses me a lot." From eveiything both Mrs. Lebrun and her daughter had said, it seems clear that Shelby is a victim of circumstances, which are presently stacked against her. One would think that she would be angry, but not once did she ever raise her voice in disgust. She remained in her slouched manner throughout the interview -making as little eye contact as possible. Fully aware of her weaknesses, Shelby works quite hard to keep on top of both her classes. "I want to succeed and do well. I work pretty hard so that I can understand it and do better." She does not seem to let her present predicament lower her self-esteem with respect to math. She strongly believes that math requires no special talent, but instead "[one has] to have the right attitude." According to Shelby's parents, she has that right attitude, because "she puts her best foot forward everyday and tries to make the most of her school situation". Now that I know a little more about her, I believe that she is a remarkable young girl, with a very determined spirit. I was looking for evidence of 77 that poor self-image she had alluded to in the math poll, but she never spoke poorly of herself during the interview. Apparently, her awareness of her ability does not limit her from being optimistic about the future. As for the nature of the subject, Shelby believes that "math is allnumbers and operations." From her own experience, she believes that practice is crucial to the learning of math. Even though there appeared to be some sadness emanating from her voice, one did not feel the need to pity her. Her parents may feel the need to shelter her, but it was obvious Shelby was a lot more capable than she was given credit for. Finally Mr. Lebrun's turn arrived. Outside the room, I could hear him asking Shelby about how the interview experience was for her. Luckily, she was not disturbed by our talk and so Mr. Lebrun strolled into the room where I was waiting. I explained my purpose for the interview and went straight into asking him questions about himself. Like me, he was a secondary.school teacher, and 1 could see that he felt a connection. We were part of the "brotherhood". He too was a native of France, but his accent was not as pronounced as his wife's own. Although he appeared quite comfortable and relaxed talking about himself, I did not want us to get off the topic. Once we got down to talking about the math, he" immediately apologised for not being able to recollect all his childhood memories. "My school days were a very long time ago... 1 just think you should keep that in mind." For starters, 1 would have to say that Mr. Lebrun was a very competent child in high school; however "[he has] trouble discussing [his] like of math, because [he has] difficulty with the word 'like'". He went on to explain: "I was certainly good at it. It was relatively easy for me, so in that sense 1 guess I liked it!" He had been 78 successful at every level of Math, but dropped it after his second year of university. "For the first time, I had to get a tutor to get me through second year, so that I could keep my grades up. I realised that I wasn't the 'hotshot' 1 thought 1 was." His math self-image was in jeopardy and he decided to end the pursuit. "Were you afraid of not being able to finish your math degree?" He would not admit to that, and instead offered this explanation. "My interests changed -I wanted to pursue other things." Mr. Lebrun claims to have mixed feelings about the usefulness of math. Based on his real life experiences with math, he has found that the math learnt up to a grade 8 level is applicable. "Anything after that is questionable". Recollecting his time teaching higher advanced math courses in India, he said: "It was frustrating teaching at that level and not knowing the [math] content as well... It was a great challenge." Even though Mr. Lebrun gets frustrated working with concepts he does not know, he^still finds that it is "fun" and "challenging". Whereas a challenge may deter some people, this is not true for Mr. Lebrun. It motivates him to push himself to the limit. Overall I would say that he had more positive than negative experiences in his school life. As far as the nature of the subject is concerned, Mr. Lebrun feels that math is veiy precise and needs to be understood at every level. He even spoke about how he would tease his math co-workers for having a seemingly easier life than his life as a social studies teacher. "If I died and came back, I would want to be a math teacher! It is so easy for them to mark papers and exams." While he joked about math teachers, he was watching my body language intently for signs of a reaction. I tried to give none. There were also two other incidents during which Mr. Lebrun tried to enlist responses from me. 79 He would periodically ask: "What would YOU say?" or "Do you AGREE with that?" Even though he tried to be persistent, 1 did not respond by indicating right or wrong answers. Mr. Lebrun does not see himself as possessing any special talent, since "there's no such thing as special talent." In his opinion "You need to be capable, and I DEFINITELY WAS CAPABLE." The stress in his intonation and the way he proudly sat upright suggested that Mr. Lebrun was quite confident about his mathematical capabilities. Toward the end of the interview, he gave his views on the gender issue. In an evasive manner he explained, "All I know is what I read, and that is, at the veiy low and veiy high ends, it tends to be quite male-dominated." At first, it seemed as though he was using what he had read to present his views, without having to take responsibility for such thinking. When asked for his own views on the matter, he said: "My gut feeling is that there is no difference... not so?" .1 felt as though Mr. Lebrun was some how making the remark because it seemed like an answer that would please me. I did not hear any conviction in his voice. Prior to our chat, Mr. ..Lebrun was entertaining guests in another part of the house. I wanted to let him resume his evening. 1 thanked him for his time, and waited for the rest of the family to see me off, but in fact 1 ended up spending another ten minutes or so meeting his guests and talking about my research. 80 CHAPTER FIVE: ANALYSIS A PICTURE PAINTS A THOUSAND WORDS Now that you have gazed at my family portraits, where I have attempted to bring into relief their mathematical beliefs and attitudes, we need to look at the questions this investigation proposed to explore. Do parents' math attitudes and beliefs still play a role in shaping their children's math attitudes and beliefs? How did a change from past methodological approaches enlighten the research on mathematical attitudes and beliefs? In answering the first question I took two approaches to assist in its analysis. One, at the end of each individual interview, I asked the interviewee to think about the influences of or on other family members. For instance, with the child, 1 asked if it was possible that either parent influenced their beliefs and attitudes; and with the parent, I asked if it was possible that his/her beliefs and attitudes influenced the child. Two, I cross-compared the interview sessions of each of the family members intending to find connections between what was said and by whom. However, before we proceed, 1 feel the need to remind the reader of Rokeach's cautionary words back in Chapter Two. There, he reminded us that in order to make sense of people's beliefs and attitudes, we must make inferences about their underlying states. As a result, it is expected that a person's beliefs and attitudes must be inferred from what he/she says, intends, or does. In the following analysis, the described relationships between the parents' and child's math beliefs and attitudes are only inferences, given that my data collection method involved only what people had to say 81 about their math beliefs and attitudes. Even though this is the case, researchers like Rokeach and Pajares still believe that inferences can help us understand more about our attitudes and beliefs. The Richards Family Vee: "No... I don't think my parents influence me." Mrs. Richards: "I could be a probable influence, but we don't really talk much about math since she does so well." Mr. Richards: "I don't see myself as influencing my daughter. I'm just grateful I can still help her with problems." In this family, Vee's experiences with math appear to be very similar to her father's high school math experiences. Both are mathematically self-confident, and in spite of their abilities, they both are modest in rating themselves. They both expressed . the sentiment that being successful at math was not particularly significant to them. Mr. Richards had said: "It is like any other subject", while Vee said: "It's no big deal, really". Mr. Richards and Vee both rated their liking of math as a 7 and both pointed to geometry as one of their favourite topics. Mr. Richards was able to say that he liked geometry because of its disciplined nature; but Vee had no specific reason for her liking of it. Vee believes that full understanding is necessaiy in learning math; however, her parents disagree with this notion. For Mr. Richards, memorization is essential. As for Mrs. Richards, she stated that "[she is] proof to the contrary!" In her opinion, she was able to successfully complete her high school math education and yet today, "[she is] still unable to grasp some of the concepts that [her] husband talks about". All three family members believe that math has little applicability beyond the basics. While Mr. and Mrs. Richards 82 spoke of their reliance on their adult experiences to shape their belief, Vee offered little explanation about her belief. Only after some probing was she able to state: "I guess we'll need it for stuff like in the grocery". Despite the reason for their beliefs, they all claimed math as being useful and, as a result, it is also necessary. In commenting on the role of having special talent, Vee and her mother agree that no talent is needed. Instead, "one needs to put in a lot of practice." On the other hand, Mr. Richards somewhat ambiguously thinks special talent is involved in learning math, although he himself possessed none. Mr. Richards and Vee both view math as a series of logical progressions; and, they also think that memorisation would be a vital part of that process. "Memorising key. facts helps you get over the simple parts to the more complex parts." However, for Mrs. Richards, math is abstract; and as a result, nothing but limitless practice is necessaiy for success. Vee's modest attitude toward her abilities in math seems to reflect those of her father. In that family, Vee is the only child highly skilled in math and the other older but less competent siblings tend to go to the mother instead of the father for help. Although, at the beginning, Mrs. Richards asserted that she had tried to portray a positive attitude to math, she also spoke of being very open and up-front about her negative high school experiences in math, with the intent of showing her children that "they too can do it if she did". Since Vee is aware of her mother's "painful" experiences, is it possible then that her modesty about her math ability is as of a result of her not wanting to cause her mother any more "pain"? Could this be one of the reasons why she turns to her father when she does need math help? Perhaps it is because Vee could be aware of her mother's contradictions, and so sees her father as less confusing. With respect to math 83 being a male-oriented subject, this family equally believes that "girls are just as smart as boys", and that if there were to be more of one group in a class, then gender would not play a role in such distribution. The Hill Family Jay: "My dad pushes me to do well. Dad is always there not necessarily [just] for homework." Mrs. Hill: "I think so.. .1 see myself in my daughter." Mr. Hill: "I hope I influence my daughter to encourage her." Analysing the parent-to-child influences in this family seems quite straightforward, because almost every one of Jay's responses involved the words: "My dad..." Since Mr. Hill is the designated "math brain" in this home, he spends a lot of time with Jay on her math (when she is not with her tutor). Like her mother, Jay rated her abilities as "average"; they are both aware of the need to work liard in order to secure good grades. However, her love of math is much stronger than her mother's was -it is more in line with her father's rating. Mrs. Hill describes Jay as once having a low tolerance for frustration, especially when working on math with her dad. "It's for this reason that we decided to get a tutor.. .1 couldn't take the arguing between the two". Mr. Hill admitted to having little patience with his daughter while working on her math homework. "I have no problem understanding math, but 1 just couldn't explain it to Jay nor did I understand why she would have problems with certain things". Hearing what Mr. Hill had to say, I wondered if Jay's desire to please her dad also played a role in her frustrations. Even though each family member believes that understanding is a necessary 84 ingredient for learning math, it is only Mr. Hill who strongly believes that one needs to understand everything in math to be successful. Mrs. Hill uses her experience to prove to herself that she does not need to know everything like her husband, because she was still able to do well in high school. As for Jay, she has similar thoughts to her'mother. She knows that she does not understand everything but she gets by. 1 was surprised she was not in fact affected by her father's insistence on the need for having full understanding at every level, since Mr. Hill claimed to constantly preach the importance of understanding all math steps to Jay. Now that Jay has the help of a tutor, she (as well as her mother) finds that she is able to accept her mother's call for having the right attitude to learning. Mr. Hill on the other hand is not sympathetic to the notion that having the right attitude affects learning; instead, he believes "math is about logic". Furthermore, Mi -. Hill has no qualms about openly blaming teachers for children's failures. In a similar fashion, Jay also would criticise some of her math teachers. In this family, the members usually attribute their failures in math to external factors -for Jay and her dad, it was the teachers, and for Mrs. Hill, it was the lack of help. Again, Jay's motivations in math as well as her confidence in her abilities both seem to have some link to her father. For instance, she spoke of "want[ing] to do well to please her dad" and "feeling good about [herself] when [her] dad was happy with [her performance]". As for math being male-oriented, each family member here stated that gender does not play a role, but the reasons for their beliefs were interesting. Mrs. Hill claims that "it has to do with what people are interested in". Jay said that "from the looks of [her] math class this year, it would seem that boys are smarter, but [she does] not believe that there are differences between boys 85 and girls". As for Mr. Hill, he had this to say: "1 can't say why there should be a difference even though there were more smart boys in my classes... Women don't seem to be as competent in math.. .This make sense because they're at home with the kids, thus they aren't exposed to [math]". Finally, both parents believe that good grades are essential, and that math may not be applicable but it is necessary. Jay also shares these thoughts. Jay and her dad were very aware of his influence on her; however, Jay did not see how her mother could also be influencing her thoughts. She admits to not spending much time with her mom; thus, it would be easy for her to overlook some of their similarities in beliefs and attitudes. The Forrester Family Sherry: "Yeah, both my parents do. They expect me to do well. My dad makes sure we stay on top.'N t Mrs. Forrester: "Yes, 1 would hope my girls see me as a positive influence." Mr. Forrester: "Yes, \ might [be] but 1 don't think so... I just like to encourage her." In the Forrester family, each member is aware of the possible influences of a parent on a child. As a result of Sherry turning to her dad for math support, one might assume that she would share more of the same math beliefs and attitudes of her father than of her mother. This however is not obviously the case. Although Mrs. Forrester rated her liking of math as a 3, she is wary about passing on her negativity to Sherry, because she does not want to discourage her daughter. It is not as if Mrs. Forrester has a poor math self-image, as her problems stem from the fact that she believes that the nature of math is not conducive to her learning style. In other words, Mrs. Forrester does not 86 blame herself for her math failures. Like her father, Sherry also has a positive math self-concept because she describes herself as being "good at it"; however she was chosen for this study because she thought herself to be poor at math and her teacher thought otherwise. After some probing, 1 discovered that this discrepancy is as of a result of some test results that Sherry had recently received prior to the poll carried out in her class. To her dismay, Sherry had done worse than anticipated and thus, she doubted her mathematical abilities at that time. From the way Sherry spoke of "liking different things at different times" it would seem that her attitudes about herself and about math tend to be unstable and change depending on the circumstances. With time to reflect upon the issue, Sherry deflected the blame of failing a recent test onto her teacher. This brings us to another interesting issue in that home - the role of teachers. .Mr. Forrester is.able to recall the positive effect his teachers had on his mathematical experiences; while Mrs. Forrester resents some of her teachers wanting her to conform to their standards. Whether.positive or negative, both parents attribute their successes and failures in math to their teachers. Sherry also believes that her teacher contributes to how well she performs; and as a result of that belief, she is able to shift blame and be less hard on herself. With respect to the nature of math, Sherry and Mr. Forrester both see it as a "straightforward" subject. This belief promotes their motivation in tackling math problems. "Because of how math is, it is easier to go along since you always know if you're right or wrong." Mrs. Forrester believes math to be quite inflexible, and her belief has an opposite effect: she is frustrated by having to operate in such an either/or manner. A belief that Mrs. Forrester and Sherry do share is that math should not be time-oriented. They both 8.7 pointed out the need to be able to "spend time with the math". This belief helps Sherry to control her frustration when working on math problems. She said: "well it depends on the problem, but I need time to make sure I do everything otherwise I'd get flustered." In contrast, Mr. Forrester believed that a person should be fairly quick in problem solving, once there is understanding. Everyone in this family believes that a math leaner needs to understand everything to do well. Whereas the father and the daughter believed this to be true for every learner, the mother thought that it was perhaps only true for her because of how she was as a student. This belief complements Sherry's other belief about math not being time-oriented. Because she sees the need to grasp all the levels of a concept, she spends as much time as necessary to ensure that she attains that full understanding. Finally, Sherry and her mother firmly do not believe that gender plays a role in one's mathematical competency, and Sherry went so far as to say that "it all depends on how hard a person tried and how capable they were". Mr. Forrester on the other hand was somewhat hesitant in his response. He eventually said: "Perhaps this was so when I was in school... but now, 1 don't feel [there is a difference]". The Lebrun Family Shelby: "Not really, my parents just want me to do well, that's all." Mrs. Lebrun: "Well we got her a tutor and tried to address her weaknesses as early as possible," Mr. Lebrun: "Probably, but 1 can't see it. ...What's important is my personality toward her [math] troubles." Shelby had been chosen for this study because of her low self-image and her poor grades. However, she portrayed a much healthier self-concept than was expected, during 88 the interview. This may be attributed to the fact that both her parents believe that math can be done by "the average person" - no special talent needed. Even though both Mr. and Mrs. Lebrun were quite successful in their math classes in high school, they do not use their experiences as a tool for rating Shelby's experiences. They were quite adamant about not letting Shelby feel that she has to be good in math because they were good in math. In fact, Shelby conies across as very optimistic for a person with difficulties in math. Her parents also believe that a person's interest in a subject plays a crucial role in how successful they will become. Shelby already knows that she will be taking on a career that requires minimum mathematical qualifications, and her parents are also aware of her aspirations. Because of their beliefs, they seemed to be encouraging Shelby to do the best she can to get through her schooling. Perhaps it is this belief that allows Shelby to take things in her stride. Shelby likes most aspects of math and even though she has a hard time grasping some concepts, she claims that she will pursue it until the end of high school. This attitude of hers could be as a result of her belief that math is important (even though she was not sure how useful it was). Her parents also seem to prescribe to the notion that math is important, as they constantly try to provide her with all the math help and support she needs. Shelby is quite diligent when it comes to her math -an attitude which parallels that of her mother rather than her father. For Mr. Lebrun, math came quite "easily", so he did not have to put in a lot of effort. However, Mrs. Lebrun described herself as being more of an "okay" person in math; she had to make a conscious attempt to keep her grades up. With respect to the nature of mathematics, Mr. and Mrs. Lebrun described it as being "rigid". For Mrs. Lebrun, being rigid was not such 89 a good quality, whereas, Mr. Lebrun found that the rigid characteristic was challenging in a positive way. As for Shelby, she did not use the precise word "rigid", but in her description of how sequential math is "with all the steps that are involved", and of the need to "memorise all the steps in an equation", perhaps she also finds math to be inflexible. However, this characteristic does not persuade her positively or negatively, in terms of the nature of mathematics - she finds that it is simply part of math's inherent nature. Finally, with respect to the way this family responded to the question of influences, I believe that Mr. and Mrs. Lebrun both want what is best for Shelby, and so they try to 'positively influence' her by encouraging her and providing her with any and all the help that she needs. Shelby for her part recognises and appreciates all that her parents do for her. She knows that they are looking out for her. 90 CHAPTER SIX: DISCUSSION DEEPENING THE CONVERSATION . What about math beliefs and attitudes? • In the review of the literature in Chapter Two, I discussed the types of beliefs and attitudes this investigation would be focusing on -namely, beliefs about math and about oneself as a learner, and attitudes toward math as a subject. According to Schoenfeld (1985), these factors make up one's "mathematical world view" because they "establish the context in which mathematics is done" (p. 45). In this chapter, I will start by discussing a few examples of beliefs and the resulting attitudes that past researchers have found to be common among math students, with the intent to see how prevalent these attitudes and beliefs were amongst four girls in the study. Math is important, difficult, and based on rules! The students interviewed here would agree that math is important, but not useful beyond basic arithmetic. As they stated, this belief is enough to motivate them to take math till the end of high school. When probed about how one could use math after school, the girls were unable to articulate how all the math they were learning could be useful in the future. Perhaps, it is their limited real-life experience that inhibits them. Or perhaps, it is because of the way mathematics is approached in schools. Math is seen as a collection of topics, and much of what is taught does in fact have little direct future use for many students. Even though the parents held the same beliefs, they claimed that it was only in retrospect that they 91 could see how their matured experiences over the years allowed them to make statements such as: "math trains the brain" and "math develops the intellect". r As for being difficult, none of the girls described math as "difficult" -not even the two that were selected for their weak mathematical abilities. Instead of being that general, they preferred to say that they found some parts hard to grasp. First off, you may recall that I had chosen to talk about math in its general sense because McDonough (1998) found little evidence of elementary-aged children being able to identify the various area of mathematics. Thus, it was a pleasant surprise to see that at least these four girls could distinguish their feelings for the different areas of math they had been exposed to. Operating under this belief, the girls were able to express their frustrations for certain aspects of math, instead offer all of math. Interestingly though, most of the mothers in the study were prone to finding math in its entirety a complex subject, even when the researcher encouraged them to talk about their feelings for the separate aspects of math. On a similar finding, the fathers also talked about their attitudes toward math in a general sense. From what researcher McDonough found, it would seem that the older we get, the better we should be at separating out our feelings for the different areas of math. Is it not therefore ironic to see that the parents in this study used a more global sense of mathematics than their children did? It should also be pointed out that the mothers here who spoke of finding math difficult, also stressed how they each fried to shield their own feelings about math from their children, for fear of discouraging them. So even though they have shared their mathematical experiences with their daughters, they expressed how they consciously try not to let their attitude affect their child. How is 92 it that the daughters do not see their mothers' negativity as an opportunity for developing their own negative attitudes? Perhaps, it has something to do with the fact that, when asked, these four mothers and their daughters stated that they do not believe math abilities to be gene-related, and that this belief mitigates against the 'automatic' social effects of parental beliefs on the construction of a child's beliefs. When asked to describe the nature of mathematics, only two of the girls stated that math involves rules, formulae,* or facts/steps. However, 1 would hazard a guess that the other two would be inclined to agree with that belief as well, because they had mentioned the need for memorising basic facts. Researchers have found that students who hold such a belief tend to approach math in a very mechanical fashion, and they also attribute little role to understanding in mathematical thinking. 1 would agree with the first part. The students in this study do tend to approach math as a series of steps; however, all four girls did express the importance of understanding in learning math. The two girls who were vety competent believed that an understanding of everything was necessary; while the other two that were less competent thought that some understanding would be sufficient for success. With respect to the parents, the fathers were more likely to be adamant about the need for full understanding of all the steps in math. In contrast, the mothers believed that some or even no understanding was necessary for being successful in math. Perhaps the fact that the fathers all expressed their conscious effort to make their daughters aware of their belief about the role of understanding is one reason why the daughters and fathers seem to share similar beliefs. 93 Only those with special talent can/earn math and making mistakes is part of the learning process: Despite her mathematical abilities, each of the girls interviewed for the study said she believed that because she could do math, then anyone could learn math! Two of the parents though, believed that learning math required some special talent although they did not see themselves as possessing such talent. It was surprising to see that (even in a sample as small as four) such a belief was not picked up by at least one of the children, especially since past researchers have found this belief in 'special talent' about math to be one of the more prevalent ones in existence. Although one child attributed her learning to natural ability, she did not see that as something unique. The girls proposed factors such as practice, hard work, paying attention, and interest in the subject as the necessary ingredients for learning math. Could it be that this 'myth' about needing special talent is finally dying,-or does this belief manifest itself as children progress through the higher math classes? Perhaps it is all just a coincidence in choosing these four families. As for making mistakes, the students and parents alike all believed that mistakes were part and parcel of the learning process in math. Sometimes it is because of "carelessness" and other times "it is due to a lack of understanding" - never is it a reflection on the ability of the student. The research says that, when students think they are dumb for making mistakes, they usually become hesitant in trying to solve problems. The four girls in the study all spoke of feeling rather comfortable in tackling their math problems. Perhaps this is so because of the belief these girls have about themselves as math learners. 94 Math is a male-oriented subject:. I was pleasantly relieved to find that these students did not think their gender played a role in how capable they were mathematically. I was also intrigued by the way they came about this belief. When asked about this issue, each girl reflected upon her own classroom environment. Because there were just as many "smart" boys as there were "smart" girls, the students concluded that gender could not be a factor in mathematical capability. Similarly, the parents also used this approach to talk about their belief about math as a male-oriented subject. However, in most of the parents' high schools, there were simply more boys than girls in the whole school. Thus, there were usually more "smart" boys than "smart" girls. As a result, the parents in this study usually reconciled their high school experience with what they saw happening in today's math classes. Given educators' past concerns about girls and math, it was encouraging to see that at least these four girls had strong images of themselves, and that being female did not prevent them from doing math. / hate math: In my own experiences as a teacher, this is a statement I hear quite often from students; however, I never heard any of the four girls express any dislike toward math on the whole. Could the social expectations of an interview have been responsible for this difference? Was 1 receiving a more thoughtful answer in the interview than I would have received in a math class? Even when a couple of the mothers expressed distaste for the subject, the respective daughters still had positive attitudes. Perhaps it had to do with the fact that the girls in the study all reported going to their fathers for math help, and so they were instead influenced by their fathers' attitudes. Perhaps it was because these girls could separate their feelings for different topics in math, and not let those feelings cloud their overall impression of the subject. Or perhaps as it turned out none of the students were really poor in math, and this would contribute to their positive attitudes. It has, however, been reported that children of this age do tend to have fairly positive attitudes toward math. It would be interesting to interview these same girls three years from now to see what their math attitudes are like, While still on the topic of math attitudes and beliefs, I remind the reader of two questions that arose while reviewing the literature, is the overall image of mathematics (in school) still a negative one for parents? And if so, has this image been passed on to today's children? From talking with these four sets of parents, 1 would say that the answer to that first question seems to be divided. It is my impression that for the four mothers, their mathematical experiences were for the most part not positive - negative seems too harsh a descriptor. However, for the four fathers, they recalled having positive experiences with math. And yet, 1 do not think that the issue is as clear cut so as to say that women have more negative experiences than men do. Some of the women here spoke of not being interested in math, or having difficulty with the rigid structure of math class. Were these factors an issue for the women because they are women? I cannot definitively say - it was simply something interesting that came about with these four mothers. With regards to the second part of the question, again there are no precise answers. All that could be said is that the four girls seem to have positive attitudes toward math, despite their mathematical abilities. They all seem to have similar attitudes to their fathers. Because there is no collective positive or negative image of math amongst the parents, the girls in the study could only show reflections of one parent. How is it that they reflect the attitudes of their fathers and not of the parent with the same sex? If this study had included boys, what would we have discovered? Again, we are leaving with more questions than answers, since these were only four families. Would the same questions arise if we were to interview more families? Now that we have mentioned a vital part of this investigation, that is, the techniques used to collect data, let us now look at what using qualitative methods did for the research process. Re-viewing the methodology and methods The use of clinical interviewing. As mentioned in Chapter Three, the clinical interview was selected as a means for data collection in keeping with the decision to re-look at past approaches to research on the affective domain. 1 wanted to take a more personal look at students' experiences, by offering the reader a chance to meet the people behind the pencil-and-paper tests. Although the clinical interview is predominantly used to provide insights into the meanings that students find in mathematical problems, I found that it was very useful in allowing me to appreciate the feelings of the students and their parents. It provided a medium for encouraging talk about mathematics. Ginsburg (1981) could not have put it more simply when he said, "If you want to know what someone is thinking, ask him" (p. 7). Similar to researchers that use any type of interviewing technique, those who use clinical interviewing have emphasised the importance of initiating the interview. This stage usually involves asking the interviewees non-threatening, personal questions, as well as.explaining the purpose of the interview and justifying the use of recording or 97 videotaping the interview. During the investigation however, two of the fathers had resented this stage of the interview, as all their verbal and nonverbal gestures suggested a preference for starting the interview process with the questions about their math beliefs and attitudes, rather than about their personal 'non-math' self. Could their reaction be as a result of the formal setting of an interview? On the other side of the issue, it takes much practice and experience to become a good interviewer. Looking back at the sessions I see how my vulnerabilities - my inexperience, my youth, my stupid questions - all revealed my own humanity. However, 1 am consoled by some advice oral historian Errante (2000) received: "being young and stupid can [sometimes] be a good thing, [because] you will ask the kinds of obvious questions you won't be able to conjure up when you become more familiar with the terrain" (p. 19). Whatever the reason for the behaviours displayed throughout the initial part of the interviews, 1 still think that this stage was extremely important to my study because of the method of data reporting that I had had in mind. Writing a portrait requires a personal context, and since .1 only had one meeting with each participant, I felt that a few personal questions about themselves would help in bringing me one step closer to getting a better feel for that person. One of the interesting issues that arose throughout several of the sessions was the parent wanting to talk about past attitudes and beliefs from a present day stance. My question about how much math was liked caused some confusion for most of the parents. They wanted to know if I was referring to their liking of math now or when they were in high school or university. A parent remarked: "I don't use [math] now in my life, so I don't have any feelings about it!" The basis of this type of concern was consistent with 98 what was discovered in the Literature Review — in that, the image of mathematics for many adults today is based on their own experiences in high school. This finding suggested to me that adults would carry this image of math they had with them throughout adulthood. Therefore, 1 was surprised by the parents' question because I thought that they would naturally refer to their own school experiences, something that came from the responsive and reactive nature of clinical interviewing that would have been obscured by quantitative methods. So when asked, I left it up to the individual to decide from which era (past or present) he or she would recall. This brings us to an important issue for future consideration - the role of memory. It is said that at any given time, an interviewee may remember, forget, or re-invent certain aspects of his or her personal past because individual memories try to validate the view of the past that has become important in the present (Errante, 2000). So, by using clinical interviewing, participants were able to immediately raise their questions and concerns and alert the researcher to their individual memories in a way that might not have been captured by an attitudinal scale. As researchers, we need to talk and critically think about the role of memory in data collection. Probing deeper, since I was able to converse with the parents, the notion arose that if a person used math in their present every day life, then that made a difference in how they felt about math. Again, because of what was previously discussed, 1 had assumed that parents' collective memory of math would be fixed. But, by allowing the parents • , 99 some freedom in where the interview would lead, I was made aware of an issue that I had not considered, because it had not arisen from previous quantitative research studies. The use of portraiture. Since I had used a new approach, in the terrain-of mathematical attitudes and beliefs of parents and children, to the collection of the data, I had to be consistent and choose an appropriate approach to reporting the data as well. After reviewing those journal articles that looked at research on math' beliefs and attitudes, I could not help but feel removed from the data. I wanted a way of letting the 'scientific facts' from in the field give voice to the people's experiences. One way of allowing readers to enter into people's lives is by telling a story commonly referred to a narrative approach to reporting. 1 wanted my reader to meet the families that I was going to meet during the investigation, but I wanted them to be seen as I had seen them. Thus, I decided to create portraits of each family. This technique suited my intentions because it has the ability to create a narrative that is complex, provocative, and inviting -all at the same time, but also honest to the researchers own biases and interpretations. If we adopt •a social constructivist lens for looking at learner, then we need to consider techniques like portraiture that embrace a sense of the complexity and intrigue that is, expected in the parent-child interaction, but which acknowledge that we as reporters can only offer our own interpretations of,the person's individually constructed beliefs and attitudes. Following on that note, by adopting aiiy type of constructivist approach, I then cannot deny that I too become a part of the research process. For this reason, I found writing portraits to be appropriate for this investigation because it provides a place for the researcher, as well as the.research.ees. In portraiture, my voice as researcher is 100 everywhere: in the assumptions, preoccupations, and the framework 1 bring to the inquiry; in the questions I ask and my personal interactions with the respondents; in the data I gather; in the choice of stories 1 tell; in the language, beat, and rhythm of my narrative. As Lawrence-Lightfoot & Davis (1997) explain, "voice is the research instrument, echoing the self of the portraitist - her eyes, her ears, her insights, her style, her aesthetic" (p. 85). With this thesis being my first attempt at this style of writing, I was unprepared for how vulnerable I was to become during the data reporting process. I often found it difficult to let go because I knew that I would be putting a lot of myself in front of the reader. As the researcher I had to be equally mindful of (mis)inteipreting what the interviewees had to say, and of how readers would interpret what I have to say. In order to familiarise researchers with this type of data reporting, 1 would have to agree with Lawrence-Lightfoot's advice when she says, "everyone should have a portrait taken". As mentioned in the earlier chapter on methodology and methods, an essential ingredient in creating a portrait is the use of "thick description" (Geertz, 1976). Being able to provide rich descriptions of the events requires the development of relationships between the portraitist/researcher and the sitter/interviewee. According to Lawrence-Lightfoot & Davis (1997), every stage of the portraiture requires us to build productive and benign relationships. The authors go on to state that; "It is through relationships ...that access is sought and given, connections made, contracts of reciprocity and responsibility (both formal and informal) developed, trust built, intimacy negotiated, data collected, and knowledge constructed" (p. 135). Although, my investigation may have lacked, this sense of developing a relationship because of the one-time interview I had with the families, 1 feel that what I have produced can still be considered as an asset to current research, because 1have been doing what the beginning of this chapter states: -that is, deepening the conversation on mathematical attitudes and beliefs! 102 CHAPTER SEVEN: REFLECTIONS MORE QUESTIONS THAN ANSWERS My journey When I started this study, 1 thought that 1 would be learning more about the mathematical beliefs and attitudes parents and children hold today, and how parents can influence their children. However, 1 also learnt how changes in methodology could affect one's research process. I expected the journey to be quite straightforward - in the way it is explained in a research methods course! Instead, I found a maze, with crossroads at every turn. Similar to McLeod (1987b), 1 discovered that research on the affective domain is notoriously difficult to communicate, and as a result, there are endless possibilities for misinterpretation. With the numerous definitions for attitudes and beliefs in existence, it was difficult to find one definitive definition of these constructs. At every junction, I tried to choose a definition that would bring me closer to the interest of my research. With respect to the factors that can influence children's beliefs and attitudes, my interests focused on the parental effect. In reviewing over twenty-five pieces of past research on parental involvement, 1 discovered that there were three basic ways in which a parent could influence a child. .1 decided to look at the interplay of parents' own beliefs and attitudes and their children's beliefs and attitudes. In my literature review, there were two other studies that looked at the math, beliefs and attitudes that parents held. To make my small contribution to the world of research, I decided to take a second look at the methodology'used in past research since 103 in the last few decades, reviewers of research on mathematics education and the affective domain have been pleading for a new approach to the research in this area. As a result, I employed a case study research design, but combined it with the novel approach of portraiture. Embarking on research can be a journey where the path and the end point are only revealed in the travelling. As Stake says (1995) reporting research "sometimes takes on a stoiy quality: And that is a story [the researcher] comes to understand as [he/she. is] writing it, not really before" (p. 124) What have we learnt here? First, in response to the claim that parents' collective image of mathematics is usually traditional and negative, I found the fust part of this claim to be true. However, although parents do recall their school math experiences as traditional, it was only after! spoke with the parents that I realised that this is true because that was the way school was in their time. In other words, it is understandable that their math image would be traditional, since math classes were more conventional then. By being able to converse with the parents, I found that they were often willing to question their image in light of what they saw happening in their children's math classes. With regards to the second part of the claim, I discovered that it could not be generalised to all parents as is found in the research because at least with the four families I interviewed, it was the mothers alone who. spoke of non-positive experiences with math in school. In the future, we could perhaps look to expand the size of the sample to see how this claim stands up in the general population. 104 Second, given my own teaching experiences and what 1 reviewed in the literature, I had expected to find more negative attitudes to math than positive ones. However, after interviewing these four families, 1 was pleasantly surprised to find an array of mostly positive beliefs about math, which are in turn leading to healthy attitudes toward math. Although it is known that students around this age do tend to have fairly positive attitudes, it is usually expected that the students' mathematical ability would override any positive feelings they may have. It was quite a relief to see that the two girls that were chosen because they were poor in math still expressed positive attitudes toward math as a whole in spite of their lack of ability. A possible consideration would be to cany out a longitudinal case study of children from their last years in elementary school through to their first years in high school, to document any changes in math beliefs and attitudes that may occur. Third, this investigation shows me that for none of the questions I chose to explore are there any clear-cut answers. One of the elements of this study that allowed me to appreciate the complexity in the research in this area is the social constructivist based portraiture approach that was undertaken. By using this lens to view the learner, I appreciate now that how students come to form their math attitudes and beliefs is far more complex than just the 'transfer' of the attitudes and beliefs of the parents. Portraiture has been shown to offer insights into this area. I am not claiming however, that all is now revealed. There is much more study that needs to be undertaken using a wide variety of methods and methodology. Even in the area of portraiture, as Lawrence-105 Lightfoot admits, "work is needed to develop and refine her new methodology" (Lawrence-Lightfoot & Davis, 1998, p. 15). What can we take from this? There were two major lessons we can take from this investigation. First, whether parents and children are aware of it or not, the mathematical beliefs and attitudes parents have cannot help but play a role in the way in which their children think and act in a mathematical context. Why? Lerman (1992, 1994) reminded us (as reviewed in Chapter Three) that parents must play a role because of the language-filled interactions that constantly take place between parents and their children, but this cannot dictate the beliefs and attitudes that children construct. Our children experience many other, language-filled interactions such as with their teachers, their friends, other family members, and 'John public'. Furthermore, 1 have demonstrated cases of parents with different views, which leave the student in a position of choice - conscious or not. Because of the varied beliefs and attitudes that parents have, we can also question the notion of the word 'parent'. From carrying out this investigation, 1 have shown that 'parents' are neither a single nor a consistent unity. As this study took on the role of parents' own beliefs and attitudes, it was discovered that the results from past positivist research had some merit. There is some interplay between the beliefs and attitudes of parents and their children, although there is no evidence of causality. By embracing social constructivism, we are made aware of the need to consider all the possible factors that could influence students' attitudes and beliefs in order to fully appreciate the whole picture. It is true that we can learn a lot by 106 examining each factor separately, which 1 did; however, we must recognise the combined role of other factors such as the school environment, and society in general. This study hopefully brings us one step closer to exploring the whole, dynamic process, so that we could one day go beyond thinking in.terms of causality and instead, talk about the relationships, between the factors. Second, the use of qualitative methods here has definitely added to the results received from past quantitative tests. Not only did the results of the investigation support the low statistical correlations usually reported in this area, they also allowed for some interesting questions to be raised. For instance, can we definitely say that it is because the girls in the study sought their fathers' help with math that they showed similar attitudes and beliefs to their fathers? In some cases, the mothers also expressed their open discussions about their feelings for math with their daughters, and yet those girls were more like their fathers than their mothers - how is that? Three of the girls also spend considerable time with a math tutor; thus from the perspective of social constructivism, this person would also play a role in the construction of attitudes and beliefs. Furthermore, all four girls are in the same math class. How much of what they say and feel about math could be attributed to their teacher? One reason the findings of this study are similar to those of quantitative tests is because we are looking at just one piece of what goes into the formation of children's math attitudes and beliefs - the parents. 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Case study research: Designs and methods. Thousand Oaks: SAGE Publications. 122 APPENDIX D GUIDING QUESTIONS These are a sample of questions used to encourage talk about beliefs about and attitudes toward mathematics. They were varied depending on the participant's responses; and, other questions were asked where needed. Math as a subject: > What's the first word that comes to mind when yon hear the word 'math'? Why? ' > Do you think it takes special talent to do well in math? Do you have such talent? How do you know this? > Do you need to understand everything you do in math to do well in it? Why is that? ^ Do you think students can discover math on their own, or does math have to be shown to them? Please explain. Enjoyment of math: > On a scale of 1 to 10, with 10 being the highest, how much do you like math? .Are there some parts of math you like and some you don't? Please explain. > How do you like math in comparison to other subjects? > What is your reason for trying to learn math? Where do such feelings come from? Value/usefulness of math: > Is the math we learn in school applicable to the real world? In what way, if any, is the math you are studying useful? > Is it important for you to take math? Why? Where do such feelings come from? Confidence in learning math: > How good are you in math? How do you know? > How can you usually tell whether you have done well in a test or not (before getting your grade back)? Motivation in math: > How hard do you work in math class? In general, what causes you to work hard in math? > Do your parents want you to do well in math? Does.that affect how hard you work? > How important is getting a good grade in math? Why do you say that? Where do such feelings come from? 123 Math as a male domain: > As a girl, do you think doing math comes easier to you than to boys? Why? > Who would need to take more advanced math classes: boys, girls or both? Please explain. Causal attributions of math achievement: > What are the reasons for your success and failures? E N T R A N C E QUESTIONS For students: > How old are you? > What grade are you in? > Tell me about yourself, for example, your hobbies, your family, etc.? > Do you know what career you would like to pursue? > Do you have any plans of attending college or university? For parents: > What do you do for a living? > What is the highest level of education you have completed? > Tell me about yourself, for example, your hobbies, your family, etc.? EXIT QUESTIONS For students: > Of all the things you have said today, do you see your parents as having any influence on your beliefs or attitudes toward math? [Check to see if both parents' influences are similar.] For parents: > Of all the things you have said today, do you see yourself as having any influence on your child's beliefs or attitudes toward math? 

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