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An analysis of teaching processes in mathematics education for adults 1995
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Title | An analysis of teaching processes in mathematics education for adults |
Creator |
Nesbit, Tom |
Date Created | 2009-04-23 |
Date Issued | 2009-04-23 |
Date | 1995 |
Description | This study explored the teaching processes in mathematics education for adults and how they are shaped by certain social and institutional forces. Teaching processes included the selection and ordering of content to be taught; the choice of such techniques as lectures or groupwork; the expectations, procedures and norms of the classroom; and the complex web of interactions between teachers and learners, and between learners themselves. The study addressed three broad questions: (1) What happens in adult mathematics classrooms? (2) What do these phenomena mean for those involved as teachers or learners? and (3) In what ways do certain factors beyond the teachers’ control affect teaching processes? The theoretical framework linked macro and micro approaches to the study of teaching, and offered an analytical perspective that showed how teachers’ thoughts and actions can be influenced and circumscribed by external factors. Further, it provided a framework for an analysis of the ways in which teaching processes were viewed, described, chosen, developed, and constrained by certain “frame” factors. The study was based in a typical setting for adult mathematics education: a community college providing a range of ABE-level mathematics courses for adults. Three introductory-level courses were selected and data collected from teachers and students in these courses, as well as material that related to the teaching and learning of mathematics within the college. The study used a variety of data collection methods in addition to document collection: surveys of teachers’ and adult learners’ attitudes, repeated semi-structured interviews with teachers and learners, and extensive ethnographic observations in several mathematics classes. The teaching of mathematics was dominated by the transmission of facts and procedures, and largely consisted of repetitious activities and tests. Teachers were pivotal in the classroom, making all the decisions that related in any way to mathematics education. They rigidly followed the set textbooks, allowing them to determine both the content and the process of mathematics education. Teachers claimed that they wished to develop motivation and responsibility for learning in their adult students, yet provided few practical opportunities for such development to occur. Few attempts were made to encourage students, or to check whether they understood what they were being asked to do. Mathematical problems were often repetitious and largely irrelevant to adult students’ daily lives. Finally, teachers “piloted” students through problem-solving situations, via a series of simple questions, designed to elicit a specific “correct” method of solution, and a single correct calculation. One major consequence of these predominant patterns was that the overall approach to mathematics education was seen as appropriate, valid, and successful. The notion of success, however, can be questioned. In sum, mathematics teaching can best be understood as situationally- constrained choice. Within their classrooms, teachers have some autonomy to act yet their actions are influenced by certain external factors. These influences act as frames, bounding and constraining classroom teaching processes and forcing teachers to adopt a conservative approach towards education. As a result, the cumulative effects of all of frame factors reproduced the status quo and ensured that the form and provision of mathematics education remained essentially unchanged. |
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Adult Education -- Research Mathematics -- Study And Teaching -- Research |
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Thesis/Dissertation |
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Text |
File Format | application/pdf |
Language | Eng |
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Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2009-04-23 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0064596 |
Degree |
Doctor of Philosophy - PhD |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1995-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/7520 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0064596/source |
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AN ANALYSIS OF TEACHING PROCESSES IN MATHEMATICS EDUCATION FOR ADULTS by TOM NESBIT B.A., The Open University, 1978 M.A., San Francisco State University, 1991 A THESIS SUBMiTTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Educational Studies) We accept this thesis as conforming to the required standard THE UNIVERSITY F BRITISH COLUMBIA May 1995 © Tom Nesbit, 1995 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. (Signature) Department of The University of British Columbia Vancouver, Canada Date l DE-6 (2/88) 11 ABSTRACT This study explored the teaching processes in mathematics education for adults and how they are shaped by certain social and institutional forces. Teaching processes included the selection and ordering of content to be taught; the choice of such techniques as lectures or groupwork; the expectations, procedures and norms of the classroom; and the complex web of interactions between teachers and learners, and between learners themselves. The study addressed three broad questions: (1) What happens in adult mathematics classrooms? (2) What do these phenomena mean for those involved as teachers or learners? and (3) In what ways do certain factors beyond the teachers’ control affect teaching processes? The theoretical framework linked macro and micro approaches to the study of teaching, and offered an analytical perspective that showed how teachers’ thoughts and actions can be influenced and circumscribed by external factors. Further, it provided a framework for an analysis of the ways in which teaching processes were viewed, described, chosen, developed, and constrained by certain “frame” factors. The study was based in a typical setting for adult mathematics education: a community college providing a range of ABE-level mathematics courses for adults. Three introductory-level courses were selected and data collected from teachers and students in these courses, as well as material that related to the teaching and learning of mathematics within the college. The study used a variety of data collection methods in addition to document collection: surveys of teachers’ and adult learners’ attitudes, repeated semi-structured interviews with teachers and learners, and extensive ethnographic observations in several mathematics classes. 111 The teaching of mathematics was dominated by the transmission of facts and procedures, and largely consisted of repetitious activities and tests. Teachers were pivotal in the classroom, making all the decisions that related in any way to mathematics education. They rigidly followed the set textbooks, allowing them to determine both the content and the process of mathematics education. Teachers claimed that they wished to develop motivation and responsibility for learning in their adult students, yet provided few practical opportunities for such development to occur. Few attempts were made to encourage students, or to check whether they understood what they were being asked to do. Mathematical problems were often repetitious and largely irrelevant to adult students’ daily lives. Finally, teachers “piloted” students through problem-solving situations, via a series of simple questions, designed to elicit a specific “correct” method of solution, and a single correct calculation. One major consequence of these predominant patterns was that the overall approach to mathematics education was seen as appropriate, valid, and successful. The notion of success, however, can be questioned. In sum, mathematics teaching can best be understood as situationally constrained choice. Within their classrooms, teachers have some autonomy to act yet their actions are influenced by certain external factors. These influences act as frames, bounding and constraining classroom teaching processes and forcing teachers to adopt a conservative approach towards education. As a result, the cumulative effects of all of frame factors reproduced the status quo and ensured that the form and provision of mathematics education remained essentially unchanged. iv TABLE OF CONTENTS Abstract. Table of Contents iv List of Figures vi Acknowledgments vii CHAPTER 1: INTRODUCTION AND PURPOSE OF STUDY 1 Background 2 Adult Numeracy 4 Research Questions 12 Structure of the Dissertation 14 CHAPTER 2: LITERATURE REVIEW AND THEORETICAL FRAMEWORK 16 Mathematics Education for Adults 17 Adult Education Research 23 Research on Teaching 26 The Behaviorist Paradigm 27 The Structuralist Paradigm 31 Frame Factor Theory 35 The Interpretivist Paradigm 43 Teachers’ Thinking 44 Summary & Discussion 54 Theoretical Framework 58 Summary 69 CHAPTER 3: METHODOLOGY 70 Selection of Research Site and Participants 70 Data Collection Procedures 73 Data Analysis Procedures 85 Criteria of Soundness 91 CHAPTER 4: CONSIDERING THE FRAMES 95 Institutional Framework 95 College 96 Departments 105 Classrooms 110 VExperiences of Students and Teachers 113 Students 114 Teachers 125 Curriculum 141 The Textbook 144 Summary & Discussion 169 CHAPTER 5: THE CLASSROOM IN ACTION 176 Planning Teaching 177 Classroom Episodes and Activities 182 Homework 206 Assessment 212 Focusing on Algebra 222 Summary 253 CHAPTER 6: ANALYZING TEACHING PROCESSES 258 Revisiting Teaching Processes 258 Frame Factors 262 Conclusion 280 CHAPTER 7: CONCLUSIONS, LIMITATIONS, RECOMMENDATIONS 283 Summary of Study 283 Summary of Results 284 Limitations of the Study 290 Implications and Recommendations 292 BIBLIOGRAPHY 300 APPENDICES 314 LIST OF FIGURES Figure 1 Understanding Teaching Processes 63 Figure 2 Distribution of Students’ Ages 116 Figure 3 Distribution of Students’ Enjoyment of Mathematics Scores 349 Figure 4 Distribution of Students’ Perceived Value of Mathematics Scores 349 vi vi’ ACKNOWLEDGMENTS This work would not have been possible without the support of many people. First, I wish to acknowledge the fundamental involvement of my research supervisor Kjell Rubenson. Throughout all stages of my research Kjell has provided constant encouragement, guidance, stimulation, and support. Further, he has demonstrated the unique skifi of offering just the right comments or asking just the right questions that provoked further thought or more rigorous analysis without ever imposing his own ideas and concerns. Kjell has been the model of a dedicated scholar and working with him has been a profoundly inspiring, enriching, and enjoyable experience. The other two members of my research supervisory committee, Klaus Hoechsmann and Tom Sork, also offered invaluable guidance and encouragement. Their perceptive suggestions helped to focus many aspects of my study and their close reading of successive drafts ensured both clarity of expression and timely completion. Several other colleagues in the Department of Educational Studies at UBC helped shape this study or provided support and encouragement throughout my research. I would like to recognize and thank the contributions of Rita Acton, Abu Bockarie, the late Pat Dyer, Bill Griffith, Garnet Grosjean, Leif Hommen, Andrea Kastner, Graham Kelsey, Liang Shou-Yu, Dan Pratt, and Jeannie Young. I also wish to acknowledge the invaluable support and help of my dissertation study group: Lyn Harper, Janice Johnson, and Tom Whalley. Their concern for, and dedication to, improving educational practices, and their regular and insistent probing of my ideas has been instrumental in this study taking the form that it does. Throughout my studies I was privileged to receive financial support from several sources at UBC. This considerably eased the burden of the non-academic aspects of doctoral study and contributed to the speedy completion of my research. I would like to acknowledge and thank those responsible for the allocation of the University Graduate Fellowships, the Killam Predoctoral Fellowships, and the Coolie Verner Prize for Research in Adult Education as well as the Faculties of Graduate Studies and Education for their awards of travel assistance. Teaching and learning mathematics can be difficult enough without having someone observe your every move. The staff and students at Acton College allowed me into their classrooms and were gracious enough to spend many hours talking to me about what they were doing. This study could never have been feasible without their participation and cooperation, and I extend my gratitude to them. Finally, no aspect of this research could have been possible without the invaluable support, advice, and encouragement of my wife Adrienne. Her descriptions of her own mathematics education helped me focus my interests, her concern for precision helped me clarify my thoughts, and her skills as an editor largely made this account readable. Above all, she showed far more patience and unfailing good humour than at times I deserved. For her: love, gratitude, and thanks of the most profound kind. CHAPTER 1: INTRODUCTION AND PURPOSE OF STUDY Why learn math? Well, I’ll tell you... .11 was] standing on the jobsite with a fellow who was the layout man for doing the carpentry work, eh. And we had a big curve to do in the front of the building, and he worked that out ....He just took a piece of wood, he measured how long the perimeter was, and then just bent it, and it bent in the curvature, and I thought this was the most incredible thing, I said, “Geez, this guy must be an engineer, you know, I mean this is incredible. He’s a genius!” And I realized that all he was doing was basic math.. . .1 want to be able to do it like that. Construction Worker, March 1994 This study concerns mathematics education for adults and seeks to explain why the teaching of mathematics takes the form that it does. In particular, it focuses on several mathematics classrooms in a typical Adult Basic Education (ABE) setting and explores how such education is viewed by those involved as teachers or learners. Further, the study examines the teaching of mathematics in light of its social context, and investigates how teaching processes are shaped by social and physical resources and constraints. My interest in the teaching of mathematics to adults arose from three sources. First, as an adult literacy/numeracy teacher, I realized how many adults regarded themselves as innumerate and avoided both numerical data and arithmetic calculations whenever they could. Of particular concern was that these people were often disenfranchised by their lack of mathematical skills from taking an active and informed role in decisions that involved either numerical data or computational skills. Second, given how widespread the problem of innumeracy is seen to be, the paucity of research on adult numeracy or mathematics education for adults is startling. Although extensive research has been conducted in the corresponding field of adult literacy, few of the approaches, assumptions, topics, or questions that have marked this research, or the insights, applications, or policies that it has generated, have been translated into research on adult numeracy. Third, such published 2 discussions that do exist often cite the predominant methods of teaching mathematics in schools as the biggest contributor to poor mathematical ability. As one noted mathematician puts it: “School mathematics is simultaneously society’s main provider of numeracy and its main source of innumeracy” (Steen, 1990, p. 222). I was interested in determining whether the teaching of mathematics in adult education reproduced that of its school counterpart. In this chapter I introduce the elements of my study. I first sketch some of the background to my study of mathematics education. Next, I provide some definitions of numeracy, discuss some of the consequences of innumeracy, and explain what steps innumerate adults can take to improve their mathematical skills. To outline the specific focus of this study, I provide a justification for, and a statement of, my research questions. Finally, I outline the structure of this dissertation. Background The mathematical abilities of adults regularly give cause for concern to government bodies, business and community leaders, and adult and mathematics educators throughout the industrialized world. There is a strong consensus, amongst these groups, that the mathematical skills, awareness, and understanding of adult learners, whether high-school leavers or college graduates, have deteriorated alarmingly in recent years. Adults “know less, understand less, have little facility with simple [mathematical] operations, and find difficulty in solving any but the shortest and simplest of mathematical problems” (Barnard & Saunders, 1994). So what? Millions of people appear to function perfectly well without ever needing to use much of the mathematics that they remember from school. No one claims to be particularly disadvantaged by a lack of mathematical abilities. In addition, many people see mathematics as an esoteric subject having little to do with their everyday lives. Indeed, mathematics commonly represents a body of ultimately abstract, objective and timeless truths, far removed from the concerns and values of humanity. If mathematics seems so tangential to everyday life, why is it such a problem if so many people can’t do math very well? Primarily it is a problem because of the societal and individual consequences of innumeracy. Numeracy--mathematical ability--is commonly recognized as a major determinant for job and career choices, and a key to economic productivity and success in modern, industrial societies. Numeracy, then, functions as “cultural capital.” Hence, the extent of mathematical ability operates as a social filter, and access to social effectiveness and privilege is restricted to those with sufficient mathematical ability. It doesn’t start out that way. Indeed, numeracy is one of the major intended outcomes of schooling, and mathematics occupies a central position in virtually every K—12 school curriculum. But somehow, mathematics teaching fails to produce numerate adults. As Western society has become increasingly informationally and technologically saturated, the innumerate are increasingly disadvantaged--confused and manipulated by numbers, unable to critically assess assumptions and logical fallacies, and unable to participate as effective and informed citizens. For example, how often are adults prepared to take statistical information and their stated conclusions at face value? How many of us feel skilled enough to look beyond the numbers to interpret what the statistics mean? Of particularly concern is the underlying pattern of inequity in adult numeracy; surveys of mathematical abilities show that performance is lower especially among working class, women, Hispanic, and Afro-American learners. So, mathematical ability is important if only because it 4 is capable of empowering so many. Why are adults’ mathematical abilities as low as they are? It has been proposed that the primary contributor is the poor teaching in school mathematics classrooms (Frankenstein, 1981; Paulos, 1988). Traditionally, mathematics education is taught as an abstract and hierarchical series of objective and decontextualized facts, rules, and answers. Further, predominant teaching methods use largely passive, authoritarian, and individualizing techniques that depend on memorization, rote calculation, and frequent testing (Bishop, 1988). Knowledge is thus portrayed as largely separate from learners’ thought processes, and mathematics education is experienced as a static, rather than dynamic process. Adults who do wish to upgrade their mathematical skills have access to a variety of courses run by local public sector educational bodies. It is unclear, however, if these courses are, in any way, adult-oriented, or merely reproduce the curricula and teaching methods so common in traditional K-12 mathematics. Given the rapid decline in adult numeracy, the nature of its social consequences, and the apparent inadequacy of current educational approaches to remedy it, this study of the teaching processes in adult mathematics classrooms is both timely and necessary. Adult Numeracy To be numerate is to function effectively mathematically in one’s daily life, at home, and at work. Being numerate is one of the major intended outcomes of schooling, and mathematics occupies a central place in the school curriculum. Indeed, mathematics is “the only subject taught in practically every school in the world” (Willis, 1990, P. 16). However, despite this privileged position of mathematics education, there is much evidence that the mathematical abilities of many adults in Britain and North America do not equip them to function effectively in their daily lives (Cockcroft, 1982; Kirsch, Jungeblut, Jenkins, & Koistad, 1993; Paulos, 1988; Statistics Canada, 1991). For example, Statistics Canada report that 38% of Canadians surveyed in 1991 did not “possess the necessary skills to meet most everyday numeracy requirements” (1991, p. 11). There are few published works that deal exclusively with adult numeracy. Occasionally, books (e.g., Dewdeney, 1993; Paulos, 1988; Tobias, 1978; Zaslavsky, 1994) are published where the authors condemn the current state of adults’ mathematical ability and suggest some alternatives for both the mathematics profession and the public. Although generating some concern at the time of their publication, these works are rarely discussed in either the mathematics education or adult education literature, and their impact on mathematics education for adults is unknown. What is clearer, however, is that overwhelmingly these books concentrate on “innumeracy” as opposed to “numeracy,” and in so doing, focus on the negative rather than the positive aspects of individual mathematical ability. This suggests the implicit classification of those who are numerate as “good” or “worthy,” and those who are innumerate as somehow “bad” or “inadequate.” Given the implications of merit in that classification, it is useful, first, to consider some definitions of numeracy. What, in practical terms, does it mean to be numerate? And, alternatively, what are some of the consequences of innumeracy at both personal and societal levels? I will discuss both of these, and finally, describe what opportunities exist for those adults who wish to improve their mathematical abilities. Definitions of Numeracy 6 Public discussion about the mathematical ability of adults is usually couched within the context of debates about adult literacy. Indeed, numeracy and literacy are often linked. For example, the Crowther Report (1959) describes numeracy as “the mirror image of literacy,” and one noted mathematician introduces his survey of contemporary approaches to mathematics education in the USA by stating that “numeracy is to mathematics as literacy is to language” (Steen, 1990, p. 211). Further, common definitions of literacy often include some reference to arithmetic skills, and numeracy as a concept is often considered a part of the wider concept of literacy. For example, UNESCO defines a literate person as one who can engage in all those activities in which literacy is required for effective functioning of his/her group and community and whose attainments in reading, writing, and arithmetic make it possible for him [sic] to continue to use these skills for his own and the community’s development. (UNESCO, 1962) More extensive definitions of numeracy are provided in the Cockcroft Report on mathematics teaching in Britain (1982). Cockcroft discusses a range of definitions from a broad conception—including familiarity with the scientific method, thinking quantitatively, avoiding statistical fallacies—to narrower ones such as the ability to perform basic arithmetic operations. Cockcroft uses the word “numerate” to mean the possession of two attributes: 1. An “at-homeness” with numbers, and an ability to make use of mathematical skills which enables an individual to cope with the practical mathematical demands of everyday life, and 2. An appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables or by reference to percentage increase or decrease. (p. 11) There are some noteworthy aspects of this definition. First, both attitudes and skills are considered important. Second, being practical is the criterion by which skills are considered important; the relevant context is provided by the demands of the person’s everyday life. Third, the appreciation of numerical information is considered important as well as the use of mathematical techniques. Hope (quoted in Keiran, 1990) provides a broader definition of numeracy; one more in tune with peoples’ everyday demands than with the narrower interests of mathematicians. After reviewing several research studies and curriculum guides, Hope identified a set of quantitative tasks that everyone should be able to perform. These included such tasks as handling money and calculating costs, reading recipes, planning renovations, using technical instruments and devices, understanding simple statistics, working with graphs, and using scoring schemes in leisure activities and games. Hope then determined five categories of essential mathematical “understandings and competencies” that he considered essential for these tasks: knowing how to use mathematics to solve problems, knowing how to perform calculations, knowing how to measure, knowing how to work with space and shape, and knowing how to analyze and interpret quantitative data and arguments based on this information. Other, broader, definitions of numeracy are beginning to emerge as educational research documents changes in school practices during the 1970’s and 1980’s. A “broad” approach regards teaching mathematics more as the development of heuristic or problem-solving skills than as the transmission of a body of concepts, facts, and skills (Baker & Street, 1994). For example, Mason, Burton, and Stacey (1985) describe numeracy as the ability to “think mathematically,” which involves the processes of conjecturing, specializing, generalizing, convincing, explaining, and describing--seen as essential to solving mathematical problems. Whatever the exact definition used, many authors claim that the kinds of mathematical skills needed by people to function effectively in daily life are changing, and are likely to continue to change. The need for certain mathematical skills such as arithmetic or algebraic computation is decreasing due to the availability of calculators and computers, while other mathematical ideas such as 8 estimation or those associated with probability and statistics are assuming greater importance (National Research Council, 1989). This continuing change in the mathematical needs of adults highlights the need for some discussion of the consequences of innumeracy. Consequences of Innumeracy In general, innumeracy is not considered as socially unacceptable as its counterpart illiteracy. One often hears statements about peoples’ mathematical inadequacies, spoken without any apparent embarrassment: “I’ve never been able to work out how much to tip”; “I never check my change from the store”; or “I’m a people person, not a numbers person.” One mathematician, John Allen Paulos, claims that part of peoples’ lack of concern about their mathematical ignorance is because the consequences of innumeracy are not as “obvious as other weaknesses” (1988, p. 4). However, regardless of popular opinion, there are several consequences of innumeracy among adults. At an individual level, there are restrictions on freedom of access to further education and training, and to higher-paying jobs. Most institutions of higher education formally require that in order to be accepted, applicants demonstrate their mathematical ability by passing certain standard examinations such as the GCSE (in Britain) or those examinations that lead to certificates of high school completion (in North America). Once enrolled in higher education, students are often required to take further mathematics courses before they can register in courses in particular disciplines (e.g., science, medicine, or economics). Numeracy can also improve the prospects for non-university level students. 9 Researchers have found that a knowledge of algebra and geometry can make the difference between a low score and a high score on most standard entry-level tests for the civil service, and for many industrial occupations (Cockcroft, 1982; Tobias, 1978). A final individual consequence of innumeracy is that adults suffering from “a low level of confidence in their constructive skills and critical insights [tend] to be dependent on the views of the ‘expert’ or ‘professional’ for their opinions” (Evans, 1989, pp. 212-213). On a societal level, the consequences of innumeracy include the loss of industrial production (in both quantity and quality), a waste of resources, the production of inaccurate or useless information, and a diminution in active citizenship (Thorstad, 1992). From a purely business perspective, numeracy and its related thinking skills are increasingly required by employers, particularly in the fast-growing high-technology fields of computers, environmental science, and biotechnology. According to the Workforce 2000 study (Johnston & Packer, 1987), the proportion of jobs requiring the equivalent of four years of high school mathematics will be 60% greater in the 1990’s than it was in the 1970’s, while the proportion where only rudimentary math skills are used will decline by as much as half. In such circumstances, merely being able to remember a few mathematical formulae--one commonly-accepted definition of mathematical ability—will no longer be enough. In a world of rapidly changing technologies, incomplete and uncertain information, and unpredictable events, all workers must be able to do more than competently apply a given mathematical formula; they must know when to apply these procedures, and which ones to use. Another aspect of innumeracy in a social context is the underlying pattern of inequity in adult numeracy; surveys of mathematical abilities indicate that performance is lower among women, working class, Hispanic, and Afro-American 10 learners. Easley and Easley (cited in Willis, 1990) argue that elitist attitudes about mathematics, and acute inequities in mathematics learning, have become part of what oppress many groups who are educationally disadvantaged on the basis of their gender, class, or race. Mathematics is powerful, but much of the power of school mathematics resides not in the mathematics but in the myth of mathematics, in the meritocratic prestige of mathematics as an intellectual discipline. Knowledge is power, particularly when that knowledge has high cultural value and is exclusive. (p. 17) Improving Numeracy In British Columbia, if adults choose to improve their mathematical knowledge and abilities, they have two main options. They can either undertake a process of individual, self-directed learning, purchasing one of several standard “refresher texts,” or they can enroll in remedial mathematics courses offered by adult education providers in their communities. Both the refresher texts and the organized courses cover much the same curriculum--both are designed to help adults prepare for and pass one of the standardized examinations (such as Math 10, 11, or 12, the General Education Development Test, or the Adult Basic Education Provincial Diploma) necessary for entrance to further education. Almost all of the locally-provided mathematics education is organized and controlled by the public education sector. For example, within the Acton area, both the Acton School Board and the community college system offer a variety of “math upgrading” courses to adults at several centres. Most adults in these courses are trying to obtain one of four certificates (Dogwood, Adult Dogwood, College Provincial, and GED) equivalent to high school completion. This pattern of provision is repeated in most urban areas across North America. Much of this provision is intended to supply opportunities for “lifelong 11 learning,” best described as “the opportunity for individuals to engage in purposeful and systematic learning throughout their lives” (Fans, 1992, p. 6). Within Canada, the federal discussion paper Learning Well...Living Well calls for the development of a lifelong learning structure and an associated learning culture that includes the provision of mathematics education for adults (Canada, 1991). For educators, this is quite significant, given the current dearth of information about the teaching of mathematics to adults. Central to the concept of lifelong learning are certain widely-held assumptions about, and practices within, adult education, that are built on ideas and theories about how adults learn and should be taught. These ideas include: teaching must be problem-centered, it must emanate from the participants’ experience of life and develop the individual socially, participants must exert definite influence on the planning of the course and the conduct of the teaching, and techniques used must be based on an interchange of experience (Knowles, 1980). Although adult education policies within Canada acknowledge the goals of lifelong learning (see, for example, Fans (1992), in relation to adult educationwithin British Columbia); how far the practice of adult mathematics education meets its ideals is currently undocumented. In addition, there is no published research in North America “relating to the unique aspects of teaching math to adults” (Gal, 1993, p. 14). Similarly, there is little international research in this area despite UNESCO’s recognition of numeracy as a key component of literacy. Some studies have been published in Australia (Foyster, 1990), Britain (ALBSU, 1983; Cockcroft, 1982; Harris, 1991; Sewell, 1981), and Sweden (Hoghielm, 1985; Lothman, 1992), but, overall, this work has been sporadic and unconnected. Given this background, my study focuses on the provision of mathematics 12 education to adults within a typical adult education setting. In the next section I present a justification for, and a statement of, my research questions. Research Questions In the background to my study, I have shown that a degree of numeracy is considered a necessary skill for adults in order for them to be engaged citizens and productive workers; further, I have shown that adults who wish to improve their mathematical abilities have access to a variety of “upgrading” courses. Despite this, the mathematical abilities of many adults continue to cause concern in many industrialized countries. Nevertheless, little research has been conducted on adult numeracy or the teaching of mathematics to adults. Much of the published material about innumeracy and the learning of mathematics by adults is written from the viewpoint of government or industry leaders (e.g., National Research Council, 1989) or university professors of mathematics (e.g., Paulos, 1988; Willoughby, 1990). These viewpoints overwhelmingly reflect either policy-making and managerial perspectives or the academic research interests of the profession. Further, they are often based on narrow technical and instrumental models of education that ignore much adult learning theory and the importance of such issues as self-concept, motivation, values, attitudes, and intentions in learning. What is missing from the published literature are the voices of those most intimately involved in mathematics education -adult teachers and the learners in their courses. Eisenhart (1988) has identified that “the meanings encoded in the language of mathematics--in the way it is presented to, 13 and used by, students--have not been a focus in mathematics education” (p. 111). She encourages researchers to seek answers to these questions, and to “use sociocultural theories to help interpret their findings” (p. 111). If both the curricula and teaching practices used in adult mathematics classrooms are based solely on those of school-based education, then there is a strong possibility that adults are expected to repeat the approach to mathematics education that they faced when they were children. This situation may persist in spite of the myriad studies which have repeatedly identified that exposure to inappropriate curricula and poor teaching practices in mathematics education is a key source of adult innumeracy. In an attempt to address this, my study explores the teaching processes in mathematics education for adults. In particular, it examines how mathematics education is viewed by those involved in it, and how such education is shaped by certain social and institutional forces. It seeks descriptive accounts of teaching processes in mathematics education from those missing perspectives, and relates those accounts to the ways in which teaching processes are influenced by external factors. By concentrating on descriptive accounts of teaching, I have been able to access “the specifics of action and of meaning-perspectives of actors [which are] often those...overlooked in other approaches to research” (Erickson, 1986, p. 124). I have used the term teaching processes generically to refer to all that “goes on” in the classroom. Thus, teaching processes include the selection of content to be taught; the choice of such techniques as lectures or groupwork; the expectations, procedures and norms of the classroom; and the complex web of interactions between teachers and learners, and between learners themselves. In an effort to illuminate the realities of mathematics education for adults, my 14 study will consider the following broad questions: (1) What happens in adult mathematics classrooms? (2) What do these phenomena mean for those involved as teachers or learners? (3) In what ways do certain factors beyond the teachers’ control affect teaching processes? Structure of the Dissertation In Chapter Two, I present, first, a survey of the literature on teaching in general, and on mathematics in particular; and second, my theoretical framework derived from a synthesis of the literature on teaching. In this chapter, I also discuss some of the implications of combining macro and micro approaches to social research. In Chapter Three, I present the methodological design of the research. I provide, in turn, details of the data sources, data gathering methods and procedures, and data analysis and interpretation procedures. In Chapter Four, I describe the background elements to my study. I discuss the settings where mathematics education takes place, the people involved in those settings, and the work that they do. In Chapter Five, I focus specifically on the teaching processes in mathematics 15 classrooms. I describe situations and episodes that are both typical and common in mathematics education in general, and on the teaching of algebra in particular. Here I include data from both my own observations, and from the perspectives of those involved as teachers and students. In Chapter Six, I analyze these teaching processes using concepts from my theoretical framework. I identify certain frame factors in adult mathematics education and examine their effects on teachers’ thoughts and actions. Finally, in Chapter Seven I summarize my study and discuss certain of its limitations. I also provide certain recommendations both for further research and for improving the teaching of mathematics to adults. CHAPTER 2: LITERATURE REVIEW AND THEORETICAL FRAMEWORK 16 Teaching is a soda! and political process, and therefore is subject to social and political influences. Consequently, a thorough explanation of teaching processes must have a conceptual framework that relates teaching processes to decisions taken in the social and political arena. Such relationships are, however, often invisible and unarticulated by the people involved. Rather than pinpoint them accurately, a researcher can discern and record their effects by both observing teaching processes as they unfold in their natural settings and by examining them from the perspectives of those involved as teachers or learners. In this chapter I describe the theoretical tools I used to so investigate teaching processes in mathematics education. First, I examine the literature on mathematics education for adults. Next, I turn to the adult education research literature, and then to the wider research on teaching in general. I categorize this research into three paradigms, and, for each, provide a brief overview of research on teaching in general, and on the teaching of mathematics in particular. During this discussion I introduce the two general domains of research which inform my own study: research on frame factors and on teachers’ thinking. These two domains can be seen as representing quite different approaches to the examination of social reality, and indicate that, if considered separately, teaching processes can be regarded as both independent of, and dependent upon, their broader social context. I next explore the differences between these approaches in terms of the macro/micro dichotomy and the related issue of structure and agency, and I indicate how this study, which combines both macro and micro approaches, resolves these issues. Finally, I provide a model of the theoretical framework of this study and briefly discuss the elements of this model. Mathematics Education for Adults 17 Despite the wealth of information available on the mathematical abilities of, and education for, school children of various ages, relatively little exists on that for adults. Within the English-speaking world, only Cockcroft (1982), ALBSU (1983), Statistics Canada (1991), and Kirsch, Jungeblut, Jenkins, and Kolstad (1993) provide detailed studies. Cockcroft’s (1982) survey was based on observations of almost 3,000 British adults taking a test on their everyday or “practical” mathematics skills. The findings were supported by further evidence from a study conducted a year later, using data from the National Child Development Study that was based on interviews from 12,50023-year olds (ALBSU, 1983). The Canadian report (Statistics Canada, 1991) was based on interviews with, and the testing of, almost 10,000 adults. Finally, the most comprehensive study (Kirsch et al., 1993) was based on interviews with, and surveys of, over 25,000 US adults. All four studies reported that a significant proportion of adults had problems with numerical calculations and cited difficulties in their everyday lives arising from these problems. These studies provide valuable information on adults’ mathematical abilities, and on their attitudes towards, beliefs about, and uses of mathematics. Much of this research, however, has had little discernible impact on mathematics education for adults, which continues to be based on research on the learning of mathematics by schoolchildren (Fans, 1992; Gal, 1992). Many adult educators make strong distinctions between adult and pre-adult education. Löthman (1992), in particular, identifies those distinctions relating to mathematics education. Two major distinctions that affect teaching processes in mathematics education for adults are those that concern adults’ beliefs about and attitudes towards mathematics, and their everyday uses of mathematics. 18 Beliefs and Attitudes about Mathematics One key skill required of an adult educator is to determine the existing concepts, beliefs and attitudes held by adult learners (Brookfield, 1986). This is no less important in mathematics education than elsewhere. Several authors (e.g., Buxton, 1981; Paulos, 1988; Quilter & Harper, 1988) stress that recognizing and acknowledging adults’ beliefs and attitudes about mathematics is key to encouraging learning. There is evidence to suggest that long before many children leave school they have adopted a view of mathematics as a cold, mechanical subject with little relation to “real life” (Paulos, 1988). Such children do their best to avoid mathematics wherever possible, and manifest anxiety when faced with even simple arithmetic problems. As these children grow into adults they “manage to organize their lives so [that] they make virtually no use of mathematics” (Steen, 1990, p. 215). In discussing the beliefs and attitudes that adults have about mathematics, several commentators (Buxton, 1981; Michael, 1981; Paulos, 1988; Tobias, 1978) have developed the concept of “mathematics anxiety.” This has been described most rigorously by Michael (1981) as “a psychological state engendered when a person experiences (or expects to experience) a loss of self-esteem in confronting a situation involving mathematics” (p. 58). Much of the discussion of mathematics anxiety locates the problem and seeks to remedy it at the individual level (for example, by suggesting that sufferers keep journals or work through self-paced learning material). There are few suggestions for practical classroom activities. Löthman’s study also found that adults were able to learn mathematics better if they could relate what they were learning to their everyday lives. Consequently, I now examine studies on the relevance of mathematics to peoples’ daily lives. Daily Uses of Mathematics 19 Because most research on the learning of mathematics is based on research in school-based education, mathematics education tends to be defined in terms of a school situation. However, in recent years, there have been a number of studies of the use of mathematics in the work and everyday life by adults of specific occupations and cultures (e.g., Carraher, Carraher, & Schliemann, 1985; Lave, Murtagh, & de la Rocha, 1984; Millroy, 1992; Scribner, 1985). For example, Miiroy (1992) studied the uses of mathematics by a group of carpenters in order to document the ideas that were “embedded” in their everyday woodworking activities. She found that, although the carpenters had received very little formal mathematics education, they demonstrated tacit mathematical understanding in their actions. They were fluent with, and made extensive use of, such conventional mathematical concepts as congruence, symmetry, and proportion, and such skills as spatial visualization and logical reasoning. These studies are part of an emerging area of study in mathematics education that adopts a more anthropological approach in order to explore and describe the mathematics that is created in different cultures and communities. D’Ambrosio (1991) uses the term “ethnomathematics” to describe “the art or technique of understanding, explaining, learning about, coping with, and managing the natural, social, and political environment by relying on processes like counting, measuring, sorting, ordering, and inferring” (1991, p. 45). Ethnomathematics, which links cultural anthropology, cognitive psychology, and mathematics, can challenge the dichotomy between “practical” and “abstract” mathematical knowledge. It forces learners to consider others’ thinking patterns, to re-examine what has been labeled “non-mathematical,” and to reconceptualize what counts as mathematical knowledge (Frankenstein & Powell, 1994). The work of several distinguished mathematics educators can illuminate 20 these different understandings of mathematics. Ascher (1991) looks at the mathematical ideas in the spatial ordering and numbering system used by the Inca people in South America. Gerdes (1988) focuses on the mathematics “frozen” in the historical and current everyday practices of traditional Mozambican craftsmen, whose baskets, weavings, houses, and fish-traps often demonstrate complex mathematical thinking, as well as the most efficient solutions to construction problems. Pinxten (1983) examines spatial concepts in the cultural traditions of the Navajo people. Unlike Western people who tend to regard the world statically and atomistically, the Navajo have a more dynamic and holistic worldview which fundamentally influences their notions of such geometrical concepts as points, distance, and space. Turning to the dominant culture within North America, Lave (1988) considers the mathematical experiences inherent in common workplace and domestic activities. In one example, she compares adults’ abilities to solve arithmetic problems arising while grocery-shopping in a supermarket with their performance on similar problems in a pencil-and-paper test. The participants’ scores on the arithmetic test averaged 59%; in the supermarket they managed to make 98%--virtually error free. Lave argues that test-taking and grocery-shopping are very different activities, and people use different methods in different situations to solve what can be seen as similar arithmetic problems. Drawing upon the mathematical traditions present in different cultures, and basing mathematical activities on adults’ day-to-day experiences of their social and physical environments broadens the traditional and often narrow approach of much mathematics education. Furthermore, this brings the learning of mathematics “into contact with a wide variety of disciplines, including art and design, history, and social studies, which it conventionally ignores. Such a holistic approach. . . serve[sl to augment rather than fragment [learners’] understanding and imagination” (Joseph, 21 1987, p. 27). Although these studies show that different forms of mathematics are generated by different cultural groups, they can still be seen as the result of broadly similar activities. Bishop (1988) identifies six fundamental mathematical activities which he regards as universal, necessary, and sufficient for the development of mathematical knowledge: Counting: the use of a systematic way to compare and order discrete objects [involving] body- or finger-counting, tallying, using objects. ..to record, or [using] special number words or names. Locating: exploring one’s spatial environment and conceptualizing and symbolizing that environment with models, diagrams, drawings, words, or other means. Measuring: quantifying amounts for the purposes of comparison and ordering, using objects or tokens as measuring devices. Designing: creating a shape for an object or for any part of one’s spatial environment. Playing: devising and engaging in games and pastimes, with more or less formal rules. Explaining: finding ways to represent relationships between phenomena. (pp. 182-183) All of these anthropological studies document the distinctive character of the mathematical skills and procedures used in work and everyday life, as compared with those taught in school mathematics. They also highlight the success of such procedures when used in particular contexts. However, the use of these procedures outside of the specific situations where they normally occur is problematic. For example, street vendors who successfully perform many relevant calculations daily in their heads find “similar” calculations to be performed with pencil and paper, outside the context of the Street market, exceedingly difficult, and make many more errors (Carraher et al., 1985). Sewell (1981) studied this phenomenon in considerable detail. Her study 22 relied upon interview data designed to cover four areas: a discussion of selected situations related to shopping and household tasks, in which mathematics might be involved; a discussion of other matters such as reading timetables and using calculators; attitudes to mathematics; and background information. Initially, 107 adults, chosen to reflect the range of expected mathematical abilities and of occupation, were interviewed. Next, follow-up interviews of greater length were conducted with about half of those who had taken part in the first stage. Those interviewed were invited to answer a series of questions about a range of mathematical situations. Of these questions, some involved calculations, others required an explanation of method but no calculation, and others required the explanation of information expressed in mathematical terms. Sewell’s findings indicate that there are many adults who are unable to cope confidently and competently with any everyday situation that requires the use of mathematics. Further, she found that the need to use mathematics could induce feelings of anxiety, helplessness, fear, and guilt. These feelings were especially marked among those with high academic qualifications, and who, consequently, felt that they ought to have a confident understanding of mathematics. Further findings included a widespread sense of inadequacy amongst those who felt they either had not used the proper method, obtained the exact answer, or performed with sufficient speed when solving mathematical problems. These studies of adults’ attitudes towards, and their daily uses of, mathematics hold rich information about how people learn and relate to mathematics; information that can have many implications for adult educators. In particular, research on teaching mathematics can be informed by what these studies reveal about learning. I now turn to the adult education research literature to examine how these issues have been studied. 23Adult Education Research Given the wealth of information on adults’ attitudes towards, and daily uses of, mathematics, research on the teaching of mathematics to adults is, surprisingly, almost non-existent. Indeed, specific adult education research on teaching in general is limited; teachers and teaching are not the main focus of discussions about adult education practice. For example, the most recent Handbook of Adult and Continuing Education (Merriam & Cunningham, 1989) cites no references to either teachers or teaching in its subject index. Further, of the key surveys of the developments in adult education research, theory, and practice (Jensen, Liveright, & Hallenback, 1964; Long, 1983; Peters, Jarvis, & Associates, 1991) only that of Long contains any discussions of research on teaching. Although the adult education literature includes a variety of approaches to teaching (e.g., Apps, 1991; Beder & Darkenwald, 1982; Brookfield; 1986; Conti, 1985; Daloz, 1986; Gaibraith, 1990; Hayes, 1989; Johnson, 1993; Pratt, 1992, in press; Renner, 1993; Rogers, 1986; Seaman & Fellenz, 1989; Wlodkowski, 1986), many are written simply as guides for practitioners and concentrate on describing strategies and tactics for improving adult learning. When describing the teaching/learning process, adult educators tend to focus largely on learners and their learning; several authors (e.g., Apps, 1991; Brookfield, 1986; Knowles, 1980; Knox, 1990) pragmatically define teaching or the teaching process solely as the process of facilitating or helping adults learn. This view has led to further empirical research that has contributed towards developing principles of good teaching practice (e.g., Ampene, 1972; Beno, 1993; Beder & Carrea, 1988, Conti, 1985, Conti & Fellenz, 1988; James, 1983; Suanmali, 1981). To take one example, Conti (1985) sought to synthesize the work on adult learning into some central principles and then designed a research instrument to examine the extent to which 24 these principles were exemplified in practical settings. Much of this work, however, although adding to the corpus of research on teaching, has remained theoretically plurative. In addition, these studies tend to regard adult education in general, and because few are based on empirically collected data in adult classrooms, downplay the influence of subject-matter or situational context. Yet, as Anyon (1981) and Stodoisky (1988) show, these factors can strongly influence teaching practices. Turning to the published studies of the teaching of mathematics to adults (e.g., Buerk, 1985; Buxton, 1981; Frankenstein, 1987, Hoghielm, 1985; Kogelman & Warren, 1978, Löthman, 1992), only those of Löthman and Hoghielm fully consider the teaching of mathematics in formal settings. (The others describe particular courses set up for specific groups of people, or to tackle specific issues.) Löthman’s study makes several theoretical and methodological contributions to my own study, and I discuss it more fully later. Hoghielm (1985) investigated mathematics teaching in Swedish municipal adult schools to determine the extent to which teaching was in accordance with certain principles of adult education. These principles were codified in a Swedish Government Bill as “the most appropriate ideals for the teaching of adults.” They include: Teaching must emanate from the participants’ experiences of life, [it] must develop the individual socially, [it] must be problem-oriented, the techniques must be based on an interchange of experience, participants must exert definite influence on the planning of the course and the conduct of teaching, and evaluation must comprise a mutual (teacher-participant) measurement of course content and planning. (Hoghielm, 1985, pp. 207-208) Apart from these latter two studies, a pragmatic approach is common (at least within North America) to adult education research in general. Further, the adult education field has also suffered from a lack of theoretical sophistication and rigor. A recent collection of different perspectives towards adult education research (Garrison, 1994) identifies how the field of adult education has suffered from a lack 25 of overall focus. As Blunt notes, Meetings of adult education researchers and their discussions about how research ought to be conducted. . . and disseminated. . .are characterized by division [and] disagreement... .The differences... also extend to disagreements over what research problems ought to be identified as priorities and the usefulness of the research results produced to date. (1994, p. 168) What is certain about recent adult education research is the diversity of its methods, approaches, topics for study, and purposes. Rubenson (1982, 1989) has identified how North American adult education research has focused more on pragmatic program needs and pedagogical concerns than it has on theory development or policy-related issues. Also, much of the research has been dominated by a narrow reliance on a psychological approach, rather than on anthropological, historical, philosophical, or sociological approaches; consequently, adult education research in general has not been well informed by these different approaches or disciplines. Much current adult education research appears to be unconcerned with developing a stronger theoretical base or with drawing upon research in other disciplines. The result of such a pragmatic and atheoretical approach within adult education is that it hasn’t contributed substantially to the wider field of educational research, or indeed, to wider economic and political issues and questions. Further, it has attempted to deal, on an individualistic and local basis, with the effects of social and political processes, but has not effectively addressed the causes of them in any meaningful way. Given that adult education is heavily influenced by social and political forces, this seems an unexplored opportunity for the field. As a source for theoretical exploration for the exploration of teaching processes, then, the adult education literature is barren. Consequently, I have to now turn to the wider literature on teaching in general. 26Research on Teaching Research on teaching has gone through several periods of change. During the past 80 or so years, this research has become successively more comprehensive and complex in its foci of study, theoretical sophistication, and methodological rigor. Rosenshine (1979) and Medley (1979) both present historical overviews of research on teaching, describing the changes in terms of cydes or phases. For Rosenshine, research on teaching initially focused on teacher personality and characteristics, then on teacher-student interactions, and finally on student attention and subject content. Medley presents a similar view, and categorizes research as focusing first on characteristics of effective teachers, then on the methods they used, next on teacher behaviors and classroom climate, and finally on teachers’ competencies. In many ways, these stages can be seen as representing different paradigms. In each stage, different schools of thought, assumptions, and conceptions have been dominant, which has led, in turn, to different goals, starting points, methods, and interpretations for research (Shulman, 1986). Three distinct paradigms can be discerned in the development of research on teaching, which I have chosen to label as behaviorist, structuralist, and interpretivist. Although these developments in approach broadly correspond to historical periods, they are not uniquely tied to them. For example, although the positivist period saw a predominant focus on psychological research models, much psychological research is currently being conducted in the more interpretive tradition, influenced by recent developments in cognitive psychology. I now discuss each of the three stages or paradigms in turn, describing a general overview of its research foci and key ideas on teaching, followed by more specific examples of research on teaching mathematics. The Behaviorist Paradigm 27 Until the 1970s, almost all research on teaching was behaviorist and empirical in nature and based on the positivist perspective. Textbooks on research strategies (e.g., Kerlinger, 1973; Travers, 1970) regarded educational research as an “objective” enterprise, and concentrated on describing appropriate research methods designed to formulate and verify particular hypotheses. It is not surprising, therefore, to discover that reports of studies on teaching from this period were substantially empirical and used such techniques as experiments and surveys to produce solely quantitative data. For example, Gage’s (1963) handbook of research on teaching contained no section on participant observational research. Further, Dunkin and Biddies (1974) comprehensive survey of studies on teaching contains only reports of research that employ quantifiable measures; it mentions no others from more qualitative or interpretivist perspectives (Shulman, 1986). Such research tended to concentrate on the development of normative laws or models about educational goals, content, and methods of instruction, and was primarily based on psychological perspectives, particularly that of behaviorism. Consequently,. theories and models about teaching in this period were principally derived from individualistic approaches, and were bound to, or reduced to, phenomena about learning and cognition (Lundgren, 1979). One key emphasis of this paradigm of research is the question of how knowledge about learning can affect teaching practices. For example, both Thorndike (1923) and Skinner (1968) argued that ideas about teaching should be based on theories about learning. Because so much of this work is conducted from a purely behaviorist perspective, it therefore focuses on the outcomes of learning rather than on how learning occurs. Consequently, most research studies have been designed to investigate what changes in teaching could produce measurable benefits in student learning. For specific examples of this, I now turn to the research on 28 teaching mathematics. Behaviorist Research on Teaching Mathematics Within what I am calling the behaviorist paradigm, the research on teaching mathematics has been based almost exclusively on theories or assumptions about how children learn. A key emphasis has been its pragmatic focus on what makes such teaching more efficient or effective; namely, what improves student achievement. Also, most of the studies in this paradigm on teaching mathematics have focused solely on the behaviors of teachers rather than on those of learners and/or on the lesson content. They have sought to illuminate student learning only in light of teachers’ actions; the “culture” of the classroom, lesson content, or student behavior or understanding have been of little consideration. Perhaps because of its predominantly behaviorist approach, most research on the teaching of mathematics within this paradigm has tended to isolate a particular “variable” and determine its effects as it was experimentally controlled. Reviewing several studies gives a flavor of the research: how children learn numerical operations (Bell, Fischbein, and Greer, 1984); the stages of children’s learning (Donaldson, 1978); time spent on task (Peterson, Swing, Stark & Waas, 1984); and seatwork (Anderson, 1981). In general, by focusing on the learning of individuals, such research has attempted to search for clusters of common characteristics from which to generalize about particular types of teachers or learners, and to offer predictions for successful ways of teaching mathematics (Nickson, 1992). Romberg and Carpenter (1986), in a summary of reviews of recent research studies on the teaching of mathematics in this behaviorist paradigm, identified several overall conceptual aspects that concern them about this research. First, they found that much research suffered from inadequate conceptualization, and was 29 theoretically weak and haphazard in its choice of which teaching behaviors to study. Researchers used “different labels for the same behavior, or the same label for different behaviors, [and] different coding procedures which yield[ed] different frequencies” (p. 860). Second, lacking substantive theories of teaching, researchers tended to focus on methodological questions. As most research was of an experimental design, researchers then concentrated on “improving research designs, providing better operational definitions of variables, or devising more adequate procedures for counting behaviors, and better techniques of statistical analysis” (p. 860). Such a concentration not only limited the kinds of problems addressed but also the ways in which they were conceptualized. Third, most studies were regarded as being too “global” in that they disregarded the content of lessons. For example, researchers tended to ignore the specific content of what was being taught to specific sorts of students, or assumed that it lay outside the scope of inquiry. Romberg and Carpenter describe an earlier study (Romberg, Small, & Carnahan, 1979) that “located hundreds of studies that assessed the effectiveness of almost every conceivable aspect of teaching behavior, but found few models of instruction that included a content component” (p. 861). Fourth, researchers tended to categorize student learning as the dependent variable. Further, in order to operationalize notions of students’ achievements and attitudes, researchers relied overwhelmingly on standardized achievement tests. However, as Romberg and Carpenter say Such tests have serious problems. They rarely reflect what was taught in any one teacher’s classroom; when used with young, bilingual, or lower socioeconomic status children, they may yield biased results; and at best, they indicate only the number of correct answers produced by a student, not how a problem was worked... .Their use merely compounds the problems when there is a lack of concern for the content being taught. (p. 861) Romberg and Carpenter also discuss four major findings of their review of 30 research. The first concerns the variability of teaching practices. As they describe it, “Every day is different in every classroom [and] every classroom is different from every other classroom” (p. 861). The variability extends to teachers’ and students’ behaviors, texts, time allocated, and content coverage. However, despite these several variations, the dominant pattern of teaching practices, in a wide range of classrooms, was “to emphasize skill development via worksheets, not to select activities that encourage discussion and exploration” (p. 862). The second finding concerns the time available for instruction. Repeatedly, studies showed that, while there were limits on the amount of time available for mathematics instruction, those teachers who consistently devoted less time to teaching mathematics than did their colleagues experienced poorer student achievement. Further, studies showed that the time available for mathematics is most effective when it is well-used in terms of its content coverage, episodic nature, and interactive engagement. “Students should be engaged in activities that are reasonable and intentional.. . .Lessons and units should have a. . . start, a development, a climax, and a summary... .Finally, [students should] be.. .interacting with ideas” (p. 863). The third finding was that student learning was increased if teachers devoted part of each lesson towards increasing students’ comprehension of skills and concepts. If teachers helped students relate new ideas to past and future ideas, then both student engagement and achievement was increased. This process was also increased if students were required to work in small groups. Those students who studied in small groups were found to be not only more cooperative and less competitive than their peers, but also to have a greater comprehension of how ideas were linked (Noddings, 1985; Weissglass, 1993). A fourth finding concerns classroom management. Although the primary 31 purposes of teachers’ behaviors were to cover the assigned content and get their students to learn something, they were also designed to maintain classroom order and control. For example, teachers would occasionally adapt materials not to increase students’ potentials for learning, but to better manage their classroom. Teachers would thereby curtail the time available for students to invent, explore, and apply mathematical relationships. Further, the teachers’ approach to textbooks was also significant. Throughout most studies, teachers would promote the textbook “as the authority on knowledge and the guide to learning” (p. 867). Although teachers could have departed from the syllabus, Romberg and Carpenter found that they chose to do so only to increase their classroom control. In sum, the behaviorist approaches to research on teaching mathematics and teaching regarded it, in general, as a predominantly one-to-one activity between a teacher and a student. Missing from this approach was any adequate conceptualization of education that linked teaching with more cultural, political, and social factors. In addition, there was no research focus on the specific nature of occurrences and events, let alone the meanings that these events had for the people involved. These two areas were developed more in subsequent research approaches. Next I consider a paradigm that sought to illuminate the links between education and social influences. The Structuralist Paradigm In the late 1960s and early 1970s, educational researchers began to adopt more sociological approaches in their studies of teaching. This research tended to fall into two contrasting positions about how issues were approached and interpreted: the consensus and conflict perspectives. Briefly, the consensus perspective is based upon 32 the notion that “societies cannot survive unless their members share at least some perceptions, attitudes and values in common” (Rubenson, 1989a, p. 53). Education is regarded as, first, an agent of socialization into the broadly-accepted values of society, and, second, a means of selection of individuals for particular societal roles based upon performance and achievement. Here, inequality is seen as inevitable, and both necessary and beneficial to society. Alternatively, the conflict perspective, with its roots in the work of Marx, Durkheim, and Weber, questions whether inequality must be inevitable or necessary. By focusing more on the interests of various groups and individuals within society (rather than on society as a unified whole), conflict theorists emphasize “competing interests, elements of domination, exploitation and coercion” (Rubenson, 1989, p. 54). The conflict perspective also promoted critical analysis of the roles and functions of education in society. Interweaving education and its function in society was hardly new; in 1916, Dewey identified educational institutions as promoting and reproducing the dominant values of society. However, the radical critics of the 60s and 70s challenged the dominant liberal view of education as merely offering opportunities for individual development, social mobility, and a redistribution of political and economic power. They argued instead that the main function of education is to reproduce the dominant cultural and political ideology, its forms of knowledge, and the social division of labor (Aronowitz & Giroux, 1993). Before turning to particular studies on teaching mathematics that adopt this approach, I first outline some analytic tools which commentators have used to describe how society influences education. Societal Influences on Education 33 To radical critics, education has several other functions that are not expressed in curricular content, and which often remain invisible to those involved. For example, within North America, several authors promote the idea that schools and other educational institutions exist to “colonize” students into accepting the culture, values, norms, purposes, and goals of the dominant class. This view is fully explored theoretically in the work of Apple (1979), Bowles and Gintis (1976), Carnoy and Levin (1985) and the early work of Giroux (1981). Most of these views about the roles and functions of educational institutions in society are based on the earlier work of Althusser (1971) and Gramsci (1971), both of whom emphasized how educational institutions transmit and maintain society’s dominant ideologies. In particular, they both identified how the needs of the dominant culture shape the provision and form of education to produce “hegemonic” knowledge and ways of thinking. From a slightly different perspective, Bourdieu (1977) argues that education is better understood in terms of more general stratifying processes. In contrast to Althusser and Gramsci, Bourdieu regards educational institutions less as agents of state control, and more as relatively autonomous bodies that are indirectly influenced by more powerful economic and political institutions. He maintains that various types of “capital”—either economic (money, objects), social (positions, networks), cultural (skills, credentials), or symbolic (legitimating codes)--are distributed unequally based on social class. For each class, there is a distinct culture--”habitus”--which is the collection of largely unconscious perceptions, choices, preferences, and behaviors or members of that class. Children learn within their habitus, acquire capital from their parents and from peers, acquire academic credentials (one form of cultural capital), and then, in turn, exchange this for other forms of capital. Thus, educational credentials become one of the key media for the purchasing and exchanging of one kind of capital for another. By seeking to align actual classroom processes with the ways that education functions within society most of these radical researchers have emerged using a largely structuralist (or “macro”) approach. This approach assumes that societal influences determine classroom behavior. As such, these researchers have focused on large-scale theoretical explanations of the relationship between schooling and society (e.g., Bowles & Gintis, 1976), or certain aspects of social structures (such as gender, ethnicity, or class) as if they were causal variables (e.g., Young, 1971). In studies such as these, the freedom of action that people have within dassroom situations, or the meanings they make about those situations, are largely downplayed or even ignored. As an alternative to these large-scale approaches, other researchers have considered small-scale studies of individual schools, teachers, or specific classroom interactions (e.g., Ball, 1981; Donovan, 1984; Hammersley & Woods, 1984). These “micro” approaches have typically focused on individual actors, regarding them as autonomous actors in situations and subject to few outside constraints. These approaches are more concerned with the subjective meanings that actors hold about the particular situations in which they find themselves, and the human actions and interactions that take place there. These two approaches have tended to be polarized and regarded as incompatible; researchers have, in general, adopted either one approach or the other. There have been few attempts to reconcile the macro-micro issue, or to design research that bridges both perspectives. Hargreaves (1985) notes that although the macro-micro issue has been the subject of a great deal of theoretical debate, it has not resulted in much empirical research. However, a comprehensive examination of any social phenomenon—such as teaching--cannot be limited tO a set of either external (macro) or internal (micro) explanations or theories. Teaching can neither be reduced to psychological principles or laws of learning, nor can it be seen as simply determined by contextual factors. To be thorough, a study must attempt to bridge these two perspectives and incorporate both macro and micro approaches. Dahllöf (1977) summarizes some characteristics of what such a model and a methodology would include: Data are curriculum-related, reflecting the goals and intentions of the instructional program as well as the ambitions of the teacher. Data are related to basic patterns of teaching.. .and reflect the cumulative character of the teaching process and its long term effects. Data mirror the teaching process as a continuous change of perceptions and behaviors over time towards certain goals. The analysis [considers] that the teaching of a certain curriculum unit generally run[s] through a series of phases like presentation, training, and control--each phase with its own characteristic pattern [of] communication and interaction. Data are dynamic.. .in that they relate in a meaningful way to the restrictions that are imposed upon most teaching situations by frame factors like space and time, [and] they try to describe and do justice to the role played by students in the teaching situation and its different phases. (p. 406-407) “Frame factor” theory is one particularly useful tool of analysis that meets Dahllöf’s criteria and integrates both the macro and micro approaches. Because it bears on the theoretical framework developed for this study, it warrants detailed examination here. Frame Factor Theory Frame factor theory (Bernstein 1971, 1975; Dahllöf, 1971; Lundgren, 1977, 1981) analyzes the ways in which teaching processes are chosen, developed, and constrained by certain frames. In contrast to research in the behaviorist paradigm that investigated teaching processes by examining how changes in teachers’ behavior 36 affected student learning, frame factor theory is more concerned with exploring how teachers’ actions are limited by external forces. Briefly, a frame is “anything that limits the teaching process and is determined outside of the control of the teacher” (Lundgren, 1981, p. 36). Examples of frames include the physical settings of teaching, curricular factors such as the syllabi or the textbooks used, and organizational influences such as the size of class or the time available for teaching. Frame factor theory claims that teaching processes are governed by “the possible scope of action which exists in a given situation” (Lundgren, 1983, p. 150). The frames mark out the limits that teaching processes have; the actual teaching is conducted within those limits. The concept of frames as a constraint on teaching processes was first developed by Bernstein (1971, 1975) and Dahllöf (1971). Bernstein refers to a frame in the “form of the context in which knowledge is transmitted and received. . . the specific pedagogical relationship of the teacher and [the] taught” (1971, p. 50). He explains that frames refer to the degree of control that teachers and learners “possess over the selection, organization, and pacing of the knowledge transmitted and received in the pedagogical relationship” (p. 50). Dahllöf describes frames more broadly, extending Bernstein’s earlier notion to include the decisions made about teaching that are outside of the teacher’s and the student’s control. Dahllöf’s usage therefore links the macro- and micro- aspects of analysis in a way that Bernstein’s does not. Lundgren (1972) conducted a study of students grouped by ability in Swedish high-school classrooms using Dahllöf’s definition of frames. He developed a model of three types of frame factors: the goals or objectives of teaching a particular subject area, the sequence of content units (lessons) through which the goals were to be achieved, and the time needed by students to master the content. Each student needed different amounts of time to learn new material, and this was related to what content was being taught and how it was taught. Lundgren found that, in order to deal with those situations in which there was insufficient time to teach the required content to all the students, teachers created a “steering group” of students. When having to choose whether to continue with a particular topic area or whether to move on to the next, even though not all of the students had fully learned the existing material, teachers would base their decision upon the demonstrations of ability from those students in the steering group. Following his study, Lundgren further developed the notion of frame factors. Recognizing that any society and the educational systems it promotes are inextricably linked, he argued that because the cultural, political, economic, and social structures of society have an effect on education, they can be regarded as frames, and therefore studied in research on teaching situations. Institutions such as schools and colleges promote learning in terms of postulated knowledge, skills, attitudes, and values. Legislation and rules prescribe the form of this institution, while the available resources in terms of personnel, teaching aides and composition of students determine how the actual teaching corresponds to the formal goals and regulations. (1979, p. 20) Hence, for Lundgren, frames are the realization of fundamental structural conditions. In his own studies he identified that time, curricula, regulations, personnel, teaching aids, and the composition and size of classes act as the most visible frames that govern and constrain the teaching processes. In his later work (noted in Elgstrom and Riis, 1992), Lundgren has also included more conceptual constraints in his notion of frame factors. Thus, personal competencies, attitudes, values, and beliefs can also be regarded as frame factors. Linking of the minutiae of classroom activity with larger social processes is integral to frame factor theory. Stable patterns of classroom interactions can be discovered by studying teaching processes, and then seen as “realizations of underlying rules that shape and steer the process.. . .As society is governed by certain 38 rules for interpersonal relations and by social perceptions, teaching is governed by frames and perceptions that functionally form the rules for the participants” (Torpor, 1994, p. 2375). Since then, particularly in Swedish educational research, frame factor theory has been regularly applied to classroom studies (e.g., Englund, 1986; Gustafsson, 1977; Kallós & Lundgren, 1979; Pedro, 1981) at both preschool and high school levels. It is ideal for research that seeks to analyze teaching processes in terms of their links with more structural elements. The factors governing, steering, and controlling teaching processes are always subject to change, so, as Torper (1994) puts it, “Frame factor theory with its wide scope and its ambition to encompass the deep structures of society, is well suited to the task of analyzing these processes” (p. 2376). Structuralist Research on Teaching Mathematics Although the structuralist orientation to research on teaching is theoretically rich, empirical studies of teaching from this approach are more rare. However, Lerman (1990) and Anyon (1981) each provide a specific example from mathematics education. Lerman initially identified several predominant views in general society about mathematics and their possible influence on mathematics education, and then conducted a field study among mathematics teachers to explore some of the issues arising from his theoretical perspective. He found that teachers’ conceptions of mathematics clearly affected their teaching. Anyon studied mathematics teaching in five schools at different socio economic levels and found that, although all the schools used the same textbooks, the teaching differed dramatically. Teachers in the two working-class schools focused on procedure without explanation or attempts at helping students understand. Teachers in the middle-class school attempted more flexibility and made some efforts towards developing student understanding. At the “professional level” school, teachers emphasized discovery and experience as a basis for the construction of mathematical knowledge. Finally, teachers at the “executive-class” school extended the discovery approach, and used enhanced instruction on problem-solving and encouraged students to justify their answers to demonstrate their mastery of the concepts. Although structuralist studies of teaching mathematics are infrequent, similar approaches to mathematics education in general are more common. Several mathematics educators (e.g., Evans, 1989, Fasheh, 1982; Frankenstein, 1981, 1987, 1989; Mellin-Olsen, 1987) are interested in the “culture” and values that are transmitted in traditional mathematics education. They note that the curricula and commonly used teaching methods are designed to reproduce the existing economic, status, and power hierarchies, and socialize learners into accepting the status quo. To these educators in particular, the traditional mathematics curriculum consists of an abstract and hierarchical series of objective and decontextualized facts, rules, and answers. Much of this curriculum covers a fixed body of knowledge and core skills largely unchanged for centuries. It is based on the assumption that learners absorb what has been covered by repetition and practice, and then become able to apply this knowledge and these skills to a variety of problems and contexts. Further, they regard teaching methods in traditional mathematics education as using largely authoritarian and individualizing techniques that depend on memorization, rote calculation, and frequent testing (Bishop, 1988). These methods convince learners that they are stupid and inferior if they can’t do simple calculations, that they have no knowledge worth sharing, and that they are cheating if they work with others. When education is so presented as a one-way transmission of knowledge from teachers, mathematics can be regarded merely as collections of facts and answers. Knowledge is seen as largely separate from learners’ thought 40 processes, and education is experienced as a static, rather than a dynamic, process. As Frankenstein describes it, much mathematics teaching is based on what Freire calls “banking” methods: “expert” teachers deposit knowledge in the blank minds of students; students memorize the required rules and expect future dividends. At best, such courses make people minimally proficient in basic math and able to get somewhat better paying jobs than those who can’t pass math skills competence tests. But they do not help people learn to think critically or to use numbers in their daily lives. At worst, they train people to follow rules obediently, without understanding, and to take their proper place in society, without questioning. (1981, P. 12) Consequently, many learners of mathematics find themselves in classes in which little effort has been made to place the subject matter in any meaningful context. For many, mathematics remains a mystery unrelated to other subjects or problems in the real world; they often come to regard mathematics as a subject largely irrelevant to their own lives. Other critics of traditional mathematics education have questioned its aims and purposes. In general, two rationales are given for why mathematics should be taught: (1) Mathematics is necessary for personal life and a prerequisite for many careers; and (2) Mathematics improves thinking, because it trains people to be analytical, logical and precise, and it provides mental exercise. Of course, these rationales do not specify what mathematics should be taught, merely that some mathematics should be. One could expect, therefore, that mathematics education would differ substantially from place to place. It is surprising, then, that one researcher discovered there was little diversity in mathematics classrooms the world over (Willis, 1990). Ernest (1990) notes that the aims of mathematics education in any location are often discussed in isolation from any social and political content. Arguing that education in society reproduces its social structure, he distinguishes three groups who have distinct aims for mathematics education: mathematics educators, mathematicians, and representatives of business and industry. To these, Howson 41 and Mellin-Olsen (1986) add further categories of parents, employers, and those in higher education. These authors daim that the aims of mathematics education are not decided on rational or educational grounds but on the basis of the power of these groups to effect change. For example, Ernest (1990) explains the changes in British mathematics education during the 1960s as a result of a struggle between certain groups he calls the “Industrial Trainers” (who emphasized a ‘back-to-basics” approach involving drills and rote learning), the “Old Humanists” (who were proponents of mathematics for its own sake, stressing its logic, rigor, and beauty), the “Public Educators” (who saw mathematics as a means to empower students to critically examine the uses of, and political and social issues surrounding, mathematics), the “Technological Pragmatists” (who believed in teaching mathematics through its applications and emphasized practical problems and utilitarian problem-solving skills), and the “Progressive Educators” (who emphasized student-centered teaching, active learning, creativity and self-expression). The authors studying the aims of mathematics education argue that it is the form rather than the content which conveys those social aims. The ways that mathematics is taught “can emphasize and reinforce the values and relationships that underlie what is produced, how it is produced, and for whose benefit” (Cooper, 1989, p. 151). Cooper also quotes two earlier researchers (Stake and Easley, 1978) who found that teachers in their study saw science and mathematics as “heavily-laden with social values”, and recognized that scientific and mathematical knowledge “may function more and more as a behavioral badge of eligibility for employment” and...wanted help in inculcating the work ethic values they saw as important in present society (p. 152). Each of these authors considers only mathematics education for children and their arguments cannot be necessarily applied to adult mathematics education. Education for children and adult education differ significantly in most Western 42 countries. Löthman (1992), in particular, highlights the differences between adult and childrens’ mathematics education and between how children and how adults wanted to be taught mathematics. She found that adults, in particular, wanted to be able to use the mathematics they learned, and, therefore, wished to be taught by practical methods. She therefore argues that there should be substantial differences between the classroom practices and the course content used in adult settings, and those used in childrens’ education. However, in most mathematics classes for adults, the curriculum appears to follow that of school-based mathematics education, itself largely determined by the requirements of college entrance boards. Within British Columbia, many adult learners of mathematics are “following the same curriculum and using the same materials as their youthful colleagues” (Fans, 1992, p. 30). Thus, it is possible that the teaching processes in mathematics education for adults closely resemble those in K- 12 education. All of these authors show how dominant views of mathematics affect how mathematics is considered and taught. In this way, the dominant conceptions of mathematics can be seen as frame factors influencing and restraining teaching processes in mathematics education. These examples from mathematics education, frame factor theory, and the structuralist approach to research on teaching all share a concern to explain how education functions in relation to social production, and how, in turn, social and political influences surface in educational settings. Lacking in this approach is much consideration for people as autonomous actors in situations. The structuralist paradigm tends to view people as being passively socialized into an institutional framework rather than “participating in their own conceptual constructions of the world and [their] own fate as a project” (Sharp & Green, 1975, p. 5). The third paradigm seeks to respond to this somewhat functionalist approach to teaching by foregrounding the roles of autonomous actors within social structures. The Interpretivist Paradigm For many researchers, the structuralist paradigm is overly deterministic. For them, teaching is not merely the result of external factors but is also heavily influenced by what teachers think and do. Some researchers (e.g., McLaren, 1989, Willis, 1977) identified a need to document specific details of classroom interactions in order to understand how immediate and local circumstances reflected broader structural forces. Other researchers, disenchanted with structuralism, but even less captivated with the predominantly behaviorist approach to studying teaching, began to focus more on the specific nature of educational occurrences and events and the meanings that these have for the people involved. Their studies adopted qualitative or “interpretive” perspectives and studied teaching from ethnographic, participant observation, case study, symbolic interactionist, phenomenological, or constructivist approaches. While these several approaches differ from each other slightly, they all share a central research interest in discovering the meanings that people (whether participants or researchers) make about aspects of human life and human interactions.The rich variety of this research can be gleaned from considering the work of, for example, Fox (1983), Samuelowicz and Bain (1992), and Pratt (1992), and the studies in Marton, Hounsell, and Entwhistle (1984). Rather than consider the general character and overall distribution of educational events and situations, interpretive studies of teaching focus more on the specifics of particular situations or events. Such studies deliberately focus on the perspectives of the people involved, and seek their meanings and interpretations about their situations. By concentrating on specific situations and actions, and on the “local” meanings actors give to these, qualitative research has attempted to uncover the “invisibility of daily life” (Erickson, 1986, p. 121). Typical of the research in the interpretivist paradigm are studies concerning teachers’ thinking. This includes such foci as teachers’ beliefs about students and teaching, their thought processes while planning instruction, and the kinds of decisions they make during teaching. Because this research also informs the theoretical framework for my own study, it warrants close examination here. Teachers’ Thinking A large part of the context of teaching consists of the thinking, planning, decision-making, and actions of teachers. Researchers from all three paradigms agree that teachers’ classroom behaviors are substantially affected by their thinking, and deliberate teaching requires choices as to what and how to teach. The term “teachers’ thinking” refers to those mental processes of teachers that involve perception, reflection, problem-solving, and the manipulation of ideas, and is concerned with how knowledge itself is acquired and used (Calderhead, 1987). This research regards teachers as active and autonomous agents in teaching situations and seeks to explore new ways of conceptualizing and understanding teaching. Research has focused on, for example, the nature of teachers’ knowledge (Zeicher, Tabachnik, & Densmore, 1987), the differences in the use of knowledge between novice and expert teachers (Berliner, 1987), teachers’ conceptions (Pratt, 1992, Samuelowicz & Bain, 1992), teachers’ planning (Clark & Yinger, 1979), teachers’ thoughts, decisions, and behaviors (Shavelson & Stern, 1981), and teachers’ theories and beliefs about students, teaching, learning, and subject matter (Clark & Peterson, 1986). Clark and Peterson (1986) have developed a model that relates teachers’ 45 thoughts to their actions, considering such aspects as teachers’ thoughts, decisions, theories, and beliefs. Their model is based on an interpretive perspective that addresses such questions as, for example, differences in meaning regarding learners’ achievements, and regarding the teacher’s role in dassroom interactions. The model consists of two domains involved in the teaching process: teachers’ thought processes and teachers’ actions and their observable effects. The domain of teachers’ thought processes includes teachers’ planning, interactive thoughts and decisions, and theories and beliefs (about teaching, learning, students, and subject matter). The domain of teachers’ actions and effects includes teachers’ classroom behavior, students’ classroom behavior, and student achievement. In both domains, the elements are seen as inter-related and their relationships as cyclical and reciprocal rather than linearly causal. For example, in the “action” domain, teacher behavior is seen as affecting student behavior, which, in turn, affects both teacher behavior and student achievement. Student achievement can cause teachers to behave differently towards the student, which then, in turn, affects student behavior and student achievement. Clark and Peterson identify a difference concerning the domains which has implications for research. Teachers’ behavior, and its effects (e.g., student behavior, and student achievement scores) are observable phenomena. In contrast, because teachers’ thought processes occur “inside teachers’ heads,” they are unobservable, and hence must be investigated by a more interpretive approach. Further, until fairly recently, the relationship between the two domains was considered unidirectional and causal; they followed a “process-product” model that assumed a causal chain between teachers’ thinking, teachers’ dassroom behavior, learners’ classroom behavior, and, finally, learners’ achievement. However, these domains are now seen as interacting in the reciprocal and cyclical way described above. Teachers’ thinking affects their actions, which in turn, influence their subsequent thinking. This reciprocity suggests that by examining the two domains together, teaching processes 46 can be more fully understood. Interpretivist Research on Teaching Mathematics Within the interpretivist paradigm, there have been attempts to draw some teaching implications from recent research in cognitive science, particularly that concerning constructivism (e.g., Resnick, 1987) or metacognition (e.g., Schoenfeld, 1985, 1987). Much of this research has focused on the belief that learners construct knowledge rather than passively absorb what they are told. This has significant implications for a subject such as mathematics, which has enjoyed a rather unusual status as a fixed body of knowledge and core skills. Views on the nature of mathematics range from “a discipline characterized by accurate results and infallible procedures” (Thompson, 1992, p. 127) somewhat “akin to a tree of knowledge [where] formulas, theorems, and results hang like ripe fruits to be plucked” (Steen, 1988, p. 611) to a human activity that “deals with ideas. Not pencil marks or chalk marks, not physical triangles or physical sets, but ideas” (Hersh, 1986, p. 22). The two poles of this range have been categorized severally as “Euclidean” and “Quasi-empirical” by Lakatos (1978), “Platonic” and “Aristotelian” by Dossey (1992), “Absolutist” and “Fallibilist” by Lerman (1990), and, perhaps most simply as “external” and “internal” by Polya (1963). Despite their appellation, the poles correspond broadly to a view of mathematics either as fixed, certain, value free, abstract, and unchallengeable, or as dynamic, relative, constructed, and negotiable. For over 2,000 years, mathematics has been dominated by an absolutist view, which regarded it as “a body of objective truths, far removed from the affairs and values of humanity” Ernest (1991, p. xi). However, in the past 20 years, mathematics has undergone a “Kunhian revolution,” in which several philosophers (e.g., Lakatos, 1976; Davis & Hersh, 1980) have regarded mathematics as more “fallible and changing, and like any other body of knowledge, the product of human inventiveness” (Ernest, 1991, p. xi). This philosophical shift has a significance that goes far beyond mathematics. For, as Ernest maintains, “mathematics is understood to be the most certain part of human knowledge, its cornerstone. If its certainty is questioned, the outcome may be that human beings have no certain knowledge at all.” (p. xi). Mathematics, as a school subject, has been largely unchanged for many years. Indeed, since the commercial and navigational needs of fifteenth century Europe began to demand an educational provision to improve arithmetic skills, much of the mathematics taught in formal settings has remained unaltered. National systems of education (that included mathematics as a school subject) were founded in France and Prussia at the beginning of the nineteenth century, in England some 50 years later, and in North America shortly after that (Howson, 1990). Within those systems, the mathematical curricula gradually expanded from commercial arithmetic to include successively algebra, geometry, trigonometry, and finally, in the early twentieth century, calculus. Since then, within North America, there has been “constant reform rhetoric but little actual reform of the school mathematics curriculum” (Stanic & Kilpatrick, 1992, p. 407). Within a particular topic area--such as, for example, that of algebra--there has also been little change in how schools have approached it during the past century. Kieran (1992) lists the topics covered in beginning algebra courses in the early 1900s as including: “the simplification of literal expressions, the forming and solving of.. .equations, the use of these techniques to find answers to problems, and practice with ratios, proportions, powers, and roots” (p. 391). These topics are identical to those in a beginning algebra course in the 1990s (see Appendix 10). Mathematics teaching, traditionally, has been based on the assumption that 48 learners absorb what has been covered by repetition and practice, and that they then become able to apply this knowledge and these skills to a variety of problems and in a variety of contexts. Recent research, however, has revealed that the commonly- used techniques do not work as well as anticipated. In fact, learners use mathematical procedures depending on context and environment, rather than, as is commonly thought, on the mathematical nature of the problems they wish to solve (Boaler, 1993; Lave, 1988). The implications of this for the teaching and learning of mathematics are only beginning to be explored. For example, within the USA, there has recently been some movement away from an overly abstract approach towards one that teaches mathematics more contextually. Modern approaches are designed to reflect the demands of real life problems and prepare learners for the mathematical requirements they might meet in their everyday lives. Recent calls for reform in mathematics education have focused on the need to promote institutional practices that facilitate what is called “meaningful learning” (National Council of Teachers of Mathematics, 1989; National Research Council, 1989). This approach is spelled out more fully in two recent NCTM documents (1989, 1991) that encourage teachers to develop school mathematics curricula and activities around promoting and enhancing mathematical understanding and skills rather than concentrating on imitation or recall. However, it is too early to determine their effect on teaching practices and mathematical ability and understanding in either school-based or adult mathematics education. How mathematics is regarded has a special significance for educators. For, if mathematics is a body of infallible, objective truths, then it has no special concern with social responsibility. Educational concerns such as the transmission of social and political values and the role of education in the distribution of wealth and power are of no relevance to mathematics. Alternatively, if mathematics is a fallible human construct, then it is not a finished product but a field of human creation and invention. Hence, mathematics education must include opportunities for learners to study mathematics in “living contexts which are meaningful and relevant to them” (Ernest, 1991, p. xii), to create their own mathematical knowledge, and to discuss the social contexts of the uses and practices of mathematics. Further, notions of what mathematics j also affect how mathematics is taught. An “externalist” view of mathematics education would stress the mastery of existing concepts and procedures; an “internalist” view would concentrate on providing “purposeful activities that grow out of problem situations, requiring reasoning and creative thinking, gathering and applying information, discovering, inventing, and communicating ideas, and testing those ideas through critical reflection and argumentation” (Thompson, 1992, p. 128). The impact of predominant worldviews of mathematics on educational practices have also interested ethnographic and anthropological researchers. In particular, these researchers have focused on the social context of mathematics education and the “culture” that is transmitted by mathematical activities, both in and out of school. Eisenhart (1988) surveys the recent research conducted from an ethnographic perspective. In particular, she draws attention to the work of Cole and Scribner (1974) and Lave (1982, 1985) who, instead of studying the learning of formal mathematics in schools, have instead tried to understand mathematical problem solving outside of schools. This work, says Eisenhart, “is predicated on the idea that by understanding existing, ‘natural’ knowledge and beliefs, researchers can bridge the gap between subjects’ capabilities and the capabilities that researchers or teachers believe students should have” (p. 111). Both Cole and Scribner’s, and Lave’s work focuses on the mathematics used by adults, and I shall discuss it in more detail below. Other recent studies on the teaching of mathematics that fall within the 50 interpretivist paradigm have concentrated on the knowledge, beliefs, and attitudes of teachers (e.g., Ernest, 1989; Thompson, 1984, 1992) and learners’ actual thought processes during mathematics education (e.g., Cobb, 1986; Desforges & Cockburn, 1987; Neuman, 1987). These studies recognize that teachers and learners perceive and interpret teaching situations differently and hence, they attempt to identify these separate interpretations. Much of this work is also based on a phenomenographic approach (Marton, 1981) that seeks the “insider” interpretations and meanings of those involved. For example, Neuman (1987) explored learners’ ways of thinking about numbers and arithmetic to see if learning was improved if teachers used learners’ concepts as a basis for teaching. In particular, Neuman was primarily interested in discovering what children’s initial concepts about numbers were and sought these “meanings” through a series of interviews with children. Löthman (1992) was also interested in discovering what students actually learned. She drew distinctions (based on the work of Bauersfeld, 1979) between the subject content that is meant (i.e., what is contained within the course syllabi and textbooks), taught (i.e., contained in the teacher’s thinking and approach), and learned (i.e., what the learners perceive they have learned). Löthman regarded teaching as ideal when all three parts combined, but recognized that the different backgrounds and perceptions of participants influenced their perceptions so that they interpreted teaching in different ways. Consequently, she noted that a dynamic social process is developed in the classroom and affects how mathematics education is regarded and constructed differently by different people. Her study focused on the conceptions of mathematics education held by two groups of learners--one of adults, one of high school children--who were studying equivalent mathematics coursework. Her purpose was to “describe conceptions of mathematics education in connection with a concrete educational course [in order to] find patterns and structures and catch their importance for the actors and the 51 education” (p. 140). She categorized her results into four “pictures” of different, but related, conceptions that “showed teaching and learning as an entirety”: MATHEMATICAL TRADITIONS, consisting of the conventional dwelling of mathematical problems in relation to the students’ experiences. MATHEMATICAL STRATEGIES, consisting of the students’ ways of understanding, reflecting on, and solving mathematical problems. MATHEMATICAL REASONINGS, consisting of the students’ ways of discussing, analyzing, and judging mathematical information. MATHEMATICAL APPLICATIONS, consisting of the students’ ways of understanding and practicing mathematical concepts outside school. (p. 148) Mathematical traditions played an integral part in all the pictures. Löthman found that both the adult group and the high-school group preferred “strong rules and formal dispositions of problems” (p. 148). The mathematical strategies of the adult group showed that they used a range of procedures due to their practical experiences of calculation and their earlier education. Mathematical reasonings differed between the two groups. Adults preferred to know why they were doing something (such as a problem-solving technique) before they did it; schoolchildren merely wanted a rapid and expedient model. Finally, because of their greater experience, adults were able to see the practical uses of mathematical applications far more clearly than were high-school learners. Löthman further found differences between adult learners’ and their teachers’ conceptions of mathematics education. They were “pointing in two directions. [The learner] was aiming at comprehension and [the teacher] was aiming at procedure” (p. 147). Löthman claims that this difference comes partly from the adult learners’ previous mathematics education and partly from their “experiences of different occupations.. ..These experiences convinced them of the necessity of understanding” (p. 148). What is meant by teaching for “comprehension” or “mathematical understanding,” or promoting “meaningful learning” can be drawn from examples of research in of instructional situations. The work of Skemp (1976), Richards (1991), 52 and Brown, Collins, and Duguid is helpful here. Skemp (1976) discusses the distinction between “relational” and “instrumental” learning. Instrumental learning involves being able to follow rules without ever developing the true ability to synthesize. In contrast, relational learning means knowing both what to do and why. In a mathematical context, this is the difference in being able to solve a textbook word problem through application of series of rules versus an on-the-spot reckoning of currency exchange while bargaining in a foreign country. Similarly, Richards (cited in Cobb, Wood, Yackel, & McNeal, 1992) developed the notions of “school mathematics” and “inquiry mathematics”. School mathematics, which corresponds to instrumental learning, is best characterized as the transmission of knowledge from the teacher to passive students. Here, teachers establish not only the content of what is to be learned, but also how it is to be regarded and interpreted. Students, in order to be successful, must adopt the teachers’ interpretations of the content. For them, learning mathematics becomes the acceptance of others’ norms rather than an active construction of knowledge. Further, because teachers promote mathematics as having its own internal logic and meanings, mathematics for students is “reduced to an activity that involves constructing associations between signifiers that do not signify anything beyond themselves” (Cobb et al., p. 587). In this way, teachers enculturate students into what Lave (1988) calls “folk beliefs” about mathematics. These include the conviction that it is impermissible to use any methods other than the standard procedures taught in schools to solve school-like tasks and that the use of these procedures is the rational and objective way to solve mathematical tasks in any situation whatsoever. (Cobb et al., p. 589) On the other hand, Richards maintains that inquiry mathematics actively seeks to promote a deeper understanding. Teachers, rather than regarding themselves as the sole validators of what counts as legitimate mathematical activity and learning, encourage and guide students to propound and discuss their own interpretations and insights. In this way, teachers promote the notion of mathematics 53 more as a legitimated set of interpretations of “activities that were intrinsically explainable and justifiable” (Cobb et al., 1992, p. 594) rather than as a set of acontextual and fixed rules and procedures. Richards observed that because teachers tended to follow either one set of practices or the other, the school and inquiry mathematics approaches took on the character of traditions. Cobb et al. (1992) found this school/inquiry dichotomy too simplistic and argued that, regardless of their tradition, teachers “initiated their students into particular interpretive stances [where] students learned which mathematics activities were acceptable, which needed to be explained or justified, and what counted as a legitimate explanation or justification” (p. 597). In the classrooms Cobb studied, regardless of the approach of the teacher, students would generally experience an activity as meaningful if it made sense to them within the classroom context rather than with reference to their individual, beliefs, values, and purposes. Brown, Collins, and Duguid (1989) further investigate this phenomenon in their contrast of “authentic” and “inauthentic” mathematical activity. To them, authentic mathematical activity is the “ordinary [mathematical] practices of the culture” (p. 34) which is coherent, meaningful, and purposeful only when it is socially- and contextually-situated. In contrast, common school mathematical activities are not authentic because they prevent students from engaging with everyday culture and impose a more “school” culture. As Brown et al. say, “although students are shown the tools of many academic cultures in the course of a school career, the persuasive cultures that they observe [and] in which they participate. . .are the cultures of school life itself” (p. 34). In other words, classroom activities take place within a school, rather than an everyday culture. The activities that students are asked to perform within this culture of school are attributed to other cultures-- such as those of mathematicians--and yet, they “would not make sense or be 54 endorsed by the cultures [and practitioners] to which they are attributed” (p. 34). In sum, the interpretivist paradigm seeks not to articulate causal relationships between teaching and its effects, nor to illuminate the interweaving of social and political influences in the minutiae of the classroom; instead it seeks to discover the meanings which participants ascribe to a situation. In terms of this study, the interpretivist research about teachers’ thinking is especially topical. Teachers have theories and beliefs that influence their perceptions, decision-making, planning, and actions. These, in a reciprocal and cyclical way affect, and are affected by, learners. The notion that knowledge may be “constructed” and “interpreted” rather than “fixed” and “transmitted” is particularly intriguing in the teaching of mathematics. Uncovering teachers’ own ideas, theories, beliefs, and “hidden” mental activities in mathematics classrooms can prove fruitful in light of actual classroom experiences. Summary & Discussion Research on teaching has tended to fall into one of three distinct paradigms and, within each paradigm, examined only “macro” or “micro” aspects of teaching. Macro studies have focused on large-scale general features of society such as organizations, institutions, and culture and used experimental and quantitative methods to derive explanations about the effects of these external influences on teaching. Alternatively, micro studies have focused on the more personal and immediate aspects of teaching and used descriptive or interpretive frameworks and methods of inquiry to understand the internal meanings and perspectives of participants. Most of this research has been theoretically dichotomous and, 55 consequently, difficult to compare. Part of the difficulty arises from the epistemological distinctions between the different approaches. Indeed, in many ways, each paradigm can be seen as reflecting different positions on the macro-micro issue. In addition, most research on teaching has focused predominantly on the relationship between teaching and learning; indeed, it has largely viewed teaching solely as the promotion of learning. Such a view has tended to reduce social phenomena, such as classroom behaviors and processes, to the behaviors of single individuals. Further, discussions of teaching have tended to regard curriculum (what to teach) and instruction (how to teach) as separate, and, as Doyle (1992) describes, “work in each domain has gone on if the other did not exist” (p. 486). Such separations are, to me, artificial and reductionist. They ignore the social nature of much classroom behavior and assume divisions between external and internal perspectives, between teaching and learning, and between curriculum and pedagogy. Although teaching and learning can be regarded as separate (although linked) processes, they can also be studied as one pedagogic or “teaching/learning” process involving content, activity, and people (Löthman, 1992; Lundgren, 1981; Pask, 1976). Further, curriculum and pedagogy can also be related: “A curriculum is intended to frame or guide teaching practice and cannot be achieved except during acts of teaching. Similarly, teaching is always about something so it cannot escape curriculum. Teaching practices, in themselves, imply curricular assumptions and consequences” (Doyle, 1992, p. 486). As the structuralist paradigm was too determinist or functionalist, the interpretive paradigm, in regarding all social life as ultimately explicable in terms of the actions and intentions of individuals, has gone too far in the opposite direction. The reconciliation of the two implies that observed behavior and its effects need to be viewed both in the context of the meanings and 56 motivations of actors and within wider social contexts and influences. Clark and Peterson (1986) accept that a complete discussion of teaching processes cannot only concern what teachers’ think, it must also include an understanding of constraints and opportunities that impinge upon them. As they see it, “teachers’ actions are often constrained by the physical setting or by external influences such as the school, the principal, the community, and the curriculum” (p. 258). Tn addition, teachers’ actual thought processes can be similarly constrained. For example, teachers may have (or think they have) less opportunity to plan their lessons in the ways that they wish because certain decisions have already been made by the institution in which they work. In this way, research on teachers’ thinking can appear to overlap with the more structuralist frame factor theory. Hence, research that attempts to combine both theories must overcome certain issues--the relationship between macro and micro approaches and the related issue of structure and agency--to which I now turn. Structure & Agency Structure and agency are concepts which attempt to explain actions in social settings as the effects of large scale structural forces or policies (structure) or small scale individual, voluntary actions and patterns of behavior (agency). As traditionally conceived, structure and agency are regarded as competing explanations of social reality. Hence, attempts to combine them lead to ontological (how social processes are generated and shaped), epistemological (what counts as knowledge), and methodological (how research should be conducted) problems. There is considerable overlap between notions of structures and macro 57 phenomena in that they both refer to the reproduction of patterns of power and social organization (Layder, 1994). Within educational settings, “structural” studies have focused on explanations of how educational practices and processes are situated in, and determined by, broad social structures. In these studies, it is considered unnecessary to gather the perspectives of actors because they can be deduced from determining the effects of social structures. Similarly, micro analyses overlap with a concern for agency. “Agency” studies have supposed that events and actions are produced by largely autonomous individuals; they therefore have concentrated on eliciting the actors’ intentions, meanings, and actions about situations. The difficulty with this dichotomy is that it makes the assumption.. . that social life exists on different levels (Shilling (1992). Both approaches are, therefore, limited and fail to capture the totality of social life in general, or educational settings in particular. People do not exist on different levels, so separating social life into hierarchical levels “makes it difficult to conceptualize change as a dynamic process involving both structure and human agents” (Shilling, 1992, p. 70). As Marx put it, in describing human activity: “Men make their own history.. .not under circumstances chosen by themselves, but under circumstances directly encountered, given, and transmitted from the past” (1950, p. 225). Consequently, research on social settings-- such as classroom teaching--that attempts to synthesize micro and macro approaches needs to include both empirical evidence of actual events in particular settings with particular actors (micro and more agency-driven explanations), and supra individual theories that provide a context for those events (macro and more structurally-driven explanations). As Giddens (1976, 1984) reminds us in his theory of structuration, people are, at the same time, creators of social systems and also created by them. In other words, instead of viewing structure and agency as separate phenomena, structuration 58 theory stresses their inter-connection. Thus, structuration theory has potential for empirical application in research. For example, it highlights how a study of teaching can consider the intermediary aspects between structure and agency both within and outside the classroom. Therefore, it is particularly useful for looking at teaching as a social and political process. “Between the rules, negotiations, and bargainings of classroom interaction, and the dynamics of the capitalist economy, or the relative autonomy of the state, lie a whole range of intermediary processes and structures” (Hargreaves, 1985, p. 41). More practically, structuration theory suggests that researchers use multi-strategy approaches to achieve a dense theoretical and empirical coverage of the topic, initiate and develop theory from fieldwork, recognize that all activities are contextually-situated and all situations are the product of human actions, and seek the relevance of pieces of empirical data to wider theoretical issues. This discussion of the macro-micro duality and the relationship between structure and agency inform my own study. I now describe the theoretical framework for my study and discuss how it addresses these issues. Theoretical Framework In adopting a social theory, rather than a psychological approach, my study is based on and develops the conceptual frameworks, theories, and methodologies of other researchers. Further, it provides a way of looking at the relationship between social interaction in mathematics classrooms and the reproduction of the major 59 structural principles which characterize society. Because it adopts a social theory perspective in examining teaching processes, this study must situate its conceptual sights on local, observable phenomena, and also explore openly the various forces constraining the educational activities within the classroom. This involves examining factors that may influence teaching (and teachers) which are not necessarily obvious or apparent to those involved. Although teachers are the main source of classroom teaching activity, they are also, themselves, part of a wider context. Thus, a study of only their choices and behaviors would be incomplete; a thorough study of teaching processes must attempt to examine these contextual factors as well. Of course it is not possible to isolate and examine these factors; one can only discern their effects by studying what teachers do, and what they say about themselves and why they’re doing it. To clarify how my theoretical framework for my research incorporates all of these considerations, I now describe it in detail. The conceptual framework of this study combines Clark and Peterson’s model of teachers’ thinking and frame factor theory to construct a lens for observing teaching processes. This combination allows me to examine the process of socialization that takes place in mathematics classrooms as well as how mathematical meaning and knowledge are formed and developed. This study concerns the whole pedagogic process of teaching and learning--a process I have chosen to name, in recognition that teachers are its chief initiators, as teaching processes. By teaching processes I do not mean only the selection of content to be taught, or the choice of such techniques as lectures or discussions or whether to use group-work. I also include in my definition the expectations, rules, procedures, and norms of the classroom, as well as the complex web of interactions between teachers and learners, and between learners themselves. In other words, studying teaching 60 processes means developing an understanding of “what goes on” within classrooms. Asking “What is going on here?” may seem at first a trivial question, but as Erickson (1986) points out: “Everyday life is largely invisible to us (because of its familiarity. . . and contradictions. We do not realize the patterns in our actions as we perform them... .The fish would be the last creature to discover water” (p. 121). Hence, asking “What is going on?” can problematize the commonplace and make the invisible visible, as is appropriate in a social theory approach. To understand what is going on, one also needs to address “local” meanings and the differing perspectives of those involved. Teachers and learners regard teaching from different perspectives, and in different ways. Also, what appears to be happening may be misleading. Events that look the same may be entirely different and have distinctly different local meanings. In other words, the question “What is going on?” can be extended to include the question “What is going on for whom?” Further, to understand why teaching processes take the shape that they do, one can also identify the forces that are acting on them, and in what ways. Teaching is subject to many forces, some of which can be traced to the political, cultural, and social structures in society. Consequently, to examine fully classroom teaching processes, a research study must seek to relate them to the political and social structures of society. Thus, this study links two strands of educational research. A model of teachers’ thoughts and actions--which examines both the internal mental processes of teachers as they plan, conduct, and evaluate their teaching, and their subsequent observable behavior--is connected to frame factor research, which examines how teaching processes are affected by external factors. In this way I examine how teaching processes are viewed from the perspectives of those involved as well as 61 consider how teaching processes are influenced by other factors. A Model for Understanding Teaching Processes As I outlined above, by teaching processes I mean “what goes on” in adult mathematics classrooms. Teaching processes include the selection and ordering of the content to be taught; the expectations, rules, and procedures of the classroom; and the nature and quality of interactions between teachers and learners, and between learners themselves. To understand why teaching processes take the shape that they do, one needs to identify what forces are acting upon them, and in what ways. Applying frame factor theory to a study of mathematics education for adults, one can determine several constraining factors. Within any society, the institutional framework of adult education provision, the particularities of educational settings, and the mathematical curricula chosen to be presented in those settings all affect teaching processes in mathematics education for adults. Further, the effects of those factors can be seen as interacting with the life and professional experiences of adult teachers and their learners. In other words, social and cultural norms and values affect the settings in which adults can learn mathematics, the mathematics they are expected to learn, and the ways they and their teachers experience and regard mathematics education. These factors are not isolated, however; they act upon and react with each other. Further, the relationships between factors are dynamic rather than static, and constellate uniquely in every classroom and every setting. For example, adult learners of mathematics in a community drop-in literacy center are markedly 62 different from their counterparts in university-level settings. As such, they expect to learn different mathematical skills and knowledge from their more academic counterparts. These differences and expectations are also not constant; they change over time, and affect (to differing degrees) the individuals concerned, the institutions they attend, and the mathematics they study. Consideration of these issues requires a framework for understanding teaching processes and the forces that shape and constrain them. A model of the factors I have identified and their interrelationships is portrayed in Figure 1. The two groups--the worldview of mathematics and the institutional framework, each influenced by social structures--represent possible frame factors. The third group— experiences of teachers and learners—represents a way of observing the effects of these frame factors on teaching processes. The solid lines represent the relationships between elements that are the focus of this study. The dashed lines represent other links that exist but are not dealt with in this study. I now discuss each element in turn. Social Structures Figure 1: Understanding Teaching Processes 63 As can be seen in Figure 1, social structures are not direct influences on teaching processes; rather, they are mediated through the woridview of mathematics, the institutional framework, and the experiences and perceptions of teachers and learners. However, given that their effects are so widespread and so readily apparent, I feel it necessary to briefly explain them here. Following Giddens (1984), I regard social structures as the rules and resources that people draw upon as they produce and reproduce society in their activities. r — Social structures D I Worldview of Institutional I mathematics I I I I I I I I I Hence, social structures are both the medium and the outcome of social activity 64 rather than a system of relationships operating “above” people. In this way, social structures affect, but do not determine, human activity; they are more the result of a “process of creative interpretations by individuals who are engaged in a vast number of concerted interactions with each other” (Sharp & Green, 1975, p. 19). Practically, then, analyses of subjective meaning need to be supplemented with some description of the actual social structures within which people live and act. For a study of teaching, this implies that teachers’ thoughts and actions should be situated within a context of social and physical rules, resources, and constraints. Although teachers may not perceive these resources and constraints, they nevertheless are bound by them. Teachers’ working situations, freedom of action, and thinking are all shaped and limited by social structures. Within the dassroom, it is clear that teachers have far more power to act, direct others, and access facilities and resources than do students. Hence, this unequal distribution of power has considerable significance in explaining the differences in perspectives on teaching processes, and classroom behaviors between teachers and learners. The Worldview of Mathematics A worldview is the set of presuppositions or conceptions of a phenomenon that is held by a particular society or group. In encompassing all the different views of people in that group, the worldview reflects their specific cultural, social, and historical contexts. Hence, the notion of a woridview emphasizes the shared and social basis of knowledge; knowledge is present in the society into which individuals are socialized, and it is a resource shared by members of that society. Knowledge is seen not as a collection of “content,” but more in the “style or pattern of thought.. . .The social basis of knowledge lies in the categories of meaning used to 65 think or perceive or understand the world’ (Dant, 1991, p. 18). As I described earlier, the worldview of mathematics, common in all industrialized countries, is that it is a logical and impersonal branch of knowledge consisting of objective truths and “theories about quantity, space, and pattern [and] the study of abstract symbolic structures used to deal with these theories” (Davis, 1992, p. 134). Mathematics is regarded as an influential and privileged subject in most schools, and possession of mathematical knowledge has a high value in many cultures (Willis, 1990). As Dossey (1992) notes, “Perceptions of the nature and role of mathematics.. .have had a major influence on the development of... [mathematics] curriculum, instruction, and research” (p. 39). Hence, how mathematics is conceptualized affects how it is taught. As it lies outside the control of the teacher, the woridview of mathematics can act as a frame factor in the mathematics dassroom. Institutional Framework Institutional factors can be of two kinds: organizational factors such as the overall provision of education within an area and the physical structures of, and administrative systems in, educational institutions; and curricular factors that specify what is to be taught and in what way, and the textbooks and teaching materials to be used. In both cases, these factors, because they lie outside the control of the teacher, act as frames. Organizational Factors. In British Columbia, almost all of the locally provided mathematics education is organized and controlled by the public education sector. For example, within the Acton area, both the Acton School Board and the Community College system offer a variety of “math upgrading” courses to adults at several centers. Most adults in these courses are trying to obtain one of four 66 certificates (Dogwood, Adult Dogwood, College Provincial, and GED) equivalent to high school completion. This pattern of provision is repeated in most urban areas across North America. Much of this provision is based on the furnishing of opportunities for “lifelong learning,” best described as “the opportunity for individuals to engage in purposeful and systematic learning throughout their lives” (Fans, 1992, P. 6). Central to this concept of lifelong learning are certain widely-held assumptions about, and practices within, adult education that are built on ideas and theories about how adults learn and should be taught. These ideas include: teaching must be problem- centered, it must emanate from the participants’ experience of life and develop the individual socially, participants must exert definite influence on the planning of the course and the conduct of the teaching, and techniques used must be based on an interchange of experience (Knowles, 1980). Another set of organizational factors that can act as frames on teaching includes the physical structures of colleges and their classrooms, and administrative arrangements within particular institutions such as the size of the class, the time available for teaching, and the evaluation system. Each of these limits, but does not determine, teaching processes. Curricular Factors. A second set of institutional factors that can act as a frame on teaching concerns the process of codifying an area of knowledge into an academic discipline and appropriate curricula. When any subject matter (such as mathematics) is taught in a formal setting, it becomes a discipline by its choice of content, teaching methods, homework assignments and evaluation procedures. Thus, the teaching of mathematics becomes bounded by, and negotiated between, the inherent qualities of the subject and the goals and dynamics of the institutions in which the teaching takes place. In the case of mathematics, these negotiations become visible through two processes: setting the aims of mathematics education, and codifying its subject 67 matter into textbooks. I described earlier how the aims for mathematics education can affect its teaching. Briefly, I explained how the aims for mathematics education were, in general, set by certain dominant groups within society, and yet did not reflect, at least overtly, any social or political content. This ensured that mathematics was not seen as a tool for questioning dominant attitudes within society, and its method of presentation reflected this: it tended to be taught in an authoritarian and hierarchical way. A second process through which curricular factors influence teaching concerns the use of textbooks. In many ways, textbooks are the most central and defining feature of mathematics education. The content and structure of most mathematics courses are determined by the content and structure of the set textbooks. In many ways, the textbooks are the curriculum, codified. Romberg and Carpenter (1986), in their survey of research on teaching and learning mathematics found that “the textbook was seen as the authority on knowledge and the guide to learning” (p. 867) in all of the studies they surveyed. They concluded that many teachers “see their job as covering the text” and that mathematics was “seldom taught as scientific inquiry [but rather] presented as what the experts had found to be true” (p. 867). These conclusions are supported by Hoghielm’s study of adult mathematics classes in Sweden, in which he found that teaching practices “were organized on a ‘cramming’ basis [where] the teachers play the part of living textbooks” (1985, p. 207). It is not unrealistic to expect teachers to use some textbook or other; it is common practice in most classrooms. However, as Bishop (1988) asks, “Whose are these books? Who writes them, and for whom, and why?” (p. 10). Basing teaching largely on the textbook has several consequences. First, it 68 sustains and promotes a “top-down” approach to education. Mathematics textbooks are written by experts who purport to know what learners need, and the order and methods they need to learn it. Second, they make no distinction between what different learners bring with them to the classroom. By treating learners as impersonal or generalizable, textbooks privilege content over process. They encourage the teaching of subject matter, rather than the teaching of people. Finally, mathematics in textbooks is presented in a supposedly value-free and decontextualized way. Mathematical knowledge is seen as impersonal; learners are not encouraged to make their own meanings, or find their own significance. Institutional factors, then, can also have a large influence on teaching. The physical structures act as tangible constraints; the curricular factors act more conceptually, limiting what counts as legitimate knowledge or as an approved way of teaching and learning. Experiences of Students and Teachers Regardless of the different settings for adult mathematics education and the different curricula that exist in these settings, two groups of people are affected by and in turn affect, these factors. Adult teachers and their students are each the focus of the factors that limit classroom teaching processes, and, simultaneously, the agents of change on those factors. To my knowledge, there is no published research in North America that focuses on the teachers of mathematics to adults. However, as was mentioned earlier, adult education theories often stress the centrality of the participants to the teaching processes. In practice, this means that effective adult education teaching should relate to participants’ needs and interests (Brookfield, 1986). Earlier, I discussed studies that had considered the needs and interests of those adult learners engaged in mathematics education. I showed how adults’ 69 attitudes towards, and their daily uses of, mathematics influenced both how they approached their mathematics education and how successful they were at learning. Consequently, these attitudes and expectations, which lie outside of the teachers’ control, can also be seen as frame factors affecting teaching processes. Summary These, then, are the elements of my model for studying teaching processes. In this chapter I have described a theoretical framework and its constituent elements that have been designed to investigate and analyze teaching processes from a social theory perspective. The elements of this model will be animated through the participants’ classroom interactions in a variety of ways. I intend to observe, record, and analyze these interactions using diverse and multiple methods in order to achieve a dense, theoretical, and empirical coverage of the topic. In the following chapter, I explain my methodology. CHAPTER 3: METHODOLOGY 70 In this chapter I describe the methodology I used in my study. I first discuss my selection of a specific research site and study participants. I next describe my data collection and analysis procedures, and conclude with a brief discussion of certain issues concerning the “criteria of soundness” of my study.. Selection of Research Site and Participants I needed to contextualize my study of teaching processes in actual mathematics classroom situations. I could not, of course, gather data from all of the many providers of formal mathematics education, even in as geographically- compact an area as the Lower Mainland region of British Columbia. It seemed appropriate to base my study in a local setting where such provision commonly took place, and within that setting, to choose courses that typically reflected the overall provision. Selection of Site Within the Acton area, both the local School Boards and the Community College system provide a range of mathematics education courses to adults. Each system’s provision is comparable: their courses are of similar levels of difficulty, and are offered at broadly similar dates and times. However, one institution in the Community College system offered easy access and was already known to me. I had 71 conducted some initial research in Acton College’s mathematics department and regarded it as an ideal site. It contained a range of informants, and it provided a high probability of finding a rich mix of the teaching processes and frame factors that I wished to study. The selected college offered a range of mathematics courses for adults during both the daytime and the evening. Their provision was organized into five distinct levels, corresponding broadly to grade levels 9 through 12, and an introductory calculus course. Their most mathematically-basic courses, corresponding to grade 9, are two half-courses (050 and 051) and a combined course (050/051) which provide “a review of basic math skills and a study of metric measurement and introductory algebra and geometry” (AC, 1992). They are deliberately designed “for the student who has never studied academic mathematics before or who is lacking a good foundation in basic algebraic skills” (AC, 1993, p. 59). I chose to study the three sections of these courses (050, 051, and 050/051) offered during the 1994 Spring term. Each course was to be taught by a different instructor and was expected to recruit between 15-20 students. The courses I chose were typical of the college’s mathematics provision and their curriculum was supposedly designed to reflect a balance between the formal and practical mathematical needs of learners. Selection of Participants The participants in the study were of two types: (a) teachers of mathematics to adults at the chosen college, and (b) students in the three introductory mathematics courses. Teachers. All of the eight teachers in the mathematics department were 72 interviewed at the beginning of the term. The reasons for interviewing all of the teachers were twofold. First, as those most concerned with teaching processes, teachers have both the power to effect change, and are among those most affected by, the factors involved. I was concerned to ensure that I gained as much data as possible on the teachers’ understandings of teaching processes, the constraints that they feel, and the reasons for their choices in relation to the specific and concrete situations of their teaching. Consequently, I determined that the fullest understanding of teaching processes at the college could only be gained by interviewing all the teachers involved. The second reason for interviewing all of the teachers was more pragmatic. At the outset of my data collection, decisions about who would teach which course had not yet been finalized. Therefore, by interviewing all eight teachers, I was able to ascertain the approaches that they were taking to planning their courses before they had begun. Subsequently, when teaching assignments had been decided, I focused on the three teachers of the introductory courses more specifically. Students. All of the adult students attending any of the three introductory mathematics courses during the first two weeks of term were given a copy of the survey protocol together with an explanatory letter that invited them to participate in further stages of the research. (Copies of the explanatory letter and the survey protocol are attached as Appendices 1 and 2.) Thirty-two students completed the survey, all of whom indicated their willingness to participate further. From these 32 respondents I selected 15 (5 from each section) to interview further. Interviewees were largely selected on the basis of several demographic characteristics (viz., gender, age, ethnic origin, and employment). My intention in using these criteria for selection was not to obtain a sample of participants that purported to be in any way representative of the wider population of adult mathematics students. Instead, it was intended to help me appreciate the range of characteristics of the group of adult learners who had enrolled in the basic mathematics courses. I was interested in exploring whether adults from different backgrounds had different attitudes and approaches towards their mathematics education, or experienced teaching processes in different ways. Data Collection Procedures The study used multiple data collection procedures, combining those from both qualitative and quantitative approaches. My data were gathered through surveys of learners’ demographic characteristics and attitudes, extensive participant observation, repeated semi-structured interviews, and document collection. Each specific procedure is described more fully below, but first, I explain the reasons for usingsuch a variety of methods. Using Multiple Methods The principle use of multiple methods was to add methodological rigor to the study. Although qualitative and quantitative data collection are often seen as coming from contradictory notions of reality, here my avowed post-empiricist approach assumes that there is one “truth” to a situation. Although this truth may not be exactly “captured” by any one means, both quantitative and qualitative methods can be seen as different ways of examining the same phenomenon, and obtaining a closer correspondence with the truth. Therefore, findings that have been derived from more than one method of investigation can be viewed with greater confidence 74 and with a greater claim to validity. Denzin (1970, P. 301) describes this combination of multiple methods as “methodological triangulation” and adds that “the flaws of one method are often the strengths of another, and by combining methods, observers can achieve the best of each, while overcoming their unique differences” (p. 308). A second reason for combining both qualitative and quantitative methods concerned the differing perspectives between the researcher (outsider) and the participants (insiders). Quantitative methods, such as the survey I used, were oriented to my own specific concerns (in this case, learners’ attitudes towards mathematics). Alternatively, the more qualitative interviews and observations were oriented more towards the participants’ perspectives. Integrating both methods in one study combined the perspectives of both insiders and outsiders, and added strength to the findings. A third reason concerned the need for triangulation in the data. Students and teachers view teaching from different standpoints. Teaching is what teachers whereas teaching is done to students. As such, they have little say in, or control over, those decisions that affect teaching. Consequently, how students and teachers regard, and respond to, teaching differs markedly. In order to capture those different perspectives fully, I needed to obtain data from both groups in several different ways. Triangulation using different methods, and from different perspectives, allowed me to better capture the totality of the phenomenon of mathematics teaching. The final reason for using the particular mix of methods was that more quantitative data derived from the surveys of learners’ attitudes could be used to supplement and focus some of the later qualitative data collection. Not only did the survey data provide information that could not have been readily gained by a reliance on participant observation or semi-structured interviews, but it also enabled its use in the subsequent interviews with both teachers and learners. I now provide an overview of the three phases of data collection, and then consider each method and procedure in turn. For each, I describe its use and discuss some of the implications of using it. Overview Data collection fell into three separate phases. In the first phase (approximately 4 weeks long), I was concerned with developing some understanding of the culture and ethos of the college in general, and of the three introductory mathematics classes in particular. Here, I distributed a survey to all learners, interviewed all of the teachers within the mathematics department about their perceptions and beliefs, and conducted a series of informal discussions and ethnographic observations (ranging from 45 minutes to 2 hours) within the three introductory mathematics dasses. During this phase I conducted 8 interviews and 19 observations. The second stage of data collection focused more specifically on one particular section of the syllabus: that content area described in the course text as ‘introduction to Algebra.” The choice of the specific content area on which to focus was based on the following criteria, drawn from Löthman (1992): (a) It should consist of topics that are dealt with in the textbook, but can also be easily discussed in class. (b) It should contain problems where the content can be derived from the learners’ everyday experiences. (c) It should contain learning tasks that require written or oral products and 76 have alternative methods of solution. (d) It should be possible to discuss and analyze the solutions to any problem. The introductory courses all covered the content areas of geometry, percentages, and equations in addition to algebra; each of which would have met the above criteria. Algebra was finally chosen because of its expediency to the research. Course 050/051 was due to begin the algebra section during the earlier part of this phase of data collection; course 050 some three or four weeks later. This interval allowed me to complete this part of my data collection in one course before repeating the procedures in the next. In this second phase (8 weeks), I interviewed learners in, and the teachers of, the introductory classes several times. These interviews focused on the specific teaching processes in the lessons concerning algebra. I also observed every lesson that covered the algebra content and made extensive field-notes of my observations. Several of these lessons were also videotaped, and the video-recordings used as the basis of “stimulated recall” interviews with the class teachers. During this phase I conducted 17 interviews with students, 9 interviews with teachers, and 27 observations. In the third and final phase (4 weeks), I completed my observations of the three introductory classes, observing the concluding days of instruction in each course and their preparations for the end-of-term class examinations. I also interviewed all the students again, ascertaining their views on mathematics education in general, and the teaching in their particular course in detail. Finally, I held in-class group interviews with each of the three introductory courses. Group interviews were chosen as “a good way of getting insights [as] subjects can often stimulate each other to talk about topics” (Bogdan & Biklen, 1992, p. 100). Further, group interviews can involve those who feel either reluctant, or that they lack 77 sufficient authority, to speak (Lancy, 1993). In these group interviews I particularly encouraged responses from those individuals who had not been selected for the individual interview processes, or who had participated, but said little. During this third phase of the research, I conducted 8 observations, 3 interviews with teachers, and 15 interviews with the students (both individually and as a group). I now describe each data collection method in more detail. Survey I distributed a simple survey, by hand, to all the learners enrolled in the three sections of the course 050/051 in the Spring 1994 term. This survey gathered demographic data of gender, age, ethnic origin, and occupation; students’ attitudes towards mathematics; and an indication of willingness to further participate by agreeing to be interviewed (see Appendix 2). The section in the survey on attitudes towards mathematics was based on two instruments devised by Aiken (1974) that measure participants’ (a) enjoyment of mathematics, and (b) their perceived importance and relevance of mathematics to the individual and to society. Aiken’s instruments each use a 12 question, 5 point Likert-type scale. In a discussion of the internal reliability of the two scales, Aiken (1974) found 10 items on the “Enjoyment” scale that had a correlation coefficient between item scores and total scores above 0.75, and 10 items on the “Value” scale with a similar correlation coefficient above 0.60. These 20 items were then randomly mixed to produce the survey protocol I used. 78Interviews Teachers. I initially conducted semi-structured interviews with all of the teachers in the department during January 1994. All interviews lasted about 1 hour, were tape-recorded, and later transcribed for subsequent analysis. My interviews concerned teachers’ understandings of teaching processes, the factors they felt affected their teaching, and their reasons for choices they made in relation to specific and concrete situations they encountered in their teaching. (A copy of the interview protocol forms Appendix 3.) Some specific question areas concerned the planning of teaching, what problems teachers foresaw, their choice of course material and instructional strategies, and what they liked to know about the learners in their class. (Note: this, and all other interview protocols, were field-tested prior to data collection). I further interviewed the teachers of the three introductory classes three more times during the term. The second and third interviews--each lasting about 15-20 minutes--took place immediately before and immediately after the lessons concerning the specific content area on which I focused. The fourth interviews-- which lasted about 45 minutes each—took place during the last week of instruction. These interview protocols form Appendices 4 and 5. Briefly, they covered such issues as: (a) Before the lesson: Specific examples of how teachers chose lesson content and instructional strategies for the particular section of the syllabus, how the lesson fit into the overall course, what learner knowledge was considered a pre-requisite, and what learner problems were anticipated. (b) After the lesson: Specific examples of what changes were made to the lesson, what mathematics difficulties showed up, how those difficulties were dealt with, and what happened that was unexpected. (c) At the end of term: Specific examples of what changes were made to the teaching throughout the term, what mathematics difficulties showed up, how they were dealt with, and what happened that was unexpected. Stimulated recall. Certain lessons in this phase had also been videotaped and the teachers were asked to participate in stimulated recall interviews: a means of collecting teachers’ retrospective reports of their thought processes. Stimulated recall is a term used to describe a variety of interview techniques designed to gain access to others’ thoughts and decision-making. Typically, it involves audio-taping or videotaping participants’ behavior, such as counseling or teaching, in situ. Participants are then asked to listen to, or view, these recordings and describe their thought processes at the time of the behavior. It is assumed that the cues provided by the audiotape or videotape will enable participants to relive the episode to the extent of being able to provide, in retrospect, accurate verbal accounts of their original thought processes (Calderhead, 1981). Stimulated recall has largely been used in three different areas: with learners, with teachers, and with other practitioners (such as doctors or counselors) engaged in skilled behavior. It has taken slightly different forms in these three different contexts. Bloom (1953), who pioneered its use, was interested in learners’ thought processes during different learning situations. He played back audiotapes of lectures and discussions to university students and recorded their commentaries of their thoughts. These reported thoughts were later categorized according to their content and their relevance to the subject matter being studied. Kagan, Krathwohl, and Miller (1963) developed Interpersonal Process Recall: a form of stimulated recall using video-recordings as a means of increasing counselors’ awareness of interpersonal interactions during counseling interviews. Elstein, Kagan, Shulman, Jason, and Loupe (1972) used stimulated recall in research on clinical decision- making attempting to identify the thought processes of clinicians in simulated 80 diagnostic situations. Leithwood and others (1993) studied the problem-solving processes of school superintendents during their meetings with senior administrative staff. In classroom-based research, stimulated recall has also been used in a variety of ways to investigate the thought processes and decision-making of teachers while teaching. Keith (1988) demonstrates the diversity found in a group of selected studies. One factor which can influence the data collected by stimulated recall concerns the way in which participants are prepared for their commentary, and how they are instructed to comment. Calderhead (1981) notes that respondents can often identify, and hence comply with, the aims of the researcher. He also describes a study by McKay and Marland (1978) in which the researchers, although avoiding the imposition of any research model on the thoughts of teachers, provided detailed instructions before the videotaped lesson on the kinds of thoughts teachers were expected to recall. Calderhead claims that the provision of explicit instructions may have influenced the teaching itself, and the procedures may also have encouraged the teachers, to place a greater degree of rationality on their behavior. Mindful of these issues, in this study I merely asked the teachers to view the videotapes of their teaching and to comment on whatever they wanted. I gave them no instructions as to what to comment on, or how to comment. The teachers were given a remote control for the video playback machine and could stop the tape at any time they chose; they then commented (in whatever way they wanted) on the section of tape they had just viewed. These comments were audio-taped and later transcribed for subsequent analysis. Students. I also interviewed four adult students in each of the three classes that I selected (a total of 12 people). These people were interviewed three times: first, immediately after the first lesson of the algebra section; second, immediately after the end of the algebra section; and third, at the end of the term. The interview 81 protocols form Appendices 6 and 7. The first and second interviews—which lasted about 15-20 minutes each—dealt with the specific content of the observed lesson. Typical questions covered such areas as the content of the lesson, the work that students were asked to perform, any difficulties they encountered, and if they found this lesson typical of others. The third interviews--each lasting between 15-75 minutes--covered broader areas and included data from the preliminary survey of learners’ attitudes towards mathematics. These interviews covered such areas as learners’ experiences of mathematics education at school, and as an adult, their involvement in classroom activities during the course, and their attitudes towards the course content and teaching processes. Observations Direct observation in classrooms allowed me to study the teaching processes as they took place in their natural setting. I was able to gather such data as the form and content of verbal interaction between participants, non-verbal behavior, patterns of actions and non-action, and references to the textbook and other instructional material. Further, by acting as an “involved interpreter” I was able to “understand the events that occur, not merely record their occurrence” (Anderson & Burns, 1989, p.138). I observed the teaching in selected classes of all three sections of the introductory mathematics courses. Two sections of this course met twice each week (Tuesday/Thursday) and a combined section met four times each week (Monday - Thursday) for the full term (17 weeks). Each class was scheduled to run for two hours (12:30 - 2:30 PM). I could not, of course, observe all the classes in each section 82 (they met at the same time). I carried out three sets of observations. The first, early in the term, allowed me to familiarize myself with the classroom and college settings. Here I observed the whole lesson several times in each course (2 hours each). During this period I was able to introduce myself to the participants, gain their trust and cooperation, and collect some general data about the physical layouts of the buildings and classrooms, typical events in the college and in the mathematics classes, and details about the participants (such as their dress, their relations with others, and their behaviors). Data collected in this way informed the structuring of the later interviews. In the second set of observations, I focused specifically on one particular section of the syllabus (Introducing Algebra). Again I observed lessons in their entirety. I was able to observe how teachers introduced the chosen subject matter, and how they structured their lessons around it. Extensive field notes were taken for each observation. Events observed each time included: whether the lesson began on time, whether anyone was late, whether the teacher appeared to be following a lesson plan, the activities students were asked to perform, the students’ attentiveness and participation, what things appeared to concern the students, and the evaluation procedures used. Videotaping. I also videotaped some of these lessons using one camera to cover both the teacher and the students. In sum, 6 complete lessons were videotaped; three from each of courses 050 and 050/051. In videotaping the lessons I concentrated on supplementing my earlier ethnographic observations which had produced elaborate, though partial, field-notes. I tended to initially concentrate largely on the teachers as they moved around the classroom. However, as much of the lesson time was given over to individual student work, I could also focus the camera (with its in-built microphone) on the students’ actions and utterances. The videotaping of lessons therefore provided a richer and more detailed record than the 83 earlier note-taking. In particular, video-recording captured the details of small movements and oral comments as well as larger physical movements and differences in behavior. Some implications of using the video-recording equipment in the classroom concerned entry into the setting, the timing of taping segments, the visual point of focus, and the analysis of the data. I will deal with the data analysis issues in the next section of this chapter, but here, discuss each of the others in turn. I was initially concerned that appearing in the classroom with the video- recording equipment--which although light and portable was, nevertheless, obvious and intrusive--would be seen as overly disruptive to the participants. Consequently, I took the equipment into the classrooms for several periods beforehand and practiced filming--without making any recording--so that participants could become acclimatized to the change. I also fully explained my purposes in taping, so that participants were aware that my intention was to capture what the teaching of mathematics “looked like” rather than a focus on one particular individual or group of individuals. As participants became used to the presence of the camera in their classroom, and were never aware when I was actually recording and when I wasn’t, they tended, over time, to ignore the presence of the equipment and they did not appear to behave in specific ways for the camera. Consequently, I feel assured that the recordings I made are accurate renditions of episodes and situations in mathematics classrooms. With respect to the timing of recording, I had initially planned to sample the total available time, and record 10 minute segments of classroom behavior. After initially trying this, I changed this approach to recording the entire class lesson. The physical movement of the camera between taping and not taping proved more disruptive to the dass than I anticipated, and I felt that by taping the whole lesson I 84 was better able to capture data that matched my research questions. Visual focus was also a point of concern. Initially,I tended to concentrate the camera on the teacher, but later, as I grew more adept as a camera operator, I moved it more around the room and was able to focus on both the teacher and the students. I also tried to capture multiple points of reference: sometimes trying to capture what was seen from the students’ perspective (by taking one of the seats reserved for students), sometimes by setting the camera up to view the class as the teacher might see it, other times trying to move around the room focusing on odd segments of behavior or snatches of conversations. Document Collection The overall content of most ABE mathematics courses in British Columbia is determined largely by provincial curriculum guides. In addition, research has shown that the content and structure of the set textbooks also determines the specific dassroom content and structure of many mathematics courses in both British Columbia (Fans, 1992) and elsewhere (Höghielm, 1985; Romberg & Carpenter, 1986). Therefore, I gathered copies of documents relevant to the teaching of mathematics to the adults in the study. These included the Provincial Update on Adult Basic Education Articulation which contains generic course outlines of all ABE courses in British Columbia, copies of specific course outlines and syllabi, course handouts, examination papers, and mathematics textbooks used as course texts. In addition, I gathered any relevant material that pertained to the adult students’ uses of mathematics (e.g., the learner’s notebooks and completed homework assignments). Finally, I was able to collect copies of the students’ examination papers, after they had been marked by the teacher. Data Analysis Procedures 85 Data analysis is the process of bringing order, structure, and meaning to a mass of collected data. Within qualitative research, analysis consists of a search for general statements about relationships among categories of data. Much of this process consists of organizing the data, sorting and coding the initial data set, generating themes and categories, testing the emerging themes and concepts against the data, searching for alternative explanations, and writing the final report. In this study, analysis of the data was ongoing and iterative, and guided throughout by my conceptual framework. Data analysis fell into two phases. The first phase of data analysis began almost as immediately as data collection, where concepts that had been identified in my theoretical framework began to appear in examples of classroom practices in the early data. This concurrent process of data collection and analysis enabled me to identify themes and patterns of teaching processes from both observations of dassroom practices and from teachers’ and students’ comments. It also served as a check that sufficient and appropriately focused information was gathered before the completion of the data collection period. Initial analysis of the observational data involved searching through the data to obtain categories and themes that would portray an overall understanding of the framework of the teaching processes. Here, I paid particular attention to the roles that teachers adopted, and the tasks that they asked students to perform. During this early analysis, I also made analytic notes about specific points to pursue with the teachers in subsequent interviews. As the analysis proceeded, I modified my initial categories and themes to 86 better refine the portrayal of teaching processes and introduce the influences of frame factors. As more data were collected, particular interpretations and concepts could be refuted or confirmed by checking them against the most recently collected data. Once data collection had ceased, the second phase of data analysis began. Now, the concepts and categories that had been developed in the first phase formed the basis for a narrowing of focus, and the beginning of a process of abstracting and theorizing. I describe below the procedures I followed when analyzing different types of data. Survey Data The survey protocol contained 20 items that measured two different variables: enjoyment of, and perceived relevance of, mathematics. Respondents had indicated their attitude towards the questions on a 5-point Likert-type scale. Each person’s survey responses were quantified and transferred into computer data. Data relating to each variable were separated and total scores for each of the variables were produced for each respondent. Histograms of the distributions of total scores for each variable were produced, and their means and standard deviations calculated. Finally, a correlation of the scores for each person was calculated and a scattergram of the distribution was produced. Observational Data I had made extensive fieldnotes throughout my observations. These notes were transcribed shortly after each observation and used to generate and test conceptual themes and categories during this first phase of data analysis as 87 described above. During the second phase, when the earlier categories and themes had become more “fixed” into theories, I reread the entire corpus of fleidnotes looking for recurrent patterns and examples that might challenge or disprove those theories. Analysis of the video-recorded data proved more complicated. I found no guides or established procedures to help me. Existing guides to coding video data tended to require elaborate and intricate coding procedures and techniques, or focused instead on sophisticated micro-analysis of small segments of data. As my purpose was to capture a wider, classroom perspective, I adopted a less rigorous approach. I viewed the video-recordings repeatedly and took extensive field-notes during each viewing. Next, I used my conceptual framework and data obtained through other collection methods to discern some examples of the themes in the video data. I identified which segments of the tape corresponded to those themes and finally fully transcribed the audio tracks of these segments, noting specific behaviors in the margins of my transcribed notes. Interview Data Interview data were treated in much the same way as the observational fieldnotes. Each interview was tape-recorded and transcribed, and its transcription was checked against the recording, and amended where necessary for greater accuracy. The interview transcripts were then fully examined and used to search for, and refine, examples of themes and categories. I had considered returning copies of the interview transcripts to each interviewee for them to clarify certain points, and comment on my initial interpretations. This proved impossible. Data collection was intensive and little time was available (for either the researcher or the respondents) for the further interviews 88 that such clarification would have necessitated. Further, because so many interviews were being conducted, the interview-transcription-checking process was also intensive, and each cycle took upwards of 3 weeks before the interview transcripts could have been returned to respondents. As so much of the data collected was contextually specific, I was afraid that the lengthy time difference would influence the ability of the respondents to adequately remember the earlier situations. In any case, I was interviewing the same people repeatedly, and could more easily verify their earlier responses in subsequent interviews. Stimulated recall interviews. The data obtained through the stimulated recall interviews provided a means of collecting teachers’ retrospective reports of their thought processes. As such, it provided a further type of data to be organized and interpreted; additional categories could be developed to analyze the kinds of thoughts that teachers report. However, such categories also reflected my interests as the researcher and, hence, differed from the interpretive frameworks of teachers. I also identified several other issues relating to the interpretation of such data: Could stimulated recall reports be taken to reflect teachers’ real thoughts during teaching? Did teachers’ reasons for their behaviors constitute adequate explanations? Did the teachers censor or distort their thoughts in order to present themselves more favorably? I was unable to adequately answer these questions. However, my purpose was not to inquire too deeply into whether what the teachers said they were thinking corresponded exactly to what they were thinking at the time. I therefore resolved that the data I obtained through the stimulated recall interviews, together with the other data on teachers’ thinking and behavior, enabled me to gain some picture of the types of decisions that teachers made, and the ways that they described their own actions. These issues have also been identified in other discussions of the use of 89 stimulated recall interviews. Calderhead (1981) identifies two types of factors that determine the significance or status of stimulated recall data. First, several factors may influence the extent to which people recall and report their thoughts. For example, viewing a videotape of one of their lessons can be, for many teachers, an anxiety-provoking experience which may influence their recall or the extent to which they report it. Additionally, Bloom (1953) suggests that each individual perceives a unique set of visual clues which may or may not be recorded by the researcher. Fuller and Manning (1973) make a similar point in suggesting that teachers viewing videotapes of their lessons are perceiving the lesson from a different perspective--as observers rather than actors. This, they claim, can affect participants’ thinking. Additionally, they note that participants can be distracted by, for example, their physical characteristics. A second category of factors concern those areas of a person’s knowledge that have never been verbalized and may not be communicable in verbal form. Calderhead (1981) describes this as tacit knowledge which, although forming a large part of everyday cognitive ability, may have been developed through experience or trial and error, and cannot be verbalized during stimulated recall. For experienced teachers, much of their classroom behavior may be unthinking and automatic: they have long since forgotten the rationale for such behavior. It would seem unlikely that stimulated recall could reveal thoughts which occur at a low level of awareness or without any awareness whatsoever. Nisbett & Wilson (1977) argue that self- reporting of such higher-order cognitive processes is impossible and that data collected by stimulated recall is not the result of introspective awareness but the result of recalling of a priori causal theories which participants may regard as appropriate explanations for the outcome of their thoughts. In this way, they may not represent the actual decision-making processes involved. 90 Documents Curriculum and textbook materials were analyzed in two ways. First I analyzed the content and style of curriculum guides and textbooks to examine what they specifically said about mathematics education. For example, questions I considered included: What do they say are the goals of mathematics education? Do they indicate how these goals can be achieved? What kinds of knowledge are represented as being important? Do textbooks encourage classroom activities that build on learners’ experiences? Do they indicate how learners can use texts to help them learn? Selander (1990) describes a theory of pedagogic text analysis and offers methodological suggestions for such analysis. Briefly, his theory suggests that a proper understanding of textbook content involves the consideration of “background data: the social, political, and economic system in which a certain. ..written curriculum is situated;....the selection of facts and themes; the style of writing.. .and the combination of facts and explanations” (p. 147). This analysis helped me determine how much the curriculum guides and set texts are appropriate for adult learners. Second, I considered the place and role of the textbooks within each course. I gathered data on how textbooks were perceived and used by adult learners and their teachers. For example, I sought to discover: How do the teachers and learners use their textbooks? How are textbooks discussed within the class? How does the textbook fit in to the course syllabus/lesson format/classroom activity? Analysis of this data helped me understand the use of textbooks from the participants’ perspectives. Finally, I considered the students’ notebooks, homework assignments, and end-of-term examination papers. Careful inspection and analysis of these enabled me to determine those areas of mathematics which students found easy, and those 91 where they had difficulty. Further, I was able to determine the areas where students did not appear to understand the work they were being asked to perform, whether they appeared to be aware of any such conceptual lack, and, if so, how they dealt with it. Criteria of Soundness It is necessary in descriptions of research to discuss certain issues that normally fall into the categories of validity, reliability, and generalizability. These categories, however, are more appropriate to research conducted from a positivistic approach, and are usually considered inappropriate (at least in their generally accepted form) for more qualitative studies (Lincoln & Guba, 1985; Merriam, 1988). That is not to say that the issues are any less important, but that they are conceptualized and described somewhat differently in qualitative research. Qualitative research still concerns itself with “issues of a studies’ conceptualization and the ways in which data have been collected, analyzed, and interpreted” (Merriam, 1988, p. 165). I examine these aspects more fully in a methodological coda to this study (Appendix 22) by considering several general standards of judging research propounded by Hammersley (1992) and Howe and Eisenhart’s (1990). However, here, I identify and briefly describe these issues using Lincoln and Cuba’s (1985) notions of criteria of soundness--considered by them as appropriate constructs for judging qualitative inquiry. Their four criteria concern the issues of credibility, 92 transferability, dependability, and confirmability. Credibility Here, the goal is “to demonstrate that the inquiry was conducted in such a manner as to ensure that the subject was accurately identified and described” (Marshall & Rossman, 1989, p. 145). My study was based solely on data derived from extensive study in adult mathematics classrooms. Descriptions of that data were only considered within the parameters of those settings, the people in those settings, and the theoretical framework of this study. My initial findings were presented to the students and teachers of the classes I studied in both individual and group interview situations. Here, participants were able to examine some of my initial concepts and explain and darify their own perspectives on them, and on teaching in general. Further, the study was, throughout, conducted under the watchful guidance of my doctoral research committee. All stages of research conceptualization, data collection, analysis, and report writing were recounted to, and discussed with, them. In this way, I can verify that the study was conducted in a credible manner. Transferability This second criterion refers to the researcher’s ability to demonstrate that his findings can be transferred, or applied, to other contexts. I make few claims on the transferability of this research to other settings; the burden of applicability seems, to me at least, to rest with those who wish to make such a transfer. However, for those researchers who may wish to replicate such a study as this, I have, throughout my study, provided details of the theoretical parameters of my research, and my data collection and analysis procedures methods. In addition, my research involved 93 triangulation of multiple sources of data. I gathered data in several situations, from multiple informants, and chose data collection techniques that provided data from several sources to “corroborate, elaborate, or illuminate” the research (Marshall & Rossman, p. 146). Dependability Here the concern lies in accounting “for changing conditions in the phenomenon chosen for study” (Marshall & Rossman, p. 146). As the classrooms and settings I studied were constantly changing, I can make no sweeping claims for dependability. However, my continued presence in the chosen settings over a period of time allowed me to recognize and respond to any such changes. My data were gathered over the complete lifespan of the phenomenon--a term-length course—and I was able to observe the teaching in all of its different phases. Further, by constantly relating the data to the theoretical framework, I minimized any effects that changing conditions could have made on my study. Finally, I kept an “audit trail” of what data was gathered and how it was gathered and can therefore account for both the process and the product of my study. Confirmability Confirmability refers to the issue of whether the findings of the study can be confirmed by others, and not overly biased by the “natural subjectivity of the researcher” (Marshall & Rossman, p. 147). I attempted to do this in three ways: by ensuring that the study’s data and protocols are available for inspection, by constantly ensuring that all aspects of the study were related to the conceptual framework and the tenets of my chosen approach, and by ensuring that my methods of data collection were responsive, and sympathetic to, the study participants’ own situations. CHAPTER 4: CONSIDERING THE FRAIvfES In this chapter I animate my research model (see Figure 1) and consider the background elements to my study: those frame factors that can influence or limit teaching processes. I first examine the institutional framework: the contexts and settings in which mathematics education takes place. I discuss, in turn, the places where such education happens: the College, its departments, and its classrooms. Next, I turn to the people involved, and discuss their backgrounds, experiences, and attitudes. Finally, I consider the work that these people do, by examining the curricula of mathematics courses and the key role played by the set textbooks. Institutional Framework Classroom research often ignores the context or settings in which education takes place. Yet such contexts can often have a major influence on educational processes. The institutional framework of educational establishments, their administrative and physical structures, and their rules and procedures can all affect teaching processes and how those processes are perceived. In this section I discuss the first element of my model that can be seen as a frame factor: the institutional aspects of mathematics education. I first describe the College in terms of its physical and administrative structures, its overall course provision, and the services it offers to students. Next, I focus on the mathematics department itself and consider its goals and operation, the mathematics courses it offers, and some of its departmental policies. Third, I describe the mathematics 96 classrooms and how they are perceived by those who use them. Finally, I consider some of the ways that these settings affect the teaching and learning of mathematics. College Acton College (a pseudonym, abbreviated throughout as AC) is a publicly- funded post-secondary educational institution established in the 1960s from several existing adult education bodies and institutions. Its mission is to Provide adults with quality, student centered educational opportunities which promote and support lifelong learning, personal development, employability and responsible citizenship. The college welcomes all members of its culturally diverse and global community irrespective of ability or previous education, induding those encountering barriers to their full participation in society. (AC, 1994, p. 1) AC offers a wide variety of academic and vocational programs and courses to several thousand students each year. Four aspects of the College warrant discussion in terms of their apparent influence on teaching processes: its physical and organizational structures, its courses offerings, and its provision of student services. Physical Structure AC is a multi-storied concrete building situated on the side of a hill in an Acton suburb. Because of its hilly setting, the college can be entered from a variety of doorways on four of its levels. The doorways open onto a series of interlocking corridors that house all the instructional and administrative facilities. On the first floor, these corridors are all linked by outdoor patios where potted trees and shrubs are interspersed with bench seats. This area is close to the college cafeteria and is a popular place for students to take a break between classes or pause to smoke, eat their lunches, or chat with friends. Within the building, classrooms and offices for each program area are grouped together. For example, the several automotive training programs are all located in the basement. The Adult Basic Education division (which includes the mathematics department) is housed on the third floor. The internal layout of the college, although making access and movement easy, is confusing. There are few signs indicating which floor one is currently on or giving directions of where one may need to go. Also, the room numbering system is complex, and one can often be stopped by bewildered students wandering the corridors searching for a missing room. Despite this, the college has a friendly, if somewhat impersonal feel. It seems designed to be largely functional: a place to be used but not to be especially visited. “It’s not a hangout sort of a place,” as one student put it (L.3.1O). The college is a place for the purposeful. If you know what you’re doing (and where you should be doing it), then the physical structure seems designed to help you; if you don’t (or are unsure), then the structure is confusing. Organizational Structure Academically, AC is organized into three main divisions: Adult Basic Education (which indudes the Mathematics Department), Career, and English as a Second Language. The first offers programs in academic and vocational upgrading and Adult Special Education. Instruction is offered in courses from a basic literacy level through to BC Provincial Adult Secondary School Completion (Grade 12). Students can attend courses on either a full-time (registered for 20 or more hours per week) or part-time basis, and, in most areas, can opt for either classroom-based education or an individualized, self-paced program of study. AC’s organizational structure naturally affects teachers more than it does 98 students. For example, Departments and Divisions compete for limited financial resources which determine staffing levels and the scope of course provision. Despite this, however, teachers in the mathematics department thought that AC provided a good working situation. As one teacher put it, “normally, [it’s] a good place to work.. . .The environment is good. . . also the colleagues. We are like [a] big family.. .we just cooperate with each other” (T.1.4). Overall, the college places few administrative restrictions on its teachers. There are no guidelines about how to teach, for example, although the college does expect its faculty to be familiar with the different learning needs and styles of adults. To aid this, it requires an “Instructor’s Diploma” of its newer faculty and provides (voluntary) refresher “instructional development workshops.” However, all the teachers in the mathematics department have either sufficient service to be exempt from the compulsory requirement or have a BC Teachers Certificate (which also exempts them). It offers refresher “instructional development workshops,” but only one teacher had participated, and he found it of little use: “I suppose it was all right.. .not much about math, though” (T.1.3). The only administrative restriction mentioned by teachers concerned the recommended minimum class size. (Each class is supposed to have fifteen registered students by the third week of dasses or it is canceled.) As the department head put it, “If there’s enough. . .to run the class, once the term is underway [then] the college stays pretty much out of it” (T.1.1). Other teachers agreed, “There’s supposed to be a hard and fast rule about [dass size] but often it depends on the department and what other [courses] are being offered... . [For instance], could students move to another section if this [class] was cut?” (T.1.8). Teachers reported that the college administrative organization did not overly influence their day-to-day teaching. “Everything’s changing all the time,” said one teacher, “although we still have a lot of freedom. The upper level [of administration] 99 is so busy with all that’s happening outside of. . .instruction [that] we’re left alone” (T.4.2). Teachers regarded their isolation as advantageous. “I don’t think anyone should be telling us how to teach,” said one, “It’s our job to know that” (T.1.8). Indeed, teachers appeared to relish being left alone and tried hard to maintain distance between themselves and the college administration. Despite the current budgetary uncertainties, teachers appreciated what they called the “lower stress levels” of college teaching as compared to high-school teaching. One teacher listed the benefits of teaching in a college: “You don’t have to deal with parents, [and] attendance isn’t an issue. If a student doesn’t want to come to class, we try to find out why.. .but it’s not essential [that we do]” (T.1.1). “You don’t have school district rules to contend with [so] it’s much more relaxed here,” agreed a second teacher. “You don’t have to plan topics to the nth degree, there are few discipline problems [and] you don’t need any classroom survival skills” (T.1.3). In sum, we can see that teachers are relatively free to choose many of their classroom teaching processes without much interference from the college administration. Indeed, they prefer it that way. Even those factors (such as minimum class size) that could affect teaching are “negotiable.” If anything, administrative factors served to encourage teachers to devote less time to teaching. As one teacher put it, “If there’s any danger that this class may be [cut]. . . then I’m going to be a little bit cautious in terms of how much preparation I do” (T.1.8). Course Provision The mathematics courses offered by the college form part of a system of requirements for completion of the ABE Provincial Diploma. This Diploma, sanctioned by the BC Government’s Ministry of Advanced Education and Job Training in 1986, is awarded to any student who completes the requirements for 100 secondary school graduation as laid out in the Provincial ABE Framework (see Appendix 8). The Diploma is recognized by colleges and universities throughout the province as an official credential for entry into university studies (Ministry of Advanced Education, Training, and Technology, 1992). The Diploma (and the ABE Framework) is overseen by a ministerial committee on Adult Basic Education comprised of representatives of those institutions that provide ABE courses throughout the province. For students who study mathematics at AC, three aspects of course provision seem to affect them most: the admissions procedures, the financial cost of the courses, and the attendance requirements. Admissions. As a “post-secondary institution committed to educating the adult learner,” AC normally only accepts Canadian or landed immigrant students who are 18 years of age or older, or who are aged between 15 and 17 with no school- attendance in the past year. However, on occasions, “a small number of international students [are accepted] on a cost recovery basis” (AC, 1993, p. 8). Students who apply for admission must attend a pre-registration interview and “present details of their previous educational attainments” (p. 8). If prospective students have been away from an educational setting for three or more years, they are normally required to take (at cost) an assessment examination to determine their “appropriate placement level” for each of their chosen subjects (p. 8). Further, AC requires that its students must have “adequate English language skills to understand class lectures, take part in class discussions, and complete written assignments” (p. 7). Consequently, all students whose first language is not English must take an English Language Proficiency assessment before they are admitted. Hence, prospective students intending to study several subjects (as most do) are faced with an array of examinations and charges before their courses start. The College admissions procedures also seem unduly complicated. “You need 101 a degree just to get in here,” said one teacher (T.1.7). In general, teachers thought that the college restrictions could be unduly harsh on students, given their often “unconventional” lives. One teacher spoke about such difficulties: I think the whole assessment procedure is a little too off-putting for some of our students. It’s more of a hurdle than a help. We put too much credibility on assessment tests. Of course, you have to be careful: we do give [more] credibility to presented academic credentials than discussed academic credentials. But getting in can be a challenge in itself. The assessment costs $15 which can be.. . daunting for unemployed, broke people. If [students] are not sure what they want to do.. .or are just shopping around, then $15 for math and $15 for English or whatever can frighten them away. (T.1.3) The admissions procedure certainly served to weed out the uncommitted. For anyone aged over 20 who wished to study full-time, the application procedure alone could cost almost $100. Additionally, acceptance into the college depends entirely on students’ past academic performance or their achievement on subject-based “assessment” examinations. Those adult learners who pass the initial examinations and are admitted to the College have already been prepared to equate success and achievement with standardized and impersonal forms of assessment. Finance. Tuition for each course costs a further $90. In addition, students are expected to purchase the textbooks and any required extra material. (For example, students in one of the introductory mathematics classes are required to buy a $15 geometrical construction set.) Many students were not paying for courses themselves. Several indicated that they were receiving financial aid, either from family members or from government grants or loans. “I could never afford to pay for this myself,” said one publicly-funded student, dismayed at the costs he did incur, “I have to buy the books as it is” (L.3.4). Obviously, the combined cost of the admissions procedure and the courses can influence the attitudes of students who enroll. Regardless of who is paying, enrolling for courses represents a sizable investment. Many students feel pressured to attend all the sessions of those courses in which they enrolled, and try to pass 102 these courses, if only to realize their investment. One student described: “I feel like this is my last chance.. . .1 totally wasted my time at school. . . and now I’m getting money for a second shot... .I’m really lucky.. .many folks don’t get this opportunity.. .so I’d better not blow it” (L.3.2). Attendance. Once admitted to the college and enrolled in courses, students are expected to attend and participate in all of its sessions. “Successful completion of and progress through courses/programs is based on.. .class assignments, examinations, participation and attendance” (AC, 1993, p. 16). Indeed, students who fail to attend the first three classes of a course or who do not regularly attend throughout the term are asked to withdraw. However, the college recognizes that it has a responsibility to assist students in overcoming problems that affect their performance and attendance. It makes such assistance principally available from either the academic department involved (for instructional and learning problems) or the college’s counseling service (for students’ vocational and personal concerns). Student Services The college provides a variety of services to “help students with their studies and assist them in completing their goals and objectives” (AC, 1993, p. 20). These include an Assessment Centre, Bookstore, Cafeteria, Counseling Service, Daycare, Health Service, a Learning Centre, and Library. For mathematics assistance, students identified the Assessment Centre, Counseling Service, and the Learning Centre as most useful. The Assessment Centre offers (for a fee) assessments of students’ abilities in reading, writing, mathematics, typing, accountancy, or English language. These tests are designed to “help students determine their appropriate placement levels” and the results “may be used in lieu of school transcripts for admission to courses” (AC, 1993, 103 p. 20). The mathematics assessment is scheduled to take 1 hour and determines students’ skills in basic arithmetic and algebra. Most students in the introductory mathematics courses had taken the assessment as part of the admissions process and found it useful. “It wasn’t too hard,” said one. “It showed where I needed help and told me which was the right course” (L.3.3). Most of these students had accepted the results of their assessments and enrolled in the courses at the suggested level. A couple of students, however, although having been “placed” in higher level courses, wanted to re-study the more basic material. “I’ve done all this before,” explained one student, “when I was at school.. . . But I thought that I’d better go over it again.. .to get my hand in” (L.3.1). The Counseling Service provides a range of services: educational, career and vocational counseling; crisis, stress management, test and mathematics anxiety intervention; instruction in life skills; and services for disabled and international students. Most counseling is provided on an appointment-only basis, although the Service also provides a limited drop-in and emergency facility. The Counseling Service also operates a self-help resource centre for current and prospective students to obtain information about the college and its facilities and services for students. The Learning Centre is a drop-in “learning support service.. .provid[ingl students with assistance with their studies” (AC, 1993, p. 24). Its services include “one-on-one tutoring, specialized small group workshops, audio tapes and listening carrels, computer software, study areas, course materials, makeup test services, and course-related worksheets for a variety of subjects” (including mathematics) (AC, 1993, p. 25). The Centre’s regular staff includes a mathematics tutor; in addition, instructors from the mathematics department are each scheduled to provide two hours extra tutoring throughout the week. Finally, the learning centre is one of the few quiet places at AC in which students can study. Teachers were very positive about the student services provided by the 104 College, and saw them as one of the main benefits for students who choose to study at AC. “We offer as much as we can.. .certainly more than other [colleges]” said one teacher. “Funding for poor students, a counseling centre for those with learning disabilities, a learning centre which is like a study hall. There’s a good math tutor there” (T.1.4). “Often, our students do not really know much about how to learn,” said another teacher, “so the counseling centre and the other fadlities can be a real benefit” (T.3.2). Teachers also saw their own work as fitting in to this network of facilities. As one put it, We help students get what they need... .We do more than just teach. There are times when students come and try to sort out their lives with us. All sorts, from people who want to know what computer to buy, to people who are crying on your shoulder because they have larger problems. (T.1.2) Most students, in contrast, although aware of the College facilities, found little reason to use them. As one put it, “I find I don’t need them at the moment. I really just stay here as short a time as [possible]. I go home and do my work... .Maybe next term when I’m taking more courses” (L.3.2). Another claimed that, “the library’s OK, but. . .1 don’t use it much for math though... .The teacher said not to bother with other books in case we got confused (L.3.6). Perhaps to counteract the rather impersonal aspects of the College, some students appreciated the opportunities for more human problem-solving help: There’s a guy at the Learning Centre. . .he’s really helpful. I can go to him and he’ll sit down and show me a way of doing it that’s really simple. Then he lets me sit there and practice ‘til I’ve got it right... .He makes it look really easy. He does it to everybody.. .even the guys that are taking calculus. (L.3.4) Overall, students compared AC favorably to their high-schools: “AC is much better.. .1 learn faster” (L.1.3). Another agreed: “That’s why I came here,” she said, A lot of people told me it’s good here, and they were right.. . .1 have a friend who took Math 11 at high-school and he took it again here, and he says that he learned a lot more here.. .in a quicker period as well. (T.3.4) Students particularly appredated the adult atmosphere at AC. ‘The teachers 105 [here] treat you like human beings,” said one student. Like when I was at school, there was gangs and that.. . .1 mean, I was never part of that, but it’s going on around you. That does something to your learning. Like, in school I was OK in math.. .1 was a “C+.” But since coming here, I’ve been an “A.” So.. .it’s the same work, but I’m learning faster. I’m in a better environment. I can concentrate more.. .it helps me out having people I can relate to around me. (L.1.3) Departments The Adult Mathematics department, together with the departments of Science, Humanities, Business, and Computer Studies are organized into an administrative section called College Foundations (CF), all part of the ABE Division. In this section students can only take semester-long, classroom-based courses. (There is a self-study program for students who wish to study on an individualized, self- study basis.) The goal of the CF mathematics department is to “enable the adult student to study mathematics in an environment where the student can make progress and experience success” (AC, 1993, p. 58). Teachers in the department assumed the responsibility for setting an appropriate climate to meet that goal. Teachers described how such climate-setting involved not only their own dassroom behavior but also the interpersonal relationships within the department. “People are very eager to help each other,” said one teacher. ‘There’s no professional jealousy. . .our personalities just sort of mesh” (T.1.7). “It all works for the students,” explained a second. “If [teachers] are getting on together, then I’m not distracted and can focus on my teaching” (T.3.3). Teachers also place expectations and responsibilities on students so that they 106 can “make progress and experience success.” Apart from the general college guidelines on appropriate student behavior, the department requires that The new student must enter the course appropriate to his/her background. Therefore when the student has not taken a mathematics course during the prior three years, an assessment is recommended. ESL students must be at the Upper Intermediate Level of English or higher. (AC, 1993, p. 58) Thus, prospective students are obliged to fit into a pre-existing structure, irrespective of their wishes. The determination of which course is “appropriate to his/her background” rests with the department; students cannot enroll for whatever courses they choose. In this system, assessment of prior knowledge is key, both for those who have been away from school for three years (presumably most adults), and for those whose first language is one other than English. Nearly all of the students enrolled in the introductory mathematics classes fall into these categories. The mathematics department has a good reputation among students and teachers. “You get a lot of support,” explained one student. “People are tempted to wallow.. .but here you get a push.. .and there are deadlines you have to keep” (L.3.1). Some students, fearful of repeating their earlier bad experiences in math classrooms, had thought about enrolling in the self-paced program: “When I came in for my initial interview,” said one student, “the teacher suggested that I give the classroom- based course a try. I’m glad I did.. .it’s not as bad as I thought. . . and I need the discipline. . . someone to hold my nose to the grindstone” (L.3.6). “You’re forced to pay attention,” said a third student. “The teacher goes so fast that you can’t afford to miss anything” (L.3.3). Teachers identified how the mathematics department also had a good reputation with other provincial colleges. “It’s been built up over the past number of years,” said one teacher, “we hear from BCIT [and] UBC.. .that our students do really well in their courses” (T.1.6). “[BCIT] are very pleased with our program,” agreed the department head. “They tend to send, or encourage students to come here to get the 107 pre-requisites” (T.1.1). When asked why she thought that was, she continued, I think it’s a number of things.. . .They recognize that we have an adult focus... and we teach the sort of things that they want.. . .We meet with them. . . and they say that they notice that their students are really weak in particular areas, so we take that into account when we’re planning our courses.. . .The amount of depth in our courses is sometimes determined by what the receiving institutions want.. . .1 mean, we want our students to be successful. (T.1.1) Notice that, in terms of reputation, students mention comments made by other students; the teachers refer to comments made by colleagues at other participating institutions. Courses AC offers mathematics courses at three levels corresponding to academic grades 10- 12 (see Appendix 9). Within each level, there are three courses: two half- courses and one “double-block” course which combines the curricula of the two half- courses. The College calendar cautions that, “double-block classes are very intensive; they are not recommended for students who have difficulty with mathematics or who have an unduly heavy workload” (AC, 1993, p. 59). My research focused on the three courses offered at the most basic (Grade 10) level: 050, 051, and 050/051. The college calendar describes them briefly: Mathematics 050 and 051 are ABE Intermediate level mathematics courses designed for the student who has never studied academic mathematics before or who is lacking a good foundation in basic algebra skills. The content includes: a review of basic math skills, a study of measurement, and introductory algebra and geometry. Mathematics 050 must be taken before Mathematics 051. (AC, 1993, p. 59) Further written information about each course is given to students during the first meetings of each course. Usually, this information concerns the instructor’s name and phone number, a list of the set books and extra material required, the 108 meeting dates for course sessions and what topics will be covered on what days, and the course assessment guidelines. Appendices 10- 12 give details on the three courses on which my research focused. Notice that the handouts are all quite - they follow the same layout and contain the same sorts of information described in similar ways. (Indeed, the handouts describing the information for the courses offered by the department follow the same structure.) Curiously, the neat regularity of this schedule allows for considerable flexibility amongst staff. The teaching of courses is shared among all eight of the full- time mathematics teachers in the department. One teacher explained how scheduling decisions were made: “The department head.. . sends round a schedule. . .usually it’s what we did last term, and teachers can make any comments or requests on it. Then, if there are objections, conflicts. . . they can be discussed at a department meeting” (T.1.4). “The day instruction runs from 8:30 am to 4:30 pm,” said one teacher. “We can usually pick the times we want to teach.. .but not which classes... .We give our preferences but it’s the head that decides” (T.1.8). However, according to some teachers, times at which specific courses are offered vary little from term to term. One teacher explained that Math 12 and 11, they’re always offered early... .Some of that has to do with the science classes.... Students taking those levels of math are also taking the physics, and biology and chemistry, so that you have to make it flexible for them. So, 050 and 051 get offered later, usually at 12:30. They’re never offered early. (T.1.7) Teachers appreciated this opportunity to influence their teaching schedules and supported the department policy of rotating teachers among its classes: To make teaching more efficient, we have to constantly change [and] revise our curriculum. It’s the changing that makes teaching interesting and challenging and [keeps] us constantly awake. . instead of teaching the same thing. Can you imagine talking about sine and cosine, sine and cosine, sine and cosine all the time? (T.1.4) Although teachers never expressed preferences for teaching certain course 109 levels, they did choose particular days or time slots to teach. So, as the times and days on which the courses were offered changed little from term to term, teachers could effectively choose which levels they wanted to teach. I also observed a departmental staff meeting where teachers chose which courses to offer during the coming summer session, and the times and days on which to offer them. Vacation decisions had already been agreed, and final decisions about course offerings were taken on the basis of who would be taking vacation (and, hence, who would be left to work) during which periods. Here, decisions were taken, not to fit the department policy of rotation, but to “fit in” with teachers’ personal arrangements. ‘We try to rotate as much as possible,” said one teacher, “but it depends on the schedule... whether everyone’s schedule fits in” (T.1.4). Policies Other than the rotation of teachers, the department made few policies that affected how courses were taught. The only policy consistently mentioned by teachers concerned the standardized term-end examinations. “Each grade. . .has one test,” explained one teacher. “One of us has responsibility for designing the test for all classes at that level. Then all students in that grade.. .take a general test” (T.1.5). This procedure serves to impose added conformity on teaching in each class which teachers claimed to find reassuring: “It stops people doing their own thing. If I’m not preparing the test for my class, I have to make sure that I cover all the areas properly” (T.1 .7). A second policy which somewhat affected the mathematics department concerned the end of term evaluation of courses by students. Students have a right (under the College regulations) to evaluate each course, and the mathematics department has devised an evaluation form for such a use (see Appendix 13). The 110 department head explained how it should be distributed: The departmental assistant [should] give it out in each class before the end of term... so that [teachers] have a chance to respond to some of the things that the students say. Its a kind of A-B-C-D-E scale and then there are some spaces for written responses. It’s completely anonymous, so the students can say what they like. (Field Notes, 931220) However, I noticed that this form was never distributed (nor even mentioned) in the three classes I observed. Further, I never observed any time allocated to an in- class discussion of evaluation. “It’s a real political issue,” explained one teacher. “Some people are scared stiff of it.. .so it’s never gotten off the ground” (T.1.2). Others had different explanations: “I’d like to do [the evaluation],” said one teacher, ‘but at the end of term there’s not enough time” (T.1.2). “I think that [it] would be too intimidating,” said another teacher, referring to in-class evaluations. “Students would feel put on the spot” (T.3.3). Evaluation was commonly forgotten. As one teacher put it, “Some [student] will say something.. . such as ‘Why do we need to do all of this homework?’ and I’ll think’Maybe I should discuss that with the class.’ But usually I don’t” (T.1.7). However, most teachers, when asked, merely shrugged. “It’s the way we do things, I guess,” said one. “If one [of us] doesn’t do it, then there’s no pressure on the rest of us” (T.4.2). Classrooms AC’s mathematics classrooms are situated close to each other (and the teachers’ offices) on the third floor of the college. Classrooms for the music, science, and ESL departments are close by. Indeed, in the corridor outside the classrooms one is constantly aware of the proximity of other departments. Trumpet solos or operatic scales can be regularly heard, the science labs emit curious chemical odors, and ESL students chatter to each other ceaselessly in other languages. The department has sole use of two of their classrooms; the third is shared with neighboring departments. Each classroom is similar to the others: they are each about lOm x 7m in size, well-lit, and with centrally-controlled heat and air- conditioning. Each is linoleum-floored, and contains 10 or so rectangular (about 2m x Im) wooden tables and 25-30 wooden chairs laid out in rows facing the teacher’s desk and the blackboard. The length of one wall in each room is taken up entirely with desk-height windows that look out onto a concrete walkway and other classroom windows beyond. Two of the rooms also contain an overhead projector and screen set up in one corner next to the blackboard. The rooms look and feel like traditional college classrooms: anonymous, businesslike, formal. Two of the classrooms have notice boards carrying a variety of posters. These have details of the College’s health and counseling services, a notice advertising a long-past college event, the library opening hours, what to do if there’s a fire, and advertisements for credit cards and magazine subscriptions. Only a few deal with mathematics. Of these, most seem to have been produced by textbook publishers and assure the reader that “MATH IS FUN” or detail “Six Steps to Problem Solving.” A poster in one room has a large photograph of Albert Einstein over the caption, “Do not worry about your difficulties in mathematics. I can assure you that mine are greater.” Another headed “MATHPHOBIA CAN COST YOU A CAREER!” lists jobs that people are supposedly unable to hold if they have mathphobia: “statistician, physicist, pilot, dental technician, accountant, surveyor, welder, chemist.” Although these posters are presumably displayed for the benefit of students, I never saw any student stop to read them; nor were they ever referred to by the teachers. Indeed, by their yellowish tinge and curled corners, the posters all looked as if they hadn’t been 112 moved for some time. Those who used the rooms differed in their reactions towards the room layout. Teachers were aware of how classroom layout could affect learning and teaching. “Some of the rooms are better set up for a kind of interactive approach,” said one teacher. “Those with hexagonal tables are great for getting small-group work going. Some of our classrooms [have] just rectangular tables and it’s much more difficult to do anything [other] than pairs” (T.1.1). Another teacher said that he preferred to have students working together but that the table layout did not encourage collaborative working: “We do our best with the rectangular tables, but the students know that [even though they sit together] they’re not doing cooperative groupwork. I mean they don’t get the same grade. . .so they’re not that committed to each other” (T.1.6). I asked the teachers if they ever changed the table layout. [Another teacher] and I tried that one term.. .we spent quite a bit of time rearranging all the desks. The next morning the students had come in and rearranged them all the way they had been before. They obviously didn’t want to work in small groups, I guess. (T.3.1) Students, conversely, were largely unconcerned about room layout; their only comments about the rooms concerned their size. “I find.., they’re a bit too large,” complained one student, “I sometimes can’t hear what other people are saying” (L.3.3). A second student explained that, “classes always start out large.. . then get whittled down... .So, you’ve got this huge room with only eight people in it” (L.3.1). I was interested to note that the layout of the two math-only rooms did not change throughout the whole term, whereas in the shared room, the layout changed regularly. Sometimes the tables would be in rows as in the other rooms, sometimes in a hollow U-shape, once in a solid block of tables, and once with the tables put together in pairs for groupwork. During my observations in this room, neither teachers nor students ever mentioned the change in the room layout. “Don’t make no difference,” said one student when asked directly, “math is math” (L.3.9). “I do like to see everyone’s face,” said a second student, “but it’s really only important in a 113 classroom where you’re going to discuss things.. .like psychology or English” (L.3.1 1). I noticed that most students preferred to pick one seat and keep it throughout the term. “I like to sit near at the front,” said one student, “1 can concentrate better if there’s not too many distractions... .You know, people coming and going.. . .If you get that big guy, Harry, in front of you, you can’t see a thing” (L.3.2). “I’m supposed to wear glasses,” admitted another student, “so I like to sit as close to the board as I can” (L.3.4). Being close to the board was clearly important: “I don’t really care where I sit, so long as I can get a good look at the board. [The teacher] likes to write all over it, so you need to be able to see even the bottom corners” (L.3.6). Only one student said that it shouldn’t matter where people sit. “We’re there to learn,” he said, “we shouldn’t try to get as close to the board as possible” (L.3.4). Experiences of Students and Teachers Studies of teaching are often limited by focusing either solely on classroom practices and dynamics or solely on the backgrounds and experiences of learners or teachers. However, in reality, these two areas are interrelated and interact to affect classroom practices and influence interpretations of those experiences. In this section, I consider the second element of my model: the experiences of students and teachers involved in three introductory-level mathematics classes. I first discuss details of students’ backgrounds and experiences, their attitudes towards mathematics education, and their reasons for enrolling in a mathematics course. Next, I consider certain characteristics of all eight teachers within the 114 mathematics department. Although my research focused most closely on three classes, there are two reasons for obtaining data from all the teachers. First, the mathematics department seeks to arrange a teacher rotation to ensure that all teachers teach introductory level courses. Second, at the outset of data collection, no decisions had been made about which courses would recruit enough students to proceed, or which teachers would be teaching which courses. Students The beginning of the term is filled with uncertainty. Some students enroll in courses and don’t show up. Some come to classes for only a few sessions and then leave; others attend without ever having registered. College policy recommends that a minimum number of 15 students be registered in each course by the third week of classes or the course is canceled. Consequently, during the early part of the Spring term, there was considerable anxiety within the department that one or more courses would not be allowed to proceed. However, by the College’s deadline in mid- January, 37 students had enrolled in the three sections of the introductory-level mathematics courses. During the fourth week of instruction I administered a survey protocol (Appendix 2) to the 32 students who were attendance that week. Their self descriptions of basic demographic data are summarized in Appendix 14. Students’ Backgrounds Students in this study have a variety of backgrounds in terms of their gender, age, ethnicity, and occupation. The literature identifies several other features of students’ backgrounds which can influence teaching: students’ English language fluency, other courses being taken, and students’ previous experiences of 115 mathematics education. I now discuss each of these background features in turn. Gender. Of the 32 students surveyed, 20 were male and 12 female. The gender balance remained the same throughout the term even though some students dropped ou of classes and others joined. When the survey was re-administered in the final two weeks of term, the proportion was unchanged. These figures represent all the students enrolled in the three introductory- level classes. When each course is considered separately, a difference in the gender balance appears. In the two classes that met only twice per week (050 and 051) the gender balance was almost even, while the four-day per week “double-block” class (050/051) contained only male students. “Pretty normal,” said the double-block class teacher, “usually [a] lot more men. [They] have more time” (T.1.5). Age. Students’ ages ranged from 18 to 45 years with a mean of slightly over 24 years. The distribution of ages is shown in Figure 2. (Note: only 30 of the 32 students gave their ages in the demographic survey.) 116 Histogram of Xi: age 5 4 Z3 o 3 P. . age Figure 2: Distribution of Student’s Ages There was no marked difference between the range of ages of the male and female students. Further, with regard to the distribution of their students’ ages, the three classes were similar: each dass contained one or two students aged 19 years or younger, five or six students in their 20s, and two or three aged 30 years or over. Most students said that they liked the range of ages in their classes. “You don’t feel so stupid,” said one, “when you see guys in their 40s in the class” (L.3.6). Ethnicity and English language ability. Only half of the students were either Canadian or part-Canadian. The others identified themselves as either First Nations members (3) or immigrants from Europe (5), Asia (4), Africa (3) or Central America (1). Most (23) students spoke English as their first language; the exceptions were the non-European immigrants, all of whom were also enrolled in college English classes. Language ability was key for many students. Although the language used in the mathematics classes was not seen as “hard”, it occasionally contained uncommon words, which, if not understood, could delay students’ understanding of the mathematics. Several of the immigrant students said that, although they had 117 previously studied mathematics in their native countries, they were re-taking Grade 10 mathematics in order to gain further familiarity with the English language. “They wanted to put me in a higher grade,” said one Chinese student when referring to her initial placement interview, “but I said I want to go over [grade 101 again--to revise...and to practice with the words” (L.1.1). Few of the foreign students identified that they had much difficulty with spoken or written language in the math class. “Sometimes the teacher goes a little fast,” said one, “but I can read it later in the book” (L.3.4). A couple of students said that they carried dictionaries with them to the mathematics class, and would occasionally look up unfamiliar words. However, this remained a private activity--often carried out under the desk (and out of sight of the teacher). Language difficulties were never addressed publicly in class, although fellow students could often determine who was struggling. “I think a couple of the non- English speakers are having difficulty,” said one Canadian student. “They sit up at the front and you can tell [that] it’s not clicking” (L.2.1). Language ability and use was also an issue for native English speakers. Several students commented on the way language was used by teachers: “Some of the teachers talk to you like you’re a 12-year-old. Enunciating everything. ..like you’re stupid” (L.3.2). Another English-speaking student, describing a similar experience, explained: One of my classmates said near the middle of the term, they’re a bit peeved because she [the teacher] seemed to speak down to them all the time, but she’s not really speaking down, and now this classmate has now said, “Oh, I’m glad she does do that, because it means she makes sure that you know.” She [takes] great pains to make sure you understand. . .almost to the point of annoyance. But I don’t mind that.. . she’s just trying to help. (L.3.1) Occupation and student status. Sixteen students said that they worked at least part-time; the others were either full-time students or unemployed. Of the 16 workers, the eight men had jobs as cook, musician, taxi driver, clerk/cashier, 118 maintenance worker, and night-watcher. The eight women had jobs as arts administrator, cook, waitress, clerk, hostess, and cashier. Several students spoke about the experience of being working students: I work about 20 hours a week in a store. I finish here [college] at quarter to three and start work at three. Then work until 8 or 9 at night. Get home about 9:30; then I spend one hour to do my homework. (L.3.8) I’m a taxi-driver...and I usually work late afternoons [so that] I can come to class during the day. That’s not so bad.. .but it’s the homework. I try to do some while I’m at work.. .but usually I have to get up early to do it. When you’ve not gotten in until 2 or 3 am.. .It’s hard. (L.3.11) I work at [a record store] stacking CDs. I’m only part-time so it’s usually it’s about 10 hours per week. Normally, they’re very good about letting me have time off to come to class...sometimes I have to switch shifts with other staff. A couple of weeks ago there was stocktaking and that was hell. It was very difficult at school —I had a lot of papers to hand in for my other courses. ..and we had a math test, so it was very stressful. I couldn’t miss any work in case I got laid off, so I had to miss a couple of classes. I’m just about caught up now, but it was pretty difficult. (L.3.1) Other students who worked found that they needed to alter their working arrangements to fit their school timetable: “Before I did the full-time school and full- time work. After a while I found it’s too much work. Now I just [work] on weekends, in a restaurant” (L.1.1). Only one student indicated that her employer gave her time off work to attend college: “I work in a glass shop.. .auto glass and window glass. My boss, he gives me time off.. .no pay mind you.. .but I tell him when I want to come and he lets me off early or changes my shifts” (L.3.6). Of the 16 non-working students, 13 attended college full-time. They either lived at home with (and were financially supported by) family members, had built up sufficient savings to fund their time at the college, or funded their studies by student loans or government grants. One young student described his financial position: I get a grant to cover my fees. I could never afford all this by myself... .Then there’s the books.. .Social Assistance has to pay for that. You have to be 19, but they will help for education. It’s not that I want to go on it, but I’m going to 119 have to to survive, to have an income to pay rent.. ..My Mom’s letting me off with the rent right now because I’m not of age, but when I’m of age she’s going to expect it... .I’ll keep getting the funding as long as I pass [the courses]. (L.3.5) Other courses. For most students, mathematics was only one part of their studies. All the students I interviewed were taking other courses at the college; all immigrants were studying English as a Second Language, and either computing or accountancy courses. Among all the students, mathematics, computing, and accountancy appealed because they were less language-based than other subjects. “I like the math class best,” said one foreign student. “There’s just one book. . . the language is easier [than in other courses]...and you don’t have to speak in class” (L.3.7). Many foreign-born students had recently committed to attend college full- time and did not want to overtax their limited language abilities at this stage. “Sometimes [in mathematics] the words are hard,” said one Afghani student, “but [there’s] not much writing” (L.3.6). Among the non-immigrant students, computing courses were also popular, and several students also studied Science and English Literature. These students were trying to gain their high school equivalency and said that they took the science courses to help them gain access into higher education courses at other colleges or local universities. Students said that, in general, they preferred the mathematics classes to the others they were taking: “[Mathematics] is easier that way. You know what you have to do and when you have to do it” (L.3.3). Another added: “I can sit down to my math homework and know that there is an end to it.. .even though it might take me all night. With other subjects. . .like English, I never feel that” (L.3.9). For many students, part of mathematics’ attractiveness as a subject was its ‘boundedness”--the way that it was treated as a fixed and permanent body of facts and procedures. “Once you get the rule,” said one student, “you’re away.. . .You don’t have to think 120 about what it means. . .or if it applies in every case.. . .You know it does” (L.3.3). Experiences of mathematics education. Almost all of the students were entering an adult mathematics class for the first time; indeed, for many it was their first experience of adult education. Several students remembered their childhood mathematics education: these were commonly described as unpleasant experiences. For some, math was just one part of an altogether negative school experience: “I detested school, period... .Where I grew up, school was not a big pastime. ..there was a lot of...law bending and stuff. I was totally alienated at times.. . .Not just the math, everything suffered” (L.3.3). When I was a kid, we moved around a lot. I didn’t do very well.. .because we were always moving. I don’t think that math was any worse than other subjects...I think you can wing it [in math] because.. .all other classes are dealing with language. (L.3.1) For others, mathematics education was worse than for other subjects: “I can look back on it now.. .on my math courses. . . and a few of my teachers were duds and didn’t make the course interesting at all. . . they had no enthusiasm” (L.3.2). Another added, ‘1 could never get the hang of it. ...For some reason the math teachers were always the worst.. .shouting, moaning.. .sometimes hitting you.... [In math] I’ve always done three months or so.. .then got kicked out” (L.1.3). In the old days, the [math] class would go at a really rapid pace. The teachers would go like a bat out of hell. You swam as fast as you could and if you. ..couldn’t keep up you went flying over the waterfall... .What got lost was...I never understood any of this stuff. (L.1.5) For many students, learning mathematics at school was a process of sitting quietly and listening to the teacher, rather than one of asking questions. One described: The thing I remember most is that I was.. .pretty frightened to raise my hand and ask a question. For two reasons: many of the teachers were of the opinion.. .that children should be seen and not heard. If you’re not listening then that’s why you didn’t get it the first time. And secondly, if you make a 121 fool of yourself as a child, other children, they’re very cruel. (L.3.1) Even foreign students had not really enjoyed their math dasses in school: “It was just [a subject] you had to do,” said one. “Not very interesting” (L.3.4). “[Math] was the same as now, but in my own language” agreed another. “When I was at school, sometimes I [found] it hard....I didn’t [find] anything interesting in math because it was for me sometimes confusing and I didn’t know anything” (L.3.12). Algebra is one of the key topics in Grade 10 mathematics. For many of the students, the AC math class was the first time they had encountered it. “I sort of dropped the [math] class before we ever got there,” said one. “I didn’t know what algebra was.. .and it just seemed so foreign. I think I missed the middle steps to where you start algebra” (L.1.7). Another student expressed the view that any previous mathematics education could prove a hindrance: What my uncle told me about algebra, he said best... .He said if you’re just starting to learn it now, it’s easier. He said if you. ..if you learn algebra before and [then] you learn it again, it’s confusing. But if you just start learning it now, you should be OK. (L.1.2) Students’ Attitudes and Aims Given their diverse backgrounds, the students had different attitudes towards mathematics and different reasons for learning it. I now consider each of these. Attitudes towards mathematics. Students gave information about their attitudes towards mathematics in two ways: in a survey completed at the beginning of the term, and in personal interviews throughout the term. The survey protocol (see Appendix 2) measured two dimensions of students’ attitudes: (a) their enjoyment of mathematics, and (b) their perception of its value. The scale of scores for each dimension ran from 0 to 40, with larger scores indicating greater enjoyment or greater perception of value. The distribution of scores for the 122 two dimensions are presented in Appendix 21. In each figure, the horizontal axis (E sum and V-sum respectively) refers to the score, the vertical axis (count) refers to the tally for each score. The mean score for the first dimension (enjoyment of mathematics) is 24.5 (with a standard deviation of 7.4); the mean score for the second (perceived value) is 30 (with a standard deviation of 4.1). A comparison of the two sets of scores shows that students (as a group) were more likely to perceive a use for mathematics than to enjoy it. However, a correlation of the two sets of scores shows that students who scored highly on the enjoyment dimension scale also had higher scores on the value dimension scale. (There was a positive correlation between the scores of 0.195.) The survey scores give only a limited picture of students’ attitudes; their comments are more descriptive. Most students were strongly convinced of the value of mathematics. For them, mathematics occupied a central position in the world: “It [math] relates to life, right? I mean it all relates back. All this relates to something” (L.1.3). Another student described mathematics as, “the rules. You have to be precise. . .right on.. .no in-betweens. It’s the logical way. . . the way things are. You’re either right or you’re wrong” (L.3.5). A third thought mathematics was, “the modern language. It is in everything. If you want to live. . .you want to live comfortably, you must know math.” After a pause, she continued, “Even if you want to live not comfortable. If you want to live in this world, you have to know math” (L.1.8). For other students, mathematics was a way of thinking: “It’s like reasoning. . . the way of figuring out problems,” said one (S.2.5). “It’s very precise,” said a second student, “It’s black and white. You can often get in a tangle with words, because they can mean different things to different people. But if you want to prove something. . .you use math” (L.3.7). “It helps me to think,” explained another student, a salesperson: When I’m in the shop.. .1 don’t necessarily use [math] all the time because it’s 123 all computerized. But if something goes wrong, you have to know “Oh, that’s not right.” I mean I occasionally press the wrong buttons [on the computer] and I get.. . the readout and it’s clearly wrong. You need to know [math] all the time. (L.3.2) Students’ views on mathematics not only referred to the topics they were studying. One student described coming into a classroom before the last class was finished. There were all these squiggles and stuff all over the board. I didn’t know what it was about. I asked the teacher and he said [calculus]. It looked really hard. . .1 don’t think we do it in this class, but I’m sure I will learn it one day. I mean, it’s all going to be relevant or helpful.. .it’s going to have some bearing sooner or later. (L.1.4) Only a couple of students were unsure of the usefulness of mathematics in general, or of certain topics in particular. One student described how she was unsure of the usefulness of algebra: I don’t know if I ever will [use algebra]. For what I’ve been involved with I have used different types of math. Certainly I can see that.... But this [algebra].. .1 don’t know how specifically I’ll use it. I can’t think of any uses right off. (L.1.7) Another student also wondered why he should bother learning algebra: It’s just how it is, I guess. I need it on my transcript. That’s the only reason I’m taking it... .It’s so ridiculous you’ve got to learn how to do it so you don’t feel you got beaten by a ridiculous concept. If it wasn’t mandatory, I bet people would be taking a lot of different kinds of math. I don’t think many people take it ‘cause they like it. . . .1 suppose that’s why it’s mandatory. . .otherwise no-one would take it. (L.3.2) Reasons for learning mathematics. Students’ stated reasons for learning mathematics were many and varied. Half of those I interviewed had clear reasons: they were trying to complete their high-school education. Whether they were Canadians who had left school before graduation or immigrants who needed to secure credentials that would be recognized in Canada, learning math and high- school completion were necessary for entrance to higher education or different (better-paying) jobs. One student explained that, for him, mathematics is one of my pre-req[uisite]s. I’m thinking of getting into BCIT or some 124 university, either a radiology or nuclear medicine program. If I get my Math 10 then I can get my sciences, which is my key to get into the program. (L.3.3) Other students cared less or were unsure about further education. “I like to continue [to take] other courses, higher and higher,” said one, “but I don’t have any [goal] in mind right now.. .so I’m just continuing my education” (L.3.4). “I want to start my own business doing massage,” said another. “So I’m going to take massage therapy, which I need my biology for, and also Shiatzu, [for] which you need biology and chemistry” (L.3.8). Some students were less certain of their future direction generally and were looking to their studies to show them a way. “That’s what I’m here to find out,” said one student, “I get lots of ideas of things to do, but then there’s lots of drawbacks to each one. So I’m just taking the courses and trying to think of what I actually want to do” (L.3.2). Several students descrIbed their reasons for learning math more personally. For them “improving self esteem” or challenging themselves intellectually was more important than a career. Some talked about the embarrassment or fear of being identified as “math anxious.” They described how they were tired of feeling unconfident or lazy. “I really didn’t feel good about math. . .or about myself,” said one, “I wanted to do something about it” (L.1.7). “I took the course because I found my brain was getting lazy,” said another. “The more you exercise it, the less lazy it gets” (L.1.2). Other students described different reasons. One student mentioned, “the horror of not being able to do basic mathematics, and not being able to admit it. . .is really depressing....It brings you down so that you feel like you shouldn’t do anything” (L.1.5). A second student described how embarrassing it is to be my age and not know.. .45% of something. . .you don’t have a bloody clue what it is. Everybody assumes because of your age that you know all of these things right off the bat. I have been doing manual labor all my life because of an embarrassment with not knowing math, too stupid to actually come, too embarrassed to come [back] to school. (L.1.5) Finally, a couple of students said they were learning mathematics in order to 125 help their children. One student who had already studied mathematics in China (her home country) explained that she came “back to learn math to help my kid. I teach them... times table, and I do some of their school math with them. It help me too” (L.1.1). Teachers In this section I describe some personal and professional characteristics of the teachers. After providing some brief demographic details, I focus on their educational and teaching experiences, and their attitudes about teaching, mathematics, and their students. The College’s adult mathematics department consists of seven full-time teachers including the department head (who teaches part-time). In addition, one teacher is shared between the mathematics department and the science department. Finally, several part-time instructors are used on an on-call basis. Of the eight full- time mathematics teachers, six are male. All the teachers are Canadian citizens, most by birth, although two are immigrants (from Hong Kong and England). The youngest teacher is 26 years old and the oldest is 56; most, however, are in their late 30s to late 40s. Educational Experience Teachers regard their work as a career; most have worked in their present jobs for over 10 years. Indeed, for the majority of teachers in the department, teaching mathematics is the only work they have ever done. Two teachers came straight to the 126 College as soon as they had completed their first university degree; the others had previously taught in other Canadian high-schools or colleges. Several teachers also indicated that they had, from time to time, tutored other people (usually children) in mathematics. All the teachers have a minimum of a bachelors degree (a college requirement); most of these are in science-related subjects (mathematics, science, or science education). Four teachers have continued their formal education with post graduate study and some also have a provincial teaching certificate which allows them to teach in any British Columbia secondary school. Two of the teachers have never studied mathematics at college level; their degrees are in music and general studies. Both of these teachers were hired because of existing relationships with the College, as either volunteer tutors or as ex-students. One described how he was hired: I had been an elementary school teacher [but left] after eight years of teaching to set up my own business. [After a while] I thought I should go back to teaching. . . .1 didn’t really relish the thought of teaching young people anymore, I wanted to teach adults. So I applied here, which is my old alma mater, [to] teach English. They said, “Oh, you’re one of our ex-students. It would be nice to have you on our staff. How about teaching mathematics first?” I didn’t really feel qualified for the position but they said, “It’s just general program math. I’m sure you could handle that.” So I started teaching general program math, business math, that sort of thing. (T.1.7) Although the teachers have, as a group, a reasonably strong background in mathematics or science education, they have markedly less training in adult education. Only two teachers have taken any formal courses in adult education. When asked about how they had learned to teach adults, most teachers said that they “picked it up as they went along.” For example, “Early on, I sensed I had to change certain styles... .1 found topics didn’t have to be pre-planned to the nth degree without any ten-second lags just for classroom management survival reasons. (T.1 .3) When I was a student myself I taught a small group of other students. I 127 perform[edl well in mathematics, so.. .a group of students approached me and asked me to help them, so I actually had some experience in teaching. It’s really informal.. .but I found it’s very interesting. So that’s a reason I tried teaching, tried out teaching in this college. (T.1.4) In any case, adult education training was not seen as important by teachers because “there really wasn’t any difference between teaching math to children and to adults” (T.1.3). Even those teachers who had studied education at a postgraduate level had found them of little value in their own subsequent teaching. Occasionally, the courses had helped with teaching techniques: I guess teaching.. .this is to quote a lot of UBC profs. . .it’s like having a shotgun and you try to get as many people as possible. So when I plan my classes what I usually do is I try to aim for the middle ground, to present it so I do not lose the lower students, but at the same time not lose the top students. Depending on.. .class interaction.. .1 could, you know, go higher or lower. (T.1.2) Only one teacher found his college mathematics education useful. He described being told about a meta-analysis of calculative research in the 1970s; the results were fascinating. The commonly-held belief is that if people can work with calculators [or] computers then there’ll be some attrition of paper and pencil skills with arithmetic. But the major finding of that analysis was that if calculators are used at the same time as paper and pencil skills, the people with calculators have a better ability not only in conceptualizing but also with the paper and pencil skills. So it’s a win-win situation for people with calculators... .1 came back to the department and said, “We should be using calculators,” and everybody said, “Yeah, you’re right.” So, since then we have. (T.1 .3) Most teachers claimed that they learned about teaching from their own experiences of learning. Sometimes, their memories significantly color their perceptions: I would think that often how we teach is affected by how we have been taught.... I have often thought about my teaching in this way. At university I had to work very hard and very independently. I’ve referred [in class] to working hard, working on your own... these are some of the things [that] I’ve adopted... .1 think that affects my own personal philosophy of striving for excellence. . .I’m a person who really likes to see excellence and organization. What bothers me is students’ lack of achievement and interest and lack of organization in their own lives. They can come in and there’s just total chaos 128in their notebooks, and I think if their notebooks look this way, what do their minds look like? (T.1.8) A second teacher spoke of how his own experiences with learning mathematics affected his current teaching practices: I tend to give a lot of notes because I find that the textbooks are usually not given in the simple terms that.. .novices to math could use. So what I usually try to do is I break things down into.. .simpler terms and so on. I guess that’s one of the things I picked up when I was going through education. (T.1.3) Often teachers remembered unpleasant memories of their own math education. One spoke of his “experience with mathematics instructors [as] almost universally horrid... .The tedium, the shiny polyester pants, the unchanged suits, the sweat stains, the jacket that never changes.. ..I try hard never to be like that” (T.1.3). Another said: There’s nothing worse than coming into a math or a science dass [and] it’s deadly silent, you don’t know anybody else in the room, you have no idea what to expect. You’re nervous or whatever, and some guy comes in and says, “Here’s the coarse outline, here’s the first chapter, get on with it.” This is what I left school for, to get away from this stuff. (T.1.6) Attitudes about Teaching Perhaps because of the similarity in their backgrounds and experience, teachers also held similar views about teaching. In general, they thought that teaching mathematics was largely a matter of conveying fixed concepts and set procedures. As the content was established, their role became one of deciding how to convey that content. Teaching became an exercise in selecting the “most efficient strategies.” Such a process could be influenced by students, but only occasionally, and only indirectly. Teachers developed their favorite strategies with experience. “I’ve built up three or four different ways of approach, “ said one teacher. “Of course, I have my favorites which I will always use unless students tell me that it doesn’t make 129 sense.. . .For me, when you teach you’re trying to sort through a whole garbage dump and see what is appropriate” (T.1.2). “I like to get a feel of a group,” explained a second teacher, “get a feel of their learning attitudes.. .So I can tell which strategy is best.. .to accommodate that, to help them achieve their learning goals” (T.1.4). In general, however, teachers subjugated the learning needs of students to their own need to “cover the material.” While they acknowledged that students had different learning styles, teachers didn’t necessarily change their teaching approach to accommodate students: “I’d like to say that it [teaching] depends on the type of students.. .but it doesn’t really. It’s almost dictated by the length of the class and what we have to cover.. .there’s so much to get through” (T.1.6). “There’s a lot of pressure here to get through the material,” agreed another teacher, “You can’t always do what might be best for the student” (T.1.8). A third added: I haven’t the time to get into learning disabilities and stuff.. .I’m not really qualified. I mean there is a structure that should help students in that sense. My job is to make sure that we cover the material. . . .If students are having difficulty they can come and see me after class.. .or go to the learning centre. (T.2.2) Most teachers also thought that students should feel a sense of personal accomplishment at the end of each course. “It’s important that [students] meet their goals, not only pass [the] test” explained one teacher (T.3.1). Teachers appreciated that some students were already highly motivated. “They’re here because they want to be. . . they’ve got their own reasons. . .but they’re very focused. It makes your job wonderful, sometimes” (T.1.6). “The more you give them, the more they give back. They respond if you try to make it interesting” (T.1.1). Encouraging motivation among the less enthusiastic students was also necessary, although these students were seen as requiring guidance towards “setting realistic goals.” “Many students are not prepared for the hard work that they have to do,” explained one teacher. “Sometimes it comes as quite a shock” (T.1.1). A second teacher explained that adult students, often with “poor study habits.. .well, no study 130 habits at all” are unprepared for the level of work expected of them. I always tell them, right at the beginning, “Listen this isn’t going to be hard, but it’s going to be fast... and if you are taking 6 classes.. . and have two kids at home, and you’re working 20 hours a week, then you should really take a look at what you’ve bitten off.” I try to tell them that the demands are great. (T.1.8) “Encouraging students to take responsibility for their own learning,” was seen as particularly important for adult learners. “When they leave this place they’re going to be very much on their own at UBC or wherever,” explained one teacher. “To get [them] ready for that means that we can’t hold their hand the whole way through the term.. .we’ve got to get them on their own feet” (T.1.6). “It’s impossible to cover everything,” said another, I have a philosophy that there comes a point where the student has to make certain connections on their own. I tell them this, “I cannot foresee every possibility and difficulty you might have. You must come and tell me of [your] problems.” Some do, some don’t. But, you know they’re adults.. .they should know what responsibility is. (T.1.2) Another teacher compared teaching children and teaching adults. “When you’re teaching children,” he explained, you have some responsibility for the actual learning that the person’s doing. When you’re working with adults you’re free of that responsibility. This [math education] is such a minor part of their lives.., they’ve got more pressing problems in their lives than learning. The choice to learn is clearly their own... .My responsibility is to remind them of that. (T.1.3) Developing responsibility in learning affects how people teach. One teacher thought it crucial to understand students’ backgrounds to ascertain their attitudes: “I like to find out who I’ve got here.. .where they’re coming from. That’s going to affect what I do. It tells me whether people are here because they’ve failed or because they haven’t had it (T.1.8). Other teachers said that they develop a sense of responsibility in students by encouraging them to ask questions: “What I like to do in the first 20 minutes or so,” explained one teacher, “is deal with any questions that they have.. .from their own 131 work and study. I encourage everybody to ask questions.. .even if they’re dumb questions. It’s important that they say what they don’t understand” (T.1.3). Encouraging a questioning approach in students was seen, by some teachers, as time-consuming yet rewarding: I find that with the lower levels.. .more time seems to be spent on individual activity. I’ll coach more.. .spend more time with each individual student... to get them to think. . .to understand it for themselves.. .so they can work more on their own in the future. (T.1.3) Other teachers were less enthusiastic: Oh, it gets very hard to maintain your level of enthusiasm over the term. You can see some students just don’t have a clue.. .even though you spend time after time with them. I find I get personally worn down. Especially at a level like 050 where there’s a lot of attrition anyway. (T.1.8) Often, you think you’ve got it just right. . . the right dimate and everything. Then students will come up with unexpected problems.. .perhaps financial or their cat is going to die.. .or they’re going to have to take off for a week to go to Toronto because they don’t want to be alone for Halloween. (T.3.3) Aims of teaching. Teachers’ attitudes about teaching influenced their instructional goals and aims. Overwhelmingly, teachers said their general goals or aims in teaching mathematics were to foster an understanding or enjoyment of mathematics. Only one teacher described his goals purely in terms of completing the course material. (For this teacher, the main goal was “helping students to get the course done. This fulfills the requirements” (T.1.5).) The others spoke of “getting students.. .to have a basic grip of what it’s about.. .and liking it” (T.1.1), “getting students to know how to do it and to understand it” (T.1.8). “The best enjoyment I get,” said one teacher, “is when some students come back and say ‘I really liked that course.. .1 really felt I understood math for the first time.’ It’s great when that happens” (T.1.2). However, encouraging such understanding was not always easy: “Some people are impatient,” said one teacher, because that’s the way that math has been taught to them, they don’t have a 132 lot of patience or tolerance. They want you to tell them what to do and when. They don’t see that it’s necessary [for them] to do some work to have some understanding. (T. 1.8) Here, it should be mentioned that, for teachers, understanding is seen as learning (and being able to reproduce) existing knowledge. The notion of understanding as “making meaning” was never mentioned. Teachers spoke of mathematics as either a fixed body of knowledge or as a way of thinking. They felt it important that students appreciate and understand the inherent logic and organization of mathematics: “Something I try to do,” said one teacher, is get students “to appreciate the logic behind the steps. . . and why to go about it that way instead of another way” (T.1.2). “It’s important for me to convey the thought of mathematics,” said another, “not just teach them math that they can use” (T.1.4). Several teachers also spoke about how they tried to encourage motivation, interest, “taking control of learning,” and “independent learning” in their students. Fostering independence in students is seen as crucial: “If we send them out still dependent on walking into a classroom and sitting down and waiting for it all to happen, we’re not doing them any favors at all,” said one teacher (T.1.6). A second described how self-motivation was far more effective at encouraging learning among adult students than any specific teaching technique: It’s that sense of responsibility. . . that learning is not my, but [the student’s] responsibility. I try to achieve that sense of internal motivation in the student, and that’s why I strongly believe that a zillion different instructional strategies work. . . .but in the end, it’s down to the student. Their responsibility, their interest, their desire. (T.1.3) Often, encouraging motivation involved building on students’ life experiences. One teacher described how students have already made some major changes in their lives.. .quit their jobs, or left their spouses or whatever. They’ve already made some major decisions; taken some responsibility. . . .It’s my job to remind them of that and to encourage them to see their learning as something [else] they can take responsibility for. (T.1 .3) Another teacher described a telling example of how “a sense that youn do 133 it” and persistence in students can pay off: You see that guy, David, that guy who came in this morning. He’s had to take the course about three times because his life was a mess, you know there was all sorts of things going on in the background. His writing was terrible, his reading was awful, but he succeeded... .1 must have spent hours with him, going over the work. [Although] most of it was his own determination to do it. Without that it wouldn’t have worked. (T.1.6) Additionally, teachers recognized that self-motivation was aided by a non- threatening atmosphere in the classroom. They spoke of trying to minimize the “math anxiety” that many adult students feel, by making their courses less threatening. To do this, they would tell jokes or encourage ‘banter” to “lighten the atmosphere.” One teacher brings classical music tapes into his classroom to play as background music. Another, a skilled artist, uses cartoons about math to get students “laughing.. .it helps to break the atmosphere right at the beginning” (T.1.7). A third described how she would use “fun activities” to “make math seem more enjoyable”: I call them “algebra adages” because we use them in the algebra level1 . They have to solve a series of simple puzzles and each answer has a letter assigned to it, like a code. When they’ve solved one question they write the letter in the answer space.. . and when they’ve done all the questions the answers spell out a saying.. .an adage. They’re a bit elementary. . .but even though they seem like children’s activities.. .the adults enjoy them.. .and are not insulted. The [activities] certainly make doing algebra more fun. (T.1.8) Talking about learning mathematics is also seen as crucial for dealing with any student anxiety: I ask them if they think they suffer from math anxiety. If I know that someone’s really anxious about math, I’ll try and jolly them a lot more. If [students] come into the class with a negative attitude towards math it affects their achievement. Like a self-fulfilling prophecy, “I can’t do math.” Well, if you keep telling yourself that, you won’t be able to, that’s for sure. (T.1.1) 1 One example of an “algebra adage” is included as Appendix 20. I try to bring it out in the open, and say “Look, you might be the only person 134 that said you were scared, but I’ll let you know that there’s at least two thirds of the people in here feel the same way. . . .Of course, sometimes people aren’t very comfortable talking in a group, so I try and get them on their own so we can talk about it there. (T.1.6) Another teacher described how he would speak to a class about his own struggles with learning: I tell [students] that when I left school I had a Grade 5 education in math. Right away I tell them that I’m on their side; I understand where they’re coming from. And I explain to them that when I went back to school I had to change my lifestyle. I was really nervous about learning because it had been so long, and I wasn’t sure I could do it. But the fact that you’re there means that you want to be there.. .and that makes up for quite a bit. (T.1.7) Attitudes about Teaching Mathematics Teachers tended to regard mathematics as either a set of thinking skills or a fixed body of knowledge that transcends context. “Most of us who have ever taught math know it’s a universal thing,” as one teacher put it (T.1.2). Another described that “the important thing [in learning mathematics] is the ability to think and reason. So it’s not so much being able to factor, but. . . [understand] the process behind factoring” (T.3.3). “Mathematics is everyday in our lives,” said a third: If you go to fill up your tank in the car, then you figure out how many litres you get.. .how much money you pay. Everyday you listen to news.. .you see percents.. .like unemployment is down, interest rates up... .Also, mathematics trains our mind. Some of the things we may not use in our daily lives, but everybody has to think and everybody has to do some kind of mathematics. (T. 1.5) Despite the ubiquity of mathematics, teachers acknowledged that for many students, learning it was hard work. “A lot of mathematics is learning how to use other means other than a laboring--a pick and shovel--approach to thinking your way out of situations,” said one teacher. I often tell the students that if you want to be lazy, you look for the easy way 135 out. You can go out there and run a measuring tape between here and Mars, or you can use your head and not have to do that labor. (T.1.3) “Our job,” said a second teacher, “is to bring [mathematics] dowrL to a simple level. We often expect our students to know [the same] things that we do. . .because of their age. I’ve found that you sometimes can’t get basic enough” (T.1.8). For teachers, the “organization of mathematics” as a subject provided a sense of mental discipline. One described his students at the beginning of the course: You won’t believe how disorganized some are. They come into the classroom with no books, nothing to write with. God knows what the insides of their brains are like. You have to show them. . .first this, then that. The books help a lot because they’re so well laid out. You can’t do this chapter before you’ve done the previous ones. You can’t understand this concept unless you’ve mastered those.... By the end of the course they’re getting the idea. (T.1.5) Another teacher gave a telling example of how to use a mathematical concept to “train students’ memories.” The day before the test I will ask them to write it in 6 or 8 digits. Then I will say, “Tomorrow, on the test you will have to write it to 8 digits.’ Then they will be worried, “How can I remember that? I remember right now, but tomorrow I forget everything.” So, then I come up with some kind of memory aide, so they can remember. And they will. (T.1.4) A third described how “in addition to explaining the structure of mathematics,” she tried to model “good organization and discipline” in her teaching. For her, the emphasis was notably physical: I make sure that all material that I give them is on 8-1/2 xli paper. I make sure that I three-hole punch every sheet that I give them. Also, I give them special sheets to keep track of their records.. .so they can see [the records] at all times. I encourage.. .record-keeping and organization through this.. ..I think some of them catch on to it; maybe those who do it well would have done it anyway, but I will sometimes come along and talk to somebody.. .if they’re supposed to be working on something and this student hasn’t got an idea where it is.. .I’ll say “How about putting these in order and filing them at the end of the day so you know where they are?” Hopefully the organization that I bring in, the preparedness that I bring into the classroom, you know, that encourages some of them to do more of that. (T.1.8) This emphasis on the neatness and order of mathematics is not the only 136 characteristic that teachers highlight. One spoke of trying to foster more positive attitudes towards mathematics among students: “I love math, I think it’s beautiful; the symmetry and the application and everything. If I can get even one or two students to develop an appreciation for the beauty of mathematics. . .1 think I’ve been successful” (T.1 .1). Even in this, however, notice that the teacher regards mathematics as fixed; its qualities (whether of order, or of beauty) already exist and await discovery. The role, here, for the teacher is that of expert/guide, leading the way while also encouraging a notion of self-reliance. Teachers commonly thought that they could increase learning by relating mathematics to students’ interests and experiences. Most teachers claimed that they tried to make mathematics seem relevant to their students. “For example,” said one, “in doing angular speed I use car examples. Or in mixture problems I’ll talk about mixing drinks and things like that” (T.1.1). Another teacher described how he would introduce the concept of a slope of a line: I’ll say, “Do we have anyone from the construction industry here? This question is like finding out the pitch of a roof.” People will usually volunteer that sort of information. It’s great when they do, because all of a sudden there’s somebody else who is saying “This is my experience.. . this mathematics is relevant to my life.” (T.1.7) Overall, however, most teachers felt that, at the introductory level at least, the mathematical topics covered contained little of intrinsic interest for either them or their students. Teachers described how students would occasionally ask them, “What use is this?” or “Why do we have to learn this stuff?” “Got to do it.. .on the test,” was one teacher’s stock reply. Another would be equally as honest: I’ll say, “I don’t know. It’s in the curriculum guide. I don’t know why it’s there.” And I don’t. Most people never use this. For example, set notation diagrams. No working mathematician, scientist, or engineer ever uses set notation diagrams. I guess it’s.. .the concepts behind the idea of set notation that are important. (T.1.6) A third teacher described how, if so asked, he would answer with this story: 137 One time a youngster got a job. The job is to make carpets. Well, the manager only asked him to make one little corner of the carpet. Just one little piece. Other people are making other part[s]. So, everyday they are doing this. The lad is getting bored only doing the same piece day after day. So the youngster asks the manager, “I don’t like this job. I’m getting bored, everyday just doing this or that. My hands are getting tired. Why [is] it like this?” The manage