AN ANALYSIS OF TEACHING PROCESSES IN MATHEMATICS EDUCATIONFOR ADULTSbyTOM NESBITB.A., The Open University, 1978M.A., San Francisco State University, 1991A THESIS SUBMiTTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE STUDIES(Department of Educational Studies)We accept this thesis as conformingto the required standardTHE UNIVERSITY F BRITISH COLUMBIAMay 1995© Tom Nesbit, 1995In presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of this thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.(Signature)Department ofThe University of British ColumbiaVancouver, CanadaDate lDE-6 (2/88)11ABSTRACTThis study explored the teaching processes in mathematics education foradults and how they are shaped by certain social and institutional forces. Teachingprocesses included the selection and ordering of content to be taught; the choice ofsuch techniques as lectures or groupwork; the expectations, procedures and normsof 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 phenomenamean for those involved as teachers or learners? and (3) In what ways do certainfactors beyond the teachers’ control affect teaching processes?The theoretical framework linked macro and micro approaches to the study ofteaching, and offered an analytical perspective that showed how teachers’ thoughtsand actions can be influenced and circumscribed by external factors. Further, itprovided a framework for an analysis of the ways in which teaching processes wereviewed, described, chosen, developed, and constrained by certain “frame” factors.The study was based in a typical setting for adult mathematics education: acommunity college providing a range of ABE-level mathematics courses for adults.Three introductory-level courses were selected and data collected from teachers andstudents in these courses, as well as material that related to the teaching andlearning of mathematics within the college. The study used a variety of datacollection methods in addition to document collection: surveys of teachers’ andadult learners’ attitudes, repeated semi-structured interviews with teachers andlearners, and extensive ethnographic observations in several mathematics classes.111The teaching of mathematics was dominated by the transmission of facts andprocedures, and largely consisted of repetitious activities and tests. Teachers werepivotal in the classroom, making all the decisions that related in any way tomathematics education. They rigidly followed the set textbooks, allowing them todetermine both the content and the process of mathematics education. Teachersclaimed that they wished to develop motivation and responsibility for learning intheir adult students, yet provided few practical opportunities for such developmentto occur. Few attempts were made to encourage students, or to check whether theyunderstood what they were being asked to do. Mathematical problems were oftenrepetitious and largely irrelevant to adult students’ daily lives. Finally, teachers“piloted” students through problem-solving situations, via a series of simplequestions, designed to elicit a specific “correct” method of solution, and a singlecorrect calculation. One major consequence of these predominant patterns was thatthe overall approach to mathematics education was seen as appropriate, valid, andsuccessful. The notion of success, however, can be questioned.In sum, mathematics teaching can best be understood as situationallyconstrained choice. Within their classrooms, teachers have some autonomy to act yettheir actions are influenced by certain external factors. These influences act asframes, bounding and constraining classroom teaching processes and forcingteachers to adopt a conservative approach towards education. As a result, thecumulative effects of all of frame factors reproduced the status quo and ensured thatthe form and provision of mathematics education remained essentially unchanged.ivTABLE OF CONTENTSAbstract.Table of Contents ivList of Figures viAcknowledgments viiCHAPTER 1: INTRODUCTION AND PURPOSE OF STUDY 1Background 2Adult Numeracy 4Research Questions 12Structure of the Dissertation 14CHAPTER 2: LITERATURE REVIEW AND THEORETICAL FRAMEWORK 16Mathematics Education for Adults 17Adult Education Research 23Research on Teaching 26The Behaviorist Paradigm 27The Structuralist Paradigm 31Frame Factor Theory 35The Interpretivist Paradigm 43Teachers’ Thinking 44Summary & Discussion 54Theoretical Framework 58Summary 69CHAPTER 3: METHODOLOGY 70Selection of Research Site and Participants 70Data Collection Procedures 73Data Analysis Procedures 85Criteria of Soundness 91CHAPTER 4: CONSIDERING THE FRAMES 95Institutional Framework 95College 96Departments 105Classrooms 110VExperiences of Students and Teachers 113Students 114Teachers 125Curriculum 141The Textbook 144Summary & Discussion 169CHAPTER 5: THE CLASSROOM IN ACTION 176Planning Teaching 177Classroom Episodes and Activities 182Homework 206Assessment 212Focusing on Algebra 222Summary 253CHAPTER 6: ANALYZING TEACHING PROCESSES 258Revisiting Teaching Processes 258Frame Factors 262Conclusion 280CHAPTER 7: CONCLUSIONS, LIMITATIONS, RECOMMENDATIONS 283Summary of Study 283Summary of Results 284Limitations of the Study 290Implications and Recommendations 292BIBLIOGRAPHY 300APPENDICES 314LIST OF FIGURESFigure 1 Understanding Teaching Processes 63Figure 2 Distribution of Students’ Ages 116Figure 3 Distribution of Students’ Enjoyment of Mathematics Scores 349Figure 4 Distribution of Students’ Perceived Value of Mathematics Scores 349vivi’ACKNOWLEDGMENTSThis work would not have been possible without the support of many people.First, I wish to acknowledge the fundamental involvement of my researchsupervisor Kjell Rubenson. Throughout all stages of my research Kjell has providedconstant encouragement, guidance, stimulation, and support. Further, he hasdemonstrated the unique skifi of offering just the right comments or asking just theright questions that provoked further thought or more rigorous analysis withoutever imposing his own ideas and concerns. Kjell has been the model of a dedicatedscholar and working with him has been a profoundly inspiring, enriching, andenjoyable experience. The other two members of my research supervisorycommittee, Klaus Hoechsmann and Tom Sork, also offered invaluable guidance andencouragement. Their perceptive suggestions helped to focus many aspects of mystudy and their close reading of successive drafts ensured both clarity of expressionand timely completion.Several other colleagues in the Department of Educational Studies at UBChelped shape this study or provided support and encouragement throughout myresearch. I would like to recognize and thank the contributions of Rita Acton, AbuBockarie, the late Pat Dyer, Bill Griffith, Garnet Grosjean, Leif Hommen, AndreaKastner, Graham Kelsey, Liang Shou-Yu, Dan Pratt, and Jeannie Young. I also wishto 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 ideashas been instrumental in this study taking the form that it does.Throughout my studies I was privileged to receive financial support fromseveral sources at UBC. This considerably eased the burden of the non-academicaspects of doctoral study and contributed to the speedy completion of my research. Iwould like to acknowledge and thank those responsible for the allocation of theUniversity Graduate Fellowships, the Killam Predoctoral Fellowships, and theCoolie Verner Prize for Research in Adult Education as well as the Faculties ofGraduate Studies and Education for their awards of travel assistance.Teaching and learning mathematics can be difficult enough without havingsomeone observe your every move. The staff and students at Acton College allowedme into their classrooms and were gracious enough to spend many hours talking tome about what they were doing. This study could never have been feasible withouttheir participation and cooperation, and I extend my gratitude to them.Finally, no aspect of this research could have been possible without theinvaluable support, advice, and encouragement of my wife Adrienne. Herdescriptions of her own mathematics education helped me focus my interests, herconcern for precision helped me clarify my thoughts, and her skills as an editorlargely made this account readable. Above all, she showed far more patience andunfailing good humour than at times I deserved. For her: love, gratitude, and thanksof the most profound kind.CHAPTER 1: INTRODUCTION AND PURPOSE OF STUDYWhy learn math? Well, I’ll tell you... .11 was] standing on the jobsite with a fellowwho was the layout man for doing the carpentry work, eh. And we had a big curve todo 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 thecurvature, and I thought this was the most incredible thing, I said, “Geez, this guymust be an engineer, you know, I mean this is incredible. He’s a genius!” And Irealized that all he was doing was basic math.. . .1 want to be able to do it like that.Construction Worker, March 1994This study concerns mathematics education for adults and seeks to explainwhy the teaching of mathematics takes the form that it does. In particular, it focuseson several mathematics classrooms in a typical Adult Basic Education (ABE) settingand explores how such education is viewed by those involved as teachers orlearners. Further, the study examines the teaching of mathematics in light of itssocial context, and investigates how teaching processes are shaped by social andphysical 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 regardedthemselves as innumerate and avoided both numerical data and arithmeticcalculations whenever they could. Of particular concern was that these people wereoften disenfranchised by their lack of mathematical skills from taking an active andinformed role in decisions that involved either numerical data or computationalskills. Second, given how widespread the problem of innumeracy is seen to be, thepaucity of research on adult numeracy or mathematics education for adults isstartling. Although extensive research has been conducted in the corresponding fieldof adult literacy, few of the approaches, assumptions, topics, or questions that havemarked this research, or the insights, applications, or policies that it has generated,have been translated into research on adult numeracy. Third, such published 2discussions that do exist often cite the predominant methods of teachingmathematics in schools as the biggest contributor to poor mathematical ability. Asone noted mathematician puts it: “School mathematics is simultaneously society’smain provider of numeracy and its main source of innumeracy” (Steen, 1990, p. 222).I was interested in determining whether the teaching of mathematics in adulteducation reproduced that of its school counterpart.In this chapter I introduce the elements of my study. I first sketch some of thebackground to my study of mathematics education. Next, I provide some definitionsof numeracy, discuss some of the consequences of innumeracy, and explain whatsteps innumerate adults can take to improve their mathematical skills. To outline thespecific focus of this study, I provide a justification for, and a statement of, myresearch questions. Finally, I outline the structure of this dissertation.BackgroundThe mathematical abilities of adults regularly give cause for concern togovernment bodies, business and community leaders, and adult and mathematicseducators throughout the industrialized world. There is a strong consensus, amongstthese groups, that the mathematical skills, awareness, and understanding of adultlearners, whether high-school leavers or college graduates, have deterioratedalarmingly in recent years. Adults “know less, understand less, have little facilitywith simple [mathematical] operations, and find difficulty in solving any but theshortest and simplest of mathematical problems” (Barnard & Saunders, 1994).So what? Millions of people appear to function perfectly well without everneeding to use much of the mathematics that they remember from school. No oneclaims to be particularly disadvantaged by a lack of mathematical abilities. Inaddition, many people see mathematics as an esoteric subject having little to do withtheir everyday lives. Indeed, mathematics commonly represents a body of ultimatelyabstract, objective and timeless truths, far removed from the concerns and values ofhumanity. If mathematics seems so tangential to everyday life, why is it such aproblem if so many people can’t do math very well?Primarily it is a problem because of the societal and individual consequencesof innumeracy. Numeracy--mathematical ability--is commonly recognized as a majordeterminant for job and career choices, and a key to economic productivity andsuccess in modern, industrial societies. Numeracy, then, functions as “culturalcapital.” Hence, the extent of mathematical ability operates as a social filter, andaccess to social effectiveness and privilege is restricted to those with sufficientmathematical ability.It doesn’t start out that way. Indeed, numeracy is one of the major intendedoutcomes of schooling, and mathematics occupies a central position in virtuallyevery K—12 school curriculum. But somehow, mathematics teaching fails to producenumerate adults. As Western society has become increasingly informationally andtechnologically saturated, the innumerate are increasingly disadvantaged--confusedand manipulated by numbers, unable to critically assess assumptions and logicalfallacies, and unable to participate as effective and informed citizens. For example,how often are adults prepared to take statistical information and their statedconclusions at face value? How many of us feel skilled enough to look beyond thenumbers to interpret what the statistics mean? Of particularly concern is theunderlying pattern of inequity in adult numeracy; surveys of mathematical abilitiesshow that performance is lower especially among working class, women, Hispanic,and Afro-American learners. So, mathematical ability is important if only because it 4is capable of empowering so many.Why are adults’ mathematical abilities as low as they are? It has beenproposed that the primary contributor is the poor teaching in school mathematicsclassrooms (Frankenstein, 1981; Paulos, 1988). Traditionally, mathematics educationis taught as an abstract and hierarchical series of objective and decontextualizedfacts, rules, and answers. Further, predominant teaching methods use largelypassive, authoritarian, and individualizing techniques that depend onmemorization, rote calculation, and frequent testing (Bishop, 1988). Knowledge isthus portrayed as largely separate from learners’ thought processes, andmathematics education is experienced as a static, rather than dynamic process.Adults who do wish to upgrade their mathematical skills have access to a variety ofcourses run by local public sector educational bodies. It is unclear, however, if thesecourses are, in any way, adult-oriented, or merely reproduce the curricula andteaching methods so common in traditional K-12 mathematics. Given the rapiddecline in adult numeracy, the nature of its social consequences, and the apparentinadequacy of current educational approaches to remedy it, this study of theteaching processes in adult mathematics classrooms is both timely and necessary.Adult NumeracyTo be numerate is to function effectively mathematically in one’s daily life, athome, and at work. Being numerate is one of the major intended outcomes ofschooling, and mathematics occupies a central place in the school curriculum.Indeed, mathematics is “the only subject taught in practically every school in theworld” (Willis, 1990, P. 16). However, despite this privileged position of mathematicseducation, there is much evidence that the mathematical abilities of many adults inBritain and North America do not equip them to function effectively in their dailylives (Cockcroft, 1982; Kirsch, Jungeblut, Jenkins, & Koistad, 1993; Paulos, 1988;Statistics Canada, 1991). For example, Statistics Canada report that 38% of Canadianssurveyed in 1991 did not “possess the necessary skills to meet most everydaynumeracy 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 mathematicsprofession and the public. Although generating some concern at the time of theirpublication, these works are rarely discussed in either the mathematics education oradult education literature, and their impact on mathematics education for adults isunknown. What is clearer, however, is that overwhelmingly these books concentrateon “innumeracy” as opposed to “numeracy,” and in so doing, focus on the negativerather than the positive aspects of individual mathematical ability. This suggests theimplicit classification of those who are numerate as “good” or “worthy,” and thosewho are innumerate as somehow “bad” or “inadequate.” Given the implications ofmerit in that classification, it is useful, first, to consider some definitions ofnumeracy. What, in practical terms, does it mean to be numerate? And, alternatively,what are some of the consequences of innumeracy at both personal and societallevels? I will discuss both of these, and finally, describe what opportunities exist forthose adults who wish to improve their mathematical abilities.Definitions of Numeracy 6Public discussion about the mathematical ability of adults is usually couchedwithin the context of debates about adult literacy. Indeed, numeracy and literacy areoften linked. For example, the Crowther Report (1959) describes numeracy as “themirror image of literacy,” and one noted mathematician introduces his survey ofcontemporary 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, andnumeracy as a concept is often considered a part of the wider concept of literacy. Forexample, UNESCO defines a literate person as one whocan engage in all those activities in which literacy is required for effectivefunctioning of his/her group and community and whose attainments inreading, writing, and arithmetic make it possible for him [sic] to continue touse these skills for his own and the community’s development. (UNESCO,1962)More extensive definitions of numeracy are provided in the Cockcroft Reporton mathematics teaching in Britain (1982). Cockcroft discusses a range of definitionsfrom a broad conception—including familiarity with the scientific method, thinkingquantitatively, avoiding statistical fallacies—to narrower ones such as the ability toperform basic arithmetic operations. Cockcroft uses the word “numerate” to meanthe possession of two attributes:1. An “at-homeness” with numbers, and an ability to make use ofmathematical skills which enables an individual to cope with the practicalmathematical demands of everyday life, and2. An appreciation and understanding of information which is presented inmathematical terms, for instance in graphs, charts or tables or by reference topercentage increase or decrease. (p. 11)There are some noteworthy aspects of this definition. First, both attitudes andskills are considered important. Second, being practical is the criterion by whichskills are considered important; the relevant context is provided by the demands ofthe person’s everyday life. Third, the appreciation of numerical information isconsidered important as well as the use of mathematical techniques.Hope (quoted in Keiran, 1990) provides a broader definition of numeracy; onemore in tune with peoples’ everyday demands than with the narrower interests ofmathematicians. 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, understandingsimple statistics, working with graphs, and using scoring schemes in leisureactivities 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 performcalculations, knowing how to measure, knowing how to work with space and shape,and knowing how to analyze and interpret quantitative data and arguments basedon this information.Other, broader, definitions of numeracy are beginning to emerge aseducational research documents changes in school practices during the 1970’s and1980’s. A “broad” approach regards teaching mathematics more as the developmentof 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 theprocesses of conjecturing, specializing, generalizing, convincing, explaining, anddescribing--seen as essential to solving mathematical problems.Whatever the exact definition used, many authors claim that the kinds ofmathematical skills needed by people to function effectively in daily life arechanging, and are likely to continue to change. The need for certain mathematicalskills such as arithmetic or algebraic computation is decreasing due to theavailability of calculators and computers, while other mathematical ideas such as 8estimation or those associated with probability and statistics are assuming greaterimportance (National Research Council, 1989). This continuing change in themathematical needs of adults highlights the need for some discussion of theconsequences of innumeracy.Consequences of InnumeracyIn general, innumeracy is not considered as socially unacceptable as itscounterpart illiteracy. One often hears statements about peoples’ mathematicalinadequacies, spoken without any apparent embarrassment: “I’ve never been able towork out how much to tip”; “I never check my change from the store”; or “I’m apeople person, not a numbers person.” One mathematician, John Allen Paulos,claims that part of peoples’ lack of concern about their mathematical ignorance isbecause the consequences of innumeracy are not as “obvious as other weaknesses”(1988, p. 4).However, regardless of popular opinion, there are several consequences ofinnumeracy among adults. At an individual level, there are restrictions on freedomof access to further education and training, and to higher-paying jobs. Mostinstitutions of higher education formally require that in order to be accepted,applicants demonstrate their mathematical ability by passing certain standardexaminations such as the GCSE (in Britain) or those examinations that lead tocertificates of high school completion (in North America). Once enrolled in highereducation, students are often required to take further mathematics courses beforethey can register in courses in particular disciplines (e.g., science, medicine, oreconomics).Numeracy can also improve the prospects for non-university level students. 9Researchers have found that a knowledge of algebra and geometry can make thedifference between a low score and a high score on most standard entry-level testsfor the civil service, and for many industrial occupations (Cockcroft, 1982; Tobias,1978). A final individual consequence of innumeracy is that adults suffering from “alow level of confidence in their constructive skills and critical insights [tend] to bedependent 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 ofindustrial production (in both quantity and quality), a waste of resources, theproduction of inaccurate or useless information, and a diminution in activecitizenship (Thorstad, 1992). From a purely business perspective, numeracy and itsrelated thinking skills are increasingly required by employers, particularly in thefast-growing high-technology fields of computers, environmental science, andbiotechnology. According to the Workforce 2000 study (Johnston & Packer, 1987),the proportion of jobs requiring the equivalent of four years of high schoolmathematics will be 60% greater in the 1990’s than it was in the 1970’s, while theproportion where only rudimentary math skills are used will decline by as much ashalf. In such circumstances, merely being able to remember a few mathematicalformulae--one commonly-accepted definition of mathematical ability—will no longerbe enough. In a world of rapidly changing technologies, incomplete and uncertaininformation, and unpredictable events, all workers must be able to do more thancompetently apply a given mathematical formula; they must know when to applythese procedures, and which ones to use.Another aspect of innumeracy in a social context is the underlying pattern ofinequity in adult numeracy; surveys of mathematical abilities indicate thatperformance is lower among women, working class, Hispanic, and Afro-American 10learners. Easley and Easley (cited in Willis, 1990) argue thatelitist attitudes about mathematics, and acute inequities in mathematicslearning, have become part of what oppress many groups who areeducationally disadvantaged on the basis of their gender, class, or race.Mathematics is powerful, but much of the power of school mathematicsresides not in the mathematics but in the myth of mathematics, in themeritocratic prestige of mathematics as an intellectual discipline. Knowledgeis power, particularly when that knowledge has high cultural value and isexclusive. (p. 17)Improving NumeracyIn British Columbia, if adults choose to improve their mathematicalknowledge and abilities, they have two main options. They can either undertake aprocess of individual, self-directed learning, purchasing one of several standard“refresher texts,” or they can enroll in remedial mathematics courses offered by adulteducation providers in their communities. Both the refresher texts and the organizedcourses cover much the same curriculum--both are designed to help adults preparefor and pass one of the standardized examinations (such as Math 10, 11, or 12, theGeneral Education Development Test, or the Adult Basic Education ProvincialDiploma) necessary for entrance to further education.Almost all of the locally-provided mathematics education is organized andcontrolled by the public education sector. For example, within the Acton area, boththe Acton School Board and the community college system offer a variety of “mathupgrading” courses to adults at several centres. Most adults in these courses aretrying to obtain one of four certificates (Dogwood, Adult Dogwood, CollegeProvincial, and GED) equivalent to high school completion. This pattern of provisionis repeated in most urban areas across North America.Much of this provision is intended to supply opportunities for “lifelong 11learning,” best described as “the opportunity for individuals to engage in purposefuland systematic learning throughout their lives” (Fans, 1992, p. 6). Within Canada,the federal discussion paper Learning Well...Living Well calls for the development ofa lifelong learning structure and an associated learning culture that includes theprovision of mathematics education for adults (Canada, 1991).For educators, this is quite significant, given the current dearth of informationabout the teaching of mathematics to adults. Central to the concept of lifelonglearning are certain widely-held assumptions about, and practices within, adulteducation, that are built on ideas and theories about how adults learn and should betaught. These ideas include: teaching must be problem-centered, it must emanatefrom the participants’ experience of life and develop the individual socially,participants must exert definite influence on the planning of the course and theconduct of the teaching, and techniques used must be based on an interchange ofexperience (Knowles, 1980). Although adult education policies within Canadaacknowledge the goals of lifelong learning (see, for example, Fans (1992), in relationto adult educationwithin British Columbia); how far the practice of adultmathematics education meets its ideals is currently undocumented.In addition, there is no published research in North America “relating to theunique aspects of teaching math to adults” (Gal, 1993, p. 14). Similarly, there is littleinternational research in this area despite UNESCO’s recognition of numeracy as akey 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 andunconnected.Given this background, my study focuses on the provision of mathematics 12education to adults within a typical adult education setting. In the next section Ipresent a justification for, and a statement of, my research questions.Research QuestionsIn the background to my study, I have shown that a degree of numeracy isconsidered a necessary skill for adults in order for them to be engaged citizens andproductive workers; further, I have shown that adults who wish to improve theirmathematical abilities have access to a variety of “upgrading” courses. Despite this,the mathematical abilities of many adults continue to cause concern in manyindustrialized countries. Nevertheless, little research has been conducted on adultnumeracy or the teaching of mathematics to adults.Much of the published material about innumeracy and the learning ofmathematics by adults is written from the viewpoint of government or industryleaders (e.g., National Research Council, 1989) or university professors ofmathematics (e.g., Paulos, 1988; Willoughby, 1990). These viewpointsoverwhelmingly reflect either policy-making and managerial perspectives or theacademic research interests of the profession. Further, they are often based onnarrow technical and instrumental models of education that ignore much adultlearning theory and the importance of such issues as self-concept, motivation,values, attitudes, and intentions in learning. What is missing from the publishedliterature 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, 13and used by, students--have not been a focus in mathematics education” (p. 111). Sheencourages researchers to seek answers to these questions, and to “use socioculturaltheories to help interpret their findings” (p. 111).If both the curricula and teaching practices used in adult mathematicsclassrooms are based solely on those of school-based education, then there is astrong possibility that adults are expected to repeat the approach to mathematicseducation that they faced when they were children. This situation may persist inspite of the myriad studies which have repeatedly identified that exposure toinappropriate curricula and poor teaching practices in mathematics education is akey source of adult innumeracy.In an attempt to address this, my study explores the teaching processes inmathematics education for adults. In particular, it examines how mathematicseducation is viewed by those involved in it, and how such education is shaped bycertain social and institutional forces. It seeks descriptive accounts of teachingprocesses in mathematics education from those missing perspectives, and relatesthose accounts to the ways in which teaching processes are influenced by externalfactors. By concentrating on descriptive accounts of teaching, I have been able toaccess “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 “goeson” in the classroom. Thus, teaching processes include the selection of content to betaught; the choice of such techniques as lectures or groupwork; the expectations,procedures and norms of the classroom; and the complex web of interactionsbetween teachers and learners, and between learners themselves.In an effort to illuminate the realities of mathematics education for adults, my 14study will consider the following broad questions:(1) What happens in adult mathematics classrooms?(2) What do these phenomena mean for those involved as teachers orlearners?(3) In what ways do certain factors beyond the teachers’ control affectteaching processes?Structure of the DissertationIn Chapter Two, I present, first, a survey of the literature on teaching ingeneral, and on mathematics in particular; and second, my theoretical frameworkderived from a synthesis of the literature on teaching. In this chapter, I also discusssome of the implications of combining macro and micro approaches to socialresearch.In Chapter Three, I present the methodological design of the research. Iprovide, 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 discussthe settings where mathematics education takes place, the people involved in thosesettings, and the work that they do.In Chapter Five, I focus specifically on the teaching processes in mathematics 15classrooms. I describe situations and episodes that are both typical and common inmathematics education in general, and on the teaching of algebra in particular. HereI include data from both my own observations, and from the perspectives of thoseinvolved as teachers and students.In Chapter Six, I analyze these teaching processes using concepts from mytheoretical framework. I identify certain frame factors in adult mathematicseducation and examine their effects on teachers’ thoughts and actions.Finally, in Chapter Seven I summarize my study and discuss certain of itslimitations. I also provide certain recommendations both for further research and forimproving the teaching of mathematics to adults.CHAPTER 2: LITERATURE REVIEW AND THEORETICAL FRAMEWORK 16Teaching is a soda! and political process, and therefore is subject to social andpolitical influences. Consequently, a thorough explanation of teaching processesmust have a conceptual framework that relates teaching processes to decisions takenin the social and political arena. Such relationships are, however, often invisible andunarticulated by the people involved. Rather than pinpoint them accurately, aresearcher can discern and record their effects by both observing teaching processesas they unfold in their natural settings and by examining them from the perspectivesof those involved as teachers or learners.In this chapter I describe the theoretical tools I used to so investigate teachingprocesses in mathematics education. First, I examine the literature on mathematicseducation for adults. Next, I turn to the adult education research literature, and thento the wider research on teaching in general. I categorize this research into threeparadigms, and, for each, provide a brief overview of research on teaching ingeneral, and on the teaching of mathematics in particular. During this discussion Iintroduce the two general domains of research which inform my own study:research on frame factors and on teachers’ thinking. These two domains can be seenas representing quite different approaches to the examination of social reality, andindicate that, if considered separately, teaching processes can be regarded as bothindependent of, and dependent upon, their broader social context. I next explore thedifferences between these approaches in terms of the macro/micro dichotomy andthe related issue of structure and agency, and I indicate how this study, whichcombines both macro and micro approaches, resolves these issues. Finally, I providea model of the theoretical framework of this study and briefly discuss the elementsof this model.Mathematics Education for Adults 17Despite the wealth of information available on the mathematical abilities of,and education for, school children of various ages, relatively little exists on that foradults. Within the English-speaking world, only Cockcroft (1982), ALBSU (1983),Statistics Canada (1991), and Kirsch, Jungeblut, Jenkins, and Kolstad (1993) providedetailed studies. Cockcroft’s (1982) survey was based on observations of almost 3,000British adults taking a test on their everyday or “practical” mathematics skills. Thefindings were supported by further evidence from a study conducted a year later,using data from the National Child Development Study that was based oninterviews from 12,50023-year olds (ALBSU, 1983). The Canadian report (StatisticsCanada, 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 interviewswith, and surveys of, over 25,000 US adults. All four studies reported that asignificant proportion of adults had problems with numerical calculations and citeddifficulties 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 thisresearch, however, has had little discernible impact on mathematics education foradults, which continues to be based on research on the learning of mathematics byschoolchildren (Fans, 1992; Gal, 1992). Many adult educators make strongdistinctions between adult and pre-adult education. Löthman (1992), in particular,identifies those distinctions relating to mathematics education. Two majordistinctions that affect teaching processes in mathematics education for adults arethose that concern adults’ beliefs about and attitudes towards mathematics, and theireveryday uses of mathematics.18Beliefs and Attitudes about MathematicsOne key skill required of an adult educator is to determine the existingconcepts, beliefs and attitudes held by adult learners (Brookfield, 1986). This is noless important in mathematics education than elsewhere. Several authors (e.g.,Buxton, 1981; Paulos, 1988; Quilter & Harper, 1988) stress that recognizing andacknowledging adults’ beliefs and attitudes about mathematics is key toencouraging learning. There is evidence to suggest that long before many childrenleave school they have adopted a view of mathematics as a cold, mechanical subjectwith little relation to “real life” (Paulos, 1988). Such children do their best to avoidmathematics wherever possible, and manifest anxiety when faced with even simplearithmetic problems. As these children grow into adults they “manage to organizetheir 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) havedeveloped the concept of “mathematics anxiety.” This has been described mostrigorously by Michael (1981) as “a psychological state engendered when a personexperiences (or expects to experience) a loss of self-esteem in confronting a situationinvolving mathematics” (p. 58). Much of the discussion of mathematics anxietylocates the problem and seeks to remedy it at the individual level (for example, bysuggesting that sufferers keep journals or work through self-paced learningmaterial). There are few suggestions for practical classroom activities.Löthman’s study also found that adults were able to learn mathematics betterif they could relate what they were learning to their everyday lives. Consequently, Inow examine studies on the relevance of mathematics to peoples’ daily lives.Daily Uses of Mathematics 19Because most research on the learning of mathematics is based on research inschool-based education, mathematics education tends to be defined in terms of aschool situation. However, in recent years, there have been a number of studies ofthe use of mathematics in the work and everyday life by adults of specificoccupations 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 todocument the ideas that were “embedded” in their everyday woodworkingactivities. She found that, although the carpenters had received very little formalmathematics education, they demonstrated tacit mathematical understanding intheir actions. They were fluent with, and made extensive use of, such conventionalmathematical concepts as congruence, symmetry, and proportion, and such skills asspatial visualization and logical reasoning.These studies are part of an emerging area of study in mathematics educationthat adopts a more anthropological approach in order to explore and describe themathematics that is created in different cultures and communities. D’Ambrosio(1991) uses the term “ethnomathematics” to describe “the art or technique ofunderstanding, 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 linkscultural anthropology, cognitive psychology, and mathematics, can challenge thedichotomy between “practical” and “abstract” mathematical knowledge. It forceslearners to consider others’ thinking patterns, to re-examine what has been labeled“non-mathematical,” and to reconceptualize what counts as mathematicalknowledge (Frankenstein & Powell, 1994).The work of several distinguished mathematics educators can illuminate 20these different understandings of mathematics. Ascher (1991) looks at themathematical ideas in the spatial ordering and numbering system used by the Incapeople in South America. Gerdes (1988) focuses on the mathematics “frozen” in thehistorical and current everyday practices of traditional Mozambican craftsmen,whose baskets, weavings, houses, and fish-traps often demonstrate complexmathematical thinking, as well as the most efficient solutions to constructionproblems. Pinxten (1983) examines spatial concepts in the cultural traditions of theNavajo people. Unlike Western people who tend to regard the world statically andatomistically, the Navajo have a more dynamic and holistic worldview whichfundamentally influences their notions of such geometrical concepts as points,distance, and space.Turning to the dominant culture within North America, Lave (1988) considersthe mathematical experiences inherent in common workplace and domesticactivities. In one example, she compares adults’ abilities to solve arithmetic problemsarising while grocery-shopping in a supermarket with their performance on similarproblems in a pencil-and-paper test. The participants’ scores on the arithmetic testaveraged 59%; in the supermarket they managed to make 98%--virtually error free.Lave argues that test-taking and grocery-shopping are very different activities, andpeople use different methods in different situations to solve what can be seen assimilar arithmetic problems.Drawing upon the mathematical traditions present in different cultures, andbasing mathematical activities on adults’ day-to-day experiences of their social andphysical environments broadens the traditional and often narrow approach of muchmathematics education. Furthermore, this brings the learning of mathematics “intocontact with a wide variety of disciplines, including art and design, history, andsocial studies, which it conventionally ignores. Such a holistic approach. . . serve[sl toaugment rather than fragment [learners’] understanding and imagination” (Joseph, 211987, p. 27).Although these studies show that different forms of mathematics aregenerated by different cultural groups, they can still be seen as the result of broadlysimilar activities. Bishop (1988) identifies six fundamental mathematical activitieswhich he regards as universal, necessary, and sufficient for the development ofmathematical 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 andsymbolizing that environment with models, diagrams, drawings, words, orother means.Measuring: quantifying amounts for the purposes of comparison andordering, using objects or tokens as measuring devices.Designing: creating a shape for an object or for any part of one’s spatialenvironment.Playing: devising and engaging in games and pastimes, with more or lessformal rules.Explaining: finding ways to represent relationships between phenomena. (pp.182-183)All of these anthropological studies document the distinctive character of themathematical skills and procedures used in work and everyday life, as comparedwith those taught in school mathematics. They also highlight the success of suchprocedures when used in particular contexts. However, the use of these proceduresoutside of the specific situations where they normally occur is problematic. Forexample, street vendors who successfully perform many relevant calculations dailyin their heads find “similar” calculations to be performed with pencil and paper,outside the context of the Street market, exceedingly difficult, and make many moreerrors (Carraher et al., 1985).Sewell (1981) studied this phenomenon in considerable detail. Her study 22relied upon interview data designed to cover four areas: a discussion of selectedsituations related to shopping and household tasks, in which mathematics might beinvolved; a discussion of other matters such as reading timetables and usingcalculators; attitudes to mathematics; and background information. Initially, 107adults, chosen to reflect the range of expected mathematical abilities and ofoccupation, were interviewed. Next, follow-up interviews of greater length wereconducted with about half of those who had taken part in the first stage. Thoseinterviewed were invited to answer a series of questions about a range ofmathematical situations. Of these questions, some involved calculations, othersrequired an explanation of method but no calculation, and others required theexplanation of information expressed in mathematical terms.Sewell’s findings indicate that there are many adults who are unable to copeconfidently and competently with any everyday situation that requires the use ofmathematics. Further, she found that the need to use mathematics could inducefeelings of anxiety, helplessness, fear, and guilt. These feelings were especiallymarked among those with high academic qualifications, and who, consequently, feltthat they ought to have a confident understanding of mathematics. Further findingsincluded a widespread sense of inadequacy amongst those who felt they either hadnot used the proper method, obtained the exact answer, or performed with sufficientspeed 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 tomathematics; information that can have many implications for adult educators. Inparticular, research on teaching mathematics can be informed by what these studiesreveal about learning. I now turn to the adult education research literature toexamine how these issues have been studied.23Adult Education ResearchGiven the wealth of information on adults’ attitudes towards, and daily usesof, mathematics, research on the teaching of mathematics to adults is, surprisingly,almost non-existent. Indeed, specific adult education research on teaching in generalis limited; teachers and teaching are not the main focus of discussions about adulteducation practice. For example, the most recent Handbook of Adult andContinuing Education (Merriam & Cunningham, 1989) cites no references to eitherteachers or teaching in its subject index. Further, of the key surveys of thedevelopments in adult education research, theory, and practice (Jensen, Liveright, &Hallenback, 1964; Long, 1983; Peters, Jarvis, & Associates, 1991) only that of Longcontains any discussions of research on teaching. Although the adult educationliterature 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 forpractitioners and concentrate on describing strategies and tactics for improvingadult learning.When describing the teaching/learning process, adult educators tend to focuslargely on learners and their learning; several authors (e.g., Apps, 1991; Brookfield,1986; Knowles, 1980; Knox, 1990) pragmatically define teaching or the teachingprocess solely as the process of facilitating or helping adults learn. This view has ledto further empirical research that has contributed towards developing principles ofgood 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 centralprinciples and then designed a research instrument to examine the extent to which 24these principles were exemplified in practical settings. Much of this work, however,although adding to the corpus of research on teaching, has remained theoreticallyplurative. In addition, these studies tend to regard adult education in general, andbecause few are based on empirically collected data in adult classrooms, downplaythe influence of subject-matter or situational context. Yet, as Anyon (1981) andStodoisky (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 considerthe teaching of mathematics in formal settings. (The others describe particularcourses set up for specific groups of people, or to tackle specific issues.) Löthman’sstudy makes several theoretical and methodological contributions to my own study,and I discuss it more fully later.Hoghielm (1985) investigated mathematics teaching in Swedish municipaladult schools to determine the extent to which teaching was in accordance withcertain principles of adult education. These principles were codified in a SwedishGovernment Bill as “the most appropriate ideals for the teaching of adults.” Theyinclude:Teaching must emanate from the participants’ experiences of life, [it] mustdevelop the individual socially, [it] must be problem-oriented, the techniquesmust be based on an interchange of experience, participants must exertdefinite influence on the planning of the course and the conduct of teaching,and evaluation must comprise a mutual (teacher-participant) measurement ofcourse content and planning. (Hoghielm, 1985, pp. 207-208)Apart from these latter two studies, a pragmatic approach is common (at leastwithin North America) to adult education research in general. Further, the adulteducation 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 25of overall focus. As Blunt notes,Meetings of adult education researchers and their discussions about howresearch ought to be conducted. . . and disseminated. . .are characterized bydivision [and] disagreement... .The differences... also extend to disagreementsover what research problems ought to be identified as priorities and theusefulness of the research results produced to date. (1994, p. 168)What is certain about recent adult education research is the diversity of itsmethods, approaches, topics for study, and purposes. Rubenson (1982, 1989) hasidentified how North American adult education research has focused more onpragmatic program needs and pedagogical concerns than it has on theorydevelopment or policy-related issues. Also, much of the research has beendominated by a narrow reliance on a psychological approach, rather than onanthropological, historical, philosophical, or sociological approaches; consequently,adult education research in general has not been well informed by these differentapproaches or disciplines. Much current adult education research appears to beunconcerned with developing a stronger theoretical base or with drawing uponresearch in other disciplines.The result of such a pragmatic and atheoretical approach within adulteducation is that it hasn’t contributed substantially to the wider field of educationalresearch, or indeed, to wider economic and political issues and questions. Further, ithas attempted to deal, on an individualistic and local basis, with the effects of socialand political processes, but has not effectively addressed the causes of them in anymeaningful way. Given that adult education is heavily influenced by social andpolitical forces, this seems an unexplored opportunity for the field.As a source for theoretical exploration for the exploration of teachingprocesses, then, the adult education literature is barren. Consequently, I have to nowturn to the wider literature on teaching in general.26Research on TeachingResearch on teaching has gone through several periods of change. During thepast 80 or so years, this research has become successively more comprehensive andcomplex in its foci of study, theoretical sophistication, and methodological rigor.Rosenshine (1979) and Medley (1979) both present historical overviews of researchon teaching, describing the changes in terms of cydes or phases. For Rosenshine,research on teaching initially focused on teacher personality and characteristics, thenon teacher-student interactions, and finally on student attention and subject content.Medley presents a similar view, and categorizes research as focusing first oncharacteristics of effective teachers, then on the methods they used, next on teacherbehaviors 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 beendominant, which has led, in turn, to different goals, starting points, methods, andinterpretations for research (Shulman, 1986). Three distinct paradigms can bediscerned in the development of research on teaching, which I have chosen to labelas behaviorist, structuralist, and interpretivist. Although these developments inapproach broadly correspond to historical periods, they are not uniquely tied tothem. For example, although the positivist period saw a predominant focus onpsychological research models, much psychological research is currently beingconducted in the more interpretive tradition, influenced by recent developments incognitive 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 27Until the 1970s, almost all research on teaching was behaviorist and empiricalin 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 designedto formulate and verify particular hypotheses. It is not surprising, therefore, todiscover that reports of studies on teaching from this period were substantiallyempirical and used such techniques as experiments and surveys to produce solelyquantitative data. For example, Gage’s (1963) handbook of research on teachingcontained no section on participant observational research. Further, Dunkin andBiddies (1974) comprehensive survey of studies on teaching contains only reports ofresearch that employ quantifiable measures; it mentions no others from morequalitative or interpretivist perspectives (Shulman, 1986).Such research tended to concentrate on the development of normative laws ormodels about educational goals, content, and methods of instruction, and wasprimarily based on psychological perspectives, particularly that of behaviorism.Consequently,. theories and models about teaching in this period were principallyderived 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 howknowledge about learning can affect teaching practices. For example, bothThorndike (1923) and Skinner (1968) argued that ideas about teaching should bebased on theories about learning. Because so much of this work is conducted from apurely behaviorist perspective, it therefore focuses on the outcomes of learningrather than on how learning occurs. Consequently, most research studies have beendesigned to investigate what changes in teaching could produce measurable benefitsin student learning. For specific examples of this, I now turn to the research on 28teaching mathematics.Behaviorist Research on Teaching MathematicsWithin what I am calling the behaviorist paradigm, the research on teachingmathematics has been based almost exclusively on theories or assumptions abouthow children learn. A key emphasis has been its pragmatic focus on what makessuch teaching more efficient or effective; namely, what improves studentachievement. Also, most of the studies in this paradigm on teaching mathematicshave focused solely on the behaviors of teachers rather than on those of learnersand/or on the lesson content. They have sought to illuminate student learning onlyin light of teachers’ actions; the “culture” of the classroom, lesson content, or studentbehavior or understanding have been of little consideration.Perhaps because of its predominantly behaviorist approach, most research onthe teaching of mathematics within this paradigm has tended to isolate a particular“variable” and determine its effects as it was experimentally controlled. Reviewingseveral studies gives a flavor of the research: how children learn numericaloperations (Bell, Fischbein, and Greer, 1984); the stages of children’s learning(Donaldson, 1978); time spent on task (Peterson, Swing, Stark & Waas, 1984); andseatwork (Anderson, 1981). In general, by focusing on the learning of individuals,such research has attempted to search for clusters of common characteristics fromwhich to generalize about particular types of teachers or learners, and to offerpredictions for successful ways of teaching mathematics (Nickson, 1992).Romberg and Carpenter (1986), in a summary of reviews of recent researchstudies on the teaching of mathematics in this behaviorist paradigm, identifiedseveral overall conceptual aspects that concern them about this research. First, theyfound that much research suffered from inadequate conceptualization, and was 29theoretically weak and haphazard in its choice of which teaching behaviors to study.Researchers used “different labels for the same behavior, or the same label fordifferent behaviors, [and] different coding procedures which yield[ed] differentfrequencies” (p. 860).Second, lacking substantive theories of teaching, researchers tended to focuson methodological questions. As most research was of an experimental design,researchers then concentrated on “improving research designs, providing betteroperational definitions of variables, or devising more adequate procedures forcounting behaviors, and better techniques of statistical analysis” (p. 860). Such aconcentration not only limited the kinds of problems addressed but also the ways inwhich they were conceptualized.Third, most studies were regarded as being too “global” in that theydisregarded the content of lessons. For example, researchers tended to ignore thespecific content of what was being taught to specific sorts of students, or assumedthat it lay outside the scope of inquiry. Romberg and Carpenter describe an earlierstudy (Romberg, Small, & Carnahan, 1979) that “located hundreds of studies thatassessed 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 dependentvariable. Further, in order to operationalize notions of students’ achievements andattitudes, researchers relied overwhelmingly on standardized achievement tests.However, as Romberg and Carpenter saySuch tests have serious problems. They rarely reflect what was taught in anyone teacher’s classroom; when used with young, bilingual, or lowersocioeconomic status children, they may yield biased results; and at best, theyindicate only the number of correct answers produced by a student, not howa problem was worked... .Their use merely compounds the problems whenthere is a lack of concern for the content being taught. (p. 861)Romberg and Carpenter also discuss four major findings of their review of 30research. The first concerns the variability of teaching practices. As they describe it,“Every day is different in every classroom [and] every classroom is different fromevery other classroom” (p. 861). The variability extends to teachers’ and students’behaviors, texts, time allocated, and content coverage. However, despite theseseveral variations, the dominant pattern of teaching practices, in a wide range ofclassrooms, was “to emphasize skill development via worksheets, not to selectactivities 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 formathematics instruction, those teachers who consistently devoted less time toteaching mathematics than did their colleagues experienced poorer studentachievement. Further, studies showed that the time available for mathematics ismost effective when it is well-used in terms of its content coverage, episodic nature,and interactive engagement. “Students should be engaged in activities that arereasonable and intentional.. . .Lessons and units should have a. . . start, adevelopment, a climax, and a summary... .Finally, [students should] be.. .interactingwith ideas” (p. 863).The third finding was that student learning was increased if teachers devotedpart of each lesson towards increasing students’ comprehension of skills andconcepts. If teachers helped students relate new ideas to past and future ideas, thenboth student engagement and achievement was increased. This process was alsoincreased if students were required to work in small groups. Those students whostudied in small groups were found to be not only more cooperative and lesscompetitive than their peers, but also to have a greater comprehension of how ideaswere linked (Noddings, 1985; Weissglass, 1993).A fourth finding concerns classroom management. Although the primary 31purposes of teachers’ behaviors were to cover the assigned content and get theirstudents to learn something, they were also designed to maintain classroom orderand control. For example, teachers would occasionally adapt materials not toincrease 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 textbookswas also significant. Throughout most studies, teachers would promote the textbook“as the authority on knowledge and the guide to learning” (p. 867). Althoughteachers could have departed from the syllabus, Romberg and Carpenter found thatthey chose to do so only to increase their classroom control.In sum, the behaviorist approaches to research on teaching mathematics andteaching regarded it, in general, as a predominantly one-to-one activity between ateacher and a student. Missing from this approach was any adequateconceptualization of education that linked teaching with more cultural, political, andsocial factors. In addition, there was no research focus on the specific nature ofoccurrences and events, let alone the meanings that these events had for the peopleinvolved. These two areas were developed more in subsequent research approaches.Next I consider a paradigm that sought to illuminate the links between educationand social influences.The Structuralist ParadigmIn the late 1960s and early 1970s, educational researchers began to adopt moresociological approaches in their studies of teaching. This research tended to fall intotwo contrasting positions about how issues were approached and interpreted: theconsensus and conflict perspectives. Briefly, the consensus perspective is based upon 32the notion that “societies cannot survive unless their members share at least someperceptions, attitudes and values in common” (Rubenson, 1989a, p. 53). Education isregarded as, first, an agent of socialization into the broadly-accepted values ofsociety, and, second, a means of selection of individuals for particular societal rolesbased 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 ornecessary. By focusing more on the interests of various groups and individualswithin society (rather than on society as a unified whole), conflict theoristsemphasize “competing interests, elements of domination, exploitation and coercion”(Rubenson, 1989, p. 54). The conflict perspective also promoted critical analysis ofthe 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 thedominant values of society. However, the radical critics of the 60s and 70schallenged the dominant liberal view of education as merely offering opportunitiesfor individual development, social mobility, and a redistribution of political andeconomic power. They argued instead that the main function of education is toreproduce the dominant cultural and political ideology, its forms of knowledge, andthe social division of labor (Aronowitz & Giroux, 1993). Before turning to particularstudies on teaching mathematics that adopt this approach, I first outline someanalytic tools which commentators have used to describe how society influenceseducation.Societal Influences on Education 33To radical critics, education has several other functions that are not expressedin curricular content, and which often remain invisible to those involved. Forexample, within North America, several authors promote the idea that schools andother educational institutions exist to “colonize” students into accepting the culture,values, norms, purposes, and goals of the dominant class. This view is fully exploredtheoretically in the work of Apple (1979), Bowles and Gintis (1976), Carnoy andLevin (1985) and the early work of Giroux (1981).Most of these views about the roles and functions of educational institutionsin society are based on the earlier work of Althusser (1971) and Gramsci (1971), bothof whom emphasized how educational institutions transmit and maintain society’sdominant ideologies. In particular, they both identified how the needs of thedominant 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 generalstratifying processes. In contrast to Althusser and Gramsci, Bourdieu regardseducational institutions less as agents of state control, and more as relativelyautonomous bodies that are indirectly influenced by more powerful economic andpolitical institutions. He maintains that various types of “capital”—either economic(money, objects), social (positions, networks), cultural (skills, credentials), orsymbolic (legitimating codes)--are distributed unequally based on social class. Foreach class, there is a distinct culture--”habitus”--which is the collection of largelyunconscious perceptions, choices, preferences, and behaviors or members of thatclass. Children learn within their habitus, acquire capital from their parents andfrom peers, acquire academic credentials (one form of cultural capital), and then, inturn, exchange this for other forms of capital. Thus, educational credentials becomeone of the key media for the purchasing and exchanging of one kind of capital foranother.By seeking to align actual classroom processes with the ways that educationfunctions within society most of these radical researchers have emerged using alargely structuralist (or “macro”) approach. This approach assumes that societalinfluences determine classroom behavior. As such, these researchers have focusedon large-scale theoretical explanations of the relationship between schooling andsociety (e.g., Bowles & Gintis, 1976), or certain aspects of social structures (such asgender, ethnicity, or class) as if they were causal variables (e.g., Young, 1971). Instudies such as these, the freedom of action that people have within dassroomsituations, or the meanings they make about those situations, are largelydownplayed or even ignored.As an alternative to these large-scale approaches, other researchers haveconsidered small-scale studies of individual schools, teachers, or specific classroominteractions (e.g., Ball, 1981; Donovan, 1984; Hammersley & Woods, 1984). These“micro” approaches have typically focused on individual actors, regarding them asautonomous actors in situations and subject to few outside constraints. Theseapproaches are more concerned with the subjective meanings that actors hold aboutthe particular situations in which they find themselves, and the human actions andinteractions that take place there.These two approaches have tended to be polarized and regarded asincompatible; researchers have, in general, adopted either one approach or the other.There have been few attempts to reconcile the macro-micro issue, or to designresearch that bridges both perspectives. Hargreaves (1985) notes that although themacro-micro issue has been the subject of a great deal of theoretical debate, it has notresulted in much empirical research.However, a comprehensive examination of any social phenomenon—such asteaching--cannot be limited tO a set of either external (macro) or internal (micro)explanations or theories. Teaching can neither be reduced to psychological principlesor laws of learning, nor can it be seen as simply determined by contextual factors. Tobe thorough, a study must attempt to bridge these two perspectives and incorporateboth macro and micro approaches. Dahllöf (1977) summarizes some characteristicsof what such a model and a methodology would include:Data are curriculum-related, reflecting the goals and intentions of theinstructional program as well as the ambitions of the teacher.Data are related to basic patterns of teaching.. .and reflect the cumulativecharacter of the teaching process and its long term effects.Data mirror the teaching process as a continuous change of perceptions andbehaviors over time towards certain goals.The analysis [considers] that the teaching of a certain curriculum unitgenerally run[s] through a series of phases like presentation, training, andcontrol--each phase with its own characteristic pattern [of] communicationand interaction.Data are dynamic.. .in that they relate in a meaningful way to the restrictionsthat are imposed upon most teaching situations by frame factors like spaceand time, [and] they try to describe and do justice to the role played bystudents in the teaching situation and its different phases. (p. 406-407)“Frame factor” theory is one particularly useful tool of analysis that meetsDahllöf’s criteria and integrates both the macro and micro approaches. Because itbears on the theoretical framework developed for this study, it warrants detailedexamination here.Frame Factor TheoryFrame factor theory (Bernstein 1971, 1975; Dahllöf, 1971; Lundgren, 1977,1981) analyzes the ways in which teaching processes are chosen, developed, andconstrained by certain frames. In contrast to research in the behaviorist paradigmthat investigated teaching processes by examining how changes in teachers’ behavior 36affected student learning, frame factor theory is more concerned with exploring howteachers’ actions are limited by external forces.Briefly, a frame is “anything that limits the teaching process and isdetermined outside of the control of the teacher” (Lundgren, 1981, p. 36). Examplesof frames include the physical settings of teaching, curricular factors such as thesyllabi or the textbooks used, and organizational influences such as the size of classor the time available for teaching. Frame factor theory claims that teaching processesare 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 processeshave; the actual teaching is conducted within those limits.The concept of frames as a constraint on teaching processes was firstdeveloped by Bernstein (1971, 1975) and Dahllöf (1971). Bernstein refers to a frame inthe “form of the context in which knowledge is transmitted and received. . . thespecific pedagogical relationship of the teacher and [the] taught” (1971, p. 50). Heexplains that frames refer to the degree of control that teachers and learners “possessover the selection, organization, and pacing of the knowledge transmitted andreceived in the pedagogical relationship” (p. 50). Dahllöf describes frames morebroadly, extending Bernstein’s earlier notion to include the decisions made aboutteaching that are outside of the teacher’s and the student’s control. Dahllöf’s usagetherefore links the macro- and micro- aspects of analysis in a way that Bernstein’sdoes not.Lundgren (1972) conducted a study of students grouped by ability in Swedishhigh-school classrooms using Dahllöf’s definition of frames. He developed a modelof three types of frame factors: the goals or objectives of teaching a particular subjectarea, the sequence of content units (lessons) through which the goals were to beachieved, and the time needed by students to master the content. Each studentneeded different amounts of time to learn new material, and this was related to whatcontent was being taught and how it was taught. Lundgren found that, in order todeal with those situations in which there was insufficient time to teach the requiredcontent to all the students, teachers created a “steering group” of students. Whenhaving to choose whether to continue with a particular topic area or whether tomove on to the next, even though not all of the students had fully learned theexisting material, teachers would base their decision upon the demonstrations ofability 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 areinextricably linked, he argued that because the cultural, political, economic, andsocial structures of society have an effect on education, they can be regarded asframes, and therefore studied in research on teaching situations. Institutions such asschools and collegespromote learning in terms of postulated knowledge, skills, attitudes, andvalues. Legislation and rules prescribe the form of this institution, while theavailable resources in terms of personnel, teaching aides and composition ofstudents determine how the actual teaching corresponds to the formal goalsand regulations. (1979, p. 20)Hence, for Lundgren, frames are the realization of fundamental structuralconditions. In his own studies he identified that time, curricula, regulations,personnel, teaching aids, and the composition and size of classes act as the mostvisible frames that govern and constrain the teaching processes. In his later work(noted in Elgstrom and Riis, 1992), Lundgren has also included more conceptualconstraints 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 isintegral to frame factor theory. Stable patterns of classroom interactions can bediscovered by studying teaching processes, and then seen as “realizations ofunderlying rules that shape and steer the process.. . .As society is governed by certain 38rules for interpersonal relations and by social perceptions, teaching is governed byframes and perceptions that functionally form the rules for the participants” (Torpor,1994, p. 2375).Since then, particularly in Swedish educational research, frame factor theoryhas been regularly applied to classroom studies (e.g., Englund, 1986; Gustafsson,1977; Kallós & Lundgren, 1979; Pedro, 1981) at both preschool and high schoollevels. It is ideal for research that seeks to analyze teaching processes in terms oftheir links with more structural elements. The factors governing, steering, andcontrolling teaching processes are always subject to change, so, as Torper (1994) putsit, “Frame factor theory with its wide scope and its ambition to encompass the deepstructures of society, is well suited to the task of analyzing these processes” (p. 2376).Structuralist Research on Teaching MathematicsAlthough the structuralist orientation to research on teaching is theoreticallyrich, empirical studies of teaching from this approach are more rare. However,Lerman (1990) and Anyon (1981) each provide a specific example from mathematicseducation. Lerman initially identified several predominant views in general societyabout mathematics and their possible influence on mathematics education, and thenconducted a field study among mathematics teachers to explore some of the issuesarising from his theoretical perspective. He found that teachers’ conceptions ofmathematics clearly affected their teaching.Anyon studied mathematics teaching in five schools at different socioeconomic levels and found that, although all the schools used the same textbooks,the teaching differed dramatically. Teachers in the two working-class schoolsfocused on procedure without explanation or attempts at helping studentsunderstand. Teachers in the middle-class school attempted more flexibility andmade some efforts towards developing student understanding. At the “professionallevel” school, teachers emphasized discovery and experience as a basis for theconstruction of mathematical knowledge. Finally, teachers at the “executive-class”school extended the discovery approach, and used enhanced instruction onproblem-solving and encouraged students to justify their answers to demonstratetheir mastery of the concepts.Although structuralist studies of teaching mathematics are infrequent, similarapproaches to mathematics education in general are more common. Severalmathematics educators (e.g., Evans, 1989, Fasheh, 1982; Frankenstein, 1981, 1987,1989; Mellin-Olsen, 1987) are interested in the “culture” and values that aretransmitted in traditional mathematics education. They note that the curricula andcommonly used teaching methods are designed to reproduce the existing economic,status, and power hierarchies, and socialize learners into accepting the status quo. Tothese educators in particular, the traditional mathematics curriculum consists of anabstract and hierarchical series of objective and decontextualized facts, rules, andanswers. Much of this curriculum covers a fixed body of knowledge and core skillslargely unchanged for centuries. It is based on the assumption that learners absorbwhat has been covered by repetition and practice, and then become able to applythis knowledge and these skills to a variety of problems and contexts.Further, they regard teaching methods in traditional mathematics educationas using largely authoritarian and individualizing techniques that depend onmemorization, rote calculation, and frequent testing (Bishop, 1988). These methodsconvince learners that they are stupid and inferior if they can’t do simplecalculations, that they have no knowledge worth sharing, and that they are cheatingif they work with others. When education is so presented as a one-way transmissionof knowledge from teachers, mathematics can be regarded merely as collections offacts and answers. Knowledge is seen as largely separate from learners’ thought 40processes, and education is experienced as a static, rather than a dynamic, process.As Frankenstein describes it, much mathematics teaching isbased on what Freire calls “banking” methods: “expert” teachers depositknowledge in the blank minds of students; students memorize the requiredrules and expect future dividends. At best, such courses make peopleminimally proficient in basic math and able to get somewhat better payingjobs than those who can’t pass math skills competence tests. But they do nothelp people learn to think critically or to use numbers in their daily lives. Atworst, 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 inwhich little effort has been made to place the subject matter in any meaningfulcontext. For many, mathematics remains a mystery unrelated to other subjects orproblems in the real world; they often come to regard mathematics as a subjectlargely irrelevant to their own lives.Other critics of traditional mathematics education have questioned its aimsand purposes. In general, two rationales are given for why mathematics should betaught: (1) Mathematics is necessary for personal life and a prerequisite for manycareers; and (2) Mathematics improves thinking, because it trains people to beanalytical, logical and precise, and it provides mental exercise. Of course, theserationales do not specify what mathematics should be taught, merely that somemathematics should be. One could expect, therefore, that mathematics educationwould differ substantially from place to place. It is surprising, then, that oneresearcher discovered there was little diversity in mathematics classrooms the worldover (Willis, 1990).Ernest (1990) notes that the aims of mathematics education in any location areoften discussed in isolation from any social and political content. Arguing thateducation in society reproduces its social structure, he distinguishes three groupswho have distinct aims for mathematics education: mathematics educators,mathematicians, and representatives of business and industry. To these, Howson 41and Mellin-Olsen (1986) add further categories of parents, employers, and those inhigher education.These authors daim that the aims of mathematics education are not decidedon rational or educational grounds but on the basis of the power of these groups toeffect change. For example, Ernest (1990) explains the changes in British mathematicseducation during the 1960s as a result of a struggle between certain groups he callsthe “Industrial Trainers” (who emphasized a ‘back-to-basics” approach involvingdrills and rote learning), the “Old Humanists” (who were proponents of mathematicsfor its own sake, stressing its logic, rigor, and beauty), the “Public Educators” (whosaw mathematics as a means to empower students to critically examine the uses of,and political and social issues surrounding, mathematics), the “TechnologicalPragmatists” (who believed in teaching mathematics through its applications andemphasized practical problems and utilitarian problem-solving skills), and the“Progressive Educators” (who emphasized student-centered teaching, activelearning, creativity and self-expression).The authors studying the aims of mathematics education argue that it is theform rather than the content which conveys those social aims. The ways thatmathematics is taught “can emphasize and reinforce the values and relationshipsthat 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 studysaw science and mathematics as “heavily-laden with social values”, andrecognized that scientific and mathematical knowledge “may function moreand more as a behavioral badge of eligibility for employment” and...wantedhelp in inculcating the work ethic values they saw as important in presentsociety (p. 152).Each of these authors considers only mathematics education for children andtheir arguments cannot be necessarily applied to adult mathematics education.Education for children and adult education differ significantly in most Western 42countries. Löthman (1992), in particular, highlights the differences between adultand childrens’ mathematics education and between how children and how adultswanted to be taught mathematics. She found that adults, in particular, wanted to beable to use the mathematics they learned, and, therefore, wished to be taught bypractical methods. She therefore argues that there should be substantial differencesbetween the classroom practices and the course content used in adult settings, andthose used in childrens’ education.However, in most mathematics classes for adults, the curriculum appears tofollow that of school-based mathematics education, itself largely determined by therequirements of college entrance boards. Within British Columbia, many adultlearners of mathematics are “following the same curriculum and using the samematerials as their youthful colleagues” (Fans, 1992, p. 30). Thus, it is possible that theteaching processes in mathematics education for adults closely resemble those in K-12 education.All of these authors show how dominant views of mathematics affect howmathematics is considered and taught. In this way, the dominant conceptions ofmathematics can be seen as frame factors influencing and restraining teachingprocesses in mathematics education. These examples from mathematics education,frame factor theory, and the structuralist approach to research on teaching all share aconcern to explain how education functions in relation to social production, andhow, in turn, social and political influences surface in educational settings. Lackingin this approach is much consideration for people as autonomous actors insituations. The structuralist paradigm tends to view people as being passivelysocialized into an institutional framework rather than “participating in their ownconceptual constructions of the world and [their] own fate as a project” (Sharp &Green, 1975, p. 5). The third paradigm seeks to respond to this somewhatfunctionalist approach to teaching by foregrounding the roles of autonomous actorswithin social structures.The Interpretivist ParadigmFor many researchers, the structuralist paradigm is overly deterministic. Forthem, teaching is not merely the result of external factors but is also heavilyinfluenced by what teachers think and do. Some researchers (e.g., McLaren, 1989,Willis, 1977) identified a need to document specific details of classroom interactionsin order to understand how immediate and local circumstances reflected broaderstructural forces. Other researchers, disenchanted with structuralism, but even lesscaptivated with the predominantly behaviorist approach to studying teaching, beganto focus more on the specific nature of educational occurrences and events and themeanings that these have for the people involved. Their studies adopted qualitativeor “interpretive” perspectives and studied teaching from ethnographic, participantobservation, case study, symbolic interactionist, phenomenological, or constructivistapproaches. While these several approaches differ from each other slightly, they allshare a central research interest in discovering the meanings that people (whetherparticipants or researchers) make about aspects of human life and humaninteractions.The rich variety of this research can be gleaned from considering thework of, for example, Fox (1983), Samuelowicz and Bain (1992), and Pratt (1992), andthe studies in Marton, Hounsell, and Entwhistle (1984).Rather than consider the general character and overall distribution ofeducational events and situations, interpretive studies of teaching focus more on thespecifics of particular situations or events. Such studies deliberately focus on theperspectives of the people involved, and seek their meanings and interpretationsabout their situations. By concentrating on specific situations and actions, and on the“local” meanings actors give to these, qualitative research has attempted to uncoverthe “invisibility of daily life” (Erickson, 1986, p. 121). Typical of the research in theinterpretivist paradigm are studies concerning teachers’ thinking. This includes suchfoci as teachers’ beliefs about students and teaching, their thought processes whileplanning instruction, and the kinds of decisions they make during teaching. Becausethis research also informs the theoretical framework for my own study, it warrantsclose examination here.Teachers’ ThinkingA large part of the context of teaching consists of the thinking, planning,decision-making, and actions of teachers. Researchers from all three paradigmsagree 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 involveperception, reflection, problem-solving, and the manipulation of ideas, and isconcerned with how knowledge itself is acquired and used (Calderhead, 1987).This research regards teachers as active and autonomous agents in teachingsituations and seeks to explore new ways of conceptualizing and understandingteaching. Research has focused on, for example, the nature of teachers’ knowledge(Zeicher, Tabachnik, & Densmore, 1987), the differences in the use of knowledgebetween 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’ theoriesand beliefs about students, teaching, learning, and subject matter (Clark & Peterson,1986).Clark and Peterson (1986) have developed a model that relates teachers’ 45thoughts to their actions, considering such aspects as teachers’ thoughts, decisions,theories, and beliefs. Their model is based on an interpretive perspective thataddresses such questions as, for example, differences in meaning regarding learners’achievements, and regarding the teacher’s role in dassroom interactions. The modelconsists of two domains involved in the teaching process: teachers’ thoughtprocesses 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). Thedomain of teachers’ actions and effects includes teachers’ classroom behavior,students’ classroom behavior, and student achievement. In both domains, theelements are seen as inter-related and their relationships as cyclical and reciprocalrather than linearly causal. For example, in the “action” domain, teacher behavior isseen as affecting student behavior, which, in turn, affects both teacher behavior andstudent achievement. Student achievement can cause teachers to behave differentlytowards the student, which then, in turn, affects student behavior and studentachievement.Clark and Peterson identify a difference concerning the domains which hasimplications for research. Teachers’ behavior, and its effects (e.g., student behavior,and student achievement scores) are observable phenomena. In contrast, becauseteachers’ thought processes occur “inside teachers’ heads,” they are unobservable,and hence must be investigated by a more interpretive approach. Further, until fairlyrecently, the relationship between the two domains was considered unidirectionaland causal; they followed a “process-product” model that assumed a causal chainbetween teachers’ thinking, teachers’ dassroom behavior, learners’ classroombehavior, and, finally, learners’ achievement. However, these domains are now seenas interacting in the reciprocal and cyclical way described above. Teachers’ thinkingaffects their actions, which in turn, influence their subsequent thinking. Thisreciprocity suggests that by examining the two domains together, teaching processes 46can be more fully understood.Interpretivist Research on Teaching MathematicsWithin the interpretivist paradigm, there have been attempts to draw someteaching implications from recent research in cognitive science, particularly thatconcerning constructivism (e.g., Resnick, 1987) or metacognition (e.g., Schoenfeld,1985, 1987). Much of this research has focused on the belief that learners constructknowledge rather than passively absorb what they are told. This has significantimplications for a subject such as mathematics, which has enjoyed a rather unusualstatus as a fixed body of knowledge and core skills.Views on the nature of mathematics range from “a discipline characterized byaccurate results and infallible procedures” (Thompson, 1992, p. 127) somewhat “akinto a tree of knowledge [where] formulas, theorems, and results hang like ripe fruitsto be plucked” (Steen, 1988, p. 611) to a human activity that “deals with ideas. Notpencil 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” byDossey (1992), “Absolutist” and “Fallibilist” by Lerman (1990), and, perhaps mostsimply as “external” and “internal” by Polya (1963). Despite their appellation, thepoles correspond broadly to a view of mathematics either as fixed, certain, valuefree, abstract, and unchallengeable, or as dynamic, relative, constructed, andnegotiable.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 andvalues of humanity” Ernest (1991, p. xi). However, in the past 20 years, mathematicshas undergone a “Kunhian revolution,” in which several philosophers (e.g., Lakatos,1976; Davis & Hersh, 1980) have regarded mathematics as more “fallible andchanging, and like any other body of knowledge, the product of humaninventiveness” (Ernest, 1991, p. xi). This philosophical shift has a significance thatgoes far beyond mathematics. For, as Ernest maintains, “mathematics is understoodto be the most certain part of human knowledge, its cornerstone. If its certainty isquestioned, the outcome may be that human beings have no certain knowledge atall.” (p. xi).Mathematics, as a school subject, has been largely unchanged for many years.Indeed, since the commercial and navigational needs of fifteenth century Europebegan to demand an educational provision to improve arithmetic skills, much of themathematics taught in formal settings has remained unaltered. National systems ofeducation (that included mathematics as a school subject) were founded in Franceand Prussia at the beginning of the nineteenth century, in England some 50 yearslater, and in North America shortly after that (Howson, 1990). Within those systems,the mathematical curricula gradually expanded from commercial arithmetic toinclude successively algebra, geometry, trigonometry, and finally, in the earlytwentieth century, calculus. Since then, within North America, there has been“constant reform rhetoric but little actual reform of the school mathematicscurriculum” (Stanic & Kilpatrick, 1992, p. 407).Within a particular topic area--such as, for example, that of algebra--there hasalso 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 1900sas including: “the simplification of literal expressions, the forming and solvingof.. .equations, the use of these techniques to find answers to problems, and practicewith ratios, proportions, powers, and roots” (p. 391). These topics are identical tothose in a beginning algebra course in the 1990s (see Appendix 10).Mathematics teaching, traditionally, has been based on the assumption that 48learners absorb what has been covered by repetition and practice, and that they thenbecome able to apply this knowledge and these skills to a variety of problems and ina variety of contexts. Recent research, however, has revealed that the commonly-used techniques do not work as well as anticipated. In fact, learners usemathematical procedures depending on context and environment, rather than, as iscommonly 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 ofmathematics are only beginning to be explored.For example, within the USA, there has recently been some movement awayfrom an overly abstract approach towards one that teaches mathematics morecontextually. Modern approaches are designed to reflect the demands of real lifeproblems and prepare learners for the mathematical requirements they might meetin their everyday lives. Recent calls for reform in mathematics education havefocused on the need to promote institutional practices that facilitate what is called“meaningful learning” (National Council of Teachers of Mathematics, 1989; NationalResearch Council, 1989). This approach is spelled out more fully in two recentNCTM documents (1989, 1991) that encourage teachers to develop schoolmathematics curricula and activities around promoting and enhancing mathematicalunderstanding and skills rather than concentrating on imitation or recall. However,it is too early to determine their effect on teaching practices and mathematical abilityand understanding in either school-based or adult mathematics education.How mathematics is regarded has a special significance for educators. For, ifmathematics is a body of infallible, objective truths, then it has no special concernwith social responsibility. Educational concerns such as the transmission of socialand political values and the role of education in the distribution of wealth andpower are of no relevance to mathematics. Alternatively, if mathematics is a falliblehuman construct, then it is not a finished product but a field of human creation andinvention. Hence, mathematics education must include opportunities for learners tostudy mathematics in “living contexts which are meaningful and relevant to them”(Ernest, 1991, p. xii), to create their own mathematical knowledge, and to discuss thesocial contexts of the uses and practices of mathematics.Further, notions of what mathematics j also affect how mathematics istaught. An “externalist” view of mathematics education would stress the mastery ofexisting concepts and procedures; an “internalist” view would concentrate onproviding “purposeful activities that grow out of problem situations, requiringreasoning and creative thinking, gathering and applying information, discovering,inventing, and communicating ideas, and testing those ideas through criticalreflection and argumentation” (Thompson, 1992, p. 128).The impact of predominant worldviews of mathematics on educationalpractices have also interested ethnographic and anthropological researchers. Inparticular, these researchers have focused on the social context of mathematicseducation and the “culture” that is transmitted by mathematical activities, both inand out of school. Eisenhart (1988) surveys the recent research conducted from anethnographic perspective. In particular, she draws attention to the work of Cole andScribner (1974) and Lave (1982, 1985) who, instead of studying the learning of formalmathematics in schools, have instead tried to understand mathematical problemsolving outside of schools. This work, says Eisenhart, “is predicated on the idea thatby understanding existing, ‘natural’ knowledge and beliefs, researchers can bridgethe gap between subjects’ capabilities and the capabilities that researchers or teachersbelieve students should have” (p. 111). Both Cole and Scribner’s, and Lave’s workfocuses on the mathematics used by adults, and I shall discuss it in more detailbelow.Other recent studies on the teaching of mathematics that fall within the 50interpretivist paradigm have concentrated on the knowledge, beliefs, and attitudesof teachers (e.g., Ernest, 1989; Thompson, 1984, 1992) and learners’ actual thoughtprocesses during mathematics education (e.g., Cobb, 1986; Desforges & Cockburn,1987; Neuman, 1987). These studies recognize that teachers and learners perceiveand interpret teaching situations differently and hence, they attempt to identify theseseparate interpretations. Much of this work is also based on a phenomenographicapproach (Marton, 1981) that seeks the “insider” interpretations and meanings ofthose involved. For example, Neuman (1987) explored learners’ ways of thinkingabout numbers and arithmetic to see if learning was improved if teachers usedlearners’ concepts as a basis for teaching. In particular, Neuman was primarilyinterested in discovering what children’s initial concepts about numbers were andsought these “meanings” through a series of interviews with children.Löthman (1992) was also interested in discovering what students actuallylearned. She drew distinctions (based on the work of Bauersfeld, 1979) between thesubject content that is meant (i.e., what is contained within the course syllabi andtextbooks), taught (i.e., contained in the teacher’s thinking and approach), andlearned (i.e., what the learners perceive they have learned). Löthman regardedteaching as ideal when all three parts combined, but recognized that the differentbackgrounds and perceptions of participants influenced their perceptions so thatthey interpreted teaching in different ways. Consequently, she noted that a dynamicsocial process is developed in the classroom and affects how mathematics educationis regarded and constructed differently by different people.Her study focused on the conceptions of mathematics education held by twogroups of learners--one of adults, one of high school children--who were studyingequivalent mathematics coursework. Her purpose was to “describe conceptions ofmathematics education in connection with a concrete educational course [in order to]find patterns and structures and catch their importance for the actors and the 51education” (p. 140). She categorized her results into four “pictures” of different, butrelated, conceptions that “showed teaching and learning as an entirety”:MATHEMATICAL TRADITIONS, consisting of the conventional dwelling ofmathematical problems in relation to the students’ experiences.MATHEMATICAL STRATEGIES, consisting of the students’ ways ofunderstanding, reflecting on, and solving mathematical problems.MATHEMATICAL REASONINGS, consisting of the students’ ways ofdiscussing, analyzing, and judging mathematical information.MATHEMATICAL APPLICATIONS, consisting of the students’ ways ofunderstanding and practicing mathematical concepts outside school. (p. 148)Mathematical traditions played an integral part in all the pictures. Löthmanfound that both the adult group and the high-school group preferred “strong rulesand formal dispositions of problems” (p. 148). The mathematical strategies of theadult group showed that they used a range of procedures due to their practicalexperiences of calculation and their earlier education. Mathematical reasoningsdiffered between the two groups. Adults preferred to know why they were doingsomething (such as a problem-solving technique) before they did it; schoolchildrenmerely wanted a rapid and expedient model. Finally, because of their greaterexperience, adults were able to see the practical uses of mathematical applicationsfar more clearly than were high-school learners. Löthman further found differencesbetween adult learners’ and their teachers’ conceptions of mathematics education.They were “pointing in two directions. [The learner] was aiming at comprehensionand [the teacher] was aiming at procedure” (p. 147). Löthman claims that thisdifference comes partly from the adult learners’ previous mathematics educationand partly from their “experiences of different occupations.. ..These experiencesconvinced them of the necessity of understanding” (p. 148).What is meant by teaching for “comprehension” or “mathematicalunderstanding,” or promoting “meaningful learning” can be drawn from examples ofresearch in of instructional situations. The work of Skemp (1976), Richards (1991), 52and Brown, Collins, and Duguid is helpful here. Skemp (1976) discusses thedistinction between “relational” and “instrumental” learning. Instrumental learninginvolves being able to follow rules without ever developing the true ability tosynthesize. 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 textbookword problem through application of series of rules versus an on-the-spot reckoningof currency exchange while bargaining in a foreign country.Similarly, Richards (cited in Cobb, Wood, Yackel, & McNeal, 1992) developedthe notions of “school mathematics” and “inquiry mathematics”. School mathematics,which corresponds to instrumental learning, is best characterized as the transmissionof knowledge from the teacher to passive students. Here, teachers establish not onlythe content of what is to be learned, but also how it is to be regarded andinterpreted. Students, in order to be successful, must adopt the teachers’interpretations of the content. For them, learning mathematics becomes theacceptance of others’ norms rather than an active construction of knowledge.Further, because teachers promote mathematics as having its own internal logic andmeanings, mathematics for students is “reduced to an activity that involvesconstructing associations between signifiers that do not signify anything beyondthemselves” (Cobb et al., p. 587). In this way, teachers enculturate students into whatLave (1988) calls “folk beliefs” about mathematics. These includethe conviction that it is impermissible to use any methods other than thestandard procedures taught in schools to solve school-like tasks and that theuse of these procedures is the rational and objective way to solvemathematical tasks in any situation whatsoever. (Cobb et al., p. 589)On the other hand, Richards maintains that inquiry mathematics activelyseeks to promote a deeper understanding. Teachers, rather than regardingthemselves as the sole validators of what counts as legitimate mathematical activityand learning, encourage and guide students to propound and discuss their owninterpretations and insights. In this way, teachers promote the notion of mathematics 53more as a legitimated set of interpretations of “activities that were intrinsicallyexplainable and justifiable” (Cobb et al., 1992, p. 594) rather than as a set ofacontextual and fixed rules and procedures.Richards observed that because teachers tended to follow either one set ofpractices or the other, the school and inquiry mathematics approaches took on thecharacter of traditions. Cobb et al. (1992) found this school/inquiry dichotomy toosimplistic and argued that, regardless of their tradition, teachers “initiated theirstudents into particular interpretive stances [where] students learned whichmathematics activities were acceptable, which needed to be explained or justified,and what counted as a legitimate explanation or justification” (p. 597). In theclassrooms Cobb studied, regardless of the approach of the teacher, students wouldgenerally experience an activity as meaningful if it made sense to them within theclassroom context rather than with reference to their individual, beliefs, values, andpurposes.Brown, Collins, and Duguid (1989) further investigate this phenomenon intheir contrast of “authentic” and “inauthentic” mathematical activity. To them,authentic mathematical activity is the “ordinary [mathematical] practices of theculture” (p. 34) which is coherent, meaningful, and purposeful only when it issocially- and contextually-situated. In contrast, common school mathematicalactivities are not authentic because they prevent students from engaging witheveryday culture and impose a more “school” culture. As Brown et al. say, “althoughstudents are shown the tools of many academic cultures in the course of a schoolcareer, the persuasive cultures that they observe [and] in which they participate. . .arethe cultures of school life itself” (p. 34). In other words, classroom activities takeplace within a school, rather than an everyday culture. The activities that studentsare 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 54endorsed by the cultures [and practitioners] to which they are attributed” (p. 34).In sum, the interpretivist paradigm seeks not to articulate causal relationshipsbetween teaching and its effects, nor to illuminate the interweaving of social andpolitical influences in the minutiae of the classroom; instead it seeks to discover themeanings which participants ascribe to a situation. In terms of this study, theinterpretivist research about teachers’ thinking is especially topical. Teachers havetheories and beliefs that influence their perceptions, decision-making, planning, andactions. 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 inmathematics classrooms can prove fruitful in light of actual classroom experiences.Summary & DiscussionResearch on teaching has tended to fall into one of three distinct paradigmsand, within each paradigm, examined only “macro” or “micro” aspects of teaching.Macro studies have focused on large-scale general features of society such asorganizations, institutions, and culture and used experimental and quantitativemethods to derive explanations about the effects of these external influences onteaching. Alternatively, micro studies have focused on the more personal andimmediate aspects of teaching and used descriptive or interpretive frameworks andmethods of inquiry to understand the internal meanings and perspectives ofparticipants. Most of this research has been theoretically dichotomous and, 55consequently, difficult to compare. Part of the difficulty arises from theepistemological distinctions between the different approaches. Indeed, in manyways, each paradigm can be seen as reflecting different positions on the macro-microissue.In addition, most research on teaching has focused predominantly on therelationship between teaching and learning; indeed, it has largely viewed teachingsolely as the promotion of learning. Such a view has tended to reduce socialphenomena, such as classroom behaviors and processes, to the behaviors of singleindividuals. 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 socialnature of much classroom behavior and assume divisions between external andinternal perspectives, between teaching and learning, and between curriculum andpedagogy. Although teaching and learning can be regarded as separate (althoughlinked) 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 isintended to frame or guide teaching practice and cannot be achieved except duringacts of teaching. Similarly, teaching is always about something so it cannot escapecurriculum. Teaching practices, in themselves, imply curricular assumptions andconsequences” (Doyle, 1992, p. 486). As the structuralist paradigm was toodeterminist or functionalist, the interpretive paradigm, in regarding all social life asultimately explicable in terms of the actions and intentions of individuals, has gonetoo far in the opposite direction. The reconciliation of the two implies that observedbehavior and its effects need to be viewed both in the context of the meanings and 56motivations of actors and within wider social contexts and influences.Clark and Peterson (1986) accept that a complete discussion of teachingprocesses cannot only concern what teachers’ think, it must also include anunderstanding of constraints and opportunities that impinge upon them. As they seeit, “teachers’ actions are often constrained by the physical setting or by externalinfluences such as the school, the principal, the community, and the curriculum” (p.258). Tn addition, teachers’ actual thought processes can be similarly constrained. Forexample, teachers may have (or think they have) less opportunity to plan theirlessons in the ways that they wish because certain decisions have already been madeby the institution in which they work. In this way, research on teachers’ thinking canappear to overlap with the more structuralist frame factor theory.Hence, research that attempts to combine both theories must overcomecertain issues--the relationship between macro and micro approaches and the relatedissue of structure and agency--to which I now turn.Structure & AgencyStructure and agency are concepts which attempt to explain actions in socialsettings as the effects of large scale structural forces or policies (structure) or smallscale individual, voluntary actions and patterns of behavior (agency). Astraditionally conceived, structure and agency are regarded as competingexplanations of social reality. Hence, attempts to combine them lead to ontological(how social processes are generated and shaped), epistemological (what counts asknowledge), and methodological (how research should be conducted) problems.There is considerable overlap between notions of structures and macro 57phenomena in that they both refer to the reproduction of patterns of power andsocial organization (Layder, 1994). Within educational settings, “structural” studieshave focused on explanations of how educational practices and processes aresituated in, and determined by, broad social structures. In these studies, it isconsidered unnecessary to gather the perspectives of actors because they can bededuced from determining the effects of social structures. Similarly, micro analysesoverlap with a concern for agency. “Agency” studies have supposed that events andactions are produced by largely autonomous individuals; they therefore haveconcentrated on eliciting the actors’ intentions, meanings, and actions aboutsituations.The difficulty with this dichotomy is that it makes the assumption.. . thatsocial 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 settingsin particular. People do not exist on different levels, so separating social life intohierarchical levels “makes it difficult to conceptualize change as a dynamic processinvolving both structure and human agents” (Shilling, 1992, p. 70). As Marx put it, indescribing human activity: “Men make their own history.. .not under circumstanceschosen by themselves, but under circumstances directly encountered, given, andtransmitted from the past” (1950, p. 225). Consequently, research on social settings--such as classroom teaching--that attempts to synthesize micro and macro approachesneeds to include both empirical evidence of actual events in particular settings withparticular actors (micro and more agency-driven explanations), and supraindividual theories that provide a context for those events (macro and morestructurally-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 58theory stresses their inter-connection. Thus, structuration theory has potential forempirical application in research. For example, it highlights how a study of teachingcan consider the intermediary aspects between structure and agency both within andoutside the classroom. Therefore, it is particularly useful for looking at teaching as asocial and political process. “Between the rules, negotiations, and bargainings ofclassroom interaction, and the dynamics of the capitalist economy, or the relativeautonomy of the state, lie a whole range of intermediary processes and structures”(Hargreaves, 1985, p. 41). More practically, structuration theory suggests thatresearchers use multi-strategy approaches to achieve a dense theoretical andempirical coverage of the topic, initiate and develop theory from fieldwork,recognize that all activities are contextually-situated and all situations are theproduct of human actions, and seek the relevance of pieces of empirical data towider theoretical issues.This discussion of the macro-micro duality and the relationship betweenstructure and agency inform my own study. I now describe the theoreticalframework for my study and discuss how it addresses these issues.Theoretical FrameworkIn adopting a social theory, rather than a psychological approach, my study isbased on and develops the conceptual frameworks, theories, and methodologies ofother researchers. Further, it provides a way of looking at the relationship betweensocial interaction in mathematics classrooms and the reproduction of the major 59structural 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, andalso explore openly the various forces constraining the educational activities withinthe classroom. This involves examining factors that may influence teaching (andteachers) which are not necessarily obvious or apparent to those involved. Althoughteachers are the main source of classroom teaching activity, they are also,themselves, part of a wider context. Thus, a study of only their choices andbehaviors would be incomplete; a thorough study of teaching processes mustattempt to examine these contextual factors as well. Of course it is not possible toisolate and examine these factors; one can only discern their effects by studying whatteachers do, and what they say about themselves and why they’re doing it. To clarifyhow my theoretical framework for my research incorporates all of theseconsiderations, I now describe it in detail.The conceptual framework of this study combines Clark and Peterson’s modelof teachers’ thinking and frame factor theory to construct a lens for observingteaching processes. This combination allows me to examine the process ofsocialization that takes place in mathematics classrooms as well as howmathematical meaning and knowledge are formed and developed. This studyconcerns the whole pedagogic process of teaching and learning--a process I havechosen to name, in recognition that teachers are its chief initiators, as teachingprocesses. By teaching processes I do not mean only the selection of content to betaught, or the choice of such techniques as lectures or discussions or whether to usegroup-work. I also include in my definition the expectations, rules, procedures, andnorms of the classroom, as well as the complex web of interactions between teachersand learners, and between learners themselves. In other words, studying teaching 60processes means developing an understanding of “what goes on” within classrooms.Asking “What is going on here?” may seem at first a trivial question, but asErickson (1986) points out: “Everyday life is largely invisible to us (because of itsfamiliarity. . . and contradictions. We do not realize the patterns in our actions as weperform 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 theinvisible visible, as is appropriate in a social theory approach.To understand what is going on, one also needs to address “local” meaningsand the differing perspectives of those involved. Teachers and learners regardteaching from different perspectives, and in different ways. Also, what appears to behappening may be misleading. Events that look the same may be entirely differentand have distinctly different local meanings. In other words, the question “What isgoing 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. Teachingis subject to many forces, some of which can be traced to the political, cultural, andsocial structures in society. Consequently, to examine fully classroom teachingprocesses, a research study must seek to relate them to the political and socialstructures of society.Thus, this study links two strands of educational research. A model ofteachers’ thoughts and actions--which examines both the internal mental processesof teachers as they plan, conduct, and evaluate their teaching, and their subsequentobservable behavior--is connected to frame factor research, which examines howteaching processes are affected by external factors. In this way I examine howteaching processes are viewed from the perspectives of those involved as well as 61consider how teaching processes are influenced by other factors.A Model for Understanding Teaching ProcessesAs I outlined above, by teaching processes I mean “what goes on” in adultmathematics classrooms. Teaching processes include the selection and ordering ofthe content to be taught; the expectations, rules, and procedures of the classroom;and the nature and quality of interactions between teachers and learners, andbetween learners themselves. To understand why teaching processes take the shapethat they do, one needs to identify what forces are acting upon them, and in whatways.Applying frame factor theory to a study of mathematics education for adults,one can determine several constraining factors. Within any society, the institutionalframework of adult education provision, the particularities of educational settings,and the mathematical curricula chosen to be presented in those settings all affectteaching processes in mathematics education for adults. Further, the effects of thosefactors can be seen as interacting with the life and professional experiences of adultteachers and their learners. In other words, social and cultural norms and valuesaffect the settings in which adults can learn mathematics, the mathematics they areexpected to learn, and the ways they and their teachers experience and regardmathematics education.These factors are not isolated, however; they act upon and react with eachother. Further, the relationships between factors are dynamic rather than static, andconstellate uniquely in every classroom and every setting. For example, adultlearners of mathematics in a community drop-in literacy center are markedly 62different from their counterparts in university-level settings. As such, they expect tolearn different mathematical skills and knowledge from their more academiccounterparts. These differences and expectations are also not constant; they changeover time, and affect (to differing degrees) the individuals concerned, the institutionsthey attend, and the mathematics they study.Consideration of these issues requires a framework for understandingteaching processes and the forces that shape and constrain them. A model of thefactors I have identified and their interrelationships is portrayed in Figure 1. The twogroups--the worldview of mathematics and the institutional framework, eachinfluenced by social structures--represent possible frame factors. The third group—experiences of teachers and learners—represents a way of observing the effects ofthese frame factors on teaching processes. The solid lines represent the relationshipsbetween elements that are the focus of this study. The dashed lines represent otherlinks that exist but are not dealt with in this study. I now discuss each element inturn.Social StructuresFigure 1: Understanding Teaching Processes63As can be seen in Figure 1, social structures are not direct influences onteaching processes; rather, they are mediated through the woridview ofmathematics, the institutional framework, and the experiences and perceptions ofteachers and learners. However, given that their effects are so widespread and soreadily apparent, I feel it necessary to briefly explain them here.Following Giddens (1984), I regard social structures as the rules and resourcesthat people draw upon as they produce and reproduce society in their activities.r —Social structures DI Worldview of InstitutionalI mathematicsIIIIIIIIIHence, social structures are both the medium and the outcome of social activity 64rather than a system of relationships operating “above” people. In this way, socialstructures 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 numberof concerted interactions with each other” (Sharp & Green, 1975, p. 19). Practically,then, analyses of subjective meaning need to be supplemented with somedescription of the actual social structures within which people live and act.For a study of teaching, this implies that teachers’ thoughts and actionsshould be situated within a context of social and physical rules, resources, andconstraints. Although teachers may not perceive these resources and constraints,they nevertheless are bound by them. Teachers’ working situations, freedom ofaction, and thinking are all shaped and limited by social structures. Within thedassroom, it is clear that teachers have far more power to act, direct others, andaccess facilities and resources than do students. Hence, this unequal distribution ofpower has considerable significance in explaining the differences in perspectives onteaching processes, and classroom behaviors between teachers and learners.The Worldview of MathematicsA worldview is the set of presuppositions or conceptions of a phenomenonthat is held by a particular society or group. In encompassing all the different viewsof people in that group, the worldview reflects their specific cultural, social, andhistorical contexts. Hence, the notion of a woridview emphasizes the shared andsocial basis of knowledge; knowledge is present in the society into which individualsare socialized, and it is a resource shared by members of that society. Knowledge isseen not as a collection of “content,” but more in the “style or pattern ofthought.. . .The social basis of knowledge lies in the categories of meaning used to 65think or perceive or understand the world’ (Dant, 1991, p. 18).As I described earlier, the worldview of mathematics, common in allindustrialized countries, is that it is a logical and impersonal branch of knowledgeconsisting 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 inmost schools, and possession of mathematical knowledge has a high value in manycultures (Willis, 1990). As Dossey (1992) notes, “Perceptions of the nature and role ofmathematics.. .have had a major influence on the development of... [mathematics]curriculum, instruction, and research” (p. 39). Hence, how mathematics isconceptualized 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 mathematicsdassroom.Institutional FrameworkInstitutional factors can be of two kinds: organizational factors such as theoverall provision of education within an area and the physical structures of, andadministrative systems in, educational institutions; and curricular factors thatspecify what is to be taught and in what way, and the textbooks and teachingmaterials to be used. In both cases, these factors, because they lie outside the controlof the teacher, act as frames.Organizational Factors. In British Columbia, almost all of the locallyprovided mathematics education is organized and controlled by the publiceducation sector. For example, within the Acton area, both the Acton School Boardand the Community College system offer a variety of “math upgrading” courses toadults at several centers. Most adults in these courses are trying to obtain one of four 66certificates (Dogwood, Adult Dogwood, College Provincial, and GED) equivalent tohigh school completion. This pattern of provision is repeated in most urban areasacross 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 inpurposeful and systematic learning throughout their lives” (Fans, 1992, P. 6). Centralto this concept of lifelong learning are certain widely-held assumptions about, andpractices within, adult education that are built on ideas and theories about howadults learn and should be taught. These ideas include: teaching must be problem-centered, it must emanate from the participants’ experience of life and develop theindividual socially, participants must exert definite influence on the planning of thecourse and the conduct of the teaching, and techniques used must be based on aninterchange of experience (Knowles, 1980).Another set of organizational factors that can act as frames on teachingincludes the physical structures of colleges and their classrooms, and administrativearrangements within particular institutions such as the size of the class, the timeavailable for teaching, and the evaluation system. Each of these limits, but does notdetermine, teaching processes.Curricular Factors. A second set of institutional factors that can act as a frameon teaching concerns the process of codifying an area of knowledge into an academicdiscipline 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, teachingmethods, homework assignments and evaluation procedures. Thus, the teaching ofmathematics becomes bounded by, and negotiated between, the inherent qualities ofthe subject and the goals and dynamics of the institutions in which the teachingtakes place. In the case of mathematics, these negotiations become visible throughtwo processes: setting the aims of mathematics education, and codifying its subject 67matter into textbooks.I described earlier how the aims for mathematics education can affect itsteaching. Briefly, I explained how the aims for mathematics education were, ingeneral, set by certain dominant groups within society, and yet did not reflect, atleast overtly, any social or political content. This ensured that mathematics was notseen as a tool for questioning dominant attitudes within society, and its method ofpresentation reflected this: it tended to be taught in an authoritarian andhierarchical way.A second process through which curricular factors influence teachingconcerns the use of textbooks. In many ways, textbooks are the most central anddefining feature of mathematics education. The content and structure of mostmathematics courses are determined by the content and structure of the settextbooks. In many ways, the textbooks are the curriculum, codified. Romberg andCarpenter (1986), in their survey of research on teaching and learning mathematicsfound that “the textbook was seen as the authority on knowledge and the guide tolearning” (p. 867) in all of the studies they surveyed. They concluded that manyteachers “see their job as covering the text” and that mathematics was “seldom taughtas 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 mathematicsclasses 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 iscommon practice in most classrooms. However, as Bishop (1988) asks, “Whose arethese books? Who writes them, and for whom, and why?” (p. 10).Basing teaching largely on the textbook has several consequences. First, it 68sustains and promotes a “top-down” approach to education. Mathematics textbooksare written by experts who purport to know what learners need, and the order andmethods they need to learn it. Second, they make no distinction between whatdifferent learners bring with them to the classroom. By treating learners asimpersonal or generalizable, textbooks privilege content over process. Theyencourage the teaching of subject matter, rather than the teaching of people. Finally,mathematics in textbooks is presented in a supposedly value-free anddecontextualized way. Mathematical knowledge is seen as impersonal; learners arenot encouraged to make their own meanings, or find their own significance.Institutional factors, then, can also have a large influence on teaching. Thephysical structures act as tangible constraints; the curricular factors act moreconceptually, limiting what counts as legitimate knowledge or as an approved wayof teaching and learning.Experiences of Students and TeachersRegardless of the different settings for adult mathematics education and thedifferent curricula that exist in these settings, two groups of people are affected byand in turn affect, these factors. Adult teachers and their students are each the focusof the factors that limit classroom teaching processes, and, simultaneously, theagents of change on those factors. To my knowledge, there is no published researchin North America that focuses on the teachers of mathematics to adults. However, aswas mentioned earlier, adult education theories often stress the centrality of theparticipants to the teaching processes. In practice, this means that effective adulteducation teaching should relate to participants’ needs and interests (Brookfield,1986). Earlier, I discussed studies that had considered the needs and interests ofthose adult learners engaged in mathematics education. I showed how adults’ 69attitudes towards, and their daily uses of, mathematics influenced both how theyapproached 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.SummaryThese, then, are the elements of my model for studying teaching processes. Inthis chapter I have described a theoretical framework and its constituent elementsthat have been designed to investigate and analyze teaching processes from a socialtheory perspective. The elements of this model will be animated through theparticipants’ classroom interactions in a variety of ways. I intend to observe, record,and analyze these interactions using diverse and multiple methods in order toachieve a dense, theoretical, and empirical coverage of the topic. In the followingchapter, I explain my methodology.CHAPTER 3: METHODOLOGY 70In this chapter I describe the methodology I used in my study. I first discussmy selection of a specific research site and study participants. I next describe mydata collection and analysis procedures, and conclude with a brief discussion ofcertain issues concerning the “criteria of soundness” of my study..Selection of Research Site and ParticipantsI needed to contextualize my study of teaching processes in actualmathematics classroom situations. I could not, of course, gather data from all of themany providers of formal mathematics education, even in as geographically-compact an area as the Lower Mainland region of British Columbia. It seemedappropriate to base my study in a local setting where such provision commonly tookplace, and within that setting, to choose courses that typically reflected the overallprovision.Selection of SiteWithin the Acton area, both the local School Boards and the CommunityCollege system provide a range of mathematics education courses to adults. Eachsystem’s provision is comparable: their courses are of similar levels of difficulty, andare offered at broadly similar dates and times. However, one institution in theCommunity College system offered easy access and was already known to me. I had 71conducted some initial research in Acton College’s mathematics department andregarded it as an ideal site. It contained a range of informants, and it provided a highprobability of finding a rich mix of the teaching processes and frame factors that Iwished to study.The selected college offered a range of mathematics courses for adults duringboth the daytime and the evening. Their provision was organized into five distinctlevels, corresponding broadly to grade levels 9 through 12, and an introductorycalculus 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 introductoryalgebra and geometry” (AC, 1992). They are deliberately designed “for the studentwho has never studied academic mathematics before or who is lacking a goodfoundation 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 differentinstructor and was expected to recruit between 15-20 students. The courses I chosewere typical of the college’s mathematics provision and their curriculum wassupposedly designed to reflect a balance between the formal and practicalmathematical needs of learners.Selection of ParticipantsThe participants in the study were of two types: (a) teachers of mathematicsto adults at the chosen college, and (b) students in the three introductorymathematics courses.Teachers. All of the eight teachers in the mathematics department were 72interviewed at the beginning of the term. The reasons for interviewing all of theteachers were twofold. First, as those most concerned with teaching processes,teachers have both the power to effect change, and are among those most affectedby, the factors involved. I was concerned to ensure that I gained as much data aspossible on the teachers’ understandings of teaching processes, the constraints thatthey feel, and the reasons for their choices in relation to the specific and concretesituations of their teaching. Consequently, I determined that the fullestunderstanding of teaching processes at the college could only be gained byinterviewing all the teachers involved.The second reason for interviewing all of the teachers was more pragmatic. Atthe outset of my data collection, decisions about who would teach which course hadnot yet been finalized. Therefore, by interviewing all eight teachers, I was able toascertain the approaches that they were taking to planning their courses before theyhad begun. Subsequently, when teaching assignments had been decided, I focusedon the three teachers of the introductory courses more specifically.Students. All of the adult students attending any of the three introductorymathematics courses during the first two weeks of term were given a copy of thesurvey protocol together with an explanatory letter that invited them to participatein further stages of the research. (Copies of the explanatory letter and the surveyprotocol are attached as Appendices 1 and 2.) Thirty-two students completed thesurvey, all of whom indicated their willingness to participate further.From these 32 respondents I selected 15 (5 from each section) to interviewfurther. Interviewees were largely selected on the basis of several demographiccharacteristics (viz., gender, age, ethnic origin, and employment). My intention inusing these criteria for selection was not to obtain a sample of participants thatpurported to be in any way representative of the wider population of adultmathematics students. Instead, it was intended to help me appreciate the range ofcharacteristics of the group of adult learners who had enrolled in the basicmathematics courses. I was interested in exploring whether adults from differentbackgrounds had different attitudes and approaches towards their mathematicseducation, or experienced teaching processes in different ways.Data Collection ProceduresThe study used multiple data collection procedures, combining those fromboth qualitative and quantitative approaches. My data were gathered throughsurveys of learners’ demographic characteristics and attitudes, extensive participantobservation, repeated semi-structured interviews, and document collection. Eachspecific procedure is described more fully below, but first, I explain the reasons forusingsuch a variety of methods.Using Multiple MethodsThe principle use of multiple methods was to add methodological rigor to thestudy. Although qualitative and quantitative data collection are often seen as comingfrom contradictory notions of reality, here my avowed post-empiricist approachassumes that there is one “truth” to a situation. Although this truth may not beexactly “captured” by any one means, both quantitative and qualitative methods canbe seen as different ways of examining the same phenomenon, and obtaining acloser correspondence with the truth. Therefore, findings that have been derivedfrom more than one method of investigation can be viewed with greater confidence 74and with a greater claim to validity. Denzin (1970, P. 301) describes this combinationof multiple methods as “methodological triangulation” and adds that “the flaws ofone method are often the strengths of another, and by combining methods, observerscan achieve the best of each, while overcoming their unique differences” (p. 308).A second reason for combining both qualitative and quantitative methodsconcerned the differing perspectives between the researcher (outsider) and theparticipants (insiders). Quantitative methods, such as the survey I used, wereoriented to my own specific concerns (in this case, learners’ attitudes towardsmathematics). Alternatively, the more qualitative interviews and observations wereoriented more towards the participants’ perspectives. Integrating both methods inone study combined the perspectives of both insiders and outsiders, and addedstrength to the findings.A third reason concerned the need for triangulation in the data. Students andteachers view teaching from different standpoints. Teaching is what teacherswhereas teaching is done to students. As such, they have little say in, or control over,those decisions that affect teaching. Consequently, how students and teachersregard, and respond to, teaching differs markedly. In order to capture those differentperspectives fully, I needed to obtain data from both groups in several differentways. Triangulation using different methods, and from different perspectives,allowed me to better capture the totality of the phenomenon of mathematicsteaching.The final reason for using the particular mix of methods was that morequantitative data derived from the surveys of learners’ attitudes could be used tosupplement and focus some of the later qualitative data collection. Not only did thesurvey data provide information that could not have been readily gained by areliance on participant observation or semi-structured interviews, but it also enabledits use in the subsequent interviews with both teachers and learners.I now provide an overview of the three phases of data collection, and thenconsider each method and procedure in turn. For each, I describe its use and discusssome of the implications of using it.OverviewData collection fell into three separate phases. In the first phase(approximately 4 weeks long), I was concerned with developing someunderstanding of the culture and ethos of the college in general, and of the threeintroductory mathematics classes in particular. Here, I distributed a survey to alllearners, interviewed all of the teachers within the mathematics department abouttheir perceptions and beliefs, and conducted a series of informal discussions andethnographic observations (ranging from 45 minutes to 2 hours) within the threeintroductory mathematics dasses. During this phase I conducted 8 interviews and 19observations.The second stage of data collection focused more specifically on oneparticular 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 focuswas based on the following criteria, drawn from Löthman (1992):(a) It should consist of topics that are dealt with in the textbook, but can alsobe easily discussed in class.(b) It should contain problems where the content can be derived from thelearners’ everyday experiences.(c) It should contain learning tasks that require written or oral products and 76have 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 theabove 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 thisphase of data collection; course 050 some three or four weeks later. This intervalallowed me to complete this part of my data collection in one course beforerepeating 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 specificteaching processes in the lessons concerning algebra. I also observed every lessonthat 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 thebasis of “stimulated recall” interviews with the class teachers. During this phase Iconducted 17 interviews with students, 9 interviews with teachers, and 27observations.In the third and final phase (4 weeks), I completed my observations of thethree introductory classes, observing the concluding days of instruction in eachcourse and their preparations for the end-of-term class examinations. I alsointerviewed all the students again, ascertaining their views on mathematicseducation in general, and the teaching in their particular course in detail. Finally, Iheld in-class group interviews with each of the three introductory courses. Groupinterviews were chosen as “a good way of getting insights [as] subjects can oftenstimulate 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 77sufficient authority, to speak (Lancy, 1993). In these group interviews I particularlyencouraged responses from those individuals who had not been selected for theindividual interview processes, or who had participated, but said little.During this third phase of the research, I conducted 8 observations, 3interviews with teachers, and 15 interviews with the students (both individually andas a group). I now describe each data collection method in more detail.SurveyI distributed a simple survey, by hand, to all the learners enrolled in the threesections of the course 050/051 in the Spring 1994 term. This survey gathereddemographic data of gender, age, ethnic origin, and occupation; students’ attitudestowards mathematics; and an indication of willingness to further participate byagreeing to be interviewed (see Appendix 2).The section in the survey on attitudes towards mathematics was based on twoinstruments devised by Aiken (1974) that measure participants’ (a) enjoyment ofmathematics, and (b) their perceived importance and relevance of mathematics tothe individual and to society. Aiken’s instruments each use a 12 question, 5 pointLikert-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 coefficientbetween item scores and total scores above 0.75, and 10 items on the “Value” scalewith a similar correlation coefficient above 0.60. These 20 items were then randomlymixed to produce the survey protocol I used.78InterviewsTeachers. I initially conducted semi-structured interviews with all of theteachers in the department during January 1994. All interviews lasted about 1 hour,were tape-recorded, and later transcribed for subsequent analysis. My interviewsconcerned teachers’ understandings of teaching processes, the factors they feltaffected their teaching, and their reasons for choices they made in relation to specificand concrete situations they encountered in their teaching. (A copy of the interviewprotocol forms Appendix 3.) Some specific question areas concerned the planning ofteaching, what problems teachers foresaw, their choice of course material andinstructional 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 datacollection).I further interviewed the teachers of the three introductory classes three moretimes during the term. The second and third interviews--each lasting about 15-20minutes--took place immediately before and immediately after the lessonsconcerning 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 suchissues as:(a) Before the lesson: Specific examples of how teachers chose lesson contentand instructional strategies for the particular section of the syllabus, how the lessonfit 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 thelesson, what mathematics difficulties showed up, how those difficulties were dealtwith, and what happened that was unexpected.(c) At the end of term: Specific examples of what changes were made to theteaching throughout the term, what mathematics difficulties showed up, how theywere dealt with, and what happened that was unexpected.Stimulated recall. Certain lessons in this phase had also been videotaped andthe teachers were asked to participate in stimulated recall interviews: a means ofcollecting teachers’ retrospective reports of their thought processes. Stimulated recallis a term used to describe a variety of interview techniques designed to gain accessto others’ thoughts and decision-making. Typically, it involves audio-taping orvideotaping participants’ behavior, such as counseling or teaching, in situ.Participants are then asked to listen to, or view, these recordings and describe theirthought processes at the time of the behavior. It is assumed that the cues providedby the audiotape or videotape will enable participants to relive the episode to theextent of being able to provide, in retrospect, accurate verbal accounts of theiroriginal 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) engagedin skilled behavior. It has taken slightly different forms in these three differentcontexts. Bloom (1953), who pioneered its use, was interested in learners’ thoughtprocesses during different learning situations. He played back audiotapes of lecturesand discussions to university students and recorded their commentaries of theirthoughts. These reported thoughts were later categorized according to their contentand their relevance to the subject matter being studied. Kagan, Krathwohl, andMiller (1963) developed Interpersonal Process Recall: a form of stimulated recallusing video-recordings as a means of increasing counselors’ awareness ofinterpersonal 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 80diagnostic situations. Leithwood and others (1993) studied the problem-solvingprocesses of school superintendents during their meetings with senioradministrative staff. In classroom-based research, stimulated recall has also beenused in a variety of ways to investigate the thought processes and decision-makingof teachers while teaching. Keith (1988) demonstrates the diversity found in a groupof selected studies.One factor which can influence the data collected by stimulated recallconcerns the way in which participants are prepared for their commentary, and howthey are instructed to comment. Calderhead (1981) notes that respondents can oftenidentify, and hence comply with, the aims of the researcher. He also describes astudy by McKay and Marland (1978) in which the researchers, although avoiding theimposition of any research model on the thoughts of teachers, provided detailedinstructions before the videotaped lesson on the kinds of thoughts teachers wereexpected to recall. Calderhead claims that the provision of explicit instructions mayhave influenced the teaching itself, and the procedures may also have encouragedthe 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 thevideotapes of their teaching and to comment on whatever they wanted. I gave themno instructions as to what to comment on, or how to comment. The teachers weregiven a remote control for the video playback machine and could stop the tape atany time they chose; they then commented (in whatever way they wanted) on thesection of tape they had just viewed. These comments were audio-taped and latertranscribed for subsequent analysis.Students. I also interviewed four adult students in each of the three classesthat I selected (a total of 12 people). These people were interviewed three times:first, immediately after the first lesson of the algebra section; second, immediatelyafter the end of the algebra section; and third, at the end of the term. The interview 81protocols form Appendices 6 and 7. The first and second interviews—which lastedabout 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 thatstudents were asked to perform, any difficulties they encountered, and if they foundthis lesson typical of others.The third interviews--each lasting between 15-75 minutes--covered broaderareas and included data from the preliminary survey of learners’ attitudes towardsmathematics. These interviews covered such areas as learners’ experiences ofmathematics education at school, and as an adult, their involvement in classroomactivities during the course, and their attitudes towards the course content andteaching processes.ObservationsDirect observation in classrooms allowed me to study the teaching processesas they took place in their natural setting. I was able to gather such data as the formand content of verbal interaction between participants, non-verbal behavior, patternsof actions and non-action, and references to the textbook and other instructionalmaterial. Further, by acting as an “involved interpreter” I was able to “understandthe 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 theintroductory 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 twohours (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 meto familiarize myself with the classroom and college settings. Here I observed thewhole lesson several times in each course (2 hours each). During this period I wasable to introduce myself to the participants, gain their trust and cooperation, andcollect 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 theparticipants (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 particularsection of the syllabus (Introducing Algebra). Again I observed lessons in theirentirety. 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 foreach observation. Events observed each time included: whether the lesson began ontime, whether anyone was late, whether the teacher appeared to be following alesson plan, the activities students were asked to perform, the students’ attentivenessand participation, what things appeared to concern the students, and the evaluationprocedures used.Videotaping. I also videotaped some of these lessons using one camera tocover both the teacher and the students. In sum, 6 complete lessons werevideotaped; three from each of courses 050 and 050/051. In videotaping the lessons Iconcentrated on supplementing my earlier ethnographic observations which hadproduced elaborate, though partial, field-notes. I tended to initially concentratelargely on the teachers as they moved around the classroom. However, as much ofthe lesson time was given over to individual student work, I could also focus thecamera (with its in-built microphone) on the students’ actions and utterances. Thevideotaping of lessons therefore provided a richer and more detailed record than the 83earlier note-taking. In particular, video-recording captured the details of smallmovements and oral comments as well as larger physical movements anddifferences in behavior.Some implications of using the video-recording equipment in the classroomconcerned entry into the setting, the timing of taping segments, the visual point offocus, and the analysis of the data. I will deal with the data analysis issues in thenext 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, obviousand intrusive--would be seen as overly disruptive to the participants. Consequently,I took the equipment into the classrooms for several periods beforehand andpracticed filming--without making any recording--so that participants could becomeacclimatized to the change. I also fully explained my purposes in taping, so thatparticipants were aware that my intention was to capture what the teaching ofmathematics “looked like” rather than a focus on one particular individual or groupof individuals. As participants became used to the presence of the camera in theirclassroom, 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 notappear to behave in specific ways for the camera. Consequently, I feel assured thatthe recordings I made are accurate renditions of episodes and situations inmathematics classrooms.With respect to the timing of recording, I had initially planned to sample thetotal available time, and record 10 minute segments of classroom behavior. Afterinitially trying this, I changed this approach to recording the entire class lesson. Thephysical movement of the camera between taping and not taping proved moredisruptive to the dass than I anticipated, and I felt that by taping the whole lesson I 84was better able to capture data that matched my research questions.Visual focus was also a point of concern. Initially,I tended to concentrate thecamera on the teacher, but later, as I grew more adept as a camera operator, I movedit 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 whatwas seen from the students’ perspective (by taking one of the seats reserved forstudents), sometimes by setting the camera up to view the class as the teacher mightsee it, other times trying to move around the room focusing on odd segments ofbehavior or snatches of conversations.Document CollectionThe overall content of most ABE mathematics courses in British Columbia isdetermined largely by provincial curriculum guides. In addition, research has shownthat the content and structure of the set textbooks also determines the specificdassroom content and structure of many mathematics courses in both BritishColumbia (Fans, 1992) and elsewhere (Höghielm, 1985; Romberg & Carpenter, 1986).Therefore, I gathered copies of documents relevant to the teaching of mathematics tothe adults in the study. These included the Provincial Update on Adult BasicEducation Articulation which contains generic course outlines of all ABE courses inBritish Columbia, copies of specific course outlines and syllabi, course handouts,examination papers, and mathematics textbooks used as course texts. In addition, Igathered any relevant material that pertained to the adult students’ uses ofmathematics (e.g., the learner’s notebooks and completed homework assignments).Finally, I was able to collect copies of the students’ examination papers, after theyhad been marked by the teacher.Data Analysis Procedures 85Data analysis is the process of bringing order, structure, and meaning to amass of collected data. Within qualitative research, analysis consists of a search forgeneral statements about relationships among categories of data. Much of thisprocess consists of organizing the data, sorting and coding the initial data set,generating themes and categories, testing the emerging themes and concepts againstthe data, searching for alternative explanations, and writing the final report.In this study, analysis of the data was ongoing and iterative, and guidedthroughout by my conceptual framework. Data analysis fell into two phases. Thefirst phase of data analysis began almost as immediately as data collection, whereconcepts that had been identified in my theoretical framework began to appear inexamples of classroom practices in the early data. This concurrent process of datacollection and analysis enabled me to identify themes and patterns of teachingprocesses from both observations of dassroom practices and from teachers’ andstudents’ comments. It also served as a check that sufficient and appropriatelyfocused information was gathered before the completion of the data collectionperiod.Initial analysis of the observational data involved searching through the datato obtain categories and themes that would portray an overall understanding of theframework of the teaching processes. Here, I paid particular attention to the rolesthat teachers adopted, and the tasks that they asked students to perform. During thisearly analysis, I also made analytic notes about specific points to pursue with theteachers in subsequent interviews.As the analysis proceeded, I modified my initial categories and themes to 86better refine the portrayal of teaching processes and introduce the influences offrame factors. As more data were collected, particular interpretations and conceptscould be refuted or confirmed by checking them against the most recently collecteddata.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 formedthe basis for a narrowing of focus, and the beginning of a process of abstracting andtheorizing. I describe below the procedures I followed when analyzing differenttypes of data.Survey DataThe survey protocol contained 20 items that measured two different variables:enjoyment of, and perceived relevance of, mathematics. Respondents had indicatedtheir attitude towards the questions on a 5-point Likert-type scale. Each person’ssurvey responses were quantified and transferred into computer data. Data relatingto each variable were separated and total scores for each of the variables wereproduced for each respondent. Histograms of the distributions of total scores foreach variable were produced, and their means and standard deviations calculated.Finally, a correlation of the scores for each person was calculated and a scattergramof the distribution was produced.Observational DataI had made extensive fieldnotes throughout my observations. These noteswere transcribed shortly after each observation and used to generate and testconceptual themes and categories during this first phase of data analysis as 87described above. During the second phase, when the earlier categories and themeshad become more “fixed” into theories, I reread the entire corpus of fleidnoteslooking for recurrent patterns and examples that might challenge or disprove thosetheories.Analysis of the video-recorded data proved more complicated. I found noguides or established procedures to help me. Existing guides to coding video datatended to require elaborate and intricate coding procedures and techniques, orfocused instead on sophisticated micro-analysis of small segments of data. As mypurpose was to capture a wider, classroom perspective, I adopted a less rigorousapproach. I viewed the video-recordings repeatedly and took extensive field-notesduring each viewing. Next, I used my conceptual framework and data obtainedthrough other collection methods to discern some examples of the themes in thevideo data. I identified which segments of the tape corresponded to those themesand finally fully transcribed the audio tracks of these segments, noting specificbehaviors in the margins of my transcribed notes.Interview DataInterview data were treated in much the same way as the observationalfieldnotes. Each interview was tape-recorded and transcribed, and its transcriptionwas checked against the recording, and amended where necessary for greateraccuracy. 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 eachinterviewee for them to clarify certain points, and comment on my initialinterpretations. This proved impossible. Data collection was intensive and little timewas available (for either the researcher or the respondents) for the further interviews 88that such clarification would have necessitated. Further, because so many interviewswere being conducted, the interview-transcription-checking process was alsointensive, and each cycle took upwards of 3 weeks before the interview transcriptscould have been returned to respondents. As so much of the data collected wascontextually specific, I was afraid that the lengthy time difference would influencethe ability of the respondents to adequately remember the earlier situations. In anycase, I was interviewing the same people repeatedly, and could more easily verifytheir earlier responses in subsequent interviews.Stimulated recall interviews. The data obtained through the stimulated recallinterviews provided a means of collecting teachers’ retrospective reports of theirthought processes. As such, it provided a further type of data to be organized andinterpreted; additional categories could be developed to analyze the kinds ofthoughts that teachers report. However, such categories also reflected my interestsas 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 duringteaching? Did teachers’ reasons for their behaviors constitute adequate explanations?Did the teachers censor or distort their thoughts in order to present themselves morefavorably? I was unable to adequately answer these questions. However, mypurpose was not to inquire too deeply into whether what the teachers said they werethinking corresponded exactly to what they were thinking at the time. I thereforeresolved that the data I obtained through the stimulated recall interviews, togetherwith the other data on teachers’ thinking and behavior, enabled me to gain somepicture of the types of decisions that teachers made, and the ways that theydescribed their own actions.These issues have also been identified in other discussions of the use of 89stimulated recall interviews. Calderhead (1981) identifies two types of factors thatdetermine the significance or status of stimulated recall data. First, several factorsmay influence the extent to which people recall and report their thoughts. Forexample, viewing a videotape of one of their lessons can be, for many teachers, ananxiety-provoking experience which may influence their recall or the extent to whichthey report it. Additionally, Bloom (1953) suggests that each individual perceives aunique 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 viewingvideotapes of their lessons are perceiving the lesson from a different perspective--asobservers rather than actors. This, they claim, can affect participants’ thinking.Additionally, they note that participants can be distracted by, for example, theirphysical characteristics.A second category of factors concern those areas of a person’s knowledge thathave never been verbalized and may not be communicable in verbal form.Calderhead (1981) describes this as tacit knowledge which, although forming a largepart of everyday cognitive ability, may have been developed through experience ortrial and error, and cannot be verbalized during stimulated recall. For experiencedteachers, much of their classroom behavior may be unthinking and automatic: theyhave long since forgotten the rationale for such behavior. It would seem unlikelythat stimulated recall could reveal thoughts which occur at a low level of awarenessor without any awareness whatsoever. Nisbett & Wilson (1977) argue that self-reporting of such higher-order cognitive processes is impossible and that datacollected by stimulated recall is not the result of introspective awareness but theresult of recalling of a priori causal theories which participants may regard asappropriate explanations for the outcome of their thoughts. In this way, they maynot represent the actual decision-making processes involved.90DocumentsCurriculum and textbook materials were analyzed in two ways. First Ianalyzed the content and style of curriculum guides and textbooks to examine whatthey specifically said about mathematics education. For example, questions Iconsidered included: What do they say are the goals of mathematics education? Dothey indicate how these goals can be achieved? What kinds of knowledge arerepresented as being important? Do textbooks encourage classroom activities thatbuild on learners’ experiences? Do they indicate how learners can use texts to helpthem learn? Selander (1990) describes a theory of pedagogic text analysis and offersmethodological suggestions for such analysis. Briefly, his theory suggests that aproper understanding of textbook content involves the consideration of “backgrounddata: the social, political, and economic system in which a certain. ..writtencurriculum is situated;....the selection of facts and themes; the style of writing.. .andthe combination of facts and explanations” (p. 147). This analysis helped medetermine how much the curriculum guides and set texts are appropriate for adultlearners.Second, I considered the place and role of the textbooks within each course. Igathered data on how textbooks were perceived and used by adult learners and theirteachers. For example, I sought to discover: How do the teachers and learners usetheir textbooks? How are textbooks discussed within the class? How does thetextbook fit in to the course syllabus/lesson format/classroom activity? Analysis ofthis data helped me understand the use of textbooks from the participants’perspectives.Finally, I considered the students’ notebooks, homework assignments, andend-of-term examination papers. Careful inspection and analysis of these enabledme to determine those areas of mathematics which students found easy, and those 91where they had difficulty. Further, I was able to determine the areas where studentsdid not appear to understand the work they were being asked to perform, whetherthey appeared to be aware of any such conceptual lack, and, if so, how they dealtwith it.Criteria of SoundnessIt is necessary in descriptions of research to discuss certain issues thatnormally fall into the categories of validity, reliability, and generalizability. Thesecategories, however, are more appropriate to research conducted from a positivisticapproach, and are usually considered inappropriate (at least in their generallyaccepted 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 areconceptualized and described somewhat differently in qualitative research.Qualitative research still concerns itself with “issues of a studies’ conceptualizationand 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 researchpropounded 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 forjudging qualitative inquiry. Their four criteria concern the issues of credibility, 92transferability, dependability, and confirmability.CredibilityHere, the goal is “to demonstrate that the inquiry was conducted in such amanner as to ensure that the subject was accurately identified and described”(Marshall & Rossman, 1989, p. 145). My study was based solely on data derived fromextensive study in adult mathematics classrooms. Descriptions of that data wereonly considered within the parameters of those settings, the people in those settings,and the theoretical framework of this study. My initial findings were presented tothe students and teachers of the classes I studied in both individual and groupinterview situations. Here, participants were able to examine some of my initialconcepts and explain and darify their own perspectives on them, and on teaching ingeneral. Further, the study was, throughout, conducted under the watchful guidanceof my doctoral research committee. All stages of research conceptualization, datacollection, 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.TransferabilityThis second criterion refers to the researcher’s ability to demonstrate that hisfindings can be transferred, or applied, to other contexts. I make few claims on thetransferability of this research to other settings; the burden of applicability seems, tome at least, to rest with those who wish to make such a transfer. However, for thoseresearchers who may wish to replicate such a study as this, I have, throughout mystudy, provided details of the theoretical parameters of my research, and my datacollection and analysis procedures methods. In addition, my research involved 93triangulation of multiple sources of data. I gathered data in several situations, frommultiple informants, and chose data collection techniques that provided data fromseveral sources to “corroborate, elaborate, or illuminate” the research (Marshall &Rossman, p. 146).DependabilityHere the concern lies in accounting “for changing conditions in thephenomenon chosen for study” (Marshall & Rossman, p. 146). As the classrooms andsettings I studied were constantly changing, I can make no sweeping claims fordependability. However, my continued presence in the chosen settings over a periodof time allowed me to recognize and respond to any such changes. My data weregathered over the complete lifespan of the phenomenon--a term-length course—and Iwas able to observe the teaching in all of its different phases. Further, by constantlyrelating the data to the theoretical framework, I minimized any effects that changingconditions could have made on my study. Finally, I kept an “audit trail” of what datawas gathered and how it was gathered and can therefore account for both theprocess and the product of my study.ConfirmabilityConfirmability refers to the issue of whether the findings of the study can beconfirmed by others, and not overly biased by the “natural subjectivity of theresearcher” (Marshall & Rossman, p. 147). I attempted to do this in three ways: byensuring that the study’s data and protocols are available for inspection, byconstantly ensuring that all aspects of the study were related to the conceptualframework and the tenets of my chosen approach, and by ensuring that my methodsof data collection were responsive, and sympathetic to, the study participants’ ownsituations.CHAPTER 4: CONSIDERING THE FRAIvfESIn this chapter I animate my research model (see Figure 1) and consider thebackground elements to my study: those frame factors that can influence or limitteaching processes. I first examine the institutional framework: the contexts andsettings in which mathematics education takes place. I discuss, in turn, the placeswhere such education happens: the College, its departments, and its classrooms.Next, I turn to the people involved, and discuss their backgrounds, experiences, andattitudes. Finally, I consider the work that these people do, by examining thecurricula of mathematics courses and the key role played by the set textbooks.Institutional FrameworkClassroom research often ignores the context or settings in which educationtakes place. Yet such contexts can often have a major influence on educationalprocesses. The institutional framework of educational establishments, theiradministrative and physical structures, and their rules and procedures can all affectteaching processes and how those processes are perceived.In this section I discuss the first element of my model that can be seen as aframe factor: the institutional aspects of mathematics education. I first describe theCollege in terms of its physical and administrative structures, its overall courseprovision, and the services it offers to students. Next, I focus on the mathematicsdepartment itself and consider its goals and operation, the mathematics courses itoffers, and some of its departmental policies. Third, I describe the mathematics 96classrooms and how they are perceived by those who use them. Finally, I considersome of the ways that these settings affect the teaching and learning of mathematics.CollegeActon College (a pseudonym, abbreviated throughout as AC) is a publicly-funded post-secondary educational institution established in the 1960s from severalexisting adult education bodies and institutions. Its mission is toProvide adults with quality, student centered educational opportunitieswhich promote and support lifelong learning, personal development,employability and responsible citizenship. The college welcomes all membersof its culturally diverse and global community irrespective of ability orprevious education, induding those encountering barriers to their fullparticipation in society. (AC, 1994, p. 1)AC offers a wide variety of academic and vocational programs and courses toseveral thousand students each year. Four aspects of the College warrant discussionin terms of their apparent influence on teaching processes: its physical andorganizational structures, its courses offerings, and its provision of student services.Physical StructureAC is a multi-storied concrete building situated on the side of a hill in anActon suburb. Because of its hilly setting, the college can be entered from a variety ofdoorways on four of its levels. The doorways open onto a series of interlockingcorridors that house all the instructional and administrative facilities. On the firstfloor, these corridors are all linked by outdoor patios where potted trees and shrubsare interspersed with bench seats. This area is close to the college cafeteria and is apopular place for students to take a break between classes or pause to smoke, eattheir lunches, or chat with friends.Within the building, classrooms and offices for each program area aregrouped together. For example, the several automotive training programs are alllocated in the basement. The Adult Basic Education division (which includes themathematics department) is housed on the third floor. The internal layout of thecollege, although making access and movement easy, is confusing. There are fewsigns indicating which floor one is currently on or giving directions of where onemay need to go. Also, the room numbering system is complex, and one can often bestopped by bewildered students wandering the corridors searching for a missingroom. Despite this, the college has a friendly, if somewhat impersonal feel. It seemsdesigned 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 (andwhere you should be doing it), then the physical structure seems designed to helpyou; if you don’t (or are unsure), then the structure is confusing.Organizational StructureAcademically, AC is organized into three main divisions: Adult BasicEducation (which indudes the Mathematics Department), Career, and English as aSecond Language. The first offers programs in academic and vocational upgradingand Adult Special Education. Instruction is offered in courses from a basic literacylevel 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 perweek) or part-time basis, and, in most areas, can opt for either classroom-basededucation or an individualized, self-paced program of study.AC’s organizational structure naturally affects teachers more than it does 98students. For example, Departments and Divisions compete for limited financialresources which determine staffing levels and the scope of course provision. Despitethis, however, teachers in the mathematics department thought that AC provided agood working situation. As one teacher put it, “normally, [it’s] a good place towork.. . .The environment is good. . . also the colleagues. We are like [a] bigfamily.. .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 doesexpect 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 theteachers in the mathematics department have either sufficient service to be exemptfrom the compulsory requirement or have a BC Teachers Certificate (which alsoexempts them). It offers refresher “instructional development workshops,” but onlyone teacher had participated, and he found it of little use: “I suppose it was allright.. .not much about math, though” (T.1.3).The only administrative restriction mentioned by teachers concerned therecommended minimum class size. (Each class is supposed to have fifteen registeredstudents by the third week of dasses or it is canceled.) As the department head putit, “If there’s enough. . .to run the class, once the term is underway [then] the collegestays pretty much out of it” (T.1.1). Other teachers agreed, “There’s supposed to be ahard and fast rule about [dass size] but often it depends on the department andwhat other [courses] are being offered... . [For instance], could students move toanother section if this [class] was cut?” (T.1.8).Teachers reported that the college administrative organization did not overlyinfluence their day-to-day teaching. “Everything’s changing all the time,” said oneteacher, “although we still have a lot of freedom. The upper level [of administration] 99is 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 anyoneshould 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 maintaindistance between themselves and the college administration.Despite the current budgetary uncertainties, teachers appreciated what theycalled the “lower stress levels” of college teaching as compared to high-schoolteaching. One teacher listed the benefits of teaching in a college: “You don’t have todeal with parents, [and] attendance isn’t an issue. If a student doesn’t want to cometo class, we try to find out why.. .but it’s not essential [that we do]” (T.1.1). “Youdon’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 arefew 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 theirclassroom teaching processes without much interference from the collegeadministration. Indeed, they prefer it that way. Even those factors (such as minimumclass size) that could affect teaching are “negotiable.” If anything, administrativefactors served to encourage teachers to devote less time to teaching. As one teacherput it, “If there’s any danger that this class may be [cut]. . . then I’m going to be a littlebit cautious in terms of how much preparation I do” (T.1.8).Course ProvisionThe mathematics courses offered by the college form part of a system ofrequirements for completion of the ABE Provincial Diploma. This Diploma,sanctioned by the BC Government’s Ministry of Advanced Education and JobTraining in 1986, is awarded to any student who completes the requirements for 100secondary school graduation as laid out in the Provincial ABE Framework (seeAppendix 8). The Diploma is recognized by colleges and universities throughout theprovince as an official credential for entry into university studies (Ministry ofAdvanced Education, Training, and Technology, 1992). The Diploma (and the ABEFramework) is overseen by a ministerial committee on Adult Basic Educationcomprised of representatives of those institutions that provide ABE coursesthroughout the province.For students who study mathematics at AC, three aspects of course provisionseem to affect them most: the admissions procedures, the financial cost of thecourses, and the attendance requirements.Admissions. As a “post-secondary institution committed to educating theadult learner,” AC normally only accepts Canadian or landed immigrant studentswho 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 internationalstudents [are accepted] on a cost recovery basis” (AC, 1993, p. 8). Students whoapply for admission must attend a pre-registration interview and “present details oftheir previous educational attainments” (p. 8). If prospective students have beenaway from an educational setting for three or more years, they are normally requiredto take (at cost) an assessment examination to determine their “appropriateplacement level” for each of their chosen subjects (p. 8). Further, AC requires that itsstudents 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 EnglishLanguage Proficiency assessment before they are admitted. Hence, prospectivestudents intending to study several subjects (as most do) are faced with an array ofexaminations and charges before their courses start.The College admissions procedures also seem unduly complicated. “You need 101a degree just to get in here,” said one teacher (T.1.7). In general, teachers thought thatthe 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 ofour students. It’s more of a hurdle than a help. We put too much credibility onassessment tests. Of course, you have to be careful: we do give [more]credibility to presented academic credentials than discussed academiccredentials. But getting in can be a challenge in itself. The assessment costs$15 which can be.. . daunting for unemployed, broke people. If [students] arenot sure what they want to do.. .or are just shopping around, then $15 formath and $15 for English or whatever can frighten them away. (T.1.3)The admissions procedure certainly served to weed out the uncommitted. Foranyone aged over 20 who wished to study full-time, the application procedure alonecould cost almost $100. Additionally, acceptance into the college depends entirely onstudents’ past academic performance or their achievement on subject-based“assessment” examinations. Those adult learners who pass the initial examinationsand are admitted to the College have already been prepared to equate success andachievement with standardized and impersonal forms of assessment.Finance. Tuition for each course costs a further $90. In addition, students areexpected to purchase the textbooks and any required extra material. (For example,students in one of the introductory mathematics classes are required to buy a $15geometrical construction set.) Many students were not paying for coursesthemselves. Several indicated that they were receiving financial aid, either fromfamily members or from government grants or loans. “I could never afford to pay forthis myself,” said one publicly-funded student, dismayed at the costs he did incur, “Ihave to buy the books as it is” (L.3.4).Obviously, the combined cost of the admissions procedure and the coursescan influence the attitudes of students who enroll. Regardless of who is paying,enrolling for courses represents a sizable investment. Many students feel pressuredto attend all the sessions of those courses in which they enrolled, and try to pass 102these courses, if only to realize their investment. One student described: “I feel likethis is my last chance.. . .1 totally wasted my time at school. . . and now I’m gettingmoney for a second shot... .I’m really lucky.. .many folks don’t get thisopportunity.. .so I’d better not blow it” (L.3.2).Attendance. Once admitted to the college and enrolled in courses, studentsare expected to attend and participate in all of its sessions. “Successful completion ofand progress through courses/programs is based on.. .class assignments,examinations, participation and attendance” (AC, 1993, p. 16). Indeed, students whofail to attend the first three classes of a course or who do not regularly attendthroughout the term are asked to withdraw. However, the college recognizes that ithas a responsibility to assist students in overcoming problems that affect theirperformance and attendance. It makes such assistance principally available fromeither the academic department involved (for instructional and learning problems)or the college’s counseling service (for students’ vocational and personal concerns).Student ServicesThe college provides a variety of services to “help students with their studiesand assist them in completing their goals and objectives” (AC, 1993, p. 20). Theseinclude an Assessment Centre, Bookstore, Cafeteria, Counseling Service, Daycare,Health Service, a Learning Centre, and Library. For mathematics assistance, studentsidentified the Assessment Centre, Counseling Service, and the Learning Centre asmost useful.The Assessment Centre offers (for a fee) assessments of students’ abilities inreading, writing, mathematics, typing, accountancy, or English language. These testsare designed to “help students determine their appropriate placement levels” and theresults “may be used in lieu of school transcripts for admission to courses” (AC, 1993, 103p. 20). The mathematics assessment is scheduled to take 1 hour and determinesstudents’ skills in basic arithmetic and algebra. Most students in the introductorymathematics courses had taken the assessment as part of the admissions process andfound it useful. “It wasn’t too hard,” said one. “It showed where I needed help andtold me which was the right course” (L.3.3). Most of these students had accepted theresults of their assessments and enrolled in the courses at the suggested level. Acouple 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 onestudent, “when I was at school.. . . But I thought that I’d better go over it again.. .to getmy hand in” (L.3.1).The Counseling Service provides a range of services: educational, career andvocational counseling; crisis, stress management, test and mathematics anxietyintervention; instruction in life skills; and services for disabled and internationalstudents. Most counseling is provided on an appointment-only basis, although theService also provides a limited drop-in and emergency facility. The CounselingService also operates a self-help resource centre for current and prospective studentsto obtain information about the college and its facilities and services for students.The Learning Centre is a drop-in “learning support service.. .provid[inglstudents with assistance with their studies” (AC, 1993, p. 24). Its services include“one-on-one tutoring, specialized small group workshops, audio tapes and listeningcarrels, computer software, study areas, course materials, makeup test services, andcourse-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 twohours extra tutoring throughout the week. Finally, the learning centre is one of thefew quiet places at AC in which students can study.Teachers were very positive about the student services provided by the 104College, and saw them as one of the main benefits for students who choose to studyat AC. “We offer as much as we can.. .certainly more than other [colleges]” said oneteacher. “Funding for poor students, a counseling centre for those with learningdisabilities, a learning centre which is like a study hall. There’s a good math tutorthere” (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 realbenefit” (T.3.2). Teachers also saw their own work as fitting in to this network offacilities. As one put it,We help students get what they need... .We do more than just teach. There aretimes 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 arecrying on your shoulder because they have larger problems. (T.1.2)Most students, in contrast, although aware of the College facilities, found littlereason to use them. As one put it, “I find I don’t need them at the moment. I reallyjust stay here as short a time as [possible]. I go home and do my work... .Maybe nextterm 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 otherbooks in case we got confused (L.3.6).Perhaps to counteract the rather impersonal aspects of the College, somestudents 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 andhe’ll sit down and show me a way of doing it that’s really simple. Then he letsme sit there and practice ‘til I’ve got it right... .He makes it look really easy. Hedoes it to everybody.. .even the guys that are taking calculus. (L.3.4)Overall, students compared AC favorably to their high-schools: “AC is muchbetter.. .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 friendwho took Math 11 at high-school and he took it again here, and he says thathe 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 neverpart of that, but it’s going on around you. That does something to yourlearning. Like, in school I was OK in math.. .1 was a “C+.” But since cominghere, I’ve been an “A.” So.. .it’s the same work, but I’m learning faster. I’m in abetter environment. I can concentrate more.. .it helps me out having people Ican relate to around me. (L.1.3)DepartmentsThe Adult Mathematics department, together with the departments ofScience, Humanities, Business, and Computer Studies are organized into anadministrative section called College Foundations (CF), all part of the ABE Division.In this section students can only take semester-long, classroom-based courses. (Thereis 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 tostudy mathematics in an environment where the student can make progress andexperience success” (AC, 1993, p. 58). Teachers in the department assumed theresponsibility for setting an appropriate climate to meet that goal. Teachersdescribed how such climate-setting involved not only their own dassroom behaviorbut also the interpersonal relationships within the department. “People are veryeager to help each other,” said one teacher. ‘There’s no professional jealousy. . .ourpersonalities just sort of mesh” (T.1.7). “It all works for the students,” explained asecond. “If [teachers] are getting on together, then I’m not distracted and can focuson my teaching” (T.3.3).Teachers also place expectations and responsibilities on students so that they 106can “make progress and experience success.” Apart from the general collegeguidelines on appropriate student behavior, the department requires thatThe new student must enter the course appropriate to his/her background.Therefore when the student has not taken a mathematics course during theprior three years, an assessment is recommended. ESL students must be at theUpper 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 tohis/her background” rests with the department; students cannot enroll for whatevercourses they choose. In this system, assessment of prior knowledge is key, both forthose 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 thestudents enrolled in the introductory mathematics classes fall into these categories.The mathematics department has a good reputation among students andteachers. “You get a lot of support,” explained one student. “People are tempted towallow.. .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 myinitial 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 thediscipline. . . someone to hold my nose to the grindstone” (L.3.6). “You’re forced topay attention,” said a third student. “The teacher goes so fast that you can’t afford tomiss anything” (L.3.3).Teachers identified how the mathematics department also had a goodreputation with other provincial colleges. “It’s been built up over the past number ofyears,” said one teacher, “we hear from BCIT [and] UBC.. .that our students do reallywell in their courses” (T.1.6). “[BCIT] are very pleased with our program,” agreed thedepartment head. “They tend to send, or encourage students to come here to get the 107pre-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. . . andthey say that they notice that their students are really weak in particularareas, so we take that into account when we’re planning our courses.. . .Theamount of depth in our courses is sometimes determined by what thereceiving institutions want.. . .1 mean, we want our students to be successful.(T.1.1)Notice that, in terms of reputation, students mention comments made byother students; the teachers refer to comments made by colleagues at otherparticipating institutions.CoursesAC offers mathematics courses at three levels corresponding to academicgrades 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 orwho 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 coursesdesigned for the student who has never studied academic mathematics beforeor who is lacking a good foundation in basic algebra skills. The contentincludes: a review of basic math skills, a study of measurement, andintroductory algebra and geometry. Mathematics 050 must be taken beforeMathematics 051. (AC, 1993, p. 59)Further written information about each course is given to students during thefirst meetings of each course. Usually, this information concerns the instructor’sname and phone number, a list of the set books and extra material required, the 108meeting dates for course sessions and what topics will be covered on what days, andthe course assessment guidelines. Appendices 10- 12 give details on the threecourses 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 insimilar ways. (Indeed, the handouts describing the information for the coursesoffered by the department follow the same structure.)Curiously, the neat regularity of this schedule allows for considerableflexibility amongst staff. The teaching of courses is shared among all eight of the full-time mathematics teachers in the department. One teacher explained howscheduling decisions were made: “The department head.. . sends round aschedule. . .usually it’s what we did last term, and teachers can make any commentsor requests on it. Then, if there are objections, conflicts. . . they can be discussed at adepartment 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 whichclasses... .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 littlefrom term to term. One teacher explained thatMath 12 and 11, they’re always offered early... .Some of that has to do with thescience classes.... Students taking those levels of math are also taking thephysics, and biology and chemistry, so that you have to make it flexible forthem. So, 050 and 051 get offered later, usually at 12:30. They’re never offeredearly. (T.1.7)Teachers appreciated this opportunity to influence their teaching schedulesand supported the department policy of rotating teachers among its classes:To make teaching more efficient, we have to constantly change [and] reviseour curriculum. It’s the changing that makes teaching interesting andchallenging and [keeps] us constantly awake. . instead of teaching the samething. Can you imagine talking about sine and cosine, sine and cosine, sineand cosine all the time? (T.1.4)Although teachers never expressed preferences for teaching certain course 109levels, they did choose particular days or time slots to teach. So, as the times anddays on which the courses were offered changed little from term to term, teacherscould effectively choose which levels they wanted to teach. I also observed adepartmental staff meeting where teachers chose which courses to offer during thecoming summer session, and the times and days on which to offer them. Vacationdecisions had already been agreed, and final decisions about course offerings weretaken on the basis of who would be taking vacation (and, hence, who would be leftto work) during which periods. Here, decisions were taken, not to fit the departmentpolicy of rotation, but to “fit in” with teachers’ personal arrangements. ‘We try torotate as much as possible,” said one teacher, “but it depends on the schedule...whether everyone’s schedule fits in” (T.1.4).PoliciesOther than the rotation of teachers, the department made few policies thataffected how courses were taught. The only policy consistently mentioned byteachers concerned the standardized term-end examinations. “Each grade. . .has onetest,” explained one teacher. “One of us has responsibility for designing the test forall 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 whichteachers claimed to find reassuring: “It stops people doing their own thing. If I’m notpreparing the test for my class, I have to make sure that I cover all the areasproperly” (T.1 .7).A second policy which somewhat affected the mathematics departmentconcerned the end of term evaluation of courses by students. Students have a right(under the College regulations) to evaluate each course, and the mathematicsdepartment has devised an evaluation form for such a use (see Appendix 13). The 110department head explained how it should be distributed:The departmental assistant [should] give it out in each class before the end ofterm... so that [teachers] have a chance to respond to some of the things thatthe students say. Its a kind of A-B-C-D-E scale and then there are some spacesfor written responses. It’s completely anonymous, so the students can saywhat 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). Othershad different explanations: “I’d like to do [the evaluation],” said one teacher, ‘but atthe end of term there’s not enough time” (T.1.2). “I think that [it] would be toointimidating,” said another teacher, referring to in-class evaluations. “Studentswould 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’llthink’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, Iguess,” said one. “If one [of us] doesn’t do it, then there’s no pressure on the rest ofus” (T.4.2).ClassroomsAC’s mathematics classrooms are situated close to each other (and theteachers’ 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 classroomsone is constantly aware of the proximity of other departments. Trumpet solos oroperatic 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 sharedwith neighboring departments. Each classroom is similar to the others: they are eachabout 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 2mx Im) wooden tables and 25-30 wooden chairs laid out in rows facing the teacher’sdesk and the blackboard. The length of one wall in each room is taken up entirelywith desk-height windows that look out onto a concrete walkway and otherclassroom windows beyond. Two of the rooms also contain an overhead projectorand screen set up in one corner next to the blackboard. The rooms look and feel liketraditional college classrooms: anonymous, businesslike, formal.Two of the classrooms have notice boards carrying a variety of posters. Thesehave details of the College’s health and counseling services, a notice advertising along-past college event, the library opening hours, what to do if there’s a fire, andadvertisements for credit cards and magazine subscriptions. Only a few deal withmathematics. Of these, most seem to have been produced by textbook publishersand 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, “Donot worry about your difficulties in mathematics. I can assure you that mine aregreater.” Another headed “MATHPHOBIA CAN COST YOU A CAREER!” lists jobsthat people are supposedly unable to hold if they have mathphobia: “statistician,physicist, pilot, dental technician, accountant, surveyor, welder, chemist.” Althoughthese posters are presumably displayed for the benefit of students, I never saw anystudent stop to read them; nor were they ever referred to by the teachers. Indeed, bytheir yellowish tinge and curled corners, the posters all looked as if they hadn’t been 112moved for some time.Those who used the rooms differed in their reactions towards the roomlayout. Teachers were aware of how classroom layout could affect learning andteaching. “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-groupwork going. Some of our classrooms [have] just rectangular tables and it’s muchmore difficult to do anything [other] than pairs” (T.1.1). Another teacher said that hepreferred to have students working together but that the table layout did notencourage collaborative working: “We do our best with the rectangular tables, butthe students know that [even though they sit together] they’re not doing cooperativegroupwork. I mean they don’t get the same grade. . .so they’re not that committed toeach 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 timerearranging all the desks. The next morning the students had come in andrearranged them all the way they had been before. They obviously didn’twant to work in small groups, I guess. (T.3.1)Students, conversely, were largely unconcerned about room layout; their onlycomments 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 getwhittled down... .So, you’ve got this huge room with only eight people in it” (L.3.1). Iwas interested to note that the layout of the two math-only rooms did not changethroughout the whole term, whereas in the shared room, the layout changedregularly. Sometimes the tables would be in rows as in the other rooms, sometimesin a hollow U-shape, once in a solid block of tables, and once with the tables puttogether in pairs for groupwork. During my observations in this room, neitherteachers nor students ever mentioned the change in the room layout. “Don’t make nodifference,” said one student when asked directly, “math is math” (L.3.9). “I do like tosee everyone’s face,” said a second student, “but it’s really only important in a 113classroom 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 throughoutthe term. “I like to sit near at the front,” said one student, “1 can concentrate better ifthere’s not too many distractions... .You know, people coming and going.. . .If you getthat big guy, Harry, in front of you, you can’t see a thing” (L.3.2). “I’m supposed towear 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 Isit, so long as I can get a good look at the board. [The teacher] likes to write all overit, so you need to be able to see even the bottom corners” (L.3.6). Only one studentsaid that it shouldn’t matter where people sit. “We’re there to learn,” he said, “weshouldn’t try to get as close to the board as possible” (L.3.4).Experiences of Students and TeachersStudies of teaching are often limited by focusing either solely on classroompractices and dynamics or solely on the backgrounds and experiences of learners orteachers. However, in reality, these two areas are interrelated and interact to affectclassroom practices and influence interpretations of those experiences. In thissection, I consider the second element of my model: the experiences of students andteachers involved in three introductory-level mathematics classes.I first discuss details of students’ backgrounds and experiences, their attitudestowards mathematics education, and their reasons for enrolling in a mathematicscourse. Next, I consider certain characteristics of all eight teachers within the 114mathematics department. Although my research focused most closely on threeclasses, there are two reasons for obtaining data from all the teachers. First, themathematics department seeks to arrange a teacher rotation to ensure that allteachers teach introductory level courses. Second, at the outset of data collection, nodecisions had been made about which courses would recruit enough students toproceed, or which teachers would be teaching which courses.StudentsThe beginning of the term is filled with uncertainty. Some students enroll incourses and don’t show up. Some come to classes for only a few sessions and thenleave; others attend without ever having registered. College policy recommends thata minimum number of 15 students be registered in each course by the third week ofclasses or the course is canceled. Consequently, during the early part of the Springterm, there was considerable anxiety within the department that one or more courseswould 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-levelmathematics courses. During the fourth week of instruction I administered a surveyprotocol (Appendix 2) to the 32 students who were attendance that week. Their selfdescriptions of basic demographic data are summarized in Appendix 14.Students’ BackgroundsStudents in this study have a variety of backgrounds in terms of their gender,age, ethnicity, and occupation. The literature identifies several other features ofstudents’ backgrounds which can influence teaching: students’ English languagefluency, other courses being taken, and students’ previous experiences of 115mathematics education. I now discuss each of these background features in turn.Gender. Of the 32 students surveyed, 20 were male and 12 female. Thegender balance remained the same throughout the term even though some studentsdropped ou of classes and others joined. When the survey was re-administered inthe 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 genderbalance appears. In the two classes that met only twice per week (050 and 051) thegender 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 classteacher, “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 over24 years. The distribution of ages is shown in Figure 2. (Note: only 30 of the 32students gave their ages in the demographic survey.)116Histogram of Xi: age54Z3o 3P..ageFigure 2: Distribution of Student’s AgesThere was no marked difference between the range of ages of the male andfemale students. Further, with regard to the distribution of their students’ ages, thethree classes were similar: each dass contained one or two students aged 19 years oryounger, 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 feelso 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 eitherCanadian or part-Canadian. The others identified themselves as either First Nationsmembers (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 thenon-European immigrants, all of whom were also enrolled in college English classes.Language ability was key for many students. Although the language used inthe mathematics classes was not seen as “hard”, it occasionally contained uncommonwords, which, if not understood, could delay students’ understanding of themathematics. Several of the immigrant students said that, although they had 117previously studied mathematics in their native countries, they were re-taking Grade10 mathematics in order to gain further familiarity with the English language. “Theywanted to put me in a higher grade,” said one Chinese student when referring to herinitial placement interview, “but I said I want to go over [grade 101 again--torevise...and to practice with the words” (L.1.1). Few of the foreign students identifiedthat 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 themathematics class, and would occasionally look up unfamiliar words. However, thisremained a private activity--often carried out under the desk (and out of sight of theteacher).Language difficulties were never addressed publicly in class, although fellowstudents 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 atthe 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 ofthe teachers talk to you like you’re a 12-year-old. Enunciating everything. ..like you’restupid” (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 peevedbecause she [the teacher] seemed to speak down to them all the time, but she’snot really speaking down, and now this classmate has now said, “Oh, I’m gladshe 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 atleast part-time; the others were either full-time students or unemployed. Of the 16workers, the eight men had jobs as cook, musician, taxi driver, clerk/cashier, 118maintenance worker, and night-watcher. The eight women had jobs as artsadministrator, 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 tothree and start work at three. Then work until 8 or 9 at night. Get home about9:30; 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 toclass during the day. That’s not so bad.. .but it’s the homework. I try to dosome while I’m at work.. .but usually I have to get up early to do it. Whenyou’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’sabout 10 hours per week. Normally, they’re very good about letting me havetime off to come to class...sometimes I have to switch shifts with other staff. Acouple of weeks ago there was stocktaking and that was hell. It was verydifficult at school —I had a lot of papers to hand in for my other courses. ..andwe had a math test, so it was very stressful. I couldn’t miss any work in case Igot 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 workingarrangements 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 timeoff 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 tocome 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 eitherlived at home with (and were financially supported by) family members, had builtup sufficient savings to fund their time at the college, or funded their studies bystudent loans or government grants. One young student described his financialposition:I get a grant to cover my fees. I could never afford all this by myself... .Thenthere’s the books.. .Social Assistance has to pay for that. You have to be 19, butthey will help for education. It’s not that I want to go on it, but I’m going to 119have to to survive, to have an income to pay rent.. ..My Mom’s letting me offwith the rent right now because I’m not of age, but when I’m of age she’sgoing to expect it... .I’ll keep getting the funding as long as I pass [thecourses]. (L.3.5)Other courses. For most students, mathematics was only one part of theirstudies. All the students I interviewed were taking other courses at the college; allimmigrants were studying English as a Second Language, and either computing oraccountancy courses. Among all the students, mathematics, computing, andaccountancy appealed because they were less language-based than other subjects. “Ilike the math class best,” said one foreign student. “There’s just one book. . . thelanguage 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 studentswere trying to gain their high school equivalency and said that they took the sciencecourses to help them gain access into higher education courses at other colleges orlocal universities.Students said that, in general, they preferred the mathematics classes to theothers they were taking: “[Mathematics] is easier that way. You know what youhave to do and when you have to do it” (L.3.3). Another added: “I can sit down tomy math homework and know that there is an end to it.. .even though it might takeme all night. With other subjects. . .like English, I never feel that” (L.3.9). For manystudents, part of mathematics’ attractiveness as a subject was its ‘boundedness”--theway 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 120about 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 wereentering an adult mathematics class for the first time; indeed, for many it was theirfirst experience of adult education. Several students remembered their childhoodmathematics education: these were commonly described as unpleasant experiences.For some, math was just one part of an altogether negative school experience: “Idetested school, period... .Where I grew up, school was not a big pastime. ..there wasa 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 wewere always moving. I don’t think that math was any worse than othersubjects...I think you can wing it [in math] because.. .all other classes aredealing with language. (L.3.1)For others, mathematics education was worse than for other subjects: “I canlook back on it now.. .on my math courses. . . and a few of my teachers were duds anddidn’t make the course interesting at all. . . they had no enthusiasm” (L.3.2). Anotheradded, ‘1 could never get the hang of it. ...For some reason the math teachers werealways the worst.. .shouting, moaning.. .sometimes hitting you.... [In math] I’ve alwaysdone 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 teacherswould go like a bat out of hell. You swam as fast as you could and ifyou. ..couldn’t keep up you went flying over the waterfall... .What got lostwas...I never understood any of this stuff. (L.1.5)For many students, learning mathematics at school was a process of sittingquietly and listening to the teacher, rather than one of asking questions. Onedescribed:The thing I remember most is that I was.. .pretty frightened to raise my handand ask a question. For two reasons: many of the teachers were of theopinion.. .that children should be seen and not heard. If you’re not listeningthen that’s why you didn’t get it the first time. And secondly, if you make a 121fool 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: “Itwas 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 atschool, sometimes I [found] it hard....I didn’t [find] anything interesting in mathbecause 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 thestudents, the AC math class was the first time they had encountered it. “I sort ofdropped the [math] class before we ever got there,” said one. “I didn’t know whatalgebra was.. .and it just seemed so foreign. I think I missed the middle steps towhere you start algebra” (L.1.7). Another student expressed the view that anyprevious mathematics education could prove a hindrance:What my uncle told me about algebra, he said best... .He said if you’re juststarting to learn it now, it’s easier. He said if you. ..if you learn algebra beforeand [then] you learn it again, it’s confusing. But if you just start learning itnow, you should be OK. (L.1.2)Students’ Attitudes and AimsGiven their diverse backgrounds, the students had different attitudes towardsmathematics and different reasons for learning it. I now consider each of these.Attitudes towards mathematics. Students gave information about theirattitudes towards mathematics in two ways: in a survey completed at the beginningof 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 indicatinggreater enjoyment or greater perception of value. The distribution of scores for the 122two dimensions are presented in Appendix 21. In each figure, the horizontal axis (Esum and V-sum respectively) refers to the score, the vertical axis (count) refers to thetally 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) is30 (with a standard deviation of 4.1). A comparison of the two sets of scores showsthat students (as a group) were more likely to perceive a use for mathematics than toenjoy it. However, a correlation of the two sets of scores shows that students whoscored highly on the enjoyment dimension scale also had higher scores on the valuedimension scale. (There was a positive correlation between the scores of 0.195.)The survey scores give only a limited picture of students’ attitudes; theircomments are more descriptive. Most students were strongly convinced of the valueof 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 beprecise. . .right on.. .no in-betweens. It’s the logical way. . . the way things are. You’reeither right or you’re wrong” (L.3.5). A third thought mathematics was, “the modernlanguage. It is in everything. If you want to live. . .you want to live comfortably, youmust know math.” After a pause, she continued, “Even if you want to live notcomfortable. 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 likereasoning. . . 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 withwords, because they can mean different things to different people. But if you want toprove something. . .you use math” (L.3.7). “It helps me to think,” explained anotherstudent, a salesperson:When I’m in the shop.. .1 don’t necessarily use [math] all the time because it’s 123all computerized. But if something goes wrong, you have to know “Oh, that’snot 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 thetime. (L.3.2)Students’ views on mathematics not only referred to the topics they werestudying. One student describedcoming into a classroom before the last class was finished. There were allthese squiggles and stuff all over the board. I didn’t know what it was about. Iasked the teacher and he said [calculus]. It looked really hard. . .1 don’t thinkwe do it in this class, but I’m sure I will learn it one day. I mean, it’s all goingto 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 ingeneral, or of certain topics in particular. One student described how she was unsureof the usefulness of algebra:I don’t know if I ever will [use algebra]. For what I’ve been involved with Ihave 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 usesright 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’mtaking it... .It’s so ridiculous you’ve got to learn how to do it so you don’t feelyou got beaten by a ridiculous concept. If it wasn’t mandatory, I bet peoplewould be taking a lot of different kinds of math. I don’t think many peopletake it ‘cause they like it. . . .1 suppose that’s why it’s mandatory. . .otherwiseno-one would take it. (L.3.2)Reasons for learning mathematics. Students’ stated reasons for learningmathematics were many and varied. Half of those I interviewed had clear reasons:they were trying to complete their high-school education. Whether they wereCanadians who had left school before graduation or immigrants who needed tosecure 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, mathematicsis one of my pre-req[uisite]s. I’m thinking of getting into BCIT or some 124university, either a radiology or nuclear medicine program. If I get my Math10 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 tocontinue [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 tostart my own business doing massage,” said another. “So I’m going to take massagetherapy, which I need my biology for, and also Shiatzu, [for] which you needbiology and chemistry” (L.3.8). Some students were less certain of their futuredirection generally and were looking to their studies to show them a way. “That’swhat I’m here to find out,” said one student, “I get lots of ideas of things to do, butthen there’s lots of drawbacks to each one. So I’m just taking the courses and tryingto 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 moreimportant than a career. Some talked about the embarrassment or fear of beingidentified as “math anxious.” They described how they were tired of feelingunconfident or lazy. “I really didn’t feel good about math. . .or about myself,” saidone, “I wanted to do something about it” (L.1.7). “I took the course because I foundmy brain was getting lazy,” said another. “The more you exercise it, the less lazy itgets” (L.1.2).Other students described different reasons. One student mentioned, “thehorror of not being able to do basic mathematics, and not being able to admit it. . .isreally depressing....It brings you down so that you feel like you shouldn’t doanything” (L.1.5). A second student described howembarrassing it is to be my age and not know.. .45% of something. . .you don’thave a bloody clue what it is. Everybody assumes because of your age thatyou know all of these things right off the bat. I have been doing manual laborall my life because of an embarrassment with not knowing math, too stupid toactually come, too embarrassed to come [back] to school. (L.1.5)Finally, a couple of students said they were learning mathematics in order to 125help their children. One student who had already studied mathematics in China (herhome country) explained that she came “back to learn math to help my kid. I teachthem... times table, and I do some of their school math with them. It help me too”(L.1.1).TeachersIn this section I describe some personal and professional characteristics of theteachers. After providing some brief demographic details, I focus on theireducational and teaching experiences, and their attitudes about teaching,mathematics, and their students.The College’s adult mathematics department consists of seven full-timeteachers including the department head (who teaches part-time). In addition, oneteacher 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, mostby birth, although two are immigrants (from Hong Kong and England). Theyoungest teacher is 26 years old and the oldest is 56; most, however, are in their late30s to late 40s.Educational ExperienceTeachers regard their work as a career; most have worked in their present jobsfor over 10 years. Indeed, for the majority of teachers in the department, teachingmathematics is the only work they have ever done. Two teachers came straight to the 126College as soon as they had completed their first university degree; the others hadpreviously taught in other Canadian high-schools or colleges. Several teachers alsoindicated that they had, from time to time, tutored other people (usually children) inmathematics.All the teachers have a minimum of a bachelors degree (a collegerequirement); most of these are in science-related subjects (mathematics, science, orscience education). Four teachers have continued their formal education with postgraduate study and some also have a provincial teaching certificate which allowsthem to teach in any British Columbia secondary school. Two of the teachers havenever studied mathematics at college level; their degrees are in music and generalstudies. Both of these teachers were hired because of existing relationships with theCollege, as either volunteer tutors or as ex-students. One described how he washired:I had been an elementary school teacher [but left] after eight years of teachingto set up my own business. [After a while] I thought I should go back toteaching. . . .1 didn’t really relish the thought of teaching young peopleanymore, I wanted to teach adults. So I applied here, which is my old almamater, [to] teach English. They said, “Oh, you’re one of our ex-students. Itwould be nice to have you on our staff. How about teaching mathematicsfirst?” I didn’t really feel qualified for the position but they said, “It’s justgeneral program math. I’m sure you could handle that.” So I started teachinggeneral program math, business math, that sort of thing. (T.1.7)Although the teachers have, as a group, a reasonably strong background inmathematics or science education, they have markedly less training in adulteducation. Only two teachers have taken any formal courses in adult education.When asked about how they had learned to teach adults, most teachers said thatthey “picked it up as they went along.” For example, “Early on, I sensed I had tochange certain styles... .1 found topics didn’t have to be pre-planned to the nthdegree 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 127perform[edl well in mathematics, so.. .a group of students approached meand asked me to help them, so I actually had some experience in teaching. It’sreally informal.. .but I found it’s very interesting. So that’s a reason I triedteaching, tried out teaching in this college. (T.1.4)In any case, adult education training was not seen as important by teachersbecause “there really wasn’t any difference between teaching math to children and toadults” (T.1.3). Even those teachers who had studied education at a postgraduatelevel 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 ashotgun and you try to get as many people as possible. So when I plan myclasses what I usually do is I try to aim for the middle ground, to present it soI do not lose the lower students, but at the same time not lose the topstudents. Depending on.. .class interaction.. .1 could, you know, go higher orlower. (T.1.2)Only one teacher found his college mathematics education useful. Hedescribed being told abouta meta-analysis of calculative research in the 1970s; the results werefascinating. The commonly-held belief is that if people can work withcalculators [or] computers then there’ll be some attrition of paper and pencilskills with arithmetic. But the major finding of that analysis was that ifcalculators are used at the same time as paper and pencil skills, the peoplewith calculators have a better ability not only in conceptualizing but also withthe paper and pencil skills. So it’s a win-win situation for people withcalculators... .1 came back to the department and said, “We should be usingcalculators,” 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 ownexperiences of learning. Sometimes, their memories significantly color theirperceptions:I would think that often how we teach is affected by how we have beentaught.... I have often thought about my teaching in this way. At university Ihad to work very hard and very independently. I’ve referred [in class] toworking hard, working on your own... these are some of the things [that] I’veadopted... .1 think that affects my own personal philosophy of striving forexcellence. . .I’m a person who really likes to see excellence and organization.What bothers me is students’ lack of achievement and interest and lack oforganization 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 theirminds look like? (T.1.8)A second teacher spoke of how his own experiences with learningmathematics affected his current teaching practices:I tend to give a lot of notes because I find that the textbooks are usually notgiven in the simple terms that.. .novices to math could use. So what I usuallytry to do is I break things down into.. .simpler terms and so on. I guess that’sone of the things I picked up when I was going through education. (T.1.3)Often teachers remembered unpleasant memories of their own matheducation. One spoke of his “experience with mathematics instructors [as] almostuniversally horrid... .The tedium, the shiny polyester pants, the unchanged suits, thesweat 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’sdeadly silent, you don’t know anybody else in the room, you have no ideawhat 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 whatI left school for, to get away from this stuff. (T.1.6)Attitudes about TeachingPerhaps because of the similarity in their backgrounds and experience,teachers also held similar views about teaching. In general, they thought thatteaching mathematics was largely a matter of conveying fixed concepts and setprocedures. As the content was established, their role became one of deciding howto convey that content. Teaching became an exercise in selecting the “most efficientstrategies.” Such a process could be influenced by students, but only occasionally,and only indirectly.Teachers developed their favorite strategies with experience. “I’ve built upthree or four different ways of approach, “ said one teacher. “Of course, I have myfavorites which I will always use unless students tell me that it doesn’t make 129sense.. . .For me, when you teach you’re trying to sort through a whole garbagedump and see what is appropriate” (T.1.2). “I like to get a feel of a group,” explaineda second teacher, “get a feel of their learning attitudes.. .So I can tell which strategy isbest.. .to accommodate that, to help them achieve their learning goals” (T.1.4).In general, however, teachers subjugated the learning needs of students totheir own need to “cover the material.” While they acknowledged that students haddifferent learning styles, teachers didn’t necessarily change their teaching approachto accommodate students: “I’d like to say that it [teaching] depends on the type ofstudents.. .but it doesn’t really. It’s almost dictated by the length of the class andwhat we have to cover.. .there’s so much to get through” (T.1.6). “There’s a lot ofpressure here to get through the material,” agreed another teacher, “You can’t alwaysdo 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 reallyqualified. 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 havingdifficulty 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 personalaccomplishment at the end of each course. “It’s important that [students] meet theirgoals, not only pass [the] test” explained one teacher (T.3.1). Teachers appreciatedthat some students were already highly motivated. “They’re here because they wantto be. . . they’ve got their own reasons. . .but they’re very focused. It makes your jobwonderful, 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 alsonecessary, although these students were seen as requiring guidance towards “settingrealistic goals.” “Many students are not prepared for the hard work that they have todo,” explained one teacher. “Sometimes it comes as quite a shock” (T.1.1). A secondteacher explained that adult students, often with “poor study habits.. .well, no study 130habits 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 kidsat home, and you’re working 20 hours a week, then you should really take alook 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 seenas particularly important for adult learners. “When they leave this place they’regoing to be very much on their own at UBC or wherever,” explained one teacher. “Toget [them] ready for that means that we can’t hold their hand the whole way throughthe term.. .we’ve got to get them on their own feet” (T.1.6). “It’s impossible to covereverything,” said another,I have a philosophy that there comes a point where the student has to makecertain connections on their own. I tell them this, “I cannot foresee everypossibility 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.. .theyshould know what responsibility is. (T.1.2)Another teacher compared teaching children and teaching adults. “Whenyou’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 morepressing problems in their lives than learning. The choice to learn is clearlytheir own... .My responsibility is to remind them of that. (T.1.3)Developing responsibility in learning affects how people teach. One teacherthought it crucial to understand students’ backgrounds to ascertain their attitudes: “Ilike to find out who I’ve got here.. .where they’re coming from. That’s going to affectwhat I do. It tells me whether people are here because they’ve failed or because theyhaven’t had it (T.1.8).Other teachers said that they develop a sense of responsibility in students byencouraging 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 131work and study. I encourage everybody to ask questions.. .even if they’re dumbquestions. 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 individualactivity. I’ll coach more.. .spend more time with each individual student... toget them to think. . .to understand it for themselves.. .so they can work moreon 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. Youcan see some students just don’t have a clue.. .even though you spend timeafter time with them. I find I get personally worn down. Especially at a levellike 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 ortheir cat is going to die.. .or they’re going to have to take off for a week to goto Toronto because they don’t want to be alone for Halloween. (T.3.3)Aims of teaching. Teachers’ attitudes about teaching influenced theirinstructional goals and aims. Overwhelmingly, teachers said their general goals oraims in teaching mathematics were to foster an understanding or enjoyment ofmathematics. Only one teacher described his goals purely in terms of completing thecourse material. (For this teacher, the main goal was “helping students to get thecourse done. This fulfills the requirements” (T.1.5).) The others spoke of “gettingstudents.. .to have a basic grip of what it’s about.. .and liking it” (T.1.1), “gettingstudents to know how to do it and to understand it” (T.1.8). “The best enjoyment Iget,” said one teacher, “is when some students come back and say ‘I really liked thatcourse.. .1 really felt I understood math for the first time.’ It’s great when thathappens” (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 132lot 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 someunderstanding. (T. 1.8)Here, it should be mentioned that, for teachers, understanding is seen aslearning (and being able to reproduce) existing knowledge. The notion ofunderstanding as “making meaning” was never mentioned. Teachers spoke ofmathematics as either a fixed body of knowledge or as a way of thinking. They felt itimportant that students appreciate and understand the inherent logic andorganization of mathematics: “Something I try to do,” said one teacher, is getstudents “to appreciate the logic behind the steps. . . and why to go about it that wayinstead of another way” (T.1.2). “It’s important for me to convey the thought ofmathematics,” 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 stilldependent on walking into a classroom and sitting down and waiting for it all tohappen, we’re not doing them any favors at all,” said one teacher (T.1.6). A seconddescribed how self-motivation was far more effective at encouraging learning amongadult 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 strategieswork. . . .but in the end, it’s down to the student. Their responsibility, theirinterest, their desire. (T.1.3)Often, encouraging motivation involved building on students’ lifeexperiences. One teacher described how students havealready made some major changes in their lives.. .quit their jobs, or left theirspouses or whatever. They’ve already made some major decisions; takensome responsibility. . . .It’s my job to remind them of that and to encouragethem 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 133it” and persistence in students can pay off:You see that guy, David, that guy who came in this morning. He’s had to takethe course about three times because his life was a mess, you know there wasall sorts of things going on in the background. His writing was terrible, hisreading 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 doit. 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 lessthreatening. To do this, they would tell jokes or encourage ‘banter” to “lighten theatmosphere.” One teacher brings classical music tapes into his classroom to play asbackground music. Another, a skilled artist, uses cartoons about math to getstudents “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 moreenjoyable”:I call them “algebra adages” because we use them in the algebra level1 . Theyhave to solve a series of simple puzzles and each answer has a letter assignedto it, like a code. When they’ve solved one question they write the letter in theanswer space.. . and when they’ve done all the questions the answers spell outa saying.. .an adage. They’re a bit elementary. . .but even though they seemlike 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 withany student anxiety:I ask them if they think they suffer from math anxiety. If I know thatsomeone’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 affectstheir achievement. Like a self-fulfilling prophecy, “I can’t do math.” Well, ifyou 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 134that said you were scared, but I’ll let you know that there’s at least two thirdsof the people in here feel the same way. . . .Of course, sometimes people aren’tvery comfortable talking in a group, so I try and get them on their own so wecan talk about it there. (T.1.6)Another teacher described how he would speak to a class about his ownstruggles 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’recoming from. And I explain to them that when I went back to school I had tochange my lifestyle. I was really nervous about learning because it had beenso long, and I wasn’t sure I could do it. But the fact that you’re there meansthat you want to be there.. .and that makes up for quite a bit. (T.1.7)Attitudes about Teaching MathematicsTeachers tended to regard mathematics as either a set of thinking skills or afixed body of knowledge that transcends context. “Most of us who have ever taughtmath know it’s a universal thing,” as one teacher put it (T.1.2). Another describedthat “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 behindfactoring” (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 litresyou get.. .how much money you pay. Everyday you listen to news.. .you seepercents.. .like unemployment is down, interest rates up... .Also, mathematicstrains our mind. Some of the things we may not use in our daily lives, buteverybody has to think and everybody has to do some kind of mathematics.(T. 1.5)Despite the ubiquity of mathematics, teachers acknowledged that for manystudents, learning it was hard work. “A lot of mathematics is learning how to useother means other than a laboring--a pick and shovel--approach to thinking yourway out of situations,” said one teacher.I often tell the students that if you want to be lazy, you look for the easy way 135out. 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 simplelevel. We often expect our students to know [the same] things that we do. . .becauseof 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 senseof mental discipline. One described his students at the beginning of the course:You won’t believe how disorganized some are. They come into the classroomwith no books, nothing to write with. God knows what the insides of theirbrains are like. You have to show them. . .first this, then that. The books help alot because they’re so well laid out. You can’t do this chapter before you’vedone the previous ones. You can’t understand this concept unless you’vemastered 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 conceptto “train students’ memories.”The day before the test I will ask them to write it in 6 or 8 digits. Then I willsay, “Tomorrow, on the test you will have to write it to 8 digits.’ Then theywill be worried, “How can I remember that? I remember right now, buttomorrow I forget everything.” So, then I come up with some kind of memoryaide, so they can remember. And they will. (T.1.4)A third described how “in addition to explaining the structure ofmathematics,” 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 makesure that I three-hole punch every sheet that I give them. Also, I give themspecial sheets to keep track of their records.. .so they can see [the records] atall times. I encourage.. .record-keeping and organization through this.. ..Ithink some of them catch on to it; maybe those who do it well would havedone it anyway, but I will sometimes come along and talk to somebody.. .ifthey’re supposed to be working on something and this student hasn’t got anidea where it is.. .I’ll say “How about putting these in order and filing them atthe end of the day so you know where they are?” Hopefully the organizationthat 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 136characteristic that teachers highlight. One spoke of trying to foster more positiveattitudes 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 twostudents to develop an appreciation for the beauty of mathematics. . .1 think I’ve beensuccessful” (T.1 .1).Even in this, however, notice that the teacher regards mathematics as fixed; itsqualities (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 encouraginga notion of self-reliance.Teachers commonly thought that they could increase learning by relatingmathematics to students’ interests and experiences. Most teachers claimed that theytried 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 aboutmixing drinks and things like that” (T.1.1). Another teacher described how he wouldintroduce the concept of a slope of a line:I’ll say, “Do we have anyone from the construction industry here? Thisquestion is like finding out the pitch of a roof.” People will usually volunteerthat sort of information. It’s great when they do, because all of a suddenthere’s somebody else who is saying “This is my experience.. . thismathematics is relevant to my life.” (T.1.7)Overall, however, most teachers felt that, at the introductory level at least, themathematical topics covered contained little of intrinsic interest for either them ortheir 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 thetest,” 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’sthere.” And I don’t. Most people never use this. For example, set notationdiagrams. No working mathematician, scientist, or engineer ever uses setnotation diagrams. I guess it’s.. .the concepts behind the idea of set notationthat are important. (T.1.6)A third teacher described how, if so asked, he would answer with this story: 137One time a youngster got a job. The job is to make carpets. Well, the manageronly 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. Thelad is getting bored only doing the same piece day after day. So the youngsterasks the manager, “I don’t like this job. I’m getting bored, everyday just doingthis or that. My hands are getting tired. Why [is] it like this?” The managertook the youngster and sent him a little bit away to look at the whole carpet.“See the whole picture... .You are making a beautiful picture on the carpet.And the piece that you are making is the most beautiful thing.” So, from thattime on, the boy say, “Oh, I want to finish this beautiful carpet now.” So, if hesee only his part, very boring. . . cannot see the whole thing. But if you can seeit in the future, the part he is doing is part of the beautiful carpet.So, I give them this illustration. If you can see it, in the future you will haveyour career, you will want to do this, this mathematics. This is part of it.(T.3.1)Finally, teachers did not really expect their students to feel positively towardsmathematics. As one put it, “Math can be very boring. Really dry, very boring. Butthe students don’t seem to mind that. They seem to expect math to be boring” (T.1.6).Teachers’ Attitudes about StudentsIrrespective of which classes they taught, teachers claimed that theydisregarded students’ characteristics and backgrounds. Teachers’ only concern forstudents’ characteristics seemed to be how much they affected a student’s attitude.“Some of them haven’t been to school in a long time,” explained one teacher. “Theyforget some things. But their spirit. . . their attitude, is much better than youngerstudents” (T.1.5). Adult students were also thought to be “more purposeful... .Mostlyit’s their second chance of education,” as one teacher put it. “They tend to be moreserious about their study” (T.1.4).Teachers recognized that students had different life experiences, and even“appreciated” them. “They have such interesting stories. . .1 really enjoy hearing abouthow they came to be here” (T.1.1). “The population of students is extremely diverse,” 138said a second teacher:Many classes are often “United Nations.” You’ll have. . .in 30 students you’llhave 30 countries represented. Often very recent arrivals. I find that doesn’tmake a difference [to teaching]. Whether they’re an international student fromHong Kong paying a zillion dollars to sit in class.. .or whether they’re aNative student or a Caucasian from the East Side.. .it doesn’t matter. The onlything I look for is motivation, and that doesn’t depend on where you comefrom. (T.1.3)However, such differences were not allowed to intrude unnecessarily into theclassroom. Most teachers agreed that the demographic make-up of the class did notmaterially affect how or what they taught. For teachers, students only differed intwo areas: their “learning characteristics” and their goals.Learning characteristics. Teachers thought that the adults in their classes hadparticular learning characteristics. As one teacher put it,Adult students seem to have poorer memory than children... .Adult[s] enjoymore with logical reasonings. Everything they do, if they can reason, they canunderstand, they can perform better. Young children they have bettermemory so they don’t necessarily want to reason that much. (T.1.4)Then, there were those whom teachers saw as “problem” students. “Some ofour students are people that other institutions don’t want,” (T.1.2) explained oneteacher. “We get all sorts at this college,” agreed another:Schizophrenics, manic depressives, people from penitentiaries, weirdos,wackos, prostitutes, everything. There are others of course. . .but we do tendto get more people who are down-and-outers and have problems, a lot ofthose. Sometimes, these people have been kicked around all their lives, andthey feel as if everyone’s out to get them.. . .So you have to be able to spotthem, in case there’s any trouble. (T.1.7)Problem students were also identified as those with certain learningdifficulties. “We also get a fair bit of those,” said one teacher:They are very dependent in their learning styles. I find that very irritating.Dependent, whining. . . the person who wants you to chew their lunch forthem. One of the first things that comes to mind is.. .is that person going tocreate any problems for me because I’ve got to deal with someone who’smiserable and whining all the time. (T.1.3)Another teacher, when asked about learning difficulties he encountered, 139described students who do not wish to learn in the way that he chooses to teach:Sometimes there are students who do not want to do math the way that youshow them. Last semester, I had one.. .he rejected doing the word problemsby setting equations. He did it by arithmetic; got everything right. So this is atough case to deal with. Because you have to convince him, otherwise hesee[s] no point. Some students, they’ve been doing it their way for 10years.. .you cannot change them in a week. (T.1.4)Finally, teachers also gave special attention to those students who may bethinking about dropping the course:Certainly at the lower levels we tend to have very high attrition rates. Ahigher-risk population; older students. Often people in your class haveattempted the course before. Like there’s one fellow in this class.. .1 must havehad him in the same class in 1984. Here he is again. He’s enthusiastic all-rightbut I’m sure he’s taken six runs at it. For some of them it’s an infinite ioop.(T.1 .3)Appreciating the students’ learning characteristics allowed the teachers tosometimes make allowances:Many of our students are single parents and working, trying to hold downpart-time jobs while going to school.. .so we try to make allowances forthat.. .we recognize that they can’t always be here to take a test. So we havedays for doing make-ups and giving extra time on tests for those that have arecognized learning disability. (T.1.1)Knowing the students’ educational history also allowed teachers to choosewhich students to focus on. As one teacher described:I know them and I know their marks. If I know their marks then I know whois good [student]. Usually the good ones I don’t pay too much attention to.And the very poor one, I cannot spend too much time because he is holding[up] the class. So then when I teach... I teach the middle. (T.1.5).I try to baby my students a bit. I lead them in one particular direction. . . andencourage them to make the next logical step. Although in some cases you dohave top students who are capable of seeing past something and goingbeyond that. But these I usually keep on a one-to-one level. (T.1.2)Three teachers gave examples of how background information on studentsaffects the language that they used in class: “If I have a lot of students who havenever done any algebra. . .sometimes you do.. . .If they have no algebra at all, then I 140will probably talk to them differently. I’ll talk more about the arithmeticbackground” (T.1 .8).Goals. Teachers also perceive that knowing students’ ultimate goals affectstheir teaching. Most teachers thought that “almost all” the students were headed forhigher education: “Most want to go on to some kind of post-secondary training,either university or at BC1T. In fact BCIT sends many students over here to get thepre-requisites” (T.1.1). “They apply for courses [at those institutions] and they go tosome form of counseling and learn that they have to get certain requirements to getin.. .prerequisites like Math 11 or Math 12. So they come here for those subjects”(T.1.5).One teacher spoke of how he might use information about students’ careergoals:If I get a lot of students interested in a certain area then I might use theterminology of that area. If say 15 out of 20 say they’re going into landscapearchitecture. . . then I might talk a bit about the area of a yard, and. . .how manybits of rolled turf would fill this. (T.3.3)“We also get a lot of students who didn’t finish high-school math,” saidanother:They need math for a particular course, so they come for that. There are [also]some who just want to change careers.. .and pick up things they’ve neverdone. We get some that are not happy with what they got previously. We getsome international students who are here to complete theireducation... .Everyone [is] looking for a change. (T.1.2)One teacher felt that the name of the College department was a clue tounderstanding the students’ goals:The name.. .College Foundations.. .it means preparation doesn’t it? Notcompletion. So, for a lot of students, what they want is to say, “Look give mewhat I need to go on and get me out of here and on to the next step.” (T.1.6)Teachers tended to regard their students as individuals who had clear goals; 141consequently, they often simply presented material and relied on students to raiseany problems. “I don’t think you can break things down into this for this kind ofgroup, because each person is so unique, said one teacher. “They know what theywant, all they want from me is some ideas of how to do it.. .usually as speedily aspossiblet’(T.3.2). Further, although much of the classroom time was spent onindividual activity, teachers tended to focus their teaching on the “average level” of agroup, arguing that those at the margins would raise problems upon encounteringthem. “I guess we’re really striving for the medium [of a group],” said one teacher.“We’re not going one way or the other.. .so we [can] cover everything” (T.1.2).Overall, however, knowledge of students’ backgrounds did not influenceteaching significantly. “I don’t really concern myself with why they’re here,” as oneteacher put it. “They are here and I just try to pass something on to them” (T.1.2).“We’re supposed to.. .pitch the course at the right level,” said another teacher, “but inreality we have to do what’s in the book, so it really doesn’t make any difference”(T.1.6).CurriculumIn this section, I discuss the curriculum of the mathematics courses. Ratherthan being a separate frame factor, the curriculum can be seen as a combination oftwo elements of my model: the woridview of mathematics (the main factor) and theinstitutional framework. I discussed the overall woridviews of mathematics earlier;here, I consider how these worldviews are translated, through the administrativemachinery of provincial articulation and college provision, into course curricula. 142first describe the curriculum that is used as a basis for the college’s introductorymathematics courses. Next, I consider the use and role of set textbooks. I detail thestructure and layout of one book in particular (a set text for the courses on which myresearch focused most closely), and the uses of, and views towards this book held byboth the teachers of those courses and by their students.The curriculum for the mathematics courses at AC follows the province-wideguidelines as described in the Provincial Update on Adult Basic EducationArticulation (BC Government, 1992). The provincial ABE program framework hasfour hierarchical levels: Fundamental, Intermediate, Advanced, and Provincial.Completion of the Provincial level leads to the award of the ABE Provincial Diplomawhich is recognized by the province’s universities and degree-granting colleges asequivalent to secondary school graduation and is, therefore, accepted as a necessarycredential for entry into university-level study. The mathematics courses at ACcorrespond to the three higher levels (Intermediate, Advanced, and Provincial). (Acopy of an AC poster advertising their course and showing which coursescorrespond to which ABE level is attached as Appendix 15.)My research focused on three courses at the Intermediate level (Math 050, 051,and 050/051). The ABE Articulation Guide details the topics to be covered at thislevel. The relevant section forms Appendix 16, but, briefly, it covers the followinggeneric topics: measurement; ratio and proportion; percentages; geometry; algebra;charts, tables, and graphs; statistics, problem-solving, and trigonometry. Further, theGuide makes it clear that the goal of courses at this level is “to enable adult learnersto acquire mathematical knowledge, skills, and strategies needed to enterappropriate higher level courses or to satisfy personal or career goals” (BCGovernment, 1992, p. 26).Teachers were well aware of the existence, content, and goals of the 143curriculum guide. One teacher described how the curriculum they follow “startedwithin the department originally, but now its a province-wide standard. Now, theyset the core curriculum and we follow it.. . .As a department, we set our content tomatch the provincial standard” (T.1.7). “It’s a decision of all the departmentinstructors,” agreed a second teacher, “everyone teaching this course has to cover thismaterial.. .because students are going on to the next course” (T.1.4).Only rarely did teachers depart from the curriculum laid down for them. Thebeginning of the introductory courses was one such instance. “We do a catch-upthing here,” explained one teacher.We do a lot more review of arithmetic in the early stages.. .which is not in theArticulation guide. It’s a college decision. . .what we call Grade 10 is not reallyGrade 10. Our students definitely need to do some revision. (T.1.1)Spending
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An analysis of teaching processes in mathematics education for adults Nesbit, Tom 1995
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Title | An analysis of teaching processes in mathematics education for adults |
Creator |
Nesbit, Tom |
Date Issued | 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. |
Extent | 8737235 bytes |
Subject |
Adult education -- Research Mathematics -- Study and teaching -- Research |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-04-23 |
Provider | Vancouver : University of British Columbia Library |
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 |
URI | http://hdl.handle.net/2429/7520 |
Degree |
Doctor of Philosophy - PhD |
Program |
Education |
Affiliation |
Education, Faculty of |
Degree Grantor | University of British Columbia |
GraduationDate | 1995-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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