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Preface by Series Editordevelopment of her ideas and in giving a rigorous, critical and reflective accountof the research. This fits with a current trend in educational and feminist researchin the interpretative paradigm towards narrative, biography, and research accountsin which the presence of the researcher is centrally acknowledged. Not many suchaccounts are yet available, especially in the field of mathematics education. I predictthat this book will be very influential in stimulating work in this orientation, andwill be much cited by researchers wishing to follow a similar path. It also makesa point which is lost in the sometimes hot debates about constructivism. Namelythat there is no essential and circumscribed fixed nature to constructivism. It is asBarbara Jaworski shows, an evolving cluster of ideas and orientations which becomes constructed and elaborated as the researcher unpacks and creates it in herunique social contexts and trajectories of enquiry.Paul ErnestUniversity of ExeterMay 1994xii

PrefaceThis book is about mathematics, teaching and learning and how these are affectedby a constructivis, philosophy. It charts my own exploration of what might beinvolved in an investigative approach to mathematics teaching. This was a concept whichhad started to form in my mind while I was still a classroom teacher, and whichmy move into university education offered me a chance to explore more fully.My introduction to constructivism was a timely experience, and much of my exploration involved rationalizing classroom issues with this theoretical perspective.The research involved a qualitative study of mathematics teaching throughclose scrutiny of the practice of a small number of mathematics teachers. It'spurpose was to try to characterize the nature of an investigative approach tomathematics teaching. It involved many hours of cbservation in classrooms andof conversations with teachers and pupils. An essential part of the study was aclarification of what an investigative approach might mean in terms of teachingpractices and issues for teachers. It led to a recognition of issues in the development of teaching, and also to a critiquing of the research process and the involvement of the researcher, myself.In 1994, as I write this, it is hard to perceive the person I was in 1985 whenthe research began. Indeed it is only possible to look back through the lens of mycurrent knowledge based on experience which includes the last nine years as wellas all those before that. As I reflect now on that research, on the questions Iaddressed about teaching and learning, and about the research process itself, Irecognize that it involved a deep learning experience for me, of which my observations of, and interactions with, the other teachers were merely a part, albeit avery important part.In 1991 I gained my PhD from the Open University with a thesis which tookthe form of a research biography. In it I tried to set out in a style acceptable tothe examiners the theory, research and conclusions of five year's wok and thinking. It proved impossible to do this from any position other than that of thecentrality of the researcher to the research. I saw the research then as being ethnographic, or interpretivist, but I realize now that an alternative term is constructivist.I took constructivism as the central tenet of my theoretical position on teachingand learning, and tentatively recognized its internal consistency with the way myresearch evolved. There are some problems in labelling the research itself as'constructivist' just as there arc in talking about constructivist teaching. Ways inwhich constructivism as a philosophy can be seen to influence research and teachingare a major focus of the book.13

PrefaceCriticisms of constructivist-based research centre around problems in justifying research conclusions, in judgments involving the validity of the research. Ifollow a position that a researcher needs to embed the research in its total situatedcontext, and that this includes his or her own experiences and thinking. In beingguided by constructivism, I see myself not so much as being drawn into anincreasingly relativistic spiral, as being challenged constantlyto be critical of my conclusions and their bases, andto relate these to my thinking and analysis and its interactions with otherthinkerseither personally, or indirectly through their written accounts.In this book I have tried where possible to include details of my thinking atall stages of the research, as encapsulated in my writings at the time, and also toportray the theoretical perspectives which prevailed when I wrote my thesis.However, I recognize also the need to set these into the context of my currentthinking. I therefore offer this account as a synthesis of all these influences.The structure of the bookI have tried to present an account which is loosely chronological, in that it followsthe development of thinking through three successive phases of research. Eachphase involved a study of teaching over a period of six to nine months. Analysistook place both during and between the phases.Chapter 1 presents a background to the study of an investigative approach tomathematics teaching. This includes both my own perspective of investigationalwork and its origins, and wider perspectives in the development of mathematicsteaching.Chapter 2 provides an account of constructivism as a philosophy of knowledge and learning, particularly as it relates to mathematics education and necessarily from my own perspective. This includes a rationale for both radical and socialconstructivism, and relationships between constructivism and knowledge, communication, and the classroom. It also addresses some of the challenges toconstructivism which seem relevant to mathematics education.Chapter 3 presents some of the very early stages of my research along withthe thinking which this involved at the time and the issues which arose. It encompasses early thinking about an investigative approach and also about research whichmight elucidate such an approach. It tries to capture the methodological beginnings of the research project.In Chapter 4, I provide a rationale for the research methods employed in mymain study for which the first phase of research, reported in Chapter 3, was apilot. This examines the choice of an ethnographic approach to the research throughparticipant observation and informal interviewing. It examines issues which arisefrom an interpretivist analysis, and the place of theory in the research. It justifiesthe use of research biography as a reflexive accounting process which contributesto the validatory processes in the research.Chapters 6, 7 and 9 are case studies taken from the second and third phasesof the research which formed my main study. In this I present in some detail acharacterization of the teaching and thinking of three teachers, Clare, Mike andxiv

PrefaceBen, and the development of theory through analysis of my observations. Incategorization arising from the teaching of Clare, in Phase II, a descriptive modelemerged which became known as the Teaching Triad. This was subsequentlytested in Phase III, through the teaching of Ben, and again through a re-analysisof the teaching of Mike from Phase II.Chapters 5 and 8, interludes between the above chapters, offer research reflections relevant to the three phases of research. These link observations with mlown development of thinking, and therefore analysis, across the phases and particular teachers.In Chapter 10 I have tried to synthesize general concepts which have arisenfrom my analyses. both in terms of characteristics of an investigative approachand tensions which it raises for teachers in its classroom implementation.A most significant outcome of the research was a recognition of the implications of reflective practice for development of both teaching gad research. Therelationship between teacher and researcher was centrz to this recognition. Chapter 11 offers a characterLation of the teacher researcher relationship which includes a descriptive model for reflective practice.Chapter 12 returns to the theoretical basis of the research. I re-ea amine therelevance of constructivism to the research outcomes in the light of questionsabout the adequacy of radical constructivism to explain the complexities of classroom interaction. The chapter considers theoretical perspectives of social andcultural dimensions in learning and teaching and their relation to cognitive processes. It justifies my own theoretical position in presenting the research from asocial constructivist perspective.lust one point finally: My classroom observations took place just before theintroduction of the National Curriculum in mathematics. Thus, the teachers studiedwere not at this stage influenced by pressures of the NC and its testing. It has notbeen possible to revisit these teachers to explore what differences this has made totheir classroom practice.Barbara JaworskiMay 1994xv

AcknowledgmentsThis study owes a great deal to the teachers who were subjects of my classroomresearch. To Felicity, Jane, Clare, Mike, Ben and Simon, my very sincere thanksfor their interest and cooperation, and for the time whicit they so generously gave.Many of my colleagues have played an important part in terms of supportand encouragement and extreme tolerance. I have appreciated their willingness tolisten, to talk over ideas and to offer their perceptions. It would be impossible tolist everyone who has been of help, but I should like particularly to thank PeterGates, Sheila Hirst, David Pimtn, Stephanie Prestage, Brian Tuck and AnneWatson.I owe special gratitude to John Mason, Christine Shiu, and John Jaworski fortheir exceptional support and inspiration, and to Paul Ernest, the series editor, forhis critical appreciation.Finally I must acknowledge Princess Borris-in-Ossory and Frankincense for theircontribution to this work. They were mainly responsible for my not getting coldfeet during its many drafts.Barbara JaworskiMay 1994xvi

In the halls of memory we bear the images of things once perceived, asmemorials which we can contemplate mentally, and can speak of witha good conscience and without lying. But these memorials belong to usprivately. If anyone hears me speak of them, provided he has seen themhimself, he does not learn from my words, but recognises the truth ofwhat I say by the images which he has in his own memory. But if he hasnot had these sensations, obviously he believes my words rather thanlearns from them When we have to do with things which we behold inthe mind . . . we speak of things which we look upon directly in theinner light of truth . . .(St. Augustine, De Magistro, 4th century AD*). . must . . . look upon human life as chiefly a vast interpretive process in which people, singly and collectively, guide themselves by definingthe objects, events and situations which they encounter . . . Any schemedesigned to analyse human group life in its general character has to fit thisWe .process of interpretation.(Blumer, 1956, p. 686)* Augustine: Earlier writings. H.S. Burleigh (Ed), Westminster Press. p. 96.xvii4

Chapter 1An Investigative Approach:Why and How?As a classroom teacher of mathematics in English secondary comprehensive schools in the 1970s and early 1980s, I struggled with ways of helpingmy students to learn mathematics. I enjoyed doing mathematics myself,and I wanted students to have the pleasure that I had in being successfulwith mathematical problems. I experienced the introduction and implementation of syllabuses involving the 'new' mathematics. This gave mea lot of pleasure as I enjoyed working with sets and functions, withBoolean algebra, with matrices, with transformation geometry. Some ofmy students enjoyed this too, but the vast majority were, I came torealize, as mystified with the 'new' maths as with any of the more traditional topics on the syllabus.Personally and HistoricallyTeaching mathematics was difficult, because students found learning mathematicsdifficult. I started to question what it actually meant to learn mathematics. Explanations or exposition seemed very limited in terms of their effect on students'learning, so I found myself seeking alternative approaches to teaching mathematical topics, especially for students who were not inclined to like or be successfulwith mathematics.I gained considerable personal enjoyment from puzzles, problems and mathematical investigations such as those offered by Martin Gardner (1965, for example) and occasionally used some of these, or modifications of them with students.This was 'extra' to my teaching of the mathematics syllabus, and I suspect, onreflection, very much in the mode of the teacher described by Stephen Lerman(1989b, p. 73) who 'went around the classroom offering advice such as 'no, notthat way, it won't lead anywhere, try this'. According to Lerman 'them !tad beenno opportunity for the teacher to discuss or examine . . how [an investigation]might differ from "normal" mathematics'. In my earlier teaching it was not somuch lack of opportunity to examine this issue as a lack of awareness on my partof how mathematical investigations might be linked to the mathematics or hesyllabus.In 1980, dissatisfaction with the mathematical achievement of many students,and a new post as head of a school mathematics department, made me start to119

I ivestigating Mathematics Teachinglook more seriously at the idea of mathematical investigation permeating 'normal'mathematics teaching. A number of my colleagues, who enjoyed getting involvedin mathematical problems, were willing to start to think, at least in theory, aboutwhat an investigative approach to mathematics teaching might involve. As a resultof a number if departmental sessions where we enjoyed ourselves in solvingproblems together, we came to a view that investigative teaching was about 'opening up' mathematics, about asking questions which were more open-ended, aboutencouraging student enquiry rather than straightforward 'learning' of facts andprocedures.'This was not well articulated, and its implications for what we would actuallydo in our classrooms was far from clear. We were very conscious of the time factorin terms of 'getting through the syllabus'. We recognized that enquiry and discussion by students took more time than exposition and explanation by the teacher.We accepted, I think with some guilt and. some relief, that we would revert toexposition and explanation whenever we felt under pressure because we weresecure with these methods and used them with confidence. To a great extent wewere floundering with enquiry methods, not being at all sure of what we weretrying to achieve, and having little confidence of their success in terms of thestudents' ultimate ability to succeed with level or CSE examinations.'At an early presidential address to the UK Association of Teachers. of Mathematics (ATM), Caieb Gattegno said:When we know why we do something in the classroom and what effectit has on our students, we shall be able to claim that we are contributingto the clarification of our activity as if it were a science. (Gattegno, 1960)This clarification was one of my main objectives when I moved from secondaryschool teaching into higher education in the mid-1980s. It was my declared intentto explore further the potential of an investigative approach to mathematics teaching from perspectives of both theory and practice. I wanted to be clearer aboutwhat such an approach meant in theoretical terns, but I wanted also to find outhow it might be implemented in the .lassroom, and what implications this had formathematics teachers.The Origins of InvestigationsInvestigational activity in mathematics teaching in the UK was introduced duringthe 1960s, primarily through publications of, and workshops organized by theATM, and through teacher-education courses in colleges and universities (Association of Teachers of Matheinatics, 1966; Association of Teachers in Colleges andDepartments of Education, 1967). A seminal sourcebook was Starting Points, byBanwell, Saunders and Tahta (1972). From the 1970s, in the UK, there are manydocumented examples of mathematical investigations in classrooms as part ofmathematics learning and teaching. For example, Favis (1975) described an occa-sion when students in a class were 'investigating rectangular numbers' and twogirls had taken off in a direction different from that proposed by the teacher, withsome particularly fruitful results. Irwin (1976), a third-year (Year 9) student,described his own findings resulting from an investigation into combinations of22

00An Investigative Approach: Why and How?functions which emerged from a question asked by his teacher. Edwards (1974),a first-year student-teacher, described an investigation which she herself undertook concerning the rotation of the numbered pins on a geoboard. Curtis (1975)described the results of two number investigations used with his lower-sixth form,non-examination, option group. In his introduction he commented:As you will notice, neither investigation is complete. We found that assoon as we had an answer to one question, there were three other newquestions to be answered. This continual pushing forward, trying toevaluate what we had already achieved and what was still to be done,gave the group an insight into how professional mathematicians spendmuch of their time . . . the fact that none of our conclusions had anydirect application meant that we worked hard simply because we foundit enjoyable. (Curtis, 1975, p. 40)Mathematical investigation seemed

AUTHOR Jaworski, Barbara TITLE Investigating Mathematics Ter.ching: A Constructivist. Enquiry. Studies in Mathematics Education Series: 5. REPORT. NO. ISBN--7507-0373-3 PUB DATE. 94. NOTE 250p.;. Part 5 of Studies in Mathematics Education. Series. AVAILABLE FROM Falmer Press, Taylor & Francis Inc., 1900 Frost Road,