D cLKDU Interpretations of a constructivist philosophy in mathematics teaching Barbara Jaworski BSc BA Thesis offered for the degree of Doctor of Philosophy of the Open University in the discipline of Mathematics Education May 1991 uYA: O325tb rSi.  11 ct r1  ii c r To John 111 Acknowledgements This study owes a great deal to the teachers who were subjects of my classroom research. To Felicity, Jane, Clare, Mike, Ben and Simon, my very sincere thanks for their interest and cooperation, and for the time which they so generously gave. Many of my colleagues have played an important part in terms of support and encouragement and extreme tolerance. I have particularly appreciated their willingness to listen, to talk over ideas and to offer their perceptions. In particular I should like to thank all members of the Centre for Mathematics Education at the Open University and my close colleagues in the School of Education at the University of Birmingham. It would be impossible to list everyone who has been of help, but I should like especially to thank Peter Gates, Sheila Hirst, David Pimm, Stephanie Prestage, Brian Tuck and Anne Watson. Three people are owed especial gratitude: John Mason, who has been an inspiration over many years and who has ever been willing to engage with ideas and offer his own particular gift of enabling me to reconstruct what I know for greater sense and coherence. Christine Shiu, who has supervised this study giving generously of her time and friendship, experience and sensitivity, and her own particular brand of care and attention to detail. John Jaworski, who has given not only his love and forbearance, humour and support, but also his time and expertise in the presentation of this manuscript I shall not be forgiven if I do not give credit to Princess Boris-in-Ossory and Frankincense for their contribution to this work. They were mainly responsible for my not getting cold feet during the many drafts of this thesis. V ABSTRACT This thesis is a research biography which reports a study of mathematics teaching. It involves research into the classroom teaching of mathematics of six teachers, and into their associated beliefs and motivations. The teachers were selected because they gave evidence of employing an investigative approach to mathematics teaching, according to the researcher's perspective. A research aim was to characterise such an approach through the practice of these teachers. An investigative approach was seen to be embedded in a radical constructivist philosophy of knowledge and learning. Observations and analysis were undertaken from a constructivist perspective and interpretations made were related to this perspective. Research methodology was ethnographic in form, using techniques of participant-observation and informal interviewing for data collection, and triangulation and respondent validation for verification of analysis. Analysis was qualitative, leading to emergent theory requiring reconciliation with a constructivist theoretical base. Rigour was sought by research being undertaken from a researcher-as-instrwnent position, with the production of a reflexive account in which interpretations were accounted for in terms of their context and the perceptions of the various participants including those of the researcher. Research showed that those teachers who could be seen to operate from a constructivist philosophy regularly made high level cognitive demands which resulted in the incidence of high level mathematical processes and thinldng skills in their pupils. Levels of interpretation within the study led to the identification of investigative teaching both as a style of mathematics teaching and as a form of reflective practice in the teaching of mathematics. These forms were synthesised as a constructivist pedagogy and as an epistemology for practice which may be seen to forge links between the theory of mathematics teaching and its practice. The research is seen to have implications for the teaching of mathematics, and for the development of mathematics teaching itself through professional development of mathematics teachers. vi' In the halls of memory we bear the images of things once perceived, as memorials which we can contemplate mentally, and can speak of with a good conscience and without lying. But these memorials belong to us privately. If anyone hears me speak of them, provided he has seen them himself, he does not learn from my words, but recognises the truth of what I say by the images which he has in his own memory. But if he has not had these sensations, obviously he believes my words rather than learns from them. When we have to do with things which we behold in the mind we speak of things which we look upon directly in the inner light of truth (St. Augustine, De Magistro, 4th century AD1) We can, and I think must, look upon human life as chiefly a vast interpretive process in which people, singly and collectively, guide themselves by defining the objects, events. and situations which they encounter. Any scheme designed to analyse human group life in its general character has to fit this process of interpretation. (Blumer, 1956, p 6862) 'The St. Augustine quotation is taken from H.S.Burleigh (ed.) Augustine: Earlier writings, Westminster Press p 96 2 Quoted in Denzin, 1978 CONTENTS  ix TABLE OF CONTENTS PART ONE - THEORY CHAPTER 1 - BACKGROUND AND RATIONALE  1 An investigative approach  2 The origins of investigations  2 The purposes of Investigations  2 The status of investigational work in mathematics teaching  5 An investigative approach to mathematics teaching  7 The research study  9 A statement of purpose  9 The structure of this thesis  10 The fieldwork  10 My own position In the research  11 the contribution of the study  12 CHAPTER 2 CONSTRUCT! VISM  13 What Constructivism is  13 Constructivism and knowledge  16 Constructivism, meaning and communication  17 Cons tructivism and the classroom  23 Challenges to cons tructivism  25 CHAPTER 3 THE TEACHING OF MATHEMATICS  31 The implications of learning for teaching  31 The influence of Piaget  31 Construction of mathematical concepts  33 Hierarchies of mathematical concepts  35 Two kinds of learning  39 Pupil construal and its recognition  41 The role of the teacher for mathematical learning  43 The Zone of Proximal Development  43 Language and the social environment  46 The trouble with mathematics teaching  54 My own study  59 x  CONTENTS PART TWO - RESEARCH CHAPTER 4 METHODOLOGY  61 in tro duct/on  61 An initial choice  61 Ethnography, or an ethnographic approach  63 The place of theory In an ethnographic approach  66 Validation and Rigour  69 My own study  71 Interpretive enquiry  73 Data collection  74 Data analysis  79 Theoretical perspectives  82 Verification  84 Terminology  86 CHAPTER 5 PHASE ONE RESEARCH  87 introduction  87 Stages of Involvement In the Phase 1 work  87 Stagel The introductory work  87 Stage2 Teaching lessons myself  88 Stage3 Pairs of lessons of the two teachers  88 Data collection and analysis across the three stages 89 Analysis and reflection  89 Stage 1 - Initial Observations 89 Stage 2 - Teaching lessons myself, observing others 91 Stage 3 - Pairs of lessons taught by Felicity and Jane 99 Conclusions  108 INTERLUDE A  111 The 'fit' with radical constructivism  112 Relating constructivism to teacher-development  114 Implications for Phase 2  115 CHAPTER 6 PHASE TWO RESEARCH  117 introduction  117 Methodology  119 Data Collection  119 Data Analysis  121 122 122 122 123 137 150 155 160 160 160 160 167 174 179 185 189 189 192 194 195 197 201 201 202 202 203 204 204 205 207 211 229 242 242 245 247 252 254 254 255 268 269 271 CONTENTS  xi The study of Clare's teaching Introduction Lessons from which data were collected Analysis of the Autumn term lessons Analysis of recorded lessons Students' and Teacher's views Concluding my characterisation of Clare's teaching The study of Mike's teaching Introduction Lessons from which data were collected Management of Learning Sensitivity to Students Mathematical Challenge Mike's own thinking - and responses from students Conclusion to the chapter INTERLUDE B Teacher and researcher awareness Recognition of an Issue - the teacher's dilemma The researcher's dilemma Significance Implications for Phase 3 CHAPTER 7 PHASE THREE RESEARCH Introduction Methodology Data Collection Data Analysis The Study of Ben's Teaching Lessons from which data were collected The teaching triad Didactic versus Investigative teaching The Moving Squares lesson The Vectors lesson Tensions and Issues The teacher's dilemma The didactic/constructivist tension The didactic tension Constructivism and the Teaching Triad The Study of Simon's Teaching Lessons from which data were collected Consolidation of graphs The teaching triad Conclusion Conclusion to the Phase 3 work XII  CONTENTS PART THREE - CONSEQUENCES AND CONCLUSIONS CHAPTER 8 REFLECTIVE PRACTICE  275 Thinking and reflection  276 The teacher-researcher relationship  278 Stage 1 - Reflecting  280 Stage 2 - Accounting for  282 Stage 3 - Critical analysis  286 Reflecting on the conceptual model  291 Teaching knowledge and Teaching wisdom  293 The reflective teacher  294 Developing Investigative teaching  297 My own development as a reflective practitioner  299 Reflective practice is 'critical' and demands 'action'  302 Conclusion  304 CHAPTER 9 CHARACTERISTICS OF AN INVESTIGATIVE APPROACH The classrooms The teaching role Establishing meaning Engendering mutual trust and respect Encouraging responsibility for own learning Establishing an investigative approach The wider Issues Building of mathematical concepts Social issues Conclusion CHAPTER 10 BRINGING THEORY CLOSER TO PRACTICE Introduction A constructivist pedagogy The teaching triad Teacher-pupil Interactions Tensions and issues An epistemology for practice A critical appreciation of this study Methodological implications Future directions Conclusion 307 307 313 315 316 317 319 320 320 321 322 323 323 324 326 328 329 330 332 333 334 335 CONTENTS  XIII REFERENCES  337 APPENDICES 1 - Chronology and conventions 2 - Constructivism - a historical perspective 3 - Phase One lessons 4 - Phase Two lessons 5 - Phase Three lessons TABLE OF FIGURES 3.1  The twit metaphor 6.1  Research chronology 6.2 Lessons observed with Glare 6.3 Glare's diagram 6.4 The teaching triad 6.5 Lessons observed with Mike B.1  Links between theory and practice 7.1  Lessons observed with Ben 7.2  Ben's view of the teaching triad 7.3 Moving Squares 7.4 Pat's diagram 7.5 Drawing a vector 7.6 Cohn's drawing 7.7 Ben's diagram 7.8 Luke's explanation to Danny 7.9 Lessons observed with Simon 7.10 Linking the triad with constructivism 7.11 Gonstwctivism and the triad 7.12 Elaborating the teaching triad 8.1  The teacher-researcher relationship 8.2 The reflective process for the teacher 9.1  Characteristics of an investigative approach 1A 5A 9A 25A 53A 42 121 123 134 155 160 199 205 207 212 231 232 233 235 239 254 271 272 272 279 296 320 [...]... see Appendix 2 Cons tructivism, meaning and communication Fundamental to teaching and learning is a consideration of how communication takes place, of how meanings are shared In the teaching of mathematics it is also fundamental to ask what meaning and whose meaning? Von Glasersfeld wrote: 17 18 CHAPTER 2 As teachers we are intent on generating knowledge in students That after all is what we are being... Education Authority CHAPTEF BACKGROUND AND RATIONALE AN INVESTIGATIVE APPROACH TO MATHEMATICS TEACHING An investigative approach to teaching mathematics might be seen as a way of approaching the traditional mathematics syllabus which emphasises process as well as content I would see it taking the advice quoted from Cockcroft above, but going beyond this to the active encouragement of questions from pupils and... WORK IN MATHEMATICS TEACHING Investigating became more widely seen as a valuable activity for the mathematics classroom, supported by the Cockcroft report (DES, 1982), which included investigational work as one of six elements which should be included in mathematics teaching at all levels (para 243) In paragraph 250, the authors wrote: The idea of investigation is fundamental both to the study of mathematics. .. teaching It involved participant observation of the classroom practice of six secondary mathematics teachers and extensive exploration of their motivations and beliefs It began as an enquiry into an investigative approach to the teaching of mathematics - the teachers studied employed a classroom approach which could be described as investigative according to popular connotations in the mathematics education... teaching, and I tried to articulate this in Jaworski (1985b) I presume that other teachers who undertake investigational work in the classroom, beyond the doing of isolated investigations, also have a sense of what an investigative approach means, not necessarily the same as mine, or of others The value in speaking of an investigative approach is not in some narrow definition, but in its dynamic sense of. .. presence of mathematization under the headings of strucruration, dependence, infinity, making distinctions, extrapolating and iterating, generating equivalence through transformation For example, he suggests that 'searching for pattern' and 'modelling a situation' are phrases which 'grope' towards structuration; that, as Poincaré pointed out, all mathematical notions are concerned with infinity - the search... undertaken by applying a practice-able set of procedures - for example by working through a number of special cases of some given scenario, looking for a pattern in what emerged and expressing this pattern in some general form, possibly as a mathematical formula Often such sets of procedures were learned as a device for tackling the investigations rather seen as part of being more generally mathematical... certain consequences of a constructivist philosophy for a teacher in the classroom In education and educational research, adopting a constructivist perspective has noteworthy consequences: 1 There will be a radical separation between educational procedures that aim at generating understanding ( 'teaching' ) and those that merely aim at the repetition of behaviours ('training') 2 The researcher's and... Telling or explaining on the part of the teacher seems a very limited way of encouraging construction However, not-telling (ever!) seems particularly perverse An investigative approach to teaching mathematics, as well as employing investigational work in the classroom, literally investigates the most appropriate ways in which a teacher can enable concept development in pupils I see it encouraging exploration,... mathematics teaching Chapter 8 is devoted to these ideas, which are linked to the various strands of my own thinking throughout the research in a model for reflective practice THE CONTRIBUTION OF THE STUDY The main contribution of the study will be to knowledge of mathematics teaching - in particular to characteristics of teaching, and issues which teachers face in enabling pupil construal of mathematics