Philosophies of Mathematics Education

PART B: EDUCATIONAL SETTING Page 32 This thesis uses the ecological model described in Chapter 9 as a tool to explain some aspects of the history and current practice of the obuchennyi1 of probability. Such an analysis cannot be done in vacuo. Education is a complex activity pur- sued in relative freedom by people from many diverse backgrounds. All members of our society have experienced it, most have opinions about it, and some are very ready to make these opinions widely known. There is no generally accepted view of the best form of education. Many would argue that humans are so diverse that there cannot be any such monolithic view. Educational professionals work within much less well-defined paradigms than do legal or medical practitioners. All require creativity and flexibility, but even marginalised medical paradigms such as chiropractic have better defined pro- cedures and controls than do many mainstream educational ones. Partly because of such a diversity of approaches this thesis is cross-disciplinary. But the conventions of one discipline need not match those of another. My super- visors are respectively an historian and a mathematician, but neither is both. My own special skill is pedagogic. I spend time in a mathematics education community whose most outspoken members are Social Reformers and Construct- ivists. The primary focuses of this thesis are history, curriculum and assessment, but its underlying approach is ecological. So it is more important than usual to spell out in detail the assumptions and conventions which I am adopting and to set them in a wider context as part of making cohesive links between all these different approaches and philosophies. No such survey can be totally comprehensive.2 But to provide a working background for the later parts of this thesis I outline here the features which are important for me and for my approach, as well as those considered important by my professional colleagues and by the community at large as reported in the media. The features covered are: • educational philosophies; • language and linguistics; • the meaning of probability; • the mathematics of probability; • general curriculum theories; • a summary of research into probability teaching and learning. 1 Learning and teaching, understood as deeply inter-related in complex ways. Vide ch. 2. 2 Vide D. Wheeler (1996)—a comment on one attempt to establish a scientific discipline of mathematics education. For a neat, brief popularising summary, vide Burton (1986). Part B Page 33 Within each chapter I focus on those aspects which most help to elucidate the underlying ecological approach, but this may not become obvious until after reading both its explication in Chapter 9 and some of the applications in Parts C to E. While I have drawn some links between the chapters, others are certainly possible, and a sequence of fairly discrete chapters seemed to be the best way of facilitating the reader’s own creative reading. Finally, as a preparation for the historical analysis in Part C, this Part finishes with a brief summary of the educational practice in SA in the 1950s, just before the time discussed in this thesis. Part B CHAPTER 3: PHILOSOPHIES OF MATHEMATICS EDUCATION To sum up, what radical constructivism may suggest to educators is this: the art of teaching has little to do with the traffic of knowledge. Its fundamental purpose must be to foster the art of learning.1 A philosophical summary is not the most exciting way to start a thesis. But it is beliefs—of teachers, parents, children, administrators and industry—which constitute the principal focus of this dissertation, so they must be well understood. We start with an entertaining and concrete background for contextualising the more abstract analysis which follows. A BRIEF HISTORY OF RECENT PEDAGOGICAL TIME [T]hat long succession of Waves of which History is chiefly composed.2 In mathematics education from the 1960s, wave has compounded upon wave, producing potentially damaging and unpredictable perturbations. “Each generat- ion’s revolution tends to become its successor’s chains.”3 The following text expresses some of the concerns which the community has had about these waves. THE NEW, NEW MATHS: THE EVOLUTION OF A MATHEMATICAL PROBLEM Teaching in the 60s A farmer sells a bag of potatoes for $45. Production costs 80% of the sales price. What was the profit? Traditional Teaching in the 70s A farmer sells a bag of potatoes for $100. Production costs used four- fifths of his sale price, that is $80. What was his profit? Modern Teaching in the 70s A farmer exchanges a set “P” of potatoes for a set “M” of money. The order of the set “M” is 100, and each element is worth $1. Make a draw- ing of 100 dots representing the elements of the set “M”. The set “C” of 1 von Glaserfeld (1995, p. 192) 2 Sellar & Yeatman (1930, p. 4) 3 P. Hughes (1966, p. 265) Page 36 the production costs has 20 dots less than the set “M”, so now answer the next question: What is the order of the set “B” of profits? (Draw its elements in red.) New Teaching System of the 80s A farmer sells a bag of potatoes for $100. Since the production costs were $80, the profit was $20. Now underline the word “potatoes” and discuss it with your neighbour. Reformed Teaching of the 80s A privileaged Kapitalist seels injustly $20 over a beg of potatous analise the text and find out grammatik, ortografy and ponctuation error and say something about did process of getting reach. Politically Correct Teaching in the 90s There was a motion before the class management committee from Judy that the class should discuss the marketing of potatoes. The class decid- ed that the method of reproduction used by potatoes was in conflict with the class’s policy on equal opportunity and rejected the motion. Judy was rather upset by this decision and took affirmative action using some of the bag of potatoes which she had brought with her. Most of the class implemented their affirmative action training very effectively, and Judy sent a letter of resignation from her hospital bed. The potatoes were sold at the Flea Market for $100 and $20 was spent on flowers for Judy. How much money has gone into the class television fund?* INTRODUCTION [P]rior to the 1970’s the issue of the nature of mathematics was not on the agenda for discussions about mathematics education. … Advances in the history and philosophy of mathematics, the sociology of knowledge, and post-modernist thought … have shown that the myth of the unchanging nature of mathematics is probably held in place by the use of a single term ‘mathematics’ for several diverse domains of knowledge and discursive practice.4 Mathematics has never been a universally popular school subject. It has been seen by many as detached from real life, excessively rigid, frequently incomprehens- * I heard at least three forms of this text in the early 1990s, one during a very expensive long- distance phone call made explicitly to read it out to me. In keeping with the genre I have added the final paragraph before passing it on to others. An alternative form is found in Australian Mathematics Teacher 50 (1): 18. Another, using the model of a logger, is found in C&EN [sic, no other details available] 23 Jan 1995, p. 84, and was circulating on an e-mail list in an extended form in March, 1998. 4 Almeida & Ernest (1996) Educational Philosophies Chapter 3 Page 37 ible. Because it has often been used as a filter to select students suitable for further education, it is frequently associated with failure.5 Perhaps because of the recent wider community access to tertiary education, mathematics has come under increasing pressure to change, both from society and from some teachers. It may be true that changes, if followed by periods of stability, “have the capacity to capitalise on human passion, bringing about the most creative and innovative passion”.6 However, too many changes, and too few periods of stability can prove highly destructive. As suggested above, this is probably what has hap- pened within mathematics education in recent years. The purpose of this chapter is to provide some understanding of the philosophies which have been present in this period, so that their influence on the teaching and learning of probability may be assessed in the historical analysis of Part C. It has been suggested that there are three critical questions in mathematics education—what, how and why?7 People holding different philosophies will answer these questions differently. Only recently have the questions and answers been subjected to wide critical examination. But the principle users of education in western society—parents and children— tend to expect a high degree of certainty in classroom practices. For teachers too, management constraints tend to encourage fairly well structured classrooms. So, not surprisingly, “[t]he concept of mathematics as a body of infallible and objective truth, whilst questioned by many mathematicians and philosophers, appears to be still widely held by society, teachers and students”.8 Mathematical aspects of probability will be discussed in Chapter 6; here I have used summary works by Steedman and the post-modernist philosopher Ernest9 as a basis for discussing some general educational philosophies of relevance to this thesis, with a special focus on how they might directly influence classroom practice. Ernest has described five types of educator distinguished by the philosophies of mathematics education which they hold and has described how each philosophy leads to different views about many different aspects of mathematics education.10 The types are: 5 Benn & Burton (1996) 6 Ponte et al.

Load more