We would like to thank all the authors below who kindly agreed to collaborate with us in the preparation of this tribute. The personal perspectives they provide in their testimony will undoubtedly help the reader gain a sense of the variety and the depth of David's involvement in mathematics education over half-a-century. We would also like to thank Mogens Niss, editor of this Bulletin, for providing a publication outlet through which this tribute to David Wheeler could be shared broadly with the mathematics education community.

Bravo and thank you, dear friend David!

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- Bernard R. Hodgson
- Universite Laval, Quebec, Canada
- bhodgson@mat.ulaval.ca
- A.J. (Sandy) Dawson
- Simon Fraser University, Vancouver, Canada
- dawson@sfu.ca

I first met David Wheeler in the late fifties at a local branch meeting of ATAM (the Association for Teaching Aids in Mathematics, later to become ATM, the Association of Teachers of Mathematics). When the meeting broke up into small discussion groups I found myself in one chaired by this amazingly impressive man with a generous laugh, a warm and inviting manner yet with a sharp eye for fraud or insincerity. Not for the last time, I found myself simultaneously encouraged and challenged by his presence in the group. Such personal influence has been the experience of many teachers who were fortunate enough to work with him in some way during his professional life in England.

David was already a key figure in the Association which had been founded by Caleb Gattegno in 1952. He became an increasingly important influence in mathematics education in Britain over the next two decades. He was tirelessly active in a number of fields: after many years teaching mathematics in London schools, he moved into post-graduate teacher education at Leicester University, where he soon became involved in the setting up of a study group with colleagues from other universities (some readers will know the sort of influence he had in this study group which was not unlike the one he initiated many years later in Canada).

He continued to be involved in the work of ATM: he wrote regularly for the journal of the association, "Mathematics teaching", contributed to various books and edited a number of these (notably "Notes on mathematics for children", Cambridge University Press, 1977). He served on the committee of the association for many years, including a particularly fertile spell as Secretary, and he was a regular seminar leader at conferences. He became an outstanding editor of the journal whose magisterial editorials still get re-printed from time to time; his own wide interests and contacts ensured that the journal moved from being something more like a local house magazine into an authoritative and internationally respected journal.

Apart from all that, and apart from his normal university teaching and administrative responsibilities, he was often running courses for teachers outside his immediate local area, serving on national committees, attending international conferences, giving radio talks on the teaching of mathematics, writing articles in other educational journals, commissioning and editing a series of textbooks, preparing courses for the Open University - this latter yielding a remarkable book, "R is for Real" (Open University Press, 1974) which deserves to be more widely known.

Lists of achievements like the above are formal pieties which do not convey the very special nature of the legacy that David left to his many friends and colleagues in Britain. Perhaps this can be best be captured in some of his own words. For many years I would ask groups of prospective teachers to read his article on the Role of the Teacher ("Mathematics Teaching", no. 50, 1970, p23) . Discussion of this article was always fruitful. For students who were on the edge of their first experience of teaching, it was exhilarating and very helpful to read his pungent reversal of the usual traditional advice that a teacher faced with a new class needs first to establish relationship. (A typically shrewd aside noted that "teacher-trainers have one sort of language for this: experienced teachers another!") For, "if the teacher takes the initiative in establishing relationships before there are any tasks, the children will know that the tasks do not have first priority; they are being thoroughly logical in subsequently working on the relationships instead of the tasks."

The article opened with the following paragraph: "If we know that ineffective teaching of mathematics is not due to the difficulty of the subject matter, and if we know that changing the classroom environment .... does not contain within itself the possibility of acting directly on the awarenesses of children, and if we then do not re-examine in the most fundamental way how as teachers we should act, we are guilty of a total failure of seriousness, for we have stopped our progress towards a better education for children just short of the point at which we can make a contribution to it."

David himself never stopped short of making a contribution, one that was serious, challenging but sympathetic, and always tinged with humour. In the words of the title of one of his ATM conference lectures, he helped "humanise mathematical education". During the sixties, he had been a central figure in the ATM research and develop- ment group. He had not himself been a student of Gattegno's like many other members of the group, but he was certainly the one who had the most understanding in theory and practice of what Gattegno was eventually to call the science of education.

It is perhaps typical of his own research in mathematics education that at the height of a professional career in England he chose to leave a tenured post and familiar ways to work with new challenges in New York.

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- Dick Tahta
- (retired) Exeter University, UK
- d.tahta@open.ac.uk

I don't have a clear recollection of my first meeting with David Wheeler, although it was certainly in the early 1970's at Caleb Gattegno's Educational Solutions. I started working for Educational Solutions in various public schools in 1970 and Wheeler's name was often mentioned (somewhat reverently) around the office. ATM (Association of Teachers of Mathematics), MT (Mathematics Teaching), David's connection to Gattegno and David, himself, were unknown to me. These were all to change in the next couple of years.

I didn't see much of David when he first arrived. He mostly worked in the downtown 5th Avenue office of Education Solutions and I in the uptown 5th district of the New York City Public Schools. I don't know how we came to know each other better, but it was undoubtedly through some combination of seminars at Educational Solutions, the many lunches at Brew Burger, and concerts at Carnegie Hall.

I appreciated David's wit and humor from our first meetings, but a true respect for his mathematical insights came a short bit later. The occurrence was a weekend workshop that he conducted using the black and white Nicolet geometry films. He guided the participants through a careful study of several of the films, revealing insight after insight. What a tour de force! And I still have my notes.

There were occasions in New York when I was leading a workshop that he was able, at a critical moment, to focus an uncertain discussion. Then, and in many working groups of CMESG/GCEDM (Canadian Mathematics Education Study Group / Groupe canadien d'etude en didactique des mathematiques) that we participated in since, I have come to value his uncanny ability to contribute when he is not the leader, not by adding more layers of detail to an existing viewpoint, but by illuminating it through the suggestion of complementary and countervailing viewpoints.

David and I used to trade what seemed to us to be interesting math problems. There must have been some discussion about their usefulness in school settings, but the memory of the joy of solving them is more prominent. After leaving New York he continued to send me problems and I have maintained them in a file which I still utilize. Recently, a discussion among members of my department on an equivalent of Wythoff's Nim sent me back to that file. There it was, clearly presented with follow-up suggestions from twenty years ago that the current discussion had yet to consider.

There was one question that I used to frequently ask David to which he did not have (or chose not to have) an unequivocal answer: "What do you do in that office?" It was asked partly out of curiosity and partly from the knowledge that I would have found it very difficult to work in such close physical proximity to Gattegno.

In retrospect I think a partial answer to the question comes from viewing his short stay in New York as a bridge spanning his significant accomplishments with ATM and MT in the UK and with FLM (For the Learning of Mathematics) and CMESG/G- CEDM in Canada. The office allowed him to cross that bridge at an ideal pace; contemplating the past, learning in the present and preparing for the future. I feel fortunate to have accompanied him for even a small part of that crossing.

Queen's College, CUNY, New York, USA

martin_hoffman@qc.edu

David Wheeler came to Canada in 1976 as professor of mathematics at Concordia University in Montreal. From that date until the time of his retirement and relocation to Vancouver a decade and a half later, he played a major role in a number of international organizations and activities. In the Canadian context he was instrumental in the formation and growth of two significant initiatives: the development of Concordia as a centre for teaching and research in mathematics education, and the creation of the Canadian Mathematics Education Study Group / Groupe canadien d'etude en didactique des mathematiques (CMESG/GCEDM).

When David Wheeler came to Concordia, the Mathematics Department's main commitment to education was through the Master's in the Teaching of Mathematics programme (M.T.M.). At the time, the M.T.M. consisted essentially of content courses in mathematics and did not provide a broader based vision of mathematics education. Wheeler brought a wider perspective to the programme by weaving in the pedagogical, psychological, historical and philosophical connection to mathematics education. He introduced faculty and students alike to Piaget's work in developmental psychology, to Polya's classical writing on heuristics and problem solving, to Lakatos' perceptive insights of the process of mathematization and proof. He brought the international mathematics education community to Concordia by attracting visiting scholars and lecturers. By co-directing the first FCAR (Fonds pour la formation de chercheurs et l'aide a la recherche du Quebec) three year research project on problem-solving, he helped launch the research aspect of the mathematics education group. Within five very short years, the group has achieved an international reputation, with a very high research profile and an active role in many national and international organizations.

CMESG/GCEDM in the 1990's has become an active and influential group involving a high percentage of the population of Canadian mathematics educators and mathematicians with a strong interest in education, as well as a few regular, 'offshore', participants. In its early days, however, it was almost exclusively Wheeler's brainchild. In the evolution of CMESG/GCEDM we have a clear picture of Wheeler at work -- imaginative, sensitive, ambitious, disciplined, diligent and determined; it is a story worth recounting in some detail.

Shortly after his arrival in Montreal, David composed a letter in which he noted his perception of the lack of any national forum for the discussion of ideas about the teaching and learning of mathematics. He went on to ask a large number of mathematicians and mathematics educators in Canada whether this perception was correct, and if it was, whether there was merit in trying to create such a forum. The response to this request was largely negative. Of the individuals who responded, the majority either did not see such a venture as particularly important, or felt that their needs were already being met adequately by the National Council of Teachers of Mathematics (USA) and its allied interest groups. In the minority group of 'positive' respondents there was a small 'cluster point' in Kingston, Ontario where, independently, two individuals had expressed some interest in Wheeler's suggestion. One was John Coleman, the long-time head of the Department of Mathematics at Queen's University, and the second was William Higginson, recently appointed as an assistant professor in the Faculty of Education at the same university. [It would later be suggested, neither unkindly nor totally inaccurately, that CMESG/GCEDM was a function of Wheeler's imagination, Coleman's influence and Higginson's energy.] With this rather thin potential base for a national organization Wheeler moved quickly and decisively taking advantage of the fact that Coleman had recently completed a major study of the "Mathematical Sciences in Canada" (Science Council of Canada, 1976) and was able to support an invitational meeting at Queen's in the summer of 1977. The format established for that gathering [invited speakers -- in this case, John Coleman, Tom Kieren of the University of Alberta, and Claude Gaulin of Universite Laval -- and working groups] has been one of the 'constants' of the organization which evolved out of that meeting. It was clear to many by the end of that first Kingston meeting [which was to be followed by three more at that location in the next three years, by which time a formally constituted organization -- whose elected president for the first ten years was David Wheeler -- had come into being] that the 'new boy' on the Canadian mathematics education block had much to offer to this previously very loosely organized community. Take, for instance, these observations from his contribution, "Reflections after the Conference" from the Conference Proceedings (pp. 56 - 61 in "Educating Teachers of Mathematics: The Universities' Responsibility", A. J. Coleman, W. C. Higginson and D. H. Wheeler, eds.; Ottawa: Science Council of Canada, 1978):

"...it would be premature to say that mathematics education is on the verge of a breakthrough comparable to that experienced by mathematics... Yet the real message of the implied parallelism is that there 'may' be a current flowing that could liberate education from its ideological constraints... It is always a possibility that those who enter with curiosity and sensitivity and persistence into a dialogue with the facts may, like Kepler or Faraday or Cantor, find themselves carried into a new world that others will inherit."

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- William Higginson
- Queen's University, Kingston, ON, Canada
- higginsw@educ.queensu.ca
- Joel Hillel
- Concordia University, Montreal, Canada
- jhillel@vax2.concordia.ca

David Wheeler's fifty years in mathematics education have left indelible marks on the international scene.

Evidence of these can be found in his exceptional contribution as writer and as editor of the internationally renowned British journal "Mathematics Teaching" as well as in his remarkable work as founder, editor, fund-raiser, administrator, and much more, of "For the Learning of Mathematics" (FLM), a journal with a well-established world-wide reputation. These are discussed by others in this Bulletin.

Other evidence is David's involvement in activities of the International Commission on Mathematical Instruction (ICMI) and International Congresses on Mathematical Education (ICMEs). Concerning ICMI, he was the first and the only Canadian official representative until his retirement from this post in 1996, and he actively participated in a number of ICMI study seminars, always providing deep insights and thoughtful reflections. On the other hand, David Wheeler has been a member of the International Programme Committees for ICME-5 (1984), ICME-6 (1988) and ICME-7 (1992). For the latter, he chaired the IPC and played other very important roles, being in the forefront organizing and developing the successful bid to host the congress in Quebec City, and sitting on the Executive Committee and the Canadian National Committee. He contributed much to the success of ICME-7 and was an important member of the Editorial Panel for the two volumes of its Proceedings. As Chair of the IPC, he insisted that members reflect and question all parts of the programme: What was the role of Working Groups, Topic Groups, etc.? Was there a proper balance between these and the more traditional lecture presentations? How could the committee facilitate real participation by those who already had and those who were new to the ICME experience? etc. Undoubtedly David Wheeler has left his mark on the evolving spirit and organization of the ICMEs. Moreover, through many invited presentations he has made during ICMEs, PME and HPM conferences, ICMI study seminars and other events around the globe, he has influenced mathematics educators from the elementary to the tertiary levels.

It is clear that David has been consistently recognised internationally not only for his thought provoking and rich articles and presentations, but perhaps even more for his brilliant, original and spontaneous interventions during meetings, often raising questions or putting in question what others assumed of no consequence or accepted without question. Always aspiring to improve knowledge and understanding, he eagerly and patiently encouraged the participation and development of others.

We take the liberty to personalize the aims which he had originally spelt out for FLM: David Wheeler... "aims to stimulate reflection on and study of the practices and theories of mathematics education at all levels; to generate productive discussion; to encourage enquiry and research; to promote criticism and evaluation of ideas and procedures current in the field." In his fifty years of activities in mathematics education, David has certainly achieved that and we are most grateful for it.

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- Claude Gaulin
- Universite Laval, Quebec, Canada
- cgaulin@fse.ulaval.ca
- Eric Muller
- Brock university, St. Catharines, ON, Canada
- emuller@spartan.ac.brocku.ca

In July 1980, the first issue of "For the Learning of Mathematics" (FLM) appeared -- conceived, edited and financed (with some support from Concordia University in Montreal) by David Wheeler. By June 1997, the fiftieth issue will have appeared, David's final one as editor of his journal. Although the journal's synchronic appearance was on occasion aleatory, its diachronic presence is now an established regularity in the academic world (reflecting the most important factor when calling the June issue the June issue).

One of the many lasting impressions David has made in this realm has been produced through the pages of this journal, despite his almost never appearing as a named presence in the pages themselves. (He had a short editorial on page 1 together with a few briefly-worded questions and comments in his interview/discussion with Caleb Gattegno in issue number 1, and a second editorial to end things off in issue number 50. And that's it.) The incoming editor might be permitted a gleam in his eye about the pieces David might finally be inveigled into writing.

There are a number of orienting influences. One is that of the Association of Teachers of Mathematics (ATM), of whose journal David Wheeler was an early editor. In FLM issue 1, ATM is "represented" by Tahta, Trivett and Gattegno. The very name of the journal is deliberately resonant of the collections of Caleb Gattegno's writings, entitled "For the Teaching of Mathematics".

The title also signals the journal editor's strong interest in learning mathematics, without necessarily delimiting this as the journal's sole or even primary focus. The editorial on page 1 of issue 1 claims: "I want to do something to serve the interests of those who have to learn mathematics." A wide range of things can be offered "for the learning of mathematics": the title signals one answer to the question of what the journal is for.

FLM, like its editor-creator, is strongly orientated toward the mathematical, including its history and philosophy, in order to offer illumination of some of the issues at work within mathematics classrooms at all levels. FLM takes mathematics seriously. This has little to do with the age of pupils or complexity of mathematical content. It is possible to take mathematics in infant schools very seriously, as authors such as Gattegno, Rotman, Tahta and Walkerdine have shown, illuminating the referential and symbolic complexity of early arithmetic.

Elsewhere, in particular regard to mathematics, David Wheeler has written: "Dewey said somewhere that subject matter is a prime source of pedagogical insights. Almost no educators really believe this, I think, except in the trivial sense of hoping that teachers, textbook writers, and curriculum designers "know their mathematics". Even many mathematicians, who ought to know better, have no interest in looking below the instrumental or formal surface of mathematics in order to get clues about how to present it more effectively."

Wheeler has published, indeed championed, some pioneering work in the use of history of mathematics in classrooms, as well as strongly underpinning by his support a continuing exploration of the notion of "ethnomathematics". There is actually something of an irony here, as this latter notion in its various manifestations has proved a source of ambivalence to him (not least in connection to his own work on the notion of mathematising). Yet, as psychoanalyst Adam Phillips has noted, "ambivalence makes us vulnerable, because we are always on the side of the enemy".

For the Learning of Mathematics has proved itself to be open to some unfamiliar and unexpected writing (not least on occasion unexpected by the editor himself, a consequence of engaging guest editors). The special issue on psychodynamic influences brought together a number of such pieces, though other writing drawing on similar elemental themes (such as by Early or Blanchard-Laville) had appeared in the journal prior to this collection. As David has often pointed out, he doesn't have to agree with his authors. Even the Radatz article on student errors in the first ever issue contained a citation by Freud.

David Wheeler's sense of the mathematical and the educational, of what is worthwhile attending to, is well represented in the pages of his journal. It reflects a disciplined eclecticism and an appreciation of a wide variety of writing, both in content and style, corralled by a clear and unflinching eye for material of value shining through a wide range of forms. The letters he wrote to authors, whether of acceptance or rejection (producing occasional difficulty in recipients of the former in not construing them as the latter), were always motivated by a desire to make the journal the best he possibly could.

For the Learning of Mathematics will no longer be confluent with David Wheeler. But in handing its management over to the Canadian Mathematics Education Study Group / Groupe canadien d'etude en didactique des mathematiques (CMESG/GCEDM) and in taking part in the choosing of a subsequent editor, he has continued the link and underlined his continued involvement with and commitment to the journal. And it is we, its readers, who benefit and it is on behalf of the readers that I offer my appreciation.

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- David Pimm
- Open University, Milton Keynes, UK
- d.j.pimm@open.ac.uk

Last modified: Sat Jun 28 17:30:40 1997