Note: This history was original written for the monograph *Current Research on the Teaching and Learning of Mathematics in Canada* — *Les Recherches en Cours sur l’Apprentissage et l’Enseignement des Mathématiques au Canada* edited by Carolyn Kieran and A. J. (Sandy) Dawson and published for ICME-7 in 1992.

#### Introduction

The Science Council of Canada sponsored a mathematics education conference at Queen’s University, Kingston, Ontario, in September 1977. Thirty mathematicians and mathematics educators from across Canada accepted an invitation to join the three organisers of the conference (Professors A. J. Coleman and W. C. Higginson of Queen’s University, and D. H. Wheeler of Concordia University, Montreal) in discussing the general theme: “Educating teachers of mathematics: the universities’ responsibility.” The encounter generated a demand from many of the participants for further opportunities to meet and talk. The Science Council supported a second invitational meeting in June 1978 at which the decision was taken to establish a continuing group, to be called the CANADIAN MATHEMATICS EDUCATION STUDY GROUP / GROUPE CANADIEN D’ÉTUDE EN DIDACTIQUE DES MATHÉMATIQUES (CMESG/GCEDM–sometimes referred to as the Study Group). The fifteenth annual meeting of CMESG/GCEDM was held at the University of New Brunswick in Fredericton in May 1991.

The history of this professional group is very short but it seems worth presenting here, partly to give some context to the accounts of research that follow, but also because the special character of CMESG/GCEDM may be found to have some instructive features.

#### Beginnings

The introduction to the programme for the 1977 meeting reads:

The Conference has been convened as part of the follow-up to the Council’s Background Study No. 37 (

Mathematical Sciences in Canada) [1] to consider the place and responsibility of Canadian universities in the education of teachers of mathematics. The participants are university mathematics educators and mathematicians, but the organisers do not intend to imply that only universities are or should be concerned in the education of teachers. Universities have traditionally played a principal role, however, and will certainly continue to be involved in teacher education for the foreseeable future even though the forms of their involvement may change. The Conference is an opportunity to make a contribution, related to one particular aspect and from one particular point of view, to the public discussion of mathematics education in Canada. The Conference has no official status and is in no sense a policy-forming or advisory body. It is not the intention of the Conference to seek consensus or to make recommendations to anyone.One purpose of the Conference is served by the mere fact of bringing participants together and the consequent pooling of ideas and information by those who have overlapping interests but seldom meet. It is meant to have other, tougher, purposes too. At a level above that of information-sharing there are questions to be formulated, problems to be isolated and tendencies identified, maybe even achievements to be acknowledged; in other words, an attempt to get a grasp on the present situation and an orientation on the future. At a still higher level belongs the task of studying together how the questions may be answered and the problems resolved. Independent of this hierarchy is the job of communicating something of value to other professionals and to the public. How much of this can be achieved in such a short time remains to be seen. At least a start can be made.

The faintly apologetic tone of all this is characteristically Canadian, but the sense it conveys that the organisers were stepping warily is quite genuine. One good reason was that the Background Study referred to had been badly received by the mathematical community, at least as represented by the Canadian Mathematical Congress (later to rename itself the Canadian Mathematical Society), which did not enjoy the many explicit and implicit criticisms made by the writers of the Study. A reviewer of*Mathematical Sciences in Canada* summarised its general argument in the following terms:

Mathematics plays a commanding role in modern technological societies, yet many professional mathematicians have little interest in its applications, and government and business are often unsure how best to use the mathematicians they employ. Mathematics is taught to Canadians in one of the most generous and accessible educational systems in the world; yet only a minority of students gain much competence in it, and only a minority of those more than a routine grasp. Mathematical research is published in daunting quantities; yet most papers do no more than dot i’s and cross t’s well inside the frontiers. The output of Canadian PhD’s in mathematics has increased tenfold in the last fifteen years; yet a large majority of them still expect to remain in academia and do little but produce more of their kind. *Mathematical Sciences in Canada* elaborates on a situation that might once have been described as productive redundancy, but which in these less easy-going times seems more like capricious and conspicuous waste. [2]

Another reason for the organisers’ caution can be found in the statutory division of responsibilities for education in Canada between the federal and provincial authorities. The provinces have total authority for the organisation and governance of primary and secondary education. To obtain federal support for the 1977 conference, which was necessary if participants were to be drawn from all parts of Canada, the organisers had to make sure that the objectives did not infringe on the application of provincial powers. Direct examination of the school curriculum, for example, had to be carefully avoided, and the conference had to refrain from making recommendations that might appear as an attempt to interfere with provincial rights.

The programme of the 1977 conference included three keynote lectures:

- The state of research in mathematics education (T. E. Kieren)
- Innovations in teacher education programmes (C. Gaulin)
- The objectives of mathematics education (A. J. Coleman)

and four working groups:

- Teacher education programmes
- Undergraduate mathematics programmes and prospective teachers
- Research in mathematics education
- Learning and teaching mathematics.

The organisers felt that it was important for the meeting to give a substantial amount of attention to mathematics education research. Without this component it would be only too easy for the discussions in the meeting and the conclusions that might emerge to do no more than recycle familiar folklore about the shortcomings of mathematics teaching in Canada.

Conference proceedings were published by the Science Council [3]. One of the organisers, penning some “Reflections after the Conference,” which are included in the Proceedings, began by quoting from the Background Study:

It no longer seems possible for any component of the mathematical ecosystem to function effectively in isolation. Awareness and communication seem to be the key issues. [1, p. 86]

and continued:

They were the underlying themes of the Conference too. Bringing university mathematicians and mathematics educators together involved an interaction between two groups which tend to be somewhat suspicious of each other. The assumption by the universities of the responsibility for training teachers has not led, in general, to greater mutual understanding or cooperation by those who teach university mathematics and those who teach would-be teachers of mathematics. Both groups have other interests and responsibilities and it may be that the lack of common ground in these other areas contributes to the suspicion. But it also extends into that part of their work where they might be expected to find a shared cause — the preparation of specialist mathematics teachers. University mathematicians look at education courses and see an apparent lack of structure and rigour together with a plenitude of non-refutable theories; university mathematics educators look at the students emerging from undergraduate mathematics programmes and see the apparently deadening effects of a training dominated by structure and rigour. Both sides, when apart, tend to stereotype each other. [3, p. 56]

The generally favourable response to the 1977 meeting led Coleman, Higginson, and Wheeler to propose a continuation. Their first plan was to work toward meetings in 1978 and 1979 which would culminate in the production of documents; these might form the basis for a Canadian contribution to the Fourth International Congress on Mathematical Education (ICME-4) to be held in Berkeley USA in August 1980. This focus on the production of documents led them to suggest meetings covering five working days, which would allow for some writing to take place during the meetings. But the overwhelming response was a rejection of five days as impossibly long and, in the event, the 1978 meeting set a pattern which has become the standard for all subsequent meetings: three full working days sandwiched between arrival and departure half-days.

The programme for the 1978 meeting included two lectures:

- The mathematician’s contribution to curriculum development (G. R. Rising)
- The mathematician’s contribution to pedagogy (A. I. Weinzweig)

and three working groups:

- Mathematics courses for prospective elementary school teachers
- Mathematization
- Research in mathematics education.

The working groups were scheduled simultaneously for a total of 18 hours. Although this proved to be too much time — it took so large a chunk of the time available that it squeezed out other activities, such as up-dating the work done at the previous meeting — it symbolised the considerable significance that the organisers gave to this activity: the working groups were always intended to be the core activity of the meetings. From the 1979 meeting onward, working groups have met for nine hours, but they have retained their centrality, in many ways setting the tone of the meetings and distinguishing them from most other scholarly conferences in Canada. (A list of the working groups for the first fifteen meetings is given in Appendix 1.) Less distinctive, perhaps, has been the effect of putting the keynote lectures in the hands of “guest” speakers, usually non-Canadians. The intention here was to enrich the input to the meetings by inviting speakers who would bring fresh perspectives to the discussion of mathematics education. The guest speakers over the years make a diverse and distinguished bunch, as the list in Appendix 2 shows.

Unfortunately, the ambition to produce significant discussion documents for ICME-4 was not realised. The published evidence of the Study Group’s activities is largely confined to the proceedings of its annual get-togethers, and even these do not always manage to convey a good idea of the real transactions of the meetings. (Appendix 3 lists the ERIC numbers of available CMESG/GCEDM proceedings.)

At the close of the 1978 meeting the participants voted to give CMESG/GCEDM a continuing existence and an acting executive committee. A formal constitution was approved at the 1979 meeting and the first elections under the terms of the constitution took place in 1980. Although a few changes in the organisational structure have occurred, and although the annual programmes have evolved to some extent, the main characteristics of the Study Group were settled in the first few years.

#### Characteristics

Canada’s size, location, and federal structure pose special problems for any organisation aiming at nationwide status. Travel distances and costs for regular face-to-face meetings are simply enormous. Whatever purpose a Canadian group might espouse, there is almost certainly a group in the USA with a similar purpose whose meetings are as easy (or difficult) to get to. The province-based organisation of primary and secondary education tends to lock up some of the money that would otherwise be available to support attendance at meetings. Given a different context, the original animators would have tried to establish a national group open to educators in all parts of the system: elementary school teachers, school administrators, university professors of mathematics, and so on. But it never seemed realistic in the Canadian setting to attempt to cast such a wide net.

Furthermore, the first meeting of what was to become CMESG/GCEDM chiefly involved university mathematicians and university mathematics educators. These populations seemed the most appropriate to target for a number of reasons. The meetings could then be kept small enough to facilitate the kind of personal interactions the organisers wanted to promote; they could focus on some of the scholarly questions in the field; and they could help to bridge the professional and ideological gaps between mathematicians and teacher educators and researchers. So with some regret the decision was made to develop a programme to attract university teachers in departments and faculties of education and in departments of mathematics. The trade-off under this restriction would be, it was hoped, a greater involvement of university professors of mathematics. CMESG/GCEDM can report some success in attracting to its ranks a number of Canadian mathematics professors (to the extent of approximately a third of the active membership). A higher rate of participation, even if desirable, is not likely given the fact that a serious involvement in education is, for university mathematicians, an additional demand on their time and energy, a commitment rarely recognised or rewarded by their departmental colleagues. In any case, the regular interaction and cooperation of professors from education *and* mathematics departments within the Study Group remain a significant and treasurable feature.

From the beginning, as can be seen from the lists of working groups and lectures in Appendices 1 and2, the two main interests of CMESG/GCEDM have been teacher education and mathematics education research, with subsidiary interests in the teaching of mathematics at the undergraduate level and in what might be called the psycho-philosophical facets of mathematics education (mathematization, imagery, the connection between mathematics and language, for instance). There are obvious overlaps with the interests of other Canadian groups. An early decision was made to resist integration with the Education Committee of the Canadian Mathematical Congress (later “Society”) even though a group bringing together university mathematicians and mathematics educators might seem to have fitted well there. The original animators felt it was important for CMESG/GCEDM to establish an identity and a professional credibility before getting too closely involved with CMC (CMS), whose Executive Committee, in the 1970s at least, was not noticeably interested in, informed about, or sympathetic to, mathematics education. Subsequently CMESG/GCEDM developed good relations with a revitalised CMS Education Committee and in 1985, 86, and 87, the Study Group met in the same locations as the CMS so that a few of its sessions could be co-sponsored by the two organisations. In 1990 CMESG/GCEDM co-sponsored a day’s activities with the Canadian Society for the History and Philosophy of Mathematics (CSHPM).

Many scholarly and academic associations in Canada hold their annual meetings on the same site during the same period, at an event called the Learned Societies Conference. Some of the people who would have liked to be involved in CMESG/GCEDM were accustomed to attend meetings of the Canadian Society for the Study of Education (CSSE), which always participated in the “Learneds,” and it was natural for them to suggest that CMESG/GCEDM should hold its meetings there too. Again the initial organising group resisted a move toward immediate integration, though for a different reason. It seemed to them that if CMESG/GCEDM was to develop a distinctive character, and particularly if it was to develop a genuine working atmosphere, it needed to be able to persuade people to commit themselves entirely to the Study Group for the whole of a meeting. Setting the meeting in a situation where *n* fascinating lectures were always on offer in adjacent buildings would make that dedication difficult if not impossible to realise. So, to the annoyance of a few, CMESG/GCEDM did not join the collection of societies in the “Learneds.” (It must be noted here, with considerable gratitude, that the Social Sciences and Humanities Research Council of Canada, which gives a block grant to the “Learneds,” has never used its muscle to insist that CMESG/GCEDM belong in order to qualify for financial help.)

Attendance at CMESG/GCEDM meetings has varied between 30 and 70, with most in the 50-60 range. This is a good size for the kind of meetings the Group organises: small enough to give a feeling of community while large enough to ensure a mix of interest and experience. Two-thirds of this number are usually regulars who attend most of the meetings. Membership is predominantly but not exclusively Canadian. The Group benefits a lot from the presence of a few non-Canadians, though it is watchful that the proportion does not grow too large.

#### Innovations

The emphasis on spending a substantial amount of time at meetings in working groups has already been mentioned. The “philosophy” behind this is more than an acknowledgment that “two heads are better than one,” or that multiple perspectives on important issues are potentially more illuminating than unitary ones. It goes further and says that it is possible for people to *work collaboratively* at a conference on a common theme and generate something fresh out of the knowledge and experience that each participant brings to it. This is not easy to achieve, it must be said, perhaps because people are not used to working this way and have not yet learned the techniques. Newcomers sometimes feel that the first 3-hour session allotted to a working group is “wasted” because the group has come together without a common view on the topic, so everyone has to sit through the expression of a lot of different opinions before the group can actually “start.” Ways have been proposed to overcome this problem: giving each member of the group papers to read before the meeting, making a clear presentation of the group’s programme before members have chosen which group to attend, and so on. But of course the ideal picture of a working group, in which everyone wants to work in exactly the same way on exactly the same questions, is a fiction. The task of the group leaders (there are usually two) is to capitalise on the diversity of expectation and experience within the group while fostering the pursuit of those emergent sub-themes which appear to be going somewhere. It is not surprising that this activity does not always lead to the sort of outcomes that can be immediately written down and polished into a conventional scholarly paper. A well-run working group handles complexity very effectively, but effective ways of recording its achievements are difficult to develop.

The emphasis on working groups influences other aspects of the CMESG/GCEDM meetings. People are not divided disjointly into a set of those who present and a set of those who sit and listen. There*are* presentations of a quite conventional kind, but in the context of the meeting they also become subjects for discussion. An innovation which symbolises this is the “discussion hour” scheduled on the day following a plenary address at which the members discuss the talk with the speaker.

CMESG/GCEDM programmes always have at least one slot in the timetable for “ad hoc groups.” Any person may volunteer to make a presentation or lead a discussion, and these items are added to the programme (subject to the availability of time and facilities).

The intention of these various opportunities is to encourage members to take an active part in the meetings. The policy would be ineffective if it did not deliver, and if it were not situated in a relatively relaxed and accepting atmosphere. As in school, people would soon stop making contributions if these kept getting shot down in flames. A CMESG/GCEDM meeting is free of the point-making and competitiveness that are features of many academic gatherings. People *listen* to other people, with respect if not necessarily agreement.

#### Impacts

With fifteen annual meetings to its credit, and a core of active members, CMESG/GCEDM now certainly exists. Although the first group of enthusiasts in 1978 may have hoped for more, they probably expected less: the Canadian environment for innovation is notoriously harsh. The Study Group hasn’t produced the discussion documents, or made the public and political pronouncements, or developed the regional and local networks, or achieved any of the partial agendas that people have at times proposed for it. But it exists. And it holds annual meetings. And these are, to judge from the comments of regulars and of newcomers, appreciably different from, and more involving than, other meetings that the same people go to. In an important sense CMESG/GCEDM* is* its annual meetings since these are where what is characteristic of CMESG/GCEDM actually takes place — its study-in-cooperative-action. (For Plato, philosophy was to be found at its best in the serious talk of friends rather than in the story of it that someone might write up afterwards.)

CMESG/GCEDM now exists in Canada alongside the CMS Education Committee, whose natural interest inclines more to the teaching of mathematics at the tertiary level. Both are small, national groups catering mainly to university teachers. Each province in Canada has its own separate association of teachers of mathematics (Quebec has *four*: three francophone, one anglophone). Two provinces, Ontario and Quebec, have associations of mathematics advisers (alternatively called “coordinators” or “consultants”). Many high school teachers and advisers belong to the National Council of Teachers of Mathematics (NCTM) and attend its annual meetings. The NCTM claims coverage of Canada, indeed, and always has a Canadian on its Board of Governors, but rarely interests itself in particularly Canadian issues. Many Canadian mathematics educators belong to the American Educational Research Association (AERA) or its subgroup SIG/RME (Special Interest Group for Research in Mathematics Education), just as many university and college professors of mathematics belong to the American Mathematical Society or the Mathematical Association of America. (And it is likely that a majority of school, college, and university teachers of mathematics are not active in any of the above.) This is a uniquely fragmented situation. There is no body in Canada able to deal with the whole of mathematics education at all levels, no national voice speaking about mathematics education to governments and the public — though perhaps this matters little in a country which has no national educational policy.

When it comes to impact and influence, though, who can be sure what Canadians lose by not having a powerful voice speaking on behalf of mathematics education? The USA and France, for example, both have very powerful professional organisations able to talk to governments, but it is by no means certain that their influence is always good, judged from the viewpoint of the “consumers” of mathematics education in the schools. (National medical associations, to consider a possible parallel, do not always seem to be arguing or advancing the cause of the sick.) CMESG/GCEDM lacks a powerful voice, but it has influenced, perhaps changed, a number of individuals.

The Study Group takes as its essential position that the teaching of mathematics and all the human activities that are connected to it can, and should, be *studied,* whether the study has the form of an individual’s reflections, the reasoned argument of professional colleagues, or the more formal questioning of empirical or scholarly research. By putting this emphasis CMESG/GCEDM has signalled to Canadian mathematics educators the importance of scholarship and research in a field that often seems dominated by folklore. The Study Group has provided a forum where research plans can be discussed and an encouraging atmosphere where novice researchers can find out how to begin. Mathematics teaching may go back to “the dawn of history,” as the journalist might say, but mathematics education as a field of study is only a few decades old. It has no traditions of research and scholarship: these are only now being developed.

In brief, through its activities CMESG/GCEDM has given some mathematics educators a taste for research and shown them how to get started. It has shown them that their puzzlement about some aspects of mathematics is shared by many mathematicians. It has shown some mathematicians that learning can be studied and that teaching might be made into something more than flying by the seat of the pants. A sufficient number of such small victories could launch a revolution.

#### Jobs to do

As suggested above, CMESG/GCEDM has already played a strengthening and encouraging role in the Canadian effort on research in mathematics education and it seems clear that it should continue and perhaps extend its work in this direction. There is a long way to go, as is generally acknowledged, before mathematics education research becomes a resource that everyone in the teaching business will be glad to be able to draw on, but nothing less should be demanded of it. Understandably most teachers find that most research to date is immature: it fails to convince because it cannot yet match the complexity of a good teacher’s professional insights. Nevertheless it is extremely important that the work go on. Research takes a significant stand against an extraordinarily widespread but destructive belief: that the teaching of mathematics is essentially unproblematic. That this might not be so was perhaps first signalled by Poincaré when he asked why some people never manage to acquire mathematical knowledge. Research, of course, in common with other theoretical positions, can as readily be used to “explain away” as to explain; nevertheless its insistence on inspection, reflection, and trial, is an important corrective to the naive view that teaching and learning are transparent activities.

On a less broad front, CMESG/GCEDM still needs to work on improving the amount and quality of the interaction between mathematicians and mathematics educators. There is a job to be done while there are still mathematics educators involved in teacher education and research who have only a tenuous acquaintance with genuine mathematical activity, and while there are still mathematicians who think that all questions belonging to the field of mathematics education are intrinsically trivial. University mathematicians as a class are not noticeably modest. It is probably not too much of a caricature to say that in general they seem happy to admit–*grace à Descartes*–the god-like character of their main activity. They are not in general reluctant to take advantage of the universities’ traditional favouring of academic over professional knowledge. Moreover, mathematicians have been deemed successful in what is recognised by everybody as a difficult intellectual discipline. Given all these advantages, they sometimes fail to recognise that the skills and sensitivities that have served them well in working on mathematics are not necessarily the ones that can meet the challenges presented by mathematics education.

There is a need in Canada to make public a more accurate picture of mathematics education, one which admits that its development has only just started, but which also shows that its heuristic is effective and its arguments capable of being made, within reason, rigorous and disciplined. If some real substance can be put into such an account, a greater respect for mathematics education must follow. CMESG/GCEDM is in a good position to work with mathematicians on improving the image of mathematics education as a field of study.

These are long-term goals — ideals, perhaps — which could point CMESG/GCEDM in a certain direction but do not spell out in detail how it might reach them. Probably the future of CMESG/GCEDM, in any case, will be shaped by a combination of internal and external forces most of which cannot now be predicted.

#### References

[1] Beltzner, K. P., Coleman, A. J., & Edwards, G. D. (1976). *Mathematical Sciences in Canada*(Background Study 37). Ottawa, ON: Science Council of Canada.

[2] Wheeler, D. (1977). Review of “Mathematical Sciences in Canada.” *Notes of the Canadian Mathematical Congress. *Ottawa, ON: Canadian Mathematical Congress.

[3] Coleman, A. J., Higginson, W. C., & Wheeler, D. H. (Eds.). (1978). *Educating teachers of mathematics: The universities’ responsibility. *Ottawa, ON: Science Council of Canada.