Jan van Maanen University of Groningen

In recent years there has been growing interest in the role of history of mathematics in improving the teaching and learning of mathematics. Educators throughout the world have been formulating and conducting research on the use of history of mathematics in mathematics education. Some of the results of this research have been communicated at meetings of interested organisations, and through papers in various journals.

A research programme is beginning to emerge, with contributions from many places over the globe. Such a programme involves a consolidated critical bibliography of work that has been done, and a programme for developing a deeper understanding of the factors involved in the relations between history and pedagogy of mathematics, in different areas of mathematics, and with pupils and students at different stages and with different environments and backgrounds. It also involves the identification and spreading of information and good practice in learning and teaching situations.

ICMI, the International Commission on Mathematics Instruction, has set up a Study on this topic, to report back in time to form part of the agenda at the next International Congress in Mathematics Education (ICME) in Japan in the year 2000. The present document sketches out some of the concerns to be addressed in the ICMI Study, in the hope that many people across the world will wish to contribute to the international discussions and the growing understandings reached in and about this area.

It is hoped that this discussion document will lead to a number of responses and intimations of interest in contributing to the Study. It will be followed by an invited conference (to be held in France in April 1998), from which a publication will be prepared to appear by 2000. The next section of the present document surveys the questions to be addressed. Your views are solicited both on the questions and on how to take the issues forward as implied in the commentary.

**1. How does the educational level of the learner bear upon the role of history of mathematics?**

The way history of mathematics can be used, and the rationale for its use, may vary according to the educational level of the class: children at elementary school and students at university (for example) do have different needs and possibilities. Questions arise about the ways in which history can address these differences. This may, again, be reflected in different training needs for teachers at these levels. (To speak about the ``use'' of the history of mathematics may seem to presuppose that history of mathematics is something external to mathematics. This assumption would not be universally agreed, however.)

**2. At what level does history of mathematics as a taught subject
become relevant?**

In analysing the role of history of mathematics, it is important to distinguish issues around using history of mathematics in a situation whose immediate purpose is the teaching of mathematics, and teaching the history of mathematics as such, in a course or a shorter session. It could be that courses in the history of mathematics, and its classroom use, should be included in a teacher training curriculum (see question 3). There is also a third area, related but separate, namely the history of mathematics education, which is a rather different kind of history.

**3. What are the particular functions of a history of mathematics
course or component for teachers?**

History of mathematics may play an especially important role in the training of future teachers, and also teachers undergoing in-service training. There are a number of reasons for including a historical component in such training, including the promotion of enthusiasm for mathematics, enabling trainees to see pupils differently, to see mathematics differently, and to develop skills of reading, library use and expository writing which can be neglected in mathematics courses. It may be useful here to distinguish the training needs for primary, secondary and higher levels (see question 1). A related issue is what kinds of history of mathematics is appropriate in teacher training and why: for example, it could be that the history of the foundations of mathematics and ideas of rigour and proof are especially important for future secondary and tertiary teachers. (This issue is also relevant for other categories than future teachers, and is picked up again in question 5)

This question focuses on the professional base from which practitioners emerge, and relates to the social fabric of today's mathematics education community as well as to issues about the nature of history. There are, gratifyingly, a number of leading historians of mathematics with an interest in educational issues, as there are leading mathematicians and mathematics educators with an interest in history. But as well as minor misapprehensions of the nature of the others' activities, there may be deeper tensions and conflicting aims which it is important to bring to the surface. For example, historians may underestimate the difficulty of transmuting the historical knowledge of the teacher into a productive classroom activity for the learner. It is important that historians and mathematics educators work co-operatively, since historical learning and classroom experience at the appropriate level do not always co-exist in the same person.

**5. Should different parts of the curriculum involve history of
mathematics in a different way?**

Already research is taking place to investigate the particularities of the role of history in the teaching of algebra, compared with the role of history in the teaching of geometry. Different parts of the syllabus make reference, of course, to different aspects of the history of mathematics, and it may be that different modes of use are relevant. Looking at the curriculum in a broad way, we may note that the histories of computing, of statistics, of core ``pure'' mathematics and of the interactions between mathematics and the world are all rather different pursuits. Even for the design of the curriculum historical knowledge may be valuable. A survey of recent trends in research, for example (bearing in mind that history extends into the future) could lead to suggestions for new topics to be taught.

**6. Does the experience of learning and teaching mathematics in
different parts of the world, or cultural groups in local contexts,
make different demands on the history of mathematics?**

A historical dimension to mathematics learning helps bring out two contrary perceptions in a dialectical way. One is that mathematical developments take place within cultural contexts and it is valid to speak of Islamic mathematics, Greek mathematics and so on, as developments whose style is characteristic of the generating culture. The antithesis to this is the realisation that all human cultures have given rise to mathematical developments which are now the heritage of everyone; this therefore acts against a narrow ethnocentric view within the educational system. The Study should explore the benefit to learners of realising both that they have a local heritage from their direct ancestors --in the way in which Moslem children in countries where they are in a minority are known to derive pride and strength from learning about Islamic mathematical achievements-- but also that every culture in the world has contributed to the knowledge and experience base made available to today's learners. There are many detailed studies of the interplay betwen history of mathematics and culture in educational contexts throughout the world, notably in Brazil, the Maghreb, Mozambique, China, Portugal etc, which should be drawn upon in analysing and responding to this question.

**7. What role can history of mathematics play in supporting special educational needs?**

The experience of teachers with responsibility for a wide variety of special educational needs is that history of mathematics can empower the students and valuably support the learning process. Among such areas are experiences with mature students, with students attending numeracy classes, with students in particular apprenticeship situations, with hitherto low-attaining students, with gifted students, and with students whose special needs arise from handicaps. Here the many different experiences need to be researched, their particular features drawn out, and an account provided in an overall framework of analysis and understanding.

**8. What are the relations between the role or roles we attribute to history and the ways of introducing or using it in education?**

This question has been the focus of considerable attention over recent decades. Every time someone reports on a classroom experience of using history and what it achieved they have been offering a response to this question. So a search of the literature is a fundamental part of researching the response to this question.---The question also involves also a listing of ways of introducing or incorporating a historical dimension: for example anecdotal, broad outline, content, dramatic etc. Then one would draw attention to the range of educational aims served by each mode of incorporation: the way that historical anecdotes are intended to change the image of mathematics and humanize it, for example. Or again, the way that mathematics is not, historically, a relentless surge of progress but can be a study in twists, turns, false paths and dead-ends both humanizes the subject and helps learners towards a more realistic appreciation of their own endeavours.

There are rich issues for discussion and research in, for example, the use of primary sources in mathematics classrooms at appropriate levels.

This question is a very broad one that could involve a large number of people: it may be wise to distinguish the taxonomic question --the range of different classroom aims and modes of activity-- from the further exploration of each issue.

The consequences of integrating history are far-reaching. In particular, there are wider opportunities for modes of assessment. Assessment can be broadened to develop different skills (such as writing and project activity), and consequences for students' interest and enjoyment have been noted. Teachers may well need practical guidance and support both in fresh areas of assessment, and in aspects of classroom organisation. This in turn may have consequences for teacher training as well as curriculum design.

**10. How can history of mathematics be useful for the mathematics education researcher?**

This question provides an opportunity for an exploration of the relations between the subject of this study and researchers in the mathematical education community (whose aims are, in turn, to provide insights into the processes of learning and teaching). One example is the use of history of mathematics to help both teacher and learner understand and overcome epistemological breaks in the development of mathematical understanding. A constructive critical analysis of the view that `ontogeny recapitulates phylogeny' --that the development of an individual's mathematical understanding follows the historical development of mathematical ideas-- may be appropriate. Another example is of research on the development of mathematical concepts. In this case the researcher applies history as possible `looking glasses' on the mechanisms that put mathematical thought into motion. Such combinations of historical and psychological perspectives deserve serious attention. These issues could be studied in teaching experiments in which the above questions are addressed, and also questions like: What is good for the learner? How do you know it is good for the learner? and so on. Even if a teaching experiment does not use history of mathematics explicitly, the elaboration of the teaching project may have made use of the results of history of mathematics. For instance, such a question as `is it good for the learner?' may be better understood in the light of the history of mathematics. So the question here is: how can research in mathematics education profit from historical knowledge? The answer to this question might deal with themes such as the historical genesis of a concept and an epistemological analysis of the interplay between history and the teaching of a subject. Moreover, history of mathematics helps to understand the distance between the way in which concepts function in the mathematics community and the way they function in the school. There are also fundamental questions about the style and evaluation of research in this area. Different styles which have been used in the past range from the anecdotal (in effect) to quasi-scientific surveys with questionnaires and statistical apparatus. A process of such considerable complexity evidently calls for a research methodology of some sophistication. Fortunately the wider mathematics education community has been studying this problem for some time: it is indeed the subject of an earlier ICMI Study (What is research in mathematics education and what are its results?). So a group could be encouraged to draw upon the wider community experience and consider its application to our area of concern.

**11. What are the national experiences of incorporating history of
mathematics in national curriculum documents and central political guidance?**

This is not so much a question for discussion as a fairly straightforward empirical question, needing input from knowledgeable people in as many countries and states as possible. But of course it has policy implications too, and could lead to a sharing of experience among members of the community about how they have reached the policy-making level in their countries to influence the content or rhetoric of public documents. Perhaps this study could be carried on in parallel with the more discursive questions, organised by a small group who could put the results (in the sense of public documents or quotations from them as well as brief historical accounts of national curriculum change) on the WorldWideWeb as they are collected.In some parts of the world a different relationship between history and mathematics may have been developed. For example, in Denmark and Sweden history of mathematics is regarded as an intrinsic part of the subject itself. There are also differences in styles of examination and assessment. If everyone with access to examples of such different approaches, from different countries and states, could pool their experience it would be a most valuable input to the Study.

**12. What work has been done on the area of this Study in the past?
**

The answer is: quite a lot. But it is all over the place and needs to be gathered together and referenced analytically. A major annotated critical bibliographical study of the field, which might well take up a sizable proportion of the final publication, would be an enormously valuable contribution that the ICMI Study could make. It should include a brief abstract of each paper or piece of work included, and indications of the categories to which the work relates in an analytical index. The organisation of this sub-project will need to be different from that of the rest of the Study. It will need to be even more pro-active to achieve a useful result. A small group should perhaps take this in hand and work out how it can be achieved collaboratively. Some progress on such a bibliography is already in hand in various places, notably by Fred Rickey in the US, John Fauvel in the UK. This seems another place where work in progress could be available on the WorldWideWeb.

Calinger, Ronald (ed), *Vita mathematica: historical research and
integration with teaching*, Mathematical Association of America 1996

Fauvel, John (ed), *History in the Mathematics Classroom. The IREM
Papers*, The Mathematical Association 1990 (translation from the French
of papers by the Committee Inter-IREM, combined with classroom
resources)

Fauvel, John (ed), *For the learning of mathematics *11 no 2 (June 1991;
special issue on using history of mathematics in the mathematics
classroom)

Fuehrer, Lutz (ed), *mathematik lehren* 19 (December 1986; special issue
entitled `Geschichte -- Geschichten')

IREM de Franche-Comte (coll.ed), *Contribution `a une approche
historique de l'enseignement des math'ematiques*, Besancon 1996
(proceedings of the 6th summer university, Besancon July 1995)

IREM de Montpellier (coll.ed), *Histoire et epistemologie dans l'
education mathematique*, Montpellier 1995 (proceedings of the first
European Summer University, Montpellier August 1993)

McKinnon, Nick (ed), *The mathematical gazette* 76 no 475 (March 1992;
special issue on using history of mathematics in the teaching of
mathematics)

Nobre, Sergio (ed), Meeting of The International Study Group on Relations Between History and Pedagogy of Mathematics. Blumenau/ Brazil 25--27 July 1994, UNESP 1994

Schoenebeck, Juergen (ed.), *mathematik lehren* 47 (August 1991; special
issue about `Historische Quellen fuer den Mathematikunterricht')

Swetz, Frank, et al (ed), *'Learn from the Masters!*', Mathematical
Association of America 1995

Veloso, Eduardo (ed), *Historia e Educacao
Matematica*. proceedings/actes/actas, Braga/Lisbon 1996

The International Programme Committee (IPC) for the study invites members of the educational and historical communities to propose or submit contributions on specific questions, problems or issues stimulated by this discussion document no later than 1 October 1997 (but earlier if possible). Contributions, in the form of research papers, discussion papers or shorter responses, may address questions raised above or questions that arise in response, or further issues relating to the content of the study. Contributions should be sent to the co-chairs (addresses below). Proposals for research that is on its way, or still to be carried out, are also welcome; questions should be carefully stated and a sketch of the outcome --actual or hoped-for-- should be presented, if possible with reference to earlier and related studies. All such contributions will be regarded as input to the planning of the study conference.

The members of the International Programme Committee are Abraham Arcavi (Israel), Evelyne Barbin (France), Jean-Luc Dorier (France), Florence Fasanelli (USA), John Fauvel (UK, co-chair), Alejandro Garciadiego (Mexico), Ewa Lakoma (Poland), Jan van Maanen (Netherlands, co-chair), Mogens Niss (Denmark) and Man-Keung Siu (Hong Kong).

This document was prepared by John Fauvel and Jan van Maanen with the help of Abraham Arcavi, Evelyne Barbin, Alphonse Buccino, Ron Calinger, Jean-Luc Dorier, Florence Fasanelli, Alejandro Garciadiego, Torkil Heiede, Victor Katz, Manfred Kronfellner, Reinhard Laubenbacher, David Robertson, Anna Sfard, and Daniele Struppa.

Contributions should be sent to the co-chairs at the following addresses:

John Fauvel, Mathematics Faculty, The Open University, Milton Keynes MK7 6AA, England UK (j.g.fauvel@open.ac.uk)

Jan van Maanen, Department of Mathematics, University of Groningen, P O Box 800, 9700 AV Groningen, The Netherlands (maanen@math.rug.nl)

VOLTAR ATRÁS