Number 2, November 1990 Editor: Dr. Stephen Lerman

This issue has been made possible by the generous support of the editor's institution, South Bank Polytechnic, Department of Computing and Mathematics, Borough Road, London SE1 0AA, U.K.

(Phone: Fax: E-mail: )



Raffaella Borasi (USA), Stephen I. Brown (USA), Leone Burton (UK), Paul Cobb (USA), Jere Confrey (USA), Kathryn Crawford (Australia), Philip J. Davis (USA), Paul Ernest, Group Chair (UK),, David Henderson (USA), Reuben Hersh (USA), Christine Keitel (FRG), Stephen Lerman (UK), Marilyn Nickson (UK), David Pimm (UK), Sal Restivo (USA), Leo Rogers (UK), Anna Sfard (Israel), Ole Skovsmose (Denmark), Hans-Georg Steiner (FRG), John Volmink (USA).

The editor of The Philosophy of Mathematics Education Newsletter 3 will be Leo Rogers. Please send any correspondence, items, comments, publication or conference announcements for issue 3 to him, at the following address.

Leo Rogers, Digby Stuart College, Roehampton Institute, Roehampton Lane, London, SW15 5PH, U.K. [Fax (0)81-392-2384]

The POME Newsletter mailing list is fast approaching 200. Since many of the addresses may have been taken from out-of-date sources, it is requested that all who wish to continue receiving it write briefly to the address below, in confirmation. This is not necessary for anyone who has made any sort of direct contact or response already, personal or written.

There has been a very positive response to the formation of the network and the first newsletter. Announcements of the formation of the group have been carried in Educational Studies in Mathematics, History, Philosophy and Teaching Science Newsletter 2, Mathematical Association Newsletter (UK) and Epistemologia della Matematica newsheet (Italy). Scholars from a number of different countries have responded by sending personal opinions and comments, details of their interests, copies of their books and papers, and the names of others to add to the mailing list. More of the same would be most welcome!


POME Group Chair, Dr. Paul Ernest (Fax 392-411274)

University of Exeter, School of Education, Exeter EX1 2LU, U.K.


There will be a Philosophy of Mathematics Education Discussion Group at the British Congress of Mathematics Education, Loughborough, July 13-16, 1991, led by Paul Ernest. The group will have 3 X 90 minute slots.

Preliminary plans for the group are now underway, and current thinking is that we should aim for some depth of treatment of issues. Consequently offers are sought for papers to be presented in this group, and for themes under which to organise them. It is likely that presentations in the sessions will brief, to allow for discussion, but that fuller written papers can be distributed. Given sufficient offers, we may organise parallel sessions.

There is no restriction to U.K. residents for either contributions or attendance. Indeed, the intellectual tone will probably be higher if the discussion group represents the international mathematics education research community.

Therefore please send in as soon as possible your outline proposals for papers for the group, as well as organising themes. Naturally these should be related to the philosophy of mathematics education, understood in the broadest sense. (See Newsletter 1). Both theoretical and empirical papers are solicited.

There is a limited total number of places at the conference (around 300), and those wishing to be guaranteed a place should send a deposit of œ25 (Sterling), or the full residential congress fee of œ145, made out to JMC (BCME 91), to Marion Keeling, BCME Bookings, 7 Shaftesbury Street, Derby, DE3 8YB, U.K. In the event of cancellation before 31 April 1991, all but œ15 will be refunded.

Requests to present posters and individual research papers outside the group should be made at the same time as any conference booking.


Currently this consists of Leone Burton, Tony Cotton, Paul Ernest (Chair), Ruth Farwell, Christopher Knee, Steve Lerman, John Mason, Marilyn Nickson, Chris Ormell, David Pimm, Leo Rogers, Europe Singh, Ole Skovsmose, Dick Tahta, Chris Weeks, Davis Wells, but others have also been invited to join.


This has been proposed for the 7th International Congress of Mathematical Education, Quebec, August 16-23, 1992. The proposal has been acknowledged, but a final decision has not yet been notified. If accepted we will have 2 X 90 minute sessions. If rejected, we will move to organise an unprogrammed session or two. A possibility being explored is that of running a short satellite conference immediately following ICME, possibly in conjunction with another group such as TME.

For details and registration write to ICME-7 Congress, Universite Laval, Quebec, QC, Canada, G1K 7P4.

The Fifth International Conference on Theory of Mathematics Education (TME-5), is to be held in Northern Italy, June 22-27, 1991. The following are topics of concentration:

(I) The role of metaphors and metonymies in mathematics, mathematics education and the mathematics classroom.

(II) Social interaction and knowledge development - Vygotskyan perspectives on teaching and learning of mathematics in the construction zone.

The TME group have led in the cooperative study of theoretical issues in mathematics education, including philosophical aspects, chaired by Hans-Georg Steiner. TME is also offering a Topic Group for ICME-7, although without the philosophical theme mentioned in Newsletter 1. Currently avenues of cooperation between POME and TME are being explored.

For more information concerning TME-5 contact Prof. H-G Steiner, IDM, Universitat Bielefeld, Postfach 8640, 4800 Bielefeld 1, Federal Republic of Germany.

The Fifteenth International Conference on the Psychology of Mathematics Education (PME-15), is to be held in Assisi, Northern Italy, June 29 - July 4, 1991. Estimated costs are $400, and the organizers would welcome a reservation fee of $100 in 1990. Research papers must be submitted by 31 January 1991. The nearest airport is Fiumicino (Rome).

For further information contact Prof. Paolo Boero, University of Genova, Via L.B. Alberti 4, Genova 16132, Italy.

The Second International History and Philosophy of Science Teaching Conference will be held at Queens University, Kingston, Ontario, Canada, K7L 3N6, 11-15 May 1992. The chair of the local organising committee is Prof. Skip Hills, Faculty of Education.

Details and a call for papers are included in the International HPS and Teaching Group Newsletter, Volume 2 (October 1990), Editor Dr. Michael M. Matthews, School of Education, UNSW, Kensington, NSW 2033, Australia.

The newsletter carries an announcement for this (POME) group, as well as details of publications associated with the first conference. These include Synthese, Vol. 80(1), 1989, with 'Cognition, Construction of Knowledge, and Teaching' by Ernst von Glasersfeld, Studies in Philosophy and Education, Vol. 10(1), 1990, with 'Arithmetic and Geometry: Some Remarks on the Concept of Complementarity' by Michael Otte, and Science Education, Vol. 74(5), 1990, with 'Analogies for Philosophy and Sociology of Science for understanding Classroom Life' by Paul Cobb et al., and 'Constructive Perspectives on Mathematics and Science Learning' by Grayson Wheatley, to name but a few. These and other dedicated journal issues are available for $10 each (or even for $8, for 10 or more copies).

The conferences, publications and newsletter represent an admirable movement to interrelate History, Philosophy, Science and Education, paralleling (on a grander scale) what is being attempted in the present newsletter and network.


Philosophy of Mathematics Education discussion group at PME 14, in Mexico, July 15-20, led by Paul Ernest. A small group, notably including Hans-Georg Steiner, discussed alternative views of mathematics and their educational implications.

Philosophy of Mathematics Education discussion group at BSRLM, Bath (UK), October 20, led by Paul Ernest.

One of the central issues discussed was the significance of absolutist philosophies of mathematics, with their assumption that mathematical knowledge is certain, incorrigible, neutral, and with a unique and fixed hierarchical structure. Such a view of mathematics may be said to underpin the new British National Curriculum in mathematics. For this fixes the mathematics curriculum for children aged 5 to 16 years as a rigid hierarchical structure of Attainment Targets, made up of items of knowledge and skill at ten discrete levels. As well as indicating its assumed absolutist conception of mathematics and the curriculum, this design feature (with other indicators) betrays a conception of mathematical ability as fixed and distributed in terms of another hierarchy. It may be conjectured that these fixed hierarchical conceptions, including that of a rigidly stratified social order, serve a social aim. Notably the reproduction of social hierarchy and the associated distribution of privileges. Even if this is not intended, it is certainly likely to be one of the outcomes. For those children who attain the higher levels of the national curriculum will largely be those labelled as of high 'mathematical ability'. They frequently come from middle class backgrounds (and are often white and male), and are those most likely to achieve salaried professional occupations. Children labelled as low in 'mathematical ability' will be stuck at the lower, basic skills orientated levels of the National Curriculum, they are more often than not working class (or female, or minority students), and regularly end up amongst the lower wage earners.

There was a fair measure of support for this provocative view. A fuller account will be published in BSRLM Proceedings.

(For details of BSRLM, the UK counterpart of PME-NA, contact Dr. S. Pirie, University of Oxford, Mathematics Education Research Centre, 15 Norham Gardens, Oxford, OX2 6PY, U.K.)

Critical Mathematics Education Conference

Cornell University, N.Y., 13-14 October, 1990.

This was a small, international invitation conference consisting of informal presentations and intensive discussion on the following three themes and related questions.

Epistemology and Philosophy of Critical Mathematics Education

How do we see mathematics knowledge itself as problematic? What are the origins of mathematical knowledge? Whose knowledge is it and in whose interest? What meaning does that have for people: for social change? How do we deal with conflicts/contradictions in knowledge?

Mathematics in its Cultural context

How will we reconceptualize mathematics to incorporate non-Eurocentric views and include the historiography of how and when the Eurocentric view became 'standard'? How can we begin to speak about mathematics that we cannot recognize through Eurocentric Eyes? What are the effects of culture, language and ideology on the mathematics people develop?

Political, Economic and Social issues in mathematics Education

How is mathematics knowledge used to understand or obscure political, economic and social issues? What is the relationship between mathematics knowledge and power? What is emancipatory mathematics knowledge? What does it mean to empower students? What are the differences between awareness and indoctrination?

Key philosophical themes discussed at the conference included:

The Philosophy of Mathematics. There is a need for a radical view of mathematics as a social phenomenon - not an absolutist body of incorrigible knowledge - to provide the foundation for a view of mathematics as created by learners and indeed by all peoples.

The Nature of Ethnomathematics. Is it the study of the mathematical ideas of non-literate peoples, or does it include all socially-situated mathematical practices and activities beyond (or including?) the formal academic discipline of mathematics? The latter is the view most consistent with the following ideas.

Cultural Imperialism, Racism and Mathematics. Both Eurocentric histories of mathematics, and the dominance of white academic Western mathematics (with the concomittent invalidation of all else) is culturally imperialist and racist. Likewise, school practices, whether overt or covert, which reproduce the disadvantages of ethnic minority students are also racist.

Critical Mathematics Education (CME). The nature and the means of implementing CME were discussed, without losing sight of the antagonism and powerful reactionary forces it arouses. Two components of the foundations of CME were implicit in the discussions, as needing mutual development and interrelation:

1. The nature of mathematics, and its unique contribution to CME. In particular, the necessity for for a humanistic, fallibilistic or social constructivist philosophy of mathematics, as found in the work of Wittgenstein, Lakatos, Davis, Hersh, Kitcher, Putnam, Tymoczko, and others.

2. The nature of critical education and critical pedagogy, building on the work of Marcuse, and other critical theorists and the critical pedagogy of Freire, Giroux, Apple and others.

The aims of CME were elaborated, including the empowerment of all individual learners mathematically, irrespective of social background and disadvantage, to facilitate their personal and social development. (This necessitates respect for learners and their knowledge and the use of respectful and dialogical pedagogy). There is also the aim of raising the consciousness of all learners to look critically at the received structures of mathematical knowledge and society; to question them; and to consider more egalitarian and liberating alternatives.

An intended outcome of the conference is a network of critical mathematics educators, led by the conference organisers: Marilyn Frankenstein, Marty Hoffman, Arthur Powell and John Volmink. Readers sympathetic to the aims of CME and wishing to be involved can contact Marilyn Frankenstein, College of Community and Public Service, University of Massachusetts, Boston, MA 02125, USA.


Ernst von Glasersfeld writes in to say he too uses the chess analogy for the social basis of the certainty of mathematics. (In fact, the reference in Newsletter 1 was inspired by von Glasersfeld's contribution to the debate on constructivism at PME-11, Montreal, 1987!) In his paper A Constructivist Approach to Teaching (draft of a chapter in a forthcoming book on constructivism in mathematics education edited by Les Steffe) he draws on the chess analogy to argue for the force - the undeniable personal reality - of conventional rules, such as those of chess, which are acquired by social interaction. Geometrical forms too, and many other concepts, are acquired and elaborated on this basis. He argues that radical constructivists, notably Piaget, do attach great importance to social interaction, although in the past it has not been the primary focus of investigation. According to radical constructivism, the 'others' with whom social interaction takes place are part of the knowing subject's environment, that is 'viable' constructions made as part of the organisation of her experience. Consequently, it is claimed that a 'social constructionism' is not needed to remedy the lack of a social component in radical constructivism.

Heinrich Bauersfeld focusses on the differences between radical constructivism (RC) and a particular social theory, Activity Theory (AT). ('Activity Theory and Radical Constructivism: What do they have in common and how do they differ?' presented at Annual Meeting of the American Society for Cybernetics, Oslo, July, 1990). One contrast is that according to AT, objectively true knowledge, reflecting reality, is produced. This contradicts the view of RC, and, it must be said, the received view in the philosophy of science. Another contradiction is that between the view that internal activity does not rise above or separate from external activity (AT), and the view that novel structures are formed purely within the subject's mind (RC). Finally, elaborating on some of the discussion in the paper, there is the concept of 'collective subject' (AT), which is made up of individual subjects conceptualized on a par. RC seems unable to accord interchangeability across the domain of individual subjects, since one occupies a distinguished place.

David Johnson of Rutgers argues that although RC claims to be ontologically neutral, it denies that we can ever have knowledge of the world, if it exists ('Constructivism and the Transmission of Knowledge', Oslo Meeting, as above). Agreeing that we can never have certain knowledge of reality, he argues that we do not need to reject the possibility of some tentative knowledge of the world, which he terms 'pragmatic realism'. This allows him to upgrade "people from the status of phenomenal objects created by my consciousness of them to individual existences that are in fact presupposed by my consciousness" (p. 4).


A.J. (Sandy) Dawson, The Implications of the Work of Popper, Polya, and Lakatos for a Model of Mathematics Instruction, unpublished Ph.D. thesis, 1969, University of Alberta, 258 pp.

This is an early contribution to the philosophy of mathematics education, as the opening paragraph of the abstract indicates.

"The purpose of the study was to describe the instructional applications of a philosophically based model of mathematical inquiry to the teaching of mathematics. The first phase of the study developed the model of inquiry of mathematics based on the philosophical position known as Critical Fallibilism. In the second phase of the study, stratagems of teaching were derived from the Fallibilistic model of mathematical inquiry. This phase of the study also included an assessment of the Madison Project as a Fallibilistic approach to the teaching of mathematics."

A strength of this thesis is its extensive treatment of philosophical foundations, Eighty pages are devoted to critically reviewing philosophies of mathematics and endorsing the Lakatosian quasi-empiricist view. Beyond this, there is a systematic application of Lakatos' heuristic - his logic of proofs and refutations - to classroom inquiry (and indeed to a set of curriculum materials). This provides a foundation for process aspects of mathematics - mathematizing - in the classroom.

A weakness of the treatment, understandable given the date of the work, is the lack of elaboration of the broader implications of the position adopted, both philosophically and educationally. These include the nature of mathematics as a social construction, the decisive part of history, its cultural embeddedness and value-ladenness. Educationally, there is little recognition of the macro-social context and the part mathematics teaching plays in this, the aims of mathematics education and their social location, the role of teacher ideologies, the teacher's authority, the hidden curriculum, and so on.

This criticism (from the vantage point of twenty years of philosophical and educational development) notwithstanding, the thesis is in many ways still remarkably current in its concerns, and deserves a wider audience. No-one has yet to my knowledge developed more extensively or systematically the use of Lakatos' heuristic for the mathematics classroom. Furthermore, the narrowness of focus may be justified in terms of the depth of treatment.

Please contact the author to find out how copies may be obtained. Dr. A.J.Dawson, Faculty of Education, Simon Fraser University, Burnaby, British Columbia, V5A 1S6, Canada.

A published account of Sandy Dawson's work is the following:

Dawson, A.J. (1971) A fallibilistic model for instruction,

Journal of Structural Learning, Vol. 3(1), 1-19 No. 1,


Neil A. Pateman Teaching Mathematics - A Tantalising Enterprise: On the Nature of Mathematics and Mathematics Teaching, Deakin University, Geelong, Victoria 3217, Australia, 1989, 43pp.

This is a slim volume which discusses philosophies of mathematics, and relates them to the teaching of mathematics. The central chapter (of three) provides an overview of the philosophy of mathematics structured in terms of the crucial dichotomy: quasi-empiricism versus foundationalism. Whether couched in these terms or others, such as fallibilism versus absolutism, many currently working in this area would agree that this is the most fundamental distinction that needs to be made in the philosophy of mathematics. There is also a brief treatment of radical constructivism, Activity theory (based on Mellin-Olsen's account), and sociological accounts of mathematics. An introduction to all of these current theoretical ideas is very welcome, which are too often difficult to access.

However, the book should be read critically, for there are a number of deficiencies. There are some over-simple connections drawn between philosophies of mathematics and education, for example the section on intuitionism moves too rapidly to asserting the dangers of false intuitions in the classroom, which is quite another matter. The account fails to respect the distinctions between philosophy, psychology and sociology, and uses some terms ambiguously. For example, quasi-empiricism is used to argue that a sociology of mathematics may be possible. But this already has a long history, and I think what is intended is a social philosophy of mathematics. This is not merely terminological, for the account repeatedly slips unremarked from the concerns of one discipline to those of another. Finally, a weakness is the lack of reference to educational literature on the significance of the philosophy of mathematics, which a quick review of back issues of For the Learning of Mathematics would begin to supply.

Many of these deficiencies are perhaps inevitable in a short treatment of this nature. The book is to be welcomed for being one of the first to venture into the area, and it is a difficult and multi-disciplinary theme, not easy to treat clearly. It offers a readable and provocative introduction for readers who might not easily access these ideas otherwise. All in all, it is a valuable brief account, usable at the undergraduate level, for which it was written, provided it is read critically.