Ó Springer 2008

Metascience (2008) 17:439–443 DOI 10.1007/s11016-008-9206-1

REVIEW

A LESSON IN INCOHERENCE

Karen Franc¸ois and Jean Paul Van Bendegem (eds), Philosophical Dimensions in Mathematics Education. Mathematics Education Library, Volume 42. New York: Springer. Pp. 240. £54.00 HB

By John Monaghan and John Threlfall Philosophical Dimensions in Mathematics Education is Volume 42 in Springer’s respected series Mathematics Education Library. It is a loosely edited collection of nine essays with linking material, all purportedly treating philosophical dimensions Ôin’ mathematics education. This review is in three parts. Part 1 summarises each chapter, and considers how they are educational/philosophical. Part 2 considers one of the chapters in greater depth, as an illustration of the problems that we found with the book. Part 3 makes some further observations about the book and its origins.

PART 1 Chapter 1, despite its title, ÔThe Untouchable and Frightening Status of Mathematics’, is a rambling consideration of differences between vocational and general education, which uses curriculum statements as its evidence base. Its claim to being philosophical appears to lie in its consideration of Ôphilosophy of mathematics’ contained in attainment targets (see also Part 2). Chapter 2, ÔPhilosophical Reflections in Mathematics Classrooms’, is more coherent and considers the extent to which philosophy can and should be part of what is considered during mathematics lessons, looking for strategies for initiating philosophical reflections for school students. In Chapter 3, ÔIntegrating the Philosophy of Mathematics in Teacher Training Courses’, the author presents units from the course he teaches which deal with the philosophy of mathematics.

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These units are there because ‘‘philosophy of mathematics per se has to be considered an indispensable component of teachers’ professional training’’ (p. 63) but the author does not subject this statement to philosophical analysis. Chapter 4, ÔLearning Concepts Through the History of Mathematics’, makes a case for an historical coverage of mathematical concepts as a means of alerting students to ontological and epistemological issues in mathematics. This is an interesting and unusual take on using history in mathematics teaching. The chapter is also interesting for the absence of any consideration of the learner in this endeavour. Chapter 5, ÔThe Meaning and Understanding of Mathematics’, summarises interpretations of probability and offers an explanation for misconceptions in probabilistic reasoning through an account of Ôsemiotic conflicts’. It probably counts as philosophical because it considers Popper’s propensity interpretation and Carnap’s degree of confirmation and because it uses the word Ômeaning’ a lot. Chapter 6, ÔThe Formalist Tradition as an Obstacle to Stochastic Reasoning’, is an impassioned plea (the word Ôshould’ occurs nine times in Section 5.1.4) to create a statistics curriculum on the basis of students’ intuitions and essential statistical ideas. Its link to philosophy is not clear to us but there are references to some philosophical sources. Chapter 7, ÔLogic and Intuition in Mathematics and Mathematical Education’, and Chapter 8, ÔA Place for Education in the Contemporary Philosophy of Mathematics’, are philosophical in as much as they refer to famous mathematicians and philosophers who have written about the foundations of mathematics. Chapter 7 addresses the invention/discovery debate as a lead in to the central place of intuition in mathematics. Universal Darwinism is invoked to conclude that proofs are ‘‘always partly an invention and partly a discovery’’ (p. 169) but the application of Darwin’s theory to mathematics is not subjected to philosophical inquiry. Chapter 8 considers themes from Lakatos and his interpreters and reaches the conclusion that ‘‘Lakatos did not bring about a radical break with past logical approaches’’ (p. 206). Both chapters mention Ôstudents’ and Ôeducation’, but neither students nor education are integral to either chapter. Chapter 9, ÔEthnomathematics in Practice’, begins with a consideration of the distinction (previously proposed by Alan Bishop of Monash University, Melbourne) between ÔMathematics’ as a scientific discipline and Ômathematics’ as a set of skills and procedures used in

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everyday life. The main section is an account of projects with Navajo Indians and with Turkish minorities in Belgium. There is certainly potential for philosophical approaches to issues of culture and knowledge in these projects but this potential was not realised in this chapter.

PART 2 The weaknesses in this book are illustrated by Chapter 1, written by the principal editor. The differences between vocational and general mathematics education are pursued by examining curriculum statements. These are summarised in tabular form with statements such as ‘‘one can see that there is no room for philosophy in vocational education’’ (p. 16). The focus of mathematics in vocational education is then linked to the distinction between ÔMathematics’ and Ômathematics’ and this distinction in linked to a criticism of mathematics education in Flanders, which is unsupported either by evidence or argument. The chapter moves on to a short review of ethnomathematics before finishing with rhetoric about what is desirable. Naively, we expected attention to argument and evidence in a chapter claiming to be philosophical. We found no way to verify tabular summaries of curriculum statements. The other main source of evidence was citing the literature, but we found this to be uncritical, e.g. ‘‘Based on the theoretical framework of . . . we will argue’’ (p. 22). With regard to reasoning we were confused by the apparent argument that: Mathematics is presented as if there exists one and only one mathematical system that is made up of absolute truths and certain knowledge. The deductive method provides the warrant for the assertion of mathematical knowledge. Hence the curriculum is strongly directed towards the performance of techniques (pp. 25–26).

Is this ‘‘hence’’ a logical Ôhence’? If so, then we cannot see the logic. Further to this what are ‘‘techniques’’? Just boring old procedures and algorithms? No, this is surface analysis. School-based techniques in mathematics are complex things and in recent years mathematics educators have explored the pragmatic (efficiency) and epistemic values of techniques; but the author does not appear to know about this important work.

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PART 3 As academics we are told that we should Ôadvance knowledge’, and two of the many ways to do this are to write academic conference papers and to write academic journal or book papers. In our experience there is a transformation of knowledge from conference paper to journal/book paper. Academic journal/book papers often start their printed life as conference papers (sometimes several papers, and prior to that as ideas presented in seminars). Seminars and conference papers allow us to sort out ideas. They form part of the development and the advancement of knowledge. Sometimes a conference has a theme, and in such conferences people present often disparate ideas. Proceedings of such conferences can be very useful in terms of disseminating ideas that like (and unlike) minded people have. But when we come to turn our conference papers into journal/book papers we refine our ideas to the best of our abilities, with peer reviewing adding another level of refinement. The expectation when reading an academic book is of this more refined knowledge, transformed from its first manifestation into something more rigorous, coherent and consistent. The chapters in this book started life as conference papers, a conference that had the central question ‘‘Is there room for a philosophy of mathematics in school practice?’’. One can imagine this question being interpreted in a host of different ways, and the contributions of the different chapters of the book, each presumably a development of a paper presented at the conference, represents a good proportion of that potential variability. However, in our view they have not been sufficiently transformed by further attention. Our central problem with this book is that they remain, to us, conference papers. The challenge of editing a book with a disparate range of contributions is to add the coherence that the diversity of individual interpretation undermines. The editorial commentary on the book, contained in a number of interludes that alternate with the chapters of the book, and are found also in the prelude, is written with that in mind; but we do not believe that it succeeds. If this book had remained Ôproceedings’ then we would have come to it with different expectations, including a lack of coherence. But this is Volume 42 of Springer’s Mathematics Education Library, which with the title and the presentation led us to expectations of coherence

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and philosophical insight into issues in mathematics education that were not met. To conclude, we do not see what a philosopher might learn about mathematics education from this book, nor what a mathematics educator might learn about philosophy, nor what anyone else might learn about the multifaceted relationship between mathematics, philosophy, and mathematics education. Centre for Studies in Science and Mathematics Education, School of Education University of Leeds Leeds, UK