Radical 
REVIEWS Ethnomathematics  challenging eurocentrism in mathematics education Arthur B. Powell and Marilyn Frankenstein (Eds.) (1997), 440 pages, ISBN 079143351X,
Review by Janet Shapiro Introduction I taught mathematics in school for several years, and found myself combating the rather discouraging approach used in mathematics education. I wanted people to acknowledge the mathematical skills applied by necessity in many practical situations. So, I welcome this publication which confirms my own convictions, and I am very glad to have the opportunity to review it. After all, UNESCO designated the year 2000 as the International Year of Mathematics, and Maths Year 2000 is a DfEE (Department of Education and Employment) funded initiative. While academic in style, the book includes practical experience. Many of the chapters describe teachers learning from their pupils' responses, gaining insight into how cultural background and priorities of purpose can influence general problem solving strategies. Frequently the message is that the teacher must have awareness and sensitivity to appreciate the train of thought used by a pupil. There are frequent reports of otherwise illiterate people demonstrating innate mathematical ability. Structure However, the book suffers from being a collection of academic papers, each having a long list of citations, which inevitably carries a great deal of repetition. Many pages are devoted to polemical battles, in which the authors refute previous work with a Eurocentrist or elitist bias. The chapters are arranged in six sections, but section headings are not emphasised throughout the text. After reading through the chapters a second time I found that these thematic headings helped me to understand the thesis better. In my own words these themes were:
The contributors come from a wide range of academic disciplines which include: mathematics, chemistry, physics, economic and social science, cognitive psychology, anthropology, philosophy, and political science and politics. A concise and coherent statement is difficult to achieve in a compilation of chapters from so many sources, but the authors are in such agreement that the main force of the argument is made clear. However, it is unlikely that a reader would read the book from cover to cover, and most people will pick and choose between the articles. Ethnomathematics The term 'Ethnomathematics' was first coined by the Brazilian Professor Ubiratan D'Ambrosio in the 1970s out of the necessity to take better account of other cultural groups, usually subordinate ones, for improved communication, respect and understanding. Marcelo Borba gives a definition of 'Ethnomathematical Knowledge' in Chapter 12. Recognising that culture has an influence on our basic mathematical concepts, resulting in completely different ways of approaching counting, ordering, sorting, measuring, inferring, classifying and modelling, he defines 'ethnomathematics' as 'mathematical knowledge expressed in the language code of a given sociological group' so that 'even the mathematics produced by professional mathematicians can be seen as a form of ethnomathematics'. The acceptance of this concept engenders a sense of humility and an ability to think flexibly. It allows one to interpret responses from people with a different cultural background with proper respect. It is consistent to look afresh at the assumptions made about the origins of mathematics in Greek culture rather than in Africa or Asia, and to question the assumption that pure mathematics is of a higher level than technological or craft mathematics. These themes are discussed throughout the book. I like to think that the book has helped me to understand how higher mathematics, rather like literacy, tends to be the reserve of the ruling classes reinforcing a classbased society, and how this tendency has not been contradicted within the national curriculum. Content The book is packed throughout with accounts, some anecdotal and some detailed, of how mathematical concepts and techniques are developed differently in other cultures. I have made a selection from the eighteen chapters and six editorial papers to give a flavour of the content. The editors dismiss the myth of the supremacy of Greek mathematics and the associated elevation of 'pure' abstract mathematics over practical technological mathematics. The first few papers give some instances of the historical origins from Africa and Asia, and some discoveries of remarkable skills passed on by oral tradition, such as the complex navigation system adopted by peoples of the Caroline Islands which enabled deadreckoning navigation over hundreds of miles of ocean. The reference is made to women being one of the subordinate groups whose mathematical voice should be heard, but there is no detailed examination of gender and mathematics. Three papers in section three deal with culture and mathematical knowledge. Herbert Ginsburg points out that standard scholastic measures do not give a reliable indication of a child's mathematical cognitive skills, rather the methods indicate a cultural divide. Brian Martin dismisses the idea that mathematical methods are neutral, and shows that mathematics, like science, is connected with social interests. Dirk Struik describes how Karl Marx found time to reconcile how differential calculus was presented in text books with his own expectation of mathematical rigour. Marx uncovered considerable differences of approach and much confusion, classifying them as 'the mystical', 'the rational' and 'the algebraic'. His evaluations and conclusions would have been in sympathy with contemporaneous mathematicians had he been in contact with them, but his work illustrates how a mathematical technique may be demonstrably useful and developed independently from several different philosophies.(1) Several papers in section four deal with the mismatch between mathematics as practised in real life and as studied in school, leaving many groups, including girls and workingclass boys, labelled as poorachievers. Mary Harris presents the complex traditional craft activities carried out by women as intrinsically mathematical, and Paulus Gerdes reads complex geometry into the African craftwork of weaving. The traditional ways of working produce optimal means of production. Section five deals with the role of dialogue and the holistic approach. Munir Fasheh describes his teaching in Palestine and his use of searching questions and class discussions about problems contemporary to the class. He justifies his pragmatic approach by quoting Einstein's remark 'As far as the laws of math relate to reality, they are not certain; as far as they are certain, they do not refer to reality.' S. E. Anderson attacks the damaging effect of teaching mathematics in isolation as an abstract and elitist subject. He recommends an active, interactive, crossdisciplinary approach, which enables the students to learn more of the nonwhite, nonwestern origins of mathematics. Claudia Zaslovsky, an expert on mathematical practice, including games and crafts of the indigenous people of Africa and the Americas brings her experience into the classroom. She observes a crossdisciplinary transference of achievement between cultural studies and mathematics for minority groups. I have heard teachers remark upon a similar association between musical activity and achievement in science and mathematics. In section six it is recognised that more research is needed. Paulus Gerdes presents a wealth of studies of the mathematics practised in traditional fashion, some involving probabilities, without any contact with standard mathematics. Rik Pinxton has identified a completely different conceptualisation of the world which is the way of thinking for the Navajo Indians, living in a reservation in the USA (SW). The Navajo Indians adopt a holistic approach, in which process and flux take the place of our primarily reductionist and stationary model of the world. Their way of thinking is echoed in dynamic topology and catastrophe theory. Conclusion The book is complex because many connecting themes are carried along together. Several contributors combat fragmentation and favour a holistic approach. Others combat racism and narrowminded prejudice in all its forms, and are influenced by such educators as Paulo Freire. All of them want to engender a reevaluation of nonwestern cultural achievements. I would recommend those engaged in mathematics education to study this text. However, to publicise the ideas more widely, a simpler book is needed. In the era of the 'common curriculum' we need to widen the debate and to critically examine the standard approach used in mathematics teaching. Such schemes as 'Impact', organised by the University of North London and established in 1985, involve pupils and parents and inevitably encourage more interactive engagement. There must be other such initiatives, which have sprung up in spite of the current mania for central control, and it would be good to hear about them. NOTES Janet Shapiro

