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Bernd Bekemeier, Martin Ohm (1792-1872): Universitiits und Sehulmathematik in der neuhumanistisehen Bildungsreform, Vandenhoeck & Ruprecht, G6ttingen, 1987; series Studien zur Wissenschafts-, Sozialund Bildungsgeschichte der Mathematik. xxiv + 326 pp. Martin Ohm is not what you would call a well known mathematician. Dieudonnr's Abrbgb d'histoire des mathbmatiques devotes a small page to him; characteristically, this is written by Dugac who is a professional historian of mathematics. Bourbaki's Elements d'histoire des mathdmatiques never mention him. It may be worthwhile to think a little about this absence. The reason i~ clear: Martin Ohm did not contribute any significant new result to mathenPatics. Still he should have drawn Bourbaki's attention for the similarity between his main work: the Versuch eines vollkommen consequenten Systems der Mathematik (Essay of a completely consequent system of mathematics) and Bourbaki's own Elements de Mathbmatique. After all, Ohm's book has been, as far as I know, the first attempt since Euclid to write down a logical exposition o f everything that was more or less basic in contemporary mathematics, starting from scratch. Moreover, Ohm has a completely formalist conception which contributed a good deal to misunderstandings with research mathematicians of his time. This must not of course make us forget the differences between these works, one of which, perhaps the most important, comes from the motivations of the two authors. Whereas Bourbaki wanted to give a completely reliable basis to mathematical research, Ohm found that a systematic exposition of mathematics was necessary to solve the difficulties that the teaching of mathematics met in early XIXth century Germany. For this reason, I think Bekemeier's book is of great value to everybody interested in the didactics of the discipline. Many of the points discussed in it sound very modern. Let me first give a very quick summary of the book. The first, and by far the largest, chapter is devoted to Martin Ohm's biography. The word biography is to be taken here in a very comprehensive sense including some indications of the historical context and a study o f the formation of Ohm's ideas, as precise as the available documents make it possible. Incidentally a non German reader might find it easier to understand if he has read the first Educational Studies in Mathematics 20: 469-474, 1989.

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26 pages of Lewis Pyenson's Neohumanism and the persistence of pure mathematics in Wilhelmian Germany (American Philosophical Society, Philadelphia 1983). So at the end of this chapter we are acquainted with Ohm's ideas and with the general lines of mathematical teaching in the Germany of his time. The second, more technical chapter is an analysis of the Versuch. The third chapter studies how Ohm's works relate to major trends in the (implicit and explicit) philosophy of mathematics. O f special interest are the controversies with Cauchy and Kummer. The fourth and final chapter tries to evaluate the importance of Martin Ohm for XIXth century mathematics. One can find evidence of his influence on the English algebraic school and on Grassmann. But his contribution is much more important for the teaching of mathematics; we shall return to this point. After the defeat of the Prussian army at Jena (1806) a group of neohumanistic reformers achieved a leading position in the government. Influenced by the philosophy of Enlightenment but rejecting a large part of it, they were very conscious of the necessity of reforming the administration (largely on the French model) and of creating moral support for the state. For this they relied very much on education (Bildung) based on a number of ideas. The word formal played a key role in these. The phrase formale Bildung came to mean education as aimed at the capacity to understand, elaborate and report, but also at a moral formation. O f course, this could be acquired only through the study of some content. Classical languages were this privileged subject of study. The ambition was to give the students in the Gymnasien a command of these languages good enough for them to enjoy Latin and Greek literatures, identify with their heroes and be penetrated by the values of the civilizations of which the German nation was thought to be the heir. Many a leading actor of the Prussian reform has been a scholar in ancient literatures. So the formal (in the usual meaning) study of Latin and Greek was the core of this type of education. This is the point where the ideas of a number of mathematics teachers, Ohm being the most important of them, came in. They pointed out that, if developed systematically, the learning of mathematics could contribute as well as Latin and Greek to this aspect of education. Ohm often insisted on the similarities between the teaching of mathematics and that of a language. For mathematics to play this role - and as a consequence for mathematicians and teachers of mathematics to get prestige comparable to one of their colleagues in the humanities - two conditions had to be fulfilled. The first was to have well trained teachers. The second was to have a body of knowledge systematically derived from a few basic concepts and ideas. This was also necessary in order to teach the teachers. Euclid's system was

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available for geometry. It did not cover some modern material and had other drawbacks, but at least it was there. Nothing of that kind existed for arithmetic, algebra, etc., the fields Ohm termed theory of numbers, or, more properly as we shall see, Zahlenzeichenlehre, the theory of symbols of numbers, as distinguished from the theory of magnitudes i.e., geometry. Starting when he was in his twenties, Martin Ohm worked to have these two conditions fulfilled. He did it by his own teaching, both to " n o r m a l " students and to would-be teachers, he did it by his organizational propositions to the governments o f Bavaria and Prussia, he did it by writing articles and textbooks and, concerning the second condition, he did it by writing the Versuch. This was to consist of ten parts; the first was published in 1822 and nine in all during his lifetime; some had two or three editions. The general ideas on which Ohm's system is built seem surprisingly modern. One of them is: start from the natural numbers and the operations on them. Then, use the signs o f operations to build more and more general expressions. These expressions have no definite meaning, but the equality between two o f them is defined, so that relations can be derived. The fact that these relations are true when natural numbers are substituted for the operands and the results are natural numbers again will warrant that they are contradiction free; this is as much as can be required for mathematical truth. These ideas were in flat contradiction with what practically every contemporary mathematician thought. For one thing, if a mathematical statement had obviously to be non-contradictory, this was not the criterion, let alone the only criterion, of truth. This point was an aspect o f the deeper question of the object of mathematics, which was magnitudes. Not for Ohm: he considered the symbols (Zeichen) to be the real object of the science. As a consequence of this assertion, the dividing line between pure and applied mathematics was shifted a long way. Computations with fractions for instance, were, in Ohm's view, considered as an application as soon as they bore on definite, and not symbolic, numbers. Incidentally, when Ohm applied for a professorship at the University of Berlin, he was blamed for not knowing enough of applied mathematics. Be that as it may, he had not the slightest contempt for applications. For one thing, he had the greatest admiration for his brother Georg Simon thanks to whom everybody has heard the family name in connection with the law of electric currents. And this admiration was not a matter of course, as Georg Simon's article Die galvanische Kette mathematisch bearbeitet drew hardly any attention from physicists for ten years. Notice also from the title that both brothers considered the discovery o f what was later to be called Ohm's law as a piece of applied mathematics. So this was seen by them as an all-invading field of

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knowledge. But there is more to it. To the objection that his conception of mathematics as a purely formal work with symbols made it meaningless and without object, Martin Ohm answered that the content and meaning of it was given by the whole of its applications. Bekemeier has not been able to make entirely clear how Ohm related the Versuch to his pedagogy. Pedagogical questions were widely discussed among teachers, government officials and philosophers of early XIXth century Germany. Some thought, as did exponents of the Enlightenment, that the scientific exposition of a field had to be used such as it was for its teaching. Martin Ohm was not one of them and his textbooks use the Versuch as a guideline, not as a pattern. What remains unknown is whether he considered it as the exposition of mathematics or only as one of them, the one which was suited to the education of teachers. One point is quite clear: Martin Ohm was a very successful teacher. On this, both the supporters and the opponents of his system agree. The supporters quite naturally argued that his success as a teacher granted the quality of his books. But the opponents as soundly pointed out that it took all the exceptional talent of the author as a teacher to make his method work. To understand mathematics as Ohm conceived it one needs of course very strong powers o f abstraction. For him, one of the objectives of instruction in mathematics, perhaps the most important, wh~ precisely to form this power o f abstraction. He was also very conscious that it was not there at the start. Whereas geometry had no place in his system of pure mathematics, he valued it very highly as a matter of teaching, especially in the lower forms. Here we find again his conception of the meaning and content of mathematics. He thought that these must not interfere with the logical development of the science but he was very much aware that they were needed for pupils to understand that development, as long as they did not have a very good grasp of it. As a science of magnitudes using the mathematical method, geometry filled that need very properly, allowing them to use mathematical reasoning in situations they had an intuitive control of. For similar reasons, he always opposed starting the teaching o f mathematics at too early an age. Reckoning was to be taught in a coherent way, not contradicting the mathematical instruction that some of the pupils would receive later. But it should be very clearly separated from that instruction and not taught as a preparation for it. Ohm's principle of teaching mathematics as a science bearing on symbols of course reminds us of "new math". This is of importance because he was not an isolated and unfollowed thinker. In a part of his book which particularly interested me, Bekemeier analyzes textbooks by authors who

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shared his ideas. Some of them were widely used for decades. So we have here a phenomenon that lasted longer than "new math" which seems to be on the decline. I want to point out one perspective from which the two phenomena should be compared, that of activity. Remember (if you are old enough) that in the beginning "new math" relied very much on the pupils' activity, its exponents referring usually to Piaget for a link between the changes of content and of method. Characteristically, Agir pour abstraire was the title of a doctorate thesis defended in 1973 (Nicole Picard, published by OCDL, Paris 1976). Now the neohumanist reformers whose ideas Martin Ohm shared also gave a tremendous importance to activity (Tiitigkeit). I am a little pedantic here in giving the German word for which there is no problem of translation, but this is to emphasize that it had for them complex and far reaching connotations referring on one side to a key notion of German idealist philosophy and on the other to the pedagogy of Pestalozzi (though the Ohm brothers found it very inadequate for mathematics). Activity could be activity of the mind; typically for Ohm an activity was for the student to derive equalities between symbolic expressions. Thinking about the comparison between Ohm's system and "new math" seems to me quite worthwhile and leads me to the point which I consider most important about Bekemeier's book. I mean the importance of the context for the teaching of mathematics. Many deny it, many others agree, but very few study it with any precision. Bekemeier shows how the general political and social context contributed to the birth of Ohm's ideas. He indicates their relation to the general features of the educational system and studies their link to the philosophy prevailing at the time. It is fair to mention parenthetically that Bekemeier relied on a number of studies by scholars of the Institut fiir Didaktik der Mathematik of Bielefeld (Niels Jahnke, Michael Otte, Bernd Schminnes, Gert Schubring, you'll find the references in the bibliography). All these contextual aspects were quite different in the time of "new math" and they are still changing now. A very significant difference lies in the fact that mathematics teachers nowadays lack the very strong general education they had in XIXth century Germany. Another difference lies in the educational context. Clearly, a very formal teaching of mathematics has a much better chance of success in a situation where the work in key disciplines, such as languages, is formal anyway. It is a difficult task to study these links between the teaching of mathematics and the material and intellectual life of the environing society and Bekemeier gives an excellent such study. It is still much more difficult to try a similar study for nowadays, but it is indispensable. You can make

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very serious studies of the image students possess of such or such mathematical concept, think out and experiment teaching situations very critically. If you are not conscious of the context (including the context of your own philosophy), the context will modify the outcome o f your work and you run a great risk it'll do it in a way you won't like. I could illustrate this by a few remarks on the recent reform of teaching in France, but I fear this would lead us a bit too far afield. Let me only say, coming back to a question which was central for Martin Ohm, that failure is likely and its most important factor is the lack of training of teachers. I have in mind their lack of methodological knowledge and of general culture. In the context of a reform which claims to change the spirit of teaching, hardly anything is done to make them aware that mathematics is taught in the framework of some philosophy, whether the teacher knows it or not. Studying Martin Ohm can help us think about problems which are indeed very much those of today.

LM.S.P., Mathbmatiques, Universitb de Nice, Parc Valrose, F06034 Nice Cedex, France

MARTIN ZERNER

Alan J. Bishop: Mathematical Enculturation: A Cultural Perspective on Mathematics Education, Dordrecht, Kluwer Academic, 1988, 195 pp., Dfl. 130.00/$69.00/£39.50. Alan Bishop's book is written in a clear style that is a pleasant mix of personal opinion and a wealth of information about research in the field. His use of the first person is effective as a means of presenting an informed, humanistic account of education in a field traditionally perceived as objective and dissociated from cultural concerns. The book contains a useful bibliography and a set of notes associated with each chapter. The book begins boldly with the following quote from Keddie (1971): It would be the failure of high-ability pupils to question what they are taught in schools that contributes in large measure to their educational achievement. (p. 1)

Bishop casts 'a critical eye over what happens in the majority of mathematics teaching situations at present' (p. 7). He discusses four major areas of Educational Studies in Mathematics 20: 474-480, 1989.

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concern about educational practice with reference to his stated goal of relating children with their mathematical culture. These are: technique oriented curricula, impersonal learning, text teaching and the false assumptions about educational activity that underlie these practices. He is particularly concerned about systematised 'top down' approaches to mathematics education. The book focusses attention on the relationship between technological development and mathematical growth in cultures. White's (1959) book The Evolution of Culture is acknowledged as an influential source of a deterministic view that technological development shapes ethical, philosophical and sentimental aspects of a culture and a view of mathematics as essentially a 'symbolic technology' (p. 18). Culture is 'objectivised' throughout most of the book. Possibly as a result of this, an assumption that technology is experienced in the same way by all members of a culture is implicit in the discussion. Less attention is paid to the relationship between culturally learned world views and technological development. Bishop clearly defines the variant o f 'Mathematics' that forms the focus of his book as that exemplified by Kline's Mathematics in Western Culture. The 'Mathematical culture' described by Bishop assumes a Cartesian world view is equally evident and desirable to all. He describes mathematics as a cultural phenomenon that 'transcends societal boundaries in the same way that music does'. He states: When we talk of the power of mathematical method, we don't see this power as being seen within narrow societal boundaries, we imagine it being visible anywhere and everywhere. (p. 16) A case is presented for consideration of six cultural activities that occur in many different cultures and are associated with mathematical development. These are: counting, locating, measuring, designing, playing and explaining. Examples o f these activities from the cross cultural research in less technologically centred cultures are provided as evidence of the pan-cultural nature of mathematical development. Cultural variations in the significance attached to these activities are mentioned. However, Bishop argues that the activities occur in some form in all cultures. An interesting and technocentric account of 'design' is given with reference to industrialised nations. Bishop suggests that 'most of us now live in societies which are more and more designed and which are more and more dependent on technological development. We could indeed think o f modern industrialised society as being based on Mathematico-technological culture' (p. 59). Bishop argues that, in addition to its function as a symbolic technology,

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Mathematics 'is also a bearer of, and product of, certain values' (p. 59). He chooses to restrict discussion to three 'complementary' sets of values. These are: rationalism and 'objectism', control and progress, and, openness and mystery. A dualistic approach is taken to the discussion of these values in what is perhaps the least satisfying section of the book. The values are generally discussed in an unproblematic way. In keeping with the structure of the rest of the book, alternative views are acknowledged and expanded in the notes. In the discussion of 'objectism' Bishop asserts that 'Mathematics favours an objective rather than a subjective view of reality' (p. 66). An absolute notion of 'objective' reality is implicit in the discussion that follows the above statement. The less absolute views of a socially constructed reality proposed by Vygotsky and other educational theorists of constructivist persuasion are not acknowledged. In contrast to Walkerdine (1988) in The Mastery of Reason, Bishop chooses not to discuss the social dynamics of the development of the social and language skills necessary for rational argument. Rationalism is presented as a powerful mode of thinking that should be encouraged. He specifically rejects suggestions by Cole and Bruner that Mathematics (and in particular mathematical reasoning) is a phenomenon of middle class culture (p. 18). He acknowledges that 'it is not always socially acceptable to be logical, precise, critical and argumentative' (p. 77). He does not mention who sets the rules for 'socially acceptable' behaviour or if the same rules apply to all members of a society. The implications of his statement for socio-cultural differences in access to and experience of rational argument within a culture are not discussed. Bishop describes a 'preferred world view of 'atoms' and objects' (p. 65) that are connected by ideas. Alternative world views from ancient times (e.g. Heroclitus) are cursorily dismissed. More importantly, questions raised recently by some scientists, particularly physicists, about the efficacy of the Cartesian mind/matter (rationalism/objectivism) duality in the light of a changing, less fragmented, view of the environment are not mentioned. It is in relation to the values of 'control' and 'progress' that the objectivism of Bishop's argument leads to an oversimplified analysis. These values, in particular, are experienced differently by individuals in Western cultures according to their status and gender. Feelings of control are discussed exclusively from the emotional perspective of 'controllers'. Also, the metaphors used in relation to power and control suggest a masculine viewpoint. Although the increased use of technology (and Mathematics) for social control is acknowledged, the perspective of the 'controlled' is not

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presented. Similarly, at societal and personal levels, technological and Mathematical 'progress' are discussed as though they are equally beneficial for all. However, in a Mathematical culture, the 'progress' of some implies a loss of status for others. Further, technological progress, such as automation, has immediate and drastic economic consequences for the families of many children attending school. This is a major issue in public education since attitudes and values are derived from cultural experience. An acknowledgement of feminine points of view and the socio-economic differences in experience of 'control' and 'progress' would have enhanced the discussion. Educators will need to deal with these issues, as well as the 'destructive potential' (p. 74) of these values for the environment, if Mathematical knowledge is indeed to be more 'open' to all. The second half of the book elaborates on Mathematical enculturation as an educational process. The hierarchical model of cultural levels proposed by Davies (i.e. the technical, formal and informal) is used as a basis of discussion of the educational needs of children. The elaborated descriptions of the three levels with respect to a mathematical sub-culture are interesting. The informal level of mathematical culture appears to approximate to ethnomathematics - the informal mathematical knowledge associated with everyday cultural activities. The formal level is associated with mathematical applications in professional and industrial fields. For example, the mathematics associated with the work of architects, engineers and cartographers. The technical level of the mathematical culture is associated with the developmental work of mathematicians. Bishop locates mathematical enculturation by schools at the formal level. He notes that many cultural activities (e.g. advertising techniques) at the informal level are likely to undermine the educational process in mathematics. Also that people working at the technical level of the culture (that is, research mathematicians) are less influential in the educational process than before. Bishop suggests that a shift in emphasis is necessary which allows 'rationalism to be stressed more than objectism, where progress can be emphasised more than control and where openness can be more significant than mystery' (p. 95). That is, a shift to a curriculum that is more representative of the values of 'Mathematical culture'. A comprehensive guide to the development of a cultural approach to the mathematics curriculum, based on the following five principles, is given: it should represent the Mathematical culture, in terms of both symbolic technology and values. it should objectify the formal level of that culture. - it should be accessible to all children.

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it should emphasise Mathematics as explanatiori. it should be relatively broad and elementary rather than narrow and demanding in its conception. (p. 98) He describes a curriculum structure consisting of three 'separate, yet overlapping and interacting' components which is consistent with the above principles. A 'symbolic component', a 'societal component' and a 'cultural component' would each be associated with different learning experiences and stress different values. For example, the Societal component exemplifies 'society's manifold uses of Mathematical explanations, and the principal values of 'control' and 'progress' which have developed with these uses' (p. 98). The content and processes of the curriculum are elaborated in some detail. The suggested relationships between the age of students and the types of learning that should be provided in the curriculum seem a little contentious in the light of recent research at the Early Childhood level. For example, early results of the C A N project in England suggest that very young children are capable of a form of mathematical investigation. Bishop chooses not to specify assessment procedures for students. He suggests that from 'the perspective of enculturation, assessment is unnecessary since enculturation is not something you pass or fail, nor is it something you are better at than someone else!' (p. 120). Bishop's reluctance to discuss assessment of such curriculum outcomes as attitudes to the 'Mathematical culture' and capabilities for mathematical reasoning and investigation, is quite understandable. Efforts to measure such capabilities have in the past been less than satisfactory. However, a detailed assessment of the educational outcomes for students of various kinds seems a necessary part of any educational innovation that aims to be accessible to all. This seems especially important in the case of the proposed curriculum for at least two reasons. Assessments of cultural identity and status form an integral, if covert, part of all social interaction. Mathematics education occurs in a social context and concerns a socially powerful form of knowlege (at least in technologically centred societies). It would seem inconsistent if these forms of assessment were not overt in an enculturating curriculum which emphasised openness. The last part of the b o o k is devoted to a discussion of the process of Mathematical enculturation - 'a process whereby concepts, meanings, processes and values are shaped according to certain criteria. It is therefore an intentional process of shaping ideas' (p. 124). Issues associated with the cultural imperialism that is implicit in the process are not discussed. Bishop argues for a change in mathematics education and suggests that there is a need to 'move away from the impersonal, instrumental and mechanistic

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ideas which dominate at present' (p. 125). He describes the process of enculturation as 'a certain kind of dynamic relationship between the constructing, idea-providing, adaptive learner, and the pressurising, encouraging, restricting or freeing social environment' (p. 127). The asymmetric nature of the relationship between teacher and learner is stressed and the 'legitimate' use of teacher power and influence is discussed. A teaching role as a facilitator of collaborative and constructive engagement among learners is suggested. Transcripts from research provide clear illustrations of the forms of discourse that are envisaged. Perhaps because of the hierarchical nature of the proposed teacher/student relationship, suggestions for teaching and teacher intentions are elaborated rather more than the learning process or the needs and goals of the learners. A detailed analysis of the implications of his curriculum proposal for teacher education and some suggested criteria for selection of 'Mathematics enculturators' are provided. Bishop suggests that the 'first important criterion that mathematical enculturators should therefore meet is that they should personify the Mathematical culture, in terms of representing both the symbolic technology and the values' (p. 165). This book is an informed, extremely rational and objective account of some aspects of enculturation and educational activity in the field of mathematics. I would recommend the book to all interested in mathematics education and curriculum design. However, in some instances the level of objectivity is not entirely convincing, particularly in relation to the 'power' and 'control'. For example, Bishop makes the following statements about the aims of the curriculum process: to enable the learner to understand more about how society is 'controlled' by Mathematics. (p. 145) and The project-environment I have in mind will treat Mathematical power and control as problematic. (p. 146) However, he avoids dealing directly with the highly personal and asymmetrical nature of power relationships in any society. Values, of the type he discusses, are commonly held by users of powerful knowledge of any kind. A richer account of the changing social dynamics of the use of mathematical knowledge in technologically centred societies may have provided further support for his claims about the supra-societal nature of the power of mathematical method and the values associated with it. Similarly, the 'Mathematico-technological culture' at the informal level is discussed as though 'we' all participate in it in an undifferentiated fashion.

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Bishop has chosen not to address issues associated with differences in cultural experience (for example, those associated with gender and socioeconomic status) both inside and outside the 'formal' education system. My own research suggests that socio-cultural differences in experience of information technologies are substantial. These qualitative differences in experience appear to affect student perceptions of whether they control or are controlled by technology. It seems necessary to acknowledge the extent that such differences in experience shape values and mathematical development if an 'enculturating curriculum' is to be equally accessible and relevant to all children. Nevertheless, a curriculum, such as the one envisaged by Bishop, which makes the values and processes associated with the use of mathematics explicit and problematic, has the potential to provide all students with knowledge of where, when and why mathematics is used in society as well as know-how about mathematical procedures. Also, experience of such a curriculum may enhance the capacity of some students, in the future, to adapt the 'symbolic technology' in the light of a changing view of the world and the place of humanity within it.

Faculty of Education The University of Sydney, NSW, 2006 Australia

KATHRYN CRAWFORD