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B. Christiansen, A. G. Howson and M. Otte (eds.): Perspectives in M a t h e matics Education, Mathematics Education Library, D. Reidel, 1986, ISBN 90-277-1929-2 (hbk.) Dfl. 172,-, $69.00, s and ISBN 90-277-2118-1 (pbk.) Dfl.74,-, $29.50, s 371 pages. This b o o k is the product of a series of meetings of the members of the international research group " B A C O M E T " (BAsic COmponents of Mathematics Education for Teachers). The founder members (Christiansen, Howson, and Otte) started meeting in 1978, and this was followed by conferences of a group of 15 members between 1980 and 1984 (the group is still meeting). The three founders define a "Basic C o m p o n e n t " by the three following points: It would be an aspect of mathematics education which was (i) fundamental in the sense that it played a decisive part in the functioning of mathematics teachers; (ii) elementary in the sense that it would be accessibleto intending teachers (that it would be of immediate interest for those aiming to become teachers of mathematics and would introduce the teacher to, and prepare him for, important didactical and practical functions, both motivating him and enabling him to become acquainted with such functions); and (iii) exemplary in the sense that it exemplifiesimportant didactical or practical functions of the teacher and their inter-relationships. The book has been written to present the knowledge required by mathematics teachers for the performance of their profession ("knowledge for action"), stressing the role of "knowledge about knowledge" in teacher training which should then consist of: (i) domain-specific information about ways to act and re-act in well-definedworking-situations to be commonly met in the classroom and in which there is a demand for action by the teacher; (ii) an overall orientation providing insight and/or awareness about more general types of situations and problems with which they have to cope and work in school. The word "perspectives" has 2 meanings: (1) Future outlook, and (2) Points of view. The meaning of the title of the book should not be taken only in the first sense, since it also contains an assessment of the past and present situation of mathematics teaching, particularly in the chapter by Howson and Mellin-Olsen, but also in the criticisms, which run through every chapter, of the present-day textbooks or classroom practice. However each Educational Studies tn Mathematics 19 (1988) 105.

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contributor presents suggestions for improvement - these assessments are not only arguments to present 'perspectives' (meaning (1)). The title of the book should also be taken to include the second sense of "perspectives", since the chapters all present a range of points of view on the teaching of mathematics. I must point out at once that it is (deliberately, I think) not an exhaustive range, omitting such out-dated ideas as the attitudes of the traditional mathematician who isolates mathematical concepts from the realities of the classroom. The different points are discussed as follows. Chapter 1 (A. G. Howson and S. Mellin-Olsen) discusses the position of mathematics in the environment of school and of society, its evolution with time and in different countries (particularly the developing countries), the expectations and the pressures experienced by the teacher, especially in making him perform the role of a "sieve" ("an obstacle course which serves to distinguish between the 'able' and the 'rest' "). This emphasises the clash with "the principle of giving all pupils success in mathematics". The authors also show how there is a conflict for teachers between "Significant and Instrumental" rationales. They put forward ten propositions for avoiding the damage which is often caused by evaluations, e.g. "(10) Do not confuse testing with teaching! It is the latter which is the more important, the former is a means and not an end." Chapters 2 (W. Dorfler and R. R. McLone) and 3 (R. Bromme and J. Brophy) return in part to 'expectations' and analyse different opinions on the reasons for teaching mathematics, the different answers to the questions "What is and what can be school mathematics?", especially in the age of computer science (here they put forward some very interesting views on the evolution of the relationship between mathematics and computer science as well as that between mathematics and other subjects). The process of mathematical modelling and the difficulties in teaching it are especially well analysed, as well as the implications for the training of a good mathematics teacher, such as his "view of himself as a mathematician", and "credibility", which are plainly at odds with the present-day situation. (An extract from a report on a student: 'It is not thought that this candidate is sufficiently able to withstand the rigour of a mathematical course, so we think a teaching qualification is more appropriate'!) They stress how important it is for teachers to have the capacity of "mathematical awareness" as well as "withitness". Bromme and Brophy discuss especially the varied cognitive activities performed by teachers and the kind of training which these require, so in Part 4 they propose 'methods to develop teaching skills and teachers' knowledge about their own teaching', and argue excellently for their ideal of training by presence in the classroom.

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Chapters 4 (J. van Dormolen) and 5 (M. Otte) deal with the subject of mathematics text books. Van Dormolen analyses people's ideas about textbooks according to their opinion on the nature of mathematics, and according to 2 dimensions- the kernels, (which "form the hard core of mathematics") and the aspects (theory, algorithm, logic, methodology, communication). Most of the comments are concerned with the use and exploitation of textbooks, while Otte makes a more descriptive but detailed study of them (an assessment rather than a perspective). I enjoyed Otte's study of the descriptions that have been used, particularly some of the metaphorical descriptions (e.g. the equation as a balance). The last 3 chapters are concerned with mathematics classes in school. Chapter 6 (G. Brousseau, R. B. Davis, and T. Werner) deals with observation of the work of school pupils (individual, in small groups, whole-class), and in particular they analyse the errors. They provide many good examples which pose just as many problems for the teacher who has to interpret them. They cite many references on the huge amount of research in the field, especially so-called "disaster studies" ("a considerable proportion of apparently successful students do not understand even the elementary, fundamental aspects of the subjects that they were thought to have learned"). If the argument of the theory stemming from cognitive science seems to me a little unconvincing, especially as regards the complexity of the reported observations throughout this book, I am sure that the reader will value the replies to the questions "How can teaching take advantage of effective personal ideas, and how can teaching try to minimise the damage from serious misconceptions?" Chapter 7 (B. Christiansen and G. Walther) describes various tasks which pupils have been set, distinguishes the ones which contain true mathematical value, and refers to an old Chinese proverb "the road is the goal". The basic point of this contribution concerns the study of the principles and methods of teaching which allow teachers to get their pupils to perform these activities. The definition of these principles and methods results from a didactical analysis of what the teacher tutor achieves when he takes, as an object of study, the consideration that the teacher must give to the pupils and their mathematical tasks. This thought is expressed well by this scheme: "Teacher educator ~ [Teacher ~ (Pupils ~ mathematical tasks)]" This summarises, moreover, the whole approach of the book. The last chapter (A. J. Bishop and F. Goffree) treats the dynamics and organisation of mathematics classes by an approach which I am accustomed to call "ethnological" and which produces interesting results. We can see very well the operation and interaction of all the phenomena cited in the previous chapters (expectations of society, educational establishment, literature and resources, the personality of the teacher, of the pupil, the

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development of classroom activities, communication...). It is a revealing chapter based on many illuminating and stimulating examples. It forms both a conclusion and an overture for the book. I hope that this short review will give the reader a taste for reading the whole book. I must also add that the book is very easy to read; all the authors refer constantly to concrete experiences; the examples are genuine quotations from the classroom and can sometimes be very amusing. Of course one may deplore a lack of homogeneity in the style of the various articles (this is inevitable in a collective work of this kind). Howson and Mellin-Olsen are the only authors to end each section with a list of didactical exercises ("activities to be undertaken during teacher-training"), although the other chapters lead up implicitly to the practicalities of teacher training. It is also regrettable that there is some repetition (for instance Neisser's scheme is discussed in two different chapters), but this does allow the reader to study each chapter as an independent entity. In spite of these small criticisms, I think that the book is admirably consistent, since I read it through with great attention to detail and found no contradictions between the authors (this is not necessarily a bad thing). Could the reason for this be that no clear-cut opinions are asserted? This is not a book which deals with categorical assertions. In my opinion it will be a fundamental reference, for years to come, for the training of mathematics teachers. However it will not give them a clear formula for success; on the contrary it sets problems and puts its readers in the position of posing many more. It is a stimulating book, which will have a position in my bookshelf next to those of H. Freudenthal, and I know that I will refer to it many times in the future. ACKNOWLEDGEMENT I am very grateful to Maureen Ashby for her translation of this paper from the French.

Universitb Paris 7, 2, Place Jussieu, Paris 75005, France

JOSETTE ADDA

F. Lowenthal and F. Vandamme (eds.), Pragmatics and Education, NewYork: Plenum Press, Coll.: Language and Language Acquisition, 1986, ISBN 0-306-42374-X, 344 pages. This book edited by F. Lowenthat and F. Vandamme is based on the results arising from the Third Language and Language Acquisition conference on Educational Studies in Mathematics 19 (1988) 108.

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Pragmatics and Education, held in March 21-25, 1983, in Ghent, Belgium. It aims to illustrate different approaches in the field of languages studies: education and cognitive aspects of discourses, non-verbal communication devices, mathematics education, native (and non-native) language acquisition and learning, and language disorders. We will focus here our attention on the part of the book devoted to mathematics education. The main question we hold in this book is that of the relation between language and construction of meaning in the learning process and in connection with teaching. Indeed, communication is one of the main activities involved in the teaching process. Teacher and students produce discourses which, as Grize emphazises it, have two main powers closely linked: an organizing power and a creative power. Thus, speaking as listening involves a cognitive activity, that of "construction of meaning" (p. 26). So, as Wells discusses it, the analysis of language phenomena in the teaching-learning process cannot be understood in terms of a transmission model as far as such a model "ignores the contribution of the learners [...] knowledge and understanding can only be constructed by each individual for him-or-herself" (p. 76). That means that a theoretical model to understand these phenomena should take into account: (i) "that learning is an active process in which the learner modifies his internal model of the world in response to the interpretation he is able to put upon the information that is presented to him" (ibid.), and (ii) that "learning is also an interactive process both in the Piagetian sense [...] and also in the requirement for collaboration by teacher and learner together in the joint construction of meaning" (ibid.). Studies concerning mathematics education are reported in five articles dealing with geometrical, arithmetical and algorithmic concepts. Grize mentions, comparing logico-mather~atical languages and natural languages, that the former "are characterized by the univocity of the precision of their definition" (p. 19). This strength, stated apriori, contrasts with the openness of the learner construction of meaning in mathematics as it appears in some of the papers: - the study reported by Guillerault and Laborde, gives evidence of the complexity of the construction of meaning even in case of elementary concepts such as those of points and segments. In a task in which pupils had to describe a given geometrical drawing, their research shows that these basic geometrical concepts were not used at once but after a cognitive construction in which they appeared to be powerful tools for description. First, pupils started with a static inventory of the objects (mainly polygons) of which the figure is composed, only later they moved onto an instructional description (measurement, location, lettering or numbering of point or segments). The

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need for an effective instructional description of the figure requires linguistic tools which lead to the recognition of the geometrical elements. There is an interactive construction of language and concepts; as the authors report: the "evolution towards the recognition of a point or a segment as an object frequently occurs interdependently with the numbering or lettering process" (p. 225). The research presented considers not only the encoding process, but also, by means of a specific experiment, the decoding process of a text (produced by pupils) describing a geometrical drawing. The last experiment raises fundamental problems about the reading activity: "the existence of expressions with double or triple meanings is not, ~a priori)), a hindrance to decoders" (ibid.), some decoders attribute to "expressions whose meaning remains constant throught the message [i.e. the text] [...] successively different meanings" (ibid.), some ambiguities (like a point/segment confusion) "does not appear to the student so inacceptable as it would do to an adult (mathematician)" (p. 236). - the research report presented by Cohors-Fresenborg deals also with the problem of the interelation between language and concept formation, that of algorithm. This study relies upon the hypothesis that "concept formation is more done by experience of acting than by (nominal) explanation or definition" (p. 207). The related mathematical concept-field considered is that of natural numbers and the related basic operation. Pupils are confronted with three kind of representation of an algorithm: (i) the handling of sticks, (ii) computing network, (iii) program words for a "Registermachine". If on the one hand the author reports that there is hardly any problem in ~translating)) from a computing network into a program word for the Registermachine (p. 215), on the other hand he emphasizes that even if they have developed a correct algorithm at the level of the sticks, some pupils nevertheless do not know how they really got the result (ibid.). In relation to the initial hypothesis, the fundamental problem raised by this research is that of the passage from knowledge in action towards its elicitation; it appears to be not a mere process of phrasing an implicit knowledge. - the report presented by Wolters concern problems of teaching-learning basic arithmetic rules. The author first mentions that "children assimilate what is taught in school to what they already know and [that] the result is an 'invented strategy'" (p. 263). The idea is to teach explicitly some efficient ~rnvented strategies)), as far as "average and low-ability pupils are less likely to discover these strategies by themselves" (ibid.). But, as a result, the author reports that (i) an obstacle to this project comes from the fact that some of the strategies used by bright children are not generally applicable (p. 269), and that (ii) "learning basic facts by rote is not very effective and very

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time-consuming" (ibid.). This study suggests that the effectiveness and reliability of the so-called ~(invented strategies~, come from the fact that they have been constructed by the "bright pupils" by themselves and their domains for validity have been recognized by them in action. These reports draw attention to the importance of the processes involved by students in their mathematical activity. The evidence is that the meaning of students productions is determined by the intellectual processes of which they are a result. A methodological problem is then: how to elicit and then to observe these processes? This problem is not only that of research, as Osser emphasizes it: "the end products of student's work cannot, by itself, provide the teacher with anything like an adequate understanding of both what and how the student is learning" (p. 255). Two papers presented in the book deal with this methodological problem: - Osser advocates for "clinical assessment practices" which represents "attempts to discover how students are structuring, and restructuring, their school experiences, and what forms their knowledge takes" (p. 252), by contrast with psychometric methods of assessment which "essentially represents the student simply as a passive absorber of lessons" (ibid.). The author illustrates his approach with a study of students' metacognitive knowledge, the reported clinical study has been done with both a teacher and some of her students focussing on the procedure of "checking" as an instance of monitoring school work. This case study clearly shows that students "conferred their own meaning on school experience which did not coincide with what had been proposed by the teacher" (p. 260). As Osser concludes "the clinical assessment procedure appears to provide a useful entry to the student's world" (ibid.). Lowenthal and Harmegnies propose a very sophisticated experimental setting to infer the intellectual process engaged by students, in a problem solving activity, from the sequence of elementary actions they perform. This setting uses a "Non-Verbal Communication Device" (NVCD) environment. A NVCD is "a means to create situations where sound structured communication can start, without ambiguity, and about a precise topic chosen by [its] creator" (p. 44). The NVCD used in the present experimental setting, has been designed by the authors to enable "the observer to better analyze and understand the subject's reasoning, simply by looking at the complete sequence of steps occurring in the construction of a final result". Actually, this paper does not present the data and the related results, they will be published in coming publications. We have focused our attention on the aspects of the book we think of the greatest interest for readers of "Educational Studies in Mathematics". -

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But other areas are explored, such as: reading processes (Boekaerts, Spoelder et al.), time and tense in English (Engels et al.), spontaneous family interaction (Huls), register theory (Simon-Vendenbergen), form and function of questions (Sinclair), theory in language instruction (Straight), and three papers about language disorder (aphasic child: De Bleser, Lowenthal et al.; deaf children: Loncke et al.). Most of these papers present stimulating hypotheses or results for both researchers and mathematics educators. The discussions of methodological and theoretical aspects in some others are challenging, but also it gives evidence that there is the need to explore this important domain of the relations between concept formation and language acquisition with a more precise problbmatique related to specific conceptual fields. Perhaps we could regret that the "synthesis and future perspective" presented by Lowenthal focus only on gathering data and methodological problems, leaving aside theoretical consideration such as: how to formulate the research problems in this domain? Which kind of theory will give tools to study phenomena related to language activity in teaching and learning a specific concept? Which conditions determine the construction of meaning by pupils, and what could be a theory in which to express them? How should the teaching situation, as a specific context for communication, be integrated in the theoretical research framework? Some of the papers contribute to such a debate, but they appear among other questions so that in some way they are lost. Perhaps to focus on methodology could have been the only way to make a real synthesis of that variety of contributions, we hope that coming conferences will focus more precisely on some fundamental question. No doubt that Lowenthal has such a project in mind. At the present time the book he and Vandamme offer to us constitutes a valuable step in a field in progress, at least by the debate that it is likely to provoke. Equipe de Didactique des Mathkmatiques et de l'Informatique, USTMG et CNRS, BP 68, 38402 St Martin d'Hkres Cedex - France

N. BALACHEFF

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I. M. Yaglom, Mathematical Structures and Mathematical Modelling. Translated from the Russian by Dana Nance. New York/London: Gordon and Breach Science Publishers, 1986, $ 76.50. This book, written by a well known mathematician and first published in Russian in 1980, belongs to the growing number of books about mathematics. As we read in the preface it was the author's intention to present a "book of a more general nature, devoted to modern viewpoints on the applications of mathematics and the concept of a mathematical model" which he thinks "most important element of popular culture" and "indispensible attributes of an educated person". As a consequence the book is oriented to a wide circle of readers across the spectrum of fields where mathematical methods are used (including the humanities). The first chapter is aimed at comparing and relating mathematics to the natural sciences and to the humanities. Contrary to the traditional view which draws sharp barriers between these fields as to their "scientific dignity" the author describes a convergence between them leading not only to a mathematization of the humanities but also to a "humanization' of mathematics. The second chapter ('When did mathematics arise?') is a very short historical account of how the systematic-deductive framework of mathematics evolved. The main emphasis is on Greek mathematics, whereas the development of modem axiomatics up to Bourbaki is dealt with on only four pages. Chapters 3 to 6 are devoted to "mathematical structures". The reader is introduced into Bourbaki's view on mathematics by means of a formal description of a "mathematical structure" and of examples from within mathematics (finite planes, groups, rings, division rings, fields, vector spaces, lattices) and from outside (kinship, elementary geometry, comparison of expert evaluations). Apart from small discursive elements in the text the presentation is not any different from that of most mathematical textbooks. Obviously the author feels at home here as a mathematician. The last chapter explains the basic message of the book: Mathematical structures do not belong to a realm of their own, isolated from reality, but they represent mathematical models of real situations or at least possible models. The development of mathematics must therefore be seen in close relationship with its fields of application. From the point of view of mathematics education it is important to state that around the world mathematicians are giving up the long held dogmatic conception of mathematics as the science of formal structures existing for themselves and providing absolute truth and are developing a more "humanistic" picture. Educational Studies in Mathematics 19 (1988) 113.

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The present book is witness to this development. But at the same time the book also shows how difficult it is for working mathematicians to articulate this new picture. The first chapter is a very interesting exposition revealing fresh ideas discussed nowadays in the circles of mathematicians. Many of these ideas, however, are mentioned only in the footnotes and unfortunately are not developed further in the subsequent chapters. On the contrary, the more the book proceeds the more the author restricts himself to "orthodox" topics presented in the "orthodox" style. As far as the relationship between axiomatics and models of real situations is concerned it would have been interesting to see a natural and convincing example. In the view of the reviewer the examples given in the book are not. For example, it seems rather unnatural to state axioms on the distance of expert evaluations first and then to derive an explicit formula for the distance from them. It is much more natural to think about a measure for the distance first and only then to investigate the properties of this function. To formulate some of its properties (which?) as axioms and to prove the uniqueness of this function would be the very last step. During the period of New Maths there have been many efforts in mathematics education to find convincing examples of applying the axiomatic method to real situations. Probably H. G. Steiner's treatment of voting bodies has been the only serious example. But is it really convincing? And is one example sufficient to prove the strength of the whole approach? From the perspective of mathematics education there is clearly no objection to viewing mathematical structures as possible models of real situations. In fact this philosophy was explicitly adopted by mathematics educators already in the seventies (cf. Ormell, Chr.: 'Mathematics- science of possibility', Int. J. Math. Educ. Sei. Technol. 3 (1972), 329-341). But here "mathematical structures" are not understood as formal structures in the sense of Bourbaki, but as concrete structures embedded into or derived from the classical topics of algebra, geometry, analysis, probability, and statistics. To summarize: From the reviewer's point of view the most interesting part of the book is the first chapter. In following the references given here mathematics educators might broaden their understanding of mathematics as a human endeavour.

Universitiit Dortmund, Fachbereich Mathematik, Institut filr Didaktik der Mathematik, Postfach 500500, D 4600 Dortmund 50, F.R.G.

ERICH WITTMANN