BOOK REVIEW

The Development of Multiplicative Reasoning in the Learning of Mathematics. Edited by Guershon Harel and Jere Confrey. Albany: State University of New York Press. Publication date: July 22, 1994; Hardcover; US$24.95. The introduction plus the 11 chapters of this book on multiplicative reasoning offer an invaluable contribution to our understanding of this topic and to thinking in mathematics education in a more general way. The joint effort of the contributors shows that thinking and research about multiplicative reasoning has developed much beyond the seminal work of Hart, Fishbein, Brown and others who started the work on this field with the categorization of multiplication and division problems. Although the classifications provided in the past remain significant for the analysis of multiplicative situations, this book offers several analyses of multiplicative reasoning carried out on a different level of theorizing. The central question posed in this book concerns the nature of the operations that lead to the construction of number meanings in the domain of multiplicative reasoning. Are these operations different from the operations that constitute number meanings in the conceptual field of additive structures? In chosing to address this sort of question, the contributors have brought a constructivist framework to the analysis of multiplicative reasoning. This allows for the originality of their ideas to come across more clearly. Three types of new meanings for numbers in the field of multiplicative reasoning in contrast to those in the additive field are proposed. Some chapters (most clearly those by Steffe, Lamon, Kaput and West, and Behr et al.) investigate number meanings in relation to the concept of complex units, or sets of sets, a meaning which is constructed through the operation of replicating (the term is used by Kieran in his final discussion chapter). Replicating is distinctive from joining, an action constitutive of number meanings in the additive field, because replicating is a ratio-preserving operation applied simultaneously on two variables. Other chapters (most markedly those by Vergnaud and by Confrey) investigate a new number meaning, that of a (linear) function relating two variables, constructed through the operation of establishing covadation. The construction of covariation may rest, for Educational Studies in Mathematics 29: 293-294, 1995.

294 example, on a notion of empirical causality or on actions such as magnifying. Although solutions to problems involving linear functions can be obtained through building up strategies, also used when problems have to do with sets of sets, the construction of number meanings based on functions must be distinguished from those related to sets of sets. The novelty in this type of number meaning relates to the notions of variable and covariation, which are distinct from sets of sets. Finally, a third meaning for numbers in the multiplicative field is investigated (mostly in the chapters by Confrey, and Smith and Confrey), which involves exponential functions, and results from the operation of splitting. Although some authors concentrate on only some of these meanings and, in some cases do not explicitly discuss them, they seem to cover the scope of the new analysis of numbers in the multiplicative field presented in the book. The analysis of multiplicative reasoning is approached in the book through analyses of problem situations, of children's reasoning, of mathematical concepts in and of themselves, and of historical developments within mathematics. There is a rich variation in the sorts of methodological approaches used which is refreshing especially when sections of the community of mathematics education researchers sometimes appear to be falling prey of methodological orthodoxy. Children's reasoning, for example, is examined through case studies in learning experiments, quantitative comparisons of performance of a large number of children on the same problem, analysis of verbal protocols, and teaching experiments using computer technology. There is also frank disagreement amongst contributors to the book, with thoughtful discussions of the different interpretations and what they mean for mathematics teaching and learning. The framework presented by Kieran in the final discussion helps bring the differences between some of these conflicting hypotheses into focus. Because some of the ideas are only in their early stages of development, the empirical work supporting them is not always there. What is lacking in empirical support presently is clearly offered to the researcher as future questions to be tackled and to the teacher as new ideas to be experimented with in the classroom. Perhaps there is no need to say it, but I am a supporter of clearly stated conclusions: this is a book no researcher in mathematics education would want to miss.

T. NUNES Institute of Education, University of London.