"This book is a most valuable contribution to scholarship in Mathematics Education and Education in general. Thanks to a remarkable organization, the book leads to discussions of arithmetic thinking, of language and of cultural issues. The resource to the most recent advances in the sciences of cognition and the amazing list of about 500 items distributed in the six chapters are an indicator of the high scholarship level of this book. I highlighted some key discussions in the book. The authors aim at discussing arithmetic skill as the ability to think about the behavior of numbers in relation to arithmetic operations. They focus on how children regard quantitative thinking. Since early childhood, children develop capability of observing their environment and comparing what they observe. This reveals quantitative perceptions. The authors carefully explain how the referential and analytical meanings of words progressively changes for children and how the nature of word meanings give them the ability to use language as a tool for thought. When children understand the analytical meaning of words, given by the relations between words in the language, this knowledge enriches their understanding of the referential meaning, because the analytical meaning leads to thinking about many relations. Children have the same motivation and use the same cognitive resources to learn the meaning of number words as to learn other words in the language. But when it comes to number words, they have an extra resource. They learn that, if they count the items, they can connect number words with quantities in the external world. Although the authors do not provide a systematic review of the vast literature on children’s counting, their conclusion is that children learn to implement the counting principles over a few years. In our view, the ability to be systematic in implementing procedures is part of their development of mathematical thinking. They also discuss rational numbers based on division rather than addition, hence the analytical meaning of rational numbers is also different from the analytical meaning of natural numbers: any natural number has a unique successor but rational numbers do not. They also discuss word problems as a way to apply mathematics in different contexts. The authors synthesize the importance of word problems as the promotion of students quantitative reasoning, the recognition of relations between quantities in the situation rather than by arithmetical operations and the explanation of students reasoning as part of solving the problem. Bringing the context to play a fundamental role in problem solving leads to very interesting discussions of how culture is intrinsic to mathematical thinking. I learned a lot from reading this book." Ubiratan D’Ambrosio is Emeritus Professor of State University of Campinas (UNICAMP), Brazil, and recipient of the Kenneth O. May Prize in History of Mathematics (2001) and of the Felix Klein Medal (2005) for pioneering work in mathematics education."In this impressively clear, coherent and thought-provoking book, two of the most eminent scholars of the past half a century of research on mathematics education, provide a unique synthesis of and reflection upon their work on children’s mathematical thinking and learning. The insightful central idea of their "opus magnum" is the need to put the referential meaning of numbers and arithmetic operations and, thus, the development of quantitative reasoning much more into the center of our educational efforts. This ambitious reconceptualization of mathematics education, which brings both culture and mathematical modelling to the foreground, will inspire every researcher and practitioner active in this field."Lieven Verschaffel is Full Professor in Educational Sciences at the University of Leuven, Belgium. Member of the Flemish Royal Academia for Sciences and Arts of Belgium and Member of the Academia Europeae.
