In the text *Understanding Numbers in Elementary School Mathematics*, Hung-Hsi Wu takes a definition-driven, highly mathematical look at elementary mathematics concepts. Throughout the book he carefully incorporates precise mathematical definitions with more user-friendly explanations and examples. He also provides the reader with a set of exercises at the end of each chapter. While the author does not directly address the pedagogical implications of the concepts contained in the book, he does consider ideas that help in developing pedagogically sound practices. For example, throughout the text Wu stresses the need for teaching for understanding and not just teaching procedurally. Additionally, he is able to weave a little history of mathematics and some real world context into his text.

While most of the concepts in this book are presented in a logical and comprehensive manner, at times the book can be a bit puzzling, especially to its intended reader, the pre-service teacher. For example, in the chapter on comparing fractions Wu was less clear than I would have liked. In this section, he discusses his dissatisfaction with the current method of finding common denominators to compare fractions but then uses the cross multiplication algorithm, a method based on finding common denominators, as a viable means for comparing them. To many readers this would seem like a complete contradiction. However, upon reading more deeply, I believe that Wu was not trying to condemn the use of common denominators but was instead trying to stress the need for conceptual understanding instead of following mindless procedures. I am afraid this idea might be lost on a less experienced pre-service teacher who is not yet familiar with our current ways of thinking about elementary mathematics.

My greatest concern is that this book may not meet the needs of its intended audience. The author intends it to be used with pre-service elementary teachers and in-service teachers participating in professional development. While the book is precise mathematically and presents ways of thinking about elementary mathematics that may be new to some, the content is presented in a less accessible manner than many of the current “mathematics for elementary teachers” texts. Much of Wu’s text is easy to read; however, the frequent use of formal mathematical language and symbols will make the ideas difficult to interpret for students and teachers with limited backgrounds in mathematics. And because of the lack of pedagogical focus throughout, this text may not help help teachers develop the mathematical knowledge necessary for teaching young students. Instead it is focused solely on enhancing the content knowledge of the reader.

This book is a comprehensive and coherent account of elementary school mathematics, but for me it missed the mark. With the intentional omission of pedagogical ideas, young readers might get the impression that teaching elementary mathematics is as simple as giving a precise definition and using a number line. Despite this problem, however, there is a lot of good information in this book. Instead of pre-service teachers, I see this text best used as a reference for mathematicians interested in elementary mathematics, mathematics educators working directly with elementary pre-service teachers, and elementary school mathematics specialists.

Janet Shiver is a mathematics educator at Central Washington University in Ellensburg, Washington. In addition to teaching mathematics courses for preservice elementary and middle grades teachers, she keeps busy working on various grants and pursuing her love of gardening.