Should children be taught computational procedures before they understand the concepts on which they are based? What does it really mean to be proficient in elementary mathematics? What is the teacher's role in helping children learn mathematics? What must teachers know in order to teach school mathematics well? These and other important questions have been raised in the debates surrounding the mathematical education of school children over the past decade. Acknowledging that there are legitimate concerns about the mathematical performance of America's students and the preparation and professional development of mathematics teachers, *Adding It Up* proposes some detailed solutions that are informed by an extensive review and synthesis of the research on how children learn mathematics.

*Adding It Up* is a product of the collaboration of the mathematicians and mathematics educators of the Mathematics Learning Study Committee of the National Research Council. The committee's charge was to synthesize the relevant research on learning and to formulate recommendations for teaching, teacher preparation, curriculum, and future research, as well as to provide guidance to educators, researchers, publishers, policy makers, and parents on the topic of mathematical education in grades K-8. A formidable task!

The following emerged from the committee's work: a working description of the notion of mathematical proficiency, recommendations for coordinating aspects of the teaching and learning environment to provide programs that develop children's mathematical proficiency, and specific proposals for ways in which the mathematical preparation and professional development of teachers can be improved.

There are many notable aspects of this informative and sometimes provocative book. For a quick overview, the reader can first consult the succinct executive summary of its contents. The report itself, while lengthy at over 400 pages, is cogent, easy to read, and free of excessive jargon. The exposition includes excerpts from instructional materials and illuminating vignettes of classroom situations. Those interested in delving deeper into any of the topics covered can consult each chapter's endnotes and extensive list of references on the research into children's thinking and teaching practice that informed the discussion in the chapter.

The centerpiece of the book appears in Chapter 4: the committee's theoretical model for what constitutes mathematical proficiency. Firmly grounded in the scholarly literature, it is used as the guiding framework for the entire report. The committee visualizes mathematical proficiency as a rope with intertwined strands, each strand signifying one of its aspects. The five interwoven and interdependent strands of mathematical proficiency are presented as follows (page 116):

*conceptual understanding* - comprehension of mathematical concepts, operations, and relations
*procedural fluency* - skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
*strategic competence* - ability to formulate, represent, and solve mathematical problems
*adaptive reasoning*- capacity for logical thought, reflection, explanation, and justification
*productive disposition* - habitual inclination to see mathematics as sensible, useful, and worthwhile,

coupled with a belief in diligence and one's own efficacy.

The authors argue convincingly that all strands must be developed throughout an individual's education, and that by developing all strands it becomes more feasible to develop any one. Particular examples from the elementary mathematics curriculum are used to illustrate the role that each strand plays in one's mathematical development and how each strand supports the development of the others. While the book focuses mainly on number and operation, brief discussions of the ways in which the same model can be applied in other domains of mathematics are also included.

It was this chapter that convinced me of the importance of this book for my own teaching of courses for future teachers. Often undergraduates have a narrow view of mathematics that must be broadened and enriched if they are going to become effective teachers. This chapter provides the most coherent and accessible description of the aspects of elementary mathematical thinking that I have seen. Several topics in the elementary curriculum are discussed in terms of the model of mathematical proficiency. These illustrations enhance the explication of the model and demonstrate the significance of each of its components. This chapter alone would provide an engaging introduction for students who are learning to plan, implement, and assess mathematics instruction at any level.

More generally, anyone who is interested in the teaching and learning of mathematics would find much of interest in this book. Without question, it should be read by all college mathematics faculty who develop and teach content or methods courses for future mathematics teachers at any level. It would also be of practical interest to teachers of beginning algebra, including those who teach developmental courses at the college level. Most relevant for this audience is the lucid discussion of the relationship between arithmetic and algebraic thinking in Chapter 8 (Developing mathematical proficiency beyond number). The authors focus on the main types of algebraic activity (representing, transforming, generalizing and justifying), and describe the importance of mathematical proficiency with number and operation in each activity. They identify the obstacles that often trip up students who are in transition from arithmetic to algebraic thinking; this information strikes me as especially important for those who teach students who have been unsuccessful in learning algebra previously. Also included are descriptions of some innovative programs in which technology has been used successfully to enhance algebra learning.

In sum, the authors of *Adding It Up* present an ambitious and well-researched blueprint for developing programs that have mathematical proficiency for all as their goal, and they challenge educators and policy makers to make this goal a reality. For those who wish to browse this valuable resource immediately, the full text of the published book is available and searchable on-line on the National Academy Press web site.

Anne Brown (abrown@iusb.edu) is an associate professor in the Department of Mathematical Sciences at Indiana University South Bend. Her research interests are in collegiate mathematics education, which she conducts in collaboration with colleagues in RUMEC. She is also interested in the scholarship of teaching and learning in general, and is a member of Indiana University's Faculty Colloquium on Excellence in Teaching.