This 454-page book is a report of the Mathematics Learning Study Committee of the National Research Council. The focus of the report is on how children in grades K-8 learn the concepts of the mathematical domain of number.

The purpose of this book is to use research to "address the conflicts in current proposals for changing school mathematics by giving a more rounded portrayal of the mathematics children need to learn, how they learn it, and how it might be taught to them effectively" (p. xiv). The book focuses on number concepts because "the heart of mathematics in [grades K-8] are concepts of number and operations with number," and "the learning of concepts associated with number also has been more thoroughly investigated than the learning of other parts of the mathematics curriculum" (p. 2). The authors clearly state, however, that the "decision to address the domain of number was a pragmatic one; in no way does it imply that the elementary and middle school curriculum should be limited to arithmetic" (p. 2). The audience for this report encompasses anyone who is concerned about mathematics education in the United States, including mathematics educators, mathematicians, mathematics teachers, researchers, publishers, policy makers, and parents.

Adding It Up provides a significant and timely contribution to the field of mathematics education. The report begins by discussing how the committee looked at research and experiences about mathematics and learning, followed by an analysis of the state of school mathematics in the United States. Also described are the concepts in the domain of number that students in grades K-8 need to learn to be successful in mathematics. Chapter 4 identifies five strands that, together, define mathematical proficiency: conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition. The next four chapters examine what research says about the mathematics knowledge children bring with them to school and how students can develop proficiency with whole numbers, other types of numbers, and beyond number. The final two content-related chapters deal with research about how teachers can teach to mathematical proficiency for students and how teachers can develop their own proficiency in teaching mathematics. The report concludes with its recommendations for mathematics education, particularly in relation to the teaching of number in grades K-8.

The authors examined a vast array of research reports about how children learn mathematics in grades K-8. This book summarizes knowledge gained through research and makes recommendations for teaching methods, teacher education, and teaching materials that lead to an improvement in the applications and conceptual understandings of school mathematics. The book is filled with excellent examples to demonstrate student misconceptions in the areas of mathematics where teachers need to be aware of and anticipate possible confusion. Classroom vignettes are used to point out strengths and weaknesses in different lessons and illustrate important points. The report is very reflective on how practices and materials promote or discourage equity among all learners, emphasizing the importance of mathematical competency for all students.

The text is written clearly and succinctly, with effective summaries of relevant research that inform the topic of number. Statements and recommendations throughout this text are backed by extensive research, and an impressive list of references follows each chapter. Recommendations are made for areas where further research is needed, including investigations into the effectiveness of strategies that would help students move from arithmetic to algebraic ways of thinking in grades pre-K through eight; research on the effects of using calculators and computers in the study of algebraic concepts; and studies conducted independently to analyze textbooks, instructional materials, and assessments.

With the current emphasis on examining and applying scientifically-based research, this report could prove a valuable resource for those designing and presenting pre-service and in-service professional development programs. One reviewer suggested that groups of mathematics educators at the elementary, middle school, and university levels could read and discuss this book and consider ways to apply the research to their work with teachers and students. This text could also be an important reference for curriculum directors and others at the school district level who guide decisions around curriculum and instruction. The report (and the materials discussed in it) could also be used as a resource for designing and delivering programs for the public in general concerning what works in mathematics education for grades K-8.

One reviewer noted that, although this report centers on the domain of number, readers can apply its ideas and recommendations throughout the mathematics curriculum.