In their preface, Arthur Baroody and Ann Dowker set out two questions about the nature of arithmetic expertise and how instruction can best promote it, linked to a third about the nature of and contribution made by adaptive expertise. Because the first two questions were known to Plato, neither is novel. But the third is more recent, with its origin in Giyoo Hatano's account summarized in a foreword about "the ability to apply meaningfully learned procedures flexibly and creatively [notably when students] invent effective procedures for solving new problems" (p. xi). Central to this, Hatano contends, is conceptual knowledge underpinning mental models generative of effective procedures, where such knowledge is seldom taught declaratively but all the same is nurtured in motivating, interactive contexts. This book includes 17 chapters, each substantial. The final two chapters are commentaries by Jeffrey Bisanz on chapters 1-8 and by David Geary on chapters 9®15. Bisanz's commentary is a tour de force: detailed, chapter-specific, incisive, fair, humorous, and telling. Geary's commentary comes from a comparable stable.