## Reviews

**Abstract**

Edited by Jeffrey Nunemacher Mathematics and Computer Science, Ohio Wesleyan University, Delaware, OH 43015 How Humans Learn to Think Mathematically. Exploring the Three Worlds of Mathematics. By David Tall. Cambridge University Press, New York, 2013, xxi + 455 pp., ISBN 978-1- 107-66854-6, $39.99 Reviewed by Reuben Hersh Educational theory is not my area of expertise. This is an important book on educa- tional theory and practice, deserving an expert and thorough review. So why did I agree to review it? In answer, I quote a friend here at the University of New Mexico. “You write a book review because you want to say something about the subject of the book. But you are required ﬁrst of all actually to tell about the book itself.” I will tell you about David Tall’s impressive new book about how humans learn to think mathematically. I will also tell you my own thoughts about how humans think mathematically to discover and prove new theorems. It turns out that these two questions—How does one learn to think mathematically, and how does one actually do research in mathematics?—are almost the same question! I will shamelessly advertise my own newest book, Experiencing Mathematics (American Mathematical Society,