The book under review is an outcome of a rare experiment: an university math professor (a high caliber professor at that from one of the best universities in the world) who has responded to a challenge to teach in elementary school shares the acquired insights about teaching young children and their mathematics. The book is a very enjoyable read, the advice proffered is sound, the pedagogy is illustrated by numerous examples. I highly recommend the book to the grown ups concerned with young children education (...)

In the Introduction, the author outlines the sequence of events that led him to taking up teaching elementary grades in a small town on the Northern outskirts of Israel and describes the surprise experiences that came out of that experiment. The greatest surprise was in that the teaching of elementary grades provided him - a professional mathematician - an opportunity to learn mathematics: not any new facts of course but the subtleties inherent in the elementary mathematics. (As an aside, this part supplies a crashing argument in the hotly discussed topic as to whether or not elementary school teachers can be expected to possess math expertise.)

The first part is a collection of interrelated essays that discuss the fundamental role of abstraction in mathematics, the mathematical beauty, the peculiar economy of thought and expression that characterizes mathematics and the its hierarchical organization. There are also chapters on the whole numbers, the decimals and general thoughts of what might be expected to be learned in elementary school. Mathematical economy is beautifully classified as being achieved in three ways: Order (by looking for patterns), Generalization (by abstracting common features from different areas) and Concise Representation (the decimal system serving as an example.)

The second part is deftly subtitled The Road to Abstraction as conveying an abstraction is a fundamental need and principle of teaching mathematics. The author's methodology is to start with familiar and diversify the examples to help students grasp the abstraction as a common feature of several examples and prevent them from attaching unintended importance to auxiliary details (...)

The third part is the largest - it takes about two thirds of the book. The third part opens with a chapter on the meaning of arithmetic operations. For example, there is a real (for children) difference between questions such as:
There are five apples of which 2 have been eaten. How many apples have been left?
In a family of 5 siblings, 2 are boys. How many girls are in the family?
Joseph has 5 toys, Reena has 2. How many more toys does Joseph have?

All are naturally solved by subtraction 5 - 2 = 3. However, in the first, subtraction means a removal of two items. In the second, it means classification of objects into two types and counting each type separately. In the third, subtractions means comparison.

With the same attention to detail and profound insights, he then talks of the nature and rules of calculations, of fractions, decimals, and ratios. It is plain fascinating how much meaningful information is hidden behind simple arithmetic facts. There is so much that children may miss! There is so much to be learned in order to acquire a working grasp of the concepts of the elementary mathematics.

The book will be helpful to and enjoyed by teachers, parents who attend to their kids' study and, of course, the home schooling parents. It's a treasure trove of ideas usually missed out in textbooks and teacher manuals.