For several years, my son went to the UCLA Math Circle run by Dr. Olga Radko and Dr. Oleg Gleizer (both mathematics PhDs), whom I've gotten to know well. It's a superb program, which teaches children ages 5 to 17 serious mathematical concepts and mathematical problem-solving, not just drills or algorithms.

Gleizer and Radko now have a second, two-volume, edition of a book based on their work with the math circle, "Breaking Numbers into Parts," aimed at parents and teachers who want to teach 5- and 6-year-olds (here's volume 2). The book isn't written to be read by children, who might not even be able to read well (or to read at all) at that age. But adults can use it very effectively to guide children through concepts such as commutativity, the number line, the properties of odd and even numbers, mathematical operations as a generalizable concept, reflections and symmetries, and more—and, more importantly, to teach children about proofs and about the use of math and logic to solve problems.

If you have young children whom you want to teach about math, definitely get these books.