What makes it possible to learn advanced math fairly quickly is that the human brain is capable of learning to follow a given set of rules without understanding them, and apply them in an intelligent and useful fashion. Given sufficient practice, the brain eventually discovers (or creates) meaning in what began as a meaningless game.

— Keith Devlin

Should Children Learn Math by Starting with Counting?

It seems obvious that our children must have a wide range of experience with real world objects before counting, addition, or subtraction mean anything to them. But are other topics, such as calculus, better learned as abstract rules — as a game that we play with symbols? And what about the topics in the middle? For instance, how best can we break our algebra students of common errors such as distributing the square or canceling out addition terms?

To teach effectively, I need to understand how students learn. Do different approaches work best with different concepts? Or at different ages or stages of development? I can think of at least 3 ways that I have learned math — what about you? How do you and your children learn?