"A grain of discovery"

[Dictated into a smart phone. I tried to catch all of the homonyms, but, hell, it's late and I'm fading.]

Having opened up what will probably be a long thread on Common Core, we really need to bring George Pólya into the conversation.

Pólya was a first rate mathematician and an enthusiastic and dedicated teacher. so dedicated that he spent somewhere in the neighborhood of 70 years in front of a classroom and he spent a large part of those years thinking long and hard about how we learn mathematics.

I'll go into more detail later about his conclusions. In addition to being insightful and interesting in their own right, they provide a useful alternative when discussing both the rigorously axiomatic approach of 'new math' and the reductionist 'deliberate practice' being pushed by advocates of common core.

For now though, I think it's important to start with the underlying assumptions of Pólya's approach to mathematics: Humans are by nature curious and drawn to interesting problems; therefore, if taught correctly, math should be both stimulating and enjoyable (as are the many popular games and puzzles based on mathematics).

In other words, if my kids hate math, I'm not teaching it right.

This is how Pólya opens the preface to How to Solve It:

A great discovery solves a great problem but there is a grain of discovery in the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play your inventive faculties, and if you solve it by your own means, you may experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime.

Thus, a teacher of mathematics has a great opportunity. lf he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. 

Here's the entire preface, preprefaced with more of my thoughts on the subject.