Here's an excerpt:
Where a topic appears on this list affects the way it should be taught and tested. Memorizing algorithms is an entirely appropriate approach to problems that fall primarily under number one [Needed for daily life]. Take long division. We would like it if all our students understood the underlying concepts behind each step but we'll settle for all of them being able to get the right answer.This is a bit of a work in progress so if you have any relevant experience in mathematics education (and, yes, experience as a student definitely counts), I would greatly appreciate it if you went by and joined the conversation.
If, however, a problem falls primarily under four [helps develop transferable skills in reasoning, pattern-recognition and problem-solving skills], this same approach is disastrous. One of my favorite examples of this comes from a high school GT text that was supposed to develop logic skills. The lesson was built around those puzzles where you have to reason out which traits go with which person (the man in the red house owns a dog, drives a Lincoln and smokes Camels -- back when people in puzzles smoked). These puzzles require some surprisingly advanced problem solving techniques but they really can be enjoyable, as demonstrated by the millions of people who have done them just for fun. (as an added bonus, problems very similar to this frequently appear on the SAT.)
The trick to doing these puzzles is figuring out an effective way of diagramming the conditions and, of course, this ability (graphically depicting information) is absolutely vital for most high level problem solving. Even though the problem itself was trivial, the skill required to find the right approach to solve it was readily transferable to any number of high value areas. The key to teaching this type of lesson is to provide as little guidance as possible while still keeping the frustration level manageable (one way to do this is to let the students work in groups or do the problem as a class, limiting the teacher's participation to leading questions and vague hints).
What you don't want to do is spell everything out and that was, unfortunately, the exact approach the book took. It presented the students with a step-by-step guide to solving this specific kind of logic problem, even providing out the ready-to-fill-in chart. It was like taking the students to the gym then lifting the weights for them.