I wonder if part of the problem that northTOmom’s twins encounter in their math classes (http://northtomom.blogspot.com/2010/10/this-math-depresses-me.html?spref=tw) is that we don’t teach students how to learn and study math?
Here’s what I’m thinking about: to really understand and make use of something in math (like long division), you have to understand the CONCEPT, have METHODS for solving problems, and have a PROCEDURE that lets you effectively and efficiently solve any problem where division is useful. I wouldn’t expect students to discover the efficient long division algorithm. But I do expect that they can and will construct an understanding of long division concepts (like what division means — how many of these are in that, place value, etc.). I also expect that they are perfectly capable of finding ways to do meaningful division tasks (like figuring out how many 1 1/4 yard-long banners they can cut from 10 2/3 yards of cloth). Then I want them, once they have a concept and meaningful methods, to learn a procedure. That can be by teacher demonstration, by building up from student methods, by rote, whatever. Then, to do the exact task that northTOmom described in which students break down the procedure and connect it to their conceptual understanding and personal methods.
For some students, those concepts and methods are easy to come up with and the mathematical symbols are as sensible as the yards of material. For those kids, I’d be happy to show them a procedure and ask them to pick it apart. But for all kids I want to be sure that the have the trifecta of CONCEPT, METHOD, and PROCEDURE.
The various reform curricula are, I think, designed to be more explicit about focusing on concepts and methods. But too often in their implementation, students are expected to get to procedures very quickly without instruction, rather than being taught how to understand math problems, strategies to solve them, and learning strategies to help them move from solving one problem to developing a procedure. I’ve worked with too many teachers who see that the topic is “long division” and try to get students to do and understand long division procedures without actually teaching them. In good spiraling curricula, there is something to master at every level, but it might not always be the efficient algorithm. It might be getting good at reading and making sense of problem contexts that involve long division. Or getting good at recording the methods you used to solve long division problems. If those learning steps are noticed and celebrated, then the frustration element decreases, and the students’ understanding of what they are learning, and why, is increased.
I could go on, but I think I’ll wait to see what others might say in the comments…