cognitive theory

    I agree with cognitive theory in that meaning and understanding does not come directly from the environment, but is constructed by our minds. In order to best remember a mathematical fact or idea over the long term, a student should construct a meaning for the concept by relating it to prior knowledge in a variety of ways. One should always understand the idea, how it relates to other ideas, and how it can be used in a variety of situations instead of just trying to memorize individual, unrelated facts. I preach this to my students all the time. They will never learn the material if they just try to memorize how to push symbols around for hundreds of different problems. If they have a meaning constructed for the idea, they will rarely have to worry about memorizing anything.
   
    I do have some skepticism about Vygotsky's theory of cognitive development, at least what little I know from the textbook. I don't know how much of a role culture plays in these cognitive processes. Sure, different cultures might have different ways of organizing information, thereby constructing different sorts of meanings. But I suspect that underlying cognitive processes have a strong biological component and are very similar across our entire species. I suspect that every human remembers information better when it is organized in his mind in a meaningful way rather than as a bunch of unrelated facts. I doubt culture plays any role in this.

 As a teacher, I will always strive to help students create connections and meaning for the mathematical concepts they learn. In fact, I would say this is my main objective on a daily basis in the classroom. It's how I think about and plan any lesson.