theoretical frameworks

I wanted to reflect on what was said last week about theoretical frameworks. My academic background is in philosophy, not the social sciences, so I'm still learning the lay of the land. One thing I know about the program at UWB is that constructivism is the official theoretical framework for what we do. As well as I understand it, I think constructivism is basically the right idea about how people learn. However, I am eager for an explicit discussion about what exactly this theory is, and what it is not. It seems that many who write as constructivists think of it as a fairly specific doctrine about how teachers ought to teach, ie, a pedagogy, rather than simply a theory about how people learn. Thus, it is said that small group discussions, or hands-on activities, are constructivist activities. But as far as I can tell, constructivism as a theory of learning does not entail that one type of classroom activity is inherently superior to others. Although I think there are sound reasons for the view that too much listening and book work in the classroom is an ineffective way to structure classroom experience, I think an equal case can be made that too much small group work and/or computer work is also ineffective.

Anyway, I am somewhat suspicious of the idea of claiming a theoretical framework, except in a broad way, when it comes to taking an intellectual stand on something. It is always preferable to state what exactly your assumptions are about a subject, rather than to bat around labels that are ambiguously used in the literature. Obviously, the discourse should not degenerate to the level of cheap political dialogue, in which saying "I'm a Democrat" somehow justifies or explains your view of a particular question. Too often, labels are a way to avoid real thought.

Visualization Techniques

This is an ugly phrase to describe what I think is an important aspect of learning math. A lot of math skills go beyond reasoning, logic, and memorization. Sometimes a student needs to be able to visualize a concept or a relationship in picturesque terms. Here is an example.

Suppose you wanted your students to understand why cos(theta) and sec(theta) are even functions, while the other four trig functions are odd. There are several ways to show them, but an especially effective way is to get them to imagine two radii of the unit circle sweeping away from theta = 0 at the same time, so that the first angle is always the opposite of the other. They should be able to grasp more or less directly that the cosines of the two angles will always be identical, and therefore cos(-theta) = cos(theta), ie, the cosine function is even. The same visualization technique will illustrate why the other trig functions must be odd.

Hence the value of math teaching software like fathom and sketchpad, which allow you to make the visualization precise and immediate. However, I think that good teachers need to be able to think of visualization tricks on the spot, to explain them in words, so that the students have to construct the visualization in their heads from time to time.