Constructivism in the Math Classroom

When one student thinks entirely in the abstract, another thinks in terms of real-world applications, and yet another thinks in terms of the least possible work required to solve the problem, how do you teach an entire classroom full of such differing personalities? Simply put, IMO, you go for the mixed bag. Take the infamous case of multiplying (x+a)(x+b): Only the abstract student is going to be able to understand that. So you give an example of a room that measures, say, (x+5)(x+2) square feet. Have them draw out what that looks like. Then have them discover--don't show them yet, just have them discover that this can further be divided into four sections: x^2, 5x, 2x, and 5*2 piece. In other words, they come up with the FOIL method instead of having it taught straight-up.

For the student who doesn't want to go through all that work (but who really can, whether or not he or she knows it), have this student attempt to find an easier way and justify it. Pull this off properly and ironically, this student will do more work, but more important is that they will discover an important truth: there comes a point with some problems where certain steps have to be done. Granted there are often various paths that can be taken, but at some point, particular ways have to be followed.