There's a fascinating little book called "A Mathematician's Lament," written by research-mathematician-turned-math-teacher Paul Lockhart, in which the author bemoans the sorry state of mathematics teaching today and offers his thoughts on how it could be done better. In some ways, it's similar to a lot of other books and articles that complain about how math is taught (the poor curriculum, ineffective teaching methods, etc.) so I was prepared to largely ignore it. I don't generally have much use for these kinds of screeds, as I think most math teachers already know what the problems are and are doing the best they can under difficult circumstances; having someone shouting "You're not doing it right!" from the sidelines isn't particularly helpful.
I was pleasantly surprised, however, as Lockhart does in fact offer some useful insights into things that can and should be done differently for many math teachers, and the fact that he's a math teacher himself, and one who used to be an actual mathematician, gives him a bit more street cred than most others. So I wholeheartedly recommend this book, especially to math teachers, as there's a lot of good stuff here.
That said, I was somewhat troubled by the book. Lockhart's main thesis is that math is a naturally fascinating and creative and joyful activity, and the way schools and math teachers present it turns it into something dull and dreary (and he uses some pretty strong language to make his points). He's a big proponent of allowing students to "play" with math so they can experience this joy themselves, and he's a big critic of treating math as a utilitarian tool that students need to master in order to be able to function well in society. He doesn't like worksheets or textbooks or memorizing formulas or learning particular topics just because "that's how we've always done it."
That was all fine; I don't have any big problem with any of that, in theory at least, and I actually agree with Lockhart's basic point of view. I try to use "discovery learning" as often as I can and do my best to allow my students to experience the joy of math. But there seemed to be something that Lockhart wasn't addressing, and it took me a while before I could put my finger on it. I finally happened across an anonymous comment from another discussion of Lockhart's book that summed up the dissonance I was experiencing.
I have an interesting situation with regards to this. The students I teach are quite difficult at most times, and are generally quite behind in maths. They can often remember ways of doing things by rote, but have troubles generalising and get completely stuck if they find themselves in an unexpected situation. Probably for this reason, they really like worksheets and textbooks. They can get the answers (one way or another) - in fact there _is_ an answer. They have no idea of what this answer means, or, for that matter what the question means or why the hell they're doing it. But that doesn't really matter if they can get the answer. On the other hand, whenever I have tried more open, exploratory or play based activities, the students tend to revolt (in a manner of speaking). Not always, but often. There's no clear end in sight, and they have difficulties working independently, so things start going awry. A lot of this, no doubt, comes from their lives outside of school, and earlier schooling.
It's a conundrum - I can work through the state's curriculum via a textbook and tick all the boxes, and the administrators will be happy and so will the students (though they won't actually have learnt anything useful), or I can tear my hair out trying something different where the students get confused, I don't cover the expected material, and the outcome might not be much better. I'm tending towards the latter option, in the hope that over time the students will become more responsive...
I think Anonymous nailed it, and I suspect that most math teachers struggle with this type of student to one degree or another. Discovery- or play-based activities are not some magic bullet, and in fact require some degree of effort to implement, not just on the part of the teacher but on the part of the student. The students have to be willing to cut the triangles out of the paper, or graph different equations on the graphing calculator, or change the different parameters on the computer program. They have to be willing to work (or play) with a math problem that is open-ended and may not have a specific solution. Many of us (including, I suspect, Mr. Lockhart) might not consider this much "effort" at all, and would certainly balk at calling it "work," but the fact is that many of my students struggle with exactly this issue. If it's anything more complicated or creative than a standard worksheet, they're completely befuddled and will often, as Anonymous says, revolt. "This is too hard!" "You never showed us how to do this, how are we supposed to know what to do?" "Why don't you give us some real math problems?"
Mr. Lockhart's book seems to assume that if math teachers would just stop presenting the material in such a dreary and boring way, the students would immediately respond with shouts of joy and begin exploring and playing with math concepts and begin really "learning" and not just memorizing a bunch of facts and formulas. But before students can experience the true joy of math (or, I would argue, any field of endeavor), they must be willing to do at least a little bit of work. Instilling this work ethic in some students is often quite a challenge.
Image by Combined Media on flickr.