Recently I had a conversation with a teacher I work with, who teaches 10th grade geometry, that led me to a clearer articulation of something I started to try to say before. Namely:
The teaching of proof needs to be connected to kids’ own sense of what they are sure of.
This is actually obvious when you think about it. How could proof – the art and science of coming to know things for sure – be learned if the distinction between what the learner does and does not know for sure is not involved in the process?
I’ve just started reading an article that appears to be suggesting that this claim is also supported by research (more below). And yet this is not typically how proof is taught.
First of all, a typical proof problem in a geometry textbook is asking you to prove something that, as I’ve discussed before, is just about as visually clear as the givens are. So there’s really nothing you’re not sure of at all; and the process must proceed totally disconnected to your own sense of what you actually feel confident about. (Hence the kids come up with these arguments that follow the two-column format but don’t make any sense. I speak from experience. If you’ve taught a “proof unit” in geometry or algebra, you know what I’m talking about.)
But more broadly, in geometry and other classes: any problem of the form “prove X” is a setup for kids to fail to understand proof. It’s a fine kind of problem for someone who already really understands what proof is all about, so go ahead with this kind of problem in graduate classes or anywhere else if you feel your students have a well-developed sense of rigor. (And telling them what to prove is a great hint to make the problem easier without giving away too much.) BUT:
If a student does not yet understand proof, in the sense that she cannot yet produce coherent proofs, problems of the form “prove X” are not what to learn on. Why? Because once you say “prove X”, this student already believes X. You said she was supposed to prove it so it must be true.
This robs the student of her ability to sense and be guided by what she is or isn’t sure of.
Without this sense as a guide, proof becomes a shell game, and one whose rules are insubstantial and shifting, because they aren’t really the rules. There is really only one rule to proof: it has to be convincing. This is the guide and the judge. If you already “know” X is true (because the teacher told you to prove it), the guide and the judge automatically take a lunch break – you are already convinced. All you can do is write some things down in the “reasons” column and hope your teacher likes them.
What I think a “well-developed sense of rigor” really is, is the habit of bracketing anything you haven’t been totally convinced of as different from anything you have. So to a student who has such a “well-developed sense of rigor” you can say “prove X” and X is still in brackets, so she can still head toward it guided by the goal of removing the brackets. She trusts the authority of her own reasoning.
But this is not the state of most kids I have taught, or seen taught, at the K-12 level. So “prove X” is the wrong problem. “Prove or disprove X” is always better. (Unfortunately it is also always harder, and may therefore be too hard when “prove X” wouldn’t have been. But the comparative easiness isn’t worth it. They need problems where they actually don’t know the truth and have to figure it out for themselves. Otherwise they don’t learn how to prove. We have to find the X so that “prove or disprove X” is at the right level of difficulty.)
The paper I alluded to is from the May 2002 Journal of Research in Mathematics Education. It’s by Patricio Herbst and is called “Engaging Students in Proving: A Double Bind on the Teacher”. I just started it, so can’t really tell you what it’s about with confidence yet. But it contains the sentence “Studies of how students prove have demonstrated the importance, from the perspective of students’ learning, of maintaining the connections between proving and knowing (Balacheff, 1987, 1990, 1991; Chazan, 1993; Senk, 1989).” (p. 177) I take this (partly based on context) to be making my exact point, but I haven’t followed up with the citations yet. Unfortunately several of Balacheff’s are in French, which I don’t read.