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.
Research in Practice 
Have you read the old NCTM Yearbook The Nature of Proof? You can download a copy from http://www.eric.ed.gov/ERICWebPortal/custom/portlets/recordDetails/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED096174&ERICExtSearch_SearchType_0=no&accno=ED096174
From reading your post, you might like this.
That’s fascinating. I’m going to go look up that paper and read it. If you come up with some actual concrete examples of good things to ask students to prove, please share!
I really appreciated your previous posts on proof, as I ran into just this issue the only time I taught HS Geometry in 1999-2000.
I ended up teaching as an adjunct at UMaine the following year and taught a MAT 101 course that incorporated some non-standard topics.
One of units I developed for this class involved having the students solve simple logic puzzles (the person in yellow came in first, Jimmy wore blue, etc. who won the race?) and then write a paragraph explaining (rigorously) their solution.
Since the focus was no longer on mathematical content, but on mathematical reasoning, I could hold every student to very high standard of proof.
I enjoyed it, most students found it challenging. I believe it was a valuable exercise in having students engage in the experience of drawing conclusions from a set of hypotheses and justifying these conclusions.
I agree – I think logic puzzles are a wonderful way to give students practice in the art of deduction. My all-time favorite puzzle source is the books of Raymond Smullyan. (His first puzzle book, What Is the Name of This Book?, which I was given as a child, was probably my first major exercize in rigor.) Smullyan’s puzzle books have a wonderful “curricular” quality – the puzzles build on each other and develop logical themes.
A propos of the argument I’m making above, I think one of the big plusses of working with logic puzzles is that the students don’t know the answer; deduction is the way to find out the answer. Thus the process of proving is riding on the process of finding out. (This is what “prove X” problems are missing – there’s no finding out, so your own sense of what you know / want to find out can’t guide your process of proving.)
I’ve since worked up problems for my classes here at Clatsop CC that are fairly involved multi-step problems. I ask the students to write up their solutions in the form of a report, in paragraph form, using complete sentences and justifying their work.
They often ask, what needs justification and what doesn’t? This is where critical thinking comes in. The students need to decide what is important to the solution of the problem and what isn’t.
If a college algebra student leaves out the justification for adding 5 to both sides of an equation, I’ll let it slide. But, if they leave out the justification for setting up the equation itself, they lose 20% credit…and so on.
These projects are content focused, but the real work lies in seeing the connections among the different parts of the problem.
Ah, this question – “what needs justification and what doesn’t?” – is at the heart of what I’m talking about! I’m making the case that the best way to teach proof – if they aren’t already good at it – is to put people in a situation where this question is as easy and natural for them to answer as possible, because before writing down a given piece of information they needed to justify it to themselves before they even felt sure it was true. So the answer to the question can simply be – “well, what did you have to justify to yourself before you felt sure of it? That’s what needs justification.” I want problems where the process the student goes through to justify something is the same process they went through to even come to believe it.
Maybe I sound like a broken record by now. Anyway, thanks for the engagement and thoughtful discussion!
There are far better places to prove with kids in arithmetic and in algebra than in geometry.
(think of loads and loads of examples. They should look like they are probably true, but leave some doubt. I like this: 4, 16, 36, 64… is every even perfect square a multiple of 4? But I like lots of others, too. In elementary school someone showed me that there was no greatest prime… )
But geometry is the first place where they are sort of using an axiomatic system. I like to start with logic proofs, but the problem is unavoidable: we are following rules – the objects turn out to be mathematical but did not have to be – and the following of rules is the point of the proofs.
And as tortuous as the exercise is for some, it is vital for others.
Jonathan