You Want Proof? I'll GIVE You Proof!

Once again, the fires of discord are raging on math-teach@mathforum.org. One of the threads I've been embroiled in revolves on several axes: one is about teaching pure mathematics in K-12. Another is about visual proofs. The one I wish to specifically deal with here is the one that links the two: what comprises the nature of proof in elementary school mathematics classrooms and how do we get students in those grades to develop their notions of what a mathematical proof actually is?

The problems in having this sort of conversation in a hostile forum like math-teach are legion. One problem is that only a few of the participants are K-12 teachers, and fewer still are K-5 teachers or spend time working with elementary students and/or their teachers. Another is the long history of enmity from the Math Wars that tends to inform most conversations there. Fortunately, the one I'm going to pull from is not being polluted by the more egregiously nasty sorts of name-calling (due in no small part to the absence of two of the more troll-like participants from that list when it comes to progressive math education, as well as a sort of detente' that I was able to reach off-list with the person to whose ideas and arguments you'll see me replying below. It isn't quite all milk and honey, but it at least doesn't reek with epithets. I take what I can get these days, as I try to moderate my own propensities for vitriol. It's a long, hard slog, and I am fallible.

For background, things had reached a point where I was arguing for letting kids develop their own ideas of proof, and Paul A. Tanner, III made the following post:

I mean it could be made available as part of the mix in the same way that things you like could be made available, like the lattice method. (You want the lattice method made available, right? You don't want it left to chance as to whether they have the opportunity to see it, right?) It could be taught to them as part of the main course or
as a supplement. It could be in some sort of enrichment context, as something to be learned at the end of a guided method or as something to be learned in a direct teaching context. It could also be in written form, in which case they could study it if and when they wished, consulting with others outside the class or with their teacher.

Here is my reply, which touches, I believe, on several really central themes for the direction effective, student-centered mathematics education must go:

Paul, you have a unfortunate propensity for mixing up issues in ways that seem to come from your drive to win arguments rather than understand what others are saying. It's not a helpful habit and really makes discourse more difficult.

We were talking about notions of proof and whether formal proofs need be introduced earlier in K-12 and if so in what manner.

I raised the suggestion that for younger children, it would benefit them more to promote conversations amongst them about what constitutes proof and to help that notion grow OVER TIME (as in, over the course of years, not weeks or months). And I expressed concern that it would be difficult for some adults to resist pushing their more sophisticated notions on kids rather than trust that they will, if nurtured intellectually, move towards more mature and precise notions of what comprises proof, and eventually will start to see a need to know and understand what the mathematics community at large considers to be proof (and as you see, there is not exactly universal agreement about what that is: note the on-going debate here about visual proofs, for instance).

You then switch, as you all-too-often do, to algorithms, specifically so you can "trap" me with the example of lattice multiplication. But your move is flawed. Here's why. First, if you recall, I am inclined to agree with the work of Kamii, among others, who suggest that one of the big errors we make is to PREMATURELY introduce formal algorithms for arithmetic to kids. Please note: that's ANY formal algorithm, which would include the standard multiplication algorithm, the Russian Peasant algorithm, lattice multiplication, partial products, area models, et al.

And before you jump on the word "prematurely," which you have done erroneously in the past, this is NOT an issue of developmental "readiness." Rather it's a concern about crushing students' confidence in and reliance upon their OWN ideas before they've had a chance to test them and compare them with those of their peers. Adults need not be so bloody worried that the kids will get everything wrong and be irreparably harmed. Using traditional teaching methods and algorithms, teachers are ALREADY eliciting student misconceptions and errors.

So what's the rush? Can't wait to see if the kids can, by interacting with peers in guided conversations by a knowledgeable teacher, find their own way and correct misconceptions? If not, why not? Where is the evidence that this method would be WORSE than what we've done for decade upon decade? Lots of kids get mired in buggy algorithms as it is. I see the evidence all the time. A much smaller percentage of kids seem to blow it when they use lattice multiplication than the traditional algorithm because they don't wind up with misaligned partial products and they don't forget to carry out all the necessary multiplications. They can, of course, still do one digit multiplications wrong, and they can still add wrong, and they can still forget to carry. So this isn't foolproof. But the incredible number of the other two kinds of errors I mention simply don't occur with this method. And yet, the forces of anti-reform refuse to look at this, refuse to consider that it not only might this true, but that it makes perfect sense that it should be true. Given that sort of rigid thinking, can we seriously expect that such people are going to trust kids to use their own algorithms, test those algorithms against problems and against those of their peers, and choose reasonably which one(s) they prefer to use? Clearly, that isn't a possibility for educational conservatives.

Getting back to that word 'prematurely" again, the issue isn't development, as I said. It's letting kids develop their sense of what works well for them, and that they can figure this out, and that they can develop, test, and refine their own ideas about mathematics. There's lots of time to then introduce, if necessary, any algorithms that we might feel kids need to see that haven't arisen. Later.

But as I wrote in my previous post, what tends to happen is that either nothing but the traditional algorithm is taught, or when it is taught it's given explicit or implicit pride of place. And this occurs so early, in lower elementary, that what I discuss about about kids' creativity, self-confidence, and judgment about what works is crushed or suppressed. And we wind up with what? Passive kids who wait for Teacher to tell them everything. Who don't take risks. Who don't think except along the narrowest possible lines. All this well entrenched, from what I've seen, by third grade. It's a tragedy.

So, getting back to PROOF: the same rush to formalism and tradition is equally ill-founded. Let kids develop their own standards and ideas about what comprises proof in general and mathematical proof in particular over the elementary grades. Guide them towards more sophisticated notions, and by the time they've gained enough mathematical maturity, they'll want to know what professional mathematicians (about whom they might actually be allowed to learn something) consider to be proof. It may not happen in K-5. Or maybe it will. But it will happen. If we trust kids' curiosity and the ability of wise teachers not to shove things down their throats at the first sign that the students aren't doing things "by the book."

Do I trust YOU, Paul, and those who think like you, to show this wisdom and patience? I do not. I have seen ample evidence of how the majority of even elementary school teachers think and work when it comes to traditional vs. other algorithms to believe that it would be any different with notions of proof. The drive that is grounded in mistrust of kids is so strong that I had a third grade teacher tell me in 2005 that she doesn't explore student errors with students in class because "the other kids will fixate on the errors and then I can't extinguish them." I wondered why they ostensibly don't fixate on the correct solutions and methods: only the wrong ones they hear. Why would it not be useful to discuss the errors, where they come from, and try to reveal why they are grounded in things that don't correspond to what kids already know to be true from experience inside AND outside of school when it comes to math. But apparently it's just too dangerous to talk about those errors. Just say, "No, dear. You do it like this."

It's the adults, not the kids, who have problems. The kids will be fine if we let them. They are not morons, but we make them into empty-headed robots in short order with our insistence upon spoon-feeding them when they're perfectly capable of a great deal of self-nurturance. All we need do is provide a safe, rich environment and keep our eyes and ears open and, much of the time, our mouths SHUT. But few of us can do any of that: as Bob Kaplan told me in Chicago in 2003, it's easy to find teachers for the Math Circle who know the mathematics. The problem is finding teachers who know how and when to keep their mouths shut.

I know you will continue to try to lawyer this into something that allows you to shove things down kids' throats that you don't trust them to want to know later. And here's the part you miss: I don't insist that any particular algorithm MUST be taught. What I want is going to happen if we let kids breathe and think. When they want to see more methods, they'll let you know. When they become discontented with their ideas of proof, they'll let you know. And it WILL happen. Because there will always be kids who ask themselves and their peers: "Why does that work? Why does that make sense? How do you know?" And that's all we need to nurture in them: their own natural curiosity, rather than suppress that and replace it with curiosity about only the following: What does the teacher think? What does the teacher want me to say or do? What do I need to do to get an A?