Nuggets II: Proof

Paul Lockhart has a lot to say. He was a research mathematician, and now he teaches kids. His essay A Mathematician’s Lament is a passionate screed against stultifying math education. (Which Lockhart sees as typical math education.) I am not alone among lovers of mathematics in saying that when I read it I experienced many, many moments of “Yesss!!! EXACTLY.” (Well, actually, “worrrd!” I was a teenager in the nineties; don’t laugh at me.) Another math lover with the same reaction was Keith Devlin, NPR’s Math Guy, who devoted his column in the MAA’s website to it in March 2008. This year, the essay was expanded into a very short book and published by Bellevue Literary Press (with a foreword by Devlin). Keith Devlin thinks everyone in math education should read it and I think I do too.

(This is not a 100% endorsement of Lockhart’s whole point of view. There were several moments in the book I found dismissive of the hard work that thousands of teachers are putting in across the country to try to teach math in a passionate way inside the constraints of traditional schooling. But the man is angry, and with good reason, so there you go.)

Anyway, amid all the things Lockhart talks about, one jumped out at me and clicked into place in my mind.

Nugget: The impulse toward rigorous proof comes about when your intuition fails you. If your intuition is never given a chance to fail you, it’s hard to see the point of proof.

From A Mathematician’s Lament (p. 72 of the book version):

“Rigorous formal proof only becomes important when there is a crisis – when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind.”

THIS IS SO TRUE. And it’s IMPORTANT. I think the issue wouldn’t be so pressing if mathematical argumentation were more intrinsically part of every math class, at every level, going back to little kids, as it should be. But in the absence of this, what happens is that when kids reach a point in their mathematical education where they are asked to prove things, they find
a) that they have no idea how to accomplish what is being asked of them, and
b) that they don’t really get why they’re being asked to do it in the first place.
The way out of this is to give them a crisis. We need to give them problems where the obvious pattern is not the real pattern. What you see is not the whole story! Then, there is a reason to prove something.

The way it typically goes is that all of a sudden in high school geometry, you’re being asked to prove something that’s just as visually obvious as the given information. Clearly it’s a pointless question. Meanwhile, you can’t do it, because everything true about the diagram seems equally true to you. The art of proof is about taking things that you’re sure of and touching them, prodding them, turning them around, pushing them against each other, until you become sure of more things, and continuing like so until you become convinced of the thing to be proved. To do it you need to feel it. When you do it well, your gut is giving you a reading on what you’re sure of and what you’re not sure of. (With students and teachers I work with, I’ve taken to calling this gut readout “your internal compass for rigor.”) But in a typical geometry proof, the thing to be proved feels just as true as the givens. You’re being told you know one and not the other, but this is not a difference you can feel. There is no internal compass for rigor to guide your path.

This is a situation exacerbated by everything that’s happened before geometry class. Every time you noticed a pattern, it was the right pattern, unless it was demonstrably wrong. For example, maybe you got a chance to experience the following awesomeness:
1+3 = 4
1+3+5 = 9
1+3+5+7 = 16
1+3+5+7+9 = 25
BLAM! The sums of consecutive odds are squares!
Now this observation is very cool when you first notice it. But where is this going? A lot of the time, it stops at the observation. The teacher gives the class a chance to see a pattern, they see it, that’s cool for 1.5 seconds, and then that’s it. Everybody moves on. The kids in such a class are being trained not to understand the need for proof. Even the teacher is acting like seeing is believing, so why, when you later get to geometry, are you suddenly being asked to “prove” things you can obviously see?

It’s much better if after the initial “aha” moment, there’s some sort of quest for an explanation. Lockhart describes such a quest, for an explanation of this exact pattern, in the book version of his essay, on pp. 106-117. But something is still missing if the only kinds of experience the students have fit this (see a pattern) – (explain it) – (see another pattern) – (explain it) cycle. The students are never getting a chance to see the wrong pattern.

I visited Paul Lockhart’s class at St. Ann’s School, where he teaches, a few weeks ago. At one point he said to his class (I’m paraphrasing because I don’t remember exactly) – “One thing that will happen this year is that your intuition will suggest something is true, and then you’ll look for a way to establish its truth, and find it, and refine it into a solid argument. That’ll be a good exercize. But far better for your mathematical development will be when your intuition will suggest something to you, and it’s wrong. You’re dead wrong. And then you see that you need a richer understanding of what’s going on.” So, a propos of this, I’m making a case that we give our kids lots of chances to have their intuition be wrong. The earlier the better. Nothing will develop the internal compass for rigor more powerfully.

Sensing a danger of being misunderstood, let me get concrete. I’m not talking about doing anything to undermine students’ trust in their reasoning. Cultivating a student’s trust in her own reasoning is what I believe math education is most centrally about. This is not about telling students they’re wrong, it’s about giving them a crisis. I’m talking about giving them problems that suggest one pattern on the surface when really something else is going on. In this way, students’ own reasoning is what puts their intuition in check.

I’m about to tell you the best problem I know like this. Tomorrow I’ll put up some more (though they’re not as good.) I’m hoping that some of you will add to the list of problems. They’re of vital importance and, though they’re easy to come by at the level of active mathematical research, I’ve encountered very few at the K-12 level. We need a repository!

The best one I know I learned from Bob and Ellen Kaplan’s book Out of the Labyrinth, which I wrote about last week.

Take a circle. Put 2 points on the circumference and connect them with a line. Into how many regions is the circle divided? Two.

Now add a 3rd point on the circumference and connect it with lines to the other two. How many regions now? Four: the points make a triangle so the interior of the triangle is one region and another one between the circle and each edge.

Add a 4th point and connect it with the other three. How many regions? Eight. Count ’em.

A 5th point? Sixteen. If you’ve never seen this problem, you should be drawing right now because you don’t want to miss the full glory of this.

Alright, I see where you’re going, you say. Does it fit the case with only one point? Oh yeah, that’s just one region. Pretty neat, but what’s the big deal?

Draw the 6th point. Connect it to the others and count the regions. How many? Thirty-tw… Thirty-ONE? What? Did I count wrong?

No, you didn’t. That power of two thing you saw, a bulls-eye for the first five cases, is a miss on the sixth. (And a tantalizingly near miss, at that.) What’s really going on here?

Now if you want to look it up, it’s discussed in Out of the Labyrinth, pp. 71-74. But I recommend, if you’ve never seen this problem before, that you try it out yourself. What’s the maximum number of regions you will get with 7 points? With n points? And why?

The beauty of this problem is that the wrongness of the initial “obvious” pattern gives the search for the truth much more urgency. And, more importantly for the present conversation, it gives the student a reason to care about proof. I can stand up here and say “you’ve given me the first five cases, but you haven’t proved it” till I turn blue, but if you’ve never seen something work five times and then fail later, there’s some level on which you don’t believe me.

So this is what I’m advocating: let’s give students problems where there’s a superficial pattern that’s not the real deal. The need for mathematical argumentation is going to spring from these problems like corn from the Iowa soil. (Forgive the corny metaphor; I’m just excited.)

And folks: what other problems like this do you know?