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?


25 thoughts on “Nuggets II: Proof

  1. I decided long ago that Lockhart creates a caricature of math education, and doesn’t show much insight as a teacher (chooses particular recommendations that resist generalization)

    However, I’ve known about the 31 regions for a long time, a professor got us to predict 32 and fail to reach it. I’ve found the 31 in Pascal’s Triangle, which I think is pretty cool.

    Speaking of which, I like to let the kids see that $latex 11^0 = 1, and 11^1 = 11, and 11^2 = 121, and try to use this to find further rows in Pascal’s Triangle…


    1. Love the powers of eleven idea – great one. Particuarly so because the initial pattern is actually on target (unlike the points-on-a-circle problem), but it breaks down, and how it breaks down is itself illuminating.

      There’s a baby/bathwater issue with Lockhart. He’s so dismissive of math education as it’s practiced everywhere but St. Ann’s that it’s very easy just to be pissed off by him. He writes as though nobody besides him ever had these thoughts before. It’s kind of infuriating.

      Nonetheless I disagree that he “doesn’t show much insight as a teacher” to use your phrase, i.e. that he’s not saying anything useful to others. (Obviously, I guess, or I wouldn’t have written a post about his essay.) For starters, the point I’m making a big deal about above is definitely something I’ve never read in any other writing about math education, even though people are writing about how to teach proof all the time. (This month’s Mathematics Teacher is a proof “focus issue,” for example.)

      And I’m convinced he’s right on this one. The development of proof over the course of mathematical history is always a response to crises of knowledge, where the certainty of formerly held knowledge comes into doubt. (For example, 18th century calculus had a serious lack of rigor by modern standards. There were persistent criticisms from outside mathematics, like the famous essay by George Berkeley, but it wasn’t until Fourier’s work on trigonometric series threw a monkey wrench into everybody’s understanding of functions that Cauchy finally began to answer these criticisms.) Looking back on my own mathematical development, my experience echoes this as well. Even though I had lots of experience with logic puzzles as a kid and loved them, there were many areas of mathematics where I had no interest in rigorous proof until I started to have experiences that caused me to need it in order to feel sure of anything. I remember as a high schooler looking at a proof of the chain rule in some textbook and feeling totally bored – c’mon, dy/dx obviously equals dy/du * du/dx. Why the fuss? It wasn’t till I took an abstract algebra course in grad school that my adult sense of rigor really began to take shape – finally I was trying to get a hold of things I knew I didn’t have a strong intuitive sense for. In other words, just as Lockhart is saying, I had to find out that my intuition sometimes wasn’t a reliable guide before I appreciated proof fully.

      Finally, Lockhart’s point explains something I’ve consistently experienced in teaching proof and in coaching teachers who were teaching proof – a sort of lack of organicness to the enterprise. This manifests in several ways: students producing work with no logical flow, for example. Or, saying things like “we know x^2-1 = (x+1)(x-1) because 10^2-1 = 100-1 which is 99, and 11*9 is 99 also.” You ask this kid “does it work for every x?” and they say “Yes. I tried 3 and 7 as well.” This used to aggravate me (especially since probably the prior week I’d had the conversation with that class that 3 cases can’t be a proof because you’re trying to show it works for every case), but after reading the discussion in Mathematician’s Lament it makes perfect sense. They’ve never seen an equation that works for 3 numbers but doesn’t work in general! So to them, that’s all the proof they need. I was trying to make them do something stilted and inorganic based on their prior experiences.

  2. I can get behind Lockhart in theory, but when I see that kind of rhetorical artillery aimed at teachers, I stop paying attention. We know. We already know. We just don’t have the power to change things, I mean really change them, in ways that matter.

    I’m interested to see what kind of response you get for your call for problems with misleading apparent patterns. I’m drawing a blank, except for how people thought the Fermat number 2^32 + 1 must be prime for a while until they figured out it wasn’t. But the fact that 4,294,967,297 isn’t prime isn’t going to smack anyone in the head.

    One thing I keep learning and relearning is that things that are obvious and uninteresting to me aren’t necessarily obvious and uninteresting to my students. Like for example you can give them 2 lines on a grid and ask them if they think they are perpendicular, and then can they prove it. Or you can give them three lengths and ask them to prove they do or don’t make a right triangle. These are nontrivial if you’ve never done such a thing before. Just keeping that word in play in a nonthreatening context is helpful.

    1. Actually I was planning (maybe tonight? although I’m also trying to get the gf to go out to a movie…) to write a followup post with a few more problems with misleading apparent patterns, and all the ones I have are about primes. I agree the Fermat primes case isn’t useful with kids for this purpose (goodness sakes, it took EULER to find the factorization!) although it’s a fun historical anecdote illustrating the need for proof.

      I love the problems you’re suggesting. I think they could be used to serve a similar goal to the one I’m talking about. (I.e. forcing kids beyond the surface by having the surface appearance be wrong.) For example, sides of length 11, 13 and 17 form a triangle that’s _almost_ but not quite right-angled – if they draw it, they’ll be wrong. They have to dig deeper. The only thing missing is a way for them to realize they’re wrong without me pointing out something connected to the pythagorean theorem. In the points-on-a-circle problem this happens when you draw the 6th dot and then count. Ideas?

  3. Not misleading but, not obvious – take any point in an equilateral triangle, and the sum of the distances to each side is constant. This was an article in NCTM Math Teacher a few years ago – a surfer washes up on an equilateral triangle shaped island and wants to get to any side of the island as quickly as possible, so where should he build his shelter? It’s surprising when you start measuring and notice that it doesn’t seem to matter where you put the point, the sum of the distances is always the same. And THEN there are at least 3 proofs of it, probably more, so there are lots of directions you can go.

    1. That’s a hot problem! Explaining the concurrence of the altitudes, medians, angle bisectors, and perpendicular bisectors are great problems too, in a similar way. (Surprising results. The quest for an explanation of the surprise motivates a journey into the land of proof.) I’m still hoping for more problems where the first impression is actually wrong, though. Keep your eye out! I will too.

  4. I’m a science teacher and I think what you’re describing resembles what we call “discrepant events.” In science it’s pretty common for us to set up a demo or lab and have students predict what will happen and then proceed to BLOW THEIR MIND! Ok…it’s usually not quite that exciting but the balloon in the bottle tends to freak them out.

  5. I’m not sure how to use this, but one of the earliest place kids tend to make an intuitive leap that doesn’t work is in adding fractions. There is a strong temptation to just add the numerators and the denominators.

      1. I started the blog when I was on sabbatical. I still add to it occasionally, but not very often.

        It changes because there are two different shapes that produce bad cases. One of those shapes–the powers of 2 shape–dominates when N is small. But then the other shape takes over when N is big.

  6. I’m not a mathematician- just the parent of a three year old who already seems to like math. So, because my own math education was so stressful-boring I’ve been reading quite a bit about ‘what is math’ with the hope that what fascinates him now won’t be killed by the education that’s coming. Anyways- a crisis moment: my kid is pretty good at arithmetic and basic number sense. He ‘knows’ that everyone ‘knows’ that 1 + 1 = 2. The other night he was playing with an old piece of gum and said, in a really excited way- “look!”, as he snapped the gum apart. 1 piece of gum plus one piece of gum equals 1 piece of gum! Isn’t that weeeeird!?” We’ve been trying to figure out why for a few days. This makes me realize that dwelling in the conflict here is greeeeat.

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