Annotated Bibliography on Proof, part I – The Double Bind

In an earlier post I made a brief reference to an article about proving that I had just begun to read. I’ve now read it.

Patricio Herbst, “Engaging Students in Proving: A Double-Bind on the Teacher” in Journal for Research in Mathematics Education, Vol. 33, Number 3, May 2002

It’s available from JSTOR but I couldn’t find it for free.

This is a very thought-provoking article. A heads-up that it is very dense. It reminds me of the social-theory-heavy stuff I used to read for my college anthropology major.

I am going to attempt to distill the arguments in a concise and readable way. Afterward, I’ll provide some commentary.

Herbst’s argument

Herbst analyzes a classroom episode he observed and videotaped, of an advanced 9th grade integrated math class, in which the class tried to create a two-column proof of a simple proposition about angles. He sees the episode as “a typical case of a teacher engaging students in proving.” (p. 185) The proof had been assigned as an exercise for homework the previous night and the students had struggled with it. The teacher asked the class to construct a proof together; they made a little progress but then floundered. They began, alternately, to offer unproductive variants on the premises, and to assert the conclusion without adequate justification. The teacher intervened and suggested the key ideas to make progress. She framed her interventions as advice for future situations. Afterward, the students complained that the proof had been too hard, and the teacher said something to the effect that she wouldn’t assign too many problems like that on a test.

Herbst argues that:
a) When the teacher and students undertook the proof exercise as a class, they entered into a sort of “contract” where they were each expected to do a certain kind of thing.
b) The specifics of this contract placed conflicting demands on the teacher, and also prevented the students from taking certain steps that were needed to complete the problem.
c) This explains why the teacher intervened to give the key ideas in the proof, why she framed her interventions as she did, and why afterward, the students saw the proof as particularly hard, and why the teacher granted this.

More broadly, Herbst is making a case that two-column proof exercises tend to be doomed to lead to this sort of situation. Even more broadly, that any task that attempts to isolate proof as the skill being learned is similarly doomed. In other words, that in learning proof, form can’t be separated from substance. Learning about proof has to be integrated with learning the actual mathematical content.

Here is how his argument goes:

1) In the class, “proof” was understood as “transforming premises step-by-step into a conclusion by a sequence of logical deductions, with a reason cited for each step in the sequence.” With this understanding, the teacher’s job on the given occasion was to provide the students an opportunity to “do a proof,” and the students’ job was to “do a proof.” Thus the teacher and students entered into a “contract” with the following clauses:

Teacher’s Job: I’ll give you a task in which the hard part, the part that requires your thinking, is organizing the information into a chain of logical deductions. The task will be fair, i.e. you should be able to do it.

Students’ Job: We’ll take the givens and transform them step-by-step into the conclusion.

2) In order to do her job according to the contract, the teacher picked a task with the following features:

*it explicitly asked for a proof
*the proposition to be proved was stated at the outset
*the “givens” were identified at the outset
*the number of concepts involved was small
*a diagram was provided with a labeling that supported reading it a certain way

(Here is the task: the problem; the diagram. This is on p. 183; the problem came from Integrated Mathematics 2 by Rubinstein et. al., published 1995 by McDougal Littell.)

The task was designed to try to keep the students’ work limited to formally organizing ideas into a proof. Its design overtly (e.g. via the statement of what was to be proved) and covertly (e.g. via the labeling of the diagram) suggested the ideas that the students were supposed to use, so they wouldn’t be responsible for coming up with these ideas and could concentrate on the formal task of organizing them into a proof.

3) In spite of the way the task was set up, the students floundered when they tried to do their job. One way to explain this is that the intended proof had two points where it required students to do something other than transform the givens step-by-step. One was to glean information from the diagram that was not explicitly stated in the givens (specifically, that angle ABC is composed of, so its measure is the sum of the measures of, angles CBD, DBE, EBF, and FBA); another was to write down an equation based on this information and manipulate it algebraically. Students did not produce these moves; this could be a product of the fact that, adhering to the letter of the contract, they thought they were supposed to just keep looking at the givens and transforming them, so these more flexible, broader-scope moves didn’t occur to them.

4) When these difficulties emerged, the teacher was in a double bind. On the one hand, if she intervened and said “you may need to look at these broader types of moves,” she would be giving away too much information; if the students then succeeded in completing the proof, it wouldn’t feel like they really did it themselves (more or less, this is what actually happened). On the other hand, if she didn’t intervene and allowed the students to keep working as they had been, the proof might never get completed. They would probably be left with frustration and the impression that the task was too hard. It would seem as though the teacher hadn’t done her job according to the contract.

5) The teacher dealt with this situation by intervening and framing her intervention as “advice for future proving endeavors.” By doing this she transformed the activity from “an opportunity for students’ joint production of a proof” to “an opportunity to learn a strategy for future use in proving.” (p. 199) This validated the time spent, but the students still didn’t really do the proving.

Herbst makes the case that the dynamics at play here are going to be at play any time the formal skill of “doing proofs” becomes its own curricular goal. Here is his general argument:

6) As long as the goal is the formal skill of “doing proofs,” the teacher needs to create proof tasks where the only thing the student is asked to do is organize ideas into a logical proof. If the task also requires the student to creatively generate the ideas for the proof, then the task isn’t fair – a kid might be perfectly able to organize a logical argument, but still fail, because they couldn’t think of these ideas. So the teacher needs to relieve the student of the burden of generating ideas for the proof by overtly and covertly giving these ideas to the student ahead of time. (For example, by stating the proposition to be proved at the outset, and/or by providing a diagram labeled a certain way.)

7) But if the task is organized in this way, then students are likely to fail to pick up on the covertly-suggested ideas, or correctly use the overtly-suggested ideas, because they are under the impression that what they are supposed to do is logically transform the premises step-by-step into the conclusion, but really they are also supposed to pick up on all these secret cues the problem is giving them. The form of a proof inevitably depends too much on the specific content for the students to be able to produce it just by trying to transform premises. When this breakdown happens, the teacher is in a double bind: s/he can’t intervene in the middle of the process to direct students to the hidden ideas without making them feel like they didn’t really do the work; but s/he can’t let them just continue barking up the wrong tree till kingdom come.

Herbst concludes that learning the skill of “proof” can’t be separated out from the mathematical content being proven. “Any sort of tools and norms that teachers can use to engage students in proving must allow room for the teacher to negotiate with the class what counts as proof in the context of the investigation of specific, substantive questions.” (p. 200)

My thoughts

A) The classroom excerpts in the article felt extremely familiar (both from classes I’ve taught and watched). In particular, the way the initial progress ground to a halt and then students either just transformed premises unhelpfully or stated the conclusion without justification.

Also familiar from my own practice, teaching a “proof” unit in an Algebra I class in 2001-2004, was the dynamic where you search specifically for problems where the ideas in the proof aren’t too hard to come up with, so students can concentrate on the task of organizing the proof and not get tripped up on generating these ideas. And how problematic this becomes if the students don’t come up with the ideas even when you tried to make them easy. At a few points while reading I felt confused by Herbst’s contention that choosing an exercise that “made the substance of the expected proof available to students” (p. 191) was something the teacher was doing in order to fulfill her “contract” – I mean, it didn’t work, right? The kids flopped anyway! So what good is it doing the teacher? But then I realized that I knew exactly what he was talking about from my own experience trying to teach a “proof unit.” I remember thinking thoughts like, “this problem doesn’t require too many difficult insights, so it’ll be good for getting them to focus on the logical structure.” Of course they flopped anyway with me too, and so it didn’t do me any good. But the important thing is that I felt compelled by that thought while choosing the problem. The fact that I turned out to be totally wrong doesn’t mean that I hadn’t used that consideration. It amounted to me choosing problems specifically to avoid asking my kids to have to demonstrate any creative insight.

With hindsight this seems like an insane way to teach proof now. Talk about playing small. Coming up with creative ideas is what makes proving things fun. How dare I claim to be teaching proof and deny kids the opportunity to touch the most exciting part of it? Of course, the alternative, to ask kids to actually develop creative justifications, will never work either if proof is ghettoized into a unit of Algebra I or Geometry. (I’ve been doing math for 9 years without being asked to generate a single original idea, and now all of a sudden you want me to be creative? WTF?) Looking for proof should be a daily or weekly part of every single math class, all the way down. (But it shouldn’t be too formal too soon. More on this in (B).)

B) I buy Herbst’s case that the kids’ overly rigid understanding of what they were supposed to be doing (“transforming the premises step-by-step into the conclusion”) probably contributed to their difficulty. But I have a lot more to say about what was in their way. The real problem the kids were having is what I’ve talked about before: in the class session described in the article, there is absolutely no connection between the kids’ attempts at proof and their actual, real-live sense of what they know and don’t know. In fact, these two things have been forcibly, violently separated. That’s why the can’t do it. To get concrete, here’s an excerpt (Andie is the teacher):

“Andie then asked, ‘What are we trying to prove?’ – a question that students could answer but then were unable to offer further ideas. So Andie asked, ‘How are we going to get to FBD? Do we know anything about FBD?’ A student’s response, ‘Maybe it’s a right angle,’ led Andie to ask whether the students knew ‘what makes up FBD.’ The answer, ‘FBE plus EBD,’ justified by the ‘whole and parts postulate,’ started a small discussion as to whether they were then entitled to say ‘ABF plus EBD equals ninety degrees’ or whether they ‘[didn’t] know it’ yet.” (p. 184)

ARE YOU KIDDING ME? In this classroom, as described, “we don’t know that yet” clearly doesn’t mean we don’t know it yet. Everybody in the room in fact does know that ABF plus EBD is 90, because they were told to prove it (well, the equivalent), so it must be true. Not only that but it looks in the diagram like it’s true. When the students say “we don’t know that yet,” what they mean is “we don’t know if the mysterious authority that certifies that proofs are correct is going to allow us to claim that yet.” What they don’t realize (why should they? it’s been forcibly hidden from them) is that the only mysterious authority that should be involved is their own sense of conviction.

After all, the ultimate source of mathematical authority in the world is the collective conviction of mathematicians. That’s why our standards of rigor have changed so much over time. Calculus was practiced for well over a hundred years before Cauchy bothered to prove its central theorems from a (fairly) precise definition of limits. It wasn’t until the late nineteenth century that Dedekind saw a need to provide a rigorous construction of the reals from the rationals, in order to be able to prove theorems about the reals (such as the fact that they are complete).

Historically, what has pushed the mathematical community’s sense of rigor forward is not an insistence on greater rigor from an outside source, but the encounter with new ideas and examples that caused a crisis of knowledge. Hippasus’ proof that the diagonal of a square is incommensurate with its side, when all the Pythagoreans thought all lengths were commensurate, led to Eudoxus’ theory of proportions. The rigorization of calculus (beginning with Cauchy) was driven by the recognition that Fourier series didn’t behave the way anybody expected them to. (At least, according to this awesome book by David Bressoud.)

How people learn about proof and rigor is the same whether we’re talking about a class or about human civilization. We learn to prove by being challenged to convince ourselves of things. We learn rigor by encountering examples and ideas that throw our assumptions into question. If you believe something, but then somebody (like your teacher) whose authority you trust says to you “you don’t know that yet,” you are stuck. Your mathematical soul is at an impasse. You believe it but you believe somebody telling you you don’t know it yet. You get the message you can’t trust your own reasoning. And in this state you will never, ever produce a coherent proof. If there’s a gap or a flaw in your reasoning, then to grow what you need is to be shown this flaw on your own terms. To see an example that contradicts your assertion; or to hear a counterargument that debunks yours – but not from the mouth of an authority you would dare trust without thinking through it.

This is why insisting on too much formality too early is bad for people who are learning how to prove. The need for formal rigor has to be earned through crises of knowledge. If we want our students to develop an appreciation for a formal proof we have to show them counterexamples to arguments they produced less carefully. If someone is insisting on formality from you when you don’t have any reason to doubt something less carefully argued, you will get the idea that proof has nothing to do with what makes sense to you, what you find convincing. But you can’t produce a proof without being guided by this.

All of this adds up to the case I’ve made before, that saying “prove that such-and-such is true” is the wrong problem for students first encountering proof. The minute you say it, they know it’s true; and this gets in the way of their natural mathematical reasoning process giving them a readout on what’s true. Herbst is arguing that we go for problems like this because of the “contract” that says we are supposed to give them a chance to work on “proof” as opposed to something else. If they also have to figure out what is even true, that could feel like we’re asking them to do more than just prove something. The problem is that they will never learn how to prove something if we don’t ask them for more.

C) Herbst writes that, in the episode he analyzed, the “contract” forced the teacher to make sure the proof was completed. If the class had not been able to complete the proof, then the students might have interpreted their difficulty as meaning that the problem was too hard for them. (Of course this happened anyway.) Herbst says that the fear of this possibility would have made the teacher feel pressed to intervene. (p. 195)

I’m not sure that the teacher’s sense that she needed to intervene to move the process along was based on this consideration, but I suspect he’s right that it was forced by the implicit “contract.” Most math classes that I have observed or taught make sure any question the class works on jointly is resolved that period. In this way we create the expectation that if a class undertakes a proof (or any other juicy math task) together, the teacher will make sure it gets finished. This trains the kiddies to be uncomfortable with irresolution, and quite possibly to experience a class segment that ends on a note of irresolution as the teacher’s failure to do her “job.”

Of course, the discipline of mathematics is rife with irresolution. Questions posed, worked on, and not yet answered, are its lifeblood. They are fertile. Recently I’ve been reading about the history of algebraic number theory, and I learned that it was Ernst Kummer’s work on Fermat’s Last Theorem that led not only to the entire field of algebraic number theory, but to the notion of an ideal that is totally fundamental to ring theory.

I don’t have a straightforward conclusion to draw but given the fertility of irresolution in the history of mathematics, doesn’t it make sense to think about how to make our students more comfortable with it?

* * * * *

This is the first of what I hope will be a series of posts that engage with provocative pieces about proof. But I have to put this off for a month or two and switch gears. For the group theory class I’ve been teaching, I have been researching the history of the theory of equation solving, from the ancient Babylonians thru the birth of Galois theory. So my next post is going to be about two landmark algebra textbooks: Muhammad ibn Musa’s Compendium on Calculating by Completion and Reduction (the original Arabic name for which is totally the origin of the word ‘algebra’ – how awesome is that?), and Girolamo Cardano’s The Great Art which is the first published discussion of the general solution of cubic equations. I’ve read the first of these, just begun the second, and am totally excited to tell you about them. (In a week or hopefully not more than two.)

Proof again

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.

Nuggets II addendum: more problems…

This is a followup to my last post. I promised some more problems in which there is an initial pattern that’s wrong. Here is one more. It is not nearly as good as the points-on-a-circle problem I discussed before, for reasons I’ll say below. But I’m brainstorming here, and hope you’ll join me, so anything’s better than nothing. (And actually I think it’s a cool problem in its own way.) Thanks Kate, jd2718, and Gilbert for contributing ideas so far.

The problem involves seeking a simple formula that produces prime numbers only. As you probably know, in spite of centuries of research no such formula is known to this day. There is some fun history around this. For example, Fermat believed that 2^k + 1 was prime whenever k was a power of 2. It is prime for k=1, 2, 4, 8 and 16. However for k=32, the number is 4,294,967,297 which was found by Euler to be equal to 641 * 6,700,417. Now, even in the age of computers, no other prime of the form 2^k + 1 has yet been found. Of course, I’d avoid putting kids in the position of having to calculate 4,294,967,297 or to show that it’s not prime.

Anyway, the idea is to get a class engaged in a search for such a formula. My idea for how the lesson goes would be to try out some examples with them to show them what is being sought. Like, maybe 4n + 3 which equals 3, 7, 11 for n = 0, 1, 2 but then fails for n = 3, or, starting with p_1 = 2 and then recursively doubling and adding 1 which gives you 2, 5, 11, 23, 47 and then the next one fails. (A closed form for this last one would be
3*2^n – 1 for n = 0, 1, 2, 3, 4.) This second formula is a good replacement for Fermat’s conjecture, because it gives you 5 primes before it fails, just like Fermat’s conjecture, but the primes are a reasonable size and the one that fails (95) is obviously not prime. Anyway, once they understand what’s being sought, the problem is to find such a formula. They will totally fail and they have no tools that will help them, so don’t let them stew too long. Then, show them a very pretty creation of Euler’s:
n^2 – n + 41. This quadratic polynomial is (amazingly) prime for n=0, 1, 2, …, 40. So there’ll be some initial excitement as this one seems to answer the question. But actually, it can’t possibly answer it. And the class may be able to see that the n=41 case will fail without actually doing the calculation. Even if the calculation is needed, it can still lead to a cool conversation.

Now this isn’t as rich as the points-on-a-circle problem because the inordinate primality of n^2-n+41 is sort of a mathematical accident; there isn’t a rich story behind it (at least not one I’ve ever heard), so once the pattern is noticed and then broken there’s nowhere to go. But it does at least give students some experience of the fact that if a rule holds for small cases, it doesn’t mean it always holds. And the breakdown at n=41 is accessible to reasoning alone, without calculating. So it’s a win for the power of mathematical reasoning over raw pattern-noticing.

Other ideas in this vein? (Problems where there is a “obvious” or “apparent” pattern or conclusion that is actually wrong?)

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?