In response to my last post, Kate Nowak raised the most important set of issues I can think of:
I will absolutely stipulate to all of this:
This is why insisting on too much formality too early is bad for people who are learning how to prove… 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… 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.
But I need some SERIOUS training in how exactly one goes about teaching that way. I wasn’t taught that way and none of my colleagues teaches that way. Sometimes I feel like I get close, because I make the kids investigate and measure and conjecture (today, for example: median of a trapezoid), but then I stop before asking them to prove it. Or I do say something like “Why should that have to be true? Can we come up with some kind of explanation?” But they have no idea how to even start and it feels unfair and scary to ask them to. It would not occur to them to draw a picture and extend the legs and think about similar triangles, in a zillion years.
Thank you Kate for getting this conversation started. There are 2 very big questions here:
1) Why don’t they know how to start?
2) Why does it feel unfair and scary to ask them to?
These questions are bigger than me, but here are my 2 cents. I’m posting my thoughts on question (1) now because I wanted to get this up. I’ll post on question (2) tomorrow or Friday.
I have 3 thoughts to offer about why they don’t know how to start.
a) Inexperience on the students’ part.
b) Failure of the question to hook into the natural processes students need to use to actually prove things.
c) The vicious cycle.
(a) A big reason they don’t know how to start is that in most cases, no one has asked them for this type of thing before (at least not with any follow-through – more in (c)). If they’re in high school, they’ve had a long time to formulate an idea of what is going to be asked of them in math class, and typically this isn’t it. The content for which they’re being asked to seek justification has 10 years of increases in sophistication since kindergarden, but most students’ development in terms of the creative act of seeking justification is still at the kindergarden level. The later the first occasion when the students are asked to find justification, the harder it will be for them.
People who have been consistently made to justify their mathematical beliefs for a long time know how to start. I know this from private tutoring. If I have sufficient time with a kid, I have the luxury of requiring them to find a reason for every piece of content they learn in school. (I acknowledge readily that this is a luxury, and I only have it if the parents and student have made a sufficient investment of time in tutoring.) The task quickly ceases to be disorienting when it’s required every time they learn any fact. In a recent comment JD2718 wrote about the need to “play math” (meaning, in context, seeking explanations and counterexamples for patterns in numbers) well before a proof course. This is the same point. Seeking explanations and counterexamples is the main activity of research mathematicians. You can’t go from 0 to 60 on this practice; you have to start out slow and ease into it.
Consequently I think it’s totally essential that justification infuse math learning from K on up. (As I said elsewhere, I think that the commutativity of multiplication should be treated as a major theorem needing a thoughtful proof.) This is going to require some major PD for elementary teachers; I actually would love to run some of this PD. Anyway, the fact that this is not the current state of the art adds up to big problems for high school teachers who try to do something more authentic and creative with proof after at least 9 years of schooling during which a typical student has never or hardly ever been asked to come up with a reason to back up their belief. (At least, not with follow-through – again, more in (c).) Why would it ever occur to them to extend the legs? Think about similar triangles? The act of coming up with a proof is essentially creative. You don’t just get creative on the spot in a domain in which you’ve never created before.
(b) Another variable is the way the question is posed and the expectation for what will count as an answer. Especially when you’re first learning about proof, a request for justification has to hook into some natural processes in order for you to respond to it effectively. If you’re an experienced prover, you can hook a problem into these processes intentionally, but when you’re starting out, you can’t. The problem has to be posed just-so to make your natural processes connect to it.
Here’s what I’m getting at: seeking an explanation for something that puzzles you is a natural act. Accounting for your belief is another one. Outside of math class, if you assert a belief and somebody says, “why do you think so?” you’ll probably answer the question fluently. Coherent or not, you won’t be disoriented by the question – you’ll have something to say. And if something is actually puzzling you, you are actually slightly irritated until you’ve made the whole thing resolve itself in one way or another. (Some of us are in the habit of shrugging and going “well, that’s just a mystery of life.” This is one way to make it resolve. Even so, I don’t believe this is anyone’s first response to puzzlement, at least in non-academic contexts, unless there is a lot of weed involved.)
The act of mathematical proof is supposed to plug into these two natural processes. They are what give you both motivation for sticking with the question, and direction in searching for an answer. But in math class, justification questions often fail to hook into these processes. The big example is what I keep harping on: if the problem is “prove X,” and you’re new to this game, you already know X because the question implicitly told you it. Your “this is bothering me” process is missed entirely since there’s no uncertainty anywhere, and your “why do you think so?” process can only be honestly answered by “you told me to prove it,” which doesn’t count as a proof. It’s better if you didn’t start out knowing the right answer, but even here, the question can fail to connect to these processes. For example, suppose you are measuring something, and you notice a pattern, and you make a conjecture about it. If you’ve never seen a pattern in math class hold for 5 cases and fail later, then all you have to do is notice the pattern and you’re satisfied. Again, nothing is bothering you because you don’t have any uncertainty. Meanwhile, the honest answer to “why do you think so?” is “it worked 5 times.” This is one reason why it’s important for teachers at all levels not to act like something is established fact once it has been noticed as a pattern, and why I’m still hoping people contribute to my earlier call for problems that involve a pattern holding for the first few cases and failing later. (Eventually if I get enough stuff I’ll put the brainstorm in a single post. Thanks to all who have contributed so far.)
The question is most effective if it taps these processes. One way to do it: pose a problem – not a proof problem, just a figure-it-out problem – that the students don’t already know a process for, but that’s easy enough for them to find a solution. Then pose another problem that can be solved the same way. Then another one. Pretty soon the students have developed an algorithm. Now, the question “how do you know that works?” taps the “why do you think so?” process. The students do think so, for some sort of mathematical reasons, because they themselves devised the algorithm, in response to the original problem. They can come up with a good answer. I had a great time doing this very thing with a tutoring client two weeks ago. She’d just learned how to FOIL. I posed to her some simple factoring problems, such as . They were all reducible monic quadratics. She came up with one of the standard methods, totally on her own: “This really isn’t that hard. You just think of all the numbers that multiply to the last number, and you see which ones add to the middle number.” I asked her why that would work and she didn’t miss a beat, since the whole thing was her idea in the first place. She was a total Clever Hans when I started working with her two years ago. Yes, I’m proud of both of us. I botched the very next move, though – more in the forthcoming followup post.
To tap the other process you have to generate some sort of cognitive dissonance for the students. Ideally, I would like my students to experience cognitive dissonance the minute they see a pattern that is not yet explained – why (the hell) is this happening? In my experience it helps to have this attitude myself. (E.g. sometimes I act kind of paranoid when something happens 2 times without being proven, and increasingly agitated if it happens a 3rd time.) But this is only cultivated over time. To generate cognitive dissonance in students who don’t already care about justification, they have to see something happen that contradicts their intuition. JD2718 recently made a recent suggestion that strikes me as along these lines (I’ve never tried this one myself):
“9 has 3 factors (1, 3, 9). 6 has four (1, 2, 3, 6). So, even numbers have even numbers of factors, and odd numbers have odd numbers of factors. Right?”
This conclusion would appeal to the intuition of just about every student I’ve taught. I’d probably have to prod them to even get them to test another example because they’d already be convinced. If I do get them to try another one, I’ll make sure the first one I ask about fits the pattern (say 10, or 25) – so they’ll be even more convinced. In that context, the first counterexample they come across is going to bother them. Then the “what (the hell) is going on here?” process is engaged.
At any rate, these two processes, or the like, are needed to orient students as they try to prove things. I think a big part of why students flop when we ask them to justify is that the question fails to hook students into these processes.
(c) One final thought about why they have a hard time proving: there’s a vicious cycle at work here. They already do have a hard time. That means we all know that if we try to get them to do it, it’s going to take a really long time, or they’re going to fail miserably, or both. In that context, it’s very hard to take this on. It’s even harder to follow through and really make sure it happens, and not to cop out in some way (for example, as in the article I wrote about last time). But this just exacerbates the situation I described in (a) above. Often the students in front of us are so inexperienced not because nobody has ever contemplated trying to get them to prove anything before but because many people have contemplated it and then opted not to, or tried it and then given up. Ultimately the task felt unfair in some way. This is the perfect segue into question (2), which I’ll engage in a followup post in the next day or two.