What’s In the Way of Making Students Prove, part II

I am continuing my thoughts about Kate’s provocative comment regarding why it’s hard to ask students to prove something significant. Kate said:

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.

As I said last time there are two big questions here – (1) why can’t they start? and (2) why does it feel unfair and scary to ask them to? Last time I talked about (1), so now –

(2): why does it feel scary and unfair to ask students (say, in 10th grade) to seek a justification?

Here I have two ideas to offer:

a) Time pressure.
b) The Contract. (Which is turning out to be my favorite idea from that article by Patricio Herbst I wrote about last time.)

(a) Bob and Ellen Kaplan talk in their book Out of the Labyrinth about the need for a sense of relaxedness or leisureliness around time in order to run their math circles. Learning proof is the same thing. Creativity can’t be rushed. Any time I’ve successfully gotten an individual or a class to prove something at all difficult, without me intervening to suggest key ideas, the one constant has been that it took longer than expected and we had that time to give it. Every time, afterward I felt sure that the time was much better spent than it would have been any other way. But if in any way I feel a pressing need for the topic or question to get wrapped up, this makes it practically impossible for me to perceive and do what needs to be done to support the class in creating the proof without giving them too much help. It has to be okay for people to sit there stumped for a while and not make any progress. It has to be okay for people to take what I know is a wrong turn and find out it’s a wrong turn on their own. And I have to be listening to the conversation the right way – listening for the direction the ideas are taking, the obstacles coming up, and searching for the least obtrusive possible thing I could say to be helpful when they grind to a halt. If time pressure is also buzzing in my head I can’t listen that way.

I’m now getting into touchy territory but because of my conviction that the most powerful math teaching I’ve ever done has been outside of time pressure, I’ve started to believe that we as a profession need to actively fight to keep our curriculum and standards not just unreasonably overloaded but actually kind of slim and when they’re not and we can’t do anything about it, to actively prioritize depth over coverage in the face of them. I have the luxury of not having a full-time classroom job anymore, so I recognize that my point of view about this is kind of facile. On the other hand, this year I have been working as a coach in a middle school/high school, and it’s clear to me in that context that everybody’s need to get through the whole curriculum is directly at odds with their desire to do something really substantive with any one of the many topics they have to cover. So I’ve been making the case that we need to choose what the most important topics are, and feel the license to: a) treat them very slowly, b) aim for the students to prove the main results, and c) take as long as that takes.

I’ll illustrate one negative effect of time pressure with the second half of the story about my tutoring client I began to tell last time. Fresh on the heels of her triumph in creating and justifying an algorithm for factoring, I botched the next move, violating my own repeated advice to you. I was getting ready to go, but I was excited about her accomplishment, and I wanted to show her its power, and I also didn’t want her to be left with the impression that every monic quadratic can be factored over the integers. So I said something like, “You’re now in a position to prove that x^2+10x+7 can’t be factored into linear factors. How?” She did it, but it took several minutes and I ended up leaving late. Most of the time was spent in a back-and-forth in which I repeatedly realized she wasn’t sure what I was going to consider as an answer to my question. Also, she first offered that it couldn’t be factored because 7 was prime, in spite of the fact that earlier she had factored at least one trinomial that had a prime constant term. (This is a classic case of what’s wrong with a “prove X” problem – by telling her it was unfactorable, I unplugged her from her own logical process to determine whether or not it was.) I feel totally sure that if I’d just said, “Factor x^2+10x+7,” she would have taken the same few minutes realizing it was a trick question, and then she would have said, “you can’t do it.” And when I asked her how she knew, she would have provided a totally coherent proof on the spot, on her own.

Now, why didn’t I just say this to begin with? Well, I was getting ready to go – I just had two or three minutes. In that context, the good pedagogical move – the trick question, asking her to factor the irreducible trinomial – didn’t feel fair. Of course, the actual time we were supposed to end wound up being irrelevant, because I stayed late to clean up the mess I’d made by being in a rush. So my sense of being in a hurry didn’t even get us done faster.

I think this is illustrative. It certainly illustrates a dynamic I’ve been part of often enough. You feel the clock or the calendar. In that context you feel like it would be unfair to put the students in the position to struggle for a long time with no guarantee of when they’ll find what you’re contemplating asking them to look for. So you push through it, and do the heavy lifting yourself, or leave it undone. But if the topic has any subtlety (for example, if students proving something is an object), this isn’t good enough. Often enough, you end up backtracking and reteaching and losing the same time you would have lost by doing it right in the first place, and you’re in a rush because you’re now behind, so you still don’t do it right. And by “you” I mean “me.”

Anyway, again this is the perfect segue –

(b) When I read that Patricio Herbst article, I was so irritated by the theory-heavy style that I cringed at every “theoretical construct” he introduced. (Goodness sakes, how did I survive college as an anthropology major?) This caused me to miss, on the first two passes, that one of them is awesome. Namely, the “didactical contract.” (But can I skip the “didactical” please?) He has got me thinking:

Whenever something feels unfair, I should be asking –

“What is the unspoken agreement between me and my students according to which it is unfair?”

And, once I have an answer –

“Do I like this agreement or do I want to change it?”

The case at hand is Kate’s scenario – you’ve just had kids explore an object and you’ve succeeded in getting them to notice a pattern and make a conjecture about it; you’ve asked them if they can account for the pattern and they are stumped; and asking them to stick with this question feels unfair and scary. What is the unspoken agreement making it unfair?

Okay this is where if you have ever been in this situation you write a comment.

I have been in this situation, so here’s mine:

Reflecting now on past experiences of having this feeling, the common theme is that in one form or another, I’d promised them success if they do what I say. Over and over, I’d reassured them that “all you have to do is do the work and you’ll a) learn the content and b) get a good grade.” The way this promise played out day-to-day was a more immediate promise that if they actually applied themselves to any given task I assigned them, they’d conquer it. In this context, the thing that makes asking them to stick with the task of explaining the pattern they’ve observed feel both unfair and scary is that it violates my promise! The fact is that if the pattern has some subtlety, it’s conceivable that they’ll all sit there forever, apply themselves diligently, and never “in a zillion years” (to quote Kate) come up with a worthwhile explanation for it.

Once this is clear, I have to ask what my reasons were for making this promise. Well, that’s simple. Every full-time classroom job I’ve had has been in an urban public school environment where it was quite hard to get the majority of students to do the work in the first place. Since getting them to do the work was obviously the first step in getting them to learn anything, it seemed totally logical to make this one act the sole key to success. How natural does it feel to take a student who’s got serious questions about the whole ‘school’ enterprise and say, “look, all you have to do is do the work and you’re golden”?

Illustratively, the one class I taught in which it was not a struggle to get students to do the work was AP Calculus, and that was also the course where I felt the most license to give them a really bad*ss open-ended, maybe-nobody-will-get-it type of problem. (E.g., check it: let b take every value from 0 to 6, and draw each line segment in the first quadrant that connects (0,b) to (6-b,0). The union of these line segments is a region bounded by the axes and a very attractive curve. Find the area of this region.) Obviously the “AP” in the title gave me this license; but the truth is that this goes hand-in-hand with the fact that in that class, there was much less of a reason for me to communicate the message that all you have to do is the work.

Anyway, retrospectively I think this contractual agreement (in all classes but AP Calc) cost me more than it bought me. It put a cap on the amount of creativity I could ask of my students, and ultimately, engaging with math creatively is what makes it rewarding. For example, it was hard to ask students to prove something subtle. I now believe that the tasks that I avoided because they felt unfair are actually central to kids achieving the type of learning I want for them. So what ended up happening was that I violated my promise anyway. Even if you did the work, it wasn’t a guarantee that you’d learn what I wanted you to learn.

I think the issues I faced are pretty general, but my big point here is not the specifics of these issues, but just the question – if a task feels unfair, what is the unspoken agreement (the Contract) making it unfair? And is this contract worth it?

Addendum, Saturday April 10, 7pm:

I didn’t mean to end on such a down note. I actually think this reflecting-on-the-contract thing creates some really powerful and exciting opportunities for us. Actually, a cool project that I invite any of you, and myself, to take on, is to write down the ideal contract between us and our students, and then make it explicit with our students. Some inspiration:

*As Kate E says in the comments, Sam Shah describes in great detail an awesome occasion where he explicitly revised the contract he had with his calculus class, and then the new contract took effect, and the instruction felt powerful and new.

*JD2718 is constantly writing about his teaching in a way that leaves me impressed with how much intention and attention he has given to the contract between him and his students. For example, his students don’t necessarily expect him to tell them the truth:

Also, for those of you who like this sort of thing, two groups finished in what I considered too short a time, so I lied and told them I thought that there answer was too small. Now, they know I lie, but they also know that I know a lot, so they have become more used to responding, “we think we are done because….” which I consider a good thing. I don’t want them to stop because I say enough, but rather because the mathematics suggests that they have finished.

8 thoughts on “What’s In the Way of Making Students Prove, part II

  1. I’m not so sure we can just decide to change the ‘contract’. Most of my students have some serious expectations about what is supposed to happen in a math class, based on their previous classes. Anything I do differently, they experience as ‘violating the contract’ to some extent.

    Any ideas about how to build a new contract together?

    1. Yes! Basically I just second everything Kate E says below. Sam Shah’s post is a great example of how it’s done. It’ll be harder with your students because they’re older and therefore more used to what they’re used to, but it’ll still work if you stick to your word about it.

      Do you have specific unspoken expectations in mind that you want to change the contract on? It might be productive to get into specifics.

      1. I think I need a full post. I’ve made progress on reducing topics (minimal) and on removing time pressure (a bit more than minimal).

        But I’ve totally shredded the “contract.”

  2. What I did here would probably be a violation of the “contract” if I were to give it in class. It is probably unnecessarily work-intensive. The benefit I was trying to confer to the student is a faith in his own ability to break a bigger problem into smaller ones he can already solve, at the time without realizing it.

    I just remembered a time when I took Calculus in high school and the teacher was going over things I already knew so I made up random “difficult” questions to challenge myself. I thought that it would be fun to give it to a student see how far he could take it.

    The benefit of giving this question to the student was that every question he subsequently dealt with involving the chain rule (single/multiple applications with other rules mixed in) was no longer challenging. He’s climbed a much higher mountain.

    In terms of promising success. In high school math there are definite answers (in my experience) to everything. It feels more like skill building to the more advanced topics kids will encounter in college. Aside from the more abstract conceptual questions, there’s nothing “difficult” in high school math that we present. Only what is doable and what’s not. If it’s doable, no matter how many steps, it’s just more of the same thing. Even proofs that we do seem to not exceed 10 steps. You try enough things, and something works. Same as verifying trig identities. Fear of failure is a problem here. Students feel that chasing a dead end is a waste of time when it informs them and may give them insight to the problem. Sometimes teachers don’t help by showing only the “correct” way so there’s an expectation that they should be able to get the correct answer the first time.

    I agree that when the time constraint is lifted, like it is when students come in after school for tutoring. We get better quality learning. The rush to beat the bell leads us to make decisions that seems to be more to CYA than it is to help students learn.

    I’d like to hear your thoughts on Sue’s question above.

  3. Hi Ben – In total agreement about advocating for paring down required content, and making intelligent choices to prioritize. But you probably knew I would agree with that, because I scream and holler about it often enough.

    I think explaining the reluctance to ask students to delve into proof with the contract violation expectation is apt. I notice when I do start asking students to think about proof, I also rapidly lose the room. Maybe it’s merely a side effect of large classes. But it also seems like deviating from the expected script gives them a license to lose focus.

    You asked for an anecdote so I can tell you just yesterday, we worked a problem given the coordinates of the vertices of a quadrilateral, and we were supposed to prove that if we connected the midpoints of the four sides, it made a parallelogram. A student wondered aloud if that works in any quadrilateral (“It was cute, he was all like, whoa, it’s like the Davinci Code!”) so I tried to seize the moment – everybody drew a different quadrilateral on graph paper, found the midpoints, connected them, everybody’s looked like a parallelogram. I opened geometer’s sketchpad and constructed it there so we could look at a dynamic example. But at this point they were convinced it was always true and lost interest in thinking about why or developing a proof. The exploration ground to a halt and conversations started up about other stuff.

    I can’t help but blame this a little on the schooliness of school – they’ve mastered doing the minimum to get a grade that will keep their parents off their backs. The contract as they understand it doesn’t require learning or curiosity, it just requires going through whatever motions we have set up that are assigned a grade. It’s difficult to ask them to do anything that obviously you can’t grade.

    Anyway, thank you for the thought-provoking posts and the conversation.

  4. Re: Time constraints: Sometimes when I look at how math is taught and the amount of time spend reviewing concepts that student supposed learned in earlier class, I think that the underlying theme of the typical math curriculum is “we don’t have time to teach it right, but we do have to time to teach it again.”

    @Sue So I’m not yet a teacher, but based on parallels with personal relationship, the only way I know to change an implicit contract is to first make it explicit. Say “this is what we’ve done in the past, but now we’re going to try something different.” What Sam Shah does at the beginning of Day One in this post is a great example of clearly stating a break from an implicit contract.

  5. Need to remove topics. That’s huge.
    Need to defeat the time pressure.
    And need to break the contract.

    It’s more than I can fit in a comment, but let’s start anyhow.

    Time pressure.

    Think, teachers. Think of the length of the period. Think of your 150 performance indicators or whatever monstrous name for “topics” you use in your state. Think of the length of the period. Is it perfect for each of those 150 topics?

    Now, in most places your administrators will truly believe that you have exactly enough minutes to do exactly what you should. Stupid, and powerful. Not our friends. Not our students’ friends.

    But there are places where, if you cover what you should cover, they’ll look the other way if you do extra. There’s even a few administrators who might like it.

    So you, with the easy-going supervisor, now look at your topics. 48 minutes each? Nah. What’s the minimum you need? Pad it up 10%, 20%. Don’t forget that on any given day a class will grind to a halt as they collectively decide that x + x = x^2 .

    So do the calculation, and see that you have a full period lesson, another with 5 minutes dangling, another with 5, another with 15… How often, even with padding, can you find a chunk of 15 minutes? Or half an hour? Once a week? Once every other week?

    Most teacher I know, if they don’t have supervisors breathing down their necks, can find 20 minutes or so once every other week. How are they going to use it?

    How are you going to use it?

    Extra practice problems? Start the homework early?

    We have a chunk free from time pressure. Could you use it to do something different?

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