# Creating Balance III / Miscellany

The Creating Balance in an Unjust World conference is back! I went a year and a half ago and it was awesome. Math education and social justice, what more could you want?

If you’re in NYC and you’re around this weekend, it’s happening right now! I’m going to try to make it to Session 3 this afternoon. It’s at Long Island University, corner of Flatbush and DeKalb in Brooklyn, right off the DeKalb stop on the Q train. I heard from one of the organizers that you can show up and register at the conference. I’m not 100% sure how that works given that it’s already begun, but I am sure you can still go.

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I’ve just had a very intense week.

I want to get some thoughts down. I’m going to try very hard to resist my natural inclinations to a) try to work them into an overall narrative, and b) take forever doing it. Let’s see how I do.

(Ed. note: apparently not very well.)

I. Last spring I wrote

20*20 is 400; how does taking away 2 from one of the factors and 3 from the other affect the product? We get kids thinking hard about this and it would support the most contrivance-free explanation for why (neg)(neg)=(pos) that I have ever seen.

Without going into contextual details, let me just say that if you try to use this to actually develop the multiplication rules in a 1-hour lesson, all that will happen is that you will be dragging kids through the biggest, clunkiest, hardest-to-swallow, easiest-to-lose-the-forest-for-the-trees, totally-mathematically-correct-but-come-now model for signed number multiplication that you have ever seen (and this includes the hot and cold cubes). This idea makes sense for building intuition about signed numbers slowly, before they’re an actual object of study. It does not make any sense at all for teaching a one-off lesson explicitly about them. (Yes, the hard way. I totally knew this five months ago – what was I thinking?)

II. I gave a workshop Wednesday night, for about 35 experienced teachers, entitled “Why Linear Algebra Is Awesome.” The idea was to reinterpret the Fibonacci recurrence as a linear transformation and use linear algebra to get a closed form for the Fibonacci numbers. Again, without going into details –

I gave a problem set to make participants notice that the transformation we were working with was linear. I used those PCMI-style tricks like giving two problems in a row that have the same answer for a mathematically significant reason. This worked totally well. Here is the problem set:

Oops I guess I failed to avoid going into details. Anyway, the question was about how to follow this up. I went over 1-4 with everyone (actually, I had individual participants come up to the front for #3 and 4) at which point the only thing I really needed out of this – the linearity of the transformation – had been noticed by pretty much the whole room. One participant had gotten to #9 where you prove it, and I had her go over her proof.

I think this was valueless for the group as a whole. The proof was just a straight computation. You kind of have to do it yourself to feel it at all. It was such a striking difference watching people work on the problem set and have all these lightbulbs go off, vs. listening to somebody prove the thing they’d noticed. It almost seemed like people didn’t see the connection between what they’d noticed and what just got proved. I told them to take 5 minutes and discuss this connection with their table, but I got the feeling that this instruction was actually further disorienting for some participants.

I’m trying to put the experience into language so I get the lesson from it.

It’s like, there was something uninspired and disconnected about watching somebody formally prove the result, and then afterward trying to find the connection between the proof and the observation. Now that I write this down, clearly that was backward. If I wanted the proof (which was really just a boring calculation) to mean anything, especially if I wanted it to be at all engaging to watch somebody else do the proof, we needed to be in suspense about whether the result was true; either because we legitimately weren’t sure, or because we were pretty sure but a lot was riding on it.

This is adding up to: next time I do it, feel no need to prove the linearity. Let them observe it from the problem set and articulate it, but if there is no sense of uncertainty about it, this is enough. Later in the workshop, when we use it to derive a closed form for the Fibonacci numbers, now a lot is riding on it. If it feels right, we could take that moment to make sure it’s true.

III. As I work on my teacher class, something that’s impressing itself upon me for the first time is that definitions are just as important as proofs. What I mean by this is two things:

a) It makes sense to put a real lot of thought into motivating a course’s key definitions,

and maybe even more importantly,

b) Students of math need practice in creating definitions. You know I think that creating proofs is an underdeveloped skill for most students of math; it strikes me that creating definitions might be even more underdeveloped.

Definitions are one of the most overtly creative products of mathematical work, but they also solve problems. Not in quite the same sense that theorems do – they don’t answer precisely stated questions. But they answer an important question nonetheless – what do we really mean? And to really test a definition, you have to try to prove theorems with it. If it helps you prove theorems, and if the picture that emerges when you prove them matches the image you had when you started trying to make the definition, then it is a “good” definition. (This got clear for me by reading Stephen Maurer’s totally entertaining 1980 article The King Chicken Theorems.)

Anyway this adds up to an activity to put students through that I’ve never explicitly thought about before, but now find myself building up to with my teacher class:

a) Pose a definitional problem. Do a lot of work to make the class understand that we have an important idea at hand for which we lack a good definition.

b) Make them try to create a definition.

c) If they come up with something at all workable, have them try to use it to prove something they already believe true. I’ve often talked in the past about how trying to prove something you already believe true is very difficult, and that will be a problem here. However, unlike in the cases I had in mind (e.g. a typical Geometry “proof exercise”), this situation has the necessary element of suspense: does our definition work?

If they don’t come up with something workable, maybe give them a not entirely precise definition to try out.

d) Refine the definition based on the experience trying to use it to prove something.

I’ll let you know how it goes. I’m excited about it because it mirrors the process that advances mathematics as a discipline. But I expect to have a much better sense of its usefulness once I’ve given it an honest whirl.