September 15, 2017

I'm Done with Two Column Proofs.

Wow.

It really has been a while since I've posted here, hasn't it?

But I suppose having a mini-crisis in my geometry class that has forced me to reject our textbook's philosophy on what "proofs" should be is a good enough reason to resurrect this blog.  I finally have the time this year, and I feel the need to share my probably overly opinionated beliefs about math education.

This is the first year I have tried to integrate proofs into our school's geometry curriculum across the board.  (In the past, proofs were only discussed in honors classes, but I felt somehow that reasoning through how you knew something was true was important for everyone.)  I was trying to justify the two-column format, as much as I hate it, as a way to "scaffold" student thinking --- yay educational buzzwords!  But when I actually did it, it got exactly the reaction that I knew it would --- it just served to overly obfuscate the material and utterly drain the life out of it.  I realized I should have stuck to my guns and listened to the likes of Paul Lockhart and Ben Orlin.

So, after some reflection and course correction, here's the email I just sent my students.

---
Hello mathematicians. I have a rather bizarre request. 
Don't do your geometry homework this weekend. 
Yes, you read that right. Don't. 
Let me explain.
We've spent the past couple of days looking at "proofs" in geometry. The reason I say "proofs" in quotations is that, in all honesty, I don't believe the two-column proofs that our book does are are all that useful. I have actually been long opposed to them, but against my better judgment, decided to give them a shot anyway and make them sound reasonable. But you know what they say ... if you put lipstick on a chazir*, it's still a chazir. (They do say that, right?) The thing is, that style of proof just ends up sounding like an overly repetitive magical incantation rather than an actual logical argument --- as some of you pointed out in class today. I truly do value that honesty, by the way, and I hope you continue to be that honest with me. 
Here is what I actually will expect of you going forward. It's quite simple: 
I expect you to be able to tell me how you know something is true, and back it up with evidence. 
That's it. 
It may be a big-picture kind of question, or it may be telling me how we get from step A to step B, but when it really comes down to it, it's all just "here's why I know this is true, based on this evidence". It doesn't have to be some stilted-sounding name like the "Congruent Supplements Theorem" either --- just explain it in your own words. That doesn't mean that any explanation is correct --- it still has to be valid mathematical reasoning. You can't tell me that two segments on a page are congruent because they're drawn in the same color, or something silly like that. But it doesn't have to be in some prescribed way --- just as long as you show me you really do understand it. 
With that in mind, by the way, I'm also not going to be giving you a quiz on Monday, either. Instead, we're going to focus on how to make arguments that are a lot more convincing than just saying the same thing in different words. I think you'll find that Monday's class will make a lot more sense than the past few classes combined. 
So, relax, take a much-deserved Shabbat, and when we come back, I hope to invite you to see geometry the way I see it --- not as a set of arbitrary rules, but as something both logical and beautiful. 
Shabbat Shalom.

*  I teach at a Jewish private school.  "Chazir" is Hebrew for "pig", which has the added bonus of being non-Kosher.  Two-column proofs are treif... at least in the context of introductory geometry.

P.S.  I am not saying that two-column proofs NEVER have a place in mathematics.  I am merely saying that introductory geometry, when kids are still getting used to much of geometry as a subject, is not the proper place to introduce the building of an axiomatic system.  Save that for later courses for the students who choose to become STEM majors.

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