Painful “Proofs”

Last Friday I spent far too long looking through my tutee’s notes searching for the “correct” way to name the property that allows one to subtract the same quantity from both sides of an equation. The ridiculousnes of spending n>0 seconds on this pursuit led to my infamous tweet of the other night:

This is a slightly inaccurate depiction of what went down. In perfect honesty, I didn’t say that I hated properties; I said that I hated “these types of problems” i.e. algebraic properties style “proofs.” But that didn’t sound as snappy, so I changed it for the interwebs.

The thing is I love proofs. I was the totally nerdy freshman in high school who got jazzed about doing two-column geometry proofs. In hindsight, two-column proofs are the red-headed stepchild of real proofs, but, cut me some slack. I didn’t know any better at the time and it was my first real exposure to the idea of proving something rigorously.

You see, I am someone who loves being right. I can be the most bull-headed, stubborn pain in the you-know-what that you’ve ever met. In fact, I so dislike being wrong that I will do just about anything to avoid having to make a guess about something. Don’t  believe me? Just ask Avery. He’ll vouch for me.

Proofs are like manna from heaven for someone who loves to be right ALL THE TIME. They let you give the intellectual “up yours” to any doubters that might be lurking around every single time you write QED, or draw that little black square at the end of the paragraph.

So I love — and I do mean LOVE — proofs.

But I HATE the way we teach proof to students. It makes me die a little on the inside when I see kids having to prove that -1x=-x. Or that multiplication distributes across subtraction, just like it does across addition. Gasp!

The problem with these “proofs” (and I use that word with reluctance) is what I am going to refer to as the Duh Factor. No one doubts that either of these ideas is true. Not even for a second. Everyone knows they are true because they’ve been using these ideas for years by this point in their mathematical careers. You don’t get that smug satisfaction of flipping someone the mathematical bird by proving that -1x=-x. It’s an obvious, accepted fact.

No one cares.

Proofs are the persuasive writing of math. Writing a letter to someone to try to convince them of all the reasons why two-year-olds shouldn’t be taking the LSAT’s isn’t going to impress your English teacher. Why then are we making students do the intellectual equivalent in math?

If we want to amaze students with the inherent beauty, the power, the sheer coolness of math, we need to stop giving them problems with the Duh Factor and start giving them concepts to grapple with that truly stretch their thinking in new ways. That blow their friggin’ minds. Because what makes proofs truly powerful is their ability to prove that something which seems at first unbelievable is actually, unequivocally true.




I’ve been catching up on my Google Reader and have noticed–perhaps it’s just the posts I’ve decided to read–that there have been several recent (well, not necessarily recent) posts about giving written feedback to students and whether or not they read these comments. I remember spending my time and energy on writing comments on student tests and quizzes in my first year of teaching, and then feeling that stab of agony when some kid raised his hand and asked if he needed to keep it. Or finding the quizzes crumpled up on the floor at the end of the day. And then, when on another assessment I decided just to mark it and not give any comments, I’d have kids come up to me and ask me what they did wrong on a particular problem. Damned if you do, damned if you don’t.

The school I taught at last year, and the one I’m teaching at right now, are really a different environment. From the get-go I noticed that students really took comments to heart. They read them, they tried to understand what had happened with their test or quiz or other graded assessment. So, I feel like a lot of what I’m about to offer may be harder to implement in a setting where this isn’t the norm.

I also feel the need to qualify my description of these super-responsible students I’ve depicted. You do need to bring the level down a step or two for my current middle schoolers. They run the gamut from very responsible, perfectionistic, no I will not let you come in again to finish your test kids today to spacey, distracted, OMG! we have a test today?-what’s it on? kids. You know, regular middle school kids.

But part of the process of getting students to the point where they look at feedback and think about how to apply it is to give them opportunities to interact with feedback. Things like test corrections, or SBG reassessments, help students to see why reading comments and talking about them with their teachers is important. The process that I started using this year (stolen from my former co-teacher, now at-home-Mommy) is called a “recycle”. A recycle is different from test corrections I ways that I really like.

The format is that student fold a piece of paper along its vertical axis (i.e. a “hotdog” fold), or they just draw a line down the middle. The left-hand side is for students to correct all of the problems they made errors on in the quiz/test, and the right-hand side is for students to explain what kind of error they made and/or why they made that error. The second part is what I really like. Students have to really think about what kind of mistakes they are making. Are they careless errors? Are they big conceptual errors? Did they make a lot of errors involving negative signs? Did they not know the meaning of a word? As they do these recycles, students are involved in a high level of metacognition. They have to tease apart what and how they were thinking at the time they took the test and think about how that thinking has or might change now.

My eighth graders, who have been doing this process for over a year at this point, have a really good understanding of what kinds of errors they make and why. My seventh graders are still learning what makes a good recycle. Some of them continue to have trouble with the format. But they are getting there.

No matter where I wind up next school year, whether I wind up staying on at this school or continuing in my roving-teacher ways, I plan on bringing the recycle with me. After all, it’s good for the planet.

Radio Silence

I’m sitting in my living room, staring off into space, thinking about what I want to write. The words just aren’t coming. I don’t know what it is about this school year, but I find myself with little to say. I have been lurking on twitter, reading other peoples’ blog posts, but not contributing much to the dialogue.

I’m trying to unpack just why that is. When I get together with other math teachers (or science teachers–Hi Jason!) I have plenty to say. I feel like an active participant in the conversations that are occurring. Why is it that I feel like I have nothing to say in the online conversations that are happening every day?

Last year I was trying out SBG and so I was thinking about that a lot. This year I am pushing myself in ways that lie outside of my daily practice. I am speaking at three different conferences, but in the classroom, I am not doing anything that is really new for me. Which is probably why I find myself logging on to my blog and then leaving without typing a word. However, that should change for the better in the next few weeks.

I will be giving my first talk in the beginning of December, so sometime in between now and then I will need to think through exactly what I plan on doing. And I would like to share what I do with those of you who won’t be at Asilomar this year.

And then in January, Geetha and I will be co-facilitating a workshop at the Creating Balance conference in SF. We are going to be talking about status in the math classroom. That one should have me thinking about all kinds of deep ideas.

And finally, I will be speaking at NCTM in Philly this April. The three-ring circus that NCTM can wind up being should be good for at least a post or five.

So, bear with me while I muddle through this bout of writer’s block (which, thankfully, hasn’t been hitting me with regards to my NaNoWriMo novel). I’ll come through it, keyboards a’blazing and knock your socks off. Just wait and see!