Sunday, September 2, 2012

Post-mortem on a mistake I made

James Tanton's latest newsletter is phenomenal. But I got stuck on this part:
What was driving me nuts was that I thought that this argument was too loose. So I tweeted my question to the author:

Then, I got a series of very helpful tweets from Justin Lanier:

But he wasn't answering my question! He didn't understand me. I had to restate the problem I was having. Justin patiently repeated his argument. Why didn't he get it? How could I explain my issue better?

Then, James Tanton offered some help as well, giving a nearly identical argument as Justin's:
That's when it hit me.

All of a sudden everything that Justin and James Tanton had said made perfect sense. My mistake was clear. I saw where I went wrong. I had messed up an argument, and not an especially tricky one. Besides, this was high school math -- the thing that I'm supposed to be teaching. I felt embarrassed.

Here are some teaching lessons that I want to take away from this experience, assuming that what I experienced is true of others too:
  1. When someone understands most of something, they're equipped to turn a misunderstanding into an objection, and it's much harder to convince a person that their objection is wrong than it is to correct a misunderstanding. The only thing that worked for me was (a) having a second person explain it to me (b) after a break from thinking about the problem. Which, by the way, means that
  2. kids shouldn't be forced to work through problems in order, unless that sequence is necessary. Moving between problems often helps when you're in a rut.
  3. Doing and thinking about math during the year is an important part of teaching. Moments like these remind me a lot of what it felt like to be clueless during high school and college in my math and science classes.
  4. This post is still true. My insecurities about learning could easily bleed into my teaching, but I shouldn't let them.

There was a more self-indulgent post that I wanted to write. I'll throw it in the afterword, though it probably belongs elsewhere.

Here's what I remember about math and science classes in high school and college:
  • Asking dumb questions that everybody else understood in the back of Algebra.
  • Finishing last on Calculus tests.
  • Having to go to office hours every night for Multi-Variable Calculus in college.
  • Trying to understand how my friends got their answers for Mechanics.
  • Not understanding any math lecture that I went to in college, ever. 
I'm slow. Some people are sharp, quick-witted, and that's just not the sort of thing that has ever really been true about me. The kind of difficult I had above is the kind of difficulty I've been having my whole life, as far as learning stuff goes.

To turn it back to students, some of them are sharp, some of them are slow. And I think it's important to remember that nearly every aspect of school celebrates the quick and sharp intelligence over the slow one. Let's not pretend that waiting 30 seconds before taking an answer to a question is enough (though it helps) to even the playing field. If you're slow you tend to do worse on timed tests and on homework. Your sharper classmates will solve more problems than you during class. If you're slow then you finish class and your notebook seems foreign.

Though, maybe being sharp and quick is part of what it means to be good at math. Thoughts?


  1. I didn't even take math in college. When I started teaching math in Kaplan test prep courses,I occasionally didn't know how to do a tricky problem, and there were often kids in the class who did know. I found this was a teaching strength, for me, because what interests me as a teacher is finding out why I didn't know something.

    I'm very good at certain kinds of math, but unless I was teaching geometric series I'd have to relearn everything in order to answer your question. But I'm not really like you, in that if I don't get something in math, explanations by someone who does get it will rarely help. The areas of math that give me a tough time are the ones that exploit my weak areas, usually involving spatial relationships, but also anything to do with symbols.

    I'm not slow, rather, I move and think extremely quickly. But my understanding is binary; I either get something or I don't. Most of everything, I get. In high school and college, when I *didn't* get something, that was the ballgame. I couldn't function. There was no middle. My big achievement in my mid-20s was learning how to find the middle, how to spend the time figuring out something I didn't undersetand.

    I actually wrote about that a while back, here:

    "And I think it's important to remember that nearly every aspect of school celebrates the quick and sharp intelligence over the slow one. "

    Not nearly as much as it used to. And it's worth remembering that you are as much of an outlier as I am, in different ways. There's a difference between being slow to grasp something and being unable to grasp something. Most people taking math are unable to grasp it--or at least, unable to grasp it in an isolated state, as we teach math today. Those who can get it, but need more time, are not a huge group.

  2. Cheesemonkey wrote recently about not feeling like an insider with math. Anyone who loves math enough to teach it probably benefits by being a 'slower' learner. You'll have more empathy for the students.

    I eventually was slow at understanding some of the high-level math, and ended up happy with a masters degree and not a phd. My experiences at U of Michigan made me think I didn't like math, and I am so lucky to have had a good program at Eastern Michigan University that made me realize I really did love math - I just didn't like being confused in class so much.

  3. I was at a math circle training this past summer and loved it. However, when I filled out the feedback at the end of the week, I'd suggested if the leaders could make it explicit in future workshops for participants to "never share an answer." While I enjoyed the problems posed and was eager to work through them, I was also very slow (apparently a lot slower than other people in my table group), and when someone in the group blurted out the answer, all the joy just bled right out of me. With Tanton's problem above, at least you were seeking an explanation, and from that experience your post drives the point home beautifully about different students' learning ilk that we as teachers must be more cognizant of. Thanks, Michael!

  4. Okay, I'm "sharp and quick" at math, but sometimes I'm just dense, or I make an assumption and don't realize it, or I have tunnel vision, or I overthink a problem.

    Slow meaning speed and slow meaning delayed understanding are not the same as slow meaning dim-witted. You learn differently from how you were taught, or you weren't ready for algebra at the arbitrary time the curriculum said you should be, or you missed some concept that was necessary for future learning. You've obviously learned to understand math better than you did in school.

    The lesson to learn from your afterword, and the first comment, is that school performance is a poor indicator of actual ability. Don't accept the idea that a student "can't", instead figure out what needs to change so he can. Do you need to explain it differently, does he need a different environment, does he need intrinsic motivation, is it just the wrong time?

    I see it over and over. When kids don't learn X at the prescribed time in the prescribed way, they get labeled and marginalized and told, overtly or covertly, that they "can't". And they start to believe it. And usually it becomes true.

    "Moving between problems often helps when you're in a rut."

    Not only that, but taking a break. Think about something not math. Go to the restroom, doodle, play with a koosh ball. The arbitrary school schedule of doing one subject for an hour or two and then stopping abruptly wasn't designed for the unpredictable human brain. So find ways to break up the class period and/or give kids some autonomy to do what they need to do. And give them koosh balls. ;o)

    Kelly Holman

  5. Giving up, going to bed, and trying again in the morning was the ONLY way I figured out half the stuff from my problem sets in college. Thanks for a reminder of the need to come back to math with fresh eyes... I definitely don't make enough note of that while I'm teaching.