Several months ago, I asked you, “You’re about to plan a lesson on concept [x] and you’d like students to find it interesting. What questions do you ask yourself as you plan?”
There were nearly 100 responses and they said a great deal about the theories of learning and motivation that hum beneath everything we do, whether or not we’d call them “theories,” or call them anything at all.
- “How can [x] help them to see math in the world around them?”
- “How can I connect [x] to something they already know?”
- “How can I explain [x] clearly?”
- “What has led up to [x] and where does [x] lead?”
You can throw a rock in the math edublogosphere and hit ten lessons teaching [x]. They might all be great but I’d bet against even one of them describing some larger theory about learning or mathematics or describing how the lesson enacts that theory.
Without that theory, you’re left with one (maybe) great lesson you found online. Add theory, though, and you start to notice other lessons that fit and don’t fit that theory. When great lessons don’t fit your theory about what makes lessons great, you modify your theory or construct another one. The wide world of lesson plans starts to shrink. It becomes easier to find great lessons and avoid not great ones. It becomes easier to create great ones. Your flywheel starts spinning and you miss your highway exit because you’re mentally constructing a great lesson.
Here is the most satisfying question I’ve asked about great lessons in the last year. It has led to some bonkers experiences with students and I want more.
- “If [x] is aspirin, then how do I create the headache?”
I’d like you to think of yourself for a moment not as a teacher or as an explainer or a caregiver though you are doubtlessly all of those things. Think of yourself as someone who sells aspirin. And realize that the best customer for your aspirin is someone who is in pain. Not a lot of pain. Not a migraine. Just a little.
Piaget called that pain “disequilibrium.” Neo-Piagetians call it “cognitive conflict.” Guershon Harel calls it “intellectual need.” I’m calling it a headache. I’m obviously not originating this idea but I’d like to advance it some more.
One of the worst things you can do is force people who don’t feel pain to take your aspirin. They may oblige you if you have some particular kind of authority in their lives but that aspirin will feel pointless. It’ll undermine their respect for medicine in general.
Math shouldn’t feel pointless. Math isn’t pointless. It may not have a point in job [y] or [z] but math has a point in math. We invented new math to resolve the limitations of old math. My challenge to all of us here is, before you offer students the new, more powerful math, put them in a place to experience the limitations of the older, less powerful math.
I’m going to take the summer and work out this theory, once per week, with ten skills in math that are a poor fit for other theories of interest and motivation. As with everything I have ever done in math education, your comments, questions, and criticism will push this project farther than I could push it on my own.
The first skill I’ll look at it is factoring trinomials with integer roots, ie. turning x2 + 7x + 10 into (x + 5)(x + 2). All real world applications of this skill are a lie. So if your theory is “math is interesting iff it’s real world,” your theory will struggle for relevance here.
Instead, ask yourself, “Why did mathematicians think this skill was worth even a little bit of our time? If the ability to factor that trinomial is aspirin for a mathematician, then how do we create the headache?”
2019 Jan 18. The Directory of Mathematical Headaches.