Dan Meyer talks a lot about motivating mathematical concepts. For instance, I would use Andrew Stade’s Cup Stacking task to motivate y = mx + b form for linear functions. I just wrapped up a unit on transformations, and I used Robert Kaplinsky’s Ms. Pac-Man to motivate the unit.
I think I’ve learned something new about creating an intellectual need for mathematical concepts–and more specifically, to create an intellectual need for motivating conceptual mastery. I’ll make an example from scientific notation earlier this year.
My first lesson was pretty lame. I introduced the challenge of communicating very large or very small numbers in an efficient way, showed the conversion, and had students practice. Not my best work. What I imagine went through a less-motivated and likely-to-struggle students’ mind during that lesson goes something like this.
Mr. Kane is really excited about something again. He keeps mentioning space and physics and chemisty. Boring. Ok now he’s explaining the thing I have to do to get a good grade. All the numbers have 10 times something and an exponent. I count the number of 0s? Yea that works for that one example. And sometimes it’s negative and sometimes it’s positive. Right is negative, left is positive? Cool, got it. I wonder what’s for lunch today…
This is the kind of thinking that gets mediocre students most of the way there, keeps them under the radar, and doesn’t lead to much in the way of conceptual mastery. And it’s hard to get kids beyond this. I gave an exit ticket that day, and lots of students displayed simple misconceptions–counting 0s rather than how many times the decimal moved, and mixing up positive and negative exponents.
The next two days we dove into operations with large numbers, with and without scientific notation. All of a sudden, if kids couldn’t convert between standard and scientific notation they would be totally lost. Asking harder questions–even if some of them were pretty lame–created an intellectual need for understanding scientific notation. My exit tickets the next two days re-assessed scientific notation in addition to operations, and mastery on scientific notation skyrocketed.
I saw something similar with transformations the last two weeks. If I focused on just the process of a translation, kids found ways to get confused. I could dig into their misconceptions and fix them, and there was absolutely value in that. Or I could introduce a tessellation project, and almost everyone will figure out translations along the way.
This intellectual need can absolutely backfire as well–I don’t want to go down the rabbit hole of higher rigor, more challenge–it’s about finding a sweet spot between rigor and accessibility. That said, I think this says something important about moving forward with content even if only 70 or 80% of kids nailed the day’s lesson.