I went to Andrew Stadel’s afternoon session at TMC yesterday: Math Mistakes and Error Analysis: Diamonds in the Rough. I got out of it a fun technique to get students thinking, and some new perspective on building deep knowledge.
Simple, and an awesome way to get students thinking beyond answer-getting to the mathematical structure. I’m excited to give this a shot in my class this year.
Building off of Andrew’s ideas, I’ve been thinking a great deal about how students come to understand broad, abstract ideas. Dan Willingham has been a source of great thinking here. I’m coming more and more firmly behind the approach that students build abstract understanding through the variety of examples of an idea they encounter, and from looking at the connections between them — it’s less about finding that “perfect example” and more the sum of all the little things we do to get kids thinking. Building this takes time and hard intellectual work. While I was in Andrew’s session, I jotted down this idea on the back of his handout:
This is inarticulate, obviously, but it was my brain trying to get out an illustration of the “pyramid of abstraction”. There’s this big idea of exponents. It starts with some simple relationships, which we like to codify into rules. It builds up through more complex relationships and different examples, and leads students (hopefully) to a broad and transferable understanding of exponents. I’m stealing from Jason Dyer’s ideas here “teaching to the negative space” — that we need to teach what a concept isn’t, as well as what it is. The pyramid of abstraction starts with simple relationships, and this includes both the positive and the negative. It reminds me of something Dylan Wiliam quoted during his conversation with folks at PCMI last week — “Our memories aren’t good enough to remember algorithms perfectly. We need to understand so we can do a kind of ‘in-flight repair'”. In the same way, as student understanding is starting to develop, it may not be enough to try to remember relationships that are true — humans forget. If students are also examining and remembering relationships that are false, it provides one more means to move their thinking forward and build strong, flexible conceptions of math.
There was a great deal more in Andrew’s talk that I am skipping over — he posted more resources on his site here and we had great conversations and looked at other techniques. But this little tidbit — one idea of what I want to bring to my classroom, and a move forward in my thinking about how students learn — is what I am taking away from his talk.