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.

Andrew talked a great deal about the value of mistakes, and using those mistakes as opportunities for students to learn. His first proposal was simple, and looked like this:

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.

BrianReblogged this on Capture Their Interest and commented:

I like what Andrew has done here, but I wonder if asking students to identify the mistake or the misconception might be useful.

howardat58Asking students to identify the mistake is probably the most useful thing that you could do in this case (and in most others as well). You may find that those who manage to get the “correct” answer have got there by accident. (full comment below)

dkane47Post authorDefinitely hear that. I’m not going into great detail on what Andrew talked about — this was put forward as a way to introduce a new topic. Definitely agree that question is a useful one.

Lee MacArthurOhhhh I needed this. I keep trying to convince my students that it is ok to make mistakes. They want to turn in papers that are 100 percent correct and will not accept it when I tell them, we learn from mistakes. This has given me some ideas to use in class. Thank you.

howardat58It is completely clear to me that the mistakes shown in Andrew Stadel’s example are due to complete misunderstanding of exponents.

Here is a suggestion:

Definition: number A raised to power n is “one multiplied by A, n times”.

So 3 to the fourth (3^4 in computer speak) is 1 x 3 x 3 x 3 x 3

Yes, I know that this has the same value as 3 x 3 x 3 x 3, but it makes 3 to the power zero equal to 1 with no hand waving, magic, tears etcetera, since it says that we are not multiplying 1 by anything.

Not only that, negative powers appear (magically) (I mean naturally) as a consequence of dividing by 3, and 3 again, and 3 again…..

With a tiny stretch of the imagination one can find meaning for fractional powers.

For example 10 to the power of one half. This is now one times 10, half a time. Yes, meaningless until we do it again, and it is totally reasonable that doing “multiply by 10, half a time” twice is equivalent to multiplying by 10 once.

It has always seemed to me that if you understand something then the rules are often blindingly obvious.

Pingback: Making Warm-Ups Meaningful | Five Twelve Thirteen