Here are some math mistakes (source: MathMistakes.org). What do you notice?
To draw things out, kids are doing the following:
- 3 ^ 9 = 27
- 5 ^ 2 = 10
- x ^ 0 = 0
- 100 ^ 1/2 = 50
Why do kids do this? Here are 3 options:
- They don't understand exponents, and are just guessing.
- They are reasoning consistently within a model, but a mistaken model. They think that 5^2 means "You have 2 5's. That gives you 10." They think that 2^0 means "You have no 2's. That gives you 0." They think that 100^1/2 means "You have half of an 100, which is 50."
- Kids are not reasoning explicitly at all, but rather have a strong intuition that exponentiation should be solved using multiplication.
In short, we could be dealing with guessing, reasoning and intuitions. Which of these is right?
I dismiss the first option. If kids were guessing, then why don't they ever add the exponent and the base? Why don't they ever subtract? At best this explanation is incomplete.
The second option is more attractive. A kid answers exponentiation problems by saying "3^2 is 9, because two 3's make a 9" and extends that incorrectly: "3^0 is 0, because no 3's make a 0." The idea is that kids are reasoning about exponentiation in an explicit way, but that this explicit way is mistaken.
The third option is that kids are not reasoning explicitly. A search for a model would be beside the point -- kids have a strong intuition, in certain contexts, that exponentiation should be treated as multiplication.
How can we distinguish the second and third options?
- If the second option is right, then kids should need to pause to reason before incorrectly evaluating an exponent such as 100^1/2 as 50. If the third option is right, then kids should just be able to shout that answer out very quickly.
- If the second option is right, then kids should be able to explain their reasoning. Relatedly, we'd expect students without any strong model for exponents to be unable to provide any answer at all to unfamiliar exponentiation problems.
- If the second option is right, then kids should operate consistently. They shouldn't sometimes reason according to the model and sometimes not. (Or, alternatively, the fact that they inconsistently apply this model would require explanation, one potentially provided by the third option.)
I don't have all the evidence that I need to knock out the second theory, the idea that kids are explicitly reasoning about exponents. But, from what I've seen, kids have answers to the exponents questions WAY too quickly for it to be explicit reasoning. I think that there's something to the theory that this is an intuition.
So what's going on? There is a strong connection between exponentiation and multiplication. Everyone learns this strong connection. And in unfamiliar contexts the brain falls back on the intuitive connections between exponentiation and multiplication, and answers the question "What's the base times the exponent?"
Why? For reasons that I've tried to articulate before, I think that kids sometimes see harder problems as easier ones.
How could we prove or disprove this specific idea? Here are some predictions of my lil' theory:
- There should be other strongly connected operations, and we should similar mistakes when we ask kids to do tough things with those operations. A likely suspect would be subtraction of negative numbers, which asks kids to take subtraction into unfamiliar territory. There's even a bit of evidence that they treat that stuff like addition in these contexts.
- We might even find evidence of more of this stuff in the early years of schooling, as kids are just learning their operations. MathMistakes.org could use some Elementary School submissions.
- The best way to support my idea would be to artificially induce the sorts of mistakes that I'm talking about in students. The idea would work like this: I would define a new operation. Kids would show proficiency with it. Then, I'd define another operation in terms of the first. Kids would show proficiency with that one too. The next part is fun. Then I'd ask kids to extend the second operation in an unusual way, and see if they spit out the value of the first operation.
I'd like to see more stuff like this
I took a bunch of examples of student errors, I tried to unify them under some sort of theory that would make sense out of them. I tried to think through the theory to consider its competitors. I'm considering what would count as evidence for and against my theory. I'm trying to find testable predictions of my theory.
In other words, I'm trying to participate in the science of how kids learn stuff. And I think that more teachers should do that. Especially since understanding student errors would be widely valuable outside the classroom, but is easiest to theorize about when in the classroom and in interaction with warm bodies. Especially since digital cameras make it easy to collect lots and lots of evidence of how kids mess stuff up while you're grading.
Especially because it's fascinating, and I want to know more about it. Come up with a theory and write about it. How do people reach mathematical decisions?