In the library today, I tried to run a little experiment on a friend.

I told him that there was a relationship between the numbers in the circle and the numbers to the right, and I asked him to try and guess what the relationship was between the numbers at the bottom.

The catch was that I didn’t have any relationship in mind when I made this “puzzle.” Really, I wanted to see if I could get him to say, with confidence, that the answer to the puzzle was “21”.

Here was the “experiment design”:

- I used sets of three numbers, so as to not literally resemble any multiplication facts.
- In the first set, I gave three numbers whose product was easily recognizable as 16. I expect that my friend would make that connection.
- Then, I give him numbers that explicitly contradict the pattern in the first bubble.
- Finally, I made up some numbers that had very little to do with the others.
- Then, I asked him a question that was really similar to the first bubble. (Though I shifted the numbers around so that it didn’t literally match the first one.)

My friend said some really interesting things, most of which I didn’t expect.

The first thing he told me was that there was a pattern with the first and third bubbles, but not the second. He said the relationship was “multiply the top number by the bottom, and then multiply that by the number on the side.” In other words, he was telling me that 12 times 9 was 78. But he knew that didn’t work for the second one, so he was puzzled. After that, I pushed him to guess, and he told me that it would be 27 for the bottom set, which he quickly corrected to 21.

There are a few things that I think are interesting about this all:

- While I didn’t succeed in prompting him to answer the question with confidence, that first, easy relationship between 4, 4 and 1 was enough to create an expectation of a multiplicative relationship between the numbers. He wasn’t really thinking about what else it could be — he was worrying about how to deal with the fact that the second set didn’t fit that pattern.
- In fact, that expectation was so strong that he made the mistake of thinking that 12 times 9 was 78. This was a computation-heavy problem, and no-doubt his mental resources were heavily taxed. But it seems to me that the expectation of a relationship was enough to prompt the error here. This isn’t a standard computational error, after all. Maybe he figured out that 8 was in the units digit of the product, and that was enough to settle his mind.
- In the end, he did think that the bottom set was 21, even though he was troubled by that second set.
- Even though this was far, far from a careful experiment, I’m starting to spend more time explicitly thinking of ways of testing our ideas about math mistakes and their origins. Coming up with a collection of made-up operations and relationships seems to me as if it would be an important part of this sort of investigation.

One small lesson that I’m taking away from this short experiment is how sensitive our expectations are. If we see 5^1 just once, is that enough to raise our expectation that future exponent calculations will also have a multiplicative relationship? And are those expectations the seeds of our intuitions about operations?

I don’t know the answer to those questions. It’s all very up in the air for me.

But it seems worth worrying about, if you find yourself introducing a new operation to students. It seems prudent to me to introduce exponents with examples that don’t resemble multiplication facts.

You’ll comment, right? And give suggestions on ways to tweak this experiment, or ways to make it more careful, or other ideas that seem worth investigating?