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

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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?

  • I agree that the preconceived notion is like priming (http://en.wikipedia.org/wiki/Priming_(psychology) ), but I wonder if they could think about it and in the absence of other good ideas, stick with that one. I also found it interesting that your subject mentioned an order in the multiplication as if that were important to getting the result.

    So, here are ways to change the experiment which might help:
    1. Put the “obvious” one 2nd or 3rd with a non-obvious one first. I wonder if it being an “obvious” relationship and first makes a difference sets expectations or if it doesn’t matter when you’re grasping at straws.

    I mean, in the absence of a real pattern, I might’ve gone with 21 as my answer for your example because I was running out of ideas and that was the only pattern that made sense. Sort of an over-reliance on the Occam’s Razor thing. My thinking would be, “There’s no way Mr. Pershan would expect me to get the pattern if it was any more complicated than [some line I set in my head involving maybe 3 orders of operation and relatively basic operations], so I’m not even going to look at it as something like ‘solutions to quadratics where the three numbers are coefficients of a polynomial in standard form.'” Does that make sense?

    I guess I’m saying that I have an expectation of the difficulty of problem you’re giving me and if I don’t see a pattern that is below that expectation, then either: a. I’m missing it, b. there’s no solution, or c. it’s not really “fair” in the sense that you are asking me to devote more time than I want for this little “puzzle.” So, once I’ve put in what seems like a “fair” amount of thought and I can’t come up with anything, then I go with the best available option that fits my expectations of “fair.” Maybe I’m even thinking, “Maybe Mr. Pershan messed up the multiplication himself on #3 and maybe #2 the numbers got copied from the book wrong” or something.

    2. Another experiment to mess with the order: Maybe have a slightly harder pattern (like multiply two of them and subtract the third or something), but make the easy example the third one. That is, maybe examples 1 and 2 could have larger numbers, so multiplying them in my head would be harder, so I’d skip right to the 3rd and then check the pattern on the first two.

    I think this could help us know if it’s the order in which the examples are presented is more or less important than using difficult or easy examples in any order. That is, at some point, I want to present the example 5^1 which teaching exponents, but how soon am I allowed to do that without corrupting their thinking?

    3. I wonder if you had some examples where the first one fit two patterns and the others fit the harder one, would it be enough to push for the easy method. Like the first bubble has the numbers 2, 2, 1 and the patterns could be “add all 3” (2+2+1 = 5) versus “a^b + c” (2^2 + 1 = 5). Especially if the other two have “almost results” like your example 3 here was for your subject, this might be an interesting result.