These mistakes are mine. I was tasked with checking the associative property with matrix multiplication, i.e. checking that (AB)C = A(BC).

This was my first attempt. See the problem?

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By my estimation, this is a fairly straightforward mistake. I was explicitly trying to remember a routine, and I remembered the wrong one. I don’t have any particularly deep appreciation for how matrix multiplication works, so there was nothing except memory of the routine for me to draw on.

(I have vague recollections of the concept that matrices represent functions from various real spaces, but nothing close to deep enough to ground this algorithm.)

I looked up a few examples, and got myself back on track. Still, I’m not particularly adept at matrix multiplication at the moment, and I found it cognitively taxing. (That’s a feeling that I can distinctively recognize. I feel it most often when I’m doing calculations or attempting to recall lots of numbers. It’s a feeling of overflowing.)

I made a mistake that I’d characterize as a working-memory issue. Do you see it below? I even circled it.

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So far we have two sorts of mistakes in this problem: a mistaken routine mistake and a overtaxed working memory mistake. (Those names aren’t great. But you know what I mean.)

What else is there to discuss?

Well, what about the particular working memory mistake that I made? In my calculations I put a “10” where an “8” ought to be. In my mind, I was attempting 1*2 + 2*3. Why did I come up with 10 instead of 8? Was it mere chance? Could I just as easily have come up with 7, or 23?

I have a pet theory, one that I have no clue how to prove. I believe that there are certain numbers that we remember as being especially connected. To our minds, I’d suggest, the numbers 100 and 1/2 are especially connected with 50 — and not with 99.5, 200 or 100.5. Certain numbers are clustered with others, with varying degrees of strength, and in various moments our responses can reveal these deep connections between the numbers themselves.

Of course, I have no evidence for this pet theory. I’m not sure how to test it, though I’m wondering if some variant of what I tried with my friend last week might be workable.

  • Funny, when I first saw the three matrices, I immediately started multiplying the left two in my head and thought, “I wonder if he puts 10 instead of 8 in the upper right”. I believe the clustering hypothesis, but have not consciously described it. Thanks!

    In doing the matrix product, I had to resist my brain thinking the two products to be added would be 4 and 6. The issue definitely is working memory — I compute the 1*2 first, then 2*3, and at that point I have to remember the first product. It is very tempting to mistakenly think it was 4 instead of 2.

  • Definitely on board with the number associations. I have a number anti-cluster. I can add most single digit numbers in my head without even thinking about them. My brain just spits out the answer. But, for some reason, 7+5 and 8+5 are not immediate for me. I have to do 7+3+2 or 8+2+3 to get their answers (or in some cases 7+6-1 or 8+6-1).

    Also, for teaching mnemonic purposes, here’s my post about multiplying matrices that may help you remember how that goes: http://longtailsofinterest.blogspot.com/2012/07/matrix-multiplication.html

  • Mistakes are bound to happen when math is done upside down! 🙂