When I was a kid, a friend asked what my dad does for a living. “He’s a dank,” 18-year old Michael said. What I meant to say was that my dad worked at a bank, but I was distracted or tired and I mixed up the two words.

I thought about this while looking through a 3rd grader’s addition work. “He’s a dank” seems a lot like saying “46+30+2=58” to me.

I’m not sure what to call this sort of mistake. I’m tempted to call it a memory overload error, but I have no idea if that’s (a) psychologically apt or (b) meaningful to other people.

The crucial thing, though, is not to simply disregard these sorts of mistakes as silly errors, or as a sign that the student is lacking some general cognitive skill like “attention to detail” or “being careful with their work.” That would be a bad misdiagnosis.

To start building the case for why, pay attention to the “stupid” arithmetic mistakes that adults (and teachers and mathematicians) make while they’re working on a problem. Here’s one I made last summer while trying a matrix multiplication, when I did 1*2+2*3 and ended up with 10.

Do I suffer from a general sloppiness in my work? A lack of attention to detail? Nah, I was just distracted by making sure that I kept track of a bunch of others things that weren’t automatic for me. My attention was elsewhere.

What causes these sorts of errors? Any sort of distraction, but it’s important not to trivialize distraction. Distraction can come from any number of places.

• Distraction can come from various non-mathematical things, like friends, chatting, not caring about the problem, etc.
• Distraction can also come from mathematical factors. If I were better at the matrix multiplication part of matrix multiplication, I would be less likely to mess up some quick arithmetic that I’d otherwise get right.

What about my 3rd grader? There are two possibilities, and both are worth considering:

• The kid might have been distracted by whatever non-mathematical thing happened to be drawing her attention away at the moment.
• She might have found keeping track of the tens and ones difficult, and paying attention to the decomposition used up the mental resources that were needed to keep track of everything. She ends up adding 2 and 3 for the tens digit, 6 and 2 for the ones digit.

One of the themes of this blog has been a desire to dig deeper than “stupid mistake.” This is one sort of error that teachers often identify as a “silly” mistake, but labeling it as “silly” probably misses out on some truth about a kid’s mathematical thinking.

Questions:

1. What do memory overload mistakes look like in geometry? In non-computational contexts?
2. What other categories of “silly” errors are there? (I’d toss “mathematical habits” into the mix. Or maybe we should call that “fluency with a falsity”?)
3. What sort of feedback would you give my 3rd grader?
• Perhaps a particularly personal response is that I would tell “her” that I am incredibly interested in whatever she can remember about what she was thinking. I know she can add those numbers and get a different answer and I am not in the least concerned about her addition. I know she can correct this to get the expected answer. But, I look for gems of math in the uncommon and I am wondering if there is one here.

Later or in some contexts I would also ask her if she cares about this sort of situation and would like to think with me about ways to quickly know whether such an answer makes sense.

• mpershan

I agree with you. T has some truly fantastic thinking on display here, and it’s important to value that.

• I just reread your interpretation, and I don’t think I agree that “She ends up adding 2 and 3 for the tens digit, 6 and 2 for the ones digit,” unless of course she told you that was the case. I think that she wrote the equation correctly, but when doing the actual adding, she added 46 + 10 + 2, the quantities in the problem, despite writing a correct equation. Maybe the two representations were completed in two entirely separate processes, which make the equation and the sum unconnected representations.

• Yes, this seems to me also the more likely scenario and if it turns out to be the case, I would be interested in the nature of the separateness. Did the calculation of 58 happen before writing the calculation or during? If before, does that mean that the writing of the calculation represents some evolving thinking that is incomplete (perhaps involving distraction of the sort mentioned and/or an unreflective incorporation of early thinking) or a relative lack of concern for consistency/details once a satisfactory understanding/insight is achieved or ??? If during, does it mean that the preferred approach is to think with the original presentation/visual rather than the mathematical summary/formalization? or ???

• A different kid in 3rd grade today was working on 54 + 32 and ended up with 76. He just had a number written down, and I’ve been trying to push my 3rd graders to make their thinking explicit with equations, so I covered his number and asked him how he thought about 54 + 32.

He started explaining, and very quickly interrupted himself. “OH wait, I did this wrong. I added the 2 to the 5 and the 2 to the 4.”

I have no way of knowing what the student did in this problem, but I feel more comfortable about the plausibility of my explanation after listening to this other kid today.

• All of those are great questions and there’s a research study in here somewhere. So, why didn’t she add the 3 as well?

• Debbie Junk

Let her explain her answers. if she says a different one ask her to reflect on what happened in her initial work. I agree she was probably just thinking +10. Also all the other answers are correct so she knows how to add and subtract multi digit numbers.

• Marissa

I don’t think there was one math exam in elementary school when I did not make a “stupid” or “silly.” That was just the nature of my brain, as I was so overwhelmed with finishing the test and the anxiety swirling around in my mind. I would say that 101+ 25 = 125, because I ignored the ones place after I just read “10” and assumed that it was 100 because it was a three digit number. I was distracted and I also found that keeping track of the tens and ones was very difficult. Thats what I feel happened in the first of the three problems. The student just started 46, which was how many stickers Maria started with, added 10, which was how many stickers are in a strip and two singles ones at the end. What the student wrote below the problem is the correct process. He or she could have been careless and not remembered to add 30 into the solution instead of 10, as Maria got three packs of ten stickers, which equals 30. I was surprised with how intuitive that step was, which made it even more surprised in regards to the “careless” error. I would remind all of my students to do all of work on paper, which I believe would have directed the student to the right answer.