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The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.  I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add.  I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.
What do we mean by “make sense of a problem”?
Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?
Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.
Thoughts?

In this case, the mistake (or whatever we call it) isn’t about what the student wrote, but what he said.

At the end of class, I asked my 3rd Graders to write a story problem for 13 x 2 and hand it in. As he was leaving, a boy handed me this slip and apologized for it.

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“Why are you apologizing?”

“Because my story is for 2 x 13, not for 13 x 2.”

Commentary:

The big lesson here is that the order matters in multiplication, as it does with addition (for most young kids 9+2 is much easier than 2+9) and as it does for algebra (4 + 2x = 10 is not the same as 10 = 2x + 4). Each of these problems has a different flavor for people who are beginning to get comfortable with these types of problems. Saying that two problems are “the same” is a substantive mathematical claim, and it needs to be taken with the seriousness that all mathematical claims require.

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.

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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?

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Yes, yes, kids multiply the base and the power. Here’s what’s remarkable about this:

  1. They do know the definition of exponents. It’s written a line above. They did it a line above.
  2. They’re doing this with confidence. There aren’t erased numbers. This isn’t slow thinking. This is just what kids think seven squared ought to be.

By defining exponents in terms of multiplication while offering no other images or models for what exponentiation does, we create a default model for exponents that sticks with people forever. When mentally taxed — either with a tough multiplication, or with an unusual power — kids revert back to this default model. They’ll do this especially in high school, and they’ll get questions wrong on tests and all sorts of other things not because they’re being sloppy, but because this default model is constantly lurking in their minds.

Incidentally, I asked these kids why they think about multiplication when they see powers, and this is what they said:

their notes

I’ve written about a lot of this stuff before. See here, especially, where I shared the high school versions of this mistake.

Now that all of this has been established, the next step needs to be finding a curricular approach that doesn’t rely as heavily on the “repeated multiplication” model for exponents. We need to build a distinctive set of images and intuitions that are native to exponents so that our kids aren’t always defaulting into multiplication when they have to think hard about math.

This is work that I’ve started, in a post titled “Exponents Without Repeated Multiplication.”  I’ll send you there for the details, but I stake out two major claims about exponents education:

  1. Much in the way that arrays support early multiplication work, geometric notions of area and volume can serve as the bedrock of an exponents education
  2. We tend to think of four, not five, major operations of arithmetic, but we need to start thinking about exponents as on par with all the others and taking care to build them thoughtfully throughout the entire curriculum.

Beyond all of this, these exponents mistakes serve as a big reminder about the nature of learning, teaching and knowledge. The big, big lesson of all of this is that knowing/not-knowing is not clean and it’s not binary. There are degrees of knowing something. Would you say that these students don’t yet understand what exponents mean? What does that even mean, given the contradictory evidence we have in front of us.

But, then, what does it mean to understand something at all?

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I think that this might be my favorite conversation since I started hanging out (as a math assistant) with 3rd Graders. File this one under “place value.”

The question that this kid was grappling with was to write “8,000 + 500 + 20 + 6” in standard form. I’ve forgotten most of the details of the conversation, but all of the interesting stuff is right there on the page.

First, note how she first writes 25,03 to that first question, at the top of the image. And though it’s harder to see, she’s done that for 60,47 as well. When I came over to her, I thought that she was just confused about the convention, and I offered a correction.

Then she started working on the “8,000 + 500 + 20 + 6” problem. First she wrote 26, and then she tried to figure out how much more it was. She ended up with what you see in the image, at which point I realized that there was something about the way she was thinking that I hadn’t anticipated. She was chunking the number into 26 and everything else.

She really struggled to figure out what to do with 8,000 and 500. The boxes around the numbers are part of my attempt to push her to tell me what each numeral represented (“5 whats? Shouldn’t this just be 8 plus 5 plus 2 plus 6, so 21?”).

But there was no way that was going to work. The fact that she was chunking “26” together means that she doesn’t really get place value for 10s either. Those of you with more experience will hopefully help me out in the comments, but I’d imagine she sees 26 as a single number, not as composed of any parts.

This was confirmed when I asked her what the 2 in the 26 meant. She thought I was nuts. She said that 26 is the number right after 25. I repeated the question, and she thought it was ridiculous.

To me, this really speaks to the value of activities that defamiliarize place value for students. See Anna’s Ba-na-na or Christopher’s Orpda for activities that do this.