Nobody asked me, but here’s my thought.

When people learn something new, they cling to a paradigmatic case. This paradigmatic image matters more for how they use this new idea than whatever rules or logic they might otherwise adhere to. For example, when kids are first learning about triangles, they don’t identify new triangles on the basis of properties. They look at their paradigmatic image of triangle and compare this new shape to it. This is why kids misidentify so many shapes, at first. Below, 10 might be declared a triangle, though 5 would be rejected.

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Now, it would be sort of besides the point to lament young children’s tendency to identify shapes in this way. This is just what the beginning of the learning curve looks like. It helps teachers to be familiar with this tendency — we can directly address it in friendly ways — but it’s totally normal. In particular, it’s not really an artifact of instruction.

(I suppose if kids make it to a weirdly old age without being able to logically identify triangles, yeah, that would be an artifact of instruction.)

I think the situation with square roots that Justin points out is pretty similar to this. When kids solve equations by taking the square roots of both sides, a lot a lot of these cases involve a square root solution. It seems totally normal for kids to start seeing this as a paradigmatic case, and to think that all solutions to such equations involve square roots. Totally normal, not something to stress too much about.

In fact, I saw this mistake in my 8th Grade class last week. A kid was using the Pythagorean Theorem, and had put little square roots over the side lengths. No stress: told him that this wasn’t necessary; reminded him of the conceptual meanings that made this move incorrect; reminded myself to include more chances for him to practice this idea; set him off to try the next problem, but without the extra square roots.

I think this is just how learning happens.

I’ve sometimes read or talked to teachers who wished kids didn’t make these sorts of mistakes. And I guess it would be nice if kids could just reach an age where they operated as logical, analytical and meaning-oriented students at the start of their learning curves. I understand why we teachers feel a bit of nervousness when kids aren’t being guided by meaning.

But ‘being guided by meaning’ is another way of saying ‘being guided by logic,’ and this is not my understanding of how beginners hold on to new ideas.

It’ll take time, practice, corrections, maybe a big ol’ worksheet, but if a kid made it that far in solving these equations, they’ll make it the rest of the way. Keep it up!

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At first, this is what I thought the student had done:

  • First, the student drew six circles to represent “out of 6 books.”
  • Then, they distributed, one-by-one, the 66 books into each of the 6 circles. (If they just put 11 in each, why tally them?)
  • Then, the student searched for a way to represent the “5 out of” that are non-fiction.
  • It follows that the remaining books are fiction. That makes six sixes, or 36 books.

But then Bridget and Julie came in with a fantastic, different interpretation. Their’s feels like an improvement on my first draft.

We then got to work trying to come up with some activities to address this work. Suppose that your class of 6th Graders try this problem, and a lot of your class has struggles that are similar to the work above. You’re planning tomorrow’s lesson. What activity would you begin class with?

This is what we came up with. Which of these activities do you think would be most helpful? Are there any changes you would make to any of them? Is there a combination and sequence of these activities that you think would work particularly well? (I took a shot at sequencing them below. Some details on activity structures are here.)

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5 out of 6 Mistake

5 out of 6 Mistake (1)

 

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When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

  • 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
  • 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
  • 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?

Fraction comparison for 4th Graders. They’ve been working a lot with representing fractions as circles and as rectangles. They’ve done some basic addition with fractions. Most aren’t generally able to find equivalent fractions.

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What mistakes do you expect to see in the class set?

Make a prediction! Mark it down somewhere. Don’t do that internet thing of just continuously scrolling through a page at half-attention. Take a moment, form a thought. Then scroll on for the full class set of 14.

In the comments, would you please answer this question: Which mistake most surprised you? Why?

Kid 1

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Kid 2

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Kid 3

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Kid 4

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Kid 5

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Kid 6

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Kid 7

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Kid 8

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Kid 9

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Kid 10

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Kid 11

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Kid 12

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Kid 13

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Kid 14 

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BlogMathMistake

I know, I know what you’re thinking. I even know what you’re about to say. “Oy! These kids, just being taught algorithms which they blindly follow without reasoning. They even sometimes can’t even remember the algorithm! This poor kid doesn’t remember the algorithm correctly. He thinks that what he/she is supposed to do is subtract the smaller number from the larger number. Boo procedural thinking.”

Or, maybe you see this and think: “This kid isn’t even thinking. Just operating blindly on numbers. A shame, really…”

A third option: “This kid learned an incorrect rule. This kid thinks that what you’re supposed to do is subtract the smaller number from the larger number.”

Each of these explanations, I think, is a little bit off.

  • The first and the third theory make predictions about what a student “thinks is right.” What could this mean, if not that the student, when asked, would say what they did was correct? But I think that students, when prompted to reflect on this work, would quickly identify the mistake.
  • The second theory predicts that the student, when prompted, couldn’t explain how to properly subtract any double-digit numbers with understanding. I’m be willing to put down money that this kid, when presented with 54 – 32, could explain how to do this with as much understanding as your average kid.

We need some language and distinctions to properly describe what’s going on here.

  • The kid wasn’t thinking slowly, deliberately, explicitly. He wasn’t under the sway of a procedure or a concept.
  • He was just doing math, not thinking about the math he was doing. He was going with the flow, doing what seemed like it should be done.

This puts us in opposition to all three of the above theories:

  1. The kid didn’t have an explicit algorithm that he was trying to follow. He wasn’t under its sway.
  2. The student wasn’t just operating blindly and randomly on the numbers. He wasn’t guessing. He was doing the math without thinking about doing the math, though.
  3. The kid didn’t have a mistaken concept of subtraction. He wasn’t under the sway of any particular concept. He was just doing what needed to be done.

There was a mistake here? Or a misconception? Or a false belief in a bad procedure? How exactly should we describe this?

  • Objectively speaking, it is a mistake. The word “mistake” doesn’t refer to a person’s thinking, but rather refers fairly objectively to the result of their thinking. Objectively speaking, this was a mistake. The kid said something that wasn’t true.
  • But there’s no evidence here of a misconception. A misconception has to do with concepts, and this kid wasn’t under the direction of any mistaken concepts. He understands what subtraction is. He understands what place value is. He could tell you about them.
  • It’s not a false belief, because there’s no evidence here that this kid believes that what he did is correct.
  • It’s not a dumb mistake, something that happens randomly and without thought.

Instead, maybe we should call this a mental bias, or a tendency towards this sort of mistake. This problem has revealed an underlying bias in this kid’s tendency to subtract a smaller number from the larger number. What’s revealed is a sort of magnetic urge to take away a smaller number from a larger one, rather than a larger one from a smaller one.

What do we do about those sorts of tendencies? I think that a certain kind of practice is called for, but I’m not sure. Thoughts on that? On any of this?

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Decimals are hard.

What would we even want the student to do here if he’s working in decimal? Like, how do standard multiplication algorithms handle something like a repeating digit?

That’s what I’m getting out of this mistake right now: the deviousness of decimal representation, and the way it can obscure numerical properties.

How about you? What do you make of all this?