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.)

5 out of 6 Mistake-page-005

5 out of 6 Mistake-page-003

5 out of 6 Mistake-page-004

5 out of 6 Mistake

5 out of 6 Mistake (1)


pic1 pic2

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.


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

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-001

Kid 2

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-002

Kid 3

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-003

Kid 4

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-004

Kid 5

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-005


Kid 6

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-007

Kid 7

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-008

Kid 8

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-009

Kid 9

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-010

Kid 10

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-011

Kid 11

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-012

Kid 12

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-013

Kid 13

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-014


Kid 14 

6thGrade-copier@saintannsny.org_20150331_124451 (1)-page-006


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?

dylan image

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?



mistake5 mistake4 mistake3  mistake2


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?