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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What feedback would you give to this student? Some considerations…

  • Would you ask a question or make a statement?
  • What written feedback would be most helpful?
  • If you were able to have a conversation with this student, how would you start it?
  • What would the student’s job be once you handed the paper back to him/her?


This child made it clear that

  • She knew that an array was a rectangle
  • That this was technically a rectangle
  • These super-long folks were not arrays, or at least she didn’t think they were, because they didn’t look like a rectangle
  • The 2 x 17 was an array

To what do you attribute this perception? (You can check your answers in the back of the book.)

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.


  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?


We chatted on twitter about this question for a bit. What predictions can you make about the predictions that were made? Click through to check your answers!


Here are the results from the 59 students who answered this question on an exam:


How would you give feedback to the students who wrote “infinity”?

Imagine that you were to give feedback to the students who wrote “-3/7.” What feedback would you give?

In a previous post, lots of commenters said that they didn’t feel that you could give helpful, written feedback because there wasn’t enough evidence of student thinking on the quiz. Given that complaint, it might be interested to see how those same teachers would give written feedback on a quiz that gives significantly more evidence of how a student is thinking.

Here’s another quiz: what sort of written feedback would you give? (The checkmarks are from the student, who was provided with an answer key and checked her own work, ala this.)




As before, imagine that you don’t have to write a grade on this paper. Some things I’m wondering about:

  • Would you give comments on every solution, or only some of them?
  • Will you ask kids to “explain why you said _______”? When is it helpful to ask for an explanation? When isn’t it?
  • Will you give your kids specific next steps, or will you mostly point out the good and the bad of their work?
  • Will you throw up your hands and say “You really need to have a conversation with the kid!” for this sort of quiz also?


In a lot of ways, it’s much easier for me to come up with helpful feedback to give on rich, juicy problems (see here) than it is for your typical quiz or test. I find it much harder to think about how to give feedback that helps a kid’s learning when (a) the quiz is full of non-open questions and (b) the kid’s solutions don’t show a lot of thinking. But a lot of classroom assessments end up like that, and it’s important to figure out how to deal with those tough situations effectively.

So: What would you write as feedback on this quiz?




Some constraints/notes, that you should feel free to reject or challenge:

  1. Assume that we’re dealing with written feedback here. Not a conversation.
  2. Assume that we don’t have to write a grade on this piece of work. (If we wrote a grade on here, some research indicates that would ruin any feedback we gave.)
  3. You might decide to give feedback on every question of this quiz, you might not.

I’ll jump in with my thoughts in the comments. Here are some questions about your choices that I’m wondering about:

  • Would you choose to mark the questions as right/wrong?
  • Would you try to find something to value about this kid’s work in your comments, or will you be all hardass instead?
  • Would you ask questions or give suggestions?
  • Would you write one, several, or many comments?
  • Would you reject the constraints in some way?
  • Would you ask the kid to explain himself?

Excited to read your thoughts!