I find this fascinating. This student clearly knows how that multiplying the base and the height of a rectangle gives you its area. She even knows how to multiply fraction. But when it comes to part (d), she adds the numbers instead of multiplying them.

In earlier writing I hypothesized that, when put in unfamiliar situations, students often default to an “easier” operation. This idea now seems problematic to me. What, after all, is an “easier” operation any way? And what exactly would trigger this default to some other operation? And how do we explain why competent adults — like me — make similar mistakes on my own work?

It now seems more likely to me that we associate certain pairs of numbers with certain operations. Think about the numbers 100 and 1/2. I’d suggest that most people have an association of “50” with 100 and 1/2. After all, how often have you been asked to add 100 and 1/2 together? How often have you been asked to subtract 1/2 from 100? In contrast, how often have you been asked to find 1/2 of 100?

How often have you been asked to multiply 5 1/2 and 2 1/4 together? My guess is that you — and the student above — have been asked to add these sorts of mixed numbers more often than multiply them.

The idea here is that the pairs of numbers themselves come with associations.

There’s a hard version of this claim that I don’t mean to make. I don’t mean to say that, no matter the context, you’d expect a student to add 5 1/2 and 2 1/4 together. I think a division problem with mixed numbers is unlikely to trigger associations with addition. Maybe I’m moving towards a two-part model? The sorts of mistakes we make with numbers depends both on the associations with the operation and also associations with the numbers? And things get really bad when these two associations point in the same direction?

This theory feels very testable, but at the moment I’m having a hard time articulating a possible test of it. But we should be able to mess with people’s associations with numbers and see if that changes the sorts of mistakes that they make. Ideas?

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?

The activity is from a Shell Center task, and the student work is from my own class. We’re missing a few kids, but this is representative of the whole group’s work.

Questions:

1. What do you notice? Anything interesting?
2. What categories of student responses do you see?
3. What sort of feedback would you give to push their mathematical thinking further?

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’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?

What feedback would you write on this kid’s paper? Why?

(Thanks, KN!)

What’s an example of some feedback that you think a teacher might consider giving, but is not the ideal response?

What feedback would you give this student on the page?

If you had five minutes to work with this kid one-on-one, what would you talk about?

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?