(Source, *The Predictive Power of Intuitive Rules*)

Spoilers: the authors of this piece aren’t super-duper into the intuitive rules theory. But it’s interesting, no?

(Source, *The Predictive Power of Intuitive Rules*)

Spoilers: the authors of this piece aren’t super-duper into the intuitive rules theory. But it’s interesting, no?

Take a moment before reading on. How many squares would be in the 7th step of this pattern? In the 43rd? In the *n*th?

Take another moment: what mistakes would you expect to see?

—

From looking closely at student work with other visual pattern problems, you’d expect kids to think about the change of this pattern in two different ways.

**Thinking about the pattern change recursively –**Students would think about the pattern as adding four squares on to the previous image at the corners.**Thinking about the pattern change relationally –**i.e. by relating the step number to some part of each picture (e.g. number of squares in diagonals, number of sets of four squares on the corners, etc.)

The relational goggles are more powerful and useful. It helps us calculate any step of the pattern efficiently. It can be generalized to linear functions. Further, most students have an easy time seeing this pattern’s recursive growth. The real learning that can happen with this pattern, for most students, happens in the move from a recursive to a relational perspective.

—

With that in mind, I want to share some mistakes that my students made on this pattern. I’ve organized the mistakes into two categories, and I’m curious if you’ll see them the way I do.

**Category 1:**

**Category 2:**

—

The way I see it, all the mistakes that I placed in Category 1 show strong evidence of seeing the pattern’s change relationally. Both of the students in Category 2 show a recursive perspective. In fact, the students in Category 2 don’t even make any mistakes!

What feedback do you think the students in Category 1 should get? What about the students in Category 2?

—

If all you care about is whether a student’s answer is right or wrong, then all the students in Category 1 will get some sort of nudge towards the right answer, while the students in Category 2 will be praised for their correct answers and maybe encouraged to keep on going.

But the students who are able to relate the step number to part of each picture are actually in pretty great shape. Yeah, they made some mistakes, but most of those mistakes are “off by 1” or “sloppy errors,” the sorts of mistakes that are almost always the result of paying attention to something besides the calculation or step number. (In this case, attention is being sucked up by the need to focus on the structure of the pattern at each step, a way of thinking that is brain-consuming when it’s new.)

On the other hand, the second group of students are getting right answers using a limited perspective. Ultimately, we’d like to help them see a relational perspective. Even though they have the right answers, they’re struggling here.

—

It’s not news that kids who get the wrong answer might be thinking in more sophisticated ways than students who got some question correct. What is news, I think, is that we ought to be as explicit as possible to ourselves about *how* those students are thinking with more sophistication. That’s the sort of thinking that can help us be strategic about the sort of feedback that we can give.

What feedback should Category 1 get? I’m inclined to use a very light touch with these students. They’re working within a powerful framework — they’ll likely be able to tease out where they went wrong. Even though they are using a strong perspective to analyze the problem, I still think it’s worthwhile to ask them to correct the calculations. First, because even though getting a correct answer isn’t all that matters, it also matters to students and to me. I want to show that I value correctness. Second, because seeing what *doesn’t *need to change in their answer is ultimately good for learning. I see this as a chance to adopt that relational view on the pattern again (“Oh wait how did I do this…Oh yeah!”).

Here are some comments I’d give Category 1 kids:

- I love the way you brought the step number into your calculation.
- Can you revisit this? Something’s wrong, but I’m not sure what.
- Your rule here is excellent. Can you check these answers again?

Some teachers will be tempted to encourage Category 2 students to continue their work, even if it’s within a recursive perspective. They might agree that the goal is ultimately for these students to adopt a relational perspective, but they’re willing to bet that kids will come to a “realization” while working recursively all on their own. Or, teachers want to affirm these students’ good thinking, so they are reluctant to offer them another way of thinking. They’re willing to defer the relational view to some other time, and maybe the kid will just pick up the relational view during a class discussion or by talking with a classmate.

Those are all legitimate moves, depending on the kid and the classroom and the course. But what if it’s important — for the kid, classroom, course — to help these students move from a recursive to a relational perspective? What feedback could they get then?

For these students, we want to offer them a new way of thinking. Here’s what I might say:

- Lovely work so far. Can you see where the step number appears in each diagram, and use that to find the 43rd step?
- I see the 4th diagram as made up of 3s. Can you see it as made up of 4s? Try to use that to find the 43rd step.
- Nice job noticing the growth pattern. Can you find a solution to the 43rd step that doesn’t involve adding 2 forty-three times?
- Can you show that there’s a counter-example to the “multiply the step number by 4” rule?

Any other ideas, people?

—

I’ve squawked a bunch about feedback. I’ve likewise done my share of squawking about student mistakes. I’m realizing now just how much that squawking has been missing out on by failing to get specific about student thinking. This isn’t the familiar complaint (familiar to me, at least) that by focusing on mistakes we only see students for their errors. Or maybe this *is *that “deficit model” complaint, but I had always interpreted as saying something about what we value in our students, and now I’m seeing how only thinking about mistakes really gives you nothing to latch the errors on to. It’s really limiting.

The flipside of this realization is that to really get at mistakes, feedback, hints or next instructional steps, we need to map out the terrain of student thinking. And there’s no way to do that without looking at sets of student work, rather than some single kid’s thinking. And there’s no way to do that without getting messy with the details of particular mathematical topics.

This is as true in my teaching as it is for my work here or anywhere else. My best feedback comes when it’s purposefully guided by some sort of explicit story about how student thinking develops for this type of problem. This is probably something I first really learned how to do with multiplication in 4th Grade, and it’s heavily influenced by the way I read the work of the Cognitively Guided Instruction team.

—

This post is a long, long way of saying that while I’d still love it if you send in individual mistakes that tickle your fancy in any way, I would LOVE it if you could send me a class set of really *anything* that your students have done, and *especially* if it’s from a geometry unit or a geometry class. I would be eternally grateful for your class scans: michael@mathmistakes.org. (I’m really good at quickly anonymizing student work.)

Next post: more on why class sets are the best.

Previously: http://mathmistakes.org/visual-patterns/

How do you predict that a group of students (9th Graders, Geometry, nearly all are comfortable with scaling) would respond to this prompt? Do you think they’ll disagree? Converge on one option? What reasons do you think they will bring to support their answers? Do you think that their responses will differ significantly from the responses that a group of teachers would give? If so, how?

Sheesh, that’s a lot of prompts. Let’s condense that:

- What do you predict students will respond?
- How do you predict that a group of teachers will respond?
- How would you respond?

Two interesting mistakes here. The first has to do with the Pythagorean Theorem, the other (more interesting) has to do with the angle of inclination.

I wonder what she’s looking at that the angle always stays the same. My guess, based on her first triangle, is that she thinks that the diagonal of a rectangle always bisects the right angle.

This might make for a nice bit of feedback for her. I could ask, “Is it possible to draw a rectangle whose diagonals don’t always make 45 degree angles? The answer matters for what you wrote here.” Or maybe the feedback I supply here should be a counterexample — a very long rectangle whose diagonals clearly don’t make 45 degrees? What’s my goal in this feedback, anyway?

I suppose my only goal is to have her know that the diagonals *don’t *bisect the angles, and to believe this in a way that she’ll remember and be able to reproduce on a new problem. So I want to equip her with the means to prove it to herself.

Given all this, I think I should probably be more direct in my feedback about the *fact *of non-bisection. I should leave the proof up to her, though. “Try to draw a rectangle whose diagonals don’t make 45 degree angles.”

One last worry. What if I’m wrong about my diagnosis of her thinking? What if she is seeing 45 degrees in these ramps in some other way? Maybe the best thing is to check in with her verbally before giving her any written feedback, to confirm that my theory is correct?

**Update (4/23/15): **Here’s the feedback and her post-feedback work. In conversation, I was able to confirm that my “every rectangle’s diagonals bisect a right angle” theory was right.

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*

*Kid 2*

* Kid 3*

* Kid 4*

* Kid 5*

*Kid 6*

* Kid 7*

* Kid 8*

* Kid 9*

* Kid 10*

* Kid 11*

* Kid 12*

* Kid 13*

*Kid 14 *

1. Isn’t this an interesting multiplication mistake?

2. I used to ask “how could we help this student?” or “why do you think this student made this mistake?” I still think that these questions are valuable to ask when looking deeply at student thinking. But, when teaching, the better question seems to be not “what mistake did this student make?” but instead “what could this student know that might help her?”

In this case, I’d say that this student could use more versatile ways of breaking numbers apart more than any sort of reflection on the errors of her ways.

Every once in a while people get in touch with me because they don’t like that this site is focused on *mistakes*. I think this is probably what they’re getting at.

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.

“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.

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:

- The kid didn’t have an explicit algorithm that he was trying to follow. He wasn’t
*under its sway.* - 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. - 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:*

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