A few months ago, I swung by Justin Reich’s classroom and showed his undergrad some math mistakes. (Read about it here or here.) In planning the session, I practically begged Justin to let me use a class set of mistakes instead of individual pieces of interesting mistakes. Here’s what I wrote:

Your first question in the protocol is “Look at three problems on the board. Predict all of the mistakes that students might make.” I love this question. But I feel lingering guilt about how mathmistakes.org usually responds to this sort of question with a single example of student work. This has felt problematic in some of the conversations that I’ve had surrounding student work, because someone might be entirely right in their predictions in a way that isn’t affirmed by the chosen work. I worry that this feeling of “gotcha” sometimes kills discussion around student work since the initial predictions aren’t entirely engaged.
My proposal is to tweak the protocol a bit. Instead of showing kids what a single student actually did, what about showing them a class set of responses? Then we can better check our initial predictions and ask a whole host of other interesting questions. (e.g. What patterns do you see? Do you think you can tell how these kids were taught? Why do kids tend to make this mistake?)
I ended up using a class set of fraction-comparison work for Justin’s class. My experience cemented my opinion: if you want to talk about teaching or student thinking, it’s gotta be the class set. Why?
Reason #1: You can’t be right or wrong with a prediction about one piece of student work.
This is what I mentioned in my email to Justin, but I want to expand on it here. A great way to learn something is to make a prediction, and then check it against reality. But knowledge about student thinking is most powerful either in the aggregate or in the very specific. Meaning, to know something about student thinking is either to know something about how kids, in the aggregate, often think about something, or it’s knowledge about how this kid, right here thinks about it.
This is a long way of saying that checking a prediction against one wacky error is inevitably a bit of a letdown:
• OK people! Here’s this math problem. What thinking do you predict you’d see from kids here?”
• You collect predictions
• “OK here’s this piece of student work. Were you right?”
• Umm. Yes? No?

It feels unfair to me to ask — as I have in the past — for people to invest themselves with a prediction that I can’t honor with a realistic response. Using a class set better respects other people’s predictions.

Reason #2: You can’t really make connections and form generalizations about student thinking from one example of student work.
When I worked with Justin’s students, I saw them making connections across different students. They noticed the prevalence of the area model in the student work, and that led to interesting observations, connections and questions on their part. Wouldn’t have happened if I just showed them the craziest one.

Reason #3: Math Mistakes discussions sometimes devolve into “Well this kid needs one-on-one tutoring/special attention/I would pull her aside and ask…” The class context nudges us away from that.
Look back on the comments from the first few years of this site, and you’ll see this line appearing over and over again. This imagines a context where you’ve got one student having a hard time, while the class as a whole mostly gets the math. This allows participants in a math mistakes discussion to shift the responsibility onto the individual student.
That’s a fine context to imagine, but offering a class set of math mistakes offers a much richer context for conversation. Here is a class, in the middle of learning something. On the whole, they know some things and struggle with others. What are those things? What could we do next?

Reason #4: It’s more authentic to the actual work of teaching.
Finally, the class set more closely resembles the actual work of teaching, so participants get to practice an aspect of that actual work. We can sort and categorize student work. We can talk about groups of students, and develop language to describe different sorts of students and different sorts of struggles.
I haven’t posted very many class sets on this site, though I’d like to do more. Feel free to continue submitting individual pieces of student work, but you’ll get a special high five if you send me a set of a class’ work ready to post.
Here are the class sets that I’ve posted so far:

http://mathmistakes.org/recursive-and-relational-thinking-and-the-feedback-each-deserves/

http://mathmistakes.org/do-these-properties-guarantee-congruence/

http://mathmistakes.org/which-fraction-is-larger/

http://mathmistakes.org/multiplication-strategies-my-students-are-starting-with/

http://mathmistakes.org/getting-better-at-multiplying-two-digit-numbers/

This is such a good question. I don’t have a great answer, and I’d like to try articulating why that is.

When people get in touch with me about this site, it’s often to talk about using the mistakes from this site in the classroom. As far as I can tell (and I can’t!), that’s how people who use this site tend to use the site. They take mistakes and ask their kids to analyze them. Why did this student make this mistake? Or, did this student make a mistake? What advice would you give them? What could they do better next time? And so on.

What’s the theory here? Why would this help learning?

Sometimes, when I’m talking to people, it sounds like people think that being aware of possible errors will safeguard students from future errors. Let’s call this type of instruction Teaching to Avoid Temptation. To teach this way is to ask students to reflect on errors, so that next time they won’t make them again. Will they be tempted to make those same mistakes? Maybe they will, but they’ll remember this conversation or their feedback on this last quiz and then they’ll know now to combine unlike terms or whatever.

As someone who spends all day working with children, I am skeptical that we can teach them to avoid temptations.

What we can do, though, is teach them some math that will help them think differently or more fluently about certain problems. Maybe analyzing and discussing math mistakes can do that?

I’m sure that some pieces of math mistakes can be great for teaching some new ways of thinking. But not all mistakes are fruitful for learning some math. What math could a kid learn from discussing how someone multiplied the base and power?

(Maybe I’m just not being imaginative enough?)

Anyway, as I was thinking about this I came up with two situations where a mistake can really liven up a whole-group conversation.

Situation 1: When there’s a wrong way of thinking that a lot of kids have, but you want an emotionally neutral setting to dispute it. So you invent a mistake (or you pull a mistake from this site) and discuss the wrongness of that mistake instead of one from your classroom.

Situation 2: When you want to isolate a strategy from the answer. Sometimes it’s hard to distinguish a strategy from a correct procedure. Drawing your students’ attention to a mistake that nonetheless tries something worthwhile might really help them focus on that worthwhile thing, maybe more than a correct attempt would.

The conversational work that kids will do would differ in those two situations. For Situation 1, kids are tasked with formulating justifications and reasons. (Is this right? If it’s wrong, why is this wrong? What would be right? Why would it be right?) For Situation 2, the work is articulating what was good about the solution attempt. That work might also involve using and practicing that helpful strategy. An easy move is to ask students to use that strategy to correctly complete the problem. Another is to ask students to use that strategy on a related problem, or a related set of problems.

That’s all I could come up with. You?

First, the mistake:

Then, the feedback with revisions in red pencil. (I love the idea of doing revisions in different ink color. Credit to Lisa for that.)

I notice that the kid didn’t write them as (x,y) but wrote them as x,y. I wonder how come he did that? Or, more precisely, I wonder if he doesn’t see much of a difference between (x,y) and x,y or if three is some other reason for leaving off the parentheses.

(By the way, before you try to nitpick the feedback check out this conversation on twitter about it.)

From Bedtime Math:

Big kids: The record distance for a thrown boomerang to travel is 1,401 feet.  If it traveled exactly 1,401 feet on the return trip too, how many feet did it travel in total?  Bonus: Meanwhile, the longest Frisbee throw is 1,333 feet – about a quarter of a mile! How much farther from the thrower did the boomerang travel than the Frisbee?

From the submitter, who sends in the thinking of two of his students:

(1) first student, having doubled the boomerang distance in the earlier question, now doubles the frisbee distance  and calculates (2801 – 2666) feet.
(2) Second student gets an 100 board and spends a short time calculating 100 – 33 = 67. Then thinks for a long time during which I’m sure he is going to say 67 + 1 = 68, but never quite does it. I stay silent until he announces: 667. No clue where the extra 600 came from. He wasn’t willing to write down or draw anything to explain his thinking.
Interesting!  I’m inclined to put the first student in the “extending the thinking you’d do in one model to a less familiar situation” category and the second student in the associational mistake (same link) category.

OK OK OK I think I’ve got where 1024 comes from but what is going on with that 11?

Update: I think banderson2 nails it in the comments. “It comes from the power of 2. 2 = 8/4 so 8/4 + 3/4 is 11/4.”

My new favorite game is trying to classify math mistakes. (See: Classifying Math Mistakes)

Right now, I see three big categories of mistakes:

1. Mistakes Due To Limited Applicability of Models
2. Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
3. Mistakes Due to Quickly Associating Something In Place Of Another

I think this is pretty clearly an example of the third category. The student’s brain was working hard, and they swapped the 10 and the x.

These sorts of mistakes are interesting to me because I think a lot of teachers see these and say, “Oy, this student thinks that you can just swap out the x with the angle.” Or others would say, “Oy, this student has no conceptual understanding of trigonometry.”

Nah. This kid needs more practice with the Law of Sines so that you’ve got enough brain power available to pay attention to all the moving parts while you’re trying to solve the problem.

There’s something else that’s interesting about these associational errors, and it’s about the associations that students make. Isn’t it interesting that the x*sin(10) is more familiar to this student than 10*sin(x)? Maybe this also points to the need for more practice that mixes up missing angle and missing sides Law of Sines problems?

The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.  I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add.  I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.
What do we mean by “make sense of a problem”?
Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?
Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.
Thoughts?

Every few years I try this. It’s gotten to the point where I can no longer tell if this is actually helpful or illuminating, but below you’ll see the categories that I created when I tried to sort a bunch of mistakes that I’d logged on this site.

Enjoy, and please share any disagreements or alternate sortings that you see in the student work.

—-

Mistakes Due To Limited Applicability of Models

Recursive rather than Relational Thinking

http://mathmistakes.org/recursive-and-relational-thinking-and-the-feedback-each-deserves/

Circular rather than Rectangular Models of Fractions

http://mathmistakes.org/which-fraction-is-larger/

Non-commutative rather than Commutative Model of Multiplication

http://mathmistakes.org/write-a-story-problem-for-13-x-2/

Acting Out the Problem rather than Using a More Efficient Strategy

http://mathmistakes.org/91-mushrooms-7-people/

Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation

Linear properties applied in Non-Linear situation

http://mathmistakes.org/the-fundamental-mistake-of-trigonometry/

http://mathmistakes.org/value-of-absolute-value/

http://mathmistakes.org/overassuming-linearity/

One-Dimensional Distance applied in a Two-Dimensional Situation

http://mathmistakes.org/the-distance-between-11-and-45-is-7/

Additive properties applied in Multiplicative situation

http://mathmistakes.org/what-else-could-she-know-on-12×14280/

http://mathmistakes.org/5-and-12-x-2-and-14-7-and-34/

http://mathmistakes.org/squaring-doesnt-make-equivalent-fractions/

http://mathmistakes.org/comparing-ratios-what-feedback-would-you-give/

http://mathmistakes.org/complex-number-mistakes-are-often-algebra-mistakes/

http://mathmistakes.org/scaling-by-12/

Side-times-Side Formula for Finding Area Applied in non-Rectangles

http://mathmistakes.org/base1-times-base2-area-of-triangle/

Area Properties Applied to Perimeter

http://mathmistakes.org/perimeter-is-the-space-outside-of-a-shape/

Properties of some paradigmatic example of a shape applied globally [1]

http://mathmistakes.org/all-ramps-are-45-degrees-pythagorean-theorem/

http://mathmistakes.org/thats-not-an-array/

http://mathmistakes.org/triangles-and-3-gons/

Properties of a Fractional Parts of a Rectangle Applied To Other Shapes

http://mathmistakes.org/three-fifths-of-a-triangle/

Mistakes Due to Quickly Associating Something In Place Of Another

http://mathmistakes.org/integrating-by-parts/

Multiplying In Place of Exponentiation

http://mathmistakes.org/two-cubed-is-eight-but-seven-squared-is-fourteen/

http://mathmistakes.org/5-and-12-x-2-and-14-7-and-34/

Changing the Numbers of the Problem

http://mathmistakes.org/mixed-up-numbers/

Operating on the “Answer” in an Open Sentence Problem

http://mathmistakes.org/1476/

[1] This is a very mushed-together category. I’ve fallen into the trap of giving geometry short-shrift in the face of arithmetic and algebra. In general, I understand geometry thinking less well than I understand arithmetic/algebraic thinking. That category of “Properties of Shapes Overextended…” needs some serious breaking-down.

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