This come via Lois Burke on twitter, and immediately Max shows up with a possible explanation.

Dave has a different idea. Maybe the student was thinking in words — “5 and minus 1” — and this turns into its homonym “5-1.”

Personally, what I have the easiest time imagining is that the student just had “combine 5 and -1” on their mental ledger. When it came time to address that ledger, there was so much other stuff they were paying attention to that they slipped into the most natural sort of way to combine numbers they had, which is adding. (I like the metaphor of slipping. You’d very rarely see a kid slip in the other direction — from 5 + (-1) to 5 x (-1) — I think. There is a direction to this error.)

Here are the activities we came up with to help develop this sort of thinking in class. Ideas for improvement? More ideas? Other explanations of the student’s thinking?

Response to 5i^2 --- -4 Mistake

Response to 5i^2 --- -4 Mistake (2)

Response to 5i^2 --- -4 Mistake (1)


Pam Harris has an idea:

Love it. Here’s a digital version.

Response to 5i^2 --- -4 Mistake (1)

John Golden point out that there might be issues with the Which One Doesn’t Belong puzzle, so I offer this as an alternative.

Response to 5i^2 --- -4 Mistake (4)

John also offers a different problem string: “I’d be curious to see 5+i, 5+i^2, 5+i^3, 5+i^4, 5i, 5i^2, 5i^3, 5i^4.”

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.)

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



Circular rather than Rectangular Models of Fractions



Non-commutative rather than Commutative Model of Multiplication



Acting Out the Problem rather than Using a More Efficient Strategy



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

Linear properties applied in Non-Linear situation





One-Dimensional Distance applied in a Two-Dimensional Situation



Additive properties applied in Multiplicative situation











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



Area Properties Applied to Perimeter



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




Maybe: http://mathmistakes.org/a-third-34/



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



Mistakes Due to Quickly Associating Something In Place Of Another


Squares Instead of Square Roots



Multiplying In Place of Exponentiation



Addition In Place of Multiplication




Changing the Numbers of the Problem



Operating on the “Answer” in an Open Sentence Problem


[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.

Jason Ermer has a really cool thing going with Collaborative Mathematics. He wrote to me with an interesting student response that he got, and asked if I’d be interested in sharing it. Good news: I am!

First, here’s the problem:

Then we’ve got the student response.

OK, bad news, I can’t embed this video. Since this is the internet, this may be the point where you’re all like “Oh yeah like I’m going to click on a thing to get to some other interesting thing. Hell, I don’t even know if this other thing is interesting.” Well I’m vouching for this thing. It is interesting. Clicky clicky.

Did you click? Science tells us that some of you didn’t. If you didn’t, then here’s my summary of what the kid said:

“8 is 1000 in binary, so you’ll land on the same finger that you’ll land on when you’re at your eighth finger.”

(See? I told you. I bet now you’re regretting not clicking on the thing.)

Jason is going to write a few posts thinking about this student’s response, but let’s give him a head start on that discussion here.

What strikes you about this kid’s response? Please share any questions that you’ve got about the response. I’m sure we can rope Jason into lurking in the comments.

We here at Math Mistakes are always happy to share thoughtful writing about mistakes and student work.

First up is Nicora Placa, whose Bridging the Gap is one of my favorite new blogs. She asks, “Is a Careless Error Really Careless?” Here’s the headline news from a 1982 experiment:

However, as he continued his analysis of the interviews, he realized there was another explanation.   Students who incorrectly answered the questions were doing something that made sense to them.   Their intuition was to place the multiplier next to the letter associated with the larger group.   Although incorrect, it was meaningful to students.

I’m a bit ambivalent about this. I worry that coherent explanations of mistakes are often post-facto explanation of intuitions in disguise. But I have very little evidence for my claim that doesn’t come in the form of extensive surveys about exponents.

One of my favorite things about her blog is how well (and simply) annotated it is. She’s added so many things to my reading list. Go check her out.

Next up is Evelyn Lamb with a nice piece about a university-level mistake. There is a precise mathematical definition for the terms “closed” and “open” as they describe sets of numbers, but blah blah anything I could do to explain this would be worse than the way Evelyn does, so just go and check out her piece:

I had underestimated the power of the English language to suggest mathematically incorrect statements to my students. In mathematics, “open” and “closed” are not antonyms. Sets can be open, closed, both, or neither. (A set that is both open and closed is sometimes called “clopen.”) [ … ] But in English, the two words are basically opposites (although for doors and lids, we have the option of “ajar” in addition to open and closed). My students used their intuition about the way the words “open” and “closed” relate to each other in English and applied that intuition to the mathematical use of the terms.

Maybe we can change that vocabulary? Is there a better way to describe those sets, at least during students earliest exposure to these concepts?

Finally we’ve got Michael Fenton who posts about an interesting interaction he had with a student who completely nailed a set of rational exponents questions and was baffled by some square roots:

2013-09-26 15.26.26

He writes lots of interesting things that are worth thinking about, including a sketch of how he thinks learning rational exponents might go better. Go check him out.

Sue reports:

One student wanted to average the two angles at 16 degrees each. Another said the observer could stand on a stool to be a little higher, so the angles would be 16 degrees each. Their answers were very close. There were other good (but wrong) methods that all came down to assuming this relationship was linear in a way that it’s not. Since their answers were very close, it was hard to help them see what was wrong with their reasoning.

She ends with “Can anyone help me here?” And she’s got some good comments. Here’s one from Kate Nowak:

This example you’ve picked is a toughie, for two reasons. As you point out, 18 and 14 degrees are both darn close to 16 degrees, so if you make that simplifying assumption 1) the problem is much easier and 2) the answer is pretty close to what it should be. But, you have to realize you are making a simplifying assumption that will throw your answer off a little bit, which I take it was not what your students were doing. They thought the result should lead to the exact, correct answer.

Anyone here have any ideas to send her way?

tina infinity


The question asked for the range of the function SQRT(x).  What are your thoughts about the way the student answered the question. What does it show about what this particular student knows.*

In the past, I’ve publicly kvetched about how the only questions that we grapple with on this blog are about the particularities, rather than the generalities, of student work. This is a time when I think that the most interesting questions are accessed through thinking about what this particular kid was thinking. I’d also be interested in hearing how you think this represents a trend in students’ thinking, in general.

Thanks to Tina for the submission!