Category: mailbag

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[Mailbag] What Do You Do with the Ideas You Used to Call “Mistakes”

Guillaume Paré, in the very interesting comments of my last post where I urged us to reconsider mistakes:

I do agree with what is written, but I am still wondering what I’m supposed to do with that information and the student’s copy.

T: Oh this is so interesting! You’ve actually answered a different question correctly. Check this out:

T: How does that help you come up with an answer to the original question? Talk about it. I’ll be back.

That’s a script right there. It works for any incorrect answer. The script is all-purpose and all-weather but it has two challenging requirements:

  1. You have to actually believe that student ideas are interesting, especially ones that don’t correctly answer the question you were trying to ask.
  2. You have to identify the question the student answered correctly.

This is why I want to learn more math and more math and more math.

The more math I know, the more power I have not just to show off at parties but also to appreciate student ideas and to identify the different interesting questions they’re answering correctly.

Barbara Pearl, via email:

Can you write about it briefly again in a simpler way so I can try and understand it? When students make a mistake or answer something incorrectly, you want to …

I want to teach in a way that honors the specific student and also the general ways people learn.

So in any interaction with students, I need to a) understand the sense they’re making of mathematics, b) celebrate that sense, saying loudly “I see you making sense!”, and then c) help them develop that sense, connecting the question they answered correctly to a question they haven’t yet answered correctly.

Rachel, in the comments:

So, if we don’t call it a mistake, then what do we call it?

I don’t have any problem saying a student’s answer is incorrect, that they didn’t correctly answer the question I was trying to ask. But my favorite mathematical questions defy categories like “correct” and “incorrect” entirely:

  • So how would you describe the pattern?
  • What do you think will happen next?
  • Would a table, equation, or graph be more useful to you here?
  • How are you thinking about the question right now?
  • What extra information do you think would be helpful?

How can you call any answer to those questions a mistake or incorrect? What would that even mean? Those descriptions feel inadequate next to the complexity of the mathematical ideas contained in those answers, which I interpret as a signal that I’m asking questions that matter.

Featured Comment

Denise Gaskins quoting WW Sawyer:

Most remarks made by children consist of correct ideas very badly expressed. A good teacher will be very wary of saying ‘No, that’s wrong.’ Rather, he will try to discover the correct idea behind the inadequate expression. This is one of the most important principles in the whole of the art of teaching.

Daniel Peter:

“So, if we don’t call it a mistake, then what do we call it?”



I find that I have to keep insisting that they restate the question in their own words. The culture of “right answer” is filled with shame and shaming, and students will try repeatedly to just give me the “correct” answer to the original question. But this is a missed opportunity for developing understanding, in my view.

Can Sports Save Math?

A Sports Illustrated editor emailed me last week:

I’d like to write a column re: how sports could be an effective tool to teach probability/fractions/ even behavioral economics to kids. Wonder if you have thoughts here….

My response, which will hopefully serve to illustrate my last post:

I tend to side with Daniel Willingham, a cognitive psychologist who wrote in his book Why Students Don’t Like School, “Trying to make the material relevant to students’ interests doesn’t work.” That’s because, with math, there are contexts like sports or shopping but then there’s the work students do in those contexts. The boredom of the work often overwhelms the interest of the context.

To give you an example, I could have my students take the NBA’s efficiency formula and calculate it for their five favorite players. But calculating — putting numbers into a formula and then working out the arithmetic — is boring work. Important but boring. The interesting work is in coming up with the formula, in asking ourselves, “If you had to take all the available stats out there, what would your formula use? Points? Steals? Turnovers? Playing time? Shoe size? How will you assemble those in a formula?” Realizing you need to subtract turnovers from points instead of adding them is the interesting work. Actually doing the subtraction isn’t all that interesting.

So using sports as a context for math could surely increase student interest in math but only if the work they’re doing in that context is interesting also.

Featured Email

Marcia Weinhold:

After my AP stats exam, I had my students come up with their own project to program into their TI-83 calculators. The only one I remember is the student who did what you suggest — some kind of sports formula for ranking. I remember it because he was so into it, and his classmates got into it, too, but I hardly knew what they were talking about.

He had good enough explanations for everything he put into the formula, and he ranked some well known players by his formula and everyone agreed with it. But it was building the formula that hooked him, and then he had his calculator crank out the numbers.

“I would eat the extra meatball.”

Simon Terrell recaps his lesson study trip to Japan with Akihiko Takahashi, who was the subject of Elizabeth Green’s American math article last week:

In one case, a teacher was teaching a lesson about division with remainders and the example was packaging meatballs in pack of 4. When faced with the problem of having 13 meatballs and needing 4 per pack, one student’s solution was “I would eat the extra meatball and then they would all fit.” It was so funny and joyful to see that all thinking was welcomed and the teacher artfully led them to the general thinking that she wanted by the end of the lesson.

I can trace my development as a teacher through the different reactions I would have had to “I would eat the extra meatball,” from panic through irritation to some kind of bemusement.

BTW. The comments here have been on another level lately, team, including Simon’s, so thanks for that. I’ve lifted a bunch of them into the main posts of Rand Paul Fixes Calculus and These Tragic “Write An Expression” Problems.

“Think About Your Favorite Problem From A Unit”

Bob Lochel, responding to commenter Jenni who wondered how, when, and where to integrate tasks into a unit:

In my years as math coach, the most efficient piece of advice I would give to teachers is this: think about your favorite problem from a unit, the problem you look forward to, or that problem which is number 158 in the last section which you know will generate all kinds of discussion. Without fail, this problem is often done last, as the summary of all ideas in the unit. Okay, why not do it first? Keep it simmering in the background, flesh it out as ideas are developed and pratice occurs. It often doesn’t take a sledgehammer to make a good unit great.

Pennies, Pearson, And The Mistakes You Never See Coming

I took machine-graded learning to task earlier this week for obscuring interesting student misconceptions. Kristen DiCerbo at Pearson’s Research and Innovation Network picked up my post and argued I was too pessimistic about machine-graded systems, posing this scenario:

Students in the class are sitting at individual computers working through a game that introduces basic algebra courses. Ms. Reynolds looks at the alert on her tablet and sees four students with the “letters misconception” sign. She taps “work sample” and the tablet brings up their work on a problem. She notes that all four seem to be thinking that there are rules for determining which number a letter stands for in an algebraic expression. She taps the four of them on the shoulder and brings them over to a small table while bringing up a discussion prompt. She proceeds to walk them through discussion of examples that lead them to conclude the value of the letters change across problems and are not determined by rules like “c = 3 because c is the third letter of the alphabet.”

My guess is we’re decades, not years, away from this kind of classroom. If it’s possible at all. Three items in this scenario seem implausible:

  • That four students in a classroom might assume “c = 3 because c is the third letter of the alphabet.” I taught Algebra for six years and never saw this conception of variables. (Okay, this isn’t a big deal.)
  • That a teacher has the mental bandwidth to manage a classroom of thirty students and keep an eye on her iPad’s Misconception Monitor. Not long ago I begged people on Twitter to tell me how they were using learning dashboards in the classroom. Everyone said they were too demanding. They used them at home for planning purposes. This isn’t because teachers are incapable but because the job demands too much attention.
  • That the machine grading is that good. The system DiCerbo proposes is scanning and analyzing handwritten student work in real-time, weighing them against a database of misconceptions, and pairing those up with a scripted discussion. Like I said: decades, if ever.

This also means you have to anticipate all the misconceptions in advance, which is tough under the best of circumstances. Take Pennies. Even though I’ve taught it several times, I still couldn’t anticipate all the interesting misconceptions.

The Desmos crew and I had students using smaller circles full of pennies to predict how many pennies fit in a 22-inch circle.


But I can see now we messed that up. We sent students straight from filling circles with pennies to plotting them and fitting a graph. We closed off some very interesting right and wrong ways to think about those circles of pennies.

Some examples from reader Karlene Steelman via e-mail:

They tried finding a pattern with the smaller circles that were given, they added up the 1 inch circle 22 times, they combine the 6, 5, 4, 3, 2, 1, and 1 circles to equal 22 inches, they figured out the area of several circles and set up proportions between the area and the number of pennies, etc. It was wonderful for them to discuss the merits and drawbacks of the different methods.

Adding the 1-inch circle 22 times! I never saw that coming. Our system closed off that path before students had the chance even to express their preference for it.

So everyone has a different, difficult job to do here, with different criteria for success. The measure of the machine-graded system is whether it makes those student ideas invisible or visible. The measure of the teacher is whether she knows what to do with them or not. Only the teacher’s job is possible now.

Featured Comments

Sue Hellman:

This doesn’t even touch the students who get questions RIGHT for the wrong reasons.

Dave Major:

Dashboards of the traditional ‘spawn of Satan & Clippy the Excel assistant’ sort throw way too much extremely specific information straight to the surface for my liking (and brain). That information is almost always things that are easy for machines (read. programmers) to work out, and likely hard or time consuming yet dubiously useful for humans to do. I wonder how many teachers, when frozen in time mid-lesson and placed in the brain deli slicer would be thinking “Jimmy has 89% of this task correct and Sally has only highlighted four sentences on this page.”