“You’d run out of room in the circle!”
How do kids come to see lines as having no thickness at all? What experiences would support that change?
Natasha had $8.72. She spent $4.89 on a gift for her mother. How much money does Natasha have left?
Take a look below, and then report back in the comments:
Predict: What responses to this prompt would you expect from my Algebra 1 students? (Prior to this problem my kids had mostly worked with integer arithmetic, solving linear equations in one-variable and graphing scenarios and equations.)
Study: What do you notice in this (small) class set of responses? Note anything that surprises you.
How did your predictions hold up? What surprised you the most? What’s something you wish you knew more about?
A meta-question: what can we learn from a mistake like this, compared to a set of classwork on simplifying this expression?
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?)
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.
— Dan Meyer (@ddmeyer) October 7, 2015
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.