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## Interpreting Graphs Where Up Isn’t Good

Where on this graph would you find the best sprinter of all time?

In class today, a bunch of kids said that the best ever would be at the top of the graph. It took a few seconds before a wise soul pointed out that the best would be at the bottom of the graph. But at first, that’s hard to see!

The hardness of this has to do with an idea that’s pervasive in our culture: up is better. In Metaphors We Live By, Lakoff and Johnson argue that status, virtue, wealth and many other positive attributes get an “up” orientation in our language.

After a student makes a point, I sometimes ask them “What’s the ‘therefore’?” In this case, the ‘therefore’ is just about awareness. To understand that “down is better” in this graph, we have to go against our conceptual tendencies. Students are going to make this mistake, but maybe we can help by pointing out that graphs often go against the “more is better” metaphor.

Or maybe something else entirely is going on here? What do you think is happening? Where else do you see issues like these arising in understanding math?

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## 2/3 as a “2 by 3”

When kids are learning to give fractions meaning, I think they often struggle to figure out how the numerator and denominator are coordinated. Here we see a middle step in understanding, maybe: it’s not that the numerator and denominator are totally disconnected. They’re just coordinated in a way that doesn’t really correspond to how they actually work together (i.e. denominator tells you the “unit” and the numerator tells you the “quantity.”)

Maybe the progression of learning looks like this:

• 2/3 means “2 and 3,” nothing to do with each other. Totally baffling notation.
• 2/3 means “2 by 3” or “2 times 3,” some more familiar situation where two numbers can be coordinated in a relation.
• 2/3 means “2 thirds,” which is a productive way to coordinate the numerator and denominator.

Thoughts? Am I overinterpreting this as a middle step in a progression, when it’s actually just a totally uncoordinated interpretation of the fraction?

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## 1,000 Lines of Symmetry

“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?

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## Mistake Analysis + Practice

Here’s an activity I just drafted for my Algebra 1 class. I’m trying to help them get very comfortable working with the distributive property and fractions. Thoughts?

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## [Class Set] – 4th Grade, Money Subtraction

Natasha had \$8.72. She spent \$4.89 on a gift for her mother. How much money does Natasha have left?

• I gave this question to my 4th Grade class. (11 kids, one absent.) It was December, I had seen them do a variety of subtraction work. I knew that a lot of them could handle subtraction using something like the standard algorithm — though certainly not everyone — and I was wondering whether a money context would be easier or harder for them. Would you predict that \$8.72 – \$4.89 would be easier or harder than 872 – 489?
• What approaches would you predict kids to take for this money problem? What mistakes do you expect to see?

Take a look below, and then report back in the comments:

• Which student’s approach surprised you the most?
• Assume that you’ve got time in the curriculum to ask students to work on precisely one question at the beginning of class the next day. What question would you ask to address some of the ideas you see in their work below?

Student 1

Student 2

Student 3

Student 4

Student 5

Student 6

Student 7

Student 8

Student 9

Student 10

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## A Class Set of Identities Work

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.

Kid 1:

Kid 2

Kid 3

Kid 4

Kid 5

Kid 6

Kid 7

Kid 8

Kid 9

Wrap Up

How did your predictions hold up? What surprised you the most? What’s something you wish you knew more about?

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Cool mistake!

A meta-question: what can we learn from a mistake like this, compared to a set of classwork on simplifying this expression?

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## How Sets of Class Work Change the Conversation

A few months ago, I swung by Justin Reich’s private math tutor 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/

Categories

## What is going on in these volume calculations?

I’m stumped. Ideas?

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## Using Math Mistakes in Whole-Group Discussions

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?