Lots of responses to this great tweet. I wanted to understand the themes in what people were replying, so I went through everything and tried to summarize it here.

Response #1: Check Your Work, Start a Conversation

Response #2: Just Check Your Work (No Conversation Mentioned in Tweet)

Response #3: Explain the Zero Product Property

Response #4: Thinking About How to Teach the ZPP Unit

Response #5: Switch to a Graphical Context

Response #6: Ask for Explanations

Response #7: Run a New Activity with the Whole Class

I’m sure I didn’t capture everyone’s response, and I don’t know what any of this means. But there you go.

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

pic2

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.

pic2

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?

pic1 pic2

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?

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

4thGradeSubtAssessment_020

Student 2

4thGradeSubtAssessment_002

Student 3

4thGradeSubtAssessment_004

Student 4

4thGradeSubtAssessment_006

Student 5

4thGradeSubtAssessment_008 - Copy

Student 6

4thGradeSubtAssessment_010

Student 7

4thGradeSubtAssessment_012

Student 8

4thGradeSubtAssessment_014

Student 9

4thGradeSubtAssessment_016

Student 10

4thGradeSubtAssessment_018

 

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.)

problem1

Study: What do you notice in this (small) class set of responses? Note anything that surprises you.

Kid 1:

kid1

Kid 2

kid2

Kid 3

kid3

Kid 4

kid4

Kid 5

kid5

 

 

Kid 6

kid6

Kid 7

kid7

Kid 8

kid8

Kid 9

kid9

Wrap Up

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

pic2

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/