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What is the thinking that led this student to make this mistake?

I’ve been teaching geometry for six years, and I figure I must have seen this mistake dozens of times. It’s so common that I have a name for it in my class — it’s a part-whole issue. Students know that AD is to DB as AE is to EC, and I think DE gets (correctly) associated with AD and AE while BC gets (correctly) associated with DB and EC. The issue, though, is that AD, DE, AE are all whole sides whereas as DB and EC are parts of sides. So while this student is correct to associate these sides, the student is comparing whole side lengths to parts rather than finding the proportion between different whole side lengths.

I’d be pretty surprised if other geometry teachers haven’t seen this mistake too, and I’d be interested to hear their explanations of why this mistake is so common.

When I shared this on twitter, the main conversation was about the quality of the problem, and especially the fact that this diagram is not to scale.

I was surprised by this response for two reasons:

  • While I wouldn’t want my students to start studying this math with this task (they didn’t) I think the wildly out-of-scale diagram is a nice way to draw students’ attention to the underlying relationships between the sides. I often encourage students to make quick sketches to help guide their thinking, and these sketches don’t have to be to scale in order to be helpful.
  • Most importantly: The student whose work we’re studying did not have an issue with the diagram! He had successfully solved the first four problems, and then he offered a reasonable (but incorrect) answer to the last one. The underlying issue this student had is easily explained without the diagram, and it’s one that I’ve seen often with accurate diagrams.

Then again, there were so many people on twitter suggesting that this problem has major issues, it’s making me pause and wonder if they have a point. I’ll have to think more about it.

In any event, I then started thinking about addressing and furthering the thinking that this student had. This wasn’t just an isolated mistake — a lot of students in class had similar issues. I wanted to start class with an activity that would help further their thinking on this type of problem. What activity could I do?

Because I wanted to help students see the subtle difference between part/whole and whole/whole comparisons, I decided to use a Matching Connecting Representations activity (see more of these here).

I came up with two different versions. Any ideas on how to improve them? Would they spur kids to think about different strategies?

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Featured Comments: 

Some dissent from S Freedman:

I really like the lack of scale in the drawings. It’s important to teach that diagrams can be misleading. The math isn’t lying, just their unconscious interpolating brains.

Max wants to tackle the ambiguity with the diagram head-on, and offers a “Which One Doesn’t Belong” activity for doing so.

wodb

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At first, this is what I thought the student had done:

  • First, the student drew six circles to represent “out of 6 books.”
  • Then, they distributed, one-by-one, the 66 books into each of the 6 circles. (If they just put 11 in each, why tally them?)
  • Then, the student searched for a way to represent the “5 out of” that are non-fiction.
  • It follows that the remaining books are fiction. That makes six sixes, or 36 books.

But then Bridget and Julie came in with a fantastic, different interpretation. Their’s feels like an improvement on my first draft.

We then got to work trying to come up with some activities to address this work. Suppose that your class of 6th Graders try this problem, and a lot of your class has struggles that are similar to the work above. You’re planning tomorrow’s lesson. What activity would you begin class with?

This is what we came up with. Which of these activities do you think would be most helpful? Are there any changes you would make to any of them? Is there a combination and sequence of these activities that you think would work particularly well? (I took a shot at sequencing them below. Some details on activity structures are here.)

5 out of 6 Mistake-page-005

5 out of 6 Mistake-page-003

5 out of 6 Mistake-page-004

5 out of 6 Mistake

5 out of 6 Mistake (1)

 

In an earlier post, I shared Michael Fenton’s scenario and categorized the responses he got on twitter.

There were at least seven distinct responses that teachers offered to Fenton’s prompt. Wow! This makes me think two things:

  1. Fenton’s scenario was so thought-provoking that it yielded an amazing variety of responses.
  2. How come there was so much disagreement about how to act in this scenario?

Part of the disagreement, I think, comes from what went unspoken in Fenton’s mistake. We didn’t know if this mistake was shouted on in a discussion or found on a piece of paper. We don’t know if this is one of those times when we can afford to have a one-on-one conversation with a kid in response to her mistake, or if our response will be scrawled on her paper and returned. Was this a common error, or an isolated mistake? Could our response be an activity for the class instead of a chat?

While one-on-one conversations are crucial in teaching, they are hard to talk about. By their nature, they’re improvisational and somewhat unstructured. I’d also argue that opportunities for one-on-one conversations can be rare, and they get rarer as the number of students in your class grows larger.

Revising the Scenario

So let’s add some details to Fenton’s scenario. This was a mistake in an Algebra 1 class. Smart kids, thoughtful teacher, but when she collects papers after an ungraded check-in she finds that about half her class made Fenton’s mistake. Oh no! She decides that she’s going to launch class the next day with a brief activity to help advance her kids’ thinking.

Her first idea is to try a string of equations. She has three different drafts. Which one would you choose, and why?

Equation String 1

Fenton's Mistake - Various Approaches (5)

Equation String 2

Fenton's Mistake - Various Approaches (1)

 

Equation String 3

Fenton's Mistake - Various Approaches (2)

Other Activities

Then, she has some other ideas. Maybe equation strings aren’t the right move? She comes up with three other activities: Working With Examples, Which One Doesn’t Belong and Connecting Representations.

Working With Examples

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Connecting Representations

Fenton's Mistake - Various Approaches

Which One Doesn’t Belong?

Fenton's Mistake - Various Approaches (6)

Commentary

The meta-question here is about the conversation. Can we have a conversation with so many options? I don’t know. I worry that maybe I should have just limited discussion to the equation strings.

What would do?

My first reaction is that I like the equation strings, because it most directly gets at the issue of overextending the zero-product property to other equations.

But what I really want to do is lay out a sequence of 3-4 activities that I could do in sequence to develop this idea for a class.

And do I know enough to answer that question? Wouldn’t that depend on the math that we’d already studied and the math that’s coming up next?

Do we learn anything from thinking about these questions?

 

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?

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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.

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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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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

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Student 2

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Student 3

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Student 4

4thGradeSubtAssessment_006

Student 5

4thGradeSubtAssessment_008 - Copy

Student 6

4thGradeSubtAssessment_010

Student 7

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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:

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Kid 2

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Kid 3

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Kid 4

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Kid 5

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Kid 6

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Kid 7

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Kid 8

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Kid 9

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Wrap Up

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