Here’s the work of a 4th Grader named Jaden. He has a lot of interesting ideas for finding the area of a rectangle. What do you notice in his work? What do you wonder?

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When I asked teachers this question as part of a Desmathmistakes activity, there were a lot of interesting responses. While all sorts of observations about student work are valuable, it can be especially valuable to transform our observations about student thinking into some next step. (Researchers look at work as an end in itself. When teachers look at student work it’s almost always to evaluate it or to figure out what to do next in class. We’re doing the latter here.)

Here were three of my favorite responses to the activity, with thanks to (in order) Mary, K, and Cindy.

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In case you’re curious, here is everybody’s rectangles:

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Finally, on twitter Kristin Gray is thinking in a different direction:

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Kristin’s idea is for a string of area calculation problems that all total to to the same area, but are partitioned in different ways:

Some meta-questions: What were people thinking about during this activity? What were they doing? Were they learning something? Could they be learning something?

Jump into the comments if you have some thoughts about Desmathmistakes Experiment #2.

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What do you notice about this student’s thinking? What do you wonder about it?

I made a little Desmos activity to see if it’s possible to use their activity builder to share and comment on student work. I asked people to circle something they noticed in this student’s work. Here is the overlay showing everything that everyone circled.

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I’m not sure what to make of all that overlaid, but I’m definitely interested. The written answers people offered were also really interesting. Here is a sampling:

Guest: “This is a common error for my students as well. They  do not recognize that this is a quadratic function and try to get a straight line.”

Kevin: “These don’t seem to be in any particular order.”

Lane: “Does he have an eraser? Does he get confused calculating with zero?  Does he know the shape of a parabola?  Does he know that a function cannot possible have one point on top of another? Does he sometimes get confused with x^2 and 2x?  Could he have analyzed his own mistakes with a calculator?  By not checking with a calculator will some of his errors snowball and cause further confusion?  Is the student feeling frustrated? I think it is good this student understands the choice of input does not have to be in a particular order.”

Jonathan: “I notice that there is a disconnect in the student’s knowledge of linear vs. quadratic equations. I wonder how come the student did not use any negative values in her table.”

While there were a lot of great observations, the one that stood out to me was that this student could probably learn to recognize that this sort of equation will produce a U shape. Knowing that this sort of equation produces a U will make it more likely that they will test negative x-values, or at least more reliably guess the rest of the shape. I agree: it seems as if this student is trying to fit a U-shaped function into a line-shaped paradigm.

What activity could we design that would help students like this one develop their thinking?

Inspired by Bridget, I put this together:

Response to Mistake Templates - Graphing y = x^2 - 4

In the Desmos activity, I asked if people could think of a way to improve my rough draft. Here were three responses that represent some of the variations people had:

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On twitter, Bridget had a second idea for an activity that would help students like this one develop their thoughts.

To wrap things up, I shared a mockup of Bridget’s alternative activity and asked people what they thought about it.

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Some selected responses:

Max: I prefer my version of the previous activity — this activity doesn’t invite students to consider why the parabola is symmetric — it’s easy enough to connect the linear and nonlinear representations and not confront the whys of the symmetry of the nonlinear representation. Maybe including y = x^3 + 4 as a third example (with both graph and equation provided) would support that sense-making?

Brian: I like the idea of having one more equation than graph. I’m also wondering about the choice to have two linear functions vs. one quadratic. This activity provides less structure than the previous since, to determine what the function’s graph looks like he would need to do it himself. The other provided the Desmos graph. On the other hand, this activity does provide the student with a more possibilities of visualizing the function, which could yield insight into how he’s thinking about the quadratic function.

Bridget: I wonder if the graphs should be discrete points instead of continuous. Not sure if it would make a difference or not. I also wonder if the missing representation should be another quadratic. I’m trying to consider if the connecting representations should include tables. I’m not sure…

Also, I think the previous slides connect more to the issue at hand. On an assessment-do you think this student could be given the equation y=x^2-3 and choose the correct graph from four multiple choice? I’m not sure…

Overall, this was fun! I’m excited to try it again.

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This come via Lois Burke on twitter, and immediately Max shows up with a possible explanation.

Dave has a different idea. Maybe the student was thinking in words — “5 and minus 1” — and this turns into its homonym “5-1.”

Personally, what I have the easiest time imagining is that the student just had “combine 5 and -1” on their mental ledger. When it came time to address that ledger, there was so much other stuff they were paying attention to that they slipped into the most natural sort of way to combine numbers they had, which is adding. (I like the metaphor of slipping. You’d very rarely see a kid slip in the other direction — from 5 + (-1) to 5 x (-1) — I think. There is a direction to this error.)

Here are the activities we came up with to help develop this sort of thinking in class. Ideas for improvement? More ideas? Other explanations of the student’s thinking?

Response to 5i^2 --- -4 Mistake

Response to 5i^2 --- -4 Mistake (2)

Response to 5i^2 --- -4 Mistake (1)

UPDATE:

Pam Harris has an idea:

Love it. Here’s a digital version.

Response to 5i^2 --- -4 Mistake (1)

John Golden point out that there might be issues with the Which One Doesn’t Belong puzzle, so I offer this as an alternative.

Response to 5i^2 --- -4 Mistake (4)

John also offers a different problem string: “I’d be curious to see 5+i, 5+i^2, 5+i^3, 5+i^4, 5i, 5i^2, 5i^3, 5i^4.”

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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.

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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.)

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5 out of 6 Mistake-page-003

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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?