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## The Mistake

This is one of those things that sneaks up on me. I never think it’s going to be as hard for students as it is.

Even though I don’t have a picture of it, some students also say that arc AB is 100 degrees. In other words, we know that something is double the central angle and that something is equal to it, but we don’t know what.

## Why?

Every mistake points to something that students don’t yet understand as well as they could. In this case, what is it? It’s such a simple relationship.

For a lot of areas of math, we don’t have strong intuitions about the topic until we get our hands messy with it. That’s not really the case with a lot of geometry, where we can see the diagram itself and often have visceral reactions to it.

What does the above diagram suggest, to the untrained eye? Maybe it’s difficult to see that the inscribed angle must be smaller than the central angle it shares an arc with?

## What I Did

This was one of our online days, so I asked students to find the error and respond to these prompts in the chat.

I asked the first question because I wanted to draw attention to the most important thing to notice — that the central angle must be larger than the inscribed angle. I figured, that would make it easier to remember that it’s exactly half the central angle. Then, I figured it would be good to draw attention to the arc and clarify that the inscribed angle is half of that as well.

During the discussion I remembered my favorite way of thinking about this: like a rubber band or a slingshot that is pulled back. If you pull it back really far, doesn’t the angle on your finger get tighter? That’s what’s going on here too.

After calling on a few students to explain their responses, I asked them to use these ideas to try a similar problem.

After that, I paired students up (in breakout rooms) and asked them to try this amazing “misconceptions” activity that I found on Jo Morgan’s Resourceaholic collection.

## Commentary

I guess the human eye isn’t really so sensitive to differences in angles! Makes me wonder whether I should incorporate some estimation activities into this unit. Would it be useful to estimate the sizes of the angles before learning any of the theorems, just to focus on the visual skill of seeing the inscribed angle as smaller? That sounds like it could help.

I don’t know much about research on geometric misconceptions. How does one train their eye to see things differently? The way I usually approach it in my teaching is to start by teaching the true relationship, then to clearly identify errors when they come up. Maybe I should identify those errors more systematically, earlier in the unit? Now that I have these little “mistake analysis” slides, I could use them soon after studying the inscribed angle relationship.

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## Comparing Parts of Sides Instead of Whole Sides

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?

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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## Naming Coordinates, Feedback and Revision

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.)

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## Children and Teachers on Feedback

How do you predict that a group of students (9th Graders, Geometry, nearly all are comfortable with scaling) would respond to this prompt? Do you think they’ll disagree? Converge on one option? What reasons do you think they will bring to support their answers? Do you think that their responses will differ significantly from the responses that a group of teachers would give? If so, how?

Sheesh, that’s a lot of prompts. Let’s condense that:

1. What do you predict students will respond?
2. How do you predict that a group of teachers will respond?
3. How would you respond?
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## All Ramps Are 45 Degrees + Pythagorean Theorem

Two interesting mistakes here. The first has to do with the Pythagorean Theorem, the other (more interesting) has to do with the angle of inclination.

I wonder what she’s looking at that the angle always stays the same. My guess, based on her first triangle, is that she thinks that the diagonal of a rectangle always bisects the right angle.

This might make for a nice bit of feedback for her. I could ask, “Is it possible to draw a rectangle whose diagonals don’t always make 45 degree angles? The answer matters for what you wrote here.” Or maybe the feedback I supply here should be a counterexample — a very long rectangle whose diagonals clearly don’t make 45 degrees? What’s my goal in this feedback, anyway?

I suppose my only goal is to have her know that the diagonals don’t bisect the angles, and to believe this in a way that she’ll remember and be able to reproduce on a new problem. So I want to equip her with the means to prove it to herself.

Given all this, I think I should probably be more direct in my feedback about the fact of non-bisection. I should leave the proof up to her, though. “Try to draw a rectangle whose diagonals don’t make 45 degree angles.”

One last worry. What if I’m wrong about my diagnosis of her thinking? What if she is seeing 45 degrees in these ramps in some other way? Maybe the best thing is to check in with her verbally before giving her any written feedback, to confirm that my theory is correct?

Update (4/23/15): Here’s the feedback and her post-feedback work. In conversation, I was able to confirm that my “every rectangle’s diagonals bisect a right angle” theory was right.

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## 5 and 1/2 x 2 and 1/4 = 7 and 3/4

I find this fascinating. This student clearly knows how that multiplying the base and the height of a rectangle gives you its area. She even knows how to multiply fraction. But when it comes to part (d), she adds the numbers instead of multiplying them.

In earlier writing I hypothesized that, when put in unfamiliar situations, students often default to an “easier” operation. This idea now seems problematic to me. What, after all, is an “easier” operation any way? And what exactly would trigger this default to some other operation? And how do we explain why competent adults — like me — make similar mistakes on my own work?

It now seems more likely to me that we associate certain pairs of numbers with certain operations. Think about the numbers 100 and 1/2. I’d suggest that most people have an association of “50” with 100 and 1/2. After all, how often have you been asked to add 100 and 1/2 together? How often have you been asked to subtract 1/2 from 100? In contrast, how often have you been asked to find 1/2 of 100?

How often have you been asked to multiply 5 1/2 and 2 1/4 together? My guess is that you — and the student above — have been asked to add these sorts of mixed numbers more often than multiply them.

The idea here is that the pairs of numbers themselves come with associations.

There’s a hard version of this claim that I don’t mean to make. I don’t mean to say that, no matter the context, you’d expect a student to add 5 1/2 and 2 1/4 together. I think a division problem with mixed numbers is unlikely to trigger associations with addition. Maybe I’m moving towards a two-part model? The sorts of mistakes we make with numbers depends both on the associations with the operation and also associations with the numbers? And things get really bad when these two associations point in the same direction?

This theory feels very testable, but at the moment I’m having a hard time articulating a possible test of it. But we should be able to mess with people’s associations with numbers and see if that changes the sorts of mistakes that they make. Ideas?

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## Do These Properties Guarantee Congruence?

The activity is from a Shell Center task, and the student work is from my own class. We’re missing a few kids, but this is representative of the whole group’s work.

Questions:

1. What do you notice? Anything interesting?
2. What categories of student responses do you see?
3. What sort of feedback would you give to push their mathematical thinking further?

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## “What feedback would you give?”, continued

In a previous post, lots of commenters said that they didn’t feel that you could give helpful, written feedback because there wasn’t enough evidence of student thinking on the quiz. Given that complaint, it might be interested to see how those same teachers would give written feedback on a quiz that gives significantly more evidence of how a student is thinking.

Here’s another quiz: what sort of written feedback would you give? (The checkmarks are from the student, who was provided with an answer key and checked her own work, ala this.)

As before, imagine that you don’t have to write a grade on this paper. Some things I’m wondering about:

• Would you give comments on every solution, or only some of them?
• Will you ask kids to “explain why you said _______”? When is it helpful to ask for an explanation? When isn’t it?
• Will you give your kids specific next steps, or will you mostly point out the good and the bad of their work?
• Will you throw up your hands and say “You really need to have a conversation with the kid!” for this sort of quiz also?

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## Troubles With Proofs

What mistakes would you expect to see in the proofs of the problem below? Take a moment and make some predictions. You might find it helpful to know that this was part of an end-of-the-year exam, and that kids were able to use whatever proof representation they wanted. In other words, they could write two-column proofs, paragraph proofs or flowchart.

Here are the mistakes, pulled from the class set. Or maybe you feel uncomfy describing these proofs as “mistakes”? Maybe it’s better to say that they contain mistakes? Or that they are proofs that aren’t where we probably want them to be?

Do you feel comfy calling these “mistakes”? What would you call them?

What are the next steps for these kids? What would you recommend that their teacher do?

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## Decimal Misconceptions? Meet similar triangles.

A note from the submitter:

Along the lines of one I sent you awhile back. This is one of my best students, and several other students gave answers with similar misconceptions. I pretty much ignored it last time it came up, thinking that it was an anomaly, but I think it’s a significant hole in my students’ understanding. Students were using calculators today.

What’s going on here in the student work? What’s the connection to the earlier post?