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.)

From Bedtime Math:

Big kids: The record distance for a thrown boomerang to travel is 1,401 feet.  If it traveled exactly 1,401 feet on the return trip too, how many feet did it travel in total?  Bonus: Meanwhile, the longest Frisbee throw is 1,333 feet – about a quarter of a mile! How much farther from the thrower did the boomerang travel than the Frisbee?

From the submitter, who sends in the thinking of two of his students:

(1) first student, having doubled the boomerang distance in the earlier question, now doubles the frisbee distance  and calculates (2801 – 2666) feet.
(2) Second student gets an 100 board and spends a short time calculating 100 – 33 = 67. Then thinks for a long time during which I’m sure he is going to say 67 + 1 = 68, but never quite does it. I stay silent until he announces: 667. No clue where the extra 600 came from. He wasn’t willing to write down or draw anything to explain his thinking.
Interesting!  I’m inclined to put the first student in the “extending the thinking you’d do in one model to a less familiar situation” category and the second student in the associational mistake (same link) category.

OK OK OK I think I’ve got where 1024 comes from but what is going on with that 11?

Update: I think banderson2 nails it in the comments. “It comes from the power of 2. 2 = 8/4 so 8/4 + 3/4 is 11/4.”

My new favorite game is trying to classify math mistakes. (See: Classifying Math Mistakes)

Right now, I see three big categories of mistakes:

1. Mistakes Due To Limited Applicability of Models
2. Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
3. Mistakes Due to Quickly Associating Something In Place Of Another

I think this is pretty clearly an example of the third category. The student’s brain was working hard, and they swapped the 10 and the x.

These sorts of mistakes are interesting to me because I think a lot of teachers see these and say, “Oy, this student thinks that you can just swap out the x with the angle.” Or others would say, “Oy, this student has no conceptual understanding of trigonometry.”

Nah. This kid needs more practice with the Law of Sines so that you’ve got enough brain power available to pay attention to all the moving parts while you’re trying to solve the problem.

There’s something else that’s interesting about these associational errors, and it’s about the associations that students make. Isn’t it interesting that the x*sin(10) is more familiar to this student than 10*sin(x)? Maybe this also points to the need for more practice that mixes up missing angle and missing sides Law of Sines problems?

The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.  I asked the student to tell me how he got his answer; he pointed to the word “total” and said that he needed to add.  I’d like to know what ideas educators have to intervene when a student is already clearly looking for keywords and is not making sense of the problem.
What do we mean by “make sense of a problem”?
Are we imagining an all-math skill, tools that can be used to make sense of any math problem no matter the topic or age of a student? Something like “read the problem carefully!” or “draw a picture!”?
Or are we imagining a local skill, some way to make sense of this problem and problems like it? Something like…well, I’m not exactly sure what would help someone make sense of this problem. Maybe, “if you see names, you might be comparing!” or “if you see a lot of numbers in a problem, rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think of keyword mistakes as a lot like applying the distributive property where it doesn’t belong or other “over-extending” mistakes. Generalizations are smart things to do, and a keyword generalization is a smart thing to think also, and it’s usually correct. Knowing that this mistake exists, I might create a set of problems all that contain the word “total” with some being summing up problems and others being compare problems and so on. My reasoning is sort of simple: this kid thinks that “total” means add ’em all up, so let’s provide him with counterexamples and then urge him to make a new generalization.
Thoughts?

Every few years I try this. It’s gotten to the point where I can no longer tell if this is actually helpful or illuminating, but below you’ll see the categories that I created when I tried to sort a bunch of mistakes that I’d logged on this site.

Enjoy, and please share any disagreements or alternate sortings that you see in the student work.

—-

Mistakes Due To Limited Applicability of Models

Recursive rather than Relational Thinking

http://mathmistakes.org/recursive-and-relational-thinking-and-the-feedback-each-deserves/

Circular rather than Rectangular Models of Fractions

http://mathmistakes.org/which-fraction-is-larger/

Non-commutative rather than Commutative Model of Multiplication

http://mathmistakes.org/write-a-story-problem-for-13-x-2/

Acting Out the Problem rather than Using a More Efficient Strategy

http://mathmistakes.org/91-mushrooms-7-people/

Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation

Linear properties applied in Non-Linear situation

http://mathmistakes.org/the-fundamental-mistake-of-trigonometry/

http://mathmistakes.org/value-of-absolute-value/

http://mathmistakes.org/overassuming-linearity/

One-Dimensional Distance applied in a Two-Dimensional Situation

http://mathmistakes.org/the-distance-between-11-and-45-is-7/

Additive properties applied in Multiplicative situation

http://mathmistakes.org/what-else-could-she-know-on-12×14280/

http://mathmistakes.org/5-and-12-x-2-and-14-7-and-34/

http://mathmistakes.org/squaring-doesnt-make-equivalent-fractions/

http://mathmistakes.org/comparing-ratios-what-feedback-would-you-give/

http://mathmistakes.org/complex-number-mistakes-are-often-algebra-mistakes/

http://mathmistakes.org/scaling-by-12/

Side-times-Side Formula for Finding Area Applied in non-Rectangles

http://mathmistakes.org/base1-times-base2-area-of-triangle/

Area Properties Applied to Perimeter

http://mathmistakes.org/perimeter-is-the-space-outside-of-a-shape/

Properties of some paradigmatic example of a shape applied globally [1]

http://mathmistakes.org/all-ramps-are-45-degrees-pythagorean-theorem/

http://mathmistakes.org/thats-not-an-array/

http://mathmistakes.org/triangles-and-3-gons/

Properties of a Fractional Parts of a Rectangle Applied To Other Shapes

http://mathmistakes.org/three-fifths-of-a-triangle/

Mistakes Due to Quickly Associating Something In Place Of Another

http://mathmistakes.org/integrating-by-parts/

Multiplying In Place of Exponentiation

http://mathmistakes.org/two-cubed-is-eight-but-seven-squared-is-fourteen/

http://mathmistakes.org/5-and-12-x-2-and-14-7-and-34/

Changing the Numbers of the Problem

http://mathmistakes.org/mixed-up-numbers/

Operating on the “Answer” in an Open Sentence Problem

http://mathmistakes.org/1476/

[1] This is a very mushed-together category. I’ve fallen into the trap of giving geometry short-shrift in the face of arithmetic and algebra. In general, I understand geometry thinking less well than I understand arithmetic/algebraic thinking. That category of “Properties of Shapes Overextended…” needs some serious breaking-down.

Spoilers: the authors of this piece aren’t super-duper into the intuitive rules theory. But it’s interesting, no?

Take a moment  before reading on. How many squares would be in the 7th step of this pattern? In the 43rd? In the nth?

Take another moment: what mistakes would you expect to see?

From looking closely at student work with other visual pattern problems, you’d expect kids to think about the change of this pattern in two different ways.

1. Thinking about the pattern change recursively – Students would think about the pattern as adding four squares on to the previous image at the corners.
2. Thinking about the pattern change relationally – i.e. by relating the step number to some part of each picture (e.g. number of squares in diagonals,  number of sets of four squares on the corners, etc.)

The relational goggles are more powerful and useful. It helps us calculate any step of the pattern efficiently. It can be generalized to linear functions. Further, most students have an easy time seeing this pattern’s recursive growth. The real learning that can happen with this pattern, for most students, happens in the move from a recursive to a relational perspective.

With that in mind, I want to share some mistakes that my students made on this pattern. I’ve organized the mistakes into two categories, and I’m curious if you’ll see them the way I do.

Category 1:

Category 2:

The way I see it, all the mistakes that I placed in Category 1 show strong evidence of seeing the pattern’s change relationally. Both of the students in Category 2 show a recursive perspective. In fact, the students in Category 2 don’t even make any mistakes!

What feedback do you think the students in Category 1 should get? What about the students in Category 2?

If all you care about is whether a student’s answer is right or wrong, then all the students in Category 1 will get some sort of nudge towards the right answer, while the students in Category 2 will be praised for their correct answers and maybe encouraged to keep on going.

But the students who are able to relate the step number to part of each picture are actually in pretty great shape. Yeah, they made some mistakes, but most of those mistakes are “off by 1” or “sloppy errors,” the sorts of mistakes that are almost always the result of paying attention to something besides the calculation or step number. (In this case, attention is being sucked up by the need to focus on the structure of the pattern at each step, a way of thinking that is brain-consuming when it’s new.)

On the other hand, the second group of students are getting right answers using a limited perspective. Ultimately, we’d like to help them see a relational perspective. Even though they have the right answers, they’re struggling here.

It’s not news that kids who get the wrong answer might be thinking in more sophisticated ways than students who got some question correct. What is news, I think, is that we ought to be as explicit as possible to ourselves about how those students are thinking with more sophistication. That’s the sort of thinking that can help us be strategic about the sort of feedback that we can give.

What feedback should Category 1 get? I’m inclined to use a very light touch with these students. They’re working within a powerful framework — they’ll likely be able to tease out where they went wrong. Even though they are using a strong perspective to analyze the problem, I still think it’s worthwhile to ask them to correct the calculations. First, because even though getting a correct answer isn’t all that matters, it also matters to students and to me. I want to show that I value correctness. Second, because seeing what doesn’t need to change in their answer is ultimately good for learning. I see this as a chance to adopt that relational view on the pattern again (“Oh wait how did I do this…Oh yeah!”).

Here are some comments I’d give Category 1 kids:

• I love the way you brought the step number into your calculation.
• Can you revisit this? Something’s wrong, but I’m not sure what.
• Your rule here is excellent. Can you check these answers again?

Some teachers will be tempted to encourage Category 2 students to continue their work, even if it’s within a recursive perspective. They might agree that the goal is ultimately for these students to adopt a relational perspective, but they’re willing to bet that kids will come to a “realization” while working recursively all on their own. Or, teachers want to affirm these students’ good thinking, so they are reluctant to offer them another way of thinking. They’re willing to defer the relational view to some other time, and maybe the kid will just pick up the relational view during a class discussion or by talking with a classmate.

Those are all legitimate moves, depending on the kid and the classroom and the course. But what if it’s important — for the kid, classroom, course — to help these students move from a recursive to a relational perspective? What feedback could they get then?

For these students, we want to offer them a new way of thinking. Here’s what I might say:

• Lovely work so far. Can you see where the step number appears in each diagram, and use that to find the 43rd step?
• I see the 4th diagram as made up of 3s. Can you see it as made up of 4s? Try to use that to find the 43rd step.
• Nice job noticing the growth pattern. Can you find a solution to the 43rd step that doesn’t involve adding 2 forty-three times?
• Can you show that there’s a counter-example to the “multiply the step number by 4” rule?

Any other ideas, people?

I’ve squawked a bunch about feedback. I’ve likewise done my share of squawking about student mistakes. I’m realizing now just how much that squawking has been missing out on by failing to get specific about student thinking. This isn’t the familiar complaint (familiar to me, at least) that by focusing on mistakes we only see students for their errors. Or maybe this is that “deficit model” complaint, but I had always interpreted as saying something about what we value in our students, and now I’m seeing how only thinking about mistakes really gives you nothing to latch the errors on to. It’s really limiting.

The flipside of this realization is that to really get at mistakes, feedback, hints or next instructional steps, we need to map out the terrain of student thinking. And there’s no way to do that without looking at sets of student work, rather than some single kid’s  thinking. And there’s no way to do that without getting messy with the details of particular mathematical topics.

This is as true in my teaching as it is for my work here or anywhere else. My best feedback comes when it’s purposefully guided by some sort of explicit story about how student thinking develops for this type of problem. This is probably something I first really learned how to do with multiplication in 4th Grade, and it’s heavily influenced by the way I read the work of the Cognitively Guided Instruction team.

This post is a long, long way of saying that while I’d still love it if you send in individual mistakes that tickle your fancy in any way, I would LOVE it if you could send me a class set of really anything that your students have done, and especially if it’s from a geometry unit or a geometry class. I would be eternally grateful for your class scans: michael@mathmistakes.org. (I’m really good at quickly anonymizing student work.)

Next post: more on why class sets are the best.

Previously: http://mathmistakes.org/visual-patterns/

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?

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.