Categories

## Counting Fifty Dollar Bills

This yielded a few really interesting mistakes, all pointing me to the same conclusion: my 3rd Graders need to learn how large numbers work. Though, as always, the student thinking itself was interesting.

Sorry that this kid used the “highlighter.” But they got 6000 for those 30 fifty dollar bills.

Their count: 100, 200, 300, 400, 500, 600, 700, 800, 900, 1000, 2000, 3000, 4000, 5000, 6000. Thousands are stupid, we should just eliminate them entirely and count “11 hundred, 12 hundred, etc.” It would solve a lot of problems! We’ll have to wait until I’m King of Math to do that one.

This same student made another mistake while working on this problem. They asked me if 1000 was made of three 50s. I asked, “wait what?” They clarified that since two 50s make 100, maybe three 50s make 1000?

Fascinating!

Categories

## The Mistake

Students don’t know how to read equations, and when they see two numbers they habitually add them together. Any blank is there as the result of an operation. The equals sign just means “make sure that you do this operation.”

My goal is to help students connect equations to the notion of equivalence — something that students in my experience already come into my classes with a decent understanding of, whether from experience or school.

I show this image, and talk about how we know that the pairs of apple buckets have the same number of apples (you can move one apple from one bucket to the other).

Then, we move to puzzles. I tried to leave different buckets “missing,” because I know that these are really four different types of problems. In particular, students are the most confused when the third bucket is missing (since they just tend to sum the first two numbers and put that in the third bucket).

Then, I want to nudge students towards connecting the arrow symbol to the equals sign and buckets to boxes of missing numbers:

This might also be a good time for this activity:

For an extra challenge, I ask students to only use the digits 0-9 each once.

## Commentary

Every year I see this mistake in 3rd Grade. I’ve tangled with it over and over again, and I’ve also tangled with the research on the equals sign:

Does Understanding the Equal SignÂ Matter?

Equations and Equivalence in 3rd Grade

Ultimately, I don’t think the problem is in interpreting the equals sign exactly. It’s more about reading an equation, which is hard.

That said, I don’t want to ram directly into their rigid understanding of equals signs and equations, so I use arrows and buckets to help describe equivalence. Then, I just casually slide into using the equals signs and abstract equations in a similar way. That’s my current approach to making a change.

I used to have a big conversation in class about what the equals sign meant, but ultimately I became dissatisfied with that and moved to this approach.

Categories

## 8 + 32 = 40 <--- wait, what's the mistake??

Greetings from Singapore!

This is from my son’s Primary 1 workbook. He had written 38 instead of 32 in the top row and 2 instead of 8 in the second row.

The submitter thinks that this teacher’s feedback was probably not so effective, and I’m inclined (from a distance, obviously) to agree:

The teacher’s marking indicates that my son should fix his mistake in the parentheses provided. Yet seeing the “mistake” made me pause. I suppose that if the focus is on simply getting the numbers transferred correctly, then it’s a mistake. Yet if the focus is on finding the sum, and stating the relationship between the two numbers, then perhaps it shouldn’t be considered a mistake. This is interesting to me not so much because it reveals what’s going on in the kid’s head as it does how often teachers have narrow ideas of what’s correct. As for what’s going on in the kid’s head, he could’ve simply been rushing. He correctly transferred the numbers in the other problems on the worksheet. This “mistake” could’ve been used to spark a discussion about why the correct answer was obtained and when it’s appropriate to shift quantities around and when it isn’t.

Categories

## How Many More Minutes? (#elemmathchat)

Here’s the mistake we started with:

Here areÂ some of the ideas we came up with:

I wasn’t able to turn all of the ideas into activities, but here are the follow-up activities I came up with. If I were addressing this error in class I think this could be a progression of activities that help address the thinking in this mistake.

What do you think?

Update:Â This post from Andrew seems relevant.

Categories

## Finding the Area of a Rectangle [Desmathmistakes Activity]

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?

When I asked teachers this question as part of aÂ Desmathmistakes activity, there wereÂ 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,Â and Cindy.

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

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

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.

Categories

## 1,000 Lines of Symmetry

“You’d run out of roomÂ in the circle!”

How do kids come to see lines as having no thickness at all? What experiences would support that change?

Categories

## [Class Set] – 4th Grade, Money Subtraction

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.Â (11 kids, one absent.)Â 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 askÂ students to work on preciselyÂ 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

Student 2

Student 3

Student 4

Student 5

Student 6

Student 7

Student 8

Student 9

Student 10

Categories

## Bedtime Math Mistakes

From Bedtime Math:

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

From the submitter, whoÂ 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 Â 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! Â I’m inclined to put the first student in the “extending the thinking you’d do in oneÂ model to a less familiar situation” category and the second student in the associational mistakeÂ (same link) category.
Categories

## How Many More Miles Total?

The submitter of this mistake notes,

This mistake brings up the concept of teaching with keywords to me.Â  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. Â 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 anÂ all-mathÂ skill,Â 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 aÂ localÂ 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,Â rewrite them in a list so you can focus on what the problem is asking”?
I suppose that I’m inclined to think ofÂ keyword mistakes as a lot like applying the distributive property where it doesn’t belongÂ 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.Â 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?
Categories

## Recursive and Relational Thinking and the Feedback each Deserves

Take a moment Â before reading on. How many squares would be in the 7th step of this pattern? In the 43rd? In theÂ 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 toÂ think about the changeÂ of this pattern in two different ways.

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

The relational goggles are more powerful and useful. Unit Chefs 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’veÂ organized the mistakes into two categories, and I’m curious if you’ll see them the way I do.

Category 1:

Category 2:

Â Â Â Â

The way I see it, all the mistakes that I placed in Category 1 show strong evidence of seeing the pattern’s change relationally.Â 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 thinkÂ the students in Category 1 should get? What about the students inÂ 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 theÂ wrong answer might be thinking in more sophisticated ways than students who got some question correct. WhatÂ 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 aÂ very lightÂ touch with these students. They’re working within a powerful framework — they’ll likely be able to tease out where they went wrong.Â 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, becauseÂ even though getting a correct answer isn’t all that matters, it also matters to students and to me.Â I want to show that I value correctness. Second, because seeing whatÂ doesn’tÂ 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 youÂ 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’Â good thinking, so they are reluctant to offer them another way of thinking. They’re willing to deferÂ 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 theseÂ 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 IÂ might say:

• Lovely work so far.Â 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’sÂ a counter-example to the “multiplyÂ 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 nowÂ just how much that squawking hasÂ 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 thisÂ isÂ that “deficit model”Â 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 thanÂ some single kid’s Â 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 theÂ Cognitively Guided InstructionÂ 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 reallyÂ anything that your students have done, and especially if it’s from aÂ geometry unit or a geometry class. I would be eternally grateful for your class scans:Â michael@mathmistakes.org. (I’m really good at quickly anonymizing student work.)

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

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