Dear Math Mistakes,

This piece of student work [below] puzzled us. What’s your take on what’s going on in the first one?

Sincerely,

Baffled at Ben-Gurion

***

For our readers who don’t speak Hebrew, here is my translation of the problems:

A) The oldest brother received 72,600 shekels, which was 33% of the inheritance. What was the total amount of the inheritance?

B) The second brother received 37% of the inheritance. How much did he receive?

C) What percentage of the inheritance did the third brother receive? How much did the third brother inherit?

***

Dear Baffled,

Fascinating stuff!

From the second two problems, this student seems to have a solid procedure for finding a given percentage of an original amount. If asked to find 20% of 5000, this student would compute (20 * 5000)/100.
But what do you do when you don’t know the original amount? This is what the first problem calls for, since the total inheritance is the unknown. One way that a particularly sophisticated algebra student might approach this would be to solve the following equation: (33 x ?)/100 = 72600.
It seems to me that this student is trying very hard to head towards something like this sophisticated approach. They’re looking at (20 x 5000)/100 and trying to figure out how to invert the procedure, to solve for the original amount. Neither attempt really lands at anything accurate or workable, but you can see an attempt to make sense out of inverting a formula.
In that first attempt — I’m assuming it’s a first attempt — the student tries to put the brother’s inheritance at the bottom of the fraction and to insert the other two brothers’ percentage at the top of the fraction. Again, nothing truly sensical here, but you see three attempts at inversion — 67 instead of 33, multiplication by 100 instead of division by 100, and putting the inheritance in the bottom of the fraction.
In the second attempt the student tries to scale 72600 by (100 + 33) to uncover the original total inheritance. I see this as attempting something like scaling up the 72600, which is not a bad idea at all, but this is still being embedded in a larger attempt at inverting the “forwards” procedure.
(It also seems to me possible that this student tried to solve (33 x ?)/100 = 72600 for “?” and somehow mushed together the 33 and 100, but then I wouldn’t know where the 67 came from.)
What to make of all this? All that really matters for the student is what they can learn next, and engineering this is always the hard, context-dependent part of looking at student work. Based only on knowing this, though, I’d say that they could be ready to learn to solve (33 x ?)/100 = 72600 for “?.” There are other less algebraic ways to go about this, but it seems to me like this student could be ready for the algebraic approach.

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.)

I know, I know what you’re thinking. I even know what you’re about to say. “Oy! These kids, just being taught algorithms which they blindly follow without reasoning. They even sometimes can’t even remember the algorithm! This poor kid doesn’t remember the algorithm correctly. He thinks that what he/she is supposed to do is subtract the smaller number from the larger number. Boo procedural thinking.”

Or, maybe you see this and think: “This kid isn’t even thinking. Just operating blindly on numbers. A shame, really…”

A third option: “This kid learned an incorrect rule. This kid thinks that what you’re supposed to do is subtract the smaller number from the larger number.”

Each of these explanations, I think, is a little bit off.

• The first and the third theory make predictions about what a student “thinks is right.” What could this mean, if not that the student, when asked, would say what they did was correct? But I think that students, when prompted to reflect on this work, would quickly identify the mistake.
• The second theory predicts that the student, when prompted, couldn’t explain how to properly subtract any double-digit numbers with understanding. I’m be willing to put down money that this kid, when presented with 54 – 32, could explain how to do this with as much understanding as your average kid.

We need some language and distinctions to properly describe what’s going on here.

• The kid wasn’t thinking slowly, deliberately, explicitly. He wasn’t under the sway of a procedure or a concept.
• He was just doing math, not thinking about the math he was doing. He was going with the flow, doing what seemed like it should be done.

This puts us in opposition to all three of the above theories:

1. The kid didn’t have an explicit algorithm that he was trying to follow. He wasn’t under its sway.
2. The student wasn’t just operating blindly and randomly on the numbers. He wasn’t guessing. He was doing the math without thinking about doing the math, though.
3. The kid didn’t have a mistaken concept of subtraction. He wasn’t under the sway of any particular concept. He was just doing what needed to be done.

There was a mistake here? Or a misconception? Or a false belief in a bad procedure? How exactly should we describe this?

• Objectively speaking, it is a mistake. The word “mistake” doesn’t refer to a person’s thinking, but rather refers fairly objectively to the result of their thinking. Objectively speaking, this was a mistake. The kid said something that wasn’t true.
• But there’s no evidence here of a misconception. A misconception has to do with concepts, and this kid wasn’t under the direction of any mistaken concepts. He understands what subtraction is. He understands what place value is. He could tell you about them.
• It’s not a false belief, because there’s no evidence here that this kid believes that what he did is correct.
• It’s not a dumb mistake, something that happens randomly and without thought.

Instead, maybe we should call this a mental bias, or a tendency towards this sort of mistake. This problem has revealed an underlying bias in this kid’s tendency to subtract a smaller number from the larger number. What’s revealed is a sort of magnetic urge to take away a smaller number from a larger one, rather than a larger one from a smaller one.

What do we do about those sorts of tendencies? I think that a certain kind of practice is called for, but I’m not sure. Thoughts on that? On any of this?

Decimals are hard!

“Kid should’ve realized that her answer needs to be smaller.”

How do you help kids monitor themselves in this way? Do you monitor yourself in this way when you’re doing math?

(Thanks, Ruth, for the submission!)

Last week I posted a short video from a tutoring session I had with a kid. We were solving equations, and he had some interesting ideas, and it was nice to have those ideas and his mental workings become explicit.

Here’s another chunk of that video:

Comment on whatever you like, but here are some prompts:

1. Help me understand his thinking. How did he devise his test for whether his solution is correct?

Or jump in with whatever you like in the comments.

These are some “Always, Sometimes, Never” questions. Like, “Is it always, sometimes or never true that a rhombus is a parallelogram.”

What’s the fastest way to help these students?

(Thanks for the submission, Tina C!)

Question: Evaluate the expression $-z^{2} + x(3-y)^2$  when $x = 10$,   \$latex y = -2\$, and  \$latex z = -2\$.