I’ve been begging people lately to send me their solving linear equations mistakes. As long as people indulge me, this will be a series of posts devoted to thinking aloud about what sorts of linear equations are hard to solve, and what it is that makes them hard, and what exactly it means to know how to handle every equation that’s out there. The goal I have in mind is a way to be able to teach a (relatively) new teacher what sorts of equations they can expect their students to struggle with and (ideally) how to systematically teach this stuff so that kids see all the cases. (Besides for the obvious, e.g. an equation with lots of stuff in its that requires a ton of steps).

I don’t have settled thoughts about this, but I do need a place to work this stuff out. These posts are unfinished and sketchy. It’s basically my math mistakes diary. You should only read this if that sounds like fun to read…

…and here we go!

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I think there are a lot of things that make this hard. I don’t know about Michelle’s students, but fraction multiplication is touch-and-go with my students, so there’s that factor.

One of the most important little “micro-skills” relating to equations that I teach my students is about scaling equations up so as to eliminate fractions. I think that would be useful here.

There’s an interesting choice about whether to do that scaling before or after distributing. That’s another interesting little choice that a student would have to make. I could imagine designing a little activity centered on a bit of strategy: when does it make sense to scale up? I think the message to kids could be that if you scale up before distributing, you might get to avoid some fraction arithmetic.

Maybe the activity could look like this. You start with that equation and then you present two options.

“Amanda took that equation and did this: 14m – 7 – 3m/5 = 24/5 – 18m/3”

“Billy’s first move looked like this: 35(2m – 1) – 3m = 6(4 – 3m)” [include annotations to make it clear what Billy did!]

And then you offer some prompts to draw attention to the choices you get to make?

Micro-skills here: knowing whether to distribute first, or whether to wait; scaling up an equation to avoid fractions

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Awesome!

Just be explicit about the mistake…we scale both sides by -6, but the left side ends up only getting scaled by 6. The fractions seem sort of incidental to this one. Maybe this is related to the thing where multiplying by a negative feels like it should make the stuff negative, rather than giving everything its opposite sign?

To what extent is this an equations mistake vs a negatives mistake?

Lots going on here. The idea that we’re scaling both sides vs. just the fractions is tricky it seems. Clearly fractions make an equation tricky to solve as it presents a number of conceptual issues. To what extent is it helpful for a teacher to understand anything about how kids handle equations with fractions besides “with difficulty”? It seems like all the interesting choices to make when handling equations with fractions are about finding ways to transform that equation so that we can avoid those fractions.

There’s an interesting choice to make about whether you try to multiply both sides of an equation by 2 or by -2. What makes that interesting is again that multiplying by a negative introduces conceptual issues that aren’t introduced by x2. I feel like I’m getting close to saying that, as a matter of coaching, it makes sense to encourage kids to scale equations by positive numbers whenever possible. (Which, I think, is always.)

Micro-skills involved: What do you choose to scale each side of the equation by? When do you choose to scale each side of the equation? (Do you distribute first, or scale first?)

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This was one of mine that I wrote about elsewhere:

Yesterday I gave students a no-grades quiz in algebra. A student who, I had been told at the start of the year, frequently struggles in math, has been having a lot of success lately. She knew exactly how to handle both of the systems of equations that were on this short quiz, but she got stumped at one of the resulting equations:

$-1.7x = 4.3x + 3.6$

I didn’t know what to say when she got stuck, exactly, but I was fairly confident that this was an example of a micro-skill that she was missing.

She and I agreed that she’d like me to write a little example on the side of her page, so I wrote this:

$-2x = 5x + 7$

[I drew some arrows going down from each side labeled “+2x.”]

$0 = 7x + 7$

My student read the example and then exclaimed (in a way I can only describe as “joyous”), Oh wait, you can make 0 there?!

Micro-skills: Knowing that you can make 0.

There seem to be a bunch of cases where you introduce and manage zeroes in solving equations. Another one of mine that I have rattling around in the archives:

Check out the bottom right. This pair of students got frozen at y = 2y + 5 because they subtracted y from each side and then weren’t sure if they were allowed to end up with 0. (Two years ago I talked to these kids about this, so I’m not guessing from their work.) This seems identical to the situation that this year’s student.

The moral of the story to me is that making zeroes is something that students need “permission” to do. Not like permission from me, the mathematical authority. I just mean that it seems to be something that beginners need help realizing is a kosher move.

As long as this picture is here, another little decision that kids need to make (see top-right) is whether to add or subtract something when trying to use a balancing move.

Microskill: “making a zero” to decide whether to add/subtract something from each side of equation

What feedback would you give to this student? Some considerations…

• Would you ask a question or make a statement?
• What written feedback would be most helpful?
• If you were able to have a conversation with this student, how would you start it?
• What would the student’s job be once you handed the paper back to him/her?

More and more these days, when I look at student work I’m just using it as a jumping off point for anything that I find interesting. When we started this project last June, I was only looking to explain how the student ended up writing what she did, but these days that requirement seems sort of restrictive. Different pieces of student work are interesting for different reasons, and what interests us is going to vary anyway.

To me, this mistake raises the possibility that it was a reading error. Reading errors tend to get poo-pooed by teachers — along with procedural errors, “stupid” mistakes, and guesses — as the results of non-mathematical issues. Either the kid was rushing, or the kid wasn’t thinking, or the kid was sloppy, etc.

Maybe that’s right. But it also seems to me that as you get better at math you get better at noticing the structure of these sorts of questions. You know what details are crucial, you eyes start to dart in different ways, you chunk the expression differently.

In other words, you learn how to read mathematically. And while some people would prefer to distinguish between mathematical knowledge and mathematical conventions and language, such distinctions don’t really do much for me. Being able to parse mathematical language seems bound up with mathematical knowledge.

In summary: A lot of the things that we call “reading errors” or “sloppiness” are really issues in mathematical thinking.

In this case I’ll offer a testable hypothesis: People who don’t really get how negative numbers work don’t see a distinction between subtraction symbols and negative signs, and will tend to elide them in reading a problem. People who do get negative numbers immediately read the numbers, along with their sign, and then read the operation between them.

(Three cheers to Andrew for the submission!)

I’m not sure that I agree with the teacher’s diagnosis that this sort of mistake is procedural. Presumably, she means that the mistake was the result of remembering some sort of algorithm, a wrong algorithm. But what sort of division algorithm would lead you to multiply 2 and 3? in this situation?

On twitter, someone suggested (was it you?) that the issue here was that 1 divided by 1/3 is 3. And then twice that would be 6.  This would then be an instance of a more common pattern of error, the “it’s always linear” error. (Was it Dave who said this? Maybe this was Dave.)

This actually fits pretty well with the student’s explanation. It’s not a bad take.

Do you agree? The first three times that I wrote this post I said something like “The ‘linearity’ hypothesis is a pretty good one, but it doesn’t quite fit with what the student said. I’d suggest that the student had an association between the numbers 1, 2, 3 and 6, and it’s that instant judgement that her explanation is aiming to justify.” But now I’m not sure if I can really find any fault with the linearity hypothesis.

People who have more experience with fractions than I have: is the linearity explanation one that resonates with your experience of kids learning to divide fractions?

(P.S. I’ve got 4th and a 5th grade assignments, along with my high school classes this year. I’m excited to bulk up my understanding of little person stuff. Any of your submissions would help me test the theory that analyzing student work is a solid way to help build pedagogical content knowledge. SUBMIT!)

In this 7th grade class, they’re studying interesting ways of counting up the boxes in the border of a square grid.

They originally start with a 10 by 10 grid, but soon after they expand to a  6 by 6 gride and a 15 by 15 grid and use their techniques to count the squares in the new border. Then, in class, they introduce the notion of variable and set about shortening one of their verbal explanations using variables.

On the board are two expressions: 6 + 6 + 4 + 4 and 15 + 15 + 13 + 13. The teacher tasks the students with writing an algebraic expression that represents the number of squares on the border. The following is a recorded interaction among students during their group work:

Sharmeen: s + s + (s-2) + (s-2). Though that’s kind of complicated. Is there any other way to put it?

Antony: What is it?

Sharmeen: Uh, mine? It was s + s + (s-2) + (s-2)

Kim: No we had to, like, um, how about we write a variable for…make a variable for thirteen.

Sharmeen: Yeah, oops. Oh, m equals… OK, so it’s s + s + m + m.

What make you of this?

[All of this is lifted from “Connecting Mathematical Ideas“.]

I just dug this up. It’s what I handed back students after a “pre-quiz” (i.e. a quiz at the end of the unit, but before their quiz). I had forgotten that during that first year I handed back these things with class performance percentages on them.

Anyway, the way those percentages break down is interesting to me. Is it surprising that kids had so much trouble with negative exponents in numerical context, but had such less trouble with variables?