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

Nobody asked me, but here’s my thought.

When people learn something new, they cling to a paradigmatic case. This paradigmatic image matters more for how they use this new idea than whatever rules or logic they might otherwise adhere to. For example, when kids are first learning about triangles, they don’t identify new triangles on the basis of properties. They look at their paradigmatic image of triangle and compare this new shape to it. This is why kids misidentify so many shapes, at first. Below, 10 might be declared a triangle, though 5 would be rejected.

Now, it would be sort of besides the point to lament young children’s tendency to identify shapes in this way. This is just what the beginning of the learning curve looks like. It helps teachers to be familiar with this tendency — we can directly address it in friendly ways — but it’s totally normal. In particular, it’s not really an artifact of instruction.

(I suppose if kids make it to a weirdly old age without being able to logically identify triangles, yeah, that would be an artifact of instruction.)

I think the situation with square roots that Justin points out is pretty similar to this. When kids solve equations by taking the square roots of both sides, a lot a lot of these cases involve a square root solution. It seems totally normal for kids to start seeing this as a paradigmatic case, and to think that all solutions to such equations involve square roots. Totally normal, not something to stress too much about.

In fact, I saw this mistake in my 8th Grade class last week. A kid was using the Pythagorean Theorem, and had put little square roots over the side lengths. No stress: told him that this wasn’t necessary; reminded him of the conceptual meanings that made this move incorrect; reminded myself to include more chances for him to practice this idea; set him off to try the next problem, but without the extra square roots.

I think this is just how learning happens.

I’ve sometimes read or talked to teachers who wished kids didn’t make these sorts of mistakes. And I guess it would be nice if kids could just reach an age where they operated as logical, analytical and meaning-oriented students at the start of their learning curves. I understand why we teachers feel a bit of nervousness when kids aren’t being guided by meaning.

But ‘being guided by meaning’ is another way of saying ‘being guided by logic,’ and this is not my understanding of how beginners hold on to new ideas.

It’ll take time, practice, corrections, maybe a big ol’ worksheet, but if a kid made it that far in solving these equations, they’ll make it the rest of the way. Keep it up!

Where on this graph would you find the best sprinter of all time?

In class today, a bunch of kids said that the best ever would be at the top of the graph. It took a few seconds before a wise soul pointed out that the best would be at the bottom of the graph. But at first, that’s hard to see!

The hardness of this has to do with an idea that’s pervasive in our culture: up is better. In Metaphors We Live By, Lakoff and Johnson argue that status, virtue, wealth and many other positive attributes get an “up” orientation in our language.

After a student makes a point, I sometimes ask them “What’s the ‘therefore’?” In this case, the ‘therefore’ is just about awareness. To understand that “down is better” in this graph, we have to go against our conceptual tendencies. Students are going to make this mistake, but maybe we can help by pointing out that graphs often go against the “more is better” metaphor.

Or maybe something else entirely is going on here? What do you think is happening? Where else do you see issues like these arising in understanding math?

Here’s an activity I just drafted for my Algebra 1 class. I’m trying to help them get very comfortable working with the distributive property and fractions. Thoughts?

Can you see how to help this kid from this picture?

Does this video help?

What are the advantages and disadvantages of pictures and video, as far as presenting student work online is concerned?

Thanks again to Jonathan for the submission.

Thoughts?

I know the pic is a bit small, but can you see the mistake? It all has to do with what the exponent applies to. Somewhere on the internet one of you wrote about how you tell kids that “the exponent only sticks to one thing.” This mistake is about just that.

Thanks to Gregory for the submission.

Presented without comment, and with thanks to Andrew.