4. How do kids think about unsimplified equations?

Balance problems provoke students to use a special sort of thinking — what I’ve been unhelpfully calling “balance reasoning” — and this sort of thinking can be helpful for solving equations. This has potential to be helpful for students in their path on the way to using equality properties to quickly solve equations.

But I really, really wish that I didn’t have to rely on balance problems for this.

That’s not because I have anything against balance problems. They’re lovely! Students should get a chance to play with them. But I’ve taught too many sad high school students who shiver when an equation shows up in class. I want these kids to be happier in those classes, to be able to see more lovely math. I really, really would like my students to be able to solve equations, and relying on balance problems to do some of this work makes me nervous.

I get nervous about relying on balance problems because it’s really a subtle thing to help students make connections between two different scenarios. It’s possible, of course, but it’s hard. This might be a melodramatic way to put things (sorry) but using these problems is a risky move.

(And the same can be said for using “pick a number” problems.)

Is there any way I can gain the benefits of balance problems without shifting contexts from equations to mobiles?

The benefit of balance problems is “balance reasoning,” a particular way-of-seeing an equation. Can this way-of-seeing be developed by students while sticking to the solving equations context? If so, what experiences would they need to have in order to develop this way-of-seeing?

I’m not sure. I’ll put out a guess, but what I (or somebody else) would really need to do is watch kids work on different sorts of linear equations and listen to them talk about how they solved them.

Here’s my guess: most students will eventually develop balance reasoning after lots of experience with unsimplified equations. By unsimplified, I mean equations like these:

• 2 + x = x + x
• a + a + a + a – 12 = a + a + a + 8
• 5 – b – b = 3 – b

Unsimplified equations are (more or less) the equations you’ll get from translating balance problems into equations.

I bet that kids who work on these equations using guess-and-check will eventually be able to understand (either by discovery, instruction from friends, discussion, direct instruction from me) how to use balance reasoning as a short-cut for their inefficient methods. And then I bet we can name it, formalize it, and extend it to less obvious situations (like 5 – b – b = 3 – b, which is tough to represent in a balance problem).

(I wonder whether these students would draw on a balance metaphor, or whether it would just be a regularity that they notice? If students do notice this pattern, how would they make sense of it? Which is another way of asking, how would they remember it?)

It’s important to note that unsimplified equations are not quite the same thing as the sorts of equations we eventually want kids to be able to solve. Not everyone who can solve an unsimplified equation will be able to solve the simplified version of that equation. Kids might not see that 4a – 12 = 3a + 8 is the same as its unsimplified version. This is the same dilemma that we had with balance problems in the first place (transfer of skills from one type of problem to another). But I’m less worried about the jump from unsimplified to simplified equations than I am about the jump from balance problems to equations. We’re all still talking about equations, it’s a smaller leap than the leap from a context that might look drastically different from an equation to a kid.

If all this is right (and who knows if it is!) then students can develop balance reasoning in an equations context, rather than a different scenario. And there are advantages to that — the leap from equation to equation is less tricky for learning than the leap from balance scenario to equation.

There still might be reasons to study balance problems in class, or even to use balance problems to develop balance reasoning.

• It often happens in my classes that kids get fed up or frustrated after a week of playing with any one scenario. They need something that isn’t just different, but it has to feel different to them. Balance problems certainly feel different than equations.
• The surface level difference between balance problems and equations is also helpful when dealing with students who have bad associations with equations, variables and numbers. Balance problems could be a sneaky way to get balance reasoning on the table for a group of kids, before we arm them with it for equations.
• Because balance problems look different, on the surface, they could make great application problems for students who understand equations in some other context. Or maybe it’s helpful to start a unit with a few balance problems, to establish a metaphor that we can later connect to the unsimplified equations?
• Maybe we think that the learning of balance reasoning will be more effective if it takes place in the balance problem context, for some reason.

I’m influenced here by the way students can get better at arithmetic in elementary years. There’s no substitute for work with actual numbers in number contexts. Students start multiplying with some basic fact knowledge, but they extend this knowledge by finding shortcuts and regularities (successive doubling, doubling and halving, finding friendly numbers, breaking apart by place value). This makes me wonder whether it’s a mistake to jump to balance problems to develop balance reasoning for solving equations. (Though balance contexts might serve an important purpose for all the reasons mentioned above.)

If any of the above is right, then the next step of this project needs to outline problem types for solving linear equations. (Analyzing problem types was the first published part of the Cognitively Guided Instruction project.) Here are some problem types I know of so far:

• Simplified vs. unsimplified equations
• Equations with addition vs. subtraction
• One-variable equations vs. multiple-variable equations

The importance of these problem types is that different linear equations allow for different sorts of reasoning. I want to get systematic about what each of those types of linear equation are, and the thinking that they each invite.

Then, I really really really need to start looking at some student work to see if the picture I’m developing resembles reality at all.

2 thoughts on “4. How do kids think about unsimplified equations?”

1. Mike Mathews says:

Have you heard of Hands on Equations? It is an attempt at solving equations using a balance model. I had a little experience using and found some success. Your posts here have reminded me of it.
Thanks for blogging your thoughts. Keep it up.

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