The last 8 posts in this series have been about working through different aspects of how kids might think about solving linear equations. In this post I want to try to synthesize what I now know about all this and ask some questions that I don’t yet know the answers to.

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There are different types of linear equations that we want students to solve. What makes it worthwhile to distinguish the different types of equations is the different sorts of reasoning that is possible for different problems.

*An aside: Some teachers balk at the idea that we should think of teaching in this case-by-case way. These teachers would rather equip students with general methods that work in any sort of equation rather than helping students develop strategies for different types of problems. I think that this sort of teaching strategy is often a mistake. Novices in many fields often have a hard time grasping general techniques, and what experts consider to be general strategies are often only comprehensible to students as a family of related techniques. In others words, general techniques are almost always something that we obtain by making sense of related, more specific techniques. I believe that this is true based on experience and the confirmation of expert/novice research, the failure of Polya’s strategies to improve problem solving in studies, and math education research like CGI that offers a description of how many students come to understand arithmetic. *

In the previous posts, I looked at problem types that differed along the structure of the equation:

- ax + b = cx + d
- ax + b = c
- b – ax = c
- a(bx + c) = d, etc.

It’s only worthwhile to distinguish problem types on the basis of the different thinking that kids can do with them. Ideally, we would want to look at kids’ work and talk to students in order to really nail down what thinking is possible, but in the previous posts we worked out something that seems reasonable. (The “we” that worked this out was me, a handful of really sharp commentators and a few sharp people on twitter. You know who you are. Respect.) The table below represents, I think, what we hashed out:

(I obviously haven’t consulted with anyone about the correctness of this table. Let me know if you disagree. This blog is always about working out ideas, not about proselytizing for them.)

This table suggests a clear hierarchy of difficulty of solving equation problems. The equation structures that allow for the most strategies will tend to be the easiest for students, since it’s more likely that a way of approaching the problem that a student will try will bear fruit. The harder equations are the ones where the familiar strategies won’t work. Balancing won’t help you with b – ax = c. Neither will unwinding. Covering up might, but “covering up” might also be harder to learn because it isn’t as useful in the types of equations that students are most likely to practice at first — the ax + b = c and ax + b = cx + d types of problems.

Notice, also, that we can tell a story about the origins of a common misconception. Students will often try to move parts of an equation from one side of the equation to the other without keeping the sides balanced. This is a model that is useful to explain the sort of things that happen in ax – b = cx – d equations, where the subtraction makes it harder to represent the helpful algebraic move in terms of balancing weights on a mobile or any other physical quantities. Instead, it looks like we’re moving a number from one side to the other. This model sticks because, though it’s false, it can also be *used *to help students solve (incorrectly) so many other types of equations. It’s useful, even though it’s totally wrong, which is what makes it so tempting.

This all suggests to me that I have to give careful attention to the trickiest problem types — the ones that involve subtraction. What are the resources students might have to make sense of these types of problems?

- We could help students develop equality properties in the “easier” problem types, and then ask them to apply that sort of reasoning to the tricker problems.
- We could try to bend the balancing model so that it incorporates negative quantities. We might do this by treating ax – b as ax + (-b), and then removing a negative quantity from both sides of the balanced equation.
- We could try to bend the unwinding model so that it too applies more broadly. This would also involve the manipulation of expressions, along the lines of b – ax = -ax + b.

My gut says that we can’t just choose one of these routes. Different approaches might make sense for different students, depending on how they’re solving the simpler equations. And though we eventually want all students to solve equations using equality properties, that might be a rocky road for some students. They might be only comfortable with unwinding when the rest of the class is working on the trickier equations, and they’ll need help using their own methods in these new cases.

That said, the goal is still to get everyone to use equality properties.

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I think this summarizes what I’ve done. Now, what haven’t we done?

We haven’t yet created a curricular approach to developing these strategies. I’m interested in using mental equation solving activities to help students practice these strategies, along the lines of the number talks that I’ve seen work in my elementary classes with arithmetic. I’d love to have a collection of “equation strings” that target these strategies.

I thought about how the different structure an equation has can prompt different thinking. But that’s only one dimension along which an equation can differ. Equations can also differ in the numbers that are used in the place of a, b, c, and d in the table above. Some of the models are ruined by fractions, decimals or negative numbers. Others aren’t. I haven’t done any of that analysis yet, mostly because I wasn’t sure where to start. Maybe someone else can do that work, or maybe I just have to wait for when I can talk to algebra students once classes start.

Another dimension for these equations is the complexity of the expression. a(bx + c) = d wasn’t part of the above analysis. Neither was ax + bx + cx = d. There are infinite possibilities, and I wasn’t sure how to wrap my head around it without the cases multiplying and multiplying so that the whole thing got out of hand. I’d love for someone to pick this up and show me how it’s done.

On twitter, Kristin Gray really made it clear to me that it’s insanely hard to identify what approach a student has taken just based on their work. This made me quite nervous — maybe balancing reasoning is an illusion, and kids don’t really use that sort of a model when they’re working on equations. Maybe balancing reasoning is really indistinguishable from using equality properties. And how could we know if a student is unwinding or using balancing? Like I said, all of these are valid concerns that make me quite nervous. I need to get back into the classroom and talking with children so that I can hear how they talk about their thinking. I also need to see larger collections of student work beyond what I have lying around my apartment or mathmistakes.org.

Finally, now that I have all this student thinking down I think I could plan ahead for giving feedback that helps students move within the table above. What do I say if a student is using balancing, but I want them to be able to see the equality properties perspective? How do you get a kid from using the “move to the other side” model to the balancing model? How can I help kids who are only using arithmetic learn to unwind an equation? I haven’t done any of this work, but I think that I (or someone else) could.

I don’t know if I’m going to continue working on solving equations right now, but if I do I have some clear steps forward:

- Write some mental equation solving activities ala equation strings.
- Look at student work and listen to kids describe their thinking about ax + b = c and ax + b = cx + d equations to confirm that balancing/unwinding/equality properties really are distinctive ways of thinking about solving an equation.
- Look at student work and listen to kids describe their thinking about equations with negatives, decimals, fractions, or more complex expressions to understand how they think about them.
- Make notes about the feedback that would help kids move from one strategy to the other within all this.

That’s it, for now.