I want to get ready for the school year. I’ll be teaching Algebra 1, and that’s where solving equations happens. These skills are seriously useful, so I want to do right by my students. I want to teach this topic well.

As far as I can tell, the most useful way to get better at teaching something is to better understand how kids will think about that something. So: how do kids think about solving equations?*

Actually, there are a few different ways of wording this question I might focus on.

- How do kids think about solving equation problems?
- How can kids think about solving equation problems?
- How will kids think about solving equation problems?

These three questions have different connotations to me. *Do *suggests the present reality. If I surveyed 1,000 students, what sorts of thinking would they in trying to solve an equation. *Can *implies future possibility. This might include useful, but unnatural models. *Will *implies future inevitability. An answer to “How will kids think?” might include common misconceptions or models that students tend to fall into.

I think I’m interested in answering all three questions. I want to know the ways that my students might think about this problem when they walk into my class, I want to know which models might be helpful for them to obtain, and I want to know what ways of thinking they are likely to fall into.

With all that said, I sit down with a pen and paper and try to get clearer on the potential thinking. A big problem quickly arises. There are a **lot **of ways that students* *do, can or will think about linear equations.

Here was the list that I started making for how to solve **2x + 3 = 10**.

**Mental Guess and Check –**“2 times 4 plus 3 is 11… 2 times 2 plus 3 is 7…”**Guess and Check on paper-**Same, but on paper so it’s easier to notice how close we’re getting.**Organized Guess and Check –**“2(0) + 3 = 3., 2(1) + 3 = 5, 2(2) + 3 = 7 …”**Make a Table**– Which is a way of organizing an organized guess and check**Use a graph**– Graph y = 2x+3 and look for where y is 10.**Use a scale model**– (Either with manipulatives or with paper.) Draw 2 circles and 3 boxes on one side of the scale, 10 boxes on the other. Proceed…**Use a number line**– Start at the origin. Draw an arrow to the right, label it “x.” Do that again. Then indicate a move to the right of three units. The whole thing lands at 10 on the number line. Proceed…**Work Backwards**– “We’re adding 3 at the end to make 10, so everything else makes 7. So what times 2 makes 7…”**Use Properties of Equality –**“Subtract 3 from both sides. Divide by 2.”

There are certainly more ways that students *might *think about solving equations that I haven’t listed here.

This listing could go on and on, but I don’t think this list-making will help me very much. What I need isn’t an encyclopedic knowledge of how kids think about equations. What I need is a framework for thinking about teaching solving equations. I need to know what my goals are, the value of different curricular choices, the feedback that would be helpful to give, the challenges I should anticipate.

So, I go back to my notebook. I know that I prioritize one of these ways of thinking above all others. The most useful, powerful way to solve equations is to use the properties of equality.

I think the most likely situation is that my students will come into my class being comfortable with using some version of guess and check to solve an equation.

I’ve got the beginning (guess and check) pinned down. I’ve got the ending (properties of equality) pinned down. Anything else is a toss-up. But this gives me some clarity. I’m essentially looking for mental models that can help shepherd my students from guess-and-check to using the properties of equality.

In the next piece in this series, I’ll look at candidates for that middle model.

You might want to take a look at the kids’ work on Teddy Bears’ Banquet or Growing Worms or Trapezoid Teatime to find kids who solved the “Extra” problem — I think they all asked some version of “at what step of the pattern will the outcome be ___”. You’ll find Tables, Guess and Check, Work Backwards, and some symbol manipulation, by kids ages 8 – 14ish. Most of our more traditional “algebra” problems don’t show much thinking after the equation is set up, or they choose to guess and check about the problem situation itself.

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This makes sense. Thanks for the tip!

I think of scales and number-line models as different than the rest, in some way. Scales:SolvingEquations::VisualPatterns:FindingFunctions. That is, scales and visual patterns are unlikely to develop out of a kid’s initial thinking for these types of problems. Those contexts are artificially helpful for reaching sophisticated ways of thinking.

I wonder what would happen if kids had an unusually long time to get better at solving equations. What sort of improvements on guess and check would they develop? I bet they’d notice patterns about halving or thirding for simple 2x = a or 3x = a equations. Maybe I should focus on those ways of thinking, instead?

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depending what they’re gonna be used for.

but if i were gonna lay out a system for

categorizing pigeon-holes. i’d want to’ve

inspected the actual pigeons before i got

real specific about things like who to order

the parts from or how much to pay.

students will ideally be encountered

one at a time (like washing dishes).

then we’ll throw all our theories to the wind

and improvise just like the student was a

living human being capable of things like

“having a conversation”. anything else?

have fun storming the castle. the

*sand*-castle in this case, i think.

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Dragonbox seems to have had reasonable success at gameifying this process. Its interesting to see how they progress through various manipulations of an “equation” (these start out more like divided screens with various objects on each side).

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