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.