This is from Does Understanding the Equal Sign Matter? Evidence from Solving Equations (Knuth, Stephens, McNeil, Alibali, 2006). The article is about an aspect of solving equations that I haven’t focused on yet — the meaning of the equals symbol — but their research is useful for me in some other way.
They were interested in the connection between conceptual understanding of the equals symbol and the ability of students to solve equations. In doing this work, though, they asked a bunch of students to solve equations and coded their strategies. I’ve been interested in getting clearer on the ways kids think about solving equations, so this is a good bit of evidence to serve as a check on what I’ve been blabbering about.
The researchers identified 6 different sorts of responses to 4m + 10 = 70 and 3m + 7 = 25:
- Answer only
- No response/don’t know
- Guess and test
It’s good to know that students are using unwind (i.e. backtracking) in solving equations, and that the researchers could distinguish this from algebraic moves. (The difference is this: “in using an unwind strategy students start with the constant value from one side of the equation and then perform arithmetic operations on that value.” They coded something as “algebra” if it operated on both sides of an equation, like “subtract 10 from both sides.”)
What about the absence of strategies such as “balance reasoning”? It’s not surprising that this strategy was absent, since the “ax + b = c” equation type doesn’t really give much of an opportunity for that sort of thinking. It’s much more likely for balance reasoning to come up in equations of the type “ax + b = cx + d.”
Still, I wonder how could these researchers would have distinguished between balance reasoning and algebra had they studied these types of equations? I suppose that some students could make the balance reasoning explicit by drawing some sort of figure. Besides for that, though, I don’t know. If you take away an x from each side, that’s going to look like algebra.
I wonder if it would come through in students explanations of why you can take away an x from each side. This makes me think that balance reasoning (as opposed to use of algebraic principles) will be harder to identify in students’ thinking, but might be identifiable in student explanations.
In sum, reading the Knuth paper gave me confidence that unwinding/backtracking really is a strategy that students use and develop as part of their equation solving repertoires. It’s not some magical way of thinking invented by math teachers that kids only use if you tell them to.
This also helped me identify two different problem types for solving equations:
- c x [ax + b] = d, where unwinding/backtracking is a potential strategy, in addition to using equality properties and guess-and-check, but balance reasoning isn’t likely to show up.
- ax + b = cx + d, where balance reasoning is a potential strategy, in addition to using equality properties and guess-and-check, but unwinding/backtracking is unlikely to help.
As long as I’m here, I might as well add a third problem type:
- b – ax = c, where neither backtracking nor balance reasoning are much help.
Are there any strategies, beyond algebra, that are helpful for students as they try to solve b – ax = c? I feel like we’re close to getting all the big picture ways of thinking about solving equations on the table. There are so many types of equations, though, and there are lots of mini-moves that kids need to learn.
I feel like digging into what kids are likely to develop out of guess-and-check with b – ax = c would be interesting. I wonder if I can find any papers that detail student work for those types of equations. I wonder if anything on mathmistakes.org can be helpful. I wonder how close I can get by just spitballing based on experience?