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This last claim of mine — it feels like that might be going too far. The conception of equality it supposed to enable students to successfully solve equations and do algebra.
Is it possible to offer students a sturdy enough conception of equality doing algebra? One thing I’ve been telling Kent is that I prefer to focus on strategies rather than conceptions. The hope of the authors of this paper seems to be that if you give kids a great way to understand equality as elementary students, they’re going to have a easier time learning algebra.
This project quickly began to question its own legitimacy. What do we mean by “conception” or “misconception”? Lately, I’ve been coming to doubt a lot of the power of conceptions to explain mathematical thinking.
But this is definitely not something I’m sure about, and a recent conversation with Kent and Avery made me even less sure of myself.
In wondering how I might test my ideas — or make more sense to Kent and Avery — I thought of a wonderfully clear piece of research by Eric Knuth, Ana Stephens, Nicole Mc Neil and Martha Alibali about conceptions of equality.
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The researchers here argue that students don’t properly understand the equal sign, and that this poor conception is responsible for troubles they have with solving equations and algebra, in general. The researchers call the trouble-inducing notion of equality the “operational view.” This isn’t nearly as evocative as their other name they have for this, which they call seeing equals as a “do something signal.” This view is limiting.You should subtract 32 from both sides, then, because if you In other words, I see no reason why “backtracking” (or “undoing the steps”) wouldn’t be available to a “do something” student.What about seeing the equals sign as “do something” would get in the way of backtracking?The conception will be strong enough to guide future work. It seems to me that a conception of equality that can support algebra develops a student’s grasp of algebra.In other words, there are no shortcuts and no inoculation possible in the elementary years.This passage summarizes two possibilities from Kieran and Carpenter.If students had a relational view of equality, this would be better. Then these students would see a 3 = 10 as saying that “a 3” has the same value as “10.” They would treat “a 3” as a complex algebraic object, not as a variable with an operation.” My answer is a strong yes, but I seem to interpret the question differently than the authors do.They are asking, “Is understanding the equal sign a prerequisite for learning to solve equations and do algebra? But I certainly think that understanding the equal sign matters.He says that kids with a “do something” conception should have trouble with solving 5x 32 = 97 by subtracting 32 from both sides.“What kind of meaning can students who exhibit misconceptions about the equals sign attribute to this equation? “Virtually all manipulations on equations require understanding that the equal sign represents a relation.” I’m stumped as to why he says this.