Here are four equations, each using similar numbers. Which one would be the most difficult for a student to solve?
Conventional wisdom is that this question is too complex to really grapple with; it depends on the kids, the class, what they’re studying, etc.
I think the conventional wisdom has a point, but I know that I’ve learned a lot about the differences between these four problems, and there ought to be a way to teach what I’ve learned to others. Of course, only other people can be the judge of whether I know something worth teaching! Here’s a rough attempt to get at some of the pedagogical structure that underlies these problems.
Part of the problem with trying to make generalizations about how kids will tend to struggle or succeed in solving equations is that there are MANY ways to successfully solve an equation. I can’t say, “when kids see bla bla bla they do bla bla bla,” because just as many kids will as won’t do that, and it largely depends on the approaches to instruction that the teacher takes with the class.
So I want to play with the idea that it can be helpful to think of an equation as having triggers that can activate various strategies that a kid might potentially use for an equation.
Here’s at attempt to explain what I mean in the context of the four equations above. There are many different ways that a child might think about solving x – 12 = 9, especially after receiving instruction on how to solve these types of equations from a teacher. Here are just a few:
The idea I’m playing with is that the equation contains a trigger for using an arithmetic strategy, i.e. a strategy that depends on no new algebraic metaphors, procedures or ideas, just whatever knowledge of the operations and arithmetic a student comes into this work with.
I want to be clear about two things. First, I’m not claiming that all students are going to approach this equation using arithmetic. Instead, I’m saying that these equations are especially well-suited for arithmetic, and a student who uses arithmetic sometimes to solve equations would very often use it in x – 12 = 9.
Second, I think that arithmetic is generally something that students have had a lot of experience in, compared to algebra. By the time an 8th Grader gets to an equation like x – 12 = 9, thinking about this equation in terms of arithmetic is likely to be productive and yield correct and helpful thoughts about the equation.
So, here’s a big generalization that I want to throw out there: equations that can trigger arithmetic are going to tend to be easier for kids.
Not all arithmetic is equal
Their earliest work identified a taxonomy of addition and subtraction word problem types. Different problem within this taxonomy are differently difficult for children even when the “arithmetic” remains constant. For example, “Tom had 11 apples, and then lost 3 of them, how many does he have left?” might be easier for a student to represent than “Tom has 11 apples and 3 of them are spoiled, how many are not spoiled?” as the former problem has an action that a student can act out using counters, paper or some other representation.
Different arithmetic word problems trigger different strategies, even when they use the same numbers. And they have a systematic way of thinking about it!
(I hope it’s clear that they’re my paradigm for the sort of systematic approach I’m trying for here…)
Here is my understanding of the way arithmetic works. Kids form meaning of the operations in terms of contexts and word problems that are naturally associated with actions. So, subtraction gets associated with taking stuff away , and that meaning is there even when the contexts aren’t. Attempts to give kids alternate ways of seeing the operations are hard precisely because the operations are built so snugly on these paradigmatic actions. This determines which strategies get triggered by various subtraction problems, even without contexts, and even when they contain the exact same numbers.
But of course most equations can’t be solved easily by thinking of their paradigmatic meaning from arithmetic. Often you can de-trigger arithmetic in an equation by messing with the numbers, and intimidating any kid’s sense that arithmetic might be productive:
x – 12.05 = 9/2
You could think about this as arithmetic, but the arithmetic would not be a friendly way to go about this since the numbers are not so amenable to the strategies and experiences of arithmetic. That’s a precise sense in which this type of equation is more difficult — arithmetic would be unavailable to students who might solve it, and they’d have to rely on newer, more algebraic approaches.
You can also de-trigger arithmetic by tossing in more variables, but here some subtlety is needed:
21 – 10x = 9
It seems likely that students would not tend to use arithmetic naturally on this because it’s so much more complex looking than a typical arithmetic problem. At the same time, kids can learn to use their arithmetic on this type of problem, and for a lot of kids this can be productive as they’re getting the hang of algebra.
That said, there are other interpretive issues that could arise for 12 – 10x = 9 if you use a more algebraic technique that wouldn’t arise for 10x – 12 = 9. In particular, the metaphor of undoing or unwinding (i.e. inverse operations) fits more naturally with 10x – 12 = 9 than 12 – 10x = 9. So that’s another precise sense in which one of these equations is trickier than the other.
In practice, I’d predict that 12 – 10x = 9 is easier for students very early in instruction who would be more likely to be triggered to use arithmetic, whereas later in instruction when kids have learned algebraic techniques 10x – 12 = 9 would be easier, because it fits better with the unwinding/undoing way of seeing an equation.
So, in short, not only are equations that have arithmetic triggers maybe easier, but equations that can be covered-up to allow arithmetic to hook-up with a student’s thinking will also be easier.
(Another subtle distinction: 12 – 2x = 9 allows for easier arithmetic than 12 – 10x = 9 because you can use your arithmetic to solve 2x = 12 but it’s tougher to use arithmetic for 10x = 12.)
