A lot of researchers think that the math kids learn at an early age makes a difference when they learn algebra — or when they try to learn algebra but have a hard time with it. Specifically, everyone points their fingers at “mathematical equivalence” as crucially important. Equivalence isn’t just one thing, it’s more like a cluster that includes:
- solving problems like 2 + 8 = ___ + 3 (it’ s not 10 and it’s not 13)
- knowing how to define the equals sign as “is the same as” (not “the answer is”)
- remembering equations like 2 + 8 = ___ + 3 after seeing them (“encoding in memory”) even if they aren’t in the most common a + b = c format (often kids reconstruct the uncommon ones incorrectly)
Taken together, these can predict a certain amount of a kid’s future success in learning algebra. And this prediction goes beyond overall math ability, IQ, or many other things that you might want to control for.
Here’s a fantastic new paper from leading researchers on all this. The intro and discussion at the end contain tons of readable, thoughtful exposition on all these things:
Looking at these relationships between early equivalence knowledge and later algebra success leads inevitably to a conclusion: it’s really important to help kids understand how equations and equality work in their early years of school. If you can improve knowledge of equivalence, more kids will learn algebra.
OK, but why does this stuff help? A lot of theories don’t add up, but Nicole McNeil writes about a “change-resistance” hypothesis that makes a lot of sense to me.
The hypothesis goes like this: It’s harder to learn a second language as you grow older. Your knowledge of the first language is so strong that you lose flexibility. Your understanding of language is highly structured by your deep and thorough experience with the first language, and it is really hard to change how you think. You may never be 100% successful, you will never sound like a native speaker, you will never feel entirely comfortable with your non-native tongue. Not because of what you haven’t learned, but because of what you already have.
Students usually encounter equations for the first time at school, and when they do it’s often a heavy dose of equations that look pretty much the same: NUMBER SOMETHING NUMBER EQUALS BLANK. Four plus three equals blank. Five minus one equals blank. Sure, sometimes you get a question mark or a box instead of a blank. Yes, eventually multiplication and division make an appearance. Either way, there is this very rigid format to the equations kids experience in their early years.
The change-hypothesis account says, this changes kids. This is their native language.
It explains why kids can’t solve equations like 10 + 2 = ___ + 3, instead answering 12 or 15. Isn’t that how equations always work? It explains why kids define the equals sign in a narrow way as “here’s the answer” — that’s how it’s being used in all the equations they’ve experienced! And it explains why their memories have a hard time holding on to the nontraditional equations, as memory has been structured around the a + b = c format.
Now, here comes a subtlety, because we haven’t explained why this impacts later algebra success. A clean story would be that these mathematical equivalence skills are lacking for algebra students. They’re clearly prerequisite for success with algebra. If you think that equations are always telling you to perform some operation with a numerical result, yeah, algebra is going to be tough. If you can’t solve equations like 3 + 10 = __ + 5, why would you expect to be able to solve 2x – 3 = 5 + x? If you never learn mathematical equivalence, of course algebra is going to be tough for you.
Here’s the thing, though:
- The best predictor of later success is solving those problems (3 + 10 = __ + 5), following by encoding, and having a good definition of the equal sign doesn’t predict much at all
- Kids pretty much learn how to solve those types of problems as they get older (in one study that we’ll get to in a moment, undergrads solved 91.8% of these problems correctly when untimed)
But McNeil and others have an explanation for all this, which is that it’s not just about the learning. Go back to the language analogy — maybe you taught yourself to conjugate correctly in French, even though it’s not your native language. Maybe you studied really hard and practiced a great deal. But what happens in stressful moments, when you aren’t able to explicitly think through the situation? What happens when you’re negotiating over the phone and trying to remember the correct suffix for the verb? Or what happens when you’re trying to read an especially tricky French text?
The change-resistance explanation says that the initial, narrow way of thinking about equations never goes away, and it impacts your ability to learn more advanced material later.
I love some of the predictions and studies McNeil has used to test this hypothesis. My favorite are when she takes adults who — as I mentioned a moment ago — can pretty much solve the 3 + 10 = __ + 5 equations when you give them enough time, and she shows that their native language is lurking beneath the surface. There are two ways that she does this:
- Rushing them with a time constraint, and showing that when you rush a competent adult they start to make the same mistakes that 2nd and 3rd Graders make — and eye-tracking data shows that they don’t look across both sides of the equals sign when analyzing the equations, consistent with the left-to-right way of reading basic traditional equations
- Asking people to solve an arithmetic problem (like 8 + 4) reduces their ability to solve equations like 3 + 10 = __ + 5 under time pressure, compared to a control condition where participants had to add colors instead of numbers (blue and green makes ____)
According to this view, what ends up making it harder to learn algebra is this strong bias towards a + b = c equations. It’s this tendency to see equations of this type more easily. It’s possible to learn algebra even if you have this “native language” but it requires a certain amount of mindful redirection of your attention. This saps your available cognitive resources — a little or a lot, depending on the strength of the a + b = c paradigm — and makes it harder for you to learn algebra.
It also explains why interventions that simply expose students to nontraditional problem solving formats (such as 4 = 2 + 2) can make a difference — you’re really trying to disrupt the strength of the a + b = c paradigm in its formative years. McNeil’s current approach though is more holistic, focusing not just on nontraditional equation formats. I suppose this makes sense — you need to give kids a way to avoid reforming that strong bias towards a + b = c even if they continue to see problems in that format after the intervention.
If the change-resistance story is right, though, I’d think that the ultimate solution to the problem would be teachers and curricula using a variety of equation formats. There’s no real reason why equations have to all look like NUMBER OPERATION NUMBER EQUALS BLANK. I don’t think anyone says changing the way equations look would magically help everyone become great at algebra, but I think there’s a very plausible explanation for why it could really help.