# The Change-Resistance Explanation for Why Kids Struggle So Much with Algebra

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:

1. 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
2. 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?

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.

# What actually even is a growth mindset?

Not so long ago it seemed that growth mindset — briefly, the idea that your beliefs about ability impact your reactions to failure — might be on the ropes. The so-called “replication crisis” called into question research findings across many different fields, but especially social psychology. “Nobel laureate challenges psychologists to clean up their act,” wrote Ed Yong in 2012 after Daniel Kahneman called out Carol Dweck’s entire field for its underpowered studies, unreplicable work, and habit of spelunking down a dataset to find noteworthy results. (The replication crisis would come for Kahneman soon enough.)

Many of Carol Dweck’s TED talk colleagues have been lost in the reproducibility wars. Remember power posing? There were a number of prominent replication attempts of growth mindset that failed, and it seemed as if mindset was about to be essentially debunked.

But the 2nd Act of growth mindset research has been intriguing, especially once David Yeager got involved. He has helped design the sort of large, pre-registered, independently-monitored experiments that emerged from the 2010s as safeguards against researcher shenanigans. And there is now a fairly plausible story in which growth mindset holds up in a real way as a robust research finding.

“What Can Be Learned From Growth Mindset Controversies?” is the question Yeager and Dweck ask in a recent research article. They list four controversies surrounding mindset:

1. Do mindsets predict student outcomes?
2. Do student mindset interventions work?
3. Are mindset intervention effect sizes too small to be interesting?
4. Do teacher mindset interventions work?

And, in the spirit of not wasting your time, here is their answers to these questions:

1. Do mindsets predict student outcomes? YES
2. Do student mindset interventions work? YES
3. Are mindset intervention effect sizes too small to be interesting? NO
4. Do teacher mindset interventions work? NOT YET

There are conflicting findings in the mindset literature, but Yeager and Dweck attribute those to either sloppy methods or heterogeneity in impact, i.e. the fact that not everyone benefits from having more of a growth mindset.

This is crucially important for them, as the controversies have allowed them to clarify who mindset interventions will work for: students who are at risk of doing poorly but whose efforts would be rewarded. If you’re in a school where your hard work won’t be rewarded, mindset interventions won’t help. If you’re not at-risk of academic failures, mindset doesn’t come into play very much.

This sort of picking and choosing about who mindset interventions actually works for may set off your “research shenanigans” radar, and Yeager and Dweck get that. “We are aware that these kind of moderation results might, in the past, have emerged from a post hoc exploratory analysis and would, therefore, be hard to believe.” That’s right, this is the data spelunking mentioned above. However:

“But these patterns emerged from a disciplined preanalysis plan that was carried out by independent Bayesian statisticians who analyzed blinded data using machine-learning methods, and the moderators were confirmed by an independent research firm’s analyses, over which we had no influence.”

In other words, it is now part of the mindset hypothesis that this only matters for these students, and this hypothesis has support from the big, well-designed studies designed by Yeager and Dweck. Mic dropped, end of paper.

***

This is all very interesting, but any skepticism I’ve had about mindset was more rooted in confusion than methodological doubt. It just has always seemed incredibly implausible to me that these interventions should be able to make any real difference at all. So the part of the paper that I was actually most interested in was the succinct and admirably clear bit titled “What is Mindset Theory?”

I’m leaving a big chunk of this section here in case you’d like to read it yourself, but feel free to skip it. What I want to do below is explain as clearly as I can what Mindset Theory is and what it predicts, and why it would only apply to students at risk of failure. I don’t know if I believe it yet, but I now realize that there are two very important ways that people misunderstand what the Mindset Theory involves:

• It has nothing to do with the belief that “you can get better with practice”
• It largely has to do with the explanations that occur to people in the heat of a failure, like what happens to pop into their heads

Here is an observation: when people fail, they react differently. Some people seek to figure out why they failed, seek to improve, etc. And others just throw up their hands and give up. What explains these different reactions?

As an example, take me and my basketball skills. I’ve started taking going to the park sometimes and shooting around. I am, objectively speaking, bad at shooting. My form is garbage. I received one year of basketball coaching when I was in 7th Grade, but the coach pinned a kid to the floor during gym class and got fired, so it was a shaky season. When I go to the park, I miss something like 90% of the shots I take (and 100% of those I don’t?). That said, I do have good days where it feels like my shot it on and things are just going in.

Here are some things that I think are basically objective facts:

• I am bad at shooting a basketball compared to, I don’t know, people who are good at it.
• If I practiced, I would get better.
• Even if I practiced, I would never be an all-time great basketball shooter

None of these beliefs have anything to do with mindset. That’s right, even the second one about practice. That’s not mindset, not relevant here.

Now, imagine that I had a particularly bad day of shooting around at the park (like yesterday). You could imagine me not really thinking very much at all about why I did so poorly. Or you could imagine me trying to figure it out. And if I did try to explain it, to what would I attribute my failures?

In my case, I attribute yesterday’s bad shooting to a bunch of factors. First, it was raining, that’s not my fault. Also, as I mentioned, I’m not very good at shooting a basketball. Like, fundamentally, if you’re not very good at something then sometimes you’re going to be really bad at it. Now, do I think of that ability as a fixed thing? Absolutely not. I understand that I could get much better with practice. But as of now, I’m just bad at it. I think that’s a pretty productive explanation for my failure. I just need to practice.

But this is me being reflective. What about in the heat of the moment, as I’m encountering the failure for the first time? Here I might get emotional — I’m just no good at this, for whatever reason. I can imagine if a lot of people were watching me shoot around or if I was around much stronger shooters, or if I was expected in some way to be better at shooting than I am, I might just give up. And I might explain it as “I’m just no good” even if I also believe that I could get better with practice.

OK, so what does mindset theory say?

Mindset theory says that this isn’t just about basketball or academic performance — that I might have general beliefs that determine the sort of explanations I give for my failures. I might be tempted to think (as they say) “I have a certain amount of intelligence (or ability -MP) and you really can’t do much to change it.” That belief is totally consistent with a belief that you can improve with practice.

But it’s somewhat confusing to call this a “belief” because if so, it’s not a belief that you necessarily will profess all the time or in any situation. Maybe this is something you say on your bad days but not your good days. Maybe it’s something you say about ability 70% of the time, but when pushed you concede that you can do a lot to change your ability. This is a more complex and subtle — maybe more accurate — way to think about what belief means.

OK, but let’s put the pieces all together. From the perspective of mindset theory, if you are doing worse at something that you’re “supposed” to be better at, you will be tempted to explain that in terms of ability being the most important factor. Sure, you might agree that you could improve with practice. But you might believe that this would take an enormous amount of practice, given your ability. The worldview where ability or intelligence is the most important factor is the so-called “fixed mindset.” To the extent that you hold this belief strongly and absolutely, you have more of a fixed mindset. If it’s weaker or there are a lot of exceptions, you have more a growth mindset.

Can you teach people not to see the world in this way? And if you teach them that ability is not the most important factor for navigating life in our world, does that change the explanations they give for their failures? Does it lead to better actions?

Mindset theory says, yeah, it does.

One thing that’s interesting to me about this is the “more or less” or “can’t do much” side of it. There are obviously things that I can’t do because of my ability. I can’t be in the NBA, and while I agree that I can get better at basketball with practice, I don’t think I can do much about that. I think that degree of excellence at a sport is more or less about ability.

But that’s not really a problem for Yeager and Dweck. They don’t have a theory about the NBA — they have a theory about academic environments. That theory is you can teach people who fail a lot but have the potential to succeed that failure isn’t mostly about ability. And their theory is that this changes how they respond to failure.

Is it crazy that you could teach people this in an hour? I don’t know. I guess this is the same as asking whether I believe their research? I think it’s not absurd to say that you could get people headed in the right direction with an hour-long intervention that attempts to teach one big idea: that success in school mostly has to do with how hard you work. And it makes sense that this would only work if it were true — kids don’t benefit from this perspective in schools that don’t reward effort.

Maybe one way to think about why this works only for kids who experience a lot of failure is because they’re super likely to experience failure not long after the intervention. I bet a lot of kids have forgotten about this one-hour mindset lesson by the time they fail for the first time.

Maybe kids who are prone to failure get a chance to exercise this new perspective not long after their lesson, and that makes it stick longer. It gets the ball rolling in the right direction, and if their efforts are rewarded? It sticks for good.

I’m still trying to make sense of it, but Mindset Theory is starting to make some sense to me.

# A Good Eye for Arithmetic

While trying to get my lovably dorky son to put his math down and go to sleep, I came across this mistake in his work:

He is seeing this equation symmetrically — good! — but in this case he’s using mirror symmetry. From this point of view, the right side of the equation needs to be read from right to left. Isn’t this nice? That’s how equations should work.

It goes to show: a certain amount of mathematical knowledge is the ability to see things in a particular way. This is most evident in areas of math that involve shapes and stuff, but it’s just as true in arithmetic and algebra.

Jeffrey K. Bye recently shared a paper with me that explores this. The title is “Mastering algebra retrains the visual system to perceive hierarchical structure in equations” and, yeah, it’s about that. “Mathematical practice is undeniably perceptual,” they write. Some people might be inclined to say that none of this matters to the mathematician. But the algebraic notation, only a few hundred years old in most cases, is often sensitive to mathematical meaning:

While low-precedence operations like addition require a full symbol (p + q), multiplication requires only an abbreviated symbol (pq) or no symbol at all (pq). If the visual system were sensitive to such regularities, then the hierarchical structure of algebra could be read off directly from an expression’s layout. And, indeed, people are sensitive to these visuospatial norms.

Part of learning algebra is learning to see certain kind of letters and numbers as especially significant. This is a cognitive thing, but it’s also a perceptual one. A trained eye looks at these expressions differently — they have structure! We see expressions as composed of “chunks” of mathematics:

The claim of the paper is that learning algebra literally retrains your eye to see strings of symbols as objects. Their premise is that you can tell the difference between a trained and an untrained eye by how much attention is necessary to answer a question. The researchers would show participants an expression like w x a + c x f, and then ask a question about it. For example, sometimes they would color two of the letters and asking if the colors were the same of different. The goal wasn’t to catch some people answering incorrectly. The idea was that the amount of time they spend answering each question could be significant.

Their results suggested that when the mathematically significant variables are involved, people familiar with that structure answer the question with less required attention. Or, in their more jargony words, “participants who had mastered the hierarchical structure of algebra exhibited object-based attention for algebraic sub-expressions.” Right, that.

They raise another interesting point, which is that teachers and others sometimes just hate when kids aren’t thinking about things when they solve problems. The mistake that my son made at the start of this post was not the result of thinking — it was just his gut “intuition,” really a perceptual error. But when people get really good at math, they don’t think about things all the time. First they think, then it becomes “automatic.” And one of these paths towards automaticity is the retraining of the visual system:

The current results suggest that relying on visual processing might be a boon, not a barrier, to mathematical reasoning. This might come as a surprise. Confronted with evidence of students’ reliance on misleading, superficial visual strategies in algebra, some have argued that mathematical training should avoid and even suppress perceptual strategies (e.g., Kirshner, 1989; Kirshner & Awtry, 2004). For example, when asked to solve 4 + 4/2 + 2, some students might be led to answer “2,” incorrectly, because of the superficially tempting, perceptually strong 4 + 4 and 2 + 2 groups. Indeed, we sometimes found evidence for perceptual grouping around addition, rather than multiplication, particularly among participants who had yet to master the hierarchical syntax of algebra. But the fact that novices use perceptual strategies to arrive at incorrect answers does not imply that experts abandon such strategies entirely. [My emphasis – MP] Instead, experts may refine those perceptual strategies so that they become reliable, robust, and rapid routes to correct solutions (Goldstone et al., 2010; cf., Hutchins, 1995, and Rumelhart et al., 1986).

Do you have to explicitly teach students to see the structure of expressions and equations? Or does it just happen on its own when kids learn a lot of algebra via the usual concepts, procedures, skills? I don’t know. I don’t think this paper knows either.

After my son made this mistake I first said, please just go to sleep. We’ll talk about it in the morning. But when he insisted, I said, this is not how we read equations. We read both sides of the equation like we read English, left to right. Oh, he said, and he started trying to erase. Yosef, I said, please go to sleep. And then I walked out.

There’s something powerful and just a little bit scary about this kind of learning. We learn what we learn, and at some point along the way it forever changes how what we see when we look out at the world. It structures it, and then we just can’t avoid it. We’re teaching ourselves to see objects where before there were none. We can do that with algebraic expressions, OK that seems fine, but we must be doing this all the time. You know how first impressions always seem to be right? We’ve done that to ourselves, man. What we see is often not so different from how we think, something good to remember.