A growth mindset is a tendency to explain things in a certain way

While I’m starting to feel more comfortable believing in growth mindset research, there’s still one thing that bugs me, which is how growth mindset is defined. Here is the definition from Mindset Works, a company founded by Carol Dweck & Co.:

When students believe they can get smarter, they understand that effort makes them stronger. Therefore they put in extra time and effort, and that leads to higher achievement.

This is fine, and I think I get it. But what does it mean for kids to “believe they can get smarter”? Not to get extremely pedantic, but I have some very pedantic questions to ask. What does “believe” mean? Does it mean that you sit a kid down over a nice glass of apple juice, start shooting the breeze about life, work, how busy we all are, and then raise the question: do you think you can get smarter?

And what does it mean to “get smarter”? Or to “understand” that effort makes one “stronger”? What if I think that working hard increases my knowledge but leaves me at the same smartness? What if I tell you over apple juice that effort makes me stronger, but then in the moment I curl up in a ball of learned helplessness?

Pedantry aside, after reading up and thinking about these things I don’t think any of these questions really matter. There’s a way of defining growth mindset that makes a lot more sense to me, and I think it’s true to research. Here goes:

When students have a tendency to explain their successes or failures in terms of their efforts, they tend to work harder. Therefore they put in extra time and effort, and that leads to higher achievement.

I think this has a few advantages over the “belief”-based definition. Here they are:

  • If you have a “belief” and an “understanding” about intelligence then you presumably carry that around with you from room to room. But a tendency to explain your experiences in certain ways can quite clearly vary from room to room, even minute to minute. (Maybe this tendency is really just what we mean by “belief”?)
  • It makes clear that mindset training does not require lying to people. Every success we have is the result of our innate abilities along with our efforts. Defined in terms of a tendency to explain, the only question is which of these factors we are paying attention to in the moment. If our attention is constantly drawn to innate abilities, that will be demotivating. If our attention is drawn to the equally valid factors that are in our control, we will likely be more motivated.
  • It explains why some of those studies have such eye-popping results. Not because they’ve instantly changed some deeply held belief concerning the nature of intelligence with a few videos and lectures about brains. It’s because they have created a context in which students have are set up to explain successes/failures in terms of effort. If you’re running an experiment and researchers make a big stink about effort, and then in the context of that same group activity you have a success/failure, you will of course be more likely to turn to that explanation.

So I’m very happy with all this.

One thing I wonder is whether it’s somehow important, for the purpose of all these materials and studies, to define mindset in the Mindset Works way. It makes less sense to me, but maybe it’s important. The interventions involve taking a thing that we think of as innate and telling people that it’s actually malleable. “Intelligence can grow”; “Smartness depends on effort”; “Your brain gets bigger”; “Talent takes hard work.” If this messaging is effective, maybe it works because people with a tendency towards innate reasons are reminded of it when they reach for those “fixed” explanations?

Probably. Probably it’s a bad messaging to say, “of course you have some innate limits, but life goes better if you don’t stew on that and instead think about the things you can control.” God knows that I’ve spent my fair share of time stewing about how life would be different if I had spectacular literary talent, a tremendous hook shot, God-level charisma and an inspiring childhood, the ability to create a search engine in 1998. Probably the innate-to-malleable messaging in its most direct form — “talent is a matter of effort” and so on — is perfectly appropriate.

But that’s the messaging that for years left me unsure whether I should believe the results of growth mindset research. They might be good messaging, but they rang untrue. But if “teach people that practice grows your brain” is just a snappy way of saying “encourage people to explain their achievement in terms of effort,” I can absolutely live with it.

What People Get Wrong About Memorizing Math Facts

I haven’t done a survey or anything, but I’m pretty sure the most common way people try to teach math facts (like 3 x 7 or 9 + 6) is by giving kids a bunch of problems on a page and asking them to solve them. Sometimes the activity is timed, sometimes not, but that’s pretty much it. I think the lesson starts with putting, like, 10 problems on the board, and then sort of asking “ok so who knows how to do 3 x 7, Timmy do you know” until all the problems are solved, and then they move to the worksheet.

One of the most popular curriculums in the US for elementary school is EngageNY/Eureka, and they have a version of this activity that they call “Sprints.” A “sprint” is 44 problems that you’re supposed to solve as quickly as you can. It’s a bunch of problems on a page.

Some people love these activities, but a lot of people in math education hate them. Many of them are haunted by memories of “Mad Minute” drills in school, where you’re supposed to answer as many math fact questions as you can in a minute — that’s mad. These opponents prefer non-stressful practice that doesn’t have a time constraint. In their classrooms the facts are maybe presented one at a time on the board, and students are asked to derive the solution any way they like. Then they talk about the strategies people used. It’s not problems on a page, it’s a problem on the board, and we’re talking about it.

A good example of a proponent of this view is Jo Boaler, whose YouCubed has a position paper titled “Fluency Without Fear.” Besides for number talks (talking about a problem on the board) she also calls for practice that is fun and engaging, stuff like dice games and puzzles. As usual, the essay talks a lot about “brain research” that supposedly proves that this stuff works. Sure it does, whatever. This is the other perspective.

And what I’d argue is that actually both of these approaches make the same mistake, and it’s a big one. Because what both approaches assume is that if kids solve a problem a bunch of times, they will commit it to memory. And the thing is that this is not true, at least not true enough.

What we need is a theory — an explanation, really — for when people remember something. Let’s not make it complicated, we can put it very simply: people remember something when they’ve successfully remembered it a bunch of times. I don’t think this is saying anything that retrieval practice advocates haven’t already said — the best practice for remembering something is practicing remembering it.

Suppose that I ask you to find the product of 12 and 5. What is going on in your head as you answer the question? Maybe you start counting by 5s. Or maybe you remember that 12 x 4 is 48, and you do 48 + 12. Or you grab a piece of paper and start adding 12s. None of this is retrieval practice — none of this is practicing pulling the fact out of memory.

(To be fair, maybe you did practice pulling 12 x 4 out of memory. And maybe, when we’re talking about this problem after you solve it, you’ll end up having to remember that your answer to 12 x 5 was 60. But neither of these are sure things, though they might help if you keep doing them for enough time.)

There is a very simple question we can ask to see if math fact practice is likely to help in the most direct way: Are kids practicing remembering? Or are they practicing something else?

Let’s apply this test to EngageNY’s Sprints. Will kids be practicing remembering when working on them? Here’s a basic point: not unless they are successful. If a kid is unsuccessful at pulling a math fact out of memory, what are they going to do? They are going to try to derive it, using some sort of strategy. Maybe they’ll try using the most basic sort of strategy, something like skip-counting for multiplication or counting on fingers for addition. But of course they’ll derive — what else are they supposed to do?

The “Sprint” aspect of this is encouraging kids to move quickly, which is really only possible if they have very efficient strategies or have many facts in memory. True, if kids are successful then they’ll get some good practice. But if kids are not successful, they will be forced to derive solutions, which will take more time and is not the sort of thing we were trying to help kids get better at. They will complete much fewer problems, especially if they thoughtfully arrive at solutions to the questions. It’s a trap. Not hard to believe that this stresses some kids out.

Which means that “problems on a page” is lousy practice that helps strong kids get stronger but leaves students who know fewer facts spending time practicing something else, a.k.a. Not Very Good News.

OK, but we have precisely the same problem with the “Fluency Without Fear” activities. What are kids thinking about during a Number Talk? They are thinking about — the whole point is to think about — the various strategies that we’ve used to derive some fact. That’s great if you’re trying to study strategies. But it’s not giving anyone a chance to practice remembering stuff.

(And I really do think it’s good to teach strategies. For some facts, like 9 + 8, a lot of successful people just use a very efficient strategy and never end up memorizing it. Plus, you can’t memorize every useful fact, at some point mental strategies come into play. But also because it’s easier to memorize 9 x 5 if you know that it’s going to end in a 5 or a 0, easier to remember 7 x 9 if you know the digits are going to sum to 9, easier to remember 9 + 8 if you know it’ll be in the teens. Strategies are useful, but practicing with strategies isn’t retrieval practice.)

In sum: the vast majority of math fact instruction doesn’t focus students on the thing that it’s purporting to teach. Good news, though, it’s far from impossible to engineer practice that does focus on memorization. But how?

The basic answer here is “flash cards,” which has two big advantages. First, if you fail you can turn over the card and stick the fact back in your memory, then try again. You aren’t left to derive the fact (though you can). Second, you can repeat problems frequently to practice the problems that you didn’t answer correctly.

There are more complicated things to say. Brian Stockus just wrote a great post showing how he is doing one-on-one flashcard work with his daughter. I’ve written about some of the ways I’ve used flashcards in my 3rd and 4th Grade classrooms. And of course there are a million computer programs that promise to help teach kids math facts…they all are basically flashcards, each and every one of them, combined with some sort of gamey practice. You want research on how to help kids with difficulty learning facts how to learn facts? You’re going to find a lot of flashcards.

Nothing is a sure thing, and I don’t mean to make this sound easy. There are no guarantees in teaching, especially when you’re working with a whole class. Follow some of the links above and you’ll find lots of practical advice on how to manage the difficulties. Given our focus on retrieval practice, it goes without saying that you should only introduce a few new facts at a time, and the goal needs to be for students to be successful at remembering them by the end of the practice session.

There’s no point bemoaning the state of discourse in education, it’s bad, everyone knows it’s bad. Stop me if it sounds like I’m bemoaning, but it seems to me that pretty much every discussion about math facts misses the point, viz. everything I said above. People don’t ask the right question, which is “how do kids remember stuff?” Or rather they do, but answer the question in clearly insufficient ways.

People do not necessarily remember the things they derive. Repeatedly deriving something in a way is practice avoiding retrieval from memory, which is (I admit!) a very mathematical thing to do. Mathematicians love talking about formulas that they derive every time and can never seem to remember. These theorems or formulas aren’t anything but upper-level math facts.

So we should remember that this is a real phenomenon, and that it’s true for little kids as well. If you want people to remember something, it’s often not enough to get them to derive it, whether on a big page of problems or as part of a number talk. As usual in education, the people with the strongest opinions have missed the point, and apologies for just a bit of bemoaning.

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?

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.

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
Feel free to read or to ignore.

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:

I rewrote it for clarity and so forth.

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:

David’s teaching strategy here is to chunk a shape. Then he represents the structure with a structured arithmetic expression. The chunks of arithmetic correspond to chunks of shape.

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.

What if we ban private schools?

In the United States, many of the wealthiest students attend private schools. What if we simply passed a law that made it illegal for such schools to exist on the grounds of promoting education equity, as many people have suggested?


The first thing that would happen is that courts would declare the law unconstitutional. We know this would happen, because it already happened. In 1922 the state of Oregon passed the Oregon Compulsory Education Act. The law’s main target was Catholic schools, but it impacted all private academies. The law made it so that one could not fulfill their compulsory education requirement through private school attendance. Unless you had some other exemption (e.g. homeschooling) you had to attend your local public school.

The Ku Klux Klan were really pushing this law in Oregon. At the time they were huge defenders of public education, and this was all part of a national panic about immigration. (The Immigration Act of 1924 was a few years away.) Cited in David Tyack’s piece, here is the statement put forth by the Oregon Klan’s Grand Dragon: “The Klan favors … The American public school, non-partisan, non-sectarian, efficient, democratic, for all the children of all the people, equal educational opportunities for all.”

Here’s the full statement. Sheesh.

Anyway, the state was sued and the case made it to the Supreme Court. The Supreme Court was, like, nah man, this is very unconstitutional. It violates the 14th Amendment, the one that extends the rights of liberty to all citizens:

The fundamental theory of liberty upon which all governments in this Union repose excludes any general power of the state to standardize its children by forcing them to accept instruction from public teachers only. The child is not the mere creature of the state; those who nurture him and direct his destiny have the right, coupled with the high duty, to recognize and prepare him for additional obligations.

The world of 2021 is of course very different from 1922. The people in favor of banning private schools today are absolutely not trying to Americanize immigrants. They are definitely not the KKK. Still, it’s very illegal in this country to ban private schools.


Oh, I don’t know how. They change the constitution. The Supreme Court reconsiders. We find some weird loophole in the 14th Amendment, and also the 1st Amendment. You reboot the universe with a magic ring. Whatever.

As of 2017, about 10% of the nation’s students are in private schools, which makes 5.7 million students. That percentage has been declining slowly, with the loss mostly felt by Catholic schools. About a quarter of private school students attend nonsectarian schools. The rest attend religious schools.

What sort of religions are we talking about? Mostly Christian. From 2015: “In fall 2015, some 5.8 million students (10.2 percent of all elementary and secondary students) were enrolled in private elementary and secondary schools. Thirty-six percent of private school students were enrolled in Catholic schools, 13 percent were enrolled in conservative Christian schools, 10 percent were enrolled in affiliated religious schools, 16 percent were enrolled in unaffiliated religious schools, and 24 percent were enrolled in nonsectarian schools.”

All of the sudden, public schools would need to change how they handle religious education. To be clear, they wouldn’t have to, or that they should. But I bet there would be pressure to change, and I bet it would succeed.

The United States of America is a bit of a global outlier when it comes to teaching religion. We are the land (supposedly) of the strict separation of Church and State. But in much of Europe some sort of non-doctrinal religious education is compulsory. Granted, if you’re secular you often usually opt-out to take an ethics class instead.

Finland is pretty interesting. A course in religious education is compulsory but you get to choose your flavor depending on your religious beliefs. According to a professional organization for Finnish religion teachers: “In 2012 91.9% of pupils in comprehensive school took part in Evangelical-Lutheran [Religious Education], 1.4% in Orthodox, 1.5% in Islam, 0.5% in other religions and 4% in secular Ethics.” Seems like most people just go with the flow and take the Lutheran course. Still, a nice gesture.

If private schools are banned, the United States is going to have to start letting schools teach religion.

They’re also going to have to make some decisions about whether to actually accommodate religious holidays or dietary restrictions. In New York City providing kosher or halal food in public schools still at the “let’s try this interesting experiment” stage and not “we do this” stage.


About 2.5% of private school students are attending non-sectarian schools. I don’t know where this 2.5% falls on the wealth spectrum, but give-or-take special programs and financial aid, yeah, let’s go ahead and assume that these kids are all pretty well-off financially.

What happens if they all go into the public school system?

The first thing is that scores, particularly math scores, will probably go up. Private schools underperform public schools on math tests, as argued in “The Public School Advantage.” This is also maybe confirmed by an international perspective, where Cuba (whose schools are all public, Castro nationalized all education) outperforms its entire region on math.

Of course, there are a lot of different ways that countries handle private schools. In this graph the red parts are truly private; the grey parts are dependent on government funding but privately operated, so sort of like charters in the US:

Japan has a lot of private schools. So does Mexico.

At least in the US, I don’t think it’s particularly mysterious why scores would go up. The government surveys schools to see how much time they spend on various subjects. With the autonomy that private school purchases, these schools spend less time on math.

Also ELA. I made this chart.

So one thing that would probably happen if private schools were banned is that these wealthier students would score better on math.

But I think the big change would have to do with segregation.

Julie Halpert wrote a really phenomenal piece about public and private schools that catalogs some of what you might expect in a pure public system. The problem is that schools in the US are assigned by neighborhood, and wealthy people can pay to live in wealthy neighborhoods:

Wealthier families tend to buy homes in more expensive areas where the steep taxes result in greater school funding—and are often correlated with higher-quality schools. A study by the Brookings Institution of the country’s 100 largest metro areas found that housing costs tend to correspond with schools’ test scores: On average, home prices near high-performing schools were $205,000 more expensive than they were in areas with low-performing ones. This suggests that in an all-public-school world, those with means would likely buy real estate in well-resourced districts—removing their resources from needier school communities and diluting from those poorer areas the pool of high achievers who experts say can benefit lower-performing students.

If you put wealthy students from powerful families back in the public school system, the main thing you’d get is more wealthy public schools.

And if that doesn’t work, wealthy parents who are sharing a district with poorer ones have another tool in their kit: they can secede, forming a newer, “higher performing” district:

Between 2000 and 2016, 63 communities split off from their existing school districts to form new ones. In just the last two years, 10 more communities have followed. Most of the new districts are more affluent, and less racially diverse, than those they left behind. Another 16 communities are actively exploring efforts to form their own school districts, according to the report.

But would this be widespread? Would there be some benefits for poorer students on the margins?

I don’t know. It’s hard for me to say.

But it’s important to remember that educational inequality in this country is primarily a product of wealth inequality, especially for younger students. The advantages of wealth don’t primarily play out in the 2.5% of students attending non-sectarian private school. The advantages of wealth are (necessarily, really) mostly playing out among the other 97.5%.

Educational inequality is a big deal, but the fantasy of eliminating private schools totally misunderstands the problem.

In short: it’s definitely illegal to ban private schools for wealthy students, but even if you could all it would do is increase their math scores and spur the creation of new wealthy schools and school districts.

What it’s like to not know something

I’ve been reading the book “Metazoa” by Peter Goddfrey-Smith, and I was struck by his description of the “molecular storm,” the setting for cellular life:

“Inside a cell, events occur on the nanoscale, the scale at which objects are measured in millionths of a millimeter, and the medium in which things happen is one of water. Matter in this environment behaves differently from anything in our midsize, dry-land world. At this scale, activity arises spontaneously, without having to be made to happen. In a phrase due to the biophysicist Peter Hoffmann, within any cell is a “molecular storm,” a ceaseless turmoil of collisions, attractions, and repulsions.”

Peter Hoffmann wrote a book that looks just as fascinating, titled “Life’s Ratchet: How Molecular Machines Extract Order from Chaos.” More on the molecular storm, from the description of his book:

“The secret of life, he says, is not some “vital force,” but the unique operations of the second law of thermodynamics at the nanoscale, where molecular machines from kinesins to DNA synthase, fueled by ATP, can harness the energy of the “molecular storm”—the random bombardment of water molecules at jet-plane speeds—to move and work. Hoffman convincingly demonstrates how such “motors” could have evolved from simpler self-assembling structures, but admits that how all these cellular components came to regulate one another so precisely is still a mystery.”

Later, Goddfrey-Smith points to the storm as a crucial difference between the functioning of a biological system like a cell and a computer:

“In the days of ubiquitous computers and AI, it is natural, almost inevitable, to ask about the relationships between living systems and these artifacts. Do organisms and computers do essentially the same thing with different materials? Similarities between the two do arise, often unexpectedly, but it’s also important to recognize dissimilarities. One difference is that much of what a cell does, its main business, is something a computer never has to do. A great deal of the activity in a cell is concerned with maintaining itself, keeping energy coming in, keeping a pattern of activity going despite decay and turnover in materials. Within living systems, the activities that look like the things computers also do — electrical switching and “information processing” — are always embedded within a sea, a mini-ecology, of other chemical processes. In cells, everything that happens takes place in a liquid medium, subject to the vicissitudes of the molecular storm and all the chemical digressions that living systems engage in. When we build a computer, we build something whose operation is more regular and uniform; we build something that will be distracted as little as possible by the undirected ruminations of its chemistry.”

All of this has me thinking about knowledge and its absence. Just as cells are constantly bombarded with molecules, we exist in a storm of images, sounds, smells, and thoughts. Of course we don’t experience it this way — thank you brains! — and instead experience it in a reasonably orderly manner. This order is thanks to two things: the biological structure of our minds, and the structure provided by what we learn.

But the chaos is still lurking beneath the surface. I think you can see this when people make mistakes, especially when they are experiencing cognitive overload. In those moments the structure of knowledge seems to bend under the stress of the challenge, and we get to see some of the random dynamic churning of association.

You can see this at times when students are pushed to apply numerical skills to a new context. You could maybe, maybe, find a way to explain how a student multiplied 0.8 by 1.6 and got 8.0. But to me, that misses the point. The point is that there is no explanation. It’s just the intellectual storm of experience and association bubbling across the mind.

CR parallelogram

Mistakes in math are usually like this: the result of having insufficient structure, not of having the wrong structure. Whatever structures are present are not strong enough to withstand the storm of experience.

This is why I bristle when people talk about student misconceptions. Not because I’m sheepish about judging student ideas as right or wrong. (I’m not into “alternate conceptions.”) And I’m not totally skeptical that misconceptions matter for learning, particularly in science or history. But with math, I think the situation is different. There aren’t a lot of ways to have conceptions about some of these abstract topics. There aren’t a lot of “folk theories” in math, I don’t think.

And when I personally don’t understand something in mathematics, it usually feels like the storm to me. Undifferentiated thoughts, slipping and sliding between different intellectual currents. I’m searching for something to hold on to! And I think our students often feel similarly.

A lot of math teachers feel that every mistake needs to be addressed and explained. If every mistake were a misconception — a faulty or limited mental structure — then I think this would make sense. But mistakes aren’t usually like this. They are chaotic, a snapshot of the storm. And because of this, they can only be addressed is by learning, which is to say building something new.

Making Long Problems Shorter

Some math questions have long complicated answers. This is a problem for teachers, because students get lost in the middle of complex stuff. They end up not learning and feeling bad about it. This is not good.

This past year I taught calculus for the first time. There’s no way around it: some of those solutions take a lot of steps. Moreover, my class was a sort of intro to calc for people who aren’t especially enthusiastic about algebra. This was a class where kids apologized for being bad at math on the first day of school.

What could I, an award-winning* worked example enthusiast, do to help these students?

* Correction 6/15/2021: I have not won any awards.

One approach, one I don’t favor, is to break a complex problem up into many mini-problems. That way, students only have to think about a smaller part at any one moment. This is called “scaffolding.” It’s the approach Active Calculus takes for some of their problems:

Look, that’s still a lot of words and questions. And you can’t forget part (c) once you solve it — you haven’t learned anything until you take all five parts and make sense of them together. I do not like this, it is bad.

What I’ve started to think is that good teaching sometimes involves turning long ideas into a series of shorter ones. But in a way so that after each relatively short task you have learned something complete, gotten a full meal. The pieces add up to something greater, but each part is a meaningfully whole thought. You can breathe in between each piece without the entire structure falling apart.

For longer calculus problems this year, here was my routine for doing that.

Obviously you should not teach young kids how to multiply like this. That is not actually a complex procedure, splitting apart by place value is a powerful idea but it is not long. Let’s be real, I picked it precisely because it’s short and could fit on the image.

First, I would warm up students to the lesson with any skills they’d need for the day’s learning. I was just hoping to help them notice and remember anything they’d need for what was coming next. For this related rates lesson, I knew that the chain rule was going to be important, so we started with that.

Does this look a bit sloppy? Sure it does, it was produced during the glorious 2020-2021 schoolyear.

Then, I would slowly develop the problem, making sure everyone understood the question. “If we’re inflating a spherical balloon, what happens to the volume as it inflates? What happens to the radius?”

“Imagine we inflate the balloon with a constant flow of air, maybe using a pump or something? Will the volume change constantly? What about the radius?”

If I want to make sure students get it, I’ll ask them to take a guess as to what the answer will be. “Will the radius grow at a constant rate? Will it grow faster at first and then slow down? Or slower, then fast? What’s your guess?” (There is a lovely animation somebody made to go along with this problem. I projected it while collecting guesses.)

When it was time to dive into the solution, I would first give a “headline news” version of it. I’d project it on the board, and then talk through it.

The plan of attack. Calculus, attack!

“Here’s the plan of attack for this problem. We can write a function that connects the radius to the volume. But we’re going to turn that into an equation that connects the rate of change of volume to the rate of change of the radius. How? The derivative! We’ll treat volume and the radius as if they are not constants but functions of time, and we’ll use the chain rule to differentiate in terms of time. Then we’ll sub in all the given info from the problem — which rate of change do we already know? that’s right! — and solve for dr/dt.”

This approach is cribbed directly from Richard Catrambone’s research on “subgoal learning.”

Then I start revealing steps in the solution, filling in the empty spaces. What function do we have that connects the volume of a sphere to the radius? How do we differentiate? etc.

I often revealed just one section at a time. If I could prepare this ahead of time, I would. Otherwise I’d produce it on the board in class. Oh, also there’s a mistake in here. I fixed it in class, but will not fix it here. LAZINESS.

I would give students time to study this, time to explain it to a partner, and time to answer some self-explanation questions about it. I had previously explained in the abstract why we’d need to use the chain rule in the second step — I’d ask students to articulate that principle on their own. And I would ask students what if the diameter were 16 inches instead of 12 — what would the answer be then?

Their job now is to understand the details of this particular solution and connect it to the generalized outline I had already presented.

And once we’re comfortable with this solution, we’ll develop another question and ask students to solve just a bit of it on their own. In doing this we’re giving them something short to think about in the context of a much larger problem. These are “completion problems,” a type of task identified and studied by van Merrienboer (for example, here).

And it keeps going:

And so on:

Anyway, I think a lot of what I do as a teacher is I make shorter cycles of learning compared to a lot of other people. I’ll do inquiry, but for a few minutes. I’ll do worked examples, but I structure it so that it’s made of several self-contained tasks that (mostly) stand on their own. Is that just what we in the biz call “scaffolding”? I guess, but it’s not “scaffolding” in the sense of giving kids lots of help to climb up a very tall structure. More like just building a bunch of smaller structures that are just as valuable as the big tall one.

I don’t know what to call it, but it seemed to work well for some of my highly anxious, algebraphobic calculus students.

Start class with the good stuff

Not much is going very well in this most cursed school year, but I like the way I rewrote this lesson today.

Here is the original activity, from the Illustrative Mathematics Algebra 1 course:

I like the idea, but I saw room for improvement. Here were my issues:

  • The problems were somewhat repetitive.
  • Students could answer the questions without using the function expressions.
  • This might make it so they’d miss the main ideas until the last two questions.
  • I like creating situations where students have a principle that they try to apply to a lot of different situations. The way this is designed, they wouldn’t be applying any sort of general mathematical principle until the very last question.

For my redesign, I wanted to get a generalization out in front. I started by showing a parabola and asking students to identify the x-intercepts and the y-intercepts.

Then, I displayed a worked example showing how someone could use the standard form formula to find the y-intercept. I asked students to read all the text that was in green and to put their thumbs up when they were done reading. Then, knowing I had some deeper questions coming up, I explained each line of the example. “Because the y-intercept is always on the y-axis, we know that x will equal … say it aloud,” I gestured for everyone to chime in. “That’s right, x will equal 0. If you input that into the function, your output will be the y-coordinate of the y-intercept.”

After briefly explaining, I revealed those deeper questions:

First, a “what if” question that swapped the numbers from the original function around. Instead of “-5x + 4” I asked about “+4x – 5.” What would the y-intercept be if the function looked like that? This question is asking students to take a step towards a generalization, and that second question goes all the way towards generalizing.

The third question, “What about the factored form formula?”, is a bit of an extension question. It was a more challenging question than I could have asked at this stage if I hadn’t lead off with the example. I didn’t expect every student would necessarily arrive at a correct answer to this question, but every student could benefit from trying to apply what they’d learned to factored form.

As it happened, there was a nice little debate about whether the y-intercept could be recovered from factored form. One student said, only if you turn factored form back into standard form by multiplying. Another chimed in to say that they noticed that -1 x -4 = 4, which connects to the y-intercept, but they weren’t sure if it were a coincidence. Then a third kid pointed out that -1 x -4 would always be part of the binomial expansion, so it wasn’t a coincidence. And then I pointed out that the same logic as before — input x = 0, your output is the rest of the y-intercept — applies to factored form, and (0 – 1)(0 – 4) = 4. Good talk.

Then, it was time to apply this knowledge. I started with a question that was probably a bit too easy, but I wanted a chance to connect factored form to the graph one more time. I asked students to put a thumb up when they’d found the x-intercepts of this function:

Good, good, I asked them to shout out the x-intercepts and I pointed out that they could have gotten this just from the equations. Ditto for the y-intercepts.

Anyway, on to the good stuff. I assigned partners and asked students to talk together about what the x- and y-intercepts were of this function:

Surprisingly challenging for the kids, even given all that we’d talked about! Most got the x-intercepts, the y-intercept required a bit more discussion. All were interested in the reveal, where I removed the grey box.

Moving along, I asked students to talk to their partners about the x- and y-intercepts of this graph:

And then this one:

This one threw kids for a loop! There was much discussion among pairs as to what the “other” x-intercept was going to be. And then there audible “ooohs” from a few kids when I revealed the graph, and there was only one x-intercept.

I feel as if it’s important to mention at this point that this was a class of pretty strong students. Their reaction to this graph says as much about them as it does about me. I want to be real, I don’t often elicit “ooohs” from my teaching. I don’t even know if I want to be a teacher that gets a lot of “ooohs.” But I want to tell you the truth, and the truth is that there were “ooohs.” I could be making that up, but I’m not.

Finally came the “Are you ready for more?” problem from Illustrative Math:

The lesson as written by Illustrative Math was pretty good, but it definitely fell into a paradigm that we might call “Notice, Notice, Generalize, Apply.” To give students enough mathematical data to make the generalization the lesson asks kids to do something a bit repetitive — to find a lot of x-intercepts and y-intercepts. Because students don’t yet know how to use formulas to do this, the graphs need to be provided. Because the graphs are provided, kids will probably just use them. Only at the end, when they’re prompted to generalize, will kids go back and try to connect their answers with the formulas. Kids then are asked to apply that generalization to a single new case where a graph wasn’t provided.

My rewrite follows a different paradigm, something I’ve sometimes called “Analyze, Explain, Apply,” but might also be called “Generalize, Apply, Apply, Apply.” The generalization is front-loaded in the lesson, which means that I was able to turn some of those “notice” problems into chances to apply the big idea about using the formulas to find the intercepts. I hope it’s clear that this hardly eliminated the challenge for students. If anything, it deepened our mathematical discourse.

The generalization is the thing we want to help students think about. Sometimes it can be nice to give students a chance to form that generalization on their own. But the tradeoff is that they then have less time to try out that new mathematical idea with new problems. If you put that big idea towards the top of the lesson, they get many more chances to apply the idea to problems. Of course, that means that you have to provide that new idea to students, either in the form of an example or an explanation. That’s a trade that I’m increasingly willing to make.

Some basic questions about college admissions

  1. Who should be allowed to go to college?
  2. Should everyone?
  3. Who benefits from attending college?
  4. Suppose there were a test that could predict perfectly whether you would successfully graduate from a particular college or not. Would that be the fairest way to determine whether you should be allowed to attend a particular school?
  5. Suppose you had a number that could perfectly summarize a student’s prior achievement. Colleges could admit students above a certain number X of prior achievement. Would that be fair?
  6. If every student who graduated high school could choose to enroll in free public college, what percentage of students would? How many students now seek to attend college but aren’t able to?
  7. What percentage of students would like to attend a more selective college but are not able to, either because of money or rejected applications?
  8. Why does it benefit students to attend more selective colleges?
  9. Should there be elite private colleges?
  10. Who should be allowed to elite private colleges?
  11. Who benefits from attending elite private colleges? And how?
  12. Are there benefits to attending elite public colleges?
  13. Should there be elite public colleges?
  14. Who should decide who goes to elite private colleges?
  15. Do the “official admissions criteria” impact who private colleges admit? Or does it just force them to use different criteria to create demographically identical classes?

I find all of this confusing. I found it more confusing after reading this and this, both of which I recommend reading. Feel free to answer these questions. Are these even the right questions? Feel free to ask better questions.

A final question: “Just like you can’t actually diversify the British royal family, you can’t make colleges tools of equality; creating hierarchy is their very function.” Is this correct?