- 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.

]]>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:

- Do mindsets predict student outcomes?
- Do student mindset interventions work?
- Are mindset intervention effect sizes too small to be interesting?
- Do teacher mindset interventions work?

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

- Do mindsets predict student outcomes?
**YES** - Do student mindset interventions work?
**YES** - Are mindset intervention effect sizes too small to be interesting?
**NO** - 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.

]]>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 (

p•q) 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 tocorrectsolutions (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.

]]>

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.”

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:

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.

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.

]]>“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.

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.

]]>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.

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.

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.

“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 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.

]]>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.

]]>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?

]]>In support of the pro-explanation camp, there is a large body of evidence that finds that it helps people when you prompt them to explain things to themselves during their learning. And this indeed does seem to settle the case, until we ask the exact same question as before: what does it mean to explain something to yourself?

Making it even more confusing? The use of “self-explanation” by researchers is a moving target. Sometimes it is supposed to be self-talk, a coherent explanation to yourself; other times it’s just any old inference you’ve made, even if it doesn’t add up to an “explanation.” Sometimes it counts as self-explanation if you say it aloud. Other times self-explanation is supposed to be an exclusively internal phenomenon.

I indeed found this extremely confusing, but felt much clearer after I read a fantastic piece by Alexander Renkl and Alexander Eitel that lays the whole situation out in a coherent way. First, they note that the situation is indeed very confusing and the meaning of “self-explanation” has changed over time:

As use of this construct has become so widespread, its meaning has changed. For example, Chi (2000) now sees (at least some) self-explanations as the process by which learners revise and improve their yet imperfect mental models. Presumably due to the widely varying use of this construct, its recent characterizations are rather general such as “inferences by the learner that go beyond the given information” (Rittle-Johnson et al., 2017) or explaining “the content of a lesson to themselves by elaborating upon the instructional material presented” (Fiorella & Mayer, 2015, p. 125). If one relies on such definitions, it is hard to draw the boundaries between other established constructs in research on learning such as “elaborative inferences” (in text learning; Singer, 1994) or “elaboration strategies” (Weinstein & Mayer, 1986). It is also questionable whether all the phenomena labeled self-explanations in previous research can be justifiably called explanation (for discussions about the concept of explanation, see, e.g., Keil, 2006; Kiel, 1999; Lombrozo, 2012).

Then they say something very sensible, which is that self-explanation is probably not one but rather many phenomena, related but distinct:

Given that even the initial characterization of self-explanation already included relatively heterogeneous phenomena (see the four findings reported by Van Lehn et al., 1992) and given that the subsequent extensions led to very general characterizations, there would appear to be little justification for discussing the learning effects of self-explanations and their use in (classroom) instruction on a general level. It is highly probable that different types of self-explanations have different functions, lead to better learning via different mechanisms, and should not be regarded as a unitary construct when providing practice recommendations.

They then announce that they are going to focus in on just one particular form of self-explanation. They call it “principle-based self-explanations” but that isn’t terribly evocative. What they’re describing is connecting the particular details of whatever it is we’re looking at to generally applicable patterns or rules. This form of self-explanation *is *a form of explanation; it’s explanation by way of connecting *th*ese specifics to *that *general rule:

We focus on principle-based self-explanations (Renkl, 1997): Learners “self-explain” a step in a problem solution (e.g., physics problem) or a feature of an object (e.g., the appearance of an animal) in reference to an underlying principle (e.g., one of Newton’s laws or mimicry). Such self-explanations were part of the initial self-explanation concept (e.g., VanLehn et al., 1992). They can also reasonably be called explanations. Although the concept of explanation can have different meanings (e.g., Keil, 2006; Lombrozo, 2012), explaining a case in terms of underlying principles is a quite prototypical case of explanation. This type of explanation fits the subsumption and unification accounts of explanation (e.g., Lombrozo, 2012). These accounts regard a case as explained when it is subsumed under a more general pattern (e.g., Newton’s laws or the general strategy of mimicry).

Let’s come back to 2 + 3 = 5. How could you connect this to some general principles? It wouldn’t just be taking two things and three more things and counting them. That’s of course valuable, but to get the benefits of what Renkl and Eitel are talking about it would have to be something more abstract and general. It probably shouldn’t involve the words “2” or “3,” or at least that’s not the important part. Maybe you’d say, “Whenever you are adding you can count on from one number by the other number.” Or maybe “You can always put both numbers on your fingers and count them together.” The point is the *always*. You are seeking to connect the particulars of *this *to a general pattern that repeats *a lot*.

The benefits of this kind of self-explanation, Renkl and Eitel write, are to be better prepared to apply that generalization to future cases. This makes sense since you are articulating a generalization, in the process abstracting some example or solution and turning it into a generally useful principle, procedure, strategy, structure.

They even have a nifty diagram:

I think this is the simplest way to think about what a mathematical explanation is, or what it should be aiming for. It’s simply asking students to make a generalization. Articulating a generalization, if one is able to, is pretty clearly going to help students handle future cases that fall under that general pattern.

Just as equally, if students aren’t yet able to articulate a generalization then they probably won’t get much out of being asked to explain. A lot of the cases that are easiest to make fun of are situations where students just don’t yet know how to articulate those generalizations. If students don’t know how to explain 2 + 3 = 5, they can be taught what those general principles are explicitly and then asked to apply that to other problems such as 4 + 5 = 9 or whatever.

One of my favorite techniques I’ve been playing with this year is giving students two generally applicable principles and asking which better explains a given solution. This removes the burden of articulation from the kids, and instead asks them to focus on deciding which abstraction best matches the particulars. And if it’s sometimes ambiguous (as arguably the choices below are) that can be OK too, since it’s all still about connecting particulars to general principles.

Practically, this means that teachers can be clearer to students about what we’re looking for. We can ask students to tell us something that is *always *true, or that will *always *work. We can teach that great explanations aren’t just about *this *problem, but that they show how a particular instance is part of a larger pattern. In this way we can teach students how to give good explanations, at first to others. But my guess is with time this habit is internalized, and you start giving better explanations to yourself.

Sure, but what about when this is very hard? If it was easy to imitate excellence, everyone would do it. It helps if you can impose some sort of structure on the effort. Sometimes it’s useful to focus on imitating just one step at a time. Other times you need to better understand what that other person did. Hopefully this is starting to sound familiar, it is typically called “teaching.”

Specifically in the context of math or science and learning procedures, there is a method for structuring this kind of learning called “fading a worked example.” I don’t remember where I first heard of it, but it’s strongly associated with Alexander Renkl’s research. Here’s how it works:

- Show someone a task and perform all of it for them.
- Then, present a new task, and perform
*almost*all of it for them, everything but the last step. They do they last step on their own. - Present
*another*new task, and let them handle a bit more of it on their own. Ask them to do the last two steps. - Etc., until the person is performing the entire task on their own.

Last week I did this sort of activity with my calculus class. It started with a fully worked example, that I prepared before class and asked students to analyze:

Then, I started knocking out steps from the procedure, starting at the end. This lets kids practice just that step, but in a meaningful context. (That is, the context of the integration problem.)

And then I knock out some more steps.

And by the end, they should be able to try the problem all on their own.

Here is another fading activity I ran in class last week, this time with my 8th Graders. I was teaching writing expressions for linear situations. At first I gave them the expression and asked them to think about why the expression fits the scenario:

But soon after I was leaving out the expression and asking them to provide it:

As I describe in my book *Teaching Math With Examples*, these sorts of tasks do much more for students than I first expected them to:

*“As my class got to work, I was surprised by how challenging the assignment was. Students were asking me good questions about mechanical issues I hadn’t even considered to be problematic for them. It turns out that lurking beneath their overall struggles had been a variety of smaller difficulties. The faded-out exercise gave them a chance to systematically work on all of them.”*

I find myself using these faded worked examples the day after a rough class. One where every student seemed to do great right up to the moment when I asked them to try the problem on their own. It’s for those days when the class seemed nervous about even starting, not because the problem was opaque but because the solution seemed like just SO MUCH.

It’s worth pointing out that this is not rocket science. Which makes sense, because teaching is not rocket science. It’s not science? It might be fair to say that teaching is *maybe *cognitive science plus other educational research plus a lot of random practical knowledge multiplied by a ton of emotional intelligence. The point is that you don’t need a researcher to invent something like fading a worked example.

One the other hand, I learned about this from reading research. And then, after I read about it in research, I sat on that knowledge for a few years. I wasn’t sure if it would fit in with the rest of my random practical knowledge about teaching. I also was worried that it would be dull and repetitive to my students, throwing off the mood in the room. (I shouldn’t have worried, it’s great.) Now that I’ve tried it a lot in my teaching, I’m happy to share the idea with enthusiasm.

Research is great, but that will only ever be a part of what it takes for a good idea to make its way into practice. The people who use the idea need to chew on it for a while and make sure it fits in with everything else. It would be good if researchers understood this, because it implies different sort of studies than what is frequently done. (“We Did This One-On-One With a Bunch of Undergrads Which Reveals Universal Teaching Techniques and Now Go Ahead and Use It.”)

Anyway, if you’ve got a long-ish procedure and it’s a topic that is easy to create many similar-ish examples for, you might consider writing a worked example and fading out the steps, starting at the finish.

]]>