The game is Sign, played for over two hours in total silence and based on actual events. These events are: as of 1977, Nicaragua had no official sign language and deaf individuals, living in isolation from each other, were left to improvise their own gestures. The Nicaraguan government invited fifty deaf children to start a school in Managua—the plan was to teach them spoken Spanish through lip reading. This plan largely failed. But on the schoolyard, children played and socialized. As they grew older, they met at ice cream parlors, markets, dances. Spontaneously, under the noses of their teachers, these children invented their own language, what is now known as Nicaraguan Sign Language.

“In Sign,” writes philosopher C. Thi Nguyen, “Your group plays as these schoolchildren. At the beginning of the game you are assigned a goal: an inner secret that you are yearing to epxress. (In my first game, it was “I miss my cat more than I miss my parents.”) But you have no means to express it. Over the course of a few hours, you will all have to invent a sign language together, in order to communicate.”

Doesn’t this sound riveting? It’s from Nguyen’s “The Score,” a book that I think a lot of educators are going to want to read as they grapple with the proper use of grades, standardized tests, and other educational metrics.

Nguyen is a philosopher of games and clearly loves them. His book seems born out of what he experiences as a painful tension. The heart of any great game—basketball, chess, Twister, poker, Mario Kart, Sign, the fictional Cones of Dunshire— is a simple scoring mechanism. The score makes it easy for anyone to start playing, as the clear goals determine success and failure in the game. Once you’re sucked in, these simple goals guide your actions and you (hopefully) have a good time. Scores, as in these games, are good.

But!

Nguyen names case after case of scores that distort incentives, spark fixations, or institutionalize unhealthy values. Think US News and World Report’s law school rankings. Think of people chasing social media likes. Think of how some people pursue wealth seemingly for the hell of it. Think of the love administrators often have for meaningless numbers, just because those numbers are what they have. Institutional metrics are everywhere. Nguyen loves scores, but he thinks these metrics are smothering us.

“The Score” develops these ideas in parallel. On the one hand, it’s an insightful account of how games work. For Nguyen, the simplicity of a score makes it easy to temporarily adopt the game’s goals. Then, for as long as you play, you’re a slightly different person—a cutthroat killer, a member of a team, a master strategist, a Nicaraguan deaf kid. Every game is a role-playing game. You’re taking a different sort of mind on a test drive.

At the same time, Nguyen argues that institutional scores—metrics—are the technology of modern bureaucracy. This isn’t all bad. Nguyen writes about his son’s recovery from a terrible asthma attack at a hospital; this sort of recovery is only possible thanks to the coordination at scale that happens in hospitals, fueled by mechanical rules and institutional metrics.

But there are tradeoffs. We pay attention, as in games, to what is measured. Since what’s measured represents what’s possible to quantify and what institutions care about, there’s a real risk that we permanently replace our own goals with the ones represented by metrics.

This is why games and play are essential to Nguyen in our world—they teach us how to pick up and drop scores reflectively. They help us avoid having our values captured by institutions. It’s summarized neatly by a student of Nguyen’s who changes the background of her phone to a question, visible every time she opens it: “Is this the game you really want to be playing?”

It’s a big, lively book that’s stuffed with stories about pizzerias, wine-tastings and the history of cookbooks. (Nguyen used to be a food critic.) Here are a few quick education-y reflections I had while reading Nguyen’s fascinating book.

Maybe Kids Should Treat School A Little Bit Like a Game

A handy criticism of schooling is that kids just limp through their academics, treating their studies as a game. But maybe this is the healthiest possible version of institutionalized learning.

A game, in Nguyen’s telling, always has a goal and a purpose. The goal is defined by the score. But the purpose is something greater. The goal of basketball is to score points. But my purpose in playing basketball is (let’s say) to experience beautifully coordinated action. The purpose is the point, but without simple scoring nobody would play. (And pickup games would be impossible.)

For kids, school has a simple scoring system—get good grades. But the purpose of schooling is more individual, personal, and subjective than what’s represented in metrics like grades. Maybe the healthiest attitude is to think of grades as something like the score of a game. The “score” gets the motivational ball rolling and sets the direction of learning, but it’s ultimately not what matters.

Against Data-Driven Schooling

Quantitative research is good. It’s important. But it perfectly fits the “value capture” dynamic that C. Thi Nguyen describes in this book, where we gradually define our values in light of what’s measurable.

This should be correctly seen as a criticism of “data-driven” or “evidence-based” schooling. The issue is the available data and metrics don’t represent everything of value because:

• Quantified data by definition strips away contextual details

• Some data can only be collected by the government, so data availability reflects the interests of powerful actors

• Our quirkiest values or most personal goals are by definition hard to get consensus on, so aren’t measured, making public metrics a record of what we all sort of agree on, even if it’s not the most important thing for any of us

The problem isn’t really with quantitative research, but in believing that this research represents the totality of what we should care about. It’s not. And yet! Sometimes policymakers seem to be the only people in the world unaware that kids and parents care about much more than academics. These people aren’t stupid; they’ve had their values captured by public metrics. They’ve gotten lost in the game.

Make Room for Play in Schooling

Nguyen argues that play is essential and I agree.

Does everybody know about Krazy Dad? It’s arguably my favorite website on the internet. (Sorry, Substack.) It’s a guy—I guess he’s a father, I suppose he’s krazy—who creates packet after packet of logic puzzles. My personal favorite puzzle lately is Skyscraper Sudoku, which I won’t even attempt to explain. But I love ‘em, and I’ve found myself making time for Loony Pa in my classes.

Sometimes, after my students work on a puzzle, or something off the beaten path of the curriculum, I’ll ask a question: “Was this math?”

Here’s my answer to that question, the one I share with kids: Math is about thinking about what has to be true, or what can’t be. If you do that sort of thinking, that’s math. If you don’t do that thinking, it’s not. That makes math, potentially, very big. But things can be math or not, depending on how you think about them.

School is the perfect site for all the things Nguyen is worried about. It’s Value Capture City. That’s one of the tradeoffs we all collectively accepted when we made education compulsory—it’s a good deal, no regrets. But…there are tradeoffs, right? We shaved down subjects as messy and vibrant as mathematics into something coherent and tidy for school learning. We assigned scores to success with this curriculum. Naturally, people end up thinking that’s all that matters. But it’s not.

Games are resistance.1 They remind us that institutional values are just one possible set of things to care about, that we can easily adopt others. In small but tangible ways, I think ungraded puzzles and games make room for discussions about taste and value. Play is where we, for a moment or two, try being different people. And isn’t that something school should be about—trying to learn exactly who you are?

Now, back to math.

Something you learn after a few years inside schools is a lot of educational “controversies” aren’t actually that controversial. The “math wars” or “reading wars” are real, but those battles are mostly waged in universities or online. Inside, even across schools, there’s not “war” as much as a mushy, often unsatisfying, consensus.

A good example of this is math fact memorization. There are endless articles explaining why math fact memorization is important. But when Education Week surveyed 300 teachers about math fact fluency, 72% said it was “essential” and 27% said it was “helpful” for future mathematical work—only 1% said it didn’t make a difference at all.

On that same survey though, EdWeek asked teachers how they taught math facts. The dominant response—just practicing the operations over and over again, hoping that they stick—matches my impression of what’s most common in classrooms. It’s unlikely to work for many kids.

Back when I started teaching 3rd and 4th Graders, I had students practice things like 3 x 6 and 8 x 7 on paper a lot, over and over. The theory being, basically, how many times can you do the same thing without remembering it?

Heh. You know the answer, don’t you?

There are paths into New Jersey that I’ve driven a dozen times but could never in a million years recreate. There are two reasons for this. First, New Jersey is a highway hellscape and its exits were designed by psychos. Second, because I never truly rely on memory when we drive—I have Google Maps feeding me each step just in time.

In math also, kids likewise have alternatives to remembering. They can calculate. To learn facts by heart, we have to get students to rely more on their memory. (That’s why “practicing different methods of calculation” is unlikely to work.1) But how do you engineer a situation where students want to rely on memory, even though they’re (currently) bad at that?

“Oh, have you tried flashcards for this?” Yes, I have. I’ve spent a decade trying to make them work. But guess what—I’ve had it! Kids are constantly losing the decks or leaving them places. They also turn over the cards too quickly or (more often) too slowly. Plus, we need all these routines to handle situations where kids don’t immediately remember a fact: “Check the answer…say it aloud…place it next in your pile, then if you get that right…”

Thankfully, I finally have a sturdier routine, one that has uniform expectations that kids can more easily follow:

• Study the Facts: If kids are going to practice retrieving a fact from memory, the facts need to be in their memory. That’s our first priority.

• Whole-Group Practice: If we don’t practice remembering those facts right away, they’ll end up forgotten. Plus, I want to set the expectation that right now we should be answering these based on memory, rather than reasoning them out.

• Individual Quizzing: Not a “quiz” quiz; just a low-stakes chance to practice remembering this stuff on your own.

Here’s what this could look like in detail.

Study the Facts

The curriculum Math Expressions makes these wonderful study sheets for math facts. They are intended for individual use. “To practice,” they write, “students can cover the products with a pencil or a strip of heavy paper. They will say the multiplications, sliding the pencil or paper down the column to see each product after saying it.”2

I tried to repurpose this for the whole group facing the board.

“I’ll say the multiplication, and you’ll say the product right back. One times five is…” We’ll go through that first column once. If you want a challenge, try not to look at the board.

Then sometimes I’ll launch a little discussion.

“What are some things you notice about the fives?” Students might mention regularities in the tens digits or that the ones digit toggles between 5 and 0. They might mention that half of 8 is 4, half of 6 is 3, and so on. There might be good math here, but I’m really just trying to keep attention on these facts a little bit longer.

There are other ways you might focus attention on these facts. You might have a more extensive choral shpiel. You might hand out this study sheet and ask students to do the Math Expressions thing — study each column, cover it with a finger or a pencil, then quiz themselves until it’s perfect.

My flashcard routines were missing this step, instead moving directly to quizzing without any studying first. That was definitely a mistake, but now I’ve fixed it.

Whole-Group Practice

If we don’t practice things right away, we often don’t remember them. One of my goals here is to give everyone a little quizzing before the memories fade. But it’s not just that—I’m also trying to set an expectation.

“We’re trying to know these by heart. I’ll show you waht I mean. Everyone, what’s 2 + 2?” Everyone shouts out 4, immediately. “OK, but what’s 14 + 9?” The response there is slower. We’re trying to learn single-digit multiplication by heart, the same way we know 2 + 2.

“Let’s see how much we remember,” moving the study sheet off the board. “Anybody willing to get put on the spot? Toby, thanks for being brave. Let’s see how much you remember.”

I ask for volunteers because I want to put on some time pressure, and they’re going to get some wrong. “Toby, what’s 5 times 5?” He answers, I give feedback. “25, good. Now, what’s 7 times 5? No, not 45. It’s 35.” I’ll loop back to questions, to show how learning and practice works. “That’s right, 2 x 5 is 10. Now let’s go back to this one. 7 x 5 is…?”

“Give yourself a pat on the back,” I’ll say to Toby. “Anyone else want to give this a shot?”

There are other ways you might do things at this stage. You might try whole-group, choral quizzing, where you basically do flashcard practice with everybody responding. You might cover up the products on that study sheet and call on students to complete them. I can imagine a lot of things working as long as there’s fast-moving practice involved.

Individual Quizzing

Everything up till now has been to make this individual practice useful. “Go to your seats and answer these questions. Let’s see how well you remember what we studied.”

But what if they don’t remember? Well, they still have the study sheets. “You can definitely check your study sheet if you can’t remember an answer. But try to check it as few times as possible, and keep it face down until you need it.” (Of course, I’m circulating the room, keeping an eye on everything.)

This whole thing takes 10-20 minutes.

**
Flashcards are a technology for individual practice. Their whole point is to be customizable and responsive to a single person’s needs.

Teaching a group is hard, so there’s a tendency to retreat to what works for individuals. Give everyone flashcards. Give everyone a computer. Every kid has a tutor. Every kid works on the app. This tendency is maybe especially strong in thinking about math facts, since they’re so heavily studied by special education and psychological researchers who tend to think in terms of individual support—the one-on-one intervention or study. The “Science of Learning” has a bias towards individual pedagogy.

This tendency should be resisted. Teaching a group is most viable when you’re able to teach them as a group. When you treat them as twenty single individuals, each on their own different path, the job also gets twenty times harder. Now, I’m not naive. I understand there are times when the different needs of students are too great for uniform expectations. But I think we’re often too eager to turn a class into a collection of individuals. Instead we can keep class vibrant, interactive, and engaging without asking everyone to retreat to their desks. The collective deserves more respect.

I just watched like three dozen promotional videos from Alpha School. I saw young kids fencing, knitting, rock climbing, biking, jogging, petting a horse, talking to Dr. Phil, golfing, painting, training with Navy SEALs, playing piano, playing guitar, doing pullups, squatting, giving TED talks, kayaking, lighting stuff on fire, baking, running in a parking lot, and recording a podcast. But I struggled to find footage of a student reading a book.

Listen: I promise that this isn’t a takedown. I just want to make an observation.

Alpha is a network of techy private schools that uses a bundle of apps to deliver core academic instruction. They are all over the news, largely fueled by boasts of using AI to teach efficiently. This is the “2 Hour Learning” promise: that math, reading, writing, and science instruction can happen in a tight two-hour window that leaves the afternoon wide open for fencing, knitting, rock climbing, biking, jogging, petting a horse, and so on.

As Dylan Kane notes, the AI claims are a distraction. Up until recently Alpha had been using iXL—a very popular and aggressively mediocre piece of software—to provide all math instruction. They were also using Khan Academy for science. But Alpha and Dylan agree that these apps aren’t the “secret sauce”—the key is Alpha’s motivational scheme, which monitors behavior and incentivizes students to hit their goals on the apps, and especially their new app “Timeback,” which orchestrates this learning.

Joe Liemandt, the software billionaire behind Alpha, is incredibly enthusiastic about Timeback. “The single best product I’ve ever built, in four decades, by far,” he says. The premise is literally that, thanks to the efficiency of the learning and their own focused efforts with personalized learning, kids get their time back. Time back for what? For biking, jogging, petting a horse, etc.

(“Joe Liemandt was good at school, and also hated it,” a profile reports.)

Alpha is such a juicy, provocative topic with so many angles: we could dig into their data to see if it matches the hype; there are questions about digital surveillance and privacy; they pay kids to do lessons; it’s an expensive private school; it’s run by a for-profit organization; they’ve rebranded teachers as “guides”; they’re hiring a cadre of learning scientists to redesign their apps; Linda McMahon is a fan; so is Bill Ackman.

But listen, all I want to do is state the obvious: Alpha is not trying to provide the best, most ambitious math or ELA education possible according to conventional understandings of that term. If they were, they’d keep studying ELA/math in the afternoon. Instead, their goal is to minimize the time spent on core academics while maximizing skills.

This is unusual! This is not what most schools are trying to do!

Take Trinity, one of Manhattan’s fanciest schools. They tell parents that they are aiming to foster a “love for literature” in their 3rd Graders. Their texts are “primarily novels and biographies” and “chosen for their literary merit.” They say students will write personal narratives and persuasive essays. There is mandatory independent reading time and also group read-alouds. I have no idea what this looks like in the classroom—but this is what they talk about.

Alpha, on the other hand, has very different aims. Through their AlphaReads software, they assign relatively short AI-generated passages with comprehension questions, similar to those that appear on standardized exams. The texts are primarily historical, scientific, or classical: “Odysseus Returns to Ithaca,” “The Golden Age of Athens,” “Light: How It Bounces and Gets Absorbed.”

Books, as noted, are not at the core—Liemandt says they read in the afternoon, but I can’t tell if this is a daily structured activity, or just something kids sometimes do.

The texts are not human-authored. Alpha is increasing their use of AI-generated texts and questions. A recent 404 Media report had former employees speaking to how bumpy this could be for students as they roll these tools out through Timeback. My point here is simply—you can’t imagine Trinity doing this! This is not what core reading instruction looks like if you’re even talking about fostering a love for literature. Alpha’s goals are simply different than Trinity’s.

Alpha claims that their students are making extraordinary progress on NWEA’s MAP exam. This is not a takedown, I’m not going to evaluate those claims, but here’s the thing—if they are making incredible growth on MAP, that might make sense, because most fancy schools aren’t trying to do that! Most schools, once students clear a certain threshold, try to expand out beyond basic skills. They’ll do a poetry unit. They’ll have kids write their own myths. A librarian will help them find a book that they might fall in love with. Alpha is unusually ambitious in one sense and unusually unambitious in another. They want to focus on skills as measured by MAP. That is of course their right—it’s certainly interesting—but it should also color our understanding of their results.

It’s the same with math. Alpha is in the process of phasing out iXL—404 Media reports that they were scraping iXL and broke the terms of service—and moving all students over to Math Academy. You know what I think about Math Academy: that they’ve narrowed the curriculum to facilitate quick movement through it. My piece on MA was admittedy a takedown—but I’m not doing a takedown here! So let me just point out that the people around Math Academy are very explicit and clear about the tool they have built and its purpose. Its creators talk frequently of “upskilling” the importance of “full-ass effort” and the “grind” of skill development. It is a tool, like AlphaRead, for mastering necessary skills in the least amount of time possible.

This is not how most people who are ambitious and serious about math education talk! My colleagues don’t talk like that. This is an unusual goal to have! If Math Academy succeeds at accelerating strong students quickly so they can rapidly move through a lattice of skills, that might make sense—very few people are trying to do this!

What’s most novel about Alpha School and Math Academy is their fundamental orientation towards K-12 schooling. The goal, quite expressly, is to minimize it and move on. Move on to what? For Math Academy it’s Differential Equations and Machine Learning. For Alpha School it’s life skills like “Grit” and “Entrepreneurship” which are inculcated by biking, jogging, and—to tease just a little bit—petting a horse.

I keep coming around to this: the interesting innovation of Alpha School is not their apps or schedule or Timeback but their relationship to core academics. This is a school that believes that the “core” of schooling should be taken care of as quickly and painlessly as possible so that the rest of the day can be opened up to things that actually matter. Most schools don’t do this! We instead tell kids that history is a way of understanding ourselves and others. Math, we say, can be an absolute joy, full of logical surprises. We tell kids that a good story can open up your heart and mind.

Alpha doesn’t. They aim to streamline and focus on the essentials for skill mastery. Maybe what they’re showing is you can learn to comprehend challenging texts without reading books. Maybe a math education composed of examples and (mostly) multiple choice questions is, in reality, all you need to ace the SAT.

If it turns out they’re succeeding at this, it’s because they’re trying.

And maybe, one day, Alpha or someone else will crack the code for good. It then will be possible to get all students to grind through the skills and move on. With all that extra time, schools will find better things for kids to do than academics. And maybe, at some point, we’ll ask, what’s the point of grinding through things we don’t care about? Do we really need to become great at mathematics when machines can do it? How important is it really to learn how to read novels or fiction? Maybe, one day, this is how books disappear from schools for good.

But that’s just an intrusive thought I keep having, not something I’d put at Alpha’s feet. I’m simply saying this: When you’re trying to figure out whether an educational program is succeeding, look at what they’re trying to do. Alpha is trying to give students their time back from academics. Their curriculum and program reflect that aim. This is worth keeping in mind.

1.

When I got involved with BEAM, Dan Zaharopol & Co. were trying to start a math camp in New York City. They already had an overnight camp where bright, motivated kids from less wealthy neighborhoods in the city took advanced classes that were heavy on proof. Now they wanted to build out the pipeline, starting with a new camp for kids entering 7th Grade.

I taught two courses at BEAM that summer. One was focused on fractions, the other, I think, about sequences. The kids were amazing and so were the adults. It was chaotic at times—the first summer of a camp has to be—but everyone involved was smart and passionate. I felt lucky to be involved. Amy Harmon wrote about it for the New York Times.

Coming back to BEAM over the next few summers was remarkable in a different way. The organization was growing rapidly. They ironed out the wrinkles in our camp (now BEAM Discovery). They built a program in Los Angeles that paralleled New York’s. BEAM started to look more and more like a big-kid serious professional non-profit instead of a wild idea built on a mountain of Google Docs.

And they are still building!

BEAM, whose mission is to create pathways for opportunities in math and science, is now thinking hard about grades 8 through 11. They want to create “majors” for students in their programs, so that these students get in-depth, advanced experience in their area of focus—this sounds like a ton of fun for a motivated student. It’s the kind of thing kids from Washington Heights or Harlem or Bed Stuy or … uh, whatever the equivalent neighborhoods are in LA? … don’t typically get to do.

If you go to the BEAM jobs page, you’ll see 21 listings. Most of those are for summer positions, and math teachers looking for summer work should absolutely consider applying.

BEAM is also hiring curriculum developers for these grades 8-11 courses. That’s why I’m writing all this. They need course materials for these high school pathways. They need people with solid advanced math/science knowledge who can translate it into fantastic lessons. It has to be the right person. It’s a part-time role. There are a lot more details here. Take a look.

2.

Is everything getting worse, or am I just getting older? An ancient question.

Well, I think the internet’s been getting worse for about 20 years. Like all of you, I’ve been chased from platform to platform. All I want is to discover interesting things and talk to like-minded people. Big Tech has other plans. They want us to form deep and meaningful connections with the “For You” tab—in other words, they want us to form deeper connections with ourselves, and they think they know who we really are based on what they can tempt us to click on or watch.

I sure do hate it!

But, getting to the point, I’ve joined a few Discords over the past couple of years. Discord, if you aren’t familiar, is a place to chat. Now, we’re talking about the internet—of course some Discords are creepy. But many are happy places for dorks with shared interests. I’m a very happy paying member of Stereogum, the music site, and as a membership perk they invite you to the Discord. It’s lovely! People share art that they love.

Because I miss chatting with you all, I have made a Discord. You can join it by clicking the invite link: https://discord.gg/UEUzN2kd

Fair warning: I’ve named it the “Pershmail” Discord. I’ve done this because, at least for now, I am the only moderator. It’s my space, and I’ll take responsibility for people using it appropriately. That means that if you show up with weird energy or seem intent on starting fights, you’ll get the boot. (It’s also a place for adults, not children or teens, and certainly not current middle or high school students.)

I’m not trying to recreate anything like the wonderful communities of educators from the online past. Those are just super-duper gone. But I still believe that, as long as we have an internet, we should use it to talk to other actual humans and share things that we know and love. Consider joining!

At a certain point last night I needed to know exactly when people started arguing about memorization in learning. A simple search landed me on a New York Times article: “Use of Memorization in Schools Fading.” That’s from 1982, but I knew I could do better.

I continued and traced it back to 1947: “City Schools Are Developing Fresh Techniques to End ‘Monkey-Rote’ Approach.” I then needed to know if “monkey-rote” was a common phrase in 1947. It was not, :-(.

Another click or two and I was all up in an 1893 report by F.E. Marshall, a person that I have no further information about: “I believe that addition and subtraction tables are given in some books to be committed to memory; would that they were committed to the flames.” Intense!

Now here’s grammarian Guarino of Verona in 1425: “I will repeat and repeat again, and recommend many, many times that you must exercise the student’s memory. Give him something to memorize, and pay more attention to repetititon than to explanation.”1

But by the time I worked my way to Quintilian, a 1st-century Roman writer—“our whole education depends upon memory”—I decided I’d had enough, and came back to the present, where things are as confused as ever.

Last summer, Education Week featured Pam Harris, a mathematics teaching consultant, who argued that “too many classrooms focus too much on memorization.” Reporter Jennifer Vilcarino also brought in Bethany Rittle-Johnson, a leading researcher in this area, who said students “need to memorize math facts” in order to “free up their minds to see the more interesting things.”

I’m generally on board with Rittle-Johnson’s way of seeing things. But Harris does have a point: “memorize-the-steps” teaching really does happen, and it’s really not great.

The core problem with “memorize-the-steps” teaching isn’t that it’s boring or that it’s taking away a chance for discovery. The problem is, basically, that it doesn’t work, because even if the knowledge of the steps sticks in your head, that doesn’t necessarily help you become fluent in the procedure.

Explaining why takes us into what I see more and more as the core challenge of education—using language and explicit direction to steer our habits.

People Who Can Only Form Habits

There are people who, due to illness or trauma, are not able to form hold on to new information. Somehow, there are things they can learn, though they have no recollection of the learning experiences.

Clive Wearing has a near total loss of short-term memory. He experiences time in several-second episodes. It’s the stuff of literature and nightmares. At 47 years old, married, with children from a previous marriage, Clive Wearing contracted a virus that caused traumatic swelling in his brain: encephalitis. He was not expected to survive. When he did recover his brain was damaged. As a result he can not remember what he does, even moments after he does it. If he looks away, he will be surprised by your presence when he turns back. He keeps a diary—a sad, awful, beautiful diary. As a former musicologist and conductor, he was (is?) still able to play the piano, but as lovely and stirring as it is to watch videos of him playing, music providing a modicum of peace and continuity in his otherwise baffling life, let’s not kid ourselves—this shit is harrowing in the extreme.

And yet, there are things that Wearing (now 85) is able to learn. As described in the documentary Living Without Memory:

The interesting thing is that even though on a minute-to-minute basis he is forgetting everything that happens…he does show signs of implicit learning. For example when he’s been watching the same video every day, he now anticipates what is going to happen on the video, though he has no conscious recollection of ever seeing the video before.

This ability to gain new skillful knowledge even as one’s capacity to form new explicit memories has been irreperably blasted was also documented in the most famous case of Henry Molaison, known as H.M., who also suffered from lobotomy-induced amnesia. Yet H.M. was also able to make typical progress on skills like tracing a star while only watching its reflection in a mirror—a difficult task for most of us.

What is going on here? How is this learning possible? The conventional explanation is that this attests to two different systems of memory, fed by two different processes for learning—declarative and procedural.

Declarative memory is supposedly explicit. It’s often represented via language. It’s facts. It’s learning a new word for a new concept. It’s remembering my middle name. It’s knowing a rule, or a technique, a recipe, steps, an event, a story, a setting, an explanation.

Procedural memory, as researchers Knowlton and Shor put it in the Oxford Handbook of Human Memory, is “a form of implicit memory that enables us to perform skills after sufficient practice and experience without necessarily having conscious access to what has been learned.”

Like a lot of these things, the declarative/procedural distinction isn’t absolute, and the two notions seem incredibly entangled in practice.2 There’s probably no skill of any importance that doesn’t require developing both explicit and implicit knowledge. And yet there’s something undeniable about the observation. Clive Wearing can learn skills but not facts, and we all do complex things without thinking about them.

What Everyone Needs to Know About Learning

At the end of a recent class, a student came up to voice a frustration. She felt as if she understood what we were doing in class but couldn’t stop making mistakes. There was no question. It was more of a confession. If she was asking anything, it was something like, What is wrong with me? Am I just bad at this?

The answer is, no—absolutely not. This is the most normal experience ever. It’s the clash between your explicit understanding and your habits.

I told this student about the difference between understanding and forming a habit. The first thing involves focus, explanation, explicit thinking. The second, forming a habit, takes time as we retrain our associations. That sometimes involves untraining previous habits. It’s hard, laborious, and there is no real shortcut.

I see this as connected to Ira Glass’ little rant about taste outpacing ability for beginners:

“Nobody tells this to people who are beginners, I wish someone told me. All of us who do creative work, we get into it because we have good taste. But there is this gap. For the first couple years you make stuff, it’s just not that good. It’s trying to be good, it has potential, but it’s not. But your taste, the thing that got you into the game, is still killer. And your taste is why your work disappoints you. A lot of people never get past this phase, they quit. Most people I know who do interesting, creative work went through years of this.”

Learning a mathematical skill ideally doesn’t take years. But there is something parallel to this in school learning. You can understand how to solve a problem without being able to successfully solve it. Conversely, you can solve problems without necessarily understanding how you solved it.

Math Wars are Over (If You Want It)

In the EdWeek article, critics of memorization are unsure how to think about procedures that have become habitual. Pam Harris seems opposed to automatic, implicit procedural knowledge entirely:

“What we don’t need is to practice algorithms over and over and over, because what that gets us is maybe good at those algorithms. ... In reality, doing math means doing the mental actions that mathematicians do.”

Whereas Latrenda Knighten, of NCTM, seems to express the (correct) position, that the only problem is memorizing procedures, instead of practicing them to the point at which they become fluent:

Knighten differentiated algorithms from memorization by their use. Algorithms allow students to solve problems efficiently, and practicing them ultimately leads to knowing them by heart. But that shouldn’t mean they don’t have a strong grasp of the mathematics underlying the steps.

“Students will memorize something just from over time, but at the same time, they still need to be able to understand what those things mean,” said Knighten.

And Rittle-Johnson is most certainly not proposing that students memorize things that they don’t understand:

“Students should really be generating explanations, trying to make sense of things, and [looking for patterns is] just a set of instructional methods and tools that, in general, can support and connect procedural fluency and conceptual understanding,” said Rittle-Johnson.

I see the issue as basically a mistrust of habit. Harris thinks that mathematicians make every decision explicitly, without relying on instict and mindless procedures—that is not correct, because we form habits and associations from noticing correlations in our experience, without even trying. There is implicit learning for this implicit knowledge.

Meanwhile, teachers who ask students to memorize steps of complex procedures have made the same mistake. They want students, weeks later, to think back and recite mnemonics and memorized steps to solve problems. That’s not what expertise looks like. We practice to the point of fluency because that is much more reliable than laboriously recalling a memorized step and then figuring out how to apply the recipe to the problem at hand. And that is fairly mindless, since so much mental fuel is spent attending to that recipe.

Part of what makes these arguments tedious is that they elide over these basic distinctions. That our minds have these different, somewhat independent capacities is a truly basic fact about humanity—trace it back to Plato’s idea of a divided soul, if you like!

But, on the other hand, I get it. The relationship between explicit and implicit knowledge is, in fact, extremely confusing. Some students seem to be able to quickly take explicit direction and translate it into solid habit. For others, the gap seems wider, and it takes endless practice to train more accurate associations. We don’t always know how to goad the unconscious, fast-acting mind into more productive habits.

And that’s not just about teaching.

A person knows they should stop smoking but just can’t quit. A child loves museums, but gets anxious about the trip anyway. We resolve to give more to charity but don’t follow through. We promise to stop doing that thing that drives our spouse crazy, but after a month of conscious effort, we fall into old ways. We have this knowledge and understanding, and we want to impose it on ourselves, but our habits—often invisible in the moment—guide us differently.

In other words, what’s most confusing about teaching is what’s most confusing about being a person: how to get a handle on our instincts and steer them in the right direction. If you want to learn or change, there’s no way to avoid engaging with the invisible, ‘mindless’ stuff; but it’s not necessarily going to be easy, so I get why you’d want to try.

It took me a while to realize just how popular these materials were—or how popular they must be, given how often I hear about them. And yet when people talk about the direction that American math teaching is going, Gina Wilson’s name never comes up. She must be the most influential person in math education that nobody knows anything about.

Because it comes from the bottom up, it’s hard to measure Wilson’s influence. Even now, I’m doubting myself—have I overstated the case? There are numbers, though: on Teachers Pay Teachers, she has over 167,000 happy customers.

We don’t talk about Teachers Pay Teachers much anymore, do we? In its heyday in the early-2010s it was the paywalled, cutesy outpost of a freewheeling ecosystem of teacher blogs. Natasha Singer covered it in 2015 for the NYTimes:

As some on the site develop sizable and devoted audiences, TeachersPayTeachers.com is fostering the growth of a hybrid profession: teacher-entrepreneur. The phenomenon has even spawned its own neologism: teacherpreneur.

“Teacherpreneur”—that’s quite a word.

While it’s tricky to figure out how widely Wilson’s curriculum has spread, we can get a bit of a hint from survey results. American Instructional Resources Surveys is a carefully designed survey performed by researchers at RAND that tries to shed light on what resources teachers are using to teach math, ELA, and science. They ask about textbooks and curricula, but also about supplementary materials.

In the most recent results, a full 43% of respondents said they use resources from Teachers Pay Teachers. Those aren’t all using Gina Wilson’s materials, but a lot, maybe most of them are.

Who is Gina Wilson? There is remarkably little public information about her. There is a blogpost from 2013 and a brief interview from 2014. She grew up outside of Buffalo, NY and is a huge Bills fan. In 2006 she graduated college, followed her boyfriend to Virginia Beach, and began teaching Grade 8 mathematics at Great Neck Middle School.

Like so many of us, Gina didn’t like the textbook she was assigned and began searching for materials. She was an avid Pinterester, collecting the free materials that others shared online. At some point, she began making her own sheets and posting them on Teachers Pay Teachers, where she found a following.

What happened next isn’t clear. But now, “All Things Algebra” is a full-fledged curriculum company. It offers full curricula from Grades 7 through Precalculus. How many people work for this company? I don’t know. Based on the pictures Wilson has posted on Instagram, she seems to write these materials entirely on her own, scanning textbooks to find and modify questions for her own sheets. Licenses for her course materials cost several hundred dollars.

Wilson is no longer listed as an employee of Great Neck Middle School. Doing a bit of quick math—hundreds of thousands of customers, hundreds of dollars per course—she must be f***ing rich. I assume she, uhh, is no longer hanging out with 8th Graders? If she is, respect.

What can we say about the quality of Wilson’s materials? Whatever—that’s actually the wrong question, because it doesn’t matter. What’s more interesting is why they’re so popular, and that’s clear to any teacher who’s seen them: they are ready to print and use, with answer keys and plenty of white space after each question.

PLENTY of white space. Oh my god. The hours I’ve lost to white space. There is currently a nationwide effort to get teachers to actually use vetted curricula—“high-quality instructional materials”—and I swear that if people were actually serious about this all they’d have to do would be to ADD WHITE SPACE to these things and make them READY TO PRINT. It doesn’t matter how good your textbook is if it’s annoying to use. The powers that be just don’t get this, but you know who does? Gina Wilson.

There are other things that Gina Wilson understands. Her curricular materials are simple and straightforward, with common structures and routines. The answer keys are important to many teachers. (Not me, but I’m a mess, figuring out answers in realtime.) When there are variations on a typical lesson—activities like “Madlibs” or “Relay Races”—they don’t ask kids to use different mathematics but vary the surface-level activity. I don’t love this personally, because I’m boring and scared of cuteness, but I get that most teachers like these twists on the daily grind.

Last summer, Sarah Schwartz wrote for Education Week about the RAND survey. “Regardless of state or district requirements,” she wrote, “teachers mix and match the lessons and resources they use in their classrooms.” But this reporting doesn’t dig into the actual nature of these supplemental resources, their qualities, or the people who make them. No reporting does, as far as I can tell.

The biggest mistake people make in following education is assuming that classrooms reflect the official requirements. People—even people who should know better—read position papers and state mandates and say things like, “New Jersey is moving in the wrong direction!” or “Finally we’re seeing some changes in Suffolk County.”

If there is anything to understand about education, it’s this: learning happens in classrooms with teachers and little oversight, except from overstretched administrators who, often despite their best efforts, cannot keep track of everything going on in their schools.

43%—that’s a very significant number. That’s a lot of space. It’s the space between curriculum and the classroom. It’s where ed school professors lose their influence, where consultants can’t reach, where state initiatives fizzle away into nothing.

American education will either live with this space, learn to engineer around it, or try to destroy it. I wouldn’t be shocked if, some years from now, the latest technology is used to automate oversight—maybe the next gen textbook will be watching the teacher. But until then, or maybe until that fails, this space will exist—right now it belongs to people like Gina Wilson, and without making much of a fuss, she’s dominating it.

Let’s reminisce! This was 2010. A blogging habit was still something to be ashamed of. But it was becoming clear that internet personalities could make the leap. Political writers like Ezra Klein and Nate Silver got big in 2009; Lil B, the “godfather of internet rap” (whatever that means) rode virality to a debut album in 2010. This wasn’t peak-blog as much as it was peak-blog-optimism. So when I began posting lesson ideas on a blog there was a whole ecosystem of teacher bloggers I was aspiring to join. Even in this tiny world of people who thought too much about worksheets the energy was palpable.

I’d eventually benefit in tangible ways from this energy—a fellowship, speaking opportunities, consulting work, encouragement, attention. If I’d been more savvy and less interested in teaching/writing, I probably could have parlayed this into a fancier job, though probably not a better one.

Professional opportunities aside, this era was very fun. I remember surprising myself by realizing—this must have been 2013 or so—that I was fully satisfied by all this writing about teaching. The itch was scratched. I felt like the smartest people in my field were engaging with my words and that what I was doing mattered, both to me and a community of readers. I can just keep doing this forever, I thought, as the monkey’s paw curled.

By 2017 or so it was clear that the moment had passed. The energy was maybe in social media, or probably it was just gone. I felt this shift in my own little world. Fewer comments. Fewer teachers writing. Less engagement. Friends raptured into consulting work. And while I’d kept writing stories during my peak-blog era, as my friends disappeared I started devoting more attention to humor and fiction.

Laying all this out is on a certain level pretty embarrassing! On so many levels! But can’t stop now!

Over the past…I don’t know, seven years? I’ve maintained two very different writing habits. And I’ve mostly aimed to keep the education and creative worlds separate. Why? You can probably guess why. In my creative work I’m sometimes drawn to weird and extreme subjects—not that extreme, but enough to raise eyebrows at a teaching conference. I’m also very aware that my writing is accessible to students and parents and this makes for a delicate balance. I worry about crossing a line. I’m not saying Mary Gaitskill couldn’t have held down a job as a high school math teacher, but it might’ve been tough. (Especially if she published online. Especially before she was, like, Mary Gaitskill.)

That said: I recently decided to cross the streams, and it felt right.

What does it mean to “cross the streams”? First, I put links to some of my stories at the top of my posts, instead of pretending they didn’t exist. But more substantially I was very excited to send two “creative” things into your inbox:

• Curriculum Design is My Passion, where I made silly jokes about math textbooks

• How They Solved the Cubic, a historical short story about the battle over the discovery of the cubic formula

Writing these was so much fun. And whatever you might say about these pieces, they’re not “blogging.” Looking back at my output—which is what we’re doing here, this is my “year in review”—what unites my writing isn’t a single topic or theme as much as my desire to do more. I want to write sad math stories. I want to make jokes. I want to chime in on the big educational discussions. I still want to share cool teaching ideas, but I also want to figure out what that idea means, what it connects to, why it matters. All this is different from what I was doing in 2010, and thank goodness.

My Education Writing Was Fairly Cranky

My favorite education post was my most pedagogical: “Understanding” Shouldn’t Be Vague or Mysterious. I described a way to think about deep learning that I find handy. It’s fairly concrete and has a research pedigree—I think it’s a valuable shift!

Beyond this, in 2025 I often found myself in kvetchmode:

• Math Academy Wants To Supercharge Your Learning but I wasn’t impressed by their product.

• AI is Maybe Sometimes Better than Nothing although the researchers behind a weak study claimed they’d shown a lot more.

• School isn’t Just About Basic Skills and outside of policy discussions everyone seems to know this.

• Practice Software is Struggling—it’s because they’re all in on personalized learning.

• I Don’t Know What to Think About America’s Declining Test Scores and Neither Should You so stop posting screenshots of graphs.

Kvetchmode is partly a choice—I’m making an effort to connect what I know to what people are talking about. But it also reflects another year where people thought about schools in all the wrong ways.

So, what’s the right way to think about schooling? Late in 2024, I tried to gather thoughts about how schools are in tension between different goals and how that plays out for educational tradeoffs:

• 56 Theses About Education, for Fun and Debate

It’s a framework that I think holds up.

My Other Writing Was a Bit All Over the Place

My first publication came when I was in high school at a now defunct humor site called The Big Jewel. My next came in The American Bystander, a print comedy magazine, where I’ve been a frequent and enthusiastic contributor since 2018. This year the Bystander released issues #30 and #31 and I had work in both.

• Yoinlax the Zewd, The American Bystander #30. A washed-up Yiddish novelist desparately wants to get back in the game so he tries to ape a trendy BookTok-style work.

• Gravedigger for the Cadaver Synod, The American Bystander #31. Based on a real event—they dug up the previous pope’s decaying corpse to put him on trial. This is the story of the gravedigger who had to do it. (Originally posted here.)

I had two other humor publications:

• Honey I Have No Clue What You’re Talking About—I Did Not Use AI To Write My Wedding Vows, McSweeney’s Internet Tendency. Self-explanatory, see title.

• Praise for My Debut Novel, Points in Case. You know those blurbs on the back of a book? What if they were really strange?

I love funny writing, but at some point I started to bristle at its limitations. (Or maybe it’s my own limitations as a funny writer, I don’t know.) Lately I’ve been trying hard to improve as a not-necessarily-funny story writer, and I (finally!) had a bunch of stories published in non-comedic litmags:

• Groucho’s Full Quote, BULL. “Outside of a dog, a book is man’s best friend. Inside of a dog it’s too dark to read. Don’t ask how I know, it’s a terrible story. And since you didn’t ask, I’ll tell…”

• The Thing in the Jar, hex. “At a certain point the bullying became impossible to ignore. Which was how Danny and I ended up in Dr. Epstein’s office…”

• The Gorgeous Fly, HAD. “When she was eight my daughter biked past a five-foot tall shrub without looking and was hit by a car coming out of the alley. Thank goodness she was fine, though she’d broken her leg in several places…”

• Stupid Wolves, Outlook Springs. “In the beginning God created heaven and earth.
  But something felt off, so he destroyed it and began again…”

• Oh Daniel, MoonPark Review. “My wife placed my hand on her leg, but when I slipped under her nightdress she pushed me away. “Are you crazy?” she said…”

• You’re Okay Champ, Farewell Transmission. Finally, a poem, pretty directly inspired by Maurice Mannging’s wonderful collection “Bucolics.”

And one last little nice thing, or actually two of them—MoonPark Review nominated “Oh Daniel” for a Pushcart Prize, and Outlook Springs nominated “Stupid Wolves” for Best Small Fictions. I don’t think awards matter but it feels great that the editors liked these stories enough to nominate them.

What’s Next?

I’m sure I won’t have this many publications in 2026. I’ve been stuck on a few projects and, honestly, feeling very frustrated about my lack of progress. These are also longer narratives, so who knows how long they’ll take to finish? I still very much feel as if I’m trying to figure this out.

My vision for this year is to keep going. I think I’m headed in the right direction, more or less. If you subscribe to this newsletter…well, first of all, thank you very much! It means a lot to me that you’re here. If you subscribe, I think you can expect more of the same in 2026. Not making any promises, but I have a cranky education piece that I’ve been thinking about, another historical math episode that I’ve been researching, and an endless supply of ridiculous textbook images.

Subscribe now

This is all, I admit, a little bit weird. But you know what? Writing a newsletter about is a little bit weird. I’m going to lean into it. Thanks for sticking around!

For the past several years, America has been using its young people as lab rats in a sweeping, if not exactly thought-out, education experiment. Schools across the country have been lowering standards and removing penalties for failure. The results are coming into focus.

Part of the problem, according to Horowitch, is we’re giving away good grades for poor performance in high school classes. But those same signs of academic decay—low standards, reduced penalties—infect all levels of American schooling. The result? Rapidly eroding skills:

The decline started about a decade ago and sharply accelerated during the coronavirus pandemic. The average eighth grader’s math skills, which rose steadily from 1990 to 2013, are now a full school year behind where they were in 2013, according to the National Assessment of Educational Progress, the gold standard for tracking academic achievement.

But there’s a problem with Horowitch’s argument. Her piece is anchored by a fascinating report from UC San Diego about their surging remedial math program. She thinks this surge reflects a decline in basic math skills. But there’s no decline on California’s state tests, which instead show increases up until the pandemic (and slow recovery since).

This isn’t to say that Horowitch is wrong—test scores have declined in California on NAEP. But when she points to a drop starting in 2013, that could only be in 8th Grade mathematics; 4th Grade math seems to have only dipped in California with the pandemic.

So—were standards only lowered for 8th Graders? And only for math? Because 8th Graders were looking pretty good at reading on NAEP in California until the pandemic.

My point isn’t to defend or focus on California. I’m also not saying that everything is fine. What I am saying is I find all of this very confusing. Some scores definitely started dropping around 2014, but not all of them. I don’t know how you can decide what’s going on in American schools based on these graphs. Whatever the cause of the decline is, it’s not obvious or simple—maybe it’s not even problematic.

It’s Mostly About the Weakest Students

Play around with NAEP scores and you’ll notice that the declines are concentrated in the weakest students. This seems to be the case also for TIMSS, an international assessment. On both the 4th and 8th grade exams, America’s lowest performing students peaked in 2011, while the strongest students continued improving through 2019.

So are American students “getting dumber,” as Matt Yglesias says?

First, that’s rude. But second—no, they aren’t! I mean, yes, even high fliers were impacted by the pandemic. Everyone was. But strong students were looking fine up until then. That cuts out some possible explanations: this isn’t about detracking, gifted education, grading in AP classes, or anything else that would primarily impact the strongest 10% of students.

So…is it test-based accountability? Economist Joshua Goodman thinks so. He points out that 2015 is when the Obama administration started issuing waivers that defanged No Child Left Behind. Up until then, high-stakes testing pushed schools to focus on lifting weaker students up.

Listen—I don’t know! I’d quibble that since NCLB was introduced in 2001, it can’t explain the decade of improvements leading up to that law. And because NCLB was focused on math and reading it can’t explain declines in civics or US history.1

But there’s another, more significant problem with this explanation…

It’s Not Just Kids

Chad Aldeman has been writing about declining test scores for years, and generally favors an explanation not unlike Goodman or Horowitch’s. But it was from him I first learned that American scores on PIAAC, a test of workplace skills for adults of ages 16 to 65, have also been on the decline, arguably also peaking around 2014.

There were declines in every cohort, in both literacy and numeracy, even in the 55-65 group that has been out of school for over forty years.

Isn’t that…deeply weird? So it’s not just kids that have lost progress, but adults. How could schools possibly be responsible for that? And not all countries experienced a decline. So something is going on in America that impacted both students and adults beginning in 2014. What could it be?

It’s Not Just Phones, and it’s Not Just Here

Adults have phones. Teens have phones. Maybe it’s phones?

Goodman thinks this is part of the story. Horowitch mentions this too. But as Aldeman points out, phones are everywhere and declines in TIMSS scores aren’t universal:

Smartphones and social media are global phenomena, and yet scores in Australia, England, Italy, Japan and Sweden have all risen over the last decade. A couple of other countries have seen some small declines (like Finland and Denmark), but no one has else seen declines like we’ve had here in the States.

(I’d also point out that 4th Graders typically don’t have phones, so they couldn’t explain declines in 4th Grade NAEP reading.)

But on the subject of international declines, and whether America is an outlier, there is another major international exam besides TIMSS—that’s PISA. And PISA scores have been down across the board since around 2014.

Weirdly, looking at this graph, America looks like it’s beating these trends in reading and science. How do you explain that? Why would countries participating in PISA have overall declines starting midway through the decade when TIMSS did not? Why would the US partly buck the trend on PISA, the test we traditionally do worse at, and then underperform the international trend on TIMSS, the exam we do much better on?

Now, there are differences between TIMSS and PISA. TIMSS is generally more of an achievement test with questions that closely resemble what students learn in school. PISA is designed to go a step further, requiring interpretation and application. Maybe that explains the different international trends…

But at this point, I want to throw my hands up—I just don’t know what’s going on!

So, what is it?

Here’s the situation: Americans are getting dumber...well, mostly not. But our lowest performing students seem to be losing ground. And, simultaneously, our adults. Some international tests show a similar decline happening in other countries. On other exams, America is on its own. What gives?

For a moment I had a theory that I liked—that it’s about the shift from paper to digital assessments. NAEP went digital in 2017. PISA went digital in 2015. TIMSS transitioned in 2019. Kids do a lot of lazy button pushing when they take these digital exams.

Maybe the issue is giving tests with the same devices we use to scroll YouTube?

This is definitely something to worry about (it’s a “mode effect”) but it seems as if the test-makers are on top of this. I guess I trust them not to mess up? Also, you’d figure that if there was a big problematic mode effect it would be more of a one-time hit. It probably wouldn’t cause a steady decline…right?

OK, another theory: is it just a coincidence that this tumult coincides with a huge influx of immigrants to the United States? Wouldn’t that change the adult workforce along with the school population? Demographic shift is the most accurate answer to so many educational questions—could it help us here?

I don’t know! This is all so speculative. I feel like I’m just making things up. And none of the explanations are completely satisfying.

Meanwhile the amount of confidence on display from our political writers is remarkable. They seem quite sure that American schools are in crisis, that students aren’t being taught, that there is some grand trick being played on the American public.

But the facts are confusing. So why the confidence?

“America has been using its young people as lab rats in a sweeping, if not exactly thought-out, education experiment,” Horowitch writes, and would it be crazy to say that she sounds a little bit jealous?

The American school system is stubborn. It evolves slowly and only in complicated ways in response to the demands of outsiders. It’s hard to reach into classrooms. Maybe that’s the point here. Writers, advocates, policymakers—many of them love the idea that schools have been steered in the wrong direction. After all, that means they can be steered, and maybe you can even take back the wheel. Climb aboard, everyone! We’re headed back to the 2000s.

But what is conceptual understanding, and how do you develop it?

According to their documents, EdReports checks to see if texts include “brief conceptual problems with low computational difficulty,” for example:

• 11 + 6 = ___ + 2

• Find a number greater than 3/5 and less than .75.

• A fraction divided by a fraction is always/sometimes/never less than the original fraction

Curricula also need activities that focus on “concrete representations.” For example, the 7th Grade review of i-Ready specifically cites the following question in their glowing evaluation:

Neva plays a video game. On her first turn, she earns 3 points. On her second turn, she loses 3 points. The expression 3 + (-3) represents her score after the two turns. You can use integer chips to find the sum of 3 and -3.

a. The sum of any number and its opposite is 0. Another term for opposites is additive inverses. Since the sum of 1 and -1 is 0, 1 and -1 form a zero pair. Circle the zero pairs in the model.

b. How many points does Neva have after her second turn?

c. What is 3 + (-3)?

The Big Ideas books apparently lack these wordy questions about semi-physical objects. And EdReports complains that in Big Ideas the “conceptual” questions (whatever those are) mainly arrive at the beginning of the lessons, whereas the practice questions are “primarily procedural.”

What does that mean?

EdReports cites all sorts of documents and rubrics. Some of these documents define conceptual understanding, others don’t, and sometimes the definitions clash. This is exactly what researchers Crooks and Alibali found when they reviewed academic papers (about a hundred) for definitions of conceptual knowledge—that “explicit definitions of conceptual knowledge are rare” and “often vague.”

With EdReports we have a relatively clear definition of conceptual understanding but…it’s not great. That video game question isn’t going to help students understand anything. And I have no idea what a “conceptual question” is. Conceptual questions are questions that…develop conceptual understanding? Augh! We’ve gone in a circle.

Fortunately, I think there’s a perfectly sensible definition of conceptual understanding. It’s precise with clear implications for teachers.

It goes like this: conceptual understanding is knowing true and useful generalizations about mathematics. Crooks and Alibali, along with other researchers, sometimes call these principles instead of generalizations, but it’s the same deal—we want students to know facts that apply to a wide variety of cases.1

Take something like 6 x 3, which kids often think of as six 3s. We might teach kids that 6 x 3 is the same as 3 x 6, so three 6s, which can help them complete the multiplication. Terrific! But we should be more ambitious. We should teach a generally true principle—you can always reverse the order when multiplying—so that they can apply that to any problem, even one like 200 x 3.

I find this an extremely useful guiding classroom principle—to always push for generalizations, facts that are always true. To illustrate, here a few ways this has cropped up in my recent teaching.

“Procedural” Practice Can Develop General Principles

The basic idea of combining like terms is fairly simple—we can always add/subtract like terms as we do numbers—but it needs to be applied in a huge variety of cases. The variety is the challenge.

Obviously there is a limit to how much you can gain from repetitive practice of only one case. But if practice carefully escalates while gradually adding new twists, you give students a chance to apply the general principle to disparate cases.

This only works if a clear principle has been articulated before the practice. If you do that, though, this kind of practice gives that generalization a workout.

The Teacher Can Articulate the Principle

There’s a sort of discovery-ish vibe associated with conceptual understanding. But there’s no reason that generally useful principles need to be invented by students.

I shared this pair of graphs (from Math Medic) of a function and its derivative in calculus today. (If you’re rusty on calc: the derivative graphs a function’s slopes.) Can you tell, from looking only at the derivative, where the original function is increasing? My students were all able to answer accurately that it’s the portion of the derivative that’s above 0.

Terrific. But when handed this graph a new derivative they weren’t able to find where the original was increasing.

It was my role to take a step back and articulate the generalization even more clearly. The principle is where the height of the derivative is positive, the function is increasing. “Is the function increasing here? How about here?” I pointed at specific points of the derivative. After this students were able to tackle the question. They extended it in the natural way to negative values too. It was good that I said this directly after they got stuck—I wish I’d done it before then.

I sometimes think of this kind of teaching as a generalizing triangle. It’s a pattern that comes up often: teacher talk bumps up the abstraction and students connect it to new particulars.

Correct Explanations are Not Enough

Here is a good question from Transition to Algebra:

When I asked my 7th Graders to answer this, one student pointed out 100% correctly that since it’s four units up from E to D, the diagonal has to be longer than that.

I was about to move on, when I caught myself. I turned back to the kid. “So is the diagonal always going to be more than the vertical distance? Why would that be?”

I’m glad I pushed, because my students responded with two smart generally true answers:

• Yes, the diagonal is always longer than the horizontal or vertical distance, because when going diagonally you’re going in two directions, not just one.

• The Pythagorean Theorem says where the diagonal length is found by adding the squares of the horizontal and vertical distances, and that’s always going to be longer than just the vertical distance squared.

The point of an explanation isn’t just to eliminate doubt—it’s about connecting this particular situation to some generally true fact about the mathematical world.

Visuals are Only Good if They Suggest Principles

I was explaining to a student how to handle the equation 12 + 6x = 7x - 35. We had talked previously about mobiles, so I drew a quick picture:

This is a useful move because it’s easy to use the mobile to come up with a true principle—you can remove equal things from both sides of a balanced equation. Our physical knowledge of balance and symmetry are to thank. But that’s only useful if you then leverage that into a true principle about equations, something abstract we have much less experience with. And you have to get rid of the mobile! Its job is to suggest a generalization, then leave.

That integer chip thing earlier…what’s the principle? What are they getting at? Why do they get credit just for using a representation? What is the point of checking to see if textbooks do that?

***

And what about how this all feels, for a student?

Possibly my most frustrating school experience was a semester of Ancient Greek I took in college. Tos theos…I was seriously unprepared. Everyone else in the class had taken either Latin or Greek before; my grasp of English grammar is still on the shaky side. There were rules to memorize but it all felt random, isolated, disconnected to anything else I knew.

I was working with a young student recently on subtraction with borrowing. I’d show him how you could cross out the four and turn it into a fourteen by regrouping. The next question he’d cross out an eight and turn that into fourteen also. It was like that for twenty frustrating minutes. I was failing to help him see the general principle. How frustrating for him! Each subtraction was a brand new problem, unrelated to the one before.

But when it finally clicked…

We all want generalization. We want to see how it all fits together. We want to know how a particular idea flows from something bigger. We want to know how to do things—we also want to know things. We want the whole picture. And when we get it—whether we land on it on our own or someone makes us think of it with words or a picture or anything else—the feeling is terrific, like it all finally makes sense.

“Before we start, I’d just like to say how happy we are with your work so far.”

“Gee, thanks.”

“I mean it. Great problems and top-notch illustrations. Everyone here at Singapore Mathematics is a big fan.”

“It means a lot to hear that.”

“So what I’m about to say about Exercise 48, you know it’s out of love.”

“Is there a problem with 48?”

“Not a problem, but we have a suggestion. I’ll just say it: we want more butt.”

“More butt?”

“More butt. Absolutely. We’ve actually been talking. We think the bear’s butt should go all the way up to its shirt.”

“All the way?”

“Well, I guess not all the way. But almost. It should be close.”

“Listen, I can do that. But I have a question. Do you think the bear should be looking back?”

“Looking back? At what? His own butt?”

“Well, yeah. Or maybe it’s something else. It could be a little unclear.”

“So a vague look backwards, that may—or may not—be directed at his own butt? I don’t know, what do you all think?”

“I like it.”

“Me too. A little bit of mystery. Like, the bear has some secrets.”

“You’ve got my blessing as well. We can’t wait to see what you come back with.”

“I forgot one thing. The problem itself…it should be about tricycles?”

“Whatever, perfect.”

“I can’t wait to tell my granddaughter that I’m a model.”

“Yep. Smile more.”

“A real model. Plus we get to ride the rides.”

“That’s perfect. Look at the camera.”

“I’m feeling a bit nauseous, is the only thing.”

“I need you to look really happy. That’s good.”

“The agency didn’t say, but this is an advertisement? For the park?”

“Right. Look here.”

“I mean, what else could it be…hey, I love the hat.”

“Yo, thank you!”

“Both of you, look here.”

“Actually, I heard it’s for a math textbook.”

“What?”

“Look here.”

“A math textbook. That’s what they said.”

“But why a math textbook?”

“LOOK HERE NOW, GENTLEMEN, OR I WILL TELL THEM TO STOP THIS RIDE.”

“—”

“—”

“Probably for a chapter on functions.”

“FIRST OF ALL, I AM HAVING A VERY DIFFICULT WEEK. NOT LIKE ANY OF YOU CARE. SECOND, I DON’T NEED TO REMIND YOU THAT WE ARE UNDER A TIGHT DEADLINE. THIRD, AS A MATTER OF FACT I AM NOT BOTHERED BY THE WAY OUR EXERCISE MAT MELTS INTO THE PAGE. IN FACT I THINK IT RAISES INTERESTING QUESTIONS. LIKE, WHERE DOES THE PAGE END AND THE EXERCISE MAT BEGIN? WHAT IS THE TROPICAL DECAL STUCK TO? IS THIS VAST, WHITE EXERCISE MAT AN EXTRA DIMENSION? THE GREAT BEYOND? WE DON’T KNOW. BUT THAT’S INTRIGUING, SO NONE OF THAT BOTHERS ME. WHAT I AM BOTHERED BY? UNILLUSTRATED WORD PROBLEMS. SO LET’S GET ILLUSTRATING—CHOP CHOP, EVERYBODY. FOURTH, I DO NOT LIKE THIS JOB, AND EFFECTIVELY IMMEDIATELY, I QUIT.”

I have a humor piece in McSweeney’s that you should check out. It’s about a guy who uses AI to write his wedding vows. Some people think it’s funny! I also want to remind you of the story about Renaissance math duels that I posted here a few weeks ago. Please read, share, and enjoy.

Holly Korbey has a good article in Education Next about how math practice software kind of sucks and everybody hates it. I’m summarizing. But she names names. We’re talking Dreambox, Zearn, i-Ready, Khan Academy, ALEKS, and IXL.

The big issue is what is sometimes called the “5 percent problem.” This is the observation that these programs work fine when used as intended but are rarely “used as intended.” Instead kids cheat, copy, click around, get bored, switch tabs, flirt, swap computers, or walk away. (5% of the time, IXL works every time.) It’s a big issue, but the industry is working on it. Just wait, you’ll see. Soon enough we’ll be talking about a 6 percent problem, or even 7.

I tease. But the situation for these companies isn’t great. Lots of users (not all!) dislike their product. Per Korbey’s article:

• “Teachers hate it and students are lost,” said one Texas math teacher.

• “The kids were screenshotting, taking pictures, and just rapidly going through the questions, and then they would go through it a second time, get 100 percent—they got their points—and move on to the next thing… It was pure chaos, and students weren’t getting anything out of it.”

• Kane said he “hasn’t touched” the ALEKS digital platform his school purchased and favors designing his own practice problems.

Hey, wait a second—that’s our buddy Dylan Kane! Nice. Later in the article, after trashing ALEKS, Dylan shows up with a recommendation: “Kane, the 7th-grade teacher from Colorado, has found his own digital solution, similar to ASSISTments—DeltaMath.”

Now, I like Deltamath and my students do too. But, like Dylan says, it’s not personalization software. There is no algorithm. It is not adaptive. It does not aim to teach students topics they don’t yet know. It offers no incentives or rewards. It is not the future of education. It will not eliminate the need for teachers. (Listen, I’m disappointed too.)

This is where I’m supposed to say something like, “personalized tutors would be nice, too bad the software isn’t there yet.” But I don’t buy personal tutors as an ideal. The dream of a digital tutor is it gives you precisely what you need to learn at a given moment. I don’t believe in “precisely.” I think there are a lot of things you’re ready to learn at any given time, and beyond a point it doesn’t really matter what you study.

I also think there can be returns to learning with your classmates—what’s called peer effects.1 There is a fun historical example of this. As detailed in “Masters of Theory,” the top competitive math coaches at Cambridge University in the 19th Century — guys like William Hopkins and Edward Routh — taught groups of students, not individuals:

“Unlike the majority of private tutors, [Hopkins] taught students in small classes–between ten and fifteen pupils–composed of men of roughly equal ability. This meant that the class could move ahead at the fast possible pace, the students learning from and competing against each other.”

“The primary method of teaching, around which Routh’s whole system was built, was the one-hour lecture to a class of not more than ten pupils using blackboard and chalk.”

These guys weren’t simply working with groups out of necessity. They chose to work this way for the sake of their students. Like them, I also find teaching to be at its most satisfying and productive when the entire class is pointed at the same target. Part of what I love about Deltamath is how I can make it help me with whole-group learning, even though that’s not really what it was built for.

The Only Good Digital Textbook

The heart of Deltamath is an enormous list of skills. There’s unlimited practice for each skill via random numbers plugged into a few question formats. The lists are organized by grade-level and topic.

When it’s time for whole-class practice, I’ll hand out personal whiteboards, probably more useful and flexible than any piece of software could ever hope to be.

Say that the class is learning to solve equations. I’ll open the “Solving Equations” tab on Deltamath, which exposes an entire sequence of skills.

“I’m going to put a problem on the board,” I’ll say. “I’d like you to solve it on the board. We’ll start with the top of the list and see how far we can go.” I’ll project a simple-ish problem from a skill near the start of the sequence.

“Now lift up your boards,” I’ll say. “That’s pretty good. You all seemed pretty confident. Everybody wipe your boards. Let’s try another one like that.”

Why another one like that? Maybe because one kid got the previous question wrong —I want to give him another shot. Maybe I just want everyone to have a few wins before moving on. I get to decide.

“Let’s turn up the difficulty a little bit.” I’ll project a problem from the next skill and ask everyone to try it.

And so on, exploring the sequence of skills, until I hit friction. I’m probing for where things break down. I want to leave with an understanding of what the class knows and what they need to work on next.

This is dynamic. Depending on how students answer, I’ll change the questions they’re served. Look at me—I’m the algorithm. And I’m getting an enormous amount of information from the kids, though thank god there’s no teacher dashboard. I can see the “data” directly and simply. It guides my instruction. It’s news I can use. (Do we still call this formative assessment?)

More good news: in my experience, it’s all very motivating. Why? I guess it’s because the expectations are clear, the teacher is watching, attention is directed, progress is tangible, feedback is frequent, there’s a bit of competition but everybody’s in on this together. Plus, nobody gets called out for messing up. It’s the class that moves on to the next skill in the sequence. I’m treating the group as a group, even as I’m giving individuals a chance to get on board. (Now compare that to individuals on Chromebooks.)

Could I do this without Deltamath? Absolutely, but it would be harder and worse. I would have to prepare a list of problems in advance. Print textbooks often don’t have many problems for each type of equation. I might make up problems on the spot that are too hard or too easy, especially as the questions get trickier. I might forget a type of problem. I bet you can think of lots of things I’d do wrong — I’m kind of a mess.

To put it differently, there is a quality textbook hidden inside this practice software. And there are a lot of uses for a good digital text. It makes whole-group practice, a winning activity to start with, even better and easier to pull off.

It’s Hard To Use These Things as Intended

My point isn’t to trash individual practice, which is essential for learning anything.

About once a week, I’ll use Deltamath in class for what it was designed for. I could go on about all the little things I do to make it work well, but a lot of that is extremely boring. Like, “hand out paper in advance” levels of boring.

But it’s interesting just how much needs to be done to get this individual learning to work for kids. All the things that those teachers using ALEKS or IXL complain about are real. Kids get stuck easily. When they’re stuck, they might just click around randomly for a while and not ask for help. They might ask their friend to just do it for them. They’ll ask to go to the bathroom. They flirt. Cheat. Open another tab. Try to walk away.

There are ways to handle all these issues. But it’s hard. When I have kids doing practice with those mini whiteboards, I have one big problem to manage; when I hand out computers, I’ve created a few dozen .

And if a district admin or a principal told me that I had to hand out those computers every day and deal with all those little problems on a daily basis, I very well might lose my mind. I’d complain to Holly Korbey. I think, tragically, my classroom might fall victim to an act of vandalism. Something destructive but easily deniable, like a collapsed shelf or a thrown drink. Maybe I’d pay a kid to do it.

It shouldn’t be surprising that practice software is flailing around, complaining that people aren’t using it right. They’re trying to tackle one of the harder parts of teaching, and while I get what they’re going for, their solutions actually make it worse.

Trust me, I’m aware of the problems with whole-group practice. I’ve tried to teach factoring to kids who can’t add. I’ve split a class in two to deal with a huge range of skills. I’ve been interrupted in most of the ways a human being can be interrupted. I know that all ideals fail spectacularly in practice. But personalized learning is the wrong ideal. It’s thinking about this the wrong way. We shouldn’t be going all-in on kids learning on their own. We should be trying to figure out how to make whole-group learning even better.

March 25, 1539.

Niccolo Tartaglia, teacher and mathematician, had been pretty sure he did not want to go to Milan. But apparently at some point he’d changed his mind, because there he was, no longer at home in Venice but standing outside the house of Gerolamo Cardano, the argumentative physician who had spent the past few weeks berating him via post.

Cardano and Tartaglia had been locked in a touchy correspondence. They’d swapped insults, though Cardano was definitely better at it. He said Tartaglia was “secretive and pathetic” and that he’d spotted “a serious error” in the mathematician’s ballistics work. Tartaglia dismissed Cardano’s arguments as “sickly and weak.” Cardano replied by questioning Tartaglia’s “ridiculous aggression,” wondering aloud what he could have done to provoke such a harsh response.

At heart, the issue was simple. Four years earlier Tartaglia had discovered a recipe for solving cubic equations.1 For whatever reason, he had kept this formula secret. And Cardano thought this was stupid. If he didn’t want to publish, he should at least tell Cardano so he could include it in his own upcoming book, titled Ars magna—"The Great art.”

The two sent letters back and forth, the argument escalating. But then Cardano changed his tune. The Milanese physician (and gambling addict) resorted to flattery, praising Tartaglia’s achievements, begging him to visit in person. He expressed hope that the two of them could find time to discuss their algebraic work. Cardano promised to introduce Tartaglia to the powerful Marquis Alfonso d’Avalos. Tartaglia surprised himself by not just making this trip but walking up to Cardano’s house and knocking on his door.

Tartaglia was greeted by a surly, acne-faced seventeen-year-old by the name of Ludovico Ferrari. As he entered Cardano’s modest home, Niccolo saw that the servant was missing the fingers on his right hand. Ferrari, as it turned out, had lost them in a brawl. Ferrari was just a kid and figuring things out. Which were the fights you could win fair and square? And when did you need to prepare for deceit? It was so hard to tell.

Welcome to Casa di Cardano, buddy.

“I’m elated you made it,” said Cardano, after they embraced. “You’re going to love Milan. The shopping can’t be beat. Maybe let’s start with a haircut or shave?”

Tartaglia, no more than five feet tall, with a big bushy beard, stared at Cardano. He didn’t know what to say.

Tartaglia rarely knew what to say. That’s why he’d taken this name—Tartaglia means “Stammerer.” Niccolo the Stammerer. The children used to mock his impairment, so he adopted their taunt as his own title—as if that makes any sense at all, as if you’d ever introduce yourself as Michael Pitsweat.

His stammer, incidentally, stemmed from a childhood wound. A French soldier had bashed him twice in the head, then sliced through his face and teeth with a saber. He and his mother had been hiding from the French in the local cathedral. His mother nursed him to health. Niccolo had grown the beard to hide the scars which ran from cheek to chin. He still spoke clumsily. Words were not his friends.

Cardano, on the other hand, would not shut up. He had thoughts on every subject: Venice versus Milan; on progress in algebra; about Tartaglia’s cloak (“It’s so…black.”); on gambling with dice; how he would write Niccolo a letter of introduction to Marquis d’Avalos who was unfortunately out of town on business (“Returning soon, I’m told!”). Cardano was drowning Tartaglia with opinions that could not be easily distinguished from facts.

But wait—what did he just say? “…though I remain most eager to hear about your amazing technique.”

“Wait,” said Tartaglia. “Stop. Wait. I...we are not discussing that.”

Cardano closed his lips tight and his eyebrows twinged. He turned to Ferrari, his servant, who refilled their glasses with wine, presumably poured with the fingery hand.

“Well, you did say that,” said Cardano. “But then you came to us in Milan. And I thought, how strange for this fellow to visit if he didn’t intend to share his discovery? I’d assumed you were looking for someone to discuss it with. But it seems that’s not what you want?”

Tartaglia didn’t know what to say.

It wasn’t just the stammer. Tartaglia often had feelings he couldn’t describe. He knew plenty of words, but none matched his mix of emotions or their depth. He needed more complex language. If he could tell a joke he would have told one about his students, the unimaginative children of merchants—then Cardano would understand how isolated he felt in Venice’s mercantile atmosphere. He didn’t know how to lay on the charm while telling a hard tale, making it difficult to talk about his wounds, or why knowledge was for him so precious. (He was entirely self-taught. They had no paper so he practiced writing on the backs of tombstones.) He was useless at gossip—too bad, because his reluctance to share made more sense in light of the enemies he’d made in mathematical duels.

Yes—mathematical duels. This was how it was done in Italy at the time: you and your opponent would (in the presence of witnesses, a crowd, or a judge) exchange questions and battle to see who could answer more in a specified amount of time. Tartaglia routinely crushed his opponents, bolstering his reputation and gaining better teaching positions, along with a measure of fame.

In his most significant duel, Tartaglia had been baited by a lesser intellect, Antonio Maria Fior. The two men exchanged questions, thirty each, and agreed to return and compare results fifty days later—the loser owed the winner dinner. Fior left brimming with confidence, because he’d arranged a cruel joke. His thirty questions were all cubic equations. Tartaglia wouldn’t be able to solve these cubics because no one could solve cubics. Sure, he’d lose the duel. But in fifty days they’d all have a nice laugh at Tartaglia’s expense.

Two hours after they’d left, Tartaglia marched back into the Venetian square. He had solved all thirty of Fior’s equations. The mathematical community was shocked.

Tartaglia passed on dinner.

If he could have discussed that with Cardano, who was waiting for his response, then maybe he’d understand why he kept his secrets but had still come to Milan. Maybe. But Tartaglia didn’t know how to say any of this, so the moment passed. And now, bizarrely, Cardano had dropped to the floor.

“I swear to you, Niccolo, on the Sacred Gospel of God and Euclid’s Five Postulates, that, if you were to share the cubic formula with me, it would never travel beyond these walls. To Jesus Christ and also our Eternal Lord Archimedes, I promise not to be the one to divulge your secret. I swear this on the bloody cross and on the holy rhombic dodecahedron. Amen.”

“Amen,” Ferrari said.

Behind his beard, Tartaglia grinned.

“Niccolo, do you respect this oath? Do you need something else?”

Tartaglia put the glass of wine to his lips. No doubt Cardano was a bit much, but the physician had gained his respect. He spoke intelligently even on matters of Tartaglia’s expertise. What the two men shared, Niccolo thought, was a deep, almost desperate, need for knowledge. They loved their puzzles. The truth was that, coming from Cardano, it felt like an honor to be begged.

Tartaglia Tartaglia, Tartaglia—what will you do?

Here is what he did.

He asked Cardano (who asked Ferrari) to supply him with a quill and something to write on. For several minutes he scribbled at a table, his back hunched. Then he rose, returned the quill, and handed the paper to Cardano.

Tartaglia, who spoke so slowly he seemed impassive, was suddenly in a hurry. Actually, he said, had made promises to his friends in Venice. Please convey his apologies to Senor Marquis. He was under certain obligations. He had to leave. Yes, now. At this very instant. Goodbye. And he left.

Once he was gone, Cardano unfolded the paper. Ferrari, who was prodigiously brilliant, not just Cardano’s servant but also his student, looked over his master’s shoulder. Together they read Tartaglia’s note.

“Look,” said Cardano. “He wrote us a poem.”

*

Quando chel cubo con le cose appresso

Se agguaglia a qualche numero discrete

Trouan dui altri differenti in esso

“When the cube and things together

are equal to some number,

find two others whose difference it makes...”

That’s how the poem starts. It ends like this:

Questi trouai, & non con passi tardi

Nel mille cinquecent’e, quatro e trenta

Con fondamenti ben sald’è gagliardi

Nella citta dal mar’intomo centa.

“With quick steps and light feet these solutions I found,

in one thousand five hundred thirty and four,

my foundations quite sure and certainly sound

in the watery city surrounded by shore.”

*

For two straight weeks, Cardano and Ferrari labor over the poem. The lines were difficult. Could there possibly be a mistake? They were able to reproduce Tartaglia’s steps in some cases but not others. Their work went well into the night.

Ferrari was not yet Cardano’s equal, but he was close. When Ferrari arrived at Cardano’s home three years before and demanded a job, Cardano had a feeling the boy was special. Cardano always trusted his feelings. He was blessed with gut intuitions which he considered to be a form of divine protection. How else had he survived the failed abortions? How had he achieved so much despite his poor health? Not to mention the near-drownings when he’d made enemies while playing dice? When the universe spoke, Cardano had learned to listen.

Take this boy as your own, the universe had told him, even though he had two sons of his own. He will help fulfill your ambitions. Plus, check out those perfect, five-fingered hands. Gorgeous, right?

And the universe was right, because the boy was a natural. He studied with a chip on his shoulder, like he was angry at hard ideas for standing in his way. There was no better partner for Cardano as he worked to unravel Tartaglia’s strange and lonesome poem. When the two men finally did crack it, they celebrated together in Milan. Meanwhile Tartaglia sat by himself in Venice, wondering what he’d been thinking—what had compelled him to give his most valuable possession away?

Six years later, a press in Nuremberg, Germany published Ars magna.

There was a bookseller that Tartaglia frequented who specialized in mathematical works. He avoided the bookseller for days after Ars magna arrived. He traced spirals through Venice so he wouldn’t have to go in. Some of Tartaglia’s students had already mentioned that he appeared prominently in the work. This could only mean one thing. Eventually, on a late afternoon, Tartaglia forced himself to enter the shop and inspect Cardano’s work.

The book opened, on its very first page, with a brief history of what had come before. Cardano mentioned three great scholars: Al-Khwarizmi, Fibonacci, and Luca Pacciolo. Then he launched into the present and—Tartaglia grew nervous—moved to a discussion of cubic equations. Pacciolo, he wrote, had believed cubics were impossible, chasing a generation of scholars off the scent. “Was this right?” wrote Cardano. “Uh, no. This was totally WRONG.”

“There’s a man who, in the early part of this century, solved the cubic,” he wrote. “His name is Scipione del Ferro of Bologna.” (Tartaglia’s heart thudded against his chest. Del Ferro? The early part of the century?) “Del Ferro’s solution is astounding—his insight goes so far beyond everyday human reasoning that it raises questions. Questions like, is prophecy real? Does God, the Universe, or some Celestial Angel favor certain individuals with protection and insight? Are horoscopes actually real? Something to chew on, dear reader.”

“Now,” he continued. “I can’t leave out of this tale my dear friend Niccolo Tartaglia—he discovered this solution on his own (after del Ferro did) during a contest with Antonio Maria Fior (a student of del Ferro). After a lot of begging, Tartaglia gave me his solution. Shout out to Niccolo, my friend, and credit to del Ferro for being first.”

Tartaglia was ready to slam the book in disgust, but his eye had accidentally caught the name “Ferrari” down the page, compelling him to read on.

“I also have something exciting to report from my pupil, Lodovico Ferrari. It happened after we…” (Tartaglia could not believe his eyes. We.)“…after we received the key to the cubic from Tartaglia. At which point Ludovico Ferrari discovered that it could be taken further. In fact, let me be the first to share that Ferrari has a method of solving QUARTIC equations!”2

Tartaglia put the book down. He pulled on his beard. His scars ached. He took a walk. He came back, went straight to his desk, picked up his pen, and wrote.

*

Tartaglia, who had been so reluctant to publish, now published. The first eight chapters of his Quesiti et inventioni diverse collected his responses to people who had solicited his mathematical views over the past several decades. The ninth contained his thoughts on Gerolamo Cardano and his work. He considers his Ars magna, already a bestseller, to be “mostly worthless.” He attacks Cardano for stealing that which he could not derive. He calls Cardano “pitiful,” “more stupid than I’d thought,” “talentless,” “treacherous,” and, most astoundingly of all, accused him of “not having much to say.”

He challenged Cardano to settle their dispute in public debate.

Cardano never responded. He now lived in a larger home. He was a university professor of medicine. He was writing a tract on astronomy, with plans to write on music, theology, ethics, and divination. He read the Quesiti, but it only depressed him. After he and Ferrari discovered that the late del Ferro had gotten there first—in a notebook preserved by del Ferro’s son-in-law, who could not understand its contents—they were under no obligation to mention Tartaglia at all. Besides, at this point Niccolo had been sitting on his result for a decade. He gave Tartaglia every chance. Truly, he liked the man; why’d he have to run away? He had no interest in fighting him.

Lodovico Ferrari, on the other hand.

One day, Tartaglia received in the post a response to his attacks. But he was shocked to see it was not from Cardano, but from Ferrari. He was struck by its harshness of tone. How dare this teacher speak in such a way of the great Cardano? Probably, Ferrari mused, it sprung from his own inadequacy. Cardano had outpaced Tartaglia even though mathematics for the physician was just a hobby. Ferrari was persistent, vicious, and mostly correct. “Have no doubt,” he added. “I would be happy to settle this in a debate.”

Tartaglia released a counter-pamphlet. Then Ferrari responded with one his own. And so. The dispute was escalating, as before, except that at no point did Ferrari pivot to flattery and friendship as Cardano once did. That was not going to happen. This argument wasn’t some scholarly flirtation for Ludovico Ferrari. Ferrari had taken Tartaglia’s knowledge and advanced beyond it. The last time he’d been with Tartaglia, he’d been serving him wine. Now he wanted nothing more than to destroy this fellow, to bring him to an end.

*

August 10, 1548. Milan. Church of Santa Maria del Giardino. The governor of Milan, Ferante Gonzaga, was set to serve as judge. “Please,” he said to the crowd. “You simply have to be quiet. Now I will make introductions and explain how we are to proceed.” Ferrari stood on one side of Ferante. Tartaglia stood on the other, searching the audience in vain.

Since the whole thing began, Tartaglia has been clear—his dispute was with Cardano, not his former servant. He’d only agreed to return to Milan and duel with Ferrari because he’d been offered a position at a university in Brescia. However, the university did not want their school embroiled in chaos. Wrap up your dispute, they’d said. Have this duel with Ferrari. When you win, come back to us.

Tartaglia had no choice but to travel to Milan.

Throughout this whole disaster, Tartaglia continued to express a desire to settle this dispute with Cardano himself. Even when he agreed to terms with Ferrari he thought it likely that Cardano would participate—though he hadn’t seen or spoken with the man in years. Even at that moment, when Gonzaga introduced the disputants, Tartaglia still searched the crowd, hoping the physician would arrive at the last minute, even if only to bear witness. He didn’t know that Cardano had skipped town, expressly to avoid being present for this sad affair.

Of all the duels that Tartaglia had participated in, this was by far the most public. The pamphlets they’d published had drummed up interest in the dispute. The church was packed. Over several days they were to confront each other here, in full view, making their technical arguments as clear as the esteemed Ferante Gonzaga wished them to be.

“Yes, Signor Tartagila?” Gonzaga said, as he indicated a desire to speak.

The crowd, on the whole, was mathematically illiterate. No matter. Tartaglia spoke slowly and stuttered. He continuously referred to the mistakes of Cardano, though Cardano was not present. He was older than Ferrari, shorter and (digits aside) far uglier. He unwisely spent hours at the outset debating the rules of the duel itself. By the end of the first day, the crowd began to jeer when he spoke, forcing him to shout over them, pleading for them to let him finish.

Ferrari, for his part, seemed content as Gonzaga declared the first day of the debate to be complete. As the crowd trickled out he called both men to his bench. “Tomorrow we won’t have any of these tedious procedural complaints. Am I clear?” He looked towards Tartaglia. Tartaglia agreed, tomorrow there would be none of that. He stomped out of the church.

The next morning, on the second of the debate, the crowd entered as on day one. Ferrari and Gonzaga took their positions. They waited, but Tartaglia did not arrive. He could not be found. The crowd grew restless. He was not in his rooms.

Niccolo was gone.

Soon they would discover that he’d left Milan. The debate was over. By default, Ferrari had won.

When they learned the outcome, Brescia withdrew their offer of employment. Tartaglia returned to Venice, where he would teach accounting to the children of merchants for the rest of his life.

Tartaglia died in 1557, poor and alone.

Not long after, Ferrari died too. He’d quit his job, then returned to his native Bologna to live with his sister, a widow. Several years later he too would be dead—poisoned by his sister.

Cardano outlived both men. In time, his older son would be beheaded in prison for poisoning his own wife. His younger son was, like his father, a gambling addict, and he ran into debts that could only be paid back with theft. This younger son was jailed and then expelled from the city.

In “Book of My Life,” his autobiography, Cardano would devote just a sentence to the episode with Tartaglia. There is a full chapter titled “The Disaster of My Sons.”

There are also chapters titled “Powers of Foreknowledge,” “Things of Worth Which I have Achieved in Various Studies,” “Books by Me,” and “Guardian Angels.”

In his final years, Tartaglia taught hundreds of students. He admonished them for their shallow thinking, causing some of them to roll the eyes and some to laugh. But others took note. They knew that for him this wasn’t just a course in book-keeping; for him it was something more, a great art.

As he aged, sometimes Tartaglia would lose his train of thought in the middle of a lesson. He had a tendency, towards the end, to ramble. He’d complain about his many troubles. Sometimes, as he rambled, he told them stories about his childhood in Brescia. He’d point to the scars behind his beard—he’d tell them how he had to steal books to read in those days, back when he and his mother had so little to eat. In this way, his students came to appreciate their teacher more. It was nice, these breaks from the lessons. They liked the stories he told.

Notes:

The two books I relied most heavily on for this story were The Secret Formula by Fabio Toscano and Cardano’s own The Book of My Life. The translation of the poem mostly comes from Arielle Saiber’s Measured Words: Computation and Writing in Renaissance Italy. (I tinkered with a word or two.)

Del Ferro and Tartaglia didn’t have graphs, but we do. Part of their solution involves the “depressed cubic,” a shifted version of the equation that has no “squared” term. John Golden made a Geogebra sketch illustrating this shift.

This is a moment in history when mathematics was going through some major changes. I think of Cardano was partly responsible for pushing for a more abstract, streamlined algebra, whereas Tartaglia is clinging to something clunkier, more wedded to the real-world, and maybe even a bit more personal. It’s clear now which way mathematics was heading—the next big step in this story comes with architect-engineer Rafael Bombelli and his work with imaginary numbers.

I am looking to potentially dive deeper into math education research, and, in general, learning research. Is there a good place to go to get a broad, wide eyed overview of this ever growing body of research? Somewhere that maybe categorizes and/or summarizes different branches of the research landscape that would help me get a lay of the land before deciding which individual pieces to explore further?

I think the answer to your question is no, a guide of the sort you’re imagining does not exist—at least not in a way that would satisfy me, if I were in your shoes. And while it’s tempting to encourage you to simply dive in, there is probably something useful to say about the “lay of the land,” as you put it.

But first…it’s incredibly flattering to be asked a question. Thank you, Harry! If anyone else wants to ask me something, don’t be shy, as long as you’re OK with it potentially getting answered in a post like this one. Reply to this email, message me through substack, or get in touch with me on bluesky or twitter.

Back to Harry’s question.

The first issue with research on education is that it belongs to too many fields. Say you’re interested in a controversial topic such as ability grouping and tracking (something I once looked into). You’ll find papers by economists. You’ll find papers from policy researchers. You’ll find things coming from psychological scientists. Then there are education researchers, or even those who identify as math education researchers. You might even catch an historian or two. And wait—how could I forget about special education? And sociologists! And anthropologists!

And it’s not always simple to compare the evidence coming from different fields. Let’s say a lab experiment finds such-and-such, but an observational study finds that it’s not working well in classrooms. Do we say the teacher mismanaged the practice? Or that the lab setting was unrealistic? Imagine further that a law is passed requiring teachers to do such-and-such, and economists find it was actually harmful for learning—how do we square all these results?

A second issue is that these fields are often ideologically polarized. You would be very unlikely to see a mathematics education researcher come down strongly in favor of ability grouping in math. I’d be equally surprised to see an economist or policy analyst come down strongly against it. Why? I guess it might come down to the characteristic lenses and perspectives in these fields, or the biases of the people who go into them. But I think the polarization is, at this point, completely self-sustaining. The fields themselves simply represent opposing perspectives on controversial topics.1

This means that if you only read work coming from one field, you’re sometimes going to get a skewed sense of the issues and the evidence.

Then there is a third issue which is that not all fields study all topics. If you search for “effective ways to memorize multiplication facts” (as I once did) you’re going to find hundreds of studies, almost exclusively from special education researchers. If on the other hand you search for “productive struggle” (another past adventure) you’ll only land on papers coming out of math education—which makes sense, as it’s a phrase invented by math edu researchers. So, depending on how you frame your search, you may end up with papers that reflect only one side of a debate.

To be fair—and I guess I’ll call this the fourth issue—this stuff is inherently complex. You may start off looking for research on how to best assign homework (what is wrong with me) and end up entangled in a whole knot of questions—what age are the students? what’s the subject? what are we trying to measure? how are we trying to measure it? are we measuring the impact of teachers who assign more homework? or time spent by the student on homework? or homework policies that come from the school? or or or or or?

And so, getting back to your question Harry!, the issue is that the only way to understand education research in my view is to read widely across fields. Otherwise you get stuck in a rut. You’ll hear math edu researchers sometimes say things like “we’ve known for fifty years that tracking harms kids!” when generally speaking that is not true at all. And I think special education researchers at times overemphasize just how essential fact memorization is for a typical young student’s development.

So, what does one do?

I don’t know, man. My path was not efficient. I went topic by topic. I got a feel for social psychology by reading a lot about growth mindset. I did a deep dive on cognitive load theory and worked examples and that’s how I learned about cognitive science. For a while I got really into the history of schooling, and actually recently acquired a very dull looking text about abacus schools. And so on. It’s not a bad way to go about things, though it’s admittedly slow, and not really something I’d ever advise any working teacher to do, since we all have other things going on. Still, I don’t regret any of these projects.2

Still: is there a faster, more organized way to proceed?

In an attempt to actually say something useful, let’s try to break things into three big categories:

• Psychology & Cognitive Science

• Policy & Economics

• Education/Math Edu

And let’s take them one at a time. As an amateur I know I might describe these fields in ways that could be a little bit off. Feel free to chime in with an objection!

Psychology & Cognitive Science

First, psychology is a huge and varied field. Carol Dweck is a social psychologist. She studies motivation, emotions, identity, things of that sort. Daniel Willingham is a cognitive psychologist. He studies how the minds learns, reasons, and more generally processes information. Martha Alibali is a cognitive and developmental psychologist with a focus on mathematics—her work on the equality concept is Must See TV for teachers. The late Lynn Fuchs, a special education researcher, studied struggling students and designed interventions.

Looking at my shelf, I have two psych textbooks that I value highly: Children’s Thinking by Alibali and Siegler, and Cognition by Willingham. Texts seem like a fine place to start for psych. There’s probably a good social pysch text out there too. At one point I had my hands on Learning Disabilities (Fuchs was a co-author) and liked it. There’s nothing wrong with textbooks—I think at the end of the day I find reading them less interesting than noodling around with papers, but maybe I’m. Just. Built. Different.

Then there are approximately a million books that aim to present cogsci research to practitioners. I wrote one about worked examples. Craig Barton wrote a good one about math teaching, more generally. You could go to Ollie Lovell’s podcast. I recently got a look at a new title by Jim Heal and Rebekah Berlin that looked quite clear—Mental Models. These books all mention worked examples, retrieval practice, spaced repetition. Things like that.

I guess it’s fair to say you’ll get more lab-based experiments than classroom studies in these fields, but that’s definitely not strictly true. (Especially historically. If you go back to the 1960s, cognitive scientists loved the think-aloud protocol, where they’d analyze a transcript of someone talking while working on a task. Nowadays a study like this would more often appear in education research than in cognitive psychology.)

One last little thing, you might check out the What Works Clearinghouse guides to see summaries that reflect the perspectives of these fields.

Policy & Economics

Teaching and learning is full of details and subtlety. But we all want to make generalizations. And that’s what policy is, isn’t it? A big, dumb generalization that we hope does more good than harm.

Take growth mindset. Whether a growth mindset intervention “works” or whatever is a question for social psychology…up until the moment when someone tries to make all the kids in a district (or a state, or a country) do an online growth mindset intervention, one that will necessarily be somewhat watered-down. At that point we’re at a scale where such a study leaves the world of psych and instead might be published in a journal called Educational Evaluation and Policy Analysis.

Along those lines, there is something new and very useful to share! It’s called LiveHandbook and it does basically what I want every field to do—collect handy little encylopedia-style articles summarizing what is known, topic by topic. I’m impressed so far—they seem to have done this in a reasonably objective way. LiveHandbook is now my go-to stop when I want to learn about charter schools or vouchers or teacher pay or any of the hot topics that non-teachers often care deeply about, instead of the weird little corners of pedagogy that I’ve often obsessed over in my own work.

(LiveHandbook supplements but does not replace my older method, which is looking for what reporter Matt Barnum has to say.)

I’m not especially knowledgable about policy, but I don’t think you’d want to start with textbooks. By its nature, this is education research of the broadest possible interest, so there are book books on every conceivable topic that will be reasonably accessible. (Example: college admissions.)

Eventually you’ll decide that you want to understand more about particular statistical and econometric methods, or you’ll do what I usually do, which is treat them as a sort of mystery math box that hopefully someone out there has checked. The important thing is to not trick yourself into thinking you understand things that you don’t, and to move cautiously in those cases.

One last little thing: you might want to keep an eye on the Annenberg Institute’s EdWorkingPapers site, which posts free preprints of new policy research.

Education & Math Education Research

I once owned a ridiculous two-volume collection of essays called “A History of School Mathematics.” They were so large! I gave them away. One of the chapters (which I no longer have access to) told how math education became a research field of its own. There were many factors, but chief among them was a desire to move beyond experimental and statistical methods. They wanted to deeply observe students and teachers using the tools of anthropology and sociology. (I remember the first time I heard the name Jean Lave.) So math edu research begins with a self-conscious move away from various psychological fields. This chapter, if I recall, ended with a lament along the lines of, “It’s all worked out fine, but I do wish I’d hear about an experiment every now and then.”

Education research is a tricky thing to make generalizations about. It’s partly a matter of professional identity—this is education research because these are professors of education. But there are a lot of professors of education! Some don’t publish a great deal in journals, as their positions emphasize teaching, and they don’t have time or the funds needed to perform major studies. They’ll often describe small-scale efforts involving current or prospective teachers. (Peter Liljedahl comes to mind.) Others work in fancier departments and have a team of graduate students who help them pursue major grants.

I’ll avoid making further generalizations. Education research encompasses a ton, and it’s plagued by what you might call a huge range of seriousness. Some papers do little more than report on a consensus. (See my piece: “Lots of Research, No Evidence.”) But I feel uneasy with the disdain some people have for education research, a dynamic that arguably goes back to the very beginnings of the field. So, I’ll say no more, except that it’s relatively easy to find teacher-friendly information from organizations like NCTM or NCTE, though these organizations certainly have their own ideological perspectives. If you want more, you could poke around the AERA website.

So! There is plenty to read. And at the risk of going on and on, I have one last thing to say about how to read it.

At the very least, no matter how weak the evidence presented is, every published paper is an opinion. Now, sometimes when I meet a person with an opinion, I don’t really care. I just move on. But if someone has a personal experience, and I trust that what they’re saying really did happen, I can learn something from it—even if I disagree with their interpretation. “Oh sure, I don’t doubt that her kids enjoyed that game. Not sure if they learned much from it, but still.” Or whatever.

I like to read all papers in this way. Certainly observational studies. Even experiments are someone’s take on a experience that they themselves designed. “We put kids in a room and gave some of them this problem. We gave other kids this one. And kids learned more from this one. And here’s what we think happened.” Data analysis too—here’s what we saw, here’s what we think. You can learn from what they saw, and decide on your own what you think.

Reading this back, I’m like, duh. Obviously.

But I find it’s helpful to keep the human side in mind. Reading research is just reading. Writing research is just writing. It’s easy to get lost in the language of objectivity. But these are people just telling you what they think. On the one hand, you have to figure out how to put the chorus of voices together, and figure out what parts of what they’re saying you trust and believe. On the other, you have to decide, before dismissing someone entirely, why they aren’t worth listening to. Let’s say they were right here in front of you, telling you what they thought—would you trust them? (Often there’s a good reason not to!)

But ultimately I think it’s good that education is studied across different fields. You should read widely though there will be disagreement and confusion along the way. It would honestly be more concerning if there weren’t.

Before we start: I have a poem in Farewell Transmission. I’m also in the latest issue of The American Bystander with a story about a washed-up Yiddish writer who tries his hand at trendy sci-fi romance. (No offense to trendy romance!) You should read and share both these things. Very grateful to the editors who published them.

I’ve been thinking lately about how school is completely useless. How the mathematics we teach beyond a certain point exists almost exclusively in classrooms. Not just math—we are a country of people who vaguely remember a few words in a foreign language, who have forgotten the meaning of “covalent bond” and what exactly the War of 1812 was and whether we won it. And it’s not just k-12—somehow you can graduate with a degree in sociology or history or philosophy, get hired, and never think about Bourdieu again.

“How many classes did you take that you really never use on the job?” asks edu-skeptic Bryan Caplan. “How many years of foreign language did you have to take? Do you use Latin on the job? How much time did you spend studying poetry? Do you use poetry?”

He’s right—I never use Latin at work.

But then I think exactly the opposite—how school is so useful. How we need engineers and doctors and teachers and scientists and writers and readers and citizens. We need a lot of them, and we need them to be smart. And then I think about how we could eke out so much more learning from schooling if we wanted to. How is it that people spend four years in college but are taught in a way that, compared to k-12, is so pedagogically casual? In high school we ask kids to practice their skills in class. In college that’s rarer. You go to lectures and cheat on homework in the privacy of your dorm. School does not come close to maximizing learning.

The answer to this contradiction is that—despite how skeptics and critics talk about it—school is about more than just specific skills. But to talk about learning that goes beyond tests we have to open ourselves to a world of messier learning that happens, if it does, as the result of a hodgepodge of experiences, with a bunch of different teachers, taking place over decades.

Economists: Non-Test Learning is Real

How do we know that these supposed benefits are real? Kirabo Jackson has made this an area of focus. His papers have titles like “What Do Test Scores Miss?” and “What is a Good School, and Can Parents Tell?” These papers find that teachers have an influence on kids’ emotional learning. (He measures this using proxies like attendance, behavior, grade repetition.)

Skeptics might think that school does improve effort, motivation, behavior and all these other “emotional” factors—but only as a result of doing well academically. So it’s especially noteworthy that Jackson finds that his measures of a teacher’s influence on test and non-test factors don’t correlate very much. If that’s right, then kids learn more than just academics at school.

You want more research? Here’s more, found on the excellent LiveHandbook:

• Blazar and Kraft: “Teachers who are effective at improving test scores often are not equally effective at improving students’ attitudes and behaviors…These findings lend empirical evidence to well-established theory on the multidimensional nature of teaching.”

• Backes, Cowan, Goldhaber, & Theobald: “We find that teacher test and nontest measures both play important roles in predicting long-run postsecondary outcomes, but they appear to operate on different margins depending on the student outcomes in question.” (Top students are more impacted by test-improving teacher skills.)

So, hooray. Teaching isn’t a waste of everyone’s time.

But how exactly does any of this happen? Did I help my students with the noncognitive, emotional stuff this year? Did my children’s teachers? How does any of this happen? Should I be giving speeches about effort to my algebra class?

In short: do we have any idea how these emotional benefits are confered?

Cognitive Scientists: General Skills Don’t Exist

Let’s make things worse. There is a line of thought in cognitive science that the only skills and knowledge that can be taught are specific. From this point of view, we learn only that which we are explicitly taught. “Transfer” of skills to new domains is rare, maybe even non-existant. As a result, there isn’t really anything that deserves to be called “critical thinking” or “problem solving” abilities—all we can ever have are growing collections of individual techniques.

“We’re not even sure the general skills exist,” writes Daniel Willingham. “But we’re quite sure there’s no proven way to teach them directly.”

I basically agree with this perspective. I was radicalized by problem solving posters: the list of tips for getting unstuck which are hopelessly vague—kids don’t know how to productively draw a picture or “solve a simpler problem.” (Write a better introduction! Try jumping higher!) What has worked for me instead is to teach the more specific version of that skill.1

But this is all very strange—Kirabo Jackson and the others are talking about longterm emotional benefits of schooling. These are, in a sense, general social-emotional skills. Why would you be able to impact a kids’ emotional life in general terms, but not their academic skills?

It’s Bigger Than the Individual

Here’s what I think is happening. I think that specific skills can be learned under specific, definable conditions. But the “fuzzier” skills are learned under fuzzier conditions.

Put it like this. General skills are abstract. Everything I know about pedagogy and learning says that there’s no shortcut for learning abstractions. It would be hard to learn what the word “mammal” means unless you’re familiar with a few different specific cases (“cow,” “human,” “dolphin”). Likewise, you can’t learn what “solve a simpler problem means” without learning a few different versions of this. Maybe “make the numbers smaller,” or “prove it for a convex polygon first.” As these specific heuristics pile up, maybe one day you’ll connect the dots and…tada, a general mathematical skill.

Maybe all those emotional, non-cognitive outcomes are like that. How do you become emotionally resiliant? How do you learn to be responsible and timely? How do you develop the ability to hold anger at bay? Probably not all at once. Probably a little bit at a time. You learn it in 1st Grade, then again in 2nd. You learn not to tell Mr. Pershan that you think his class is stupid, then you learn not to yell at Ms. Smith in music. Next year in science you get in trouble for pushing Susanna when she takes a pencil from your desk. You learn not to push Susanna in science. Eventually, these things pile up. One day you’re able to generalize—you can manage your anger. But there’s no shortcut. How could there be?

Here’s how it all might fit together. Teachers don’t teach “emotional resiliance” or “effort” or “responsibility.” We also don’t teach “problem solving” or “critical thinking” or “listening to others” or “respective disagreement” or anything like that. By their very magnitude they are skills that need to be nurtured over years and across many different contexts. It’s like if a teacher tried to teach “maturity.”

To be clear—please don’t do this. You can’t teach maturity.

The cognitive scientists are, however, are wrong to be skeptical of general skills. True, they are learned over a longer period than is convenient to measure. But those individual skills pile up, painfully, over many years. You can try giving the process a little nudge, but you can’t rush. It’s going to take time.2

Which means that the individual teacher needs to think about their work with some humility. We participate in a process. We have no idea what we ultimately contribute to it. We can teach what effort, behavior, reasoning, listening looks like in our classroom, for our subject. And hopefully one day that piles up into something significant, something a person takes when them into work and life.

Some of the roughest teaching I’ve seen comes when teachers have a hard time handling their own smallness in this process—they reach for more and try to teach something that feels weighty and significant. They’re trying to find a shortcut for that abstract learning. There is no shortcut.

But it also means that those who see school as, essentially, about individual academic skills are badly mistaken. Tests are wonderful but there is plenty of evidence that they are incomplete. What’s missing is an entire dimension of schooling that, honestly, most people recognize as essential, but those who think too much about tech or tests often end up eliminating, either out of misguided skepticism or economic convenience. If you define school as just about the War of 1812 or covalent bonds, you’re missing out on a lot. And as a personal favor, it would be great if everybody talking about the supposed “future of education” could recognize what it is that schools actually presently do.

What They Did

The researchers created an after-school program in Benin City, Nigeria. The program met twice a week for 30 minute sessions. It took place over six weeks during summer 2024, at the end of the Nigerian school year. (Their 180-day academic calendar stretches from September through July.)

You can get a sense of how the after-school program worked from a video shared on World Bank Africa’s Facebook page. There’s a teacher in the room. There are about 30 kids, paired up, sitting in front of computers. They have been given English grammar assignments that require interacting with Microsoft Copilot.

The activites began by typing in a lengthy prompt to Copilot, explaining that they want the model to act as a “well-seasoned English grammar tutor.” They tell the model that the current lesson is about “adjectives” or “clauses” or whatever else it happened to be about. Then the kids were told to constructively interact with the model—for example, by asking it to create a sentence with an independent clause that describes Benin Kingdom’s history.

Now, could learning be happening here? Absolutely. Students are reviewing grammar taught in their normal English class. A teacher is there to supervise and help. Kids are working with classmates. They’re practicing writing, reading, and speaking English, which is not their first language.

This all seems more than fine, a worthy teaching experiment. The issues I see aren’t with the program but with the experiment attempting to capture its impact.

“Randomized Controlled Experiment”

The after-school program took place in nine schools, but most students didn’t take it. “All first-year senior secondary school students in these schools were informed about the program,” the researchers write. Kids had ten days to sign up. Then, randomly, 657 of those kids actually admitted into the program and 671 went about their business as usual.

Well, to be honest, this ruins everything.

I actually think the researchers are pretty convincing that their after-school program was good for the kids who took it. The researchers got access to the kids’ third term exam scores and the treatment group did 0.206 standard deviations better than the control, even when their performance on the second final is controlled for. The program wasn’t a waste of their time.

But that’s all this study is capable of showing—that the program wasn’t literally a waste of time. Because the control group, academically speaking, didn’t do anything. I mean presumably they did lots of things—played soccer, hung out with friends, cooked food, whatever. But fundamentally the after-school group studied more and the control group did not.

It didn’t have to be this way. The control group could have been assigned sections of a text to read with partners. Or they could have watched instructional videos from YouTube on their computers, putting ChatGPT head to head with older forms of digital media. Listen, they could have had the teachers just lecture those 30 kids for the same cost. As is, they compared their after-school program to nothing. They therefore only proved that their program is better than nothing.

That’s the big problem, but there are others, all of which are common with brief edu interventions:

• To be eligible for the study, kids had to volunteer. So all the kids in either the treatment or the control were highly motivated to study with ChatGPT.1

• They also report effects from their intervention on assessments the researchers themselves designed and administered. The effect sizes are larger for the researcher-designed assessment, and the big ridiculous claims (“equivalent to 1.5 to 2 years of business-as-usual schooling”) rely on these results. Hilariously, this includes “AI literacy,” and they report that their intervention had a big impact on kids’ AI knowledge. Well yeah, I sure hope it did!

• Dropout. Only 422 students from the treatment group completed the final assessment. It was even harder for them to get the control group to complete the final test—only 337 control kids completed that. This is better than if kids were dropping out only of the treatment group, but this is a classic way that effect sizes get boosted in edu studies, by only having the most successful kids from an intervention take the final exams. You really want to reassure the readers that this isn’t what is happening, but they don’t do a thorough job of this.

Each of these issues strips away our ability to generalize from this experiment. Is sitting with CoPilot a good use of time? Would it work with typical students? How large of an impact did it have? Is it better than discussing a text with a classmate? Does AI tutoring work?

The answer: we don’t know, we don’t know, we don’t know, we don’t know.

“AI tutor”

There’s another thing bothering me about all this. The researchers told kids to prompt the model to act as a “well-seasoned English grammar tutor.” They instructed the model to “reply to my questions in a motivational and engaging tone.” The model was told to ask questions or propose exercises on the grammar topic of the day, to provide hints or corrective feedback when needed.

OK, fine. But is this tutoring?

I mean…not really? It’s just interacting with an LLM, which is a really cool dynamic text generator, but isn’t acting as a teacher. ChatGPT didn’t decide what students were ready to learn. It didn’t create the lesson. It didn’t even do all the instruction: “At the end of each session, the students were encouraged to reflect and discuss lessons learned and challenges encountered during session to facilitate knowledge sharing among the group,” the researchers write. That sounds like teaching! The human is doing it.

This is not school with a robot teacher. It’s more like when I tell my own students to take out Chromebooks and use a math practice app like Deltamath—except that in this case it’s an LLM, which is functioning like a magic, dynamic exercise book.

The world seems set on calling this sort of thing a “tutor” but I think that’s absurd. This is not a tutor. If you insist, call it AI-driven practice software.

Now, what’s cool is that Microsoft Copilot wasn’t designed to be practice software. It was designed to write emails and to let kids cheat on essays. That it helps highly-motivated Nigerian kids learn grammar is a cool bonus. I think that’s awesome. I’m optimistic that this means LLMs could be used to create more kinds of practice software for kids. Don’t ask me if that’s possible, I’m just a guy who gets paid to watch children calculate. But at least it’s a technological vision I can get behind—practice software for anything you want to learn.

But that’s not enough, is it? I guess every incentive is aligned to pretend that our sci-fi dreams and nightmares are coming true, right now, and that you’d better get on board or be left behind. It’s how you take a lovely little teaching experiment and present it as a “randomized controlled study” that presents “big results” on the “huge impact” of “AI tutoring.” In reality, it’s none of that.

Listen, the world is weird enough as it is. Please, can everyone try not to inflate this stuff further? I’m begging you.

Jason and Sandy Roberts launched Math Academy as an accelerated math program for middle schoolers, housed in Pasadena Unified School District. The software arrived after Jason, an early Uber employee, got tired of collecting and marking homework. Once the pandemic hit Jason developed the software further, transforming Math Academy into an edtech company with ambitions to fix math education, which they believe to be badly broken.

Well, sure. Everyone in edtech says school is broken and they can fix it. I was prepared to give Math Academy no further thought.

But I was intrigued, mainly because of a dynamic, constantly growing 457-page manifesto created by their Director of Analytics, titled “The Math Academy Way.” This manifesto is subtitled “Using the Power of Science to Supercharge Student Learning.” In it they claim to have designed a system that “emulates the decisions of an expert tutor.” I was definitely skeptical, but they’re thoughtful and have experience with kids. So I shelled out $50, completed my diagnostic exam, and entered the world of accelerated learning.

Explanations are Optional

Math Academy is unabashedly procedural. They believe above all in practice that closely follows given examples. Here is how lessons are described in “Math Academy Way”:

Each lesson starts out with an introduction, and then moves to a worked example, followed by 2-5 practice questions on the same type of problem as the worked example.

Introduction. Example. Practice, largely multiple choice. This is the core of the Math Academy experience.

Now, I’m a big fan of practice. But consider what it would look like for me, a human tutor, possibly an expert, to teach as this program does:

I hand my student a page from a textbook and ask her to read. We don’t talk. She tells me when she’s finished. Then I hand her another piece of paper, this time with an example on it. She looks at it as long as she likes. Then I give her two multiple-choice problems. If she solves them correctly I say “Congratulations! You’ve earned +5 XP.”

Is that great tutoring? For a high school student? For a 4th Grader? No, no, no. Good teachers don’t sit around and watch kids read. They ask probing questions. They ask for explanations. They build understanding.

Given the choice, most people will skim over technical explanations. During my month on MA, enrolled in Probability & Statistics, I was no different, especially since the worked examples were briefer, clearer, and always sufficient for passing the lesson.

Here is one source of that 4x acceleration: Math Academy has created a system where explanations are optional.

In classrooms, we don’t let kids hit fast-forward. We prompt with questions. We command attention in real-time. Admittedly, that slows things down. I know a lot of kids would skip it if they could—that’s why you hand out the worksheet after the discussion, you know?

In most cases, “Math Academy Way” comes down strongly in favor of teacher guidance. They are against group work and unguided practice. Fair enough. But their app gives students total control over time on task and depth of engagement. No expert tutor would do this; researchers certainly don’t recommend it.

But you only get that sweet XP for solving problems…so yeah, students are going to go faster, and learn shallowly as a result.

Never Forget, Never Understand

You might expect that this would all take care of itself. MA is after all a “spaced repetition” program, constantly reviewing earlier content. Poor learning will eventually lead to incorrect answers. Users will be forced to revisit earlier material and improve.

But during my time with the program, there was a problem with this: Math Academy never let me forget my shallow learning. I kept answering questions correctly, and MA kept pushing me onward, without a chance to deepen my understanding.

I ended up thinking about multiplication facts. A definite finding of cognitive science is that people store facts in a connected network. A prompt of “12 x 8” triggers a whole host of associations—6 x 8 = 48, 12 x 4 = 48, 10 x 8 = 80, 8 + 8 = 16, and so on. A significant part of what we call “conceptual understanding” is this network of learned associations.

In remedial situations, you’ll often encounter students whose network of associations is weak. For them, 12 x 8 triggers 12 + 8 = 20, or even twelve 8s. In intervention situations it sometimes makes sense to focus on memorizing facts and formulas first, without worrying as much about the network. As MA puts it, if you practice retrieving this fact often enough, you’ll “retain it indefinitely.” In some cases, that can help.

In my time with MA, I felt myself turning into this kind of remedial student. Spaced reptition works—I really was able to hold on to formulas for variance and expected value long enough to easily ace their quizzes. But I felt the weakness of my own understanding, and it honestly bummed me out. I still don’t know what a geometric distribution is.

My retention was only possible because they were targeting it so precisely. Their probability course is composed not of units but of 180 topics, each itself composed of 2-3 skills. The problems I practiced were identical to the examples, just with different numbers. Same with review and assessment—I never encountered a question in practice that I hadn’t seen explained before.

In an actual classroom, this sort of micro-targeting is impossible.1 I would have forgotten the variance formula, and had to go back and relearn it—perhaps more deeply. My weak understanding would have been exposed by unfamiliar questions which didn’t directly resemble ones I had already studied. I’d need to go back and relearn, or ask for help. It would require, no doubt, a certain amount of resilience. I guess you could call that a flaw.

But in Math Academy, learning was smooth and flawless, to a fault.

I hated this. It made me feel sad and stupid to know how to answer questions but only in this shallow way. It also meant that, for all the criticism in “Math Academy Way” of unguided instruction, I was essentially left on my own to flesh out my understanding. Math Academy offers direct instruction for procedures, discovery learning for concepts.

But, yes: this flawless learning will indeed go faster than a typical classroom.

Who Is Math Academy For?

Friends with bright kids sometimes ask me for advice. Their children are bored in math and looking for more. It’s a hard question to answer. I’m always looking for products or courses to recommend.

Man, I sure wish I could recommend Math Academy.

I will say that as a supplement to an actual course, MA could be powerful. I don’t know of another site that offers this much basic practice for university mathematics, something that is sorely needed.

But in the absence of a teacher, I simply can’t recommend this program, even for strong students. The rapid gains are somewhat illusory. Unless you’re going deeper into something familiar, you’ll have to go back and study it all again. I was on track to pass Probability & Stats in a few months based on my accumulated ability to answer multiple-choice questions I didn’t understand. Is this what strong math students deserve?

At the end of my month on MA, I went looking for information about the original Pasadena program. I found a student article titled “Math Academy, A Decade Later,” and was surprised to see my perspective echoed by Pasadena high school students:

“Math Academy sucks and genuinely should not exist. The dropout rate is absolutely ridiculous and the stress it puts on students is just unparalleled,” said one student who left the program after 8th grade.

So what are the problems that students have with the program? For one, students say that going through content at such a fast pace results in very high standards and only a surface-level coverage of many important concepts. “We do too much in not enough time, and by the time we finish learning a topic, it doesn’t stick with us,” said another student who left after middle school.

“The XP system is just dumb. It isn’t relevant to anything we learn in class and the negative XP discourages learning on the website because the goal isn’t to learn, it’s to get as much XP as possible,” said a sophomore currently in Math Academy.

I truly have no ill will to the people who have made this thing. They all seem sincerely devoted to improving math education. There are things that they do well. They’re only in “Beta,” and will surely make further tweaks as they grow.

But I have to call it like I see it: Math Academy is fundamentally broken, and I don’t think it can be fixed.

I don’t know a single thing about scholastic disputation. Well, I’m aware of the Ninety-Five Theses. But beyond this I’m clueless, except for the things I hastily searched for and read just now in hopes I could write an impressive-sounding sentence or two in this paragraph, before realizing that I don’t really care about any of that stuff—I just like lists of strongly worded takes.

So, here is a list of things I believe about education. I couldn’t get to ninety-five. Taken collectively there is a perspective here, one that I sometimes find myself wanting to share with others. It would be fun and interesting to discuss and debate these.

So, here we go—the beliefs:

1. To understand a person, look for their contradictions.

2. Ditto for institutions.

3. Schools are asked to do many contradictory things.

4. School is supposed to host a fair competition for academic distinction.

5. There is no such thing as a fair competition.

6. To a very real extent, you get the education you pay for.

7. If you incentivize schools to focus on, say, improving math and reading abilities, then that will take away time and money from things like science and social studies.

8. Every minute spent on art and music is a minute not spent teaching students to read.

9. Schools are expected to reflect the values of their local community. They are also expected to be open to those who don’t share those values. My son’s public preschool has a huge paper Santa Claus standing in the auditorium. For his sake, they also taught the class a Hanukkah song.

10. Different children need different things, and schools can’t do all of them equally well.

11. As a consequence of all this, schools are a bit dysfunctional. They don’t do anything as well as it seems like they could. David Labaree says it: “The educational system is an abject failure in achieving any one of its primary social goals… The apparently dysfunctional outcomes of the school system, therefore, are not necessarily the result of bad planning, bad administration, or bad teaching; they are an expression of the contradictions in the liberal democratic mind.”

12. Said differently, schools are sites of compromise. They are institutions that, more or less successfully, paper over the contradictory needs of parents, students, and voters.

13. This is a strength, not a weakness, of schools.

    ---

14. In any conflict, there is rarely just one possible compromise.

15. The history of education could be told as a story of shifting compromises.

16. Larger schools tend towards the same compromises, because their size exposes them to a greater cross-section of the contradictory desires of parents, students, and voters.

17. Smaller schools can more easily reach compromises than larger schools.

18. Arrangements that work for younger students often fail when extended to older students. An instructive example from Success Academy’s attempts to build a model high school: “Another concern, according to a former Success official who spoke on condition of anonymity, was that the network wanted to cement policies that it could replicate in its future high schools, and the first high school didn’t look like the example Moskowitz wanted it to be.”

19. It is functionally impossible to create a large school that has (say) no grades or features (say) an intense focus on discipline, since these “innovations” would go against the will of many parents in the larger population. But in small schools, anything is possible.

20. Parents are the most powerful educational group, not teachers or unions.

21. Homework would have been abolished by educators in the early decades of the 20th century in many places if not for parent opposition. “In 1913, one principal who had abolished homework admitted that he “incurred considerable opposition on the part of parents, who were so indoctrinated with the idea of home study in their own school days that they protested vigorously against its prohibition” (quoted in Bok, 1913).

22. Most “innovative” ideas in education are unsavory to large segments of the parent, teacher, or student population. But smaller communities can be formed where these ideas are seen favorably (or where students, parents, teachers have less influence).

23. “Nothing scales” in education.

24. Scaling involves, along with everything else, exposing your “innovative” compromise to something more like the full force of parental desires. This is one way that scaling pushes institutions towards business as usual.

25. Countries, states, districts, schools, grades, classrooms, students—education happens at multiple nested levels.

26. Classroom pedagogy is also about contradictions and compromises.

27. Ideology is the refusal to compromise.

28. Teachers love theories, in part because of a real desire to slip free from painful compromises.

29. What works for students of certain ages, subjects, or schools, may very well not work for others. Novel pedagogies are often “born” in one educational context and then attempt to migrate to others. This almost never works.

30. “Nothing works” in teaching. Which is to say, there are no universal “best practices.”

31. “Two entrenched dogmas currently divide thinking about teaching... The first, which we have characterized as 'best practice,' views teaching primarily as technique or method, and educational improvement as a problem of identifying, disseminating, and mandating the most effective methods. The second, opposing dogma views teaching as fundamentally personal and particular, and teachers' professional autonomy as a key condition for educational improvement… Our perspective seeks to avoid the extremes of both dogmas.” —Adam Lefstein and Julia Snell in “Better Than Best Practice.” Also me, I say this too.

32. A pedagogy is a teaching system. What I describe as “fussy teaching” is a pedagogy where the teacher asks the entire class to respond to questions that are delivered at a fairly steady clip. For variety, students switch modes a few times in class, between paper worksheets, mini-whiteboards, turn-and-talks, etc. The benefits are reduction of disruptions and active participation.

33. Every pedagogy fails. The drawbacks of “fussy teaching” include: boredom from strong students, monotony, students spend the same amount of time on every question.

34. Every successful teacher has multiple pedagogies. I don’t believe there exists a good articulation yet of what teaching with multiple pedagogies looks like.

35. One way of measuring the success of a teacher isn’t how effective they are in their role, but how easily they can be placed in many different roles. They can adapt what they do for older or younger students, struggling or ambitious students, and so on. This reflects an ability to deploy multiple pedagogies.

36. Pedagogical variety, thoughtfully deployed, is itself the most successful pedagogy.

37. We lack the ability to measure most educational outcomes.

38. “Teacher effects on test scores capture only a fraction of their effect on human capital,” finds Kirabo Jackson.

39. Schools are not in the test-score maximization business. This is not what schools are being asked to do.

40. A “successful” reform could encourage schools to focus more on improving test scores. There would, predictably, be tradeoffs.

41. Almost nobody thinks that test scores are the only thing that matter in education. But there is little consensus about what those other things are.

42. Policy expresses itself in the language of consensus.

43. Test scores and academic skills are at the center of a giant Venn Diagram. They are the things that everybody, pretty much, can agree on. Partly that reflects their importance. Partly that reflects their status as a common denominator across educational disagreements.

44. Goodhart's law: “When a measure becomes a target, it ceases to be a good measure.”

45. Labaree, again: “A key to understanding the American school syndrome is to recognize that our schools have never really been about learning. The impact of school on society over the years has come more from the form of the school system than from the substance of the school curriculum.”

46. If you could wave a magic want and improve the nation’s math scores by ten points—what would you have accomplished, exactly?

46. I am not cynical about education. Learning is good. Teaching is important. Schooling is great.

47. The wealthiest people in the world could easily hire an army of private tutors to educate their children. Instead they send their kids to private schools and hire tutors to reinforce the school curriculum on the side.

48. If you increase the resources available to schools (or classrooms), new compromises become possible. (“Does money matter for schools? Most studies say yes.”)

49. “I gave him a call and basically said, ‘I’ve looked at your results, and you’re getting forty or fifty percent of kids reading, but it’s not ‘success for all,'” Greenblatt recalled telling Slavin. “‘I’m just a guy with money. Is it money?'” Yes, Slavin replied; its just money–if the money is spent in the right way. But it’s difficult to maintain tight control in most public schools. The two agreed to try to find a school they could convince to put “Success for All on steroids,” as Greenblatt put it.” From Robert Pondiscio’s book about Success Academy.

50. Decreasing the number of influential stakeholders also makes new compromises possible, for instance by creating smaller schools (or classrooms).

51. Smart, thoughtful, personable individuals can themselves make new compromises possible by attracting the trust of others.

52. It is hard to measure school quality, but relatively easy to observe it.

53. To understand schools people would need better numbers, but they are unlikely to get them.

54. In good schools, students are safe and reasonably happy while focused on academic content. Schools that can pull this off are more or less capable of figuring out what else the children and their families need.

55. Schools are poorly understood institutions.

56. “So nearly a half-century of experience in schools and the sustained research I have done have made me allergic to utopian rhetoric. Both my experience and research have changed my mind about the role of schools in society. I have become skeptical of anyone spouting words about schools being in the vanguard of social reform… Yet, I must confess that in my heart, I still believe that content-smart and classroom-savvy teachers who know their students well can make significant differences in their students’ lives even if they cannot cure societal ills.” —Larry Cuban, but also me, I say this too.

As if we didn’t know! But there is something of significance here anyway, in this weird waterfall/sieve. The goal is engaged academic time. Sometimes that comes easily. But other times it feels like a daily battle to get a group to focus on anything even remotely related to the thing we’re supposed to be teaching. The question is, is there anything we can do about this?

I think we can. The general principles are:

• Ask students to do less, more frequently.

• Every student answers every question.

• Teacher controls the pace.

• Switch modes often.

What it means to teach this way, in practice, is that lessons get fussier. Here’s an example of the sort of fussy teaching that I mean:

Geometry students enter and are told to start on an area worksheet, which is review. Gathering the group, I offer a brief explanation of how to solve a problem, asking the entire class at points to respond to prompts all together, chorally. Then I distribute a worksheet and tell students to try the first three questions on their own—even if they have to guess. After sharing answers, I ask everyone to try the next three. After completing the worksheet, I distribute mini-whiteboards and ask students to solve review problems one at a time as I write them on the board.

That’s pretty fussy. If your eyes glossed over in that too-dense paragraph, the teacher (me) carefully controlled the pace. Questions came pretty frequently; when they did, every student was supposed to answer them at the same time. (There was a tiny bit of independent time at the start of the lesson.) There were four different modes: (1) independent worksheet, (2) choral response, (3) three-at-a-time worksheet work and (4) mini-whiteboards.

Contrast this with an unfussy lesson:

I ask my Calculus students to take out their homework, going over any questions. I then explain the Product Rule to the class, frequently pausing to ask questions. I give a clear example of how to solve a problem using the Product Rule and distribute a practice worksheet that they work on with partners.

Here’s an entirely different sort of unfussy lesson:

I display a pair of staircases and ask students to decide which is steeper. I then distribute a collection of lines and ask the class to work in pairs to rank them from least to most steep. After ten minutes, I bring the group together to share ideas. During the discussion I highlight mathematical ideas that relate to the concept of “slope.”

These lesson sketches are each fine. But the unfussy lessons have big blocks of time when students are supposed to be doing stuff. Stuff like: “keeping focused” and “asking for help” or “not talking about the dumb thing your friend Becky said.” If kids are supposed to work on a worksheet for ten minutes, that’s ten minutes of self-management. A lot of the time, that’s totally fine. But, to put it bluntly, some kids are just awful at that sort of stuff.

Some of it is situational. At the start of a new topic, when the math is less famliar, kids might have a harder time maintaining stamina on a set of problems. Or maybe, in a block schedule, there’s that one day during the week when kids just seem unable to stick with a task for five minutes. Or it’s a very immature 6th Grade. Or it’s that one group of kids that just…they just...sigh.

I remember, in my first year of teaching, there was a class that was giving me trouble. I felt like I couldn’t get through an explanation at the board without interruption. My reaction was: OK, so no more explanations at the start of class. I labored over making engaging problem sets that they could work on with pairs while I circulated and helped. I tried to make my lessons even less fussy. (Forcing me to make the assignments far more fussy, ironically.) It was inefficient, but felt like improvement.

These days, in that exact same situation, I pull out the fussy teaching playbook. I tell kids do this, do that. I ask lots of questions, and everyone has to answer each one—no, you can’t sit this one out. It’s harder, I’ll admit. But the upside is that expectations are clear, feedback is constant. Engagement, I believe, is often way higher.

This sort of teaching, as Dylan notes, often is seen as anti-lecture. Well, it is. But it also doesn’t fit easily into the main ideological categories through which we talk about teaching. It’s certainly nothing like inquiry. It also doesn’t seem right to call this an explanation-heavy style of teaching. Noticing, wondering, listening, discussing, practicing, discovering—the particular activity doesn’t matter for fussy teaching. It’s all fair game, as long as kids only spend a few minutes at a time doing it.

I wish there was a catchy ideological formula for this style of teaching. “Active” comes close, I guess. “Direct Instruction,” if you look at the official capital-D, capital-I description, is in fact calling for something like this. “Fussy” is a transparently awful title, and I’ve always been bad at naming things. (Reyzl took the lead on naming our children, thank God.)

Someone smarter than me at selling things needs to tackle this. Or maybe it’s unsellable. Maybe the pedagogy of being really on top of things is too non-ideological to really spread on its own. Maybe it’s too basic, in a way too obvious for teachers to get excited about.

“I was accustomed to actors losing their tempers and shouting,” she tells LIFE. “It’s just the way theatrical people behave. But one day I thought, ‘My God, for him it’s real.’”

Two years later, in 1968, Gerd would be found dead in her kitchen, next to the stove, only 36 years old. She left behind two sons, ages 12 and 5, and her husband, age 63.

Hein is known in America mainly to mathematical enthusiasts, but remains prominent in his native Denmark, mainly for his poetry. Those poems, known as “grooks,” have been translated into 20 languages with over a million and a half copies sold. He maintained relationships with Bohr, Einstein, Norbert Wiener, and Charlie Chaplin.

Last year marked the publication of Peter Borberg’s 551-page Danish biography of Hein—and it’s only the first half. Around its release Gerd’s older son, Jotun, now a mathematics professor at Oxford, sat for an interview with Bo Bjørnvig of Weekendavisen to speak about his parents. Most of it is incredibly damning. I wish I didn’t have to rely on Google Translate, but it’s painful either way:

She was often beaten during some of our tantrums. I remember how one night, when I was 8-9, I was woken up by noise and screaming. When I came out into the hallway, mother was driving away in a taxi, and I stayed out in the hallway until she came back with the doctor. Her face was severely swollen and she had had a nosebleed from the blow [Hein] had given her. The doctor was begging to be allowed to go to the police.

This was not an isolated incident. In her autobiography, Tove Ditlevsen (who Hein had an affair with) quotes an earlier lover of Hein’s: “He is a dangerous man, created to make many women unhappy.” Birgit Tengroth, a Swedish friend of Gerd, likewise describes an upsetting visit to the Hein home in her memoir “Jag trodde du var död!” The two boys try to get Tengroth to sleep over, while Gerd tries to reassure her friend that she’s really alright: “Oona (Chaplin) has it ten times worse than me.”

Gerd was hospitalized with what sounds like a major depressive episode just weeks before her death, an apparent suicide. The doctor’s notes, according to Jotun, describe Hein as overbearing, controlling, disliked by his own sons. Jotun was sent to live with a foster family soon after his mother’s death. Hein wouldn’t pay them. He was stingy, abusive, self-obsessed.

None of this, to my knowledge, appears yet in English.

I started looking into all this while researching a shape that brought Hein fame in America. It began with Martin Gardner, who in 1956 parlayed a Scientific American article (on flexagons) into a monthly column on mathematical games. He grew interested in HEX, an out-of-production Parker Brothers game that Hein invented. (Hex is terrific.) Gardner reached out to Hein about the game. Their correspondence and a friendship flourished.1

Gardner revived interest in Hex and would cover other Hein inventions over the years, bringing him a measure of fame. (“I shall never forget that I am your discovery, perhaps even creation” Hein wrote to Gardner in 1966.) But nothing struck a chord with the audience quite like the superellipse, largely thanks to the accompanying origin story.

That story goes like this. In 1959, Stockholm razed a bunch of old homes in the name of urban renewal. They ran traffic arteries through the rubble, creating a rectangular patch of city and an urban-planning problem—what shape to make the public fountain inside the rectangle? They didn’t want to make it circular (or elliptical), since that would waste space in the frame. They wanted the fountain (and underground shopping center) to be shaped somewhere between an ellipse and rectangle. What shape could that possibly be?

The chief architect was an old friend of Hein’s and knew he liked this sort of thing. Hein took the phone call and reportedly said, “I think a curve with the same equation as an ellipse but with an exponent of two and a half would do it.” The graph of that equation does, in fact, look like something halfway between a rectangle and an ellipse. Hein called it the superellipse.

I don’t mean to be a jerk—but couldn’t anyone have thought of this? Maybe not the equation. But who needs the math at all? Go ask literally any artist for something between a circle and a rectangle, I promise you’ll get it. It is not that hard. Whatever. They asked Hein, so he gets the credit.

After Gardner’s column came the LIFE profile, then another huge, even more ridiculous profile in Esquire. It was titled “King of the Supershape” and it opened with an overheard photo of Hein splayed out on a huge blue superelliptical table—fully clothed, thank God.2 The whole thing has an Austin Powers vibe. “Civilization's oldest war is that between round and rectangular forms,” Esquire declares. “But now, peace. Behold the super-ellipse.” Give me a break.

Hein was not a research scientist or mathematician. He made money a few different ways: from mathematical toys—like Hex, or his brilliant Soma Cube—from his books of poems, from his patents, and as a designer. He had sleek, minimilistic tastes, some physics training, an obsession with abstract clarity, an interest in curves, and a knack for lamps.

Hein soon created and sold superelliptical dishes, tables, rugs. He spun the superellipse into three dimensions, yielding a toy for your desk—the superegg. The superegg is stable on its ends, though it looks just as goofy and inelegant as the superellipse. It’s not surprising that it’s stable, right? I mean, it’s pretty flat on the ends. I guess it’s supposed to be beautiful.

For the record, I also dislike Hein’s poetry. His “grooks” are tidy and clean like his games, but also self-satisfied and cloying. To be fair, he started publishing them when Denmark was under Nazi occupation and he was head of the anti-Nazi union, trying to sneak anti-Nazi messages past the occupying censors—that’s cool. But, personally, I just don’t get much out of this:

LOSING FACE

The noble art of losing face
may one day save the human race
     and turn into eternal merit
what weaker minds would call disgrace.

“Undiluted Hocus Pocus” is Martin Gardner’s autobiography, written late in his long life. In it he briefly touches on his relationship with Hein. “Piet Hein visited me twice, and we became good friends. On his second visit he brought along his beautiful wife. A native of Iceland, she had become one of Denmark’s famous actresses.” I don’t know why he said “Iceland,” but that’s all that’s been written in English on Gerd—at least until I sent Jotun’s interview through Google Translate and quoted it here.

In 2015, Numberphile, the popular math YouTube channel, released an episode about the superegg. It tells the familiar Stockholm story first told by Gardner. It mentions nothing about Gerd or Hein’s behavior. Of course not—it’s a mathematics video. It would be weird to jump into all that dark stuff, right?

It’s at least partly a question of context. I’m not about to start talking about Piet Hein’s marital life while chatting up students about funny shapes. I’m not sure how I’d approach it in something like an online video, where the context is fuzzier and out of my control. Maybe Martin Gardner should have said something. I don’t know.

There’s a larger cultural conversation about famous people who are beloved for their work but are monstrous in their personal lives, and this is probably part of that. Anyway, I’m struck by something else—knowing about his personal life makes Hein’s intellectual story make more sense.

Think about it. If you gave me a rectangle and a crayon, I’d get close to what Hein made, though I probably wouldn’t write an equation. But Hein liked precision, structure, a kind of cleanliness. He took an urban planning question and turned it into a puzzle. It’s the same sort of tidiness-in-the-extreme that his poetry exhibits. That’s not what I like in my art—I like big messy feelings and huge dramatic gestures. I don’t like aphorisms very much. But Hein sure did.

In art and life, Hein thrived when the rules of the game conformed to his expectations. “If you were alone with him and discussed something that interested him, you would get a lot of attention.” But step outside those lines, and Hein would fight to fence you back in.

There is a poem by Hein that mathematicians are particularly fond of:

PROBLEMS

Problems worthy
  of attack
prove their worth
  by hitting back.

Along with everyone else, I’ve taken it as neat couplet about problem solving. Given what I know now about Hein I’m inclined to expand it. What happens when people become the problems? Does intellectual rigor have a tendency to make us demanding of others? Do they “hit back”? Isn’t it, quite possibly, a more interesting (if disturbing) poem knowing the harm Hein inflicted on the ones he loved?

I can’t say for sure. But God bless Jotun for surviving and telling his tale.

What makes these books so popular? One thing they do—I think it might be a characteristic of all world-beating ideas—is combine a big, explanatory theory with a few handy techniques that anybody can try.

Take Mindset. It posits that people with similar skills are differently motivated. For Dweck this flows from one’s theory of intelligence. You either have a growth or fixed mindset. A growth mindset is a tendency to explain failure in terms of things you, the individual, can control. The tendency orients the individual towards learning goals. The result is that challenges, seen as an opportunity to learn, are motivating. Whatever. My point is that this is a theory, and teachers (along with the media, bosses, parents) loved it.

But it wasn’t just theory. Mindset only got huge when people learned that the theory generated some simple, distinctive actions:

• Don’t praise someone for “being smart.” Praise them for working hard.

• Motivate people by telling them about the brain’s ability to grow.

Without theory this practical stuff would feel inert. (What if Dweck had given a TED talk called “10 Motivation Hacks”?) Combine theory with practice, though, and it’s magic. The easy stuff stands in for the entire set of mindset ideas, and are imbued with this tremendous shiny meaningful energy.

When you praise your kids correctly, you are doing mindset. Or, at least, that’s how it can feel. Of course you’re not really doing it all. You’re just doing the easy stuff! But it might feel as if you’re thereby making some radical and comprehensive change by praising students for their efforts—even if, say, your classroom is structured in a way that makes it difficult to learn and grow.

Peter Liljedahl’s comparatively modest hit Building Thinking Classrooms follows a similar playbook. I doubt that this was intentional, but it’s perfectly engineered for popularity.

BTC has a theory as to why students aren’t learning. “Thinking is a necessary precursor to learning, and if students are not thinking, they are not learning,” he writes. So what are kids doing in school, if not thinking? Skating by with false engagement (“studenting”).

The BTC approach aims to break this culture of “studenting.” It’s sort of a shock-and-awe shift meant to confront kids with a radically different set of classroom expectations. Everything looks and sounds different in BTC: defronting, thin-slicing, notes-to-future-forgetful-selves, mobilizing knowledge, check-your-understanding questions (in place of homework), consolidating, and much more.

BTC strongly encourages teachers to take the whole plunge at once. The book emphasizes the need to do all this stuff together. (See excerpt below.)

But two BTC practices are by far the most popular among teachers:

• Group work at vertical whiteboards (or as BTC calls them “vertical non-permanent surfaces”)

• Assigning group partners randomly

Liljedahl could not be clearer that he is offering a comprehensive theory and radically different approach. But there are tons of teachers whose use of BTC extends to almost nothing beyond these two practices. Even if teachers are doing everything else exactly the same, they feel as if they are doing BTC if they have kids working at whiteboards in random groups. It’s the “praising effort” of Liljedahl’s approach.

“So what?” a lot of teachers say. “I take the parts that I like and don’t care about the rest. I’m pragmatic.” This is true—teachers (and everyone else) do this a lot.

But the irony is that you need to go big and comprehensive if you want your small practical ideas to spread. They need to be part of a theory, if only so that people can scrap the system for parts. We want our useful hacks to be part of something bigger.

***

At least in teaching, theories tend to fade, leaving behind the practices. David Tyack and Larry Cuban in Tinkering Toward Utopia called these “instructional add-ons.”1 The big radical ideas get digested by the teaching profession until all that’s left are those catnip practices. In thirty years, BTC will be unknown, but there will still be schools that swear by vertical non-permanent surfaces. It’s just how it goes.

You can see it happening now. In online support groups for BTC practitioners, teachers ask for help enacting the system’s somewhat demanding aspects. They may cheat a bit and offer explanations at the start of the lesson (not allowed in BTC). Or they may be unsure how to “thin-slice,” which is the BTC-approved method for learning new skills. After well-intentioned efforts to make the whole thing work, most people will end up delivering entirely conventional math lessons, just with whiteboards.

But take a moment and admire the power of theories.

Teaching (or parenting, or bossing, or social-working, etc) is confusing work. There are days when you think you’ve figured it out, only to run into humiliating defeat. At it’s worst it feels random, chaotic, and unpredictable. Well, sure. People are random, chaotic, and unpredictable. Anyone who works with humans either routinely struggles with this, or is not paying enough attention.2

We need theories to help make sense of this mess. We want labels and categories and explanations to help us make order out of the chaos. We want to understand our options and act more consistently. Of course, it also has to be practical. Nobody likes irrelevant ideas that can’t be easily brought into action.

Is this healthy? I don’t know. Maybe it is, and what we need are better books and systems. But look around at what resonates with people—there is a desperate, aching need for ideas, the bigger the better, and that’s not going to change any time soon at all. People absolutely love theories.