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Why the Tuesday baby puzzle is so frustrating by Adam Kucharski
Wednesday October 15th, 2025 at 2:26 PM

Understanding the unseen
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There’s a puzzle that’s bothered me for a long time. You may well know it. It goes something like this:

I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?

Now you might initially think ‘well, it’s 50/50’. Or you might think a bit more, and write out the possibilities in birth order:

  • boy/boy

  • boy/girl

  • girl/boy

  • girl/girl.

Then you might conclude that it’s not 50/50. Given boy/boy is one of the three possibilities that include at least one boy, the chance of two boys must be 1/3.

But that’s not the answer either. Apparently the correct answer is 13/27, or about 48.1%. And that’s what has bothered me, just like it’s bothered many others. Surely the information about the Tuesday is irrelevant?

Many types of proof

As I’ve written about previously, there are various ways of proving the solution to a probability-based puzzle. One approach is proof by exhaustion: simply write out all the combinations and select the ones you’re interested in. Given a puzzle with a boy/girl split and 7 days of the week, there are 2 x 7 = 14 possibilities for each child and hence 14 x 14 = 196 possible combinations for both children. If you write them all out, 27 of the combinations have at least one boy born on Tuesday, and 13 of these have two boys. Hence 13/27.

Except that’s not particularly convincing, is it? Instead, we could try proof by simulation: randomly simulate lots of families with two children, and see how many of those with a boy born on Tuesday have two boys. If we simulate 100,000 families, for example, and select those with a Tuesday boy, we find that around 48.1% of them will have two boys:

Again, it’s not that satisfying as an explanation. For a long time, this is as far as I’d got, putting the puzzle in a ‘there is an answer, but not a very convincing one’ mental box. The same box in which I put things like string theory and cellular immunology.

Recently, though, I decided to properly get my head around the problem. Why does the Tuesday data point make a difference?

Getting some intuition

I started, as is often helpful, by considering a simpler example. Suppose the additional event isn’t calendar based. Instead, it’s the epitome of random: a coin flip at the birth. Here’s a simplified puzzle:

I have two children. One is a boy; we flipped a coin after he was born and it came up heads. What is the probability I have two boys?

Next, I made a table of all the combinations (there are only 16, so this was a bit easier):

As you can see, there are 7 rows with at least one boy/heads, and 3 of these have two boys, i.e. 3/7 = 43%. So here’s the first piece of intuition: when there are fewer combinations, we stay closer to the original 1/3 result. As we increase the number of possible outcomes per child (e.g. from 2 coin outcomes to 7 days), the probability rises toward 1/2.

But again, why does the coin toss make a difference? That’s when I hit on the second piece of intuition: the top row in table matters (i.e. two of the same outcome). As a more extreme example, suppose I told you that one child is a boy born on Christmas Day. Of all the possible days the other child could be born on, it’s very unlikely it will be Christmas Day too.

What’s much more likely is that I have one child born on Christmas and another not. In which case it’s a 50/50 chance the non-Christmas baby is a boy or girl. So the resulting probability of having two boys is nearly 50%. It’s slightly less because there’s still a tiny chance both children share the same rare situation (i.e. both boys born on Christmas), which pulls the probability down a bit.

Or let’s go even more extreme. Imagine I tell you one child is a boy and I give you their full name. It’s near-impossible that I’d give my other child the exact same name, so in this scenario, the probability of having two boys with same name (i.e. the top row in the table) is vanishingly small. In this case, it effectively means I have one boy with a specific name plus another child, so there’s simply a 50% chance that the other child is a boy.

Goodbye Tuesday

The Tuesday puzzle isn’t really about Tuesdays at all. Or coins. Or Christmas. It’s about how adding a rare piece of information can skew what ‘at least one’ means. When you hear ‘at least one boy’ in a puzzle, you might picture all the families that could fit that description (i.e. one boy and one girl, or two boys).

But if you hear ‘at least one boy born on a Tuesday’, the description becomes rarer. So it becomes closer to saying ‘I have one specific child you know quite a lot about, and another child that is random’.

Which I think has helped the puzzle stop bothering me. (But I’d be interested to know if it has helped convince you too.)

If you’re interested in reading more about proof and probability, you might like my latest book Proof: The Uncertain Science of Certainty.

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Tech Workers Versus Enshittification by Cory Doctorow
Wednesday October 15th, 2025 at 2:04 PM

Communications of the ACM
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The men who appeared on the dais behind President Trump at his inauguration were never good men. Take Tim Cook. Cook owes his elevation to Apple CEO—which made him a billionaire twice over—to his leadership in moving Apple manufacturing to “iPhone City” in south China.3 Other Apple execs had tried this, but they could never get the precision and reliability Apple demanded. Cook managed it, but the brutal working conditions created by Foxconn, Apple’s contract manufacturer there, led to the installation of suicide nets5 to catch workers who leapt to their death rather than facing another shift.

Or look at Jeff Bezos. Workers in Bezos’s warehouses are injured (often critically) at three times the rate of workers in other companies’ warehouses.7 Bezos’s contract drivers have to relieve themselves in bottles because they are not given toilet breaks. They are also monitored by AI cameras that score their eyeball movements.11

Google’s top bosses covered up for a senior manager who allegedly preyed on a female subordinate, whom he subjected to violent sex acts. He was given millions of dollars to go away quietly12 while his victim was prohibited from suing by the “binding arbitration” waivers in her contract.

These men have always abused anyone they could get away with abusing. If they seemed to treat some people well—for example, the pampered coders who get gourmet cafeterias, free massages, and all the kombucha they can drink—it was because they feared the difficulties of replacing those workers. Now that U.S. Big Tech has shed more than 400,000 jobs in just two years, there’s a reserve army of desperate techies who can be given the suicide-net-and-urine-bottle treatment.

One of the first official acts of the current administration was to cancel every single labor investigation and enforcement action in the entire country.8 The administration’s antitrust boss canceled investigations into predatory pricing (selling goods below cost to drive out competitors) and surveillance pricing (using commercial surveillance data to charge you more (for example, by increasing the price of your breakfast sandwich on payday) or pay you less (for example, offering nurses lower wages if they are known to have overdue credit card bills).10 Instead of helping workers and shoppers resist predatory assaults fueled and perpetrated by Big Tech, Trump’s Federal Trade Commission chair is focusing all his energies on a new snitch line where FTC employees can report “wokeism.”4

The men who run Big Tech have always wanted to “enshittify” their services—to shift as much as they can from users, workers, suppliers, and business customers to themselves. The forces that prevented that—militant, scarce workers; competitors; regulators; and rival technologies—were critically weakened and even eliminated over the past 40 years. Biden’s administration was the first in two generations to take fighting corporate abuses seriously.1 Is it any wonder that Big Tech were all in for his rival?

Enshittification would be much, much worse if not for tech workers. Tech workers have historically enjoyed remarkable workplace status—not just the material perks, from dry cleaning to personal trainers, but also the power to tell their bosses to go to hell. When your skills are in such high demand that you can quit your job, walk across the street, and get a better one later that same day, your boss has a real incentive to make you feel like you are their social equal, empowered to say and do whatever feels technically right, or just, you know, show up for work with facial piercings, a green mohawk, and a black t-shirt from Defcon with writing on it that your boss doesn’t understand but still feels uncomfortable about.

Tech workers may have given their bosses tsuris and demanded sky-high wages, but at least they were wildly productive. The per-worker revenue for successful tech companies is unfathomable—tens or even hundreds of times their wages and stock compensation packages.2 What’s more, many tech businesses had more work than they could hire skilled workers to perform, so bosses’ strategy focused on getting the workers they could hire to put in longer hours.

That strategy has a name: “vocational awe,” a term coined by the librarian-theorist Fobazi Ettarh to describe the grueling hours endured by workers who believe in the social value of their work. Teachers, nurses, librarians—any profession whose practitioners are motivated by a sense of duty is a profession vulnerable to vocational awe.

Tech companies have been widely derided for their grandiose mission statements, such as Google’s “organize all the world’s information and make it useful” or Facebook’s “making the world more open and connected.” But whether the management of these companies believed in these high-minded words, they were absolutely internalized by employees, who—despite their job security—worked every hour god sent, sleeping under their desks and missing their parents’ funerals to hit their bosses’ arbitrary deadlines.

As a motivational tactic, vocational awe (or, as Elon Musk puts it, “being extremely hardcore”) works very well, but it fails very badly. It turns out that when you actually care about your job, and when your boss can’t replace you, and when you can get a new job whenever you want, it’s damned hard to convince you to wreck the things you built and harm your users, even if that will make your boss more money.

For many years, tech workers have been the last line of defense between Internet users and tech bosses, bravely holding the line on enshittification. But once tech started to shed workers en masse—260,000 layoffs in 2023,6 another 160,000 in 20249—the power of workers was shattered. Scarcity can give workers power, but scarcity is temporary: Eventually, enough people train to work in your industry, or your boss figures out how to outsource your job to overseas workers with fewer protections and rights, and your power is shattered. No wonder tech bosses are so excited about AI coding tools, which promise to turn skilled programmers from creative problem-solvers to mere code reviewers for AI as it produces tech debt at scale. Code reviewers never tell their bosses to go to hell, and they are a lot easier to replace.

Now that tech workers are as disposable as Amazon warehouse workers and drivers, as disposable as the factory workers in iPhone City, it’s only a matter of time until the job conditions are harmonized downward. Jeff Bezos doesn’t force his delivery drivers to relieve themselves in bottles because he hates delivery drivers. Jeff Bezos doesn’t allow his coders to use a restroom whenever they need to because he loves hackers. The factor that determines how Jeff Bezos treats workers is “What is the worst treatment those workers can be forced to accept?”

Throughout the entire history of human civilization, there has only ever been one way to guarantee fair wages and decent conditions for workers: unions. Even non-union workers benefit from unions, because strong unions are the force that causes labor protection laws to be passed, which protect all workers.

Tech workers have historically been monumentally uninterested in unionization, and it’s not hard to see why. Why go to all those meetings and pay those dues when you could tell your boss to go to hell on Tuesday and have a new job by Wednesday?

That’s not the case anymore. It will likely never be the case again. Interest in tech unions is at an all-time high. Groups such as Tech Solidarity and the Tech Workers Coalition are doing a land-office business, and copies of Ethan Marcotte’s You Deserve a Tech Union are flying off the shelves.

Now is the time to get organized. Your boss has made it clear how you’d be treated if they had their way. They’re about to get it. Walking a picket line is a slog, to be sure, but picket lines beat piss bottles, hands down.

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In Search of the Yakult Lady by Nina Li Coomes
Wednesday October 15th, 2025 at 9:49 AM

Eater
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When my sister and I were growing up in Nagoya, Japan, our mother would sometimes bring us to Tsuruma Park after school to let us run free. The playground, fringed by dark trees that cast shade on the Showa-era slides, always felt a little hidden, a separate place where only mothers and children could go.

But then, breaking through the unspoken barrier, there she would be: the Yakult lady, sailing toward us on her bicycle with a cool-blue cooler box mounted on the back. A discernible excitement would descend over the crowd of kids as we rushed like hungry pigeons to her feet, waiting for our turn to buy the sweet yogurt drinks she pulled from the cooler. 

There was no waste in her movements: a neat twist of the waist, a flip of the latch. My mother would press a few coins into the woman’s palm, and the Yakult lady would hand us our bottles. Her bicycle, her bonnet, the drink’s sweet tang — all seemed to be indicators that the world around me was safe and sensible. But I also remember feeling briefly confused in that moment, as the Yakult lady delivered this essential bit of maternal care. Who was taking care of us: our mother or the Yakult lady? But the moment would pass, and the next child would step up for their turn.

If you live in a major U.S. city, you’ve likely seen Yakult in your supermarket. The telltale squat bottle with a shiny aluminum cap is stocked in my local grocery store in Chicago next to tofu and other refrigerated Asian goods. The company sells 30 million bottles per day across 40 countries. Or maybe you’d recognize the drink from the 2018 film To All the Boys I’ve Loved Before, when Lara Jean (Lana Condor) tells Peter Kavinsky (Noah Centineo) about the “Korean yogurt smoothie” she’s drinking. (That tiny mention is thought to have increased the company’s shares by 2.8 percent that year.) 

Dr. Minoru Shirota invented the sweet, milk-based probiotic at Kyoto University in 1931. Formulated from Dr. Shirota’s own strain of lactobacillus — a strain of Lacticaseibacillus casei called Shirota — the drink was designed to promote a healthy digestive tract, quality sleep, and immunity. Today, that halo of health still surrounds the brand. My sister works in a Chicago hospital where a doctor sponsored by Yakult researches the benefits of lactobacillus on immune health. Influencers push it on followers as a health supplement for their families. 

While the majority of Yakult is sold in stores, the company employs thousands of Yakult ladies to sell bottles in person. As of 2024, there were about 32,000 Yakult ladies operating in Japan, and another 50,000 outside of the country, mostly in China, South Asia, Brazil, and Mexico (where the uniforms can differ from the iconic blue color scheme in Japan). They deliver their wares on foot, bicycle, and motorcycle. With powder blue visors and matching pantsuits, they cut recognizable figures wherever they make their rounds. 

Kazuhiro Noguchi, a Yakult franchise owner in Hiroshima, first introduced the delivery women in the mid-’50s. Believing it was easier for a housewife to enter a fellow housewife’s kitchen than for a salesman to get in the front door, he created a job for working women that fit with midcentury gender expectations. 

“Especially for women with young children, this type of thing — where they could work while the children were at school — was a very attractive job because it allowed the household sphere to still hold their attention,” says Hilary Holbrow, a sociologist at Indiana University who specializes in labor and gender in Japan. The Yakult lady proved to be a success, and by 1963, she became formalized as a company-wide sales system. Today, Yakult still tries to attract working mothers with flexible schedules and on-site day care facilities. 

Before having children, I assumed I would be the kind of mother who would race back to the office. When I got pregnant, I remember looking at my maternity leave — nearly four months, generous by American standards and paltry by the rest of the world — and feeling like that seemed like a long time to be alone with a baby. As someone who measured her worth through report cards and performance reviews, I felt uneasy with parenthood.

But when I had the baby, the creature I expected to feel like a distraction instead reframed my life. I didn’t want to rush back to the office. I wasn’t bored. She crystallized for me what mattered in my life, why I worked, who I wanted to provide for. I could never resent her. Instead I became resentful that I had to choose between being a working mother who histrionically complains about her children or being a devoted mother who abandons work and throws herself into caregiving. What I wanted — a working environment that would understand my temporary priority shift from career to motherhood, without negating my future ambition or need to earn money for my household — did not exist. 

Though I’m not a 1950s housewife trying to make ends meet, I can sense the attraction of Yakult’s job offer from here, nearly eight decades away. The idea that employment would be tailor-made for a mother who works, constructed with her unique needs in mind, is a heady, tantalizing thing.

To understand the day-to-day routines of these women, I set out to interview them. But, like Mordor, it turns out one does not simply walk into Yakult. To date, the company has turned down two years’ worth of interview requests. Anytime I’ve approached Yakult ladies across Japan, they’re too busy or reluctant to talk. Eventually I secured one interview the only way I knew how: I walked into the Osu Kannon Yakult Center in Nagoya with my then-6-month-old daughter strapped to my chest. 

The center was humming when I walked in, as Yakult ladies filled out delivery reports and packed their coolers with ice. But my daughter’s chubby cooing caught their attention, eventually distracting enough employees to draw a manager, who declined to share her name but agreed to speak. 

“Most Yakult ladies start their shifts around 8:30 a.m. and finish by 1 p.m., unless they work a full-time shift, in which case they work till 5,” she told me. Yakult ladies bike everywhere, cycling around town for hours, sometimes going out of their way to make a single delivery. Though most Yakult ladies deliver to private homes, the team at the Osu Kannon center have corporate routes, meaning they’re dipping in and out of office buildings to hand off drinks to clients. They’re also supposed to stop and chat with anyone who strikes up a conversation on the street. 

Yakult ladies aren’t classified as full-time employees, but kojin jigyo usha (roughly “sole proprietors”), essentially making them owners of bicycle-sized franchises. They purchase product from Yakult and make a profit based on what they can sell. Yakult says the average earnings of a Yakult lady are roughly $682 USD a month, compared to an average of $1,774 per month for Japanese women broadly. In Yahoo Answers forums, Yakult ladies claim wildly different profits: Some say they work only three hours a day and make more than the company average. Others claim to work far more, selling roughly $2,700 worth of product in a month to take home about $600, roughly a 22 percent cut. 

They may decide their own schedules, but Yakult ladies don’t have designated holidays or sick days. The company encourages them to manipulate their schedules to accommodate time off. Even with an electric bike, it can all take a physical toll, and the manager told me Yakult ladies need to “get creative” to make it work. That might not be worth it to some mothers.

“The relative appeal of these Yakult jobs — that very much fit into these older models of how people’s families and life were supposed to look — is drastically reduced compared to the past,” says Holbrow. 

Company-sponsored day care may have been attractive to midcentury housewives, but today Japan heavily subsidizes national day care, and paid maternity leaves from other companies regularly stretch for a year or more. Holbrow adds, “Something like 30 percent of women are never married by age 50. So the number of women who are responsible for raising children or taking care of their husbands is a much smaller share than it was in the past.” 

At the Yakult Center, many of the Yakult ladies seemed to be in their 60s, which tracks with national trends. Nearly a third of Japanese citizens are 65 and older, and the national fertility rate is at an all-time low. The women working at the Osu Kannon center may have been some of the same women I saw delivering drinks when I was a kid. I didn’t see new mothers — mothers who look like me — among their ranks.  

Still, Yakult positions its delivery workers more as mothers than salespeople. “We don’t do sales because everyone knows what Yakult is,” the manager told me. “The price hasn’t changed in years, so we don’t negotiate.” Instead, Yakult ladies are mijika na sonzai, roughly “known entities,” implying they’re not only familiar to their customers but close to them.

There are several company-approved “Day in the Life of a Yakult Lady” videos online, which feature the many different community members the women deliver to. A promotional video made by Yakult UK & Ireland provides a glossy, almost Yasujirō Ozu-esque fantasy of the Yakult lady’s function as a key part of a community’s social life. In recent years, the company has formalized this reputation into the “Ai no Houmon” (Love’s Visit) program, wherein Yakult ladies do what are essentially wellness checks on older adults while selling them probiotic beverages.

As I left the Yakult center, my baby clamoring for her nap, I felt oddly disillusioned — not by the women themselves, or even the no-nonsense manager, but by the corporate trappings of their work. Before I looked into it, I had swallowed the lighthearted, easy glow of Yakult’s promotional videos, which recalled my own experience when I was a kid. I would like to believe selling probiotic milk drinks is just an aside to Yakult ladies’ main mission of maternal care in the community. In the fluorescent lighting of the Yakult center, I saw their labor.

As a mother, I know creating a sense of comfort and safety requires constant work. To engender trust and calm through food, through milk specifically — in my case, produced by my own body — these are tasks that might seem effortless on the surface but in fact require effort and intention. The glassy, undisturbed surface Yakult ladies project is hard-won after hours each day spent on a bicycle loaded with drinks and selling a product hand-to-hand for little overhead. 

Even more than drinks, the image of the unflappable Yakult lady is the product that the company sells; it’s also a product formed in a different Japan and inflected by 70 years of inequity. Like many working mothers, the Yakult lady today hides a complex network of compromises and conundrums. 

A billboard for Yakult featuring a woman’s face.

The day after going to the Yakult Center I walked through a residential neighborhood. In the hushed, respectable quiet, it felt as if the rules of the 1950s still applied. The salarymen straightened their ties and boarded their packed subways. The children donned their bright yellow hats on their way to school. Laundry hung in neat, productive rows from crisply snapped laundry lines. 

Through an open window, I heard the satisfying, high-pitched whine of a vacuum cleaner. I imagined the woman behind the machine: Is she a working mother squeezing in a chore before heading to the office? Is she a stay-at-home mother, busy at the unpaid, unseen work of running her house? Is she doing what I do, running the vacuum in nonsensical circles, hoping the drone of the machine will put her fussing infant to sleep so that she might steal 10 minutes of work?

Then, disturbing the stillness, I saw the one I was looking for. She swam down the empty street on her telltale bicycle, mounted with its light-teal refrigerated boxes. She wore a face mask, businesslike and hygienic. She neared the door of someone’s home, slowed, dismounted. After checking a clipboard, she pulled out two packs of Yakult, squat little bottles in clear plastic. She rang the doorbell, called out in a friendly, knowing way to the voice that answered, and made her delivery. Then she checked her clipboard, mounted her bike, and pushed off the asphalt with one, industrious kick of her heeled foot. As she sailed off, I felt like I was seeing a familiar ghost. There she goes, off on her next delivery: the Yakult lady.

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Why Some Students Learn Faster by Dylan Kane
Tuesday October 14th, 2025 at 12:49 PM

Five Twelve Thirteen
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Students are different from one another in all sorts of ways. The overwhelming experience of teachers is that students need different things to learn. But why? What are those differences? Are there systematic ways that students are different, or is it all kindof random?

I have a hypothesis about one major difference between students. I don’t think this is the only way that students are different, but it’s an important one and has concrete implications for how we teach.

This post is based on a lot of research I’ve read, research on teaching as well as intelligence. But this isn’t a “research says” post. I’m connecting some dots, trying to take a few different threads and tie them together in a way that’s useful for teachers. This post is informed by research, but also by my experience as a teacher and my observations about what works and for which students. Here goes.

What Does It Mean To Be Smart?

We throw around the word “smart” in education. But what does it mean to be smart? The details are often glossed over. It’s easy to fall into a circular definition. Some students are smart because they do well in school, and they do well in school because they’re smart.

Researchers divide intelligence into two parts: crystallized intelligence and fluid intelligence. Crystallized intelligence refers to how much you know: knowing things makes you smarter. I’m going to put crystallized intelligence to the side for the moment. I’ll come back to it later, but for now crystallized intelligence is what we want for students. Fluid intelligence is a broad term for how well we can think and reason when we don’t have crystallized intelligence to rely on. There’s plenty of disagreement as to what exactly fluid intelligence means. I think conceptualizing fluid intelligence as some sort of vague measure of cognitive horsepower isn’t very helpful: we end up with “intelligence means being smarter” again which isn’t much use for teachers. In the debate about what exactly fluid intelligence means, two big components that most researchers agree on are working memory capacity and processing speed. Working memory capacity refers to how many ideas we can hold in our mind at once, and processing speed is a measure of how quickly we can think. Serious researchers suggest that those two may make up the vast majority of fluid intelligence. That’s my mental model for this post: some students have more working memory capacity and faster processing speed, others less working memory capacity and slower processing speed.1 When we talk about students being more or less smart, we’re often referring to these differences.

I think this is a good mental model for teachers because it’s practical. In general, teachers have an intuition that some students are smarter than others. But what does that mean? If being “smart” means that some students will always achieve at a higher level, or other students can never learn a topic because it’s too abstract, we end up with self-fulfilling prophecies for which students will be successful in school. But thinking in terms of working memory and processing speed narrows in on specific, empirical elements of thinking and learning. I can see those differences in my students: all teachers can observe that some students are quicker thinkers, and some students can juggle more ideas in their mind at once. And we can teach in ways that mitigate those differences, that help all students learn.

I want to emphasize what this mental model doesn’t say. Both working memory and processing speed influence how new information comes into our minds. They don’t imply limits on long-term memory. All of the evidence we have suggests that, for all practical purposes, long-term memory is unlimited. For the vast majority of our students there is no hard ceiling on what they are capable of learning. Instead, there are constraints in terms of how new information comes in. I’ll return to this idea at the end of the post, but for now my mental model looks something like this:

I realize this image isn’t that great. The goal is to emphasize that working memory and processing speed act as a bottleneck between learning and long-term memory.

Let’s say you accept this premise: a major difference between students is fluid intelligence, which we can think of as working memory capacity + processing speed.2 What does this mean for teachers?

Teaching

Having more working memory capacity and a faster processing speed makes learning easier. But it makes learning easier in very specific ways. I’ll repeat a quote I used in last week’s post. Talking about phonics instruction, Snow & Juel say: phonics is “helpful for all children, harmful for none, and crucial for some.”3 That’s the idea behind the teaching strategies I’ll share below. If you don’t use these, plenty of students will still learn. Teaching in this way won’t hurt any students with more fluid intelligence, but without using these strategies some students with less fluid intelligence won’t learn much at all.4

Check Prerequisite Knowledge and Reteach if Necessary

All learning builds on prior knowledge. Vocabulary, foundational skills like multiplication facts, concepts like the relationship between multiplication and division.

If students don’t have this knowledge, it’s harder to learn. But students with greater working memory capacity and faster processing speed can compensate: they are better able to quickly figure out a prerequisite skill and fill in those gaps as they learn. Students with less fluid intelligence have a harder time doing so. As teachers, we can mitigate this by thinking carefully about the prerequisite knowledge for each skill we teach, checking that students have it, and reteaching when necessary.

Time is a limiting factor here. We can’t take forever to reteach prerequisite knowledge. But in many cases, we can quickly remind students of the meaning of a vocab word or review a skill from a previous year that students are rusty on. Those quick tweaks can make a big difference in student learning. And when we have more time, there are often larger skills like math fact fluency that are worth reviewing and practicing.

Break Learning Down Into Small Steps

Whenever we teach, we ask students to make leaps from what they know to what we want them to learn. Students with more fluid intelligence can make larger leaps, moving from one idea to a much bigger idea. Students with less fluid intelligence struggle to make those large leaps, and learn more smoothly when learning is broken down into small steps.

image] The importance of small steps : r/GetMotivated

My mental model here: students with more working memory capacity and faster processing speed aren’t actually taking larger steps. They are using their own cognitive resources to fill in those rungs and build the ladder as they’re climbing it.

However you think about it, the teaching tip is straightforward. Take complex ideas, break them into small, manageable chunks, and teach one chunk at a time. There are lots of ways to do this. Teachers can take big ideas and break them down into small steps. We can also modify tasks to focus on the most important pieces we want students to learn and reduce cognitive overload. We can use questioning to model the thinking we want students to practice. We can provide more and faster feedback. We can help students gradually apply what they know in different contexts. We can give extra support when we ask students to try more challenging tasks.

The details are tricky, and they depend on the specific learning objective, but there are always smaller, more manageable steps we can provide for students.

Connect New Learning to Prior Knowledge

Learning isn’t about jamming students’ minds full of facts. It’s about building schemas — connections between different ideas, abstractions that build on concrete examples, and mental structures that allow students to make sense of new information more easily.

One important element of learning is connecting what we want to teach with what students already know. Those connections help students to build effective schemas, and make learning stickier and easier to apply in new contexts.

Students with more working memory capacity and faster processing speed can often make these connections on their own. I see this all the time. Some students are constantly saying, “oh, this is like that other thing we learned.” Others struggle to make these connections on their own because they have less spare cognitive bandwidth.

That’s where teachers come in. Our job is to make these connections clear to students. To sequence concepts together to help students see the connections. To ask questions that get students thinking about similarities and differences. To highlight the underlying structures that tie ideas together, not just the surface details that make them look different. To help students see the patterns, relationships, and principles that make knowledge transferable rather than isolated.

Obtain a High Success Rate

Learning is harder for some students than others. That’s obvious to any observant teacher. And it’s harder for lots of reasons — attention, motivation, interpersonal conflicts, lots of other reasons. One reason learning can be harder is simply that, for students with less fluid intelligence, they have to think harder in school each day. They experience more cognitive overload, more stress on their working memories, more fatigue as they take longer to think through new ideas. The specific consequences depend on the student, but in many cases students with less fluid intelligence feel frustrated and dumb in school, and that fatigue creates negative feelings about learning.

One concrete way teachers can help is to support students with more practice until they reach a high success rate. A high success rate is important for learning: when students are reliably successful with a skill, it’s more likely they’ll retain what they’ve learned. But it’s also important for motivation. For students with less fluid intelligence, who feel more strain from learning and are more likely to become unmotivated, helping those students achieve a high success rate when learning new ideas helps to motivate them, helps them to see the progress they’re making and recognize that the effort they are putting in is going somewhere fruitful.

Provide Spaced Retrieval Practice

One confusing element of working memory and processing speed is that they aren’t numbers that are set in stone. The more we know about a concept, and the more fluently we know it, the less strain that concept puts on working memory. The more we know, the faster we can think. This is true across all contexts: we read faster when reading about a familiar topic. We speak more fluidly in our first language than a language we are learning. We write more coherently when we have a lot of knowledge to draw on. This brings us back to crystallized intelligence. Crystallized intelligence supports fluid intelligence. The more we know, the better thinkers we are. Students with a lot of fluid intelligence can get away with knowing less, and compensate for a lack of crystallized intelligence because they have more cognitive resources to draw on. Others can’t.

One of the most important things we can do for those students is to help them become fluent with the skills we teach that come up most often in the future. Fluency frees up cognitive resources for new learning. The best way we can help students become fluent is regular retrieval practice, and a spacing schedule that ensures students get retrieval practice at increasing intervals to improve long-term retention. Spaced practice helps students develop automaticity with the skills we want them to learn. That automaticity frees up cognitive resources for learning more abstract and challenging skills. Then we help students develop automaticity with those more challenging skills, and bootstrap upwards.

There’s again a time constraint here. We can’t provide unlimited practice for every skill. But we can identify the most essential content that will come up over and over again in the future, things like math facts in elementary school, one-step equations in middle school, and more, and practice those until students are automatic.5

Don’t Blame Students

I’ve encountered a lot of people who tell a similar story about their math learning. Math learning went fine until one specific point. Sometimes it’s fractions, or equations, or Algebra II, or calculus, or something else. And at that point they just got stuck. Math didn’t make sense any more. These people are often convinced that their brains are incapable of learning any more math. They locate the problem inside of their brain, rather than with the instruction they received.

The teaching strategies I described above are pretty common in elementary school. I bet plenty of elementary teachers reading this post are thinking, “yea of course, we do this every day, young kids need this type of teaching.” In middle school, it’s more hit-or-miss. Some teachers do all of these. But others don’t, and this type of teaching often isn’t a big priority. In high school this type of teaching becomes rare, and in college it’s more or less nonexistent.

So when I hear people talk about how they hit a wall and just couldn’t learn math after a certain point, my interpretation is that it’s not something innate in their brain, it’s a type of teaching that we stopped providing. We left gaps in prerequisite knowledge to fester. We stopped breaking learning into small steps. We didn’t make explicit connections with prior knowledge. We didn’t push for a high success rate. We didn’t provide enough spaced retrieval practice.

It’s easy to blame students when they don’t learn. They just can’t handle the material, or they’re not motivated enough. It’s easy to look at other students who are learning well with the same teaching and assume the problem is with the students and not with the teaching. But in many cases, what we’re seeing is the differential effect of the cognitive resources our students bring to the classroom. Sure, some students can learn without these strategies. But more students can learn with the right instruction.6

I don’t want to be unrealistic. There isn’t some fantasy world of equal outcomes for all students out there if we do a better job teaching. There are also lots of other differences between students that matter in addition to this mental model for fluid intelligence. But this mental model is one approach to shrinking the gap between our most successful students and the students who struggle the most in school.

This isn’t easy. These are all teaching strategies I’m constantly working on. There’s no instruction manual lying in the teacher’s lounge that says “here is how to break skill A down into X, Y, and Z,” or “here’s how to connect today’s lesson to what students learned last year.” It takes effort and trial and error. But over time, as I find new ways to put all of these strategies into practice, they make a bigger and bigger difference for students, and help students who often struggle find success.

I want to reiterate my thesis. My mental model is that in general, students are not very different from one another in their long-term memory. There aren’t major differences relevant to K-12 education in what most students are capable of learning. The differences lie in how students need to learn it. Fluid intelligence — like any other human trait — differs from one student to the next. The five teaching strategies I lay out above mitigate those differences. Not to zero, not so that every student magically learns everything we have to teach. But they narrow the differences between students and raise the floor, so more students can access the content we intend to teach. None of the strategies are easy. They take expertise, hard-won by teachers improving their practice. They take deep knowledge of that content we teach. But the more I work on these elements of my teaching, the more I’m convinced that with the right pedagogical tools we can do a lot to shrink the gaps in achievement between students.

1

This isn’t a post about where differences in fluid intelligence come from. At a basic level, these differences are the same as any other differences between humans: humans are diverse in all sorts of ways, and cognitive capacities are no different from any other human trait in that respect. But it’s worth noting that those differences are not evenly distributed. To be clear: there are plenty of rich kids who have less fluid intelligence, and plenty of poor kids who are brilliant and have more. I’ve taught both. But on average, there are differences between those two groups. This feels a bit controversial to say but it shouldn’t be surprising. The brain is an incredibly complex organ. We know a bunch of different ways that brain development can be interrupted. Exposure to lead or air pollution are just two common examples, and these types of environmental influences are much more common for people living in poverty. There are more that we understand, and many more that we don’t understand. I won’t go on too long about this except to say that if our goal is to close achievement gaps in education, we should focus on factors outside of school in addition to classroom teaching and learning.

2

I’m not an intelligence researcher. I’ve read a fair amount about intelligence. (If you’re looking for a quick primer, the book Intelligence: All That Matters by Stuart Ritchie is a good place to start.) One tricky thing about fluid intelligence is that it’s often divided into pieces that are kindof circular. Researchers will divide it into elements like abstract reasoning, problem solving, and cognitive flexibility. But those are hard to measure, and hard to disentangle. Working memory and processing speed are two things we know how to measure in education (and often do measure, especially when testing students for special education services). There are serious intelligence researchers who suggest that those two are the underlying basis for the rest of fluid intelligence. Others would disagree. Some might add attentional control or something abstract like problem solving or something else. I’ll just say that maybe this mental model of fluid intelligence = working memory + processing speed isn’t perfect, but I think it’s a good approximation that’s concrete enough to be useful to teachers, and it’s much better than the circular “smart because learning is easier, and learning is easier because smart.”

3

There’s one clarification I want to make on the idea of “helpful for all, harmful for none, crucial for some.” Using these five teaching strategies doesn’t harm anyone’s learning. But they do take time, and there is an opportunity cost to that time. Some students don’t need as much time checking and reteaching prerequisite knowledge, or time on retrieval practice, etc. Judging how much time to spend on these elements of teaching is a practical challenge and there’s no easy answer. This is where some folks will come in advocating for tracking, and I will direct you to my recent post on tracking which emphasizes again that there’s no easy answer.

4

These teaching practices are drawn in part from Rosenshine’s principles of instruction.

5

Something that drives me absolutely fucking crazy. I’ve heard a number of prominent edu-consultants in the math world brag about how they never memorized all of their math facts. First, I always wonder if they’re lying. Second, I bet they’ve memorized the vast majority. they know 3x5=15, but they’re a bit rusty with 7x8=56. Third, they often have multiple degrees and have led successful careers as teachers and then consultants. They have the cognitive resources to learn just fine while re-deriving 7x8 whenever they need it. That’s nice for them. But plenty of our students don’t. It’s a shame that “I never memorized all of my math facts” is an applause line at math education conferences, and it reflects a fundamental misunderstanding of what students need to be successful in math class.

6

Some people here will mention that there are students who have profound intellectual disabilities, whose learning needs are very different and who this argument doesn’t apply to. Sure, I don’t disagree. But the vast majority of students — including many with mild to moderate learning disabilities — can learn the vast majority of what we intend to teach them in K-12 schooling if we provide the right instruction. Again, I don’t want to be unrealistic, but I absolutely believe we could achieve far better outcomes than what we achieve now. I think many students who could see more success in school are let down because we see other students learning from the same instruction, assume the problem is with the student, and blame them for not making enough connections or not being motivated enough.

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Books to Recommend to Maths Students by Katie Steckles
Tuesday October 14th, 2025 at 9:55 AM

The Aperiodical
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Photo of the spines of a bunch of books, seen from below
Image by Hermann Traub from Pixabay

I was asked recently by a first-year maths undergrad student if I could recommend any books on problem-solving, as they were hoping to develop their problem-solving skills. Asking around some maths communication colleagues has resulted in an impressive list of recommendations for books for maths undergraduate students, which I’m sharing here.

All of these should be available to buy from bookshops and are mostly in paperback – but many will also be available in university libraries, and it’s worth a check before shelling out for expensive copies!

Problem Solving

The classic text on mathematical thinking is George Pólya’s How to Solve It. Originally written in 1945, the book has been reprinted dozens of times, and includes some great advice on how to break down problems into steps or restate them more simply, make a plan for solving, and assess your work afterwards. Paperback £10.44

Fields Medalist Terence Tao’s Solving Mathematical Problems has lots of practical strategies for problem solving, accompanied by examples. Paperback £29.49

People also suggested Peter Eccles’ An introduction to mathematical reasoning : numbers, sets, and functions (paperback £36), and Techniques of problem solving by Steven G. Krantz (available to borrow from the Internet Archive).

Another classic of the genre is Keith Devlin’s Introduction to Mathematical Thinking – written to accompany a ‘transition course’ from school to university, it aims not to give a crash course in mathematical topics, but instead to get people thinking in more mathematical ways.

And on that subject, how could we forget friend of the site Kevin Houston’s How to think like a mathematician: a companion to undergraduate mathematics. Kevin’s website also has more information and solutions. Paperback £31

For some practice at problem-solving, Stephen Siklos’ Advanced Problems in Mathematics is published using OpenBook, and is available as in PDF or HTML format – containing many great examples of maths problems, along with comments and solutions. It’s particularly good for STEP Practice, but also for general problem-solving.

Proofs

More on the side of proofs, we had a recommendation for How to Read and Do Proofs: An Introduction to Mathematical Thought Processes, by Daniel Solow, which gives an outline of proof methods and exercises to work through. Paperback £75.95

Other recommendations going more into proof included Lakatos’ Proofs and Refutations – published in four parts in The British Journal for the Philosophy of Science (requires institutional login). It’s written as a dialogue between teacher and students, and focusses on a single problem for the entire book, conjecturing, refuting with counterexamples, then studying the counterexample to systematically improve the next conjecture. It was actually part of his PhD thesis!

More General Advice

If you want to help someone prepare for studying maths at uni – or advise someone who’s thinking about doing that – Vicky Neale wrote an excellent book called Why Study Mathematics?, which gives an idea of what to expect from maths at uni, and some wonderful advice. It’s available in paperback, and Vicky’s website has some links to excerpts, and her interview on the Numberphile podcast which covers the same content (transcript). Paperback £12.99

Another recommendation in this area is Lara Alcock’s How to study for a mathematics degree (one of her many excellent books). Paperback £22.99, and available in all good university libraries.

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Beyond Enshittification: Hostile by Ibrahim Diallo @dialloibu
Tuesday October 14th, 2025 at 9:54 AM

iDiallo.com
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The computer is not just working less well. Instead, it is actively trying to undermine you. And there is nothing you can do about it. When Windows wants to update, you don't get to say "no." You get "Update now" or "Remind me later." When Twitter shows you notifications from people you don't follow, you can't dismiss them, only "see less often." When LinkedIn changes your email preferences, you'll reset them, only to find they've reverted a few months later.

These aren't bugs. They aren't oversights. They're deliberate design choices that remove your ability to say no. It's not dark patterns anymore. It's not even enshittification. It's pure hostility.

The Two Types of Users

As developers, there are two types of users we find extremely annoying.

The first is the user who refuses to get on the latest version of the app. They're not taking advantage of the latest bug fixes we've developed. We're forced to maintain the old API because this user doesn't want to update. They're stubborn, they're stuck in their ways, and they're holding everyone back.

The second type of user is the one who's clueless about updates. It's not that they don't want to update, they don't even know there is such a thing as an update. They can be annoying because they'll eventually start complaining that the app doesn't work. But they'll do everything short of actually updating it.

Well, I fall into the first category. I understand it's annoying, but I also know that developers will often change the app in ways that don't suit me. I download an app when it's brand new and has no ads, when the developer is still passionate about the project, pouring their heart and soul into it, making sure the user experience is a priority. That's the version I like.

Because shortly after, as the metrics settle in and they want to monetize, the focus switches from being user-centric to business-centric. In Cory Doctorow's words, this is where "enshittification" starts.

Now, I'm not against a developer trying to make a buck, or millions for that matter. But I am against degrading the user experience to maximize profit.

Companies have figured out how to eliminate the first type of user entirely. They've weaponized updates to force compliance. Apps that won't launch without updating. Operating systems that update despite your settings. Games that require online connection to play single-player campaigns. Software that stops working if you don't agree to new terms of service.

The philosophy of "if it ain't broke, don't fix it" is dead. They killed it. And they can get away with it because of the network effect. We are trapped in it.

The Network Effect Trap

You use Windows because your workplace uses Windows. You use Excel because your colleagues use Excel. You use Slack because your team uses Slack. You use WhatsApp because your family uses WhatsApp.

When Windows suddenly requires you to have a Microsoft account (an online account) just to log into your local computer, what are your options? Switch to Apple? After twenty years of Windows shortcuts, file systems, and muscle memory? Switch to Linux? When you need to share files with colleagues who use proprietary Microsoft formats?

You can't. And they know you can't.

They're not competing on quality anymore. They're leveraging your professional dependency, your colleagues' software choices, your decade of learned workflows. You're not a customer who might leave if the product gets worse. You're a captive audience. This is why the hostility is possible. This is why they can get away with it.

Hostile Software

Enshittification, as Doctorow describes it, is a process of degradation. First, platforms are good to users to build market share. Then they abuse users to favor business customers. Finally, they abuse those business customers to claw back all the value for themselves.

But what we're seeing now is different. This isn't neglect or the natural decay of a profit-maximizing business. This is the deliberate, systematic removal of user agency.

You are presented with the illusion of choice. You can update now or update later, but you cannot choose to never update. You can see less often, but you cannot choose to never see it. You can accept all cookies instantly, or you can navigate through a deliberately complex maze of toggles and submenus to reject them one by one.

They borrow ransomware patterns. Notifications you can't dismiss, only snooze. Warnings that your system is "at risk" if you don't update immediately. Except once you update, the computer is restarted and you are presented with new terms you have to agree in order to access your computer.

Every Windows update that turns Bing back on and forces all links to open with Edge. Every app update that re-enables notifications you turned off. Every platform that opts you back into marketing emails and makes you opt out again.

Updates are now scary because they can take you from a version that serves your interest, to a version that services the company's. The update that adds telemetry. The update that removes features you relied on. The update that makes the app slower, more bloated, more aggressive about upselling you.

These aren't accidents. They're not the result of developers who don't care or designers who don't know better. They're the result of product meetings where someone said "users are rejecting this, how do we force them to accept it?" and someone else said "remove the 'no' button."

As a developer, and someone who has been using computers since I was 5 years old, I don't really care about the operating system. I can use them interchangeably. In fact, I don't care about Twitter, or any of these platforms.

When I log into my computer it's to write a document. When I use my mobile device, it's to talk to my friends or family. When I access my dev machine, it's to do my job. The operating systems or the platforms are secondary to the task at hand.

The software is supposed to be the tool, not the obstacle. But now the tool demands tribute. It demands your data, your attention, your compliance with whatever new terms it has decided to impose.

You can't switch because switching costs everything. Your time, your muscle memory, your compatibility with everyone else who's also trapped. The network effect isn't just about other people using the same platform. It's about your own accumulated investment in learning, customization, and integration.

So when they add hostile features, when they remove your ability to say no, when they force you to have an online account for offline work, when they interrupt you with notifications you can't dismiss, when they change interfaces you've spent years mastering, you can only accept it.

Not because you want to. Not because it's better. Because you have no choice. And that's not enshittification. That's hostility.

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