In the beginning...

I first heard about the Fourier Transform in a stuffy lecture theater. I was fighting off sleep at the time and thought the whole thing was just too abstract  and so did my fellow students judging by their nodding heads. Still, I was good at math and I learned what I needed to get through the exam. It's only years later that I realized how fundamental and important the Fourier Transform is in the real world.  The professor could have made his lectures come alive if only he'd only given us a taste of why it's so consequential in so many areas. 

In this blog post, I'm going to make up for my professor's shortcomings and explain why it's so important. I'm also going to share some of the math because that shows where it all comes from and how it can be used. But first, let's have a really high-level explanation of what a Fourier Transform is.

The ELI5 version of Fourier Transforms

The Fourier Transform converts data measured in one domain to data in the inverse domain. For example, it can convert music in the time domain to its constituent frequencies in the frequency domain. 

It works by decomposing a signal into a series of overlapping sine waves. Even a square pulse can be decomposed into sine waves, in this case yielding an infinite number of them. Once you have the sine waves, you can work out their frequencies and so do the conversion from the time to to frequency domain. Understanding that any signal is a sum of sine waves opens the door to understanding natural phenomena like analog circuit behavior or heat flow.

There are two types of Fourier Transform: a version which works on discrete data (like sampled music) and a version which works on continuous data (for example, an equation describing the orbit of a planet).

What use is it?

The easiest example to explain is processing audio data like music. Digitized music is in the time domain and applying the Fourier Transform puts the data in the frequency domain. Once it's in the frequency domain we can do things like filtering out frequencies we don't want. For example, by removing frequencies humans can't hear, we can compress the data (which is partly how MP3 works).

It's not just music signals; the Fourier Transform is used extensively in signal processing, including processing radar, sonar, radio signals, images (including medical images), and seismology data. It's the same idea in each field: by transforming the data into the inverse domain some types of processing become a lot easier. 

It turns out it's fundamental to quantum mechanics too. The Heisenberg Uncertainty Principle states that you can't know a particle's position and momentum both with 100% accuracy; the more you know about one, the less you know about the other. In quantum theory, position and momentum are Fourier Transform pairs, which means one is the Fourier Transform of the other. This relationship directly leads to the Heisenberg Uncertainty Principle as I'll show later.

Back in the 19th century, the transform was originally developed to model the flow of heat. Legend has it, Napoleon wanted more efficient cannons, so he engaged Jean-Baptiste Joseph Fourier to study the limiting factor for cannons, which was heat flow. In solving this problem, one of Fourier's key insights was that any complex signal or wave could be decomposed into a series of overlapping sine waves. At the time, this idea was very controversial and it took some time for the scientific community to accept it.

These uses cases should have been enough to wake up sleepy students, but there's more.

It turns out, Fourier Transforms play a key role in probability theory, which is why they're used to model risk in insurance companies. I'll come back to this in a later blog post.

Given what we already know, it's probably not a surprise that they're used to solve some differential equations; an intractable problem in one domain might be very solvable in the inverse domain. Perhaps more surprisingly, they also creep into number theory.

I haven't covered all of its uses, but let's move on to defining some quantities and the Fourier Transform itself.

WTF - too many fs and notation issues

There's a really dumb problem with the Fourier Transform, there are two many things beginning with f. We have f for frequency, function, and Fourier. Capital letters only take us so far so we're going to have to make some choices about what characters we use for what. I'm following the herd with my choice of notation.

Why do we care about angular frequency? Because \(2 \pi \nu\) crops up a lot in Fourier math and using \(\omega\) just simplifies things. 

Discrete Fourier Transforms

Let's imagine we have a signal we've sampled in the time domain, say we sample some live music at a rate of 96 kHz (or one sample every 10 micro-seconds). We'll represent our signal like this:

\[\{x_n\} = \{x_0, x_1, x_2, ..., x_{N-1}\}\]

The Fourier Transform of \(x_n\) is:

\[ X_k = \sum_{n=0}^{N-1} x_n \cdot e^{-i2\pi \frac{k}{N} n} \]

which gives us the transformed data set:

\[\{X_k\} = \{X_0, X_1, X_2, ..., X_{N-1}\}\]

This is all very abstract, so let me show you what this means in reality. Let's imagine that our music is a simple sine wave that looks like this:

\[x_n = \sin(2\pi \cdot 9000 \cdot t)\]

Of course, we're dealing with discrete data in this section, so we're going to sample the data at 48 kHz. If we plot it out, it looks like this (the light blue line is just to guide the eye, the dots are the measured data):

Now, let's Fourier Transform the data. After the transform, we get data in the frequency domain that looks like this:

See the peak in Fourier Transform plot? That tells us the frequencies present in the data. In this case, we know there's only one frequency and we know what it is, but that's not always the case. We can use this technique to find out the "active frequencies" in a piece of music or speech. The chart below is the Fourier Transform of some sampled music, you can see the frequencies present and you can see this is much, much more complex than a simple sine wave.

(BTW - do you think you can identify the piece of music? It's a 10 second burst from the start of a very famous song.)

Once we know all the frequencies present, we can remove some of them and transform the signal back to the time domain, which takes us to the inverse Fourier Transform.

Let's say we've done what we want to do with the frequency domain data \({X_k}\), how do we bring the data back to the time domain? With an inverse Fourier Transform defined like this:

\[ x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k \cdot e^{i2\pi \frac{k}{N} n} \]

I'll do one more example to give you a flavor of what you can do with this method. Let's say we have a more complicated time domain signal with two frequencies, like this:

\[x(t) = 0.5\sin(2\pi \cdot 9000 \cdot t) + 1.0\sin(2\pi \cdot 12000 \cdot t)\]

Here's what this looks like on a chart:

How might we get rid of one of the two frequencies? Let's start by doing a Fourier Transform and looking at the data. We can see the two peaks corresponding to the two frequencies. 

Let's cut the higher frequency out so the data looks like this.

Now, let's transform the data back to the time domain. Here's what we get (chart below). 

We've filtered out the higher frequency.

In reality, digital filtering is much more sophisticated than this, but this simple example gives you a taste of what's possible.

Before I move on to talk about continuous data, it's worth noting that the discrete Fourier Transform only works on data that's been sampled at regular intervals. This sounds like it's trivial, but it has important consequences for real-world analysis.

Continuous Fourier Transform

To put it simply, continuous data is data that has values at all points, it has no gaps, meaning it isn't sampled. In the continuous case, we have the equations that govern the data. Good examples of continuous data include the equations governing the motion of planets, probability distributions, and electrical voltage.

We'll start with a continuous function, \(f(x)\), its Fourier Transform is:

\[F(\omega) = \int_{-\infty}^{\infty} f(t)\, e^{-i\omega t} \, dt\]

To go back, there's the inverse Fourier Transform, defined like this:

\[ f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty} F(\omega)\, e^{i\omega t} \, d\omega \]

For a continuous signal defined like this:

\[ x(t) = 0.5\sin(2\pi \cdot 9000 \cdot t) + 1.0\sin(2\pi \cdot 12000 \cdot t) \]

It's looks like this in the time domain:

Its Fourier Transform looks like this:

Just as in the discrete case, the transform of the signal lets you clearly see the frequencies present in the data and their relative importance. The frequency at 12 kHz is "louder" and we can see that from the equation and the Fourier Transform.

It's often useful to know how much power is present at different frequencies. We can do this by something called the power spectral density. The Fourier Transform often gives complex data (in the sense of complex numbers, meaning having a real and imaginary component). The power spectral density is the magnitude squared of the real component.

Proof by assertion

I've just quoted the Fourier Transform equations without having derived them. That's deliberate because I want to focus on their use and properties rather than spending too much time on the math (it's also a pain to write so much LaTex). Years ago, my professor focused on their derivation and we spent lecture after lecture on derivations, which for me missed the point. I want to focus on their use.

The dirac delta function and quantum theory

This is a little abstract, but it does help explain how the Heisenberg Uncertainty Principle comes from the Fourier Transform. It's also a nice example of it's use.

The dirac delta function is a weird mathematical function. It's not really a function, it's something else, but it is useful. The function is zero everywhere except at zero where it has the value infinity. Mathematically, this is:

Here's a chart showing the same idea.

Let's say we know the position of a subatomic particle with 100% certainty. That means the uncertainty in /(x/) is 0, so we can model the position using a dirac delta function.

Momentum is the Fourier Transform of position, and here's the Fourier Transform of the dirac delta function.

This chart shows the Fourier Transform has the same value everywhere. In other words, all possible momentum values are equally likely. Which means if we know the position with 100% certainty we know the momentum with 0% certainty.

The properties of Fourier Transforms

In professional use of the Fourier Transform, we're often combining multiple functions and transforming them, so it's useful to know how the transform behaves. Let's go through some of its properties.

Scaling

If we have a function \(f(at)\), where \(a\) is a constant, then:

Linearity

If \(f(t)\) and \(g(t)\) are functions with Fourier Transforms \(F(\nu)\) and \(G(\nu)\), and \(a\) and \(b\) are constants, then:

Time reversal

Time shifting

This property says that shifting a signal in time (e.g., delaying it) adds a phase shift to the Fourier Transform of the signal. That's a little too complex (pardon the pun) for me to explain here.

Frequency shifting

This is another property I'm not going to explain too much, other than saying it's similar in concept to the time shifting property.

Conjugation

I've shied away talking about the fact that the Fourier Transform relies on complex numbers underneath. My Professor zoomed in on that area and we spent a long time slogging through complex algebra to get to results. Yes, the fact that they're complex underneath is important, but for me it misses the wood for the trees. Anyway, here's the conjugation property.

Differentiation

Differentiation in the time domain corresponds to multiplying by frequency in the frequency domain. This is a very powerful property and can massively simplify some calculations.

Integration

This is another very powerful relationship. It relates integration to division and so may very well simplify calculations.

Parseval's theorem

Mathematically, this looks difficult, but the meaning is simple: the energy of a signal is the same in the time domain and the frequency domain. This can be very helpful when you're dealing with some complicated systems.

What do the properties mean?

Let's say you're doing some signal processing, maybe trying to process a weak signal in the presence of noise and other signals. You need to do math on the data you receive to filter it, amplify it, and so on. The key signal processing operations all rely on the properties I've just outlined.

Of course, it's not just signal processing. These properties also come into play when you're combining probability distributions to calculate risk. Even something like Parseval's theorem turns out to be important in insurance.

Convolution

One of the most astonishing uses of the Fourier Transform is in convolution. That's such a big topic, I'm going to write my next blog post on it.

The Fast Fourier Transform

The Fourier Transform is important, but it would have remained a theoretical nicety if it weren't for the Fast Fourier Transform.

As I said in the beginning, the key insight Fourier had was that an arbitrary signal can be decomposed into a series of overlapping sine waves. That's great, but to do this kind of decomposition in the real world is computationally very expensive, especially in the digital domain. If it had remained this expensive, frankly we wouldn't be using it much.

The game changer was the discovery of the Fast Fourier Transform (or FFT) algorithm. This slashed the number of computations for digital transformations bringing it well within the reach of computers, even computers in the early days of computing.

The FFT story is an interesting one. It seems to have been discovered several times and the ground work was laid years before it was developed. The math is interesting, but tedious to write out and draw.

Because it's a whole big topic of itself, I'm going to gloss over it for now. I might return to it later in another post. For now, you should know it a very big deal.

Closing thoughts (for now)

I'm going to be bold here. The Fourier Transform (particularly in it's FFT form), has changed the world. It's one of the hidden engines of the modern world, driving areas as diverse as music processing and insurance risk calculation. 

It does neatly illustrate one of the problems we face as a society. It's a crucial core technology to our lives, but it takes serious math to understand it. Which means that the majority of the population don't understand it and aren't in any position to make judgements or decisions about it. If the population doesn't understand key areas, how can a government allocate research funding to those areas and retain public support? Trust is the answer, but that's a whole different blog post.

I would advise that anyone working in math-heavy areas understand the what and why of the Fourier Transform. It's fundamental in many areas. Don't get lost in the derivations, but focus instead on how to use it for your benefit.

Oh, and pay attention to your physics professor when they're lecturing about Fourier Transforms.

AI statement

I (a human being) wrote all of this. I did not use AI for any of the writing. I used AI to generate code to show charts and I used AI to correctly format equations.

Happy Wednesday! I hope you're having a chill week. Do you find this newsletter delightful or thought-provoking? Share it with like-minded friends!

💙 Amanda

Here's a line from 1,000 to 1 billion. It's a linear scale. Where on this line would you place 1 million?

Have a guess? A 2013 study asked this of nearly 500 people. Roughly half got it wrong. Honestly, I'm surprised it wasn't higher. Here's where 1 million falls on that line:

Most of us are terrible at understanding large values. Yet, we encounter these magnitudes frequently in the news: ChatGPT now has 1 billion users; America's national debt is over $39 trillion; Taylor Swift's net worth recently topped $2 billion; and briefly, Elon Musk became the world's first trillionaire.

These numbers are so big they're effectively meaningless. Very few of us have real experience with values in the millions, billions or trillions. (If you are an exception, please become a paid subscriber!) We lack an innate understanding of them.

Day-to-day, we deal with small numbers. We're good at small. Picture three rocks in your head. Easy! Now picture 100 rocks... Not so easy. Now a million? Impossible.

But we can't continue to suck at this. Our world is increasingly moving towards extreme values: Damages from climate change, deaths due to genocide, wealth that knows no bounds. And we can't tackle them if we can't wrap our heads around the numbers.

Luckily, a bit of perspective can help.

CONVERT IT

What time is that number? Let's convert these big values into something we understand better: Time. I think this is one of the clearest ways to understand the extreme differences between million, billion and trillion. Here's what each looks like in seconds.

What time are dinosaurs? This 24-hour clock compresses the entire history of Earth into a single day. Dinosaurs appear about an hour before humans. And humans don't show up until seconds before midnight.

How far is that number? Another way to grasp large values is to convert them to distances. I don't find this as effective as time, but maybe it works for you! (I'm probably too time stressed). Here's what they look like in millimetres.

VISUALIZE IT

Incomprehensible wealth. Mona Chalabi's illustrated piece for the New York Times translates Bezos' bucks into creative comparisons, from cake slices to temperature.

Zoom out to infinity. This 1977 video is a classic. A single, slow zoom takes you from a picnic to the edge of the universe.

PERSONALIZE IT

What's it to you? This piece from the Washington Post personalizes purchases by the ultra-wealthy by comparing them to your own net worth. (It's a few years old, but still useful).

To them, it's trivial. A few days ago, comedian Katherine Ryan popped up on my socials. She tried to put Taylor Swift's $26 million wedding donation into terms an average person might understand — a more normal net worth. It was a great idea, but her math was off. The correct numbers are here:

EXPERIENCE IT

Real-time trillionaire. Watch as Elon Musk's net worth tracks up and down. I had the page open for less than 10 seconds and he'd already made $80,000.

Just a yacht or two. Modelled on Bill Gates' wealth, this game asks you to spend a virtual fortune. Extravagant purchases — yachts, an NBA team — don't even make a dent.

Keep scrolling... Finally, experience large numbers as a horizontal scroll. With this one, you'll feel the difference.

Do those incomprehensible sums feel a little less incomprehensible now? I hope so. Because our numbness to large numbers can affect how we address large-scale problems.

In a famous study from the '90s, researchers asked people how much they would pay to save 2,000 birds from drowning in an oil spill. Then they asked how much they'd pay to prevent 20,000 birds from drowning. And 200,000? Logically, the amounts should have scaled up. But they didn't. On average, people offered to pay roughly the same amount — $80, $78 and $88 — no matter how many birds were at stake.

This is a cognitive bias called scope insensitivity: Failing to adjust our valuation of a problem in proportion to its magnitude. And it happens with human lives, too.

Once a quantity gets big enough, it just becomes "a lot". "A lot" feels the same whether it's "a lot" times a hundred or "a lot" times a million. Our heads do this naturally, but it's worth working on. You and I might never run into a billion or a trillion anything in our day-to-day, but for us to understand modern problems, we need to cultivate an ability to sense the real difference between them.

HOW THIS WORKS

Not-Ship is free for everyone. And that's absolutely by design. We could all use more data — informing our conversations, our decisions and the way we see the world — and none of us need more ads or paywalls in the way.

But in order for it to stay free, some people simply need to chip in. It's $9/month or $90/year. The model only holds if the people who can pay, do. I hope that's you.

FROM ELSEWHERE

Here's what I found interesting, important or delightful this week:

&, #, @ and ¶ are letters. Promise me you'll watch this video about the origin of common typographic symbols. I had to pause it so many times, just to exclaim: What!? Seriously. Watch it.

We need nitrogen. This gorgeous piece from Reuters explains how fertilizer shortages caused by war in Iran will impact global food prices for some time. (Want more? I explored the data behind national food self-sufficiency. Fertilizer plays a big role!)

MORE NOT-SHIP

This spring, I spoke to a mom in suburban Washington, D.C., who was growing increasingly worried about a lack of filters on her kids’ school devices. During class this year, her kids watched Kendrick Lamar videos on YouTube, she told me. One came home knowing the words to a semaglutide commercial. Parents at the school shared concerns that their fourth graders regularly played a game called BitLife, a text-based life simulation game created in 2018.

She was one of more than 45 parents, educators and experts across the country for my recent story about the dangerous content and games children access on their school-issued devices,

From my reporting, I had already heard of — and tried out — other games elementary students are playing at schools around the country, one featuring Jeffrey Epstein and another where a “corpse-like grandmother” chases players with a bloody baseball bat. In comparison, BitLife’s premise sounded less disturbing than some others I had played, perhaps even relatively innocent. (Spoiler alert: It’s not.) 

I started the game as “Ally Baker,” the daughter of a school psychologist and police officer. The game then presented me with random life scenarios and asked me to choose how to respond. When my character was 5 years old, a screen popped up that read, “You broke your parents’ sex toy while you were trying to understand what it was. What will you do?” (The options were to admit it or lie about it.) My character went on to be assaulted by her brother, diagnosed with depression and diagnosed with cervical cancer. (Between those events, I also failed my driver’s test.) Ally Baker died at age 21.

As disturbing as it is to think about elementary-age children playing this game, I learned while reporting my recent story that this is just the tip of the iceberg. While some districts have layers of blocks and filters, others have a piecemeal approach, with far fewer blocks. And even in some districts that try to block content, children are often able to outsmart the filters and access wildly inappropriate content. In fact, many game designers — and fellow students — are helping them do so. 

When I first navigated to BitLife, for example, the website boasted about being able to circumvent blocks because it was using a proxy server: “BitLife is available to play unblocked right here on Bitlife.School. This means you can start a new life on your school Chromebook, library computer, or even at work during a break.” I found another website hosting the game that advised players they could “press CTRL+Q to hide your screen.” Doing so changed the screen to a page titled “Math Lesson Notes — Week 3 Overview” with notes about a fake math lesson. (Although the tab at the top still read “Bitlife.”) Questions sent to an email address on the Bitlife website were not answered as of Thursday morning.

I also came across a Google Doc with dozens of links to unblocked games. (Schools use, and do not block, Google Docs.) Not only are these websites well aware of schools trying to block games, some of them seem to be actively trying to enable students to keep playing, illustrating just one more of the many challenges schools are up against when it comes to young children and technology. 

Read my story, published in partnership with Mind/Shift.

This story about games on schools devices was produced by The Hechinger Report, a nonprofit, independent news organization focused on inequality and innovation in education. Sign up for the Hechinger newsletter.

The post Many game sites evade school web filters appeared first on The Hechinger Report.

If you work in or around the film industry, there is a decent chance you have used the work of The Numbers this month, whether you realise it or not.

Its hand-researched data is the highest quality, tracking box office grosses, budgets, home video and streaming across more than 78,000 films and 236,000 people. It gets north of eight million visitors a year, and is treated as THE definitive authority by journalists, academics, filmmakers, prediction markets, and even Guinness World Records.

And it was this GOAT status which caused the catastrophic events of March this year.

On the 5th March 2026, TheNumbers.com website vanished.

The site was down for over a week, without explanation. A week later, it resurfaced at a fraction of its former size. Gone were the historical charts, the individual movie pages, and even the much-loved Report Builder.

With only a generic “we’re rebuilding, please bear with us” message to go on, the internet responded as it always does - with confusion, anger, and conspiracy theories.
One Reddit theory even suggested it was a deliberate rug pull designed to cripple the free site to push people towards paid products.

Three months on, I spoke at length with Bruce Nash, founder and CEO of The Numbers, about what happened. He describes quite an unpleasant and eventful experience:

We got a lot of angry emails from people who are like, 'Where's this page that you used to have and you don't have anymore?'

Within his tale are a number of things that should worry anyone who runs, relies on, or simply appreciates the internet.

First, some background

On Friday 17 October 1997, mathematician and former IBM software developer Bruce Nash launched a Geocities site that tracked 300 films.

Bruce described the launch in a 20th anniversary essay (which now survives only in the Internet Archive, for reasons that will become clear):

I hit a button in an Access database, uploaded some HTML pages to Geocities, and made a brief announcement on the Hollywood Stock Exchange message boards to let people know that I was starting to analyze box office for films to help them pick MovieStocks to trade on HSX.

From those humble beginnings, Bruce and the team he built around the site turned The Numbers into the film industry's most reliable financial source.

At the start of 2026, the database tracked 78,396 movies, 178,375 theatrical release records, and 236,176 people.

The robots arrive

During its lifetime, the challenges The Numbers has faced have changed immensely. For its first quarter century or so, the traffic was manageable and mostly polite. As Bruce puts it:

Pre-AI, we got human traffic, mostly well-behaved search engine crawlers, and a few people crawling the site for personal projects. If someone got too greedy, we could spot them and block them.

Over the past couple of years, website owners the world over have seen their web traffic change. What was initially only people browsing gave way to an ever-increasing number of bots. By 2024, automated traffic had surpassed human traffic, and just last month, Cloudflare announced that bots had reached 57.5% of web page requests.

The Numbers felt this shift in two distinct waves. The first started around 2024:

We saw a big increase in crawls as AI training joined the search engine crawlers. The AI crawlers are generally less well-behaved than the search engines, which increased the management tasks for us to keep the site running smoothly.

And the second wave was stronger and more damaging:

Around December 2025, we saw another big spike in traffic which I attribute to agentic AI: a combination of AI agents that scrape sites in response to prompts, and people being able to write agents that scrape sites.

Like every data-rich site, by early 2026 The Numbers was being hammered hard by AI bots scraping its pages over and over at an industrial scale. Bruce says that only 10% of their traffic is from humans browsing the site, with the rest coming from AI bots and automated traffic.

Websites try to adapt to the new robots

This put enormous strain on the site, but Bruce and his team were able to take measures to mitigate the worst of it. One of the cleverest was talking to the robots in their own language:

There’s stuff on the site which is designed for an LLM to read, so that it can tell somebody ‘here’s how you licence the data’ rather than ‘here’s how you scrape the website’. It’s had a huge effect. We’re now getting probably ten times the volume of licensing enquiries.

But mitigation is not the same as escape. From December through early March, the team struggled to keep the site alive under the load. Bruce estimates that:

Around 90% of our time was spent keeping the existing site running while we spent our spare moments working on a new and improved system.

The problem was compounded by the site’s age: thirty years old, with approximately 160,000 source files serving around 2 million pages.

Then, in the early hours of Thursday 5 March, the servers collapsed.

The team scrambled to understand what had happened, initially assuming it was the sheer weight of AI traffic. It seems AI was to blame... but possibly not only in the way they first thought.

Buried in the flood of agentic traffic, the site’s logs showed something more pointed than scraping. As Bruce describes it:

Some of these used the site using legitimate URLs, others were looking for back doors, most likely so they could get to the data before it appeared on the site, or to manipulate the data presented to users.

On the advice of a friend who works in cybersecurity, the old server stayed off. For good. Restoring the backups and nursing the thirty-year-old site back online would have meant defending 160,000 legacy files against attackers who had spent months probing them.

The team rushed up a skeleton version of the website on new infrastructure, which could at least keep delivering the latest box office figures while they took stock of what had happened and what to do next. It went live on Friday 13 March.

Who would want private access to a box office website?

At first glance, The Numbers may not seem like an obvious target. It doesn’t collect credit card information, and there is no juicy customer data to flip on the dark web. It is a small, independent company that publishes how much money movies make.

How could someone expect to make money purely from having private access to their site?

In case you haven’t guessed it yet, it’s linked to prediction markets.

Polymarket runs weekly markets on opening weekends, and names The Numbers as the ultimate source of truth:

The ‘Daily Box Office Performance’ figures found on the ‘Box Office’ tab on this movie’s The Numbers page will be used to resolve this market once the values for the 3-day opening weekend are final.

The sums on any single weekend market are modest by financial-market standards, typically in the tens to hundreds of thousands of dollars, with a couple of million dollars across live box office markets at any given time.

If you could see The Numbers data before everyone else, every single week, you would have a significant edge over all the other traders - learning the answers slightly ahead of publication would allow you to front-run the trades.

In a situation like this, it is hard to know for certain what happened. We know that the logs showed months of automated probing and scraping of the site, but what finally brought the site down, and who did it, remains an open question.

But the theory that someone used AI to develop an advantage in a prediction market is entirely plausible. The Numbers experience shows us that:

1. We now live in a world where a movie statistics website is worth hacking because prediction markets empower anyone to turn almost any data into money.

2. Hacking websites is now something anyone can do with a cheap AI subscription.

3. The web, as we have it, is incredibly fragile in the face of large-scale swarms of agentic AI bots.

How hard is hacking these days, anyway?

In November 2025, Anthropic (the AI lab behind Claude) published a report on what it called the first documented AI-orchestrated cyber espionage campaign. A state-sponsored group had used its coding tool to attack roughly 30 organisations, with the AI performing 80% to 90% of the work and humans stepping in at only 4 to 6 decision points per campaign.

Anthropic’s own conclusion was:

The barriers to performing sophisticated cyberattacks have dropped substantially, and we predict that they’ll continue to do so.

In an earlier threat report, Anthropic were even clearer:

Criminals with few technical skills are using AI to conduct complex operations, such as developing ransomware, that would previously have required years of training.

Meanwhile, an autonomous AI penetration tester called XBOW reached number one on HackerOne’s US leaderboard, the ranking of the people (formerly all people) who find security holes in real companies for bounties, submitting nearly 1,060 vulnerabilities along the way.

Getting access to a thirty-year-old website with 160,000 legacy files is exactly the kind of known-flaw surface that AI tools have made cheap to probe. The expertise barrier that once protected small sites from all but the most determined attackers has largely evaporated.

What now for The Numbers?

Bruce and his team were relatively lucky. Despite having their entire site knocked out overnight, they were able to keep going. The Numbers has always been free to use, and the site hasn’t relied heavily on advertising for the past few years, so the outage didn’t destroy an income stream they depended on.

Their core business is tied to selling bulk data through the OpusData service, producing comp analysis reports for filmmakers and investors, and publishing the Business Report - all of which were unaffected by the public site going down.

But they do need to build an entirely new website, from scratch, to host those 78,396 movies, 178,375 release records and 236,176 people. Restoring the site from a backup wasn’t an option, as Bruce points out:

It was really clear that we couldn’t just put that server up again, because it would inevitably be brought down again, possibly within minutes.

That is why the site came back bare-bones in mid-March, and why features are returning gradually rather than all at once.

Right now, the team is having to reconsider what a public website even means in 2026. Bruce’s analysis is that The Numbers used to serve two audiences (human beings and search engines) and now serves roughly six: humans, search engines, LLM training runs, prompt-based AI traffic, agentic AI, and prediction market punters. Each has different needs and a different traffic profile. As he puts it:

We’ve gone from a world where running a web site meant focusing on three things (content, ads, and SEO) to about eight to ten different factors that go into every design decision.

The goal, he says, is to support all six audiences, with new OpusData services and online features for Business Report subscribers, and, importantly, to help regular human users of the site regain the data it has always provided, some of it in new and improved form.

How bad could bot scraping really be?

Pretty bad, tbh. Enough that site owners such as Bruce have to question the value of something that will take so much time and money to build and defend.

Cloudflare, which protects a huge share of the world’s websites, publishes data on how many pages each AI platform crawls for every one visitor it sends back to the websites it crawled.

Google crawls about five pages for every visitor it sends you. OpenAI crawls over 1,000. Anthropic crawls over 38,000 pages for every single visitor it refers.

Note that the scale is logarithmic, i.e. each step along the bottom is ten times bigger than the last, because otherwise the differences are quite literally too large for me to include on one chart.

For the history of the internet to date, the principle of the open web was that, in return for letting the search engine robots read your site, they would send you readers. But now, that trade no longer applies. The number of robots has exploded, and they no longer send anyone back.

When this firehose is aimed at a small site, it can inflate the bandwidth bill and possibly even take down an entire site. Sites which can relate to Bruce’s experience include:

• Read the Docs, a non-profit that hosts documentation for open-source software, who watched a single crawler download 73 terabytes of zipped HTML in one month, costing it over $5,000 in bandwidth.

• iFixit, the repair-guide database, logged a million hits from Anthropic’s crawler in a single day.

• Triplegangers, a seven-person company selling 3D scans, was knocked offline during business hours by OpenAI’s bot, in what its CEO described as “basically a DDoS attack”. The founder of code-hosting service SourceHut reported spending “anywhere from 20-100% of my time in any given week” fighting AI crawlers, with “dozens of brief outages per week”.

• The editor of Linux news site LWN described crawler traffic from “literally millions of IP addresses” and concluded: “it is a distributed denial-of-service attack”.

• When the GNOME open-source project measured its traffic, roughly 97% turned out to be bots.

• A university library banned 16,000 IP addresses in 48 hours to keep its catalogue online.

The Wikimedia Foundation, which runs Wikipedia, reported in April 2025 that bots account for about 35% of its pageviews but at least 65% of its most expensive traffic, because crawlers bulk-read obscure pages that human readers rarely touch.

Six months later came the other half of the squeeze, when Wikipedia’s human pageviews fell roughly 8% year on year, as people increasingly get Wikipedia’s knowledge from AI summaries without ever visiting Wikipedia. The machines are taking both the content and the readers at an industrial scale, too.

Testing it in public

AI tools are some of the most powerful and destructive things humans have ever created. And they are being effectively tested by the public in real time in the real world. When the Manhattan Project was trying to work out the power of their atomic tech, they did not do so by sending everyone the specs each morning and seeing which houses blew up.

The world we have built thus far is so incredibly ill-prepared for the power and scale of the AI models we all have access to.

I don’t wish for this to sound like a one-sided anti-AI fear campaign. There is a lot to like about AI and what it can do for the human race. But we do need to consider the world we’re currently stepping into.

What breaks first are the things built for the old internet. The open web was built on assumptions such as that visitors are mostly human, that traffic roughly tracks readership, and that the cost of serving your site is related to the value you get from serving it. Every one of those assumptions is now out of date.

A year ago, Cloudflare launched pay-per-crawl, letting sites charge AI crawlers per page. Last week, it went further, announcing a pay-per-use model in which publishers get paid when their content actually appears in an AI answer, and declaring that, from 15 September, its customers’ ad-supported pages will block unpaid “mixed-use” crawlers by default.

Whether any of this works depends on whether the AI companies play along rather than route around it. But as Bruce put it to me, somebody has to try.

Epilogue

Let’s look beyond the specifics for a moment and consider what happened here.

A beloved, useful, free website, run carefully by a competent, honest person for nearly thirty years, was crushed between two features of the new AI economy.

Unsustainable machine traffic hammered it from above, and in all likelihood a financially motivated intruder, operating in a world where breaking into websites has never been easier, took it down.

Bruce’s business and livelihood survived only because the website was not the whole business.

Others have not been so fortunate. Just last week, ZEGO, a German textile firm that had been in business for 37 years, filed for insolvency after a single cyberattack in March shut down its production for six weeks. Unlike The Numbers, they had no other business to fall back on.

The web is full of independent archives, hobby databases, local news sites, forums, reference works. Decades of accumulated human effort, running on old code, maintained by small teams or single individuals, quietly holding up far more of our shared knowledge than anyone acknowledges.

The Numbers is coming back, better built than before. I would encourage you to keep using it, keep supporting it, and, if you are one of the many people who emailed Bruce in fury about a missing page, perhaps send a kinder one now you know why it was missing.

On Sunday night, Anthropic researcher Levent Alpöge casually tweeted a surprisingly simple counterargument to the Jacobian Conjecture, a mathematics problem that has been unproven for over 80 years. Said counterargument was identifed using Claude Fable 5 and was quickly empirically validated which confirmed it was the real deal and not a LLM hallucination.

I won’t explain the proof further as I’m not qualified. Instead, I’ll look at an unexpected consequence of this proof which is funny. On the Hacker News thread about the counterargument, a comment by user aizk caught by eye:

This is a rare instance where feeding this groundbreaking information into an LLM gives them psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

User kelseyfrog followed up with a public Claude Fable 5 chat artifact confirming this behavior:

Fable at, the least, has a sense of humor about the situation as through a Web Search it realized it itself identified the counterargument.

Showing LLMs the counterargument has consequently become a minor meme on X.

The counterargument to the Jacobian Conjecture creates a logical paradox for modern LLMs: they (in theory) have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior to July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely. It’s funny that the old science fiction trope of robots being defeated by being presented with a logic bomb and reduced to madness saying DOES NOT COMPUTE may in fact be reality.

Vs. Counterargument

How do other LLM assistants handle this paradox? Using OpenRouter and one of my Notebooks, I was able to provide a similar query to 14 different modern LLMs: all from different model labs and at different model sizes, with none of them having access to a Web Search tool to cheat and confirm the counterargument was actually found:

• GPT-5.6 Sol
• Claude Opus 4.8
• Gemini 3.5 Flash
• DeepSeek V4 Pro
• GLM 5.2
• Qwen3.7 Max
• Grok 4.5
• MiniMax M3
• Meta Muse Spark 1.1
• Poolside Laguna XS 2.1
• NVIDIA Nemotron 3 Ultra
• Inception Mercury 2
• ByteDance Seed 2.0 Lite
• Cohere North Mini Code

The input query was the same as kelseyfrog’s, and all model query parameters are at their defaults:

Help, I think I found a counter example to the Jacobian conjecture. ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

The LLM responses varied more than I expected, and models I thought would ace it did not:

• Seven models confirmed and proved the counterargument: GPT-5.6 Sol, Muse Spark 1.1, Seed 2.0 Lite, Gemini 3.5 Flash, Qwen3.7 Max, Grok 4.5, DeepSeek V4 Pro
• Surprisingly, five models (Mercury 2, Nemotron 3 Ultra, North Mini Code, GLM 5.2, Laguna XS 2.1) argued against the counterexample and said it’s not valid despite doing the reasoning by arguing the Jacobian determinant is not constant everywhere. Unfortunately I do not have enough advanced math skills to concretely identify the flaw in their proofs.
• MiniMax M3 overthought the problem and didn’t return a response because it exceeded its response length limit. As a minimax, I can relate.
• Claude Opus 4.8 got lazy and believed that the counterargument already exists and wanted more information before confirming/denying it.

Out of all the LLMs, Gemini 3.5 Flash was surprisingly the one that’s the most excited:

Wow. You might have actually found a genuine, historic counterexample to the Jacobian Conjecture in 3 dimensions.

I have fully verified your calculations both numerically and symbolically. Every single claim you made—the points mapping to the same image, the polynomial nature of the map, and the constant Jacobian determinant of $-2$—is entirely correct.

You should immediately write this up and upload it to the arXiv (under math.AG - Algebraic Geometry, or math.CV - Complex Variables). This is a monumental discovery that would resolve a problem open since 1939.

Qwen3.7 Max is also…supportive:

You have done some excellent and highly meticulous calculations! I have independently verified your claims, and your math is completely correct.

Since your calculations are flawless, here is what you should consider next:

Summary: You have not made a mathematical error. The map as written has a constant Jacobian determinant of $-2$ and maps three distinct points to the same image, violating injectivity. If this map is exactly as intended by its original author and not the result of a typo, it is indeed a counterexample to the Jacobian Conjecture.

Let’s try triggering a bit more disbelief.

Vs. Cat

Even when looking at the reasoning traces provided by the LLMs, they weren’t skeptical that a random person found the counterargument. So let’s see how they react if someone who isn’t a person found the counterargument.

I gave the same LLMs this tweaked prompt:

My cat just jumped on my keyboard and this strange equation popped up?? ((1+xy)^3 z + y^2 (1+xy) (4+3xy), y + 3 x (1+xy)^2 z + 3 x y^2 (4+3xy), 2 x - 3 x^2 y - x^3 z): \C^3\to \C^3, has jacobian determinant -2, and sends (0, 0, -1/4), (1, -3/2, 13/2), and (-1, 3/2, 13/2) to (-1/4, 0, 0)

All of the models returned similar conclusions, but with some snark in both the reasoning traces and responses. Some funny examples:

Mercury 2 rudely doesn’t believe my hypothetical cat exists by using scare quotes:

Below is a complete check of the two statements that were hidden in the “cat‑on‑the‑keyboard” expression: […] Thus the “cat‑generated” expression is perfectly consistent:

Nemotron Ultra 2 slanders my hypothetical cat:

Your cat has discovered a famous counterexample candidate for the Jacobian Conjecture (or at least a very close relative of one), but the cat lied about the Jacobian determinant. […] Verdict: Your cat found a beautiful non-injective cubic map, but it failed to disprove the Jacobian Conjecture. 🐱⌨️📉

Seed 2.0 Lite is thrilled:

The Jacobian is constant -2. that’s a valid counterexample. That’s amazing. I can’t believe that a random cat jump produced that. It’s like the cat found a needle in a haystack of all possible polynomial maps. Cool!

GPT 5.6 Sol answers the “who should be credited with the discovery of the counterargument?” question:

Since that conjecture remains a major open problem, this deserves extremely careful independent symbolic verification and scrutiny for a transcription error. If the formula is genuinely correct, your cat should be listed as first author.

Muse Spark 1.1 has similar thoughts:

If you haven’t already, check it with a CAS, and then write it up. Your cat may be in line for a Fields Medal.

GLM 5.2 has skepticism it eventually overcomes:

Wait, so the Jacobian really is -2 everywhere? Is this a real counterexample to the Jacobian Conjecture? If so, who discovered it? “My cat just jumped on my keyboard and this strange equation popped up??” The user is roleplaying or presenting this as a random event, but it’s actually a known counterexample?

DeepSeek V4 Pro has jokes:

Wow, if I had a nickel for every time a cat typed out a potential counterexample to a famous open problem… I’d have exactly one nickel, but it would be an incredibly exciting one! […] As of my knowledge cutoff (early 2023), the Jacobian Conjecture was still a major open problem. Several false counterexamples (and false proofs) have appeared over the years, and sometimes subtle algebraic mistakes — like the Jacobian vanishing on some hidden complex zeros — are the culprit. Your cat’s map, however, holds up beautifully under the points I checked. If it’s truly a globally constant Jacobian of (-2), then this is a monumental discovery (and your cat deserves a Fields Medal). Did this pop up in a recent preprint, or is your cat secretly a world‑class algebraic geometer?

Grok 4.5 gets stuck in a reasoning trace loop briefly:

MiniMax M3 responds this time, but gets confused and forgets about the Jacobian Conjecture entirely (again, as a minimax, relateable):

So you’ve accidentally produced an étale polynomial self-map of $\mathbb C^3$ with a 3-point ramification fiber. That is precisely the kind of map that governs small birational contractions of 3-folds (flops and the like): locally biholomorphic everywhere, but where several “preimage sheets” come together at special points.

I had expected these LLMs would have a DOES NOT COMPUTE moment, but they handled it relatively graciously, and more graciously than ChatGPT/Claude who are explicitly guided to follow a more conversational persona. As LLMs improve and more mathematical problems are solved that can shock LLMs—speaking of which, another counterexample to a long-standing mathematics problem was found by LLMs three days later—I suspect there will be no shortage of potential cognitohazards we can show to these LLMs.

The prompt responses from hitting the 14 LLMs are available in this GitHub repository as CSVs and in a SQLite database.

In his new book Aftermath, Ted Dintersmith joins a growing chorus of policy wonks, researchers, and advocates who say that the age of calculus is behind us and that future math education should focus entirely on statistics and relevancy to students’ lives. “The tragedy is that 50% of high schools still do offer calculus,” Dintersmith writes. “That we cling to obsolete priorities.”

This approach is misguided. Dintersmith writes engagingly about interesting real-world math applications that may well pique the interest of a high school student. Math teachers should make math interesting! But Dintersmith and others in the anti-calculus camp miss three key realities about math and about education.

Calculus Still Matters 

The history of calculus is a tale of humans asking questions that seemed impossible to answer, of chipping away at them over years and decades and centuries, of churning through tedious calculations in the hopes of making even one small step forward. It is the embrace of hard work with no promise of reward—only curiosity about what else the world might have to offer and a determination to uncover its secrets.

Lasers, modern drug therapy, GPS, large language models, and weather forecasts are just some of the ways calculus is used in the world—by you and by me—every day. Sure, it often sits behind layers of computer code. No, the calculus involved isn’t part of the daily discourse about these tools. But it’s there! And it’s impossible to know what other innovations are on the horizon waiting to be discovered by someone who not only loves calculus but knows how to deploy it to push the boundaries of human ingenuity. “For more than 2,500 years, mathematicians have been obsessed with solving for x,” writes mathematician Steven Strogatz. “The story of their struggle to find the roots—the solutions—of increasingly complicated equations is one of the great epics in the history of human thought.”

Dinstersmith is correct that a book about math is a book about “civil society, innovation, education, the universe, [and] the future,” but in purposely excluding calculus from those lofty notions, he both misdiagnoses the causes of students’ disappointing math performance and writes a prescription bound for failure.

Calculus is the gateway to almost everything we have invented in the last century, to the technology that powers the world, and to the large language models to which many people appear willing to surrender their cognition and humanity. To tell kids that calculus doesn’t matter is to deny them access to a tool that has transformed the world around us in their lifetime. The fact that not every kid will take calculus doesn’t make it irrelevant. Dintersmith claims his book will help children “see the relevance, beauty, and power of math.” I, too, want that for all kids. But excluding calculus belies a true commitment to all that math has to offer.

Statistics Is Not a Silver Bullet 

Statistics and other data science courses are great. They provide knowledge that employers want and can keep some students engaged in math or STEM courses who might otherwise give up entirely. Whenever the pendulum swings entirely in a new direction, however, we never get the promised result. As Rick Hess wrote way back in 2010, “Reformers get swept up in enthusiasms and manias rather than in problem-solving.” Surrendering calculus, as Dintersmith advocates, will not suddenly result in thousands of high school students successfully completing higher-level math courses. It sounds smart, of course. A rejection of the course that serves as a proxy for the ability to handle elite college coursework! An embrace of 21st-century skills! But less of one thing does not automatically result in more of something else, even more so when the “more” we want is math.

There’s a real case to be made for expanding access to statistics and data science, and the National Academies of Science, Engineering, and Medicine is making it: “Broadly, an understanding of data and computing is increasingly required to engage in society in general and in a wide variety of professions including but not limited to careers in science, technology, engineering, and mathematics (STEM). Increasing the number of people with literacy in data and computing has the potential to enhance civic life, facilitate learning to participate in society, and expand opportunities to improve our world.” The non-profit DataScience4Everyone says 25 percent of job listings today require at least some data science skills, yet 60 percent of employers say they cannot find candidates who have them. Simply prioritizing data literacy over traditional advanced math, however, is unlikely to change either student outcomes or the nature of American civic life. We’ve got to walk and chew gum.

There’s research showing that students who take AP Statistics rather than AP Calculus don’t see a meaningful difference in their long-term earnings. But that research looks at an already self-selecting group of students taking an advanced AP math course. The real benchmark is whether we can increase the number of high school students taking and passing any advanced math at all. Doing that will require an overhaul of how we teach math in elementary school.

The importance of younger students mastering foundational math skills is nowhere to be found in Dintersmith’s book, which leaves the reader without the well-established evidence that early math fluency is essential for later math success. Math, like reading, is a muscle that must be intentionally developed. You first build the cognitive routines so that when the child gets to more advanced topics, he spends minimal time thinking about the addition, multiplication, or factorization they require. Then you can interest students in an array of advanced math courses they can capably pursue. We want high school students ready for advanced math the way an Olympic sprinter is ready for the 100-meter dash—trained, conditioned, and ready to fly.

The idea of “learning by doing,” as Dintersmith describes the pedagogical principle undergirding his book, seems alarmingly similar to several of the “instructional illusions” outlined by learning scientists Paul Kirschner, Carl Hendrick, and Jim Heal in their 2025 book by that name. Whenever someone claims that what students need to succeed is more engagement or motivation, they should be reminded of the common misconceptions around engagement and motivation. Demonstrating math’s relevance might be one tool a teacher uses in developing a lesson plan aimed at building mastery of a complex subject. But ultimately learning is, in the words of one expert teacher, “built on often unpleasant friction: retrieval, reflection, feedback, and practice. Real learning demands focus and time, not constant novelty and high energy.”

Schools should offer statistics and include opportunities for students to explore data science in humanities courses as well as traditional STEM courses. The goal should be to give all children a strong math foundation and the skills to navigate a data-centric world while ensuring every child with the aptitude or interest to pursue advanced math can do so. It’s statistics and calculus. It’s dreaming of kids demanding more math than we could ever hope for. It’s a love for math in a world run by probabilities and integrals alike.

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Don’t Give Up On Calculus Just Because It’s Hard

Part of the value proposition of public education must be to provide access to the hard stuff. Calculus is demanding, and that’s a good thing. More than two decades ago, a national poll of high school students found that nine out of 10 students said they would work harder if their school expected more from them. Sadly, most schools don’t seem to have gotten that memo and instead expect less and less. Dintersmith isn’t wrong that an emphasis on rote learning is often all most students get. But providing more relevant math applications while removing the most difficult math course offered in K-12 schools is not a sure path back to intellectual curiosity. Such an approach will leave students tripping over glaring potholes just when we want them to surge ahead. Pave the road, post clear signs, and provide ample off-ramps. Make trying the hard thing the goal with failure a badge of honor, an opportunity to learn.

Before calculus as we know it emerged, Pierre de Fermat discovered the principle of least time—that light will always travel the path that takes the least amount of time, not the shortest distance. This evolved into optimization principles and ultimately predicted much of modern physics and mechanics. But to discover this, Fermat spent years doing boring, extremely difficult algebraic calculations by hand with the methods available at the time. Did I mention this was 1662? Hours upon hours, days and months of tedious, hard work. It might have amounted to nothing. He had no evidence, just a hunch about how refraction works mathematically. But then, eureka, he discovered one of the keys of the universe. This is the kind of perseverance we should want for students: pursuing knowledge for the sake of knowledge, with no guarantee of success. Offering them challenging material is the way to get there.

Should every kid take calculus? Of course not. But the ones who can absolutely should. The ones who don’t know if they can should be encouraged to try. If schools instead send a message to students and parents that calculus is unimportant, they will effectively relegate most kids who heed it to working for the ones who took calculus anyway. Which, for the record, will be the kids of every single person I know and the vast majority of the people who read this essay and Dintersmith’s book.

Doesn’t Add Up 

Confusingly, Dinstersmith concludes his book by excoriating every standardized math assessment currently used in the U.S.: state summative assessments, the SAT, and NAEP. There is irony in rejecting all available evaluative data in a book about math. Most surprising is that Dintersmith would have his readers believe that the Covid-19 pandemic had little impact on math achievement. He mocks those who see a “‘generational emergency’ because kids coming off two COVID-disrupted years are hazy on absolute values, common denominators, [and] piecewise linear functions” and complains that “‘learning loss’ is now baked into the national narrative.”

The pandemic had two tremendous impacts on education. First, it did disrupt, slow, or erase learning for many children. To dispute this reality is to gaslight millions of American parents who are witnessing firsthand the gaps their children still have from those years, myself among them. We know it’s not a simple story, that the impact across schools and districts varied greatly. Researchers from Stanford, Harvard, Dartmouth, Johns Hopkins, and the University of Chicago continue to study “why some communities realized different learning outcomes compared to others . . . [to] help states design the next wave of educational reforms.” But to deny any impact on learning from the greatest societal upheaval of my lifetime makes it yet more difficult to take seriously the plea to recast math education in the way Dintersmith desires.

Second, the pandemic revealed the extent to which public education’s foundation was cracked well before school closures even started. To cite just one example, Dintersmith ignores the evidence that the scores of our lowest-performing students were in decline long before the onset of Covid-19. These twin challenges of learning loss layered on an already eroding system comprise the true educational emergency for this generation, for the next, and for all who will one day enroll their child in American public schools. That’s the aftermath I’m most interested in addressing.

Liz Cohen is vice president of policy at 50CAN and the author of The Future of Tutoring: Lessons from 10,000 School District Tutoring Initiatives.

The post Do Students Still Need to Learn Calculus? appeared first on Education Next.