I occasionally find problems to solve or scroll back through my camera roll for screenshots of problems shared online. I use these problems not to keep my mathematical skills sharp, but to rekindle my joy for mathematics and the continuous wonder it provides my mind. Sometimes I get to do this through my work, and passion and joy get to intersect, but rarely do I find enough time to truly follow my nose – to get to, what I believe, is the essence of what it means to think and work mathematically… to explore my intuition.

I’ve written about this before, but the more I think about it, the more strongly I feel that there is not enough time and support for students to explore their intuition through mathematics at school. I sometimes wonder if debates about how students best learn mathematics eclipses how students do mathematics and I am left to question whether students ever do their own mathematics, or if it’s just always somebody else’s – answering questions that they didn’t ask and solving problems that have already been solved.

Let me try to explain what I mean with an example.
(as always, try it yourself first)

Consider two whole numbers (for example 3 & 6). These will be the first two numbers. The third number is the sum of the first two (9). The fourth is the sum of the previous two (15), and so on (3, 6, 9, 15, 24, 39, …). What do the first two numbers have to be such that the fifth number is 100?

Seed Numbers, featured on Peter Liljedahl’s website

This was a problem to solve for me. I didn’t know what to do, so I did my usual first step when I problem solve – I tried to make sense of it. I needed to know more about the problem, perhaps by finding a relationship or pattern that existed, something that might illuminate a path towards a solution. Since I had nothing to generalise from, I began specialising by trying some examples.

2 and 7
2, 7, 9, 16, 25, 41, 66

2 and 8
2, 8, 10, 18, 28, 36, 64

2 and 5
2, 5, 7, 12, 19, 31, 50

Hazaa! 50! Wait, I think I was supposed to get to 50 on my fifth number…
“What do the first two numbers have to be such that the fifth number is 100?”

You idiot. The goal is 100, not 50.

2 and 7
2, 7, 9, 16, 25, 41, 66, 107

2 and 8
2, 8, 10, 18, 28, 36, 64, 100

Wait, that’s my eighth number
Wait again! It could be my fifth number, if I count back!

2 and 8
2, 8, 10, 18, 28, 36, 64, 100

Take that, Peter Liljedahl.

Ok, is 18 and 28 the only correct response? I bet it’s not.

At this moment, my time was cut short by the cry of a baby who was supposed to be asleep for another 7 hours. So, as I rocked her back to sleep in the dark, I fired up Desmos on my phone. Game on.

Here was the correct answer I stumbled upon by chance:

18 and 28
18, 28, 36, 64, 100

Setting up two sliders on Desmos, I could quickly test out values to see what ones added to 100 on the fifth number.

Wait, what? 120?? Oh you double idiot. 18 + 28 = 46, not 36

So, it turned out I had no correct response, but with a baby still stirring in my arms and not ready to put down yet, I decided to venture onwards using my phone – brightness dimmed all the way down by the way #suchAGoodDad

Adjusting it to 8 and 28 seemed to do the trick.

8, 28, 36, 64, 100

Dragging the slider below 8 only seemed to decrease the fifth number below 100, and above 8 increased it above 100.

7 and 28
7, 28, 35, 63, 98

9 and 28
9, 28, 37, 65, 102

So I figured that 8 was the only solution for 28, but what about 29?

8 and 29
8, 29, 37, 66, 103

7 and 29
7, 29, 36, 65, 101

Ok, still a little too high. How about 6 and 29?

6, 29, 35, 64, 99

Ok, now it’s too low. 5 will only take it lower, but what if we increase 29 to 30?

Aha! 5 and 30
5, 30, 35, 65, 100

I noticed that the next one that worked decreased the first number by 3 and increased the second number by 2. 

8 and 28
5 and 30

So, my next attempt was…

2 and 32
2, 32, 34, 66, 100

This would mean that -1 and 34 would be…

-1, 34, 33, 67, 100

So my solutions so far were:

8 and 28
5 and 30
2 and 32
-1 and 34

Graphing these out on my phone (as I rocked my daughter back to sleep) looked like this.

If negatives weren’t freaky enough, I thought I’d try to figure out what it would be if the first number was 0. According to the line that was forming on my graph, it should be…

0 and 33⅓
0, 33⅓, 33⅓, 66⅔, 100
 

This all really got me thinking.. 33⅓ was quite significant as it’s ⅓ of 100. What if we were trying to get to 100 on the sixth number instead?

I adjusted my sliders and the first answer I found was 5 and 17

5, 17, 22, 39, 61, 100

Increasing the second number, I noticed that neither 18 or 19 worked.

4 and 18
4, 18, 22, 40, 62, 102

3 and 18
3, 18, 21, 49, 60, 99

2 and 19
2, 19, 21, 40, 61, 101

1 and 19
1, 19, 20, 39, 59, 98

Leaving 20, which was peculiar, because it only worked if I used 0 as the first number.

0 and 20
0, 20, 20, 40, 60, 100

This result also struck a chord with me – 0 and 33⅓ was a solution for the original problem, and 0 and 20 was a solution to this new one. In terms of 100, that’s ⅓ then ⅕. I was surprised it wasn’t 25, which would’ve been ¼, but this is how it turned out and I was interested to see what fraction might pop out if we wanted the seventh number to be 100.

On this occasion I was met with a little roadblock. Adjusting the slider for the second number led to the two following responses.

0 and 13
0, 13, 13, 26, 39, 65, 104

0 and 12
0, 12, 12, 24, 36, 60, 96

Which led me to try 0 and 12.5
0, 12.5, 12.5, 25, 37.5, 62.5, 100

12.5, which is ⅛ of 100, was also a bit surprising to me, and it wasn’t until I lined it up with the others that I realised what was going on…

⅓, ⅕, ⅛

I was so excited to test out my next guess, for when we wanted the eighth number to be 100.

Writing it out as fractions, however, lifts the veil.

0 and 100/13
0, 100/13, 100/13, 200/13, 300/13, 500/13, 800/13, 1300/13 = 100

As it does for the others:

0 and 12.5
0, 100/8, 100/8, 200/8, 300/8, 500/8, 800/8

0 and 20
0, 100/5, 100/5, 200/5, 300/5, 500/5 = 100

0 and 33⅓
0, 100/3, 100/3, 200/3, 300/3 = 100

The only thing left was to put my daughter in her cot and record my thoughts.

This was utter fun. What did I do? Sure, I solved the problem, well not really. I thought I did, but then realised I read the question wrong. Then, I thought I solved it again, but learned that I actually just read my answer wrong. Upon reflection, I think there were a couple key moments.

1. I didn’t pack it up at 8 and 28. A gripe I have with problem solving is that it’s too answer focussed. I don’t actually care too much if my answer is correct (or, evidently, if I’ve read the question correctly). I care more about the questions I ask myself along the way or after I’ve solved it. My main aim is to find something interesting, something I didn’t expect, something profound.

2. I explored my intuition and chased my curiosity. By exploring my intuition, I’m determining the path I take, following what piques my interest, and ultimately, finding joy. For me, joy is not finding out that my answer is correct when I check the back of the textbook. I find joy in the beginning – when I don’t know where my first few steps will take me. I find joy at the first plot twist, then the second, and if I’m lucky enough the third – when my intuition is proven wrong. I find joy at the end – when I look back and reflect.

Since leaving the classroom, perhaps my perspective of what it means to do mathematics has ventured even further away from the content-highway students and teachers are faced with in schools. I’m not suggesting that all problems lend themselves to such rich exploration, nor am I suggesting that this is the only way to do mathematics, but what I am suggesting is that all students get an opportunity to explore their intuition, chase their curiosity, and answer their own questions.

The first time I saw Michelangelo’s David in person, I wept. Never in my life have I witnessed a more beautiful piece of content.

I wasn’t the only one who felt that way. Museum patrons all around me were looking up from their phones to admire David as well. Despite it being legacy content, David was so scrollstoppingly captivating that it generated stronger engagement than TikTok and Instagram combined, despite those platforms offering fresher, more personally relevant content.

Imagine a pottery wheel that turns clay on an axis. A potter forms the sides and the shape goes all around. Karim Douieb made an interactive that shows how different curves look as a family of pots.

Potters don’t design bowls and vases, they design one curve. That single line, rotated around the axis, becomes the entire form.

In Line, you draw that curve. One side, shaped with a handful of control points, and the app grows a family of proportions around it: each variant keeping the radii and angles that make your line recognizably yours. Taller, wider, slender, grand: different vessels, same gesture.

Tags: curve, Karim Douïeb, pottery

You may have seen news reports lately about the ways in which AI is taking over what were once generally considered white collar duties. People are using it to write emails, to take notes, to compile presentations. Doctors are using it to record your symptoms, lawyers are using it to 'research' cases and real estate agents are using it for...everything.

Yet as more bosses begin to mandate AI's use in the office, and more professionals bring it ever more into their working lives (whether of their own free will, or because of company orders), have any of them ever stopped to wonder just what it is they do, exactly?

A white collar job does not generally involve manual tasks. It is, for the most part (and sorry if this sounds like an ad slogan), a job of the mind. From managers to HR, events to compliance, accountants to physicians, people are in those jobs and are getting paid for them because of their (supposed) ability to communicate with others, analyse things and interpret meaning from events. To draw upon their training, their experiences and their beliefs to do their work.

Let's say I'm an office manager or a HR rep, and my job is to settle a dispute between two workers. To settle that dispute is part of my job, it's why I am there. My skills at talking to people and conveying my thoughts and opinions via email are why I am being paid. If I open up ChatGPT and ask a machine how to do it--a machine that does not know how to do it, it only knows how to synthesise one million examples of this, because it is a machine--I am not doing that work anymore. Anyone could have asked the machine that question.

Or, if I'm a lawyer and my job is to prepare for a case, I am about to get paid a lot of money, because legal work isn't just hard, it requires a lot of training and has a lot of responsibility attached to it. If I start using ChatGPT to cut corners and research things for me, or even help me write things, why should I keep getting lawyer money if some kid off the street could have punched in the same prompts?

These people are hastening their own obsolescence, not because AI can do their work (it can't), but because by thinking it can they've shown themselves to be some of the dumbest people in the workforce, a collection of rubes who will throw decades of best practice out the window just because some Silicon Valley billionaire is jangling a shiny set of keys at them.

For all of the defence of AI from its boosters, cries that it's just a 'tool', it's anything but. We are years into this supposed revolution and AI is still hallucinating, making basic citation errors and failing to understand the nuance of a situation because it will never be able to. Robots might be able to do the work of a human arm, but the human brain is clearly a different story. 

A combine harvester that does the work of ten farmhands is a tool. A nailgun that lets a builder put a frame up in a fraction of the time it would have taken with a hammer is a tool. Whatever the effect of these technologies on labour or capital in those contexts, it is indisputable that those tools can do the exact same work, only faster.

A machine that tries to do your job and fails (in sometimes spectacular, embarrassing and even dangerous ways) but which you keep using anyway isn't a tool. It's a scam, one you and your bosses have fallen for. In racing to make use of it in an office job you're not optimising workflow or not getting left behind, you're showing that you either cannot or are now simply choosing not to do your job anymore.

Letting a machine write your emails or a talk or spin up a PowerPoint presentation isn't cutting corners, those tasks are literally what you're being paid for. That work is the work.

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Paige Wyatt heard plenty from administrators about using AI to generate lesson plans at the school where she taught high school English.

Her students used AI to cheat so frequently she resorted to spending her own money on detection tools. Yet she never got help with what she really wanted: how to effectively catch and stop the cheaters, some of whom were very clearly just copying and pasting from Google’s AI summaries.

Teachers like Wyatt consistently identify cheating as a major concern in the AI era, and 70% of teens report using AI for schoolwork, a Common Sense Media survey published last month found. But that concern is getting little recognition from states. While at least 13 states have passed laws about AI in education, the laws most often direct state agencies and districts to create their own AI guidance, meaning lawmakers have frequently absolved themselves from making tough decisions.

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Meanwhile, roughly three out of four states have published information to help guide AI use in education. But a Chalkbeat review found these documents tend to skirt the issue of cheating, offering little in the way of concrete steps to halt or deter students from offloading work to AI. Instead, they dedicate more real estate to AI literacy and ways teachers can use AI for efficiency.

All of that beating around the bush has left educators and districts to their own devices to come up with ways to deter and catch AI-enabled cheaters. It’s become an exhausting endeavor that hasn’t turned up any definitive solutions, educators and researchers said.

There’s no surefire way to detect AI-generated content. Nor is there an agreed-upon definition for cheating: Can students use AI to outline an essay but not write it? Can they use it to copy edit? Some states have crafted acceptable use policies which make some, but not all, of these distinctions.

Wyatt quit her teaching job last year. She didn’t leave solely due to AI, but its impact on her work was a factor.

“We really need real consequences,” she said. “We need workshops for teachers to detect cheating. We need workshops for kids.”

Most of the nation’s 10 largest school districts are developing or have developed more specific policies to try to prevent using generative AI to cheat. But it hasn’t always been easy: New York City delayed the rollout of its AI guidance for months amid fierce backlash, before announcing a new policy this week. Half of those districts explicitly discourage using AI detection tools.

Many of the official responses might underscore the urgency of the problem without helping to address it, which could only increase teachers’ anxiety.

“I think the situation teachers are in feels bad, and feels like they ought to have a solution now that they don’t have,” said Justin Reich, a professor at the Massachusetts Institute of Technology and the host of a podcast about how AI is shaping K-12 education.

Letting AI ‘run amok’ could ruin student learning

Historically, state policymakers haven’t rushed to create laws governing new technology and education.

States didn’t really start to adopt computer science requirements until the 2010s. And while the iPhone launched a smartphone revolution in 2007, the first state restrictions on cell phone use in schools didn’t come until 16 years later, in 2023.

States may be moving faster when it comes to mainstream generative AI, which took off with the advent of ChatGPT in late 2022.

“Folks can’t just hope this will pass. This is a major change, and students are already using it,” said John King, former U.S. education secretary and current chancellor of the State University of New York system. “This isn’t tomorrow’s problem. This is today’s problem.”

The District of Columbia and at least 37 states have issued recommendations for public schools about AI. Their documents range from short webpages to Illinois’ 400-page behemoth. Many define different types of AI, stress that any AI use should still be led by humans, and link to outside resources — some of which have been developed with the support of big AI companies.

But they often devote just a page or a few paragraphs to academic integrity.The documents also seem light on actual guidance and don’t include many citations, Reich said. That’s because there’s little evidence to show how effective AI tools are in the classroom, or what policies would effectively stop cheating.

“If you just let this run amok in secondary schools, all kinds of learning activities will stop,” he said. “We do not know what to do. There’s no evidence-based policy. There’s no basis for development and professional development. Like we’re just making stuff up.”

More than a third of states urge schools not to rely solely on tools that promise to detect the use of AI, citing studies showing they are not effective and incorrectly flag genuine student writing as AI.

But many states then offer little else other than acceptable use policies. In California’s case, the state offers a list of questions for districts to mull when making policies around integrity in addition to an acceptable use policy.

In the absence of hard and fast rules, teachers have reported requiring students to complete more graded work in class, using pencil and paper.

Christopher Harris, the director of the school library system for the Genesee Valley BOCES, a regional district in western New York, wants educators to go further than that and reconsider how they measure learning. Writing may be less of a valid tool to assess if students are learning, now that AI “can suddenly write anything.”

He said students could submit an oral assignment, or teachers could focus on the learning process more than the product by asking students to explain how they respond to feedback when developing a project or presentation.

New York is one of the 13 states that hasn’t published AI resources for educators. Harris’ library system has been developing resources about AI for the state’s educators using grant money.

“I really wanted to help teachers understand they can’t control what kids do with AI,” he said. “The only control they have is their agency in developing assignments.”

Adopting AI policies not straightforward for school districts

Sara Brescia thought she had a no-brainer policy proposal for her fellow Manassas City Public Schools board members: adding AI misuse to the district’s code of student conduct.

The policy she proposed for the district in northern Virginia — a hotbed for the data centers AI relies on — would prohibit students from trying to pass off AI-generated content as their own. It specified that AI could only be used with a teacher’s permission, and required students to cite AI use. Students would be punished, up to and including with expulsion, for violating any of those provisions.

But other board members voted her idea down in late July. Lisa Stevens, another school board member who is also a teacher according to her LinkedIn page, worried the policy would add a burden onto teachers without giving them any tools to identify AI-generated work.

Brescia is skeptical about the value of those tools.

“We do not have a comprehensive approach,’’ she said. “We have the student code of conduct that says nothing.”

Wyatt said when she left the classroom, students were only becoming more fluent in methods to cheat. They were finding tools like QuillBot to disguise AI-generated writing and make it sound more like student writing.

She’s open to going back to the classroom, if she gets more administrative support to help detect the kind of AI misconduct that felt out of control when she left.

“As a community, we need a support network,” she said.

Lily Altavena is a national reporter at Chalkbeat. Contact Lily at laltavena@chalkbeat.org. Lily is on Instagram, Tik Tok, and Facebook.

Wendell Berry, who died at his home in Kentucky yesterday, was a rare combination of poet and farmer. That wasn’t always an unusual thing in the literary world, but the authenticity of agricultural and pastoral poetry has been in decline since the days of Hesiod. Even Virgil was a bit of a poser. But Berry was the real deal in every way.

That would set him apart in any age, but especially in our own time, when so many of us (writers included) are cut off from nature and the holistic ecosystems of agrarian life.

To mark his passing, I’m sharing this article from my archive on Berry’s nine rules for technology. It’s a simple list, but a profound one—and even more relevant today than when it was first published in 1987.

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Back then, Wendell Berry was living on a farm in Kentucky. He did his writing with pen and paper, and his wife Tanya would create typewritten drafts of his manuscripts on a Royal standard typewriter purchased in 1956. That manual tool was, he insisted, “as good now as it was then.”

I’m sure he said the same thing about the implements he used to sow and reap. But friends told Berry he needed a computer. It would make it easier to write, they insisted.

In response, Berry came up with his list of nine reasons to embrace new technology. Let’s revisit them, one by one.

Nine Standards for Technological Innovation

(1) The new tool should be cheaper than the one it replaces.

This is a very persuasive selling point for new technology. And for most of my life, tech companies worked hard to lower prices.

I still recall my parents scrimping and saving in order to buy a color television when I was seven years old. It cost almost $500—a huge amount in those days.

They probably should have waited. A few months later, RCA dropped prices to $399. Prices continued to drop in later years. You can buy a high tech TV today at Best Buy for less than what my parents paid in the 1960s.

Computers also got more affordable—at least until recently. I got my first computer (an Apple IIE) when I was in graduate school—it was an expensive gift from the Boston Consulting Group in exchange for accepting their job offer.

The list price back then was $1,400. I could never have afforded to buy it on my tight student budget.

But, over a period of many years, each subsequent computer I acquired was better and cheaper than my previous model. Alas, that happy trend has now ended.

When I buy a new computer now, I pay more. And the performance is not always better. I recently had to scrap a new desktop after only a few months, and go back to my previous model. The new computer didn’t work as well as my five-year-old one.

When did new tech stop getting cheaper and better?

I have a cranky answer, but an accurate one. It happened the day Steve Jobs died. Maybe not exactly on that date—but shortly afterwards.

Look at this chart of iPhone prices, adjusted for inflation, and you can see what I mean.

Now let’s go to the second reason to adopt new tech from Wendell Berry’s list.

(2) It should be at least as small in scale as the one it replaces.

This is another good reason to upgrade your setup. And tech did get smaller for many decades.

Guess who played a key role in that? Yes, Steve Jobs again. Because of his obsession with product design, we now carry a huge amount of advanced tech in our pocket.

Just consider this remarkable fact: Every device featured in this Radio Shack advertisement from 1991 has been replaced by your tiny phone.

But this, too, changed soon after Jobs died. (Are you noticing a pattern here?)

The thinnest iPhone ever was the iPhone 6 (2014)—at a slim 6.9mm. The company continued to launch ‘mini’ models for a few years, but stopped after iPhone 13.

Tech is now bulking up. It’s not just the devices—wait until you see those AI data centers. A single facility can spread over two kilometers.

(3) It should do work that is clearly and demonstrably better than the one it replaces.

This is the most obvious requirement for new tech. It needs to work better than old tech.

But Silicon Valley has totally abandoned this ideal. Every web interface I use has gotten worse over time—from search engines to social media to software to shopping apps.

Google is worse than ever. Social media is worse than ever. Amazon is worse than ever. Facebook is worse than ever. Everything I get from Microsoft is worse than ever.

So here, too, we see that new tech previously fulfilled Berry’s requirement—but stopped doing so around the time Steve Jobs died.

(4) It should use less energy than the one it replaces.

Here, again, we see an ominous reversal. With the rise of AI, tech companies now use up more energy than ever before. They are sucking the power grid dry in many places.

And it’s going to get worse—much worse.

What makes this especially revealing is the fact the public intensely dislikes AI—surveys make this absolutely clear. So tech companies are destroying the environment solely to increase their dominance and control—not to please you and me.

(5) If possible, it should use some form of solar energy, such as that of the body.

Now Berry is asking for something our technocracy has never delivered. And here we encounter the exact opposite of the AI situation described above. The hot new plan is to construct nuclear reactors on site. What could possibly go wrong?

There’s another striking contrast here. AI depends on huge investment from corporations, while consumers are mostly indifferent. Solar energy is the opposite: It’s supported by investment from consumers—who use it to heat their homes, water, etc.—while corporations are mostly indifferent.

What a sad state of affairs. Private citizens have more prudent approaches to tech than the tech companies themselves.

(6) It should be repairable by a person of ordinary intelligence, provided that he or she has the necessary tools.

This, too, has changed during my lifetime. I once saw my father unscrew the back of our home TV set, and fix a malfunctioning part. Nowadays you can’t even open up those bad boys.

Tech providers create all sorts of obstacles to prevent repairs—unusual screws, arcane software, special tools, etc.

Consider the case of John Deere tractors, which wouldn’t start until a company-trained technician cleared out the error code. The company also refused to sell spare parts. Their practices got so abusive that politicians passed right-to-repair bills to protect farmers.

But the worst example happened during the COVID pandemic, when companies tried to prevent hospitals from fixing their malfunctioning ventilators. Manufacturers put software locks on this life-saving equipment to prevent repairs.

This represents a total failure on the part of the technocracy—and actual malfeasance by the executives who run these companies.

(7) It should be purchasable and repairable as near to home as possible.

Finally I can give some tiny credit to our tech titans. They do offer home delivery—even if the product is made in a sweatshop far, far away.

(8) It should come from a small, privately owned shop or store that will take it back for maintenance and repair.

This is a pipe dream. The tech product lifecycle is built on planned obsolescence, not simple repairs.

When your device or software stops working, you replace or upgrade—whether you want to or not.

In some instances, you aren’t even allowed to own, let alone fix, your tech—you just license or lease or subscribe. It’s almost like capitalism in Silicon Valley has turned into communism. You will own nothing, and will love it.

(9) It should not replace or disrupt anything good that already exists, and this includes family and community relationships.

This may be the biggest tech failure of them all.

The leading tech companies have deliberately promoted dysfunctional apps that destroy lives. And they know it.

• Leaked internal documents from TikTok show that they were aware that teens get addicted to their app in just 35 minutes. They built it that way.

• Facebook knew that Instagram use leads to depression, anxiety, eating disorders, and other problems.

• Spotify insiders have confirmed the company’s systematic plan to reduce royalties to musicians by manipulating passive listeners.

• For more examples, see my list of 52 indicators that technological progress is reversing.

This is the new normal for tech: It deliberately makes things worse, not better.

Here’s the entire list of Wendell Berry’s criteria. If this were a report card, your tech leaders would all get failing grades.

The curious fact is that the most up-to-date and forward-looking thing is this whole article is Berry’s list from 1987. Nothing on it is obsolescent or inappropriate or dysfunctional or harmful.

I wish our tech companies could say the same for their work. Maybe they could learn a thing or two from our poet-farmer—or at least take a look at his list. We can only hope.