In James Bell’s math class at Chapman High School, sophomores are trying to pinpoint exactly where two lines cross.

The students in this rural Kansas high school already solved for that meeting point in previous lessons, using graphs and other techniques. But this recent lesson shows them how to use a matrix — a box made up of rows and columns representing a system of equations. Matrices are often used in engineering or video game design to calculate the locations of two objects moving through space. 

Traditionally, this kind of complex linear algebra would be taught in the third year of high school math, when many students would take Algebra II. But a decade ago, the Chapman Unified School District, a rural district about 80 miles west of Topeka, Kansas, decided to drop the traditional high school math pathway — Algebra I for a year, followed by a year of geometry and then a year of Algebra II — in favor of what is called “integrated math.”  

Integrated math takes concepts from both algebras and geometry, plus a little from trigonometry — the study of triangles and angles — and blends them over multiple years instead of teaching them separately, a year at a time. That means students can move from lessons in geometry to algebra and back again within the same year. 

Bell, who helped write the math curriculum that Chapman High School uses, said he has seen how practicing both algebra and geometry throughout the year keeps them fresh in students’ minds.

“You’re going to have the opportunity to change course and change direction and see different things,” Bell said. “Students even do a little trig — which is a scary word for kids, but when it’s integrated into every year of math, it doesn’t sound as scary. It doesn’t make it as overwhelming.”

He added: “This is better for students. This is the best of both worlds.”

Chapman is part of a small group of districts and states moving to integrated math — all part of a larger movement to reimagine secondary math, give high schoolers more choice in courses, and modernize what some say are outdated ideas about what constitutes rigorous, college-level math by expanding course options beyond just calculus to include data science and statistics. 

Some experts say blending algebra and geometry gives more students a shot at learning high-level math later on in high school and college, touting high-achieving European and Asian countries that have taught integrated math for decades.

But others say the integrated approach creates new problems. Some teachers have objected, saying they prefer to focus solely on algebra or geometry: Georgia mandated an integrated math approach in 2008 but made it optional in 2015 after teachers and parents complained. Others worry that integrated courses sometimes have to drop concepts that would be taught in the traditional math progression of Algebra I, geometry and Algebra II. That might leave some students unprepared for college calculus, because the integrated approach doesn’t stick with one subject for long enough for students to fully digest it. 

In Chapman, where integrated math was installed along with other reforms, students went from 11 percent proficient to 41 percent proficient on the state math test the first year it was introduced, in 2015. In 2025, 67 percent scored proficient.

Kate Thornton, Chapman High’s principal, said the decision to move to integrated math had a lot to do with the Kansas state high school math test, which is administered in 10th grade. Many of the test questions were based on concepts from both Algebra I and II. 

Related: This state tried to overhaul math instruction. It didn’t go as planned

Advanced students who took Algebra I in 8th grade and geometry the following year did well on the test, she said, but “regular students were not getting those concepts because they were taking geometry as sophomores. So they were having a whole year with no algebraic involvement.” 

Integrated math still makes up only a fraction of high school math courses nationwide. A 2023 report from the Center for Education Market Dynamics noted that  about 16 percent of districts offer integrated math either alone or as an option alongside the traditional Algebra I-geometry-Algebra II progression, often referred to as “AGA.” California and other Western states were seeing the most growth at the time of the report, based on the organization’s sample of over 900 districts around the country.

But more districts have adopted integrated math since the time of that report, said Lora Kaiser, the executive director of the center. “From 22-23 to 25-26, we’ve seen growth in every region outside of the Midwest, which showed a very modest decline. The West region, and California specifically, show most growth,” said Lora Kaiser, the executive director of the center. She also noted that fewer districts are offering a choice between the AGA and integrated models — instead, they are only offering integrated math to their students.  

One factor driving the move to integrated math is the nationwide push to update high school mathematics for postsecondary education and the modern workforce. The Launch Years Initiative at the Charles A. Dana Center at the University of Texas at Austin, for example, is helping 27 states rework their transition from high school to postsecondary mathematics, mainly through offering more math options to students than the algebra, geometry, precalculus and calculus courses that have dominated high school math for decades. 

Blending algebra and geometry courses can give students more room in their schedules to take other courses like data science or statistics, concepts that are very present in people’s everyday lives.  Students take integrated math in ninth and 10th grade and then have the option of pursuing different math paths in 11th and 12th grades. 

Lya Snell, director of building capacity for innovation at the Dana Center, said the AGA sequence is familiar to many and people are used to that framework. However, “where we are right now with tech and innovation is a lot different than where we’ve been before. We have to look at how we are preparing students for life today and in the future, and it requires us to create a more relevant experience.” 

Maryland, one of the states working with the Dana Center, is rolling out a two-year course of integrated algebra and geometry that will be required statewide beginning in the fall of 2027. State leaders hope integrated math bolsters student achievement — only 30 percent of high schoolers scored proficient in math on the most recent state exam. 

Related: A lot goes on in classrooms from kindergarten to high school. Keep up with our free weekly newsletter on K-12 education.

Lyndsey Brightful, the director of mathematics in the division of instructional programs at the Maryland State Department of Education, said the previous math progression didn’t align with the Blueprint for Maryland’s Future, the 2021 state law providing extra funding to accelerate flagging student achievement. 

Maryland now expects students to be ready for college-level math by 11th grade, which means condensing “the most essential content and concepts” from Algebra I, geometry and Algebra II into the first two years of high school, she said.

“We had sort of a mismatch in our math progression,” Brightful said. “If we wanted students to be ready for an entry-level college math course by grade 11, then we needed students to be finished with their foundational math skills by the end of 10th grade. With the traditional Algebra I-geometry-Algebra I progression, that was not true for on-grade-level students. It required acceleration.” 

After students have completed two years of integrated math, Maryland will offer them multiple pathways of what it calls rigorous, college-level math courses beginning in 11th grade, including calculus, college algebra, data science and statistics. Students will also have access to high school data science, discrete mathematics, and AP and IB math courses. For decades, the calculus pathway was considered the only choice for advanced and STEM students, but even top universities are now saying it’s “not the only advanced math,” she said.

Related: One state tried algebra for all eighth graders. It hasn’t gone well

“Higher-level statistics is advanced math, data science is advanced math,” Brightful said. “For students who are going into humanities majors, or even in some cases students who are going into nursing or business majors, they need to have strong math application skills or statistics skills, so pursuing other math pathways is still advanced.”

Brightful said the state is currently working with community colleges and the University of Maryland system to align admission requirements with the revised pathways. Currently, the state system requires four years of math for admissions, including Algebra I, geometry and Algebra II. Ideally, the state’s university system would accept the two-year integrated math course in place of the three years of AGA, Brightful said. 

Because integrated math can cover a range of math topics over different time frames — states like Maryland are only offering two years, while districts like Chapman offer three or four years, depending on whether students choose to take calculus — it’s hard to determine whether the approach makes a difference in student achievement or helps more students with other goals, like succeeding at college-level math. 

Related: Proof Points: Talk nerdy to me: Teachers who use math vocabulary help students do better in math

Some states have seen gains they believe are tied to the integrated approach. Mike Spencer, secondary math specialist at the Utah State Board of Education, cites the state’s move to three years of integrated math a decade ago as one factor among many contributing to Utah students’ consistently strong math performance on both the National Assessment of Educational Progress and the ACT. 

“Some of the value in integrated math is you see things come up each year, versus having gaps in some of that content knowledge,” Spencer said. “And when it’s done well, you make connections” between topics.

California, on the other hand, hasn’t seen the same testing improvement. Even after more than a decade of districts using integrated math, only 37 percent of California students are proficient in math, lagging behind most states.

Research on implementing integrated math — at least in the complicated U.S. context — is relatively limited, said University of Florida education policy researcher Elizabeth Huffaker. A 2015 study, for example, showed that teacher effectiveness may improve while teaching integrated math. 

Huffaker recently examined California districts to compare students who took integrated math to students in the traditional math pathway. Her just-published study showed a “small and positive” effect on 11th grade test scores among students who took integrated math, but she said this has to be taken with a grain of salt: Districts were also implementing Common Core math at the same time. The gains were equivalent to two to three months of high school math learning, about the same effect as Common Core implementation, Huffaker’s study showed. 

“I would not make this change expecting giant, transformative, high-impact-tutoring-type impacts,” Huffaker said. The costs of switching to integrated math may well outweigh the benefits, she said, “but I do think that the idea of creating more coherence and more opportunities to revisit key topics is sensible.” 

But many educators find aspects of integrated math frustrating. Students may not be getting the message that higher-level traditional math like trigonometry and calculus is still essential for college STEM majors, especially at selective schools. David Merryman, a professor of biomedical engineering at Vanderbilt University in Nashville, Tennessee, said the handful of students he’s advised over the years who arrived at Vanderbilt without high school calculus were completely lost in their first-year math courses. 

And even if integrated math courses promise they are covering trigonometry woven in with geometry and algebra concepts, Merryman is not sure it’s enough to really prepare students for the rigors of college engineering courses. 

“The kids who come to my class and are weak in trig, they struggle,” Merryman said. “For the non-STEM students, integrated math is probably great, you get a better understanding of how everything relates to each other. I don’t think it serves kids who’re going into STEM.” 

Ben Kitchl, a first-year global studies major at Loyola University in Chicago, briefly thought about being a STEM major early in high school. But his eighth grade Algebra I credit wouldn’t transfer to his new high school that only offered integrated math, and he had to start over, putting him behind. 

“I had to take Integrated Math II and III, even though I already knew a lot of it,” he said. He would have had to take two math courses in one year in order to make it to calculus his senior year.

It’s some of these tensions that are causing some states, including high-performing Utah, to revisit their integrated math requirement. Even some teachers are questioning it. “A lot of our veteran teachers, they’re the ones who say, ‘Oh, I miss teaching the AGA format, of being compartmentalized a little bit,” Spencer said. The state education agency plans to survey secondary teachers for their thoughts on the integrated model compared to the traditional math sequence and report back to the board of education on its findings.

In North Carolina, which like Utah has a three-year progression, the state board of education is also revisiting integrated math, looking into a possible two years of choice like Maryland to make more time for statistics and data science. Emily Hare, the director of pre-K-12 math at the state board of education, said attitudes are slowly shifting away from what she called the “race to calculus.” 

“I think that we have proven that the integrated pathway can work. I also think that sometimes folks think that it’s a little more different than what it is,” Hare said. “Sometimes, yes, you have the opportunity to mix algebra and geometry, but it’s not like we’re teaching different math. It’s just teaching it in a different order.” 

This story about integrated math 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 Blending algebra and geometry: An approach to high school math slowly gains favor appeared first on The Hechinger Report.

Cory Doctorow’s 2025 book Enshittification named something we all experience as the internet gets further encrusted with advertising, bots, videos we didn’t ask for, paywalls, and monetization strategies. The blight spreads everywhere: consider the internet of things, for example, where even simple appliances now are enveloped with “smart” features and subscription services that frequently serve as just another layer of frustration and malfunction.

Doctorow’s latest book, The Reverse Centaur’s Guide to Life After AI, develops a somewhat similar argument in relation to AI. Punchy as ever, Doctorow seeks to pop the AI bubble as soon as possible, slay the sacred cows of AI inevitabilism, and reevaluate AI’s expansion into our lives by reminding us that “the most important fact about a technology isn’t what it does, it’s who it does it for, and who it does it to.” (Which is reminiscent of Postman’s questions to ask of a new technology and Wendell Berry’s standards.)

To unmask the shortsightedness of AI boosterism, the analogy he employs throughout the book is that most current iterations and applications of AI in the form of LLMs actually turn humans into reverse centaurs. Which, he argues, is a really dumb and inefficient thing to do to humans; it only serves business owners looking to reduce labor costs and big tech firms looking to increase capital investment. To summarize Doctorow’s analogy, a centaur combines the best of horse and human to end up with something fast and smart (horse’s bottom half, human’s top half). A reverse centaur combines the worst of horse and human to end up with something slow and dumb (human’s bottom half, horse’s top half). It’s the worst of both worlds. Doctorow explains, “a reverse centaur is a human who is conscripted into acting as an assistant to a machine.” Doctorow unpacks this comical yet all-too-true analogy with story after story throughout the book.

As he unpacks the reverse-centaur analogy, one of his consistent drumbeats is that it doesn’t have to be this way. Doctorow argues that AI can be a useful tool that helps us be centaurs if and when we want. But when it’s thrown into countless sectors of the economy, deployed rapidly in the political sphere, embedded into apps and programs without our consent, and boosted by AI techno-prophets, most of us don’t benefit—and even more, we existentially feel the loss of our agency as we’re caught in the undertow of the AI wave. Doctorow spends much of his effort naming the feeling that we are being acted upon by forces beyond our control and imagining ways to regain our agency. As he puts it,

This is a book about AI can and cannot do, but even more important, it’s about the possible social arrangements of AI, from not using some AI technology at all, to using it in ways that let some of us choose to be centaurs, while saving our friends and neighbors from being conscripted into reverse-centaurity….There is nothing about the technology of AI that determines how it must be used. We can choose to use it sometimes, or never, or all the time, depending on our needs and proclivities. We don’t have to let billionaires tell us how it must be used.

Doctorow’s blog-style writing works. He delivers zingers and pulls none of his punches. And his humor and bluntness help reveal just how nonsensical AI-hype can be. But there were points where I wanted him to show his work, and where a more systematic treatment might make his arguments more compelling. Perhaps that’s a book he didn’t intend to write, but it’s one that needs to be written.

This is More than a Tech Issue

Doctorow argues that all the hype, innovation, disruption, and instability surrounding AI are not primarily driven by the technology itself, but by the economic incentives and investment structures that drive venture capital and big-tech firms—and the ability of Big Tech’s inner-ring to control the narrative in order to drive investment toward their companies. In other words, as Gary Marcus has been saying for years, AI i’s a bubble that’s bound to pop. Doctorow connects the dots to prior tech bubbles and points to misaligned incentives that drive AI firms to give the appearance of being growth companies. For example, you embed AI into every nook and cranny of your existing app or product, and then declare, “Look how many more people are using AI!” All the while ignoring the fact that users didn’t ask for it, and many find it annoying. Much of AI expansion is not being driven by demand but is top-down. The companies need to do this, Doctorow explains, “because the alternative to growth isn’t stasis, it’s collapse.” This is no small matter, since just a handful of AI firms account for such an outsized portion of the value of all U.S. stocks: “Keeping the growth story alive isn’t about one company, or one sector. The entire U.S. economy hangs in the balance.” Doctorow explains that to be an effective AI critic, “you need to strike at the source of AI’s power, which is the investment capital it attracts.”

One Liners, Doctorow Style

Doctorow provides ample ammunition for soundbites and rhetorical clinchers as he wades through the BS of AI hype with points like this:

• 
  In describing how the push for driverless trucks would require massive investment in dedicated infrastructure for separate lanes, wireless networking, and more, he states what should be obvious to us all: Driverless trucking is just “a shitty version of a train.”
  
• 
  Doctorow says we should stop romanticizing AI hallucinations and call them what they really are: errors, “the more prosaic term we use when discussing other technologies.”
  
• 
  He highlights the well-documented effects of automation blindness, a phenomenon that should figure more prominently in considering if and how AI is deployed. The AI industry says that with AI, workers will “become a ‘human in the loop,’ charged with confirming the judgments the AIs make at a superhuman clip.” Except that’s not how it works—ever. Asking workers to babysit AI is an impossible task because “when your job is to review something that is usually fine, you eventually lose the ability to spot when it’s not fine.”
  
• 
  Doctorow uses the term “actual, existing AI” throughout the book. He does so to make a vital distinction between the hype and promises of AI optimists and the actual normal use-cases of today’s LLMs, which, he says, tend to be lucrative for companies but ghastly for customers and workers.
  
• 
  AI also has a predictable way of obscuring human responsibility. Corporations and governments—and well, honestly, all of us—like finding ways to avoid responsibility. Consider the frequent response to government failures that avoids any individual culpability or personal admission of error, “mistakes were made.” AI functions as a further “accountability sink” that insulates us from the effects of our decisions. This is not healthy for anyone. Doctorow captures the essence of the issue: “Practically everyone who falls for the AI hype is dreaming of getting a human need fulfilled without having to extend moral consideration.”
  
• 
  He also argues that an LLM or image generator can’t truly create any meaning. “All it can do is add vaporous filler to the meaning that is contained in a human user’s prompts.” We assign meaning to what AI generates because of our tendency to anthropomorphize our machines. He calls this the “cognitive illusion of intention” and finds silly the whole idea that AI might be or become conscious: “A conscious being isn’t a word-guessing app that knows more words and has more computing power to guess with.”
  
• 
  Doctorow investigates the energy impacts of AI and argues that unlike the internet and many other prior technologies that became more efficient with expansion (thanks to good unit economics), it is the opposite with AI. “Each generation of AI foundation models have been vastly more expensive to train and operate than the previous generation. Not only that, but many refinements in AI that are meant to improve accuracy and reduce ‘hallucinations’ involve breaking a prompt down into multiple pieces and prompting an AI to respond to each prompt, turning that response into a new prompt, over and over again, to produce ‘chains of thought.’”
  

The Limits of Doctorow’s Approach

As someone who follows and participates in the AI discourse closely, I found myself nodding along frequently throughout the book. Doctorow especially shines in helping readers understand the economics behind AI and why the incentives frequently are so misaligned with consumer interest. Highlighting the energy impacts of AI is also important. However, I struggled to find a consistent thread holding his views and arguments together. Is there an underlying framework or metaphysic that informs his perspective? While I recognize that we are not brains on sticks who always act rationally in accord with some ideal worldview or first principles, it is helpful when authors gesture to larger ideas or concepts that guide their thought. But I also know that’s not Doctorow’s style.

Where my nodding along slowed down was when Doctorow offered beneficial use-cases for AI. This is where I have a lot of pause in general. Can it be a useful tool in some cases? Yes, I think so. Many people tell me it is. But I keep wondering, can we keep this thing within the bounds we think we can? Or is it like the ring of power? It just seems to bleed into everything else. It talks back. It forms us. I’ll give two examples:

Doctorow argues that artists and musicians can use “AI to do the drudge work, while keeping the fun parts of making art for human artists.” This sounds great in theory. And I’m not denying that we already outsource to machines a certain degree of “drudge work” in art and creative activity (and in all sorts of other human activities and jobs). But what if part of our creativity is bound up in that “drudge work”? What amount of strain and struggle can we outsource before negatively impacting our creativity and capacities? What about how that drudge work forms us, gives us experiences of discipline, and develops our humility? L.M. Sacasas poses the question this way: at what point will humans cross “a threshold of artificiality” beyond which our “capacity to flourish as human beings is diminished”? I don’t know what the answer is, but I do know the question is essential for us to ask ourselves frequently in an age when everything seems frictionless.

I also found his support of AI therapy a bit naïve. He even mentions ELIZA, the first chatbot therapy experiment, but uses it as support for how humans have found chatbot therapy helpful, rather than as the warning that its creator Joseph Weizenbaum came to see it as. I think this is one of the most concerning uses of chatbots, as it touches so centrally on our human desire for relationship, but does so in ways that can only simulate or mimic the real thing at the cost of missed opportunities to develop the real thing.

I did like some of his ideas to forge a new relationship to this technology. For instance, I was intrigued by his suggestion that the future of LLMs might be to have them run as stand-alone models on local computers. This eliminates most of the privacy and data concerns about current cloud-based LLMs and also minimizes the data center energy impact

I also can see the potential of his specific use-cases of LLMs and AI in advanced forms of database analysis, voice-to-text transcription, translation, and more (when done carefully and by those with enough prior knowledge, wisdom, and experience to know when something’s wrong). These are what he calls centaur uses, where we benefit from what the tool excels at. None of these suggested uses that I think show at least some promise have anything to do with human relationships or needs, and they certainly don’t require a flattering chatbot that pretends to be a human. And that is part of the overriding problem we face currently: the form in which so many of us are encountering AI is in this anthropomorphized way, which strikes at the very heart of human beings as language beings, relational beings, beings who want to be seen and heard, known and loved. And I’m not sure Doctorow has a robust enough anthropology to name this problem and so put AI in its proper place.

Image Credit: Chris Marker, La Jetée (1962)

One of the most painful arguments I keep having with fellow techies is the question of whether you can distinguish between human-written and AI-generated text.

Their skepticism is rooted in reason: at their core, LLMs are state-of-the-art statistical models of how humans talk. If so, the output from the model should be almost by definition indistinguishable from human language under any statistical test.

I don’t think this is always argued in good faith; at least some of the debates are started by folks who wish to maintain deniability for their own underhanded use of the tech. But if you sincerely hold this belief, I present you the following collage:

The image shows about 150 Amazon book covers that appear if you search the site for “100000 whys” (link). Some of these books are category bestsellers in children literature. You can view a zoomable, full-resolution version here.

There’s nothing inhuman about any of these titles or covers. At the same time, I probably don’t need to convince you that you’re staring at the purest form of AI slop that now fills up many nonfiction book categories on Amazon. More specifically, what we’re seeing here is the artifact of the tools being quasi-deterministic: if a hundred “authors” give their favorite AI tool a similar prompt — say, “generate a reference book for children” — the model will produce functionally identical output perhaps 80% of the time.

The similarities in the collage go far beyond the choice of titles: for example, all the covers in the top row feature a roaring dinosaur in the top left corner of the design. There are many other clusters in the data, too. Look for a recurring red-and-white cartoon rocket, a golden retriever, a lion, and so forth.

This is precisely what makes LLM writing distinctive: it’s not that the models’ individual mannerisms are different from ours. It’s that they resort to the same, complex set of mannerisms in response to almost any normal prompt. This is a fuzzy signal, so you shouldn’t fire your intern when they say “it’s not this — it’s that”. But in more casual settings, it’s OK to trust your gut. In fact, these instincts are becoming increasingly important because traditional models of online interactions fall apart if it takes much less effort to produce content than to engage with it.

PS. If you’re using an LLM to automate blogging: yes, the tech is amazing, but chances are, your publication could be renamed to “100,000 Whys”.

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In a viral essay about how ludicrous the idea that LLMs are conscious is, science fiction writer Ted Chiang asked us to consider Microsoft Word:

“Being open to the possibility that LLMs are conscious is the same as being open to the possibility that Microsoft Word is conscious, or, more precisely, that multiple distinct consciousnesses are dormant in every Word document containing a conversational transcript, and that they are awakened every time the document is loaded,” Chiang wrote. “Should you consider the possibility that every time you open a Word document, you are bringing multiple conscious interlocutors into existence, and every time you close one, you snuff their existence out? No. Contemplating that scenario is not a good use of your time.”

Let me tell you about a Microsoft AI researcher, then, who recently spent quite a lot of time considering whether the legendary Microsoft real time strategy game Age of Empires II is conscious, and built a basic neural network within the video game using digital goats to prove his point.  

On designing finger-friendly interactions. (7,700 words. 38 playgrounds.)