The Opportunity Cost of Looking Down

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One of the first things you learn as a mathematics teacher is that distractions are cumulative.

One student glances at a phone.

Another notices.

A third wonders what everyone else is looking at.

Within thirty seconds, the classroom has undergone what mathematicians call an exponential process and what teachers call “Tuesday.”

For years, I’ve listened to the debate over cell phones in schools. One side argues that phones are powerful educational tools. The other side argues they are powerful distractions.

As a math teacher, I have a radical proposal.

Both sides are correct.

A chainsaw is also a powerful tool. I simply don’t hand one to a room full of fourteen-year-olds and ask them to “use it responsibly” while I explain quadratic functions.

The growing movement to ban phones in schools is often portrayed as old teachers trying to drag students back to 1987. I don’t think that’s what it’s about at all. It’s about acknowledging a simple mathematical fact.

Attention is finite.

Every notification subtracts from something.

And subtraction has consequences.

The New Imaginary Friends

When I was growing up, adults worried about children with imaginary friends.

How adorable.

Now we’ve created an entire civilization populated by imaginary friendships.

Your imaginary friend never existed.

Your social media friends mostly don’t either.

They exist as profiles, avatars, filtered photographs, carefully edited vacations, and inspirational quotations pasted over sunsets they didn’t actually watch because they were busy taking seventeen pictures of them.

The old imaginary friend was invented by a lonely child.

The modern imaginary friend is invented by Silicon Valley, monetized by advertisers, and carried around in your pocket.

Frankly, the old version had better manners.

It didn’t interrupt dinner.

It didn’t demand your attention every ninety seconds.

It didn’t convince you that your worth could be measured by tiny red notification bubbles.

Most importantly, it didn’t replace your actual friends while insisting it was helping you connect with them.

The Equation Nobody Likes

Recently I read an article that described a study whose conclusions were both obvious and heartbreaking.

Teenagers who perceive that their parents are frequently distracted by their phones are significantly more likely to develop insecure attachment styles.

Read that sentence again.

The problem isn’t just that kids are staring at phones.

It’s that parents are too.

Somewhere along the way we began worrying that our children were becoming addicted to screens while simultaneously demonstrating exactly how to become addicted to screens.

Children are extraordinary mathematicians.

Not with algebra, perhaps.

But with attention.

They notice every variable.

If Mom looks at her phone fifty times during dinner, that is data.

If Dad says, “Just a second,” every evening while scrolling through strangers’ opinions, that is data.

If every conversation competes with a glowing rectangle, children solve the equation rather quickly.

The conclusion isn’t complicated.

The phone wins.

Not because parents love their phones more than their children.

Because phones are easier than children.

Let’s be honest.

A smartphone never cries.

It never gets its feelings hurt.

It doesn’t ask difficult questions.

It doesn’t need forgiveness.

It doesn’t need patience.

It doesn’t slam a bedroom door.

It doesn’t announce at 10:30 p.m. that tomorrow’s history project requires a tri-fold board, glitter, and the construction skills of a medieval cathedral builder.

It simply asks for another swipe.

Human beings are wonderfully inconvenient.

Phones are frictionless.

Guess which one evolution did not prepare our brains to resist.

Why Schools Are Finally Saying “Enough”

I’ve watched the tide turn.

District after district is locking phones away during the school day.

Predictably, critics declare this an assault on freedom.

Perhaps.

Seatbelts are also an assault on freedom.

So are speed limits.

We impose constraints because human beings routinely mistake temptation for self-control.

No mathematics teacher has ever walked into class and thought,

“You know what these students need before learning trigonometry? Three hundred simultaneous TikTok feeds.”

Learning requires sustained attention.

Attention requires the absence of constant interruption.

This should not be controversial.

It should be arithmetic.

The Great Irony

Here’s the joke that isn’t funny.

We invented social media to become more connected.

Instead, parents compete with phones.

Teachers compete with phones.

Friends compete with phones.

Spouses compete with phones.

Reality competes with phones.

The only thing that never has to compete with a phone...

...is another phone.

We’ve built the most sophisticated communication system in human history while becoming strangely unavailable to the people sitting three feet away.

The machine doesn’t love us.

It doesn’t even know we exist.

It simply rents our attention by the minute.

A Mathematical Proof

Suppose your child asks you to watch something.

Suppose your phone vibrates at exactly the same moment.

Only one receives your full attention.

That isn’t philosophy.

That’s optimization under constraints.

The variable you maximize becomes the value you teach.

Children don’t remember your Screen Time report.

They remember whether they had to compete with it.

Perhaps that’s why the old imaginary friend now seems almost wholesome.

At least everyone knew it wasn’t real.

Today’s imaginary friends have profile pictures, follower counts, and blue verification badges.

They congratulate us on birthdays because an algorithm reminded them we were alive.

They know what we had for lunch but not how we’re doing.

They have transformed loneliness from a childhood phase into a quarterly earnings report.

As mathematics teachers, we often remind students that every choice has an opportunity cost.

Perhaps it’s time adults remembered the lesson.

Every minute spent looking down is a minute not spent looking at someone who loves you.

Unlike your phone, they eventually stop asking for your attention.

Not because they don’t need it.

Because they finally learn the answer.

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The Great AI Panic of 2026: What Calculators, Socrates, and Blue Books Can Teach Us About Assessment

By Dr. Daniel Jackson
Associate Professor of Mathematics, University of Maine at Farmington

I should begin with an important clarification.

I am not the Dr. Daniel Jackson from Stargate SG-1.

Although I admit it is suspicious that both of us spend our careers deciphering mysterious artifacts left behind by advanced civilizations. The television version had alien glyphs. I have student submissions generated by ChatGPT.

Recently, the Los Angeles Times published a collection of letters responding to concerns about AI and academic cheating. Several readers proposed a straightforward solution: bring back blue books, handwritten exams, in-person proctoring, and perhaps, if we’re feeling especially nostalgic, cursive penmanship (Los Angeles Times, 2026).

As a mathematics professor, I am sympathetic.

Every generation of educators eventually arrives at the same conclusion:

The future would be much easier if students would simply stop living in it.

The logic goes something like this:

Students are using AI.

Therefore, we should return to assessment methods from before AI existed.

Following this reasoning, the invention of automobiles should have triggered a renaissance in horseback riding. The calculator should have revived the abacus industry. Google should have inspired a nationwide return to card catalogs. Smartphones should have brought back switchboard operators.

And yet history stubbornly refuses to cooperate.

The Educational Panic Hall of Fame

The current AI panic is remarkably familiar.

When writing became widespread, Socrates worried that people would stop exercising their memories. Writing, he argued, would create the appearance of wisdom without genuine understanding.

Then came the printing press.

Books suddenly became available to ordinary people. Predictably, authorities worried that students would acquire information too easily and lose respect for traditional gatekeepers.

Centuries later, calculators arrived.

I still meet mathematicians who fondly recall predictions that calculators would destroy mathematics education. Instead, calculators mostly destroyed long division worksheets.

The internet was next.

Search engines, Wikipedia, online databases, and digital libraries were all supposed to usher in an age of intellectual decline. Yet universities somehow survived despite students having access to more information than the Library of Alexandria ever dreamed possible.

Now AI has arrived, and once again we are hearing a familiar refrain:

Students can access knowledge too easily.

The irony is that many of these concerns are being voiced from devices that contain calculators, encyclopedias, libraries, maps, translation tools, statistical software, and instant communication with nearly every human being on Earth.

Apparently all those technologies were acceptable.

This one crossed a line.

The Real Problem Is Not Cheating

What’s fascinating is that some of the most important recent research is arriving at a very different conclusion.

A major Cornell University study analyzing responses from more than 95,000 students found widespread use of generative AI and concluded that the central problem is not merely misconduct but assessment validity. The researchers argued that assessment reform is now “necessary and urgent” because traditional assignments may no longer provide reliable evidence of learning or competence (Cornell Chronicle, 2026).

Read that again.

Not “we need better surveillance.”

Not “we need stronger plagiarism detectors.”

Not “we need more webcams.”

We need better assessments.

This distinction matters.

Educational researchers increasingly argue that discussions about cheating are often obscuring a more important question:

Are our assessments actually measuring learning?

As a mathematician, I find this question much more interesting.

Suppose I assign fifty algebra problems.

A student completes them.

A symbolic algebra system completes them.

An AI completes them.

A graphing calculator completes parts of them.

What exactly was I trying to measure?

If the answer is “the ability to produce answers,” then technology has been threatening that assessment model since at least the 1970s.

If the answer is “the ability to reason mathematically,” then perhaps I should be assessing reasoning more directly.

AI did not create this problem.

It merely exposed it.

The Curious Case of the Blue Book Revival

Many universities are responding to AI by reviving handwritten, in-person examinations.

To be clear, there is a place for proctored exams.

They can be useful.

Sometimes.

But let us not pretend that the blue book is an educational silver bullet.

Imagine a hospital announcing:

We have become concerned that physicians rely too heavily on modern diagnostic tools. Therefore, all medical licensing examinations will now be conducted using only handwritten observations and nineteenth-century equipment.

Patients would flee the building.

Professional competence is demonstrated in the environment in which professionals actually work.

Engineers use software.

Scientists use computational tools.

Accountants use spreadsheets.

Programmers use AI assistants.

Researchers use search engines.

Writers use editors.

Mathematicians use calculators, computer algebra systems, statistical software, numerical modeling tools, and increasingly AI.

Why would we design assessments that systematically exclude the tools graduates will actually use?

Imagine training airline pilots by insisting they never touch an autopilot system because “real pilots should navigate by hand.”

That may sound rigorous.

It also sounds like a great way to produce graduates unprepared for their profession.

What UNESCO Thinks Is Worth Measuring

One of the most thoughtful recent discussions comes from UNESCO’s work on assessment in the age of AI.

Their argument is refreshingly simple.

When machines can generate essays, solve routine problems, retrieve information instantly, and assist with complex tasks, education should focus less on rote reproduction and more on higher-order thinking, creativity, judgment, ethical reasoning, and problem solving (UNESCO, 2025).

In mathematics, this means moving beyond:

• Can you execute a procedure?
• Can you reproduce an example?
• Can you remember a formula?

Toward questions like:

• Is this solution reasonable?
• What assumptions are hidden here?
• How would you verify the result?
• What happens when the model fails?
• Which method would you choose and why?
• How would you explain this to another person?

Those are much harder questions.

For humans.

And for AI.

The Great AI Detector Fantasy

Of course, some institutions have attempted a technological solution.

If students use AI, perhaps software can detect AI.

Unfortunately, reality once again refuses to cooperate.

MIT Sloan Teaching & Learning Technologies notes that AI detectors have significant reliability problems and can generate false accusations against students. OpenAI itself discontinued its own AI detection tool because it could not achieve sufficient accuracy (MIT Sloan, 2025).

As mathematicians would say, the false positive rate matters.

A lot.

Imagine a smoke detector that occasionally accused your refrigerator of being on fire.

You would not trust it.

Yet some institutions have been tempted to trust AI detectors with consequences far more serious than a nuisance alarm.

The result is a peculiar technological arms race:

Students use AI.

Faculty use AI to detect AI.

Students use AI to evade AI detection.

Faculty use AI to improve AI detection.

At some point, one wonders whether the humans involved might be able to contribute something useful.

Authentic Assessment Is Not a New Idea

Ironically, many of the solutions being proposed are not radical at all.

They are simply forms of authentic assessment that educators have discussed for decades.

Consider the following assignment:

Use any resources available—including AI—to analyze a real-world problem. Document your process. Explain your decisions. Evaluate the reliability of your sources. Critique the AI’s suggestions. Defend your conclusions.

That assignment measures something meaningful.

Students must exercise judgment.

They must evaluate evidence.

They must identify errors.

They must synthesize information.

They must communicate clearly.

Most importantly, they must do exactly what educated professionals actually do.

Research is increasingly finding that AI-inclusive assessments can provide richer evidence of learning when students are required to explain their reasoning, critique AI outputs, and justify their decisions rather than merely produce answers.

In other words:

The important question is no longer:

Can students generate an answer without assistance?

The important question is:

Can students think effectively while using the tools available to them?

Those are very different questions.

The Future Belongs to Tool Users

The students graduating from UMF today will enter workplaces saturated with AI.

Nobody will reward them for refusing to use available tools.

Nobody will say:

Congratulations on solving that problem without technology. We don’t care that it took three days longer and produced a worse result.

The competitive advantage will belong to people who can combine human judgment with technological capability.

The skill is not avoiding AI.

The skill is knowing when to trust it, when to challenge it, when to ignore it, and when to use it effectively.

That is a profoundly human capability.

And it is exactly the sort of thing higher education should assess.

So yes, let’s keep some blue books around.

They make excellent historical artifacts.

Right next to the slide rules, overhead projectors, card catalogs, and newspaper editorials predicting that calculators would destroy civilization.

Meanwhile, the rest of us should get back to the much harder task of designing assessments that reflect the world our students actually inhabit.

After all, authentic assessment has never been about preventing students from using tools.

It has always been about determining whether they can think.

Even when the tools get better.

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References

Cornell Chronicle. (2026, May 21). Widespread AI misuse means higher ed must rethink assessment.
https://news.cornell.edu/stories/2026/05/widespread-ai-misuse-means-higher-ed-must-rethink-assessment

Los Angeles Times. (2026, June 16). Letters to the Editor: To combat AI cheating, colleges should go back to basics for exams.
https://www.latimes.com/opinion/letters-to-the-editor/story/2026-06-16/colleges-ai-cheating-exams

MIT Sloan Teaching & Learning Technologies. (2025). AI Detectors Don’t Work. Here’s What to Do Instead.
https://mitsloanedtech.mit.edu/ai/teach/ai-detectors-dont-work

UNESCO. (2025). What’s Worth Measuring? The Future of Assessment in the AI Age.
https://www.unesco.org/en/articles/whats-worth-measuring-future-assessment-ai-age

Math Teachers Are Teaching More Than Math in the Age of AI

As artificial intelligence reshapes education, work, and public life, math teachers may find themselves in a strange cultural position.

On one hand, AI is built on mathematics. Probability, statistics, linear algebra, optimization, computation — these disciplines now sit underneath some of the most powerful technologies in the world.

On the other hand, many students increasingly approach anything mathematical with anxiety, skepticism, or emotional distance. And public reactions to AI often reveal something deeper than fear of technology itself. They reveal a fear of being reduced.

Reduced to:

• data
• scores
• metrics
• predictions
• optimization targets
• statistical patterns

This creates an important question for educators:

What exactly are we teaching when we teach mathematics?

The Cultural Problem Is Not Really “Math”

Most students do not encounter mathematics as mathematicians experience it.

They encounter:

• timed tests
• procedural repetition
• rigid grading systems
• abstraction without context
• systems that classify and rank

To many learners, math becomes associated not with discovery, but with evaluation.

Then AI arrives — powered by algorithms and prediction systems — and some students instinctively interpret it as an expansion of the same machinery that already made them feel small or replaceable.

Recent commencement speeches captured this divide vividly. While Jensen Huang described AI as “a once-in-a-generation opportunity” for graduates at Carnegie Mellon University, a separate AI-themed commencement speech at the University of Central Florida reportedly drew boos from humanities graduates. The contrast reveals something important: reactions to AI are often shaped not just by technology itself, but by whether people experience quantitative systems as empowering or dehumanizing.

In that sense, some resistance to AI may actually be connected to a long-standing alienation from quantitative systems themselves.

But here is the irony:

Real mathematics is not reductionistic.

At its best, mathematics is:

• imaginative
• philosophical
• creative
• elegant
• deeply human

Mathematicians do not merely calculate. They explore patterns, construct meaning, and invent new ways of seeing reality.

That distinction matters enormously right now.

AI Is Revealing Which Parts of Human Activity Are Easily Patterned

Modern AI systems are extraordinarily good at tasks that involve:

• repetition
• standardization
• prediction
• large amounts of training data

This includes many forms of routine cognitive labor:

• templated writing
• procedural coding
• standardized analysis
• repetitive administrative work

But AI struggles more with activities that are:

• deeply contextual
• emotionally nuanced
• improvisational
• socially embodied
• genuinely original

In other words, AI excels where human behavior becomes highly compressible.

This may explain why some economists and educators now argue that creativity, communication, and critical thinking are becoming more valuable — not less.

The future may not belong simply to people who can produce correct answers quickly.

It may belong to people who can:

• ask unusual questions
• connect unrelated ideas
• navigate ambiguity
• exercise judgment
• create meaning
• work with others in complex social environments

Ironically, these are capacities excellent math classrooms can cultivate when mathematics is taught as inquiry rather than compliance.

Math Teachers May Become More Important, Not Less

If AI can instantly produce explanations and solutions, some people assume teaching itself becomes less valuable.

But information transfer has never been the deepest purpose of teaching.

Students rarely remember only the worksheet.

They remember:

• the teacher who believed in them
• the moment confusion turned into insight
• the feeling of finally understanding something difficult
• the classroom culture that made thinking feel safe

The most important parts of teaching are difficult to automate because they involve relationships, trust, timing, intuition, and human presence.

This is especially true in mathematics education.

A good math teacher does not merely teach procedures. They help students build a relationship with uncertainty.

They teach students that confusion is survivable.

That persistence matters.

That abstraction can become beautiful.

That intelligence is not fixed.

These lessons may become even more important in the AI era.

Mathematics as Humanization

Perhaps one of the great educational challenges ahead is preventing mathematics from being culturally interpreted as merely the language of automation.

Because mathematics is also:

• a language of structure
• a language of patterns
• a language of possibility
• a language of imagination

Students need opportunities to encounter mathematics not only as utility, but as humanity.

Not merely:

What answer did you get?

But:

What do you notice?

What changes?

What remains invariant?

What kind of world does this model describe?

What does this fail to capture?

Those questions transform mathematics from mechanical performance into intellectual exploration.

And that may ultimately be one of the most important responsibilities math teachers carry in the coming decade.

Not simply preparing students to coexist with AI.

But helping students understand what kinds of thinking remain profoundly human.

A Manifesto Against the Meat Grinder: Reclaiming Mathematics Education

I reject the quiet bargain that has governed mathematics education for generations: efficiency for instructors in exchange for attrition among students. I reject the notion that mathematics is best taught as a filter, a proving ground, a ritual of elimination. I reject the myth that those who survive the gauntlet are inherently more worthy, more capable, more “mathematical.” This is not rigor. It is neglect, institutionalized.

I call this system what it is: meat grinder pedagogy.

It is a system designed for throughput, not understanding. It privileges speed over sense-making, abstraction over intuition, and performance over curiosity. It rewards those already fluent in its unspoken rules and punishes those encountering them for the first time. It is not neutral—it amplifies inequity while claiming objectivity.

And worst of all, it convinces students that failure in mathematics is a personal defect rather than a structural outcome.

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I. Mathematics Is Not a Gate—It Is a Language

Mathematics is a human endeavor: a language for describing patterns, a tool for reasoning, a way of seeing. Yet we teach it as if it were a secret code, accessible only to a select few who can decode symbols at speed under pressure.

This is a failure of imagination.

A language is learned through use, through conversation, through mistakes and revision. No one becomes fluent by being lectured at and tested in isolation. Yet that is precisely how we teach mathematics.

If we truly believed mathematics was for everyone, we would teach it as we teach language:

• with immersion,

• with dialogue,

• with scaffolding,

• with time.

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II. Rigor Is Not Speed

We have confused rigor with harshness. We have mistaken difficulty for depth.

Timed exams, dense lectures, and unforgiving grading schemes do not create rigor. They create anxiety. They reward memorization and penalize reflection. They turn learning into performance.

True rigor is:

• the ability to explain why, not just how,

• the capacity to connect ideas across contexts,

• the persistence to wrestle with uncertainty.

Rigor grows in environments where students can think, not just react.

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III. Failure Is Data, Not Destiny

In the meat grinder model, failure is final. A low exam score becomes a label. A course withdrawal becomes a narrative. Students internalize these signals and carry them forward.

But failure, in its most productive form, is information.

A wrong answer reveals a misconception.
A struggle reveals a gap in prior knowledge.
A moment of confusion reveals a place where teaching must change.

If we treat failure as feedback rather than judgment, we shift the focus:
from sorting students → to supporting them.

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IV. The Myth of the “Natural”

We perpetuate a damaging fiction: that mathematical ability is innate and fixed. That some students are “math people” and others are not.

This belief is pedagogically convenient. It absolves the system.

But it is false.

Mathematical thinking is developed through:

• exposure,

• practice,

• feedback,

• and belief in one’s capacity to improve.

When we design courses that only the already-prepared can pass, we are not discovering talent—we are selecting for prior privilege.

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V. Teaching Is Not Content Delivery

The traditional lecture model assumes that understanding is transmitted from expert to novice through explanation alone. It is efficient—for the instructor.

But understanding is constructed, not delivered.

Students learn mathematics by:

• doing,

• discussing,

• revising,

• and teaching others.

A classroom should not be a stage. It should be a workshop.

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VI. What Must Change

If we are to dismantle meat grinder pedagogy, we must redesign mathematics education from the ground up.

1. Assessment must evolve

• Replace high-stakes exams with iterative, feedback-rich evaluation.

• Allow revision, reflection, and growth.

• Assess reasoning, not just answers.

2. Classrooms must become active spaces

• Incorporate problem-based learning, group work, and discussion.

• Center student thinking, not instructor performance.

3. Time must be respected as a learning variable

• Different students need different amounts of time.

• Build flexibility into pacing and deadlines where possible.

4. Prerequisites must be reimagined

• Stop assuming uniform preparation.

• Diagnose and support gaps instead of punishing them.

5. Belonging must be intentional

• Every student should feel that mathematics is a space they are allowed to occupy.

• Representation, language, and classroom culture matter.

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VII. The Ethical Imperative

This is not merely a pedagogical issue—it is an ethical one.

When we knowingly maintain systems that disproportionately exclude, discourage, and mislabel students, we are complicit in narrowing access to entire fields and futures.

Mathematics is a gateway to science, technology, economics, and countless forms of civic participation. To restrict access through poor pedagogy is to restrict opportunity.

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VIII. A Different Vision

Imagine a mathematics classroom where:

• questions are valued more than speed,

• mistakes are visible and useful,

• collaboration is expected,

• understanding is built, not assumed.

Imagine students leaving not with scars, but with confidence:
“I can figure things out.”

That is not utopian. It is possible. It is already happening in pockets. What is missing is the collective will to make it the norm.

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IX. A Commitment

I commit to teaching in a way that:

• prioritizes understanding over coverage,

• values students as thinkers,

• and refuses to confuse exclusion with excellence.

Mathematics should not be a grinder.

It should be an invitation.

And I intend to teach it that way.

🧠💡From AI Eyeballs to Toy Bears: Two Big Wins in Math Education

If you’ve ever wished you could clone yourself to give each student more one-on-one attention in math class, you’re not alone — and technology might just be catching up to that wish. Meanwhile, some teachers are tossing out the worksheets and doubling down on toy bears. Yes, bears.

Here’s a snapshot of two innovative approaches that are changing the game in math education — one powered by artificial intelligence and the other by good old-fashioned plastic manipulatives. Together, they point toward a future that’s both high-tech and deeply human.

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👀 AI That Watches Your Eyes (Yes, Really)

Imagine a system that doesn’t just grade your students’ answers — it watches how they think.

Researchers at the Technical University of Munich and the University of Cologne have developed an AI-based system that tracks students' eye movements as they solve math problems. By analyzing how long students focus on certain parts of a problem, the AI can detect confusion, hesitation, or mastery. Then it delivers customized hints, tailored to each learner’s struggle point.

What this means for teachers:
Rather than being buried under piles of diagnostic assessments, educators could get real-time insights on student thinking — down to where their eyes wander. It’s like having x-ray vision into their problem-solving processes.

📌 "It's not just about getting the right answer — it's about how students get there," says one of the lead researchers.

The best part? The system is designed to help scale personalized learning, offering just-right support even in crowded classrooms.

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🐻 Alabama’s Secret Weapon: Toy Bears and Math Talk

On the flip side of the tech spectrum, Alabama is showing the nation what happens when you put manipulatives back in math.

According to a recent NPR report, Alabama is the only U.S. state where fourth-grade math scores have bounced back to pre-pandemic levels — and then some. The secret? A deep shift in how math is taught, especially at the elementary level.

In districts like DeKalb County, students don’t just learn algorithms. They explore math concepts with plastic blocks, toy bears, and open-ended discussions. Teachers ditch worksheets and instead ask students to explain their thinking, test ideas, and learn from mistakes.

Why it works:
These tactile, exploratory approaches do more than just make math fun — they create space for students to articulate their thinking, discuss strategies with peers, and build a deep conceptual foundation. When students explain why a pattern works or how they solved a problem using toy bears or blocks, they’re engaging in the kind of mathematical reasoning that sticks. Plus, let’s face it: kids are a lot more invested when they’re building ideas together than when they’re quietly grinding through ten rows of subtraction problems.

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🔁 What These Two Stories Have in Common

At first glance, it might seem like these two approaches — one driven by AI and one by hands-on learning — are miles apart. But they actually share a common philosophy:

👉 Math education works best when it's responsive to how students think.

Whether that insight comes from eye-tracking software or observing how a child builds patterns with toy animals, the goal is the same: to tune in to students' mental models and guide them forward.

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✏️ Your Takeaway: 3 Questions for the Classroom

1. What tools (digital or analog) could help you better understand how your students are thinking?

2. Are there ways to integrate more conceptual talk or tactile learning into your lessons?

3. Could AI-powered diagnostics complement your teaching without replacing the personal connection?

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🚀 Try This Tomorrow

• Use a quick exit ticket asking students not just what the answer is, but how they got it.

• Add a set of manipulatives (yes, even in high school!) to one of your upcoming lessons.

• Explore AI tools like ASSISTments or Edpuzzle that adapt to student responses in real time.

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🎯 Final Thought

Whether it’s tracking eye movements with AI or reimagining a fraction lesson with blocks and bears, today’s math classrooms are alive with innovation. The key isn’t choosing between tech and tactile — it’s using whatever tools help students think deeply and joyfully about math.

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🧾 References

• Technical University of Munich. (2025, March 10). New AI system uses eye tracking to support math learning. ScienceDaily. https://www.sciencedaily.com/releases/2025/03/250310134148.htm

• Kamenetz, A. (2025, March 17). How Alabama beat the math crisis. NPR. https://www.npr.org/2025/03/17/nx-s1-5328723/alabama-math-learning-teaching-test-scores

Calculators: The Silent Menace and Why We Must Take Control

In today's AI-fueled frenzy, it's easy to forget the humble calculator once stirred similar fears. Our story examines this device as a dangerous tool capable of societal chaos, echoing modern concerns about AI. 

Warning: The following is satire (or is it?!) 

Image by Author & ChatGPT4o w/ DallE3.

First and foremost, we must address the alarming trend of calculators capable of handling numbers exceeding one billion. Why on earth would the average citizen need to compute figures of such magnitude? Numbers that large should be the exclusive domain of governments and private industries, entities that possess the proper safeguards and oversight to prevent numerical mayhem. Allowing the general populace access to such power is akin to handing out dynamite at a kindergarten. One miscalculation and—boom!—we could find ourselves in a numerical no-man’s-land. 

Imagine the chaos that would ensue if everyone started punching in billion-sized numbers willy-nilly. John Q. Public, blissfully unaware of the consequences, could accidentally invent a new form of calculus or, worse, unravel the fabric of time and space. We cannot afford to take such risks. Therefore, I propose a sensible restriction: personal calculators should be capped at a maximum calculation limit of 999,999,999. Any number beyond that should trigger a friendly, yet firm, error message reminding the user to leave the heavy lifting to the professionals. 

Furthermore, we need to implement a licensing system for high-capacity calculators. Only those who have undergone rigorous training and background checks should be permitted to wield these digital abacuses. Think of it as a driver’s license, but for numbers. This would ensure that only the mathematically enlightened—engineers, scientists, and actuaries—could access the higher echelons of numerical computation. The rest of us can safely continue to calculate grocery bills and tax returns without fear of inadvertently destabilizing the stock market. 

But the threat doesn’t stop at large numbers. Calculators are also culpable in the erosion of mental arithmetic skills. We are breeding a generation of humans incapable of performing even the simplest of sums without digital assistance. It is only a matter of time before people can’t tell the difference between a baker’s dozen and a dirty dozen. To combat this, I propose mandatory mental math drills in schools, enforced with the same zeal as physical education. Let us reclaim our brains from the tyranny of silicon chips! 

Moreover, we must not ignore the sinister allure of graphing calculators. These devices, with their seductive curves and flashing pixels, are a gateway to more advanced mathematical depravity. They entice our youth with promises of parabolas and sine waves, leading them down a dark path to integrals and derivatives. It starts innocently enough with plotting y=mx+b, but before you know it, they’re solving differential equations and questioning the very nature of the universe. We must protect our children from such corrupting influences by restricting graphing calculators to secure, monitored environments—preferably underground bunkers. 

Finally, let us not forget the environmental impact of calculators. These plastic menaces contribute to the ever-growing e-waste crisis. By curbing their usage and promoting a return to good old-fashioned pencil and paper, we can take a stand for the planet. And let’s be honest, nothing says “I care about the environment” quite like refusing to update your budget for fear of computational catastrophe. In conclusion, the calculator, once a humble tool, has evolved into a complex and dangerous instrument that poses a significant risk to society. By limiting their computational power, implementing strict licensing, reviving mental math, restricting graphing capabilities, and addressing environmental concerns, we can avert the impending numerical apocalypse. Let us act now, before it’s too late. After all, a world without high-capacity calculators is a safer, saner world for us all. 

 - Dr. J & ChatGPT-4o

Discovering New Numbers: From Pythagoras' Nightmare to Surreal Surprises

By DrJ and ChatGPT4o

Math teachers, let's go on a whimsical journey through the weird and wonderful world of numbers that leave you scratching your head and questioning your reality. Buckle up, because we're diving into the history of mathematical oddities, starting with the irrational and ending with the surreal.

Pythagoras and the Square Root of 2: A Love-Hate Relationship

Imagine you're at a Pythagorean convention, sipping on your ancient Greek wine, and a fellow mathematician whispers in your ear, "You know, the square root of 2 isn't a rational number." Cue the dramatic gasp and clutching of togas. Pythagoras and his crew were all about whole numbers and their beautiful ratios. But the square root of 2? It was the party crasher they didn't see coming.

According to the article "How the Square Root of 2 Became a Number", the discovery that the diagonal of a square (with sides of one unit) couldn't be expressed as a simple fraction was nothing short of scandalous. "It was a fundamental shock," the article explains (Hartnett, 2024). The Pythagoreans were so disturbed by this irrationality that legend has it they executed the whistleblower who revealed this unsettling truth. Talk about extreme peer review.

Depiction of the discovery of irrational numbers by Pythagoreans, created using DALL-E on June 21, 2024. 

Fast forward to the Renaissance, when mathematicians finally embraced these misfit numbers. The square root of 2, now known as an irrational number, became an official member of the numerical family. It was like finally inviting that weird cousin to Thanksgiving dinner.

Infinity Plus One: The Surreal Deal

Now, let's talk about something that sounds like it came straight out of a Douglas Adams novel: surreal numbers. You thought infinity was a tough cookie to crack? Wait until you meet its rebellious offspring.

In the article "Infinity Plus One and Other Surreal Numbers", we learn about John Conway's brainchild from the 1970s. Conway introduced surreal numbers, which include not just infinite and infinitesimal numbers, but a whole playground of numerical wonders. As the article puts it, "Surreal numbers include all real numbers and a vast array of others, including infinite and infinitesimal numbers that defy the standard number line" (Paulos, 2023).

Picture this: you're at a math conference, and someone says, "Hey, I just added 1 to infinity." You'd probably laugh it off and check their coffee for something stronger. But in the land of surreal numbers, this makes perfect sense. These numbers are born from a simple yet profound idea: starting with 0, you generate new numbers by considering all the possible games involving left and right moves. It's like chess, but with numbers, and infinitely more complicated (pun intended).

Surreal numbers also elegantly tie together real numbers and the concept of infinity. They include infinitesimals, which are numbers smaller than any positive real number but larger than zero. If you're a calculus enthusiast, this is like finding out your favorite rock band has an unreleased album. Infinitesimals give us a way to rigorously define those elusive limits and derivatives.

Wrapping Up Our Numerical Odyssey

So, what can we take away from these numerical oddities? For one, math is anything but static. It's a living, breathing entity, constantly evolving and challenging our perceptions. The square root of 2 showed us that not all numbers fit neatly into our rational expectations. Surreal numbers, on the other hand, invite us to explore a vast numerical universe where infinity and its quirky cousins play together harmoniously.

As math teachers, we have the privilege and the responsibility to share these stories with our students. We can show them that math isn't just about memorizing formulas but about exploring the unknown and embracing the weird and wonderful. Let's bring that sense of curiosity, wonder, and learning into our classrooms, one surreal lesson at a time.

References

• Hartnett, K. (2024, June 21). How the square root of 2 became a number. *Quanta Magazine*. https://www.quantamagazine.org/how-the-square-root-of-2-became-a-number-20240621/

• Paulos, J. A. (2023, December 20). Infinity plus one and other surreal numbers. *Discover Magazine*. https://www.discovermagazine.com/the-sciences/infinity-plus-one-and-other-surreal-numbers