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*Tune in next week for more wacky hijinks as Young Gottfried tries to build a horizontal windmill!*

[NOTE: No, I am not actually doing a serialized webcomic on the adventures of a teenaged Gottfried Leibniz, c0-inventor of the calculus. You are welcome to do so yourself; I promise I won’t litigate.]

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So here are some ideas of other ways to slice, dice, and rearrange the mathematics currently taught in high schools. (And hey, maybe we’ll want to switch out an ingredient here or there, too.)

I’ll lay out four proposals.

First up…

This is an approach with a simple goal: Make Math Useful.

You’d begin with a condensed version of the Geometry/Algebra sequence (** Mathematical Reasoning**). We’d cut some of the dregs (compass-and-ruler constructions, circle theorems, conic sections, fussy technical proofs). Instead, you’d do a semester of geometric proof, focusing on the greatest hits, and a semester of algebra, focusing on expressing linear relationships.

Then, we’d get a year of probability and statistics (** Data Mathematics**). You’d learn about randomness, standard deviation, conditional probability, p-value, and how to build a spreadsheet. All that tasty, jobs-y goodness.

For junior year, you’d get a course that covers some traditional content (functions, graphing skills, exponential growth, derivatives) but with a focus on financial and economic applications (** Financial Mathematics**). How does interest work? How does debt work? Emphasizing the real-world relevance of these ideas wouldn’t have to undercut their theoretical beauty; rather, it would add to it.

And finally, you’d get a course in slightly more advanced topics relevant to the digital age (** Computer Mathematics**): first-order logic; expressing transformations with position vectors and matrices; binary; writing and understanding algorithms.

Yes, we lose some stuff. (Goodbye, arcsecant—and good riddance!) We even lose some good stuff. (I love teaching about limits in depth, but there’s no space for it here.) But I think we gain in coherence and purpose. Instead of organizing the curriculum around traditional divisions within mathematics, we organize them around goals that make actual sense for actual students.

Or does it worry you to blur the historic boundaries between subfields? Well, if traditional disciplinary structure is your thing, then you might go for…

Our second proposal:

Personally, I’d be jazzed to teach any of these courses.

But perhaps you’re not. Perhaps you’re a dreary Eeyore soul, and you’ve observed that this is a forbiddingly large number of different classes for a high school to teach.

Well, you conservative stickler, here’s a menu for you.

Our third proposal:

Wait—isn’t this what we have now?

More or less, yeah. But I’m picturing one big change.

Currently, there are two years of AP Calculus available. Lots of students take the first one as juniors, and the second as seniors, meaning that half of their high school math time is spent on calculus.

This stuns and baffles the Brits I teach with. They approach calculus very differently, scattering it throughout 11^{th} and 12^{th} grade. (You might even encounter easy bits of differentiation in 9^{th} and 10^{th}.) Some topics get cut entirely—for example, they never really tackle limits, and frankly, they don’t seem to miss them.

So my suggestion: ditch the second year of calculus.

This requires some cutting (goodbye, partial fractions and trig substitution). So why is it worth it?

Because ambitious college-bound students would still want two years of AP mathematics. And, without the impressive two years of calculus on the table, they’d turn to the alternative, a course that’s useful for citizenship and which everyone should probably be taking anyway:

Statistics.

Of course, if this modest proposal is too stodgy and incremental to get your blood flowing, I’ve got one final suggestion.

The fourth proposal:

Teachers like me urge each other to think in terms of verbs. “Education is not about helping them *know* stuff,” we say. “It’s about helping them *do* stuff.”

So let’s make the “doing” central.

Begin with lots of modeling activities. Explore how proportions, linear relationships, and trig ratios can help us express a whole host of real-world relationships. Do three-acts. Build inclinometers. Go wild.

Then, in 10^{th} grade, hit the disciplinary core of mathematics: proof. Build systematically from Euclid’s axioms (the roots of geometry) all the way to subtle proofs about planar figures and Platonic solids (the canopy of geometry). Hit up algebraic classics, too—the proof that root-2 is irrational, and that the primes never end. Get kids arguing, justifying, deploying logic.

Next, in 11^{th} grade, dive into the hot topic in mathematics: statistical analysis. The focus here is on describing the messy, multifaceted world we live in. How can probability capture the breadth of possibilities in life? How can statistics tease apart true claims from spurious ones?

Finally, in 12^{th} grade, reconquer the terrain where mathematics education currently spends most of its time: calculation. Explore the rules of calculus, contrasting the systematic completeness of differentiation rules with the scattershot, ad hoc methods that integration allows. Don’t just carry out algorithms; master the idea of an “algorithm” itself. Investigate numerical techniques like Newton’s Method. Write computer programs to execute rote steps so you don’t have to do it yourself.

Would any of my ideas be strictly better than the existing status quo? Maybe, maybe not. As I’ve written before, the devil’s in the details.

But it’s always fun to imagine a new recipe.

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But sometimes, it takes a little translating…

Half of my classroom conversations go like this.

Student: “I don’t get the question.”

Me: [*longwinded, exhaustive explanation of what the question is asking*]

Student: “Yeah, I knew that. But I don’t get the question.

Me: “Oh. This is one of *those* conversations.”

Kids may expect to know how to tackle a question upon first glance. Feeling unsure what to do is tantamount to “not getting” the question.

But sometimes, they might know everything they’ll need. All that’s missing is the experience of applying that knowledge, of forging connections between ideas. And that’s not something I should (or can!) tell them how to do.

This is dangerous ground. We math teachers often find ourselves defending the indefensible, trying to argue, “Ah yes, you’re going to need rational functions all the time in your career as… uh…. an HR representative.”

I try to remember that the real battleground isn’t “useful”; it’s “meaningful” or “worthwhile.” If a lesson nourishes our curiosity and makes us feel like masters, then we don’t care so much whether it’ll help us compute our taxes.

Sometimes, the work I assign really is too hard: I’ve misjudged the class’s background, overestimated the impact of yesterday’s lesson, or forgotten about the intricacies of the puzzle at hand.

But sometimes, “too hard” is actually “just right.” The problem isn’t really the problem itself. It’s the human fear of making mistakes, of taking wrong turns. The students don’t need an easier task; they need the courage (and the encouragement) to take risks on this one.

Sure, I’ve been guilty of assigning excessive work.

But far more often, I’ve been guilty of assigning *tedious* work. Knowing the importance of practice, I forget that not all practice is created equal. Practice that’s folded naturally into a meaningful task is better than a string of decontextualized problems any day of the week.

Some kids really connect with certain subjects, regardless of their scores and grades. Power to ‘em.

For the rest of us, our favorites tend to be the ones where we feel most successful. Earned the top mark in your class? Then that class is likely to earn a special spot in your heart. It feels great to be great.

In this way, grades can actually work against us as teachers, both for their artificial scarcity (“I can’t give *all* of you A’s!”) and for the sense of hierarchy they engender (“Sure, I got an A-, but my friends all got A’s”).

They create a climate where not everyone can feel successful.

What we need are ways to give kids the chance to feel successful not in comparison to others, but to themselves. It’s no easy task. But it’s *the *task.

It’s tempting to write off grade-grubbing as groundless whining, like a frivolous and opportunistic lawsuit that any decent judge will chuck out of court.

But for some students, it’s not about the grade itself. It’s about the sense of disconnect: the feeling that the effort they put in was greater than the subjective judgment I’ve rendered.

When that happens, I need to provide channels for their effort. It’s true that some effort bears little fruit. But well-directed effort, almost by definition, will yield success—and it’s my job as a teacher to act as director.

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*As always, there’s lots of thoughtful replies in the comments below, including many who disagree with me. Worth a read!*

*UPDATE 11/3/2016: I’ve now closed comments on this thread. As far as I can tell the productive discussion had been exhausted, and it had turned into a reiteration of disagreements.*

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At a conference like the HLF—bringing together researchers from across diverse fields—you’re bound to run into a few turf wars.

Mathematician vs. computer scientist.

Mathematician vs. physicist.

Even—in one delicious exchange on Tuesday—mathematician vs. mathematician.

In his morning talk, Sir Andrew Wiles emphasized a fundamental change in his field of number theory over the last half-century: its move from *abelian* to *non-abelian *realms.

Afterwards, Michael Atiyah—fellow mathematician and fellow Sir—rose to comment. After praising a “brilliant talk,” he started to redraw the intellectual boundaries.

“The whole idea of doing non-abelian theory permeates not just number theory,” Atiyah said, “but physics and geometry and vast parts of mathematics. What we’re really looking for is an overall unification in some distant future.”

Wiles mostly agreed, then laughed: “We’ve had this discussion before.”

“When I gave a lecture, probably 25 years ago,” Wiles said, “Michael told me that the future of number theory was to be subsumed by physics.”

Wiles smiled at the memory. “I was a little taken aback by this. It wasn’t what I planned for my future.”

But Wiles got the last laugh. “[Physicist] Cumrun Vafa came up to me afterwards,” Wiles explained, “and said, ‘Don’t worry about it. It’s the other way around.’”

*This is my last post from the Heidelberg Laureate Forum.
Back to your regularly scheduled bad drawings next week!*

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