If the Food Network has taught me one thing, it’s that how you plate a meal matters almost as much as what you’re serving.
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.
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 11th and 12th grade. (You might even encounter easy bits of differentiation in 9th and 10th.) 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:
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 10th 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 11th 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 12th 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.