Other Ways to Carve Up the Math Curriculum

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.

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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…

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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.

20161024085458_00030Then, 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.

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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.

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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.

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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:

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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:

20161024085458_00036Wait—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:

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:

20161024085458_00038Teachers 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.

20161024085458_00039Then, 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.

20161024085458_00040Next, 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?

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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.

20161024085458_00042Would 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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35 thoughts on “Other Ways to Carve Up the Math Curriculum

  1. That “Go Forth and Prosper” curriculum sound absolutely AMAZING! The high school math department I work with has actually tried to move in this direction with what limited resources we have in the way of teachers and time.

  2. How does it make any sense to have 2 years of AP Math???? The AP Curriculum isn’t even written like that!

    Anyone who wants to put the work into building one of these proposals out into a full blown plan, or just writing a textbook for one of the non-traditional courses I’m on board.

    • I definitely like proposal 2 the best, as it would allow for collaboration among departments and give students a real opportunity to try things and go for what they are interested. As a mathematician first and a teacher second and as a hobby programmer I like proposal 2 the most.

      Teaching a class on mathematical programming is crazy interesting and math for mathematicians would be right up my alley.

      Proposal 2 also seems like it could easily satisfy all of the Common Core requirements that those of us in the States have to deal with.

  3. hi Ben ,
    My daughter is a musician and we have been searching for great ways to teach her Math, I saw that you had a section-Math for Artists, could you point me into some interesting books, or ways that would be good to teach Math for people like this. I love your blog, it is the first time someone’s thinking about Math has made sense.

  4. Given a diversity of high schools for parents and students to choose among, there would be no problem in implementing any or all of these ideas — not all at any one school, of course.

    Given a federal mandate for “common core” — or any similar requirement — and only one curricular sequence per school district (in cities, one district encompasses several high schools) only one idea will “win” — and all the parents and students who might prefer and benefit from an alternative will LOSE.

    Even if any one of these ideas is better than status quo, and even if a district makes that “improvement”, all those who would choose otherwise lose because they have no choice.

  5. Interesting. I come from a high school which didn’t do calculus at all. Instead, third year algebra included probability, linear algebra and analytic geometry. Linear algebra was the worst — no applications whatsoever. And why is Cramer’s Rule ever taught? Gaussian elimination follows easily enough from systems of equations.

    The other most useless subject was going in depth with complex numbers. Unless you go into quantum mechanics, circuit theory, or other dynamic modeling that involves Laplace transforms, complex numbers are mostly a curiosity.

    Love your idea of making high school math more applied. Let me throw in one more application that used to be part of the core liberal arts curriculum back in the middle ages: statics. Statics has applications for any who would build. By going in depth with statics you can teach vectors without also having to teach calculus concepts at the same time, as is traditional in physics. The first weeks of physics are overwhelming because vectors and calculus concepts are taught in parallel. It’s like teaching juggling while riding a unicycle.

    Another reform would be to teach calculus numerically. Numerical methods are to calculus what counting on your fingers is to arithmetic. Imagine teaching first graders the topology of real numbers before learning how to count! That’s sort of what we do with calculus. With numerical methods, we can introduce interesting problems early.

    My math track would look something like this:

    basic algebra -> financial applications -> statistics
    geometry -> statics -> linear algebra for manipulating vectors
    computer programming -> beginning calculus and differential equations using numerical methods

    I leave it as an exercise for the reader as to how to turn these tracks into actual curricula. I would note that for brighter students, the beginning algebra could start in eighth grade.

    • Funny to hear you say that there are no applications to linear algebra. To me it is one the the most practical fields of mathematics.

      The whole mathematica software program is basically a big linear-algebra engine.

      A complex number is a vector. Oh, but you don’t like complex numbers either, more on that later

      Differentiation is a linear operator.

      The covariance in statistics is an inner product. (This is my life)

      The Heisenberg uncertainty principle falls-out from the Cauchy-Schwartz inequality.

      Complex numbers appear whenever there is a wave a vibration, an oscillation, and even a rotation. Quite a lot more applications than just signal processing and quantum mechanics.

      You want to teach vectors, but don’t want to teach linear algebra? The two go hand-in-hand.

      • I said there was no applications of linear algebra as it was taught to me in high school. If you look at my proposed curriculum, it includes linear algebra. The lead in is vector geometry and statics. And yes, you could come back to matrices when doing my proposed numerical calculus course. They are pretty handy for solving differential equations.

        I personally use complex numbers and get annoyed at computer languages which don’t include them. (Java!) But I’m a physicist.

        How many people do wave mechanics? Seriously? Now we are talking partial differential equations! Maybe this can wait until college. For those who do, going deeper into complex numbers can be handled at that point.

        The question before us is what to teach high school students, the majority of which will never become mathematicians, engineers, or physicists. Only a tiny fraction of such students will even need to know how to get all the imaginary bits out of the denominator. I had limited use for such things until I took up designing loudspeakers as a hobby. Complex polynomial fractions figure in crossover design big time. (Quantum mechanics has complex numbers but they don’t generally show up as polynomials in a denominator.)

        The simpler bits of linear algebra are quite generally useful. Anyone who wants to make a computer game needs basic numerical differential equations and the ability to manipulate vectors. Linear algebra is needed for analyzing trusses and the like. Statics makes for a clear use case for linear algebra.

        Everyone who votes should have some knowledge of probability and statistics. It is also handy to have some intuition of dynamical systems, to realize that cause and effect can be quite separate in time. The nascent central planner needs to experience chaos as well. It only takes a system of three equations to produce chaos, if I recall correctly. (It’s been a quarter century…)

        • Thanks for the feedback.

          I suppose if there were one unit I would add to the math curriculum, it would be Excel.

          Functions, financial calculations, numerical methods, etc. Handy tool, and something that a whole lot of people use “in the real world.”

  6. Are there really 2 years of AP calculus. As I a remember there is the AB and the BC test. The BC test covers everything that the AB test covers plus a little bit extra. But that little bit extra is not a full year’s wort of material. After AB the student should be ready for differential Equations, Linear Algebra and multi-variate calculus.

    Regarding Stats — I think that knowing calculus makes it much easier to learn the the subtleties of probability and statistics. How can you teach the central limit theorem if you haven’t defined a limit?

    In my personal experience, I learned the theory of probability by playing Dungeons & Dragons.

    And what I have learned in statistics, is that if you torture your statistics, they will confess to anything.

  7. I see one crucial problem here: the power of math is that a single language is used to tackle diverse practical applications. You are suggesting to go the other way around, and focus on the applications.

    The fact is that most people don’t use math in their jobs. The main utility of math for the general public should be to teach kids to think, to ask “why”, and be able to answer such questions.

    Your focus is more on the “how”, and it calls for math technology more than anything else. For applications of math, you don’t need much understanding of the math concepts. What’s missing more and more is the understanding of math–and the test centered education are to be blamed.

    I think a better approach to teach math is to introduce a math concept via a real life example, so kids see that the concept is not just some abstract nonsense, but is rooted in their lives (like talk about 6 apples). Then understand the new concept in the mathematical framework (like what is 6 as a number). Then show diverse applications (like we can take not just 6 apples but 6 TVs or 6 whales).

    The other problem with your proposal is that it is just too much stuff. Even the best high school students would have difficulty completing it. Kids would be able to calculate % without understanding what they are doing. There would be no time to really understand % (and hence fractions).

    To tell you the truth, I never understood the enormous focus on calculus. It’s a very complicated subject, and going beyond, say, an intuitive understanding of instantaneous speed makes little sense for the average high school student. On the other hand, a good, interesting, slow paced, playful exposition of combinatorics would fit those teenager minds and the applications of math much better.

    Here is the, imo, right amount of math and “application”. The joy and the understanding of combinatorics clearly doesn’t come from the usefulness of the practical application.

    • Calculus is enormously useful for understanding dynamical systems. Calculus is mind-numbing as it is generally taught because of the mathematicians’ love of rigor. Go lighter on rigor on the first pass and calculus makes sense.

      But the real key is to go straight to numerical methods. Most of the work in a typical calculus class is working out closed form solutions — which is a lot of work and only applies to some special cases. Such work does belong in the curriculum at some point, but maybe it should come a bit later — after students experience Euler’s method blowing up in their faces.

      • “Calculus is enormously useful for understanding dynamical systems. ”

        Even from a utilitarian viewpoint: How many people will use dynamical systems?

        But the main goal of teaching anything in K-8 and in high school is not about meeting possible applications. Despite the hype, the vast majority of kids do not use math in their lives.

        Teaching kids about instantaneous speed makes sense, but beyond that? Why should the average student need to be able to differentiate crazy functions, learn the product rule, etc?

        In general, teaching something which is hard to motivate needs to be avoided in K-12. Saying “Yeah, calculating all kinds of derivatives, numerical errors, standard deviations seems pointless to you now, but you’ll find it important later in life” is a poor argument, imo.

        • People misunderstand dynamical systems every time they blame current conditions on current leadership. Building an intuition that systems do not respond instantaneously to “forces” is important.

          Going straight to numeric allows one to look at interesting problems without the product rule, etc. Start with Euler’s method and when it breaks down dig deeper.

          Mix it with computer programming and you can relate the material with something very familiar to students: video games.

        • “Building an intuition that systems do not respond instantaneously to “forces” is important.”

          For the general student audience?

          As it is, the math curriculum (and most other curriculum) is way overstuffed with “important” things for kids to learn. The main reason for math anxiety is that the speed with which the material needs to be learned leaves most kids behind.

          What you guys recommend here is exciting, but would effectively double the stress on kids. And stress results in burnt out kids and math haters.

  8. If I were to change the curriculum, I would start in grade 7 and focus on the connections between topics rather than teaching them seperately

    We could start with the big topic of “Mathematical reasoning” in grade 7, but split into two parts. Combinatorics (with “ad hoc”-solutions” first) and geometry (with a nice and elegant theory), possibly presenting both as views on algebra (which will therefore implicitly be taught), e.q. rationals could easily be introduced using geometry rather than the usual cake-cutting routine, also we can introduce coordinate systems here and start with some linear equations from a geometric point of view.

    8th grade will then start with reviewing linear equations from a more abstract point of view: We can now introduce the concept of functions as ways to transform problems from one subject into another (e.g. geometry algebra). We continue the geometric studies with these new tools and discover the interscept theorem and the Pythagorean theorem and get more functions out of them (rational functions, quadratics, square roots).

    In 9th grade, we will leave the subject of flat, algebraic geometry. As we now have a basic understanding of functions and geometry, we can get further in abstraction: For geometry, we can study three dimensional objects and in terms of functions we try to generalize our understanding to discrete, combinatorial problems to see, that we need to develop a concept of probability (and do so).

    10th grade can now focus on two things: Basic calculus and statistics. Thereby basic calculus will include introducing trigonometric, exponential and logarithmic functions (the first in the context of reviewing geometry), describing graphs based on their term and related topics which _don’t_ need integration or differentiation (but give a basic understanding of limits in an intuitive way). Statistics on the other hand will be discussing concepts of probability, reliability etc. on a basic, discrete level.

    11th grade will aim to bring the now separated topics of algebra, geometry, Calculus and Statistics back together. We will now discuss graphs of functions as geometric objects and in view of geometry try to define limits more precisely and then introduce derivation motivated by geometry, get back to three dimensional objects and their algebraic representation (in terms of coordinates) and combine Calculus and statistics by looking at distributions for random variables in a large set.

    12th grade will then be mostly focused on computations and some integration, which is good to combine, since in many cases integration is not easily possible. However, it will be about applying all of the previously discussed topics, which means that the problems will include many different aspects. We could probably do a pratical project during this last year to see the connection between all of the topics.

    (This curriculum would not be so different from the one I had in school, apart from switching some topics between 8 and 9 and the practical focus in grade 12, however in all years the aim should be to show the connections between the subjects rather than teaching them seperately)

    • “focus on the connections between topics rather than teaching them seperately”

      Yes, this is exactly the basic role of math: a common language. I think the most important guiding principle in designing a curriculum is not what society wants or expects from the kids, not what is considered important or exciting to the adult world, but what kids are capable of enjoying and comprehending, and what they are curious about.

      If you say, this condition very much depends on the individual student and the place, you are correct, and this is why a general curriculum for the whole US makes little sense. Teachers need their autonomy back more than any guidance about what to teach.

  9. This plan leaves financial education to the 11th grade. This is too late. It needs to be moved earlier, say 8th grade, to reach as many students and as many households as possible. The students that will drop out need financial education more than the students who reach the 12th grade. Putting financial education/information into homes that are prone to have drop outs may benefit the whole family.

        • I read it, and I don’t see it as a math curriculum. It’s an applications of math curriculum. Those who prefer the applications of math curriculum over math seem to think that it’s always preferable to talk about 6 apples, and don’t like to talk about just the number 6.

          It of course is interesting to be able to talk about, say, compounded interest. But in a math curriculum, its interest is secondary; it’s more important to learn that that the same math can be used to handle radioactive decay, age of ancient artifacts, population increase, spread of diseases, seed arrangements in a sunflower.

          The math is much simpler than these, and hence it is worthwhile to explore it on its own.

  10. Pingback: Carnival of Mathematics 140 – mathematicsandcoding

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