This presentation has been around for a few years, but I’ve never highlighted it here on my blog. It’s a Prezi presentation by Alison Blank and it’s called Math is not linear.
It brings up an excellent point that math is often talked about as being a sequence with one class coming before another, when in fact math isn’t always like this. It is absolutely true that there’s room for prerequisites (it’s not a great idea to take Calculus 3 before Calculus 1), but much of mathematics can be approached at any time.
math courses–perhaps more like a tree then a line?
Math majors realize this when they get to college. Once you take a few basic classes like Calculus, Linear Algebra, and Statistics, you can take almost any other course you want. Occasionally there are other prerequisites (a two-course sequence in Differential Equations might require students to take them in order). But generally, you can order your college math courses in many different ways.
I’ve noticed this in grad school too. Here’s the order I’ve taken my classes so far with my comments:
- Differential Equations (took it in undergrad too, which helped!)
- Cryptography (might have been good to do abstract algebra first, but we learned the practical things we needed to know along the way)
- Abstract Algebra (when we talked about cryptography and elliptic curves, it was total review!)
- Real Analysis (took it in undergrad, but boy I understood it a lot better the second time around)
- Statistics (took it in undergrad, but I loved learning it from a more theoretical, Calculus-based approach)
- Queueing Theory (stat required, for good reason; just started this course, so I can’t tell you much more)
Did I have to take these courses in this order? Not necessarily. I’ve found that cross-references are made between courses constantly. In my queuing theory class this past week, we solved a differential equation. So there! 🙂
Could we do this in high school? To some extent, yes. I think it’s still important to have prerequisites, however. And to get to Calculus, you have to take a *somewhat* linear path. (Do you agree that Calculus is the pinnacle of high school mathematics?) Along the way, though, we should indulge in lots of mathematical tangents, as the Prezi suggests.
In some ways we already allow our students a bit of latitude. We offer AP Statistics, which can can be taken almost anytime after Algebra 2. And we offer Higher Level math, which gives a nice taste of college-level math.
As Alison Blank suggests, could we teach a little topology to high school students? Certainly we could. Weren’t we all interested in the advanced math topics before we actually took the class? I remember learning all the fun and interesting results from topology way before I ever read through a topology textbook. I still haven’t taken a class in topology! (But I read a lot of Munkre’s book and did a lot of the exercises, and I think that counts for something :-).)
The point is, I think we can talk about fun and interesting math ideas way before we’re “allowed” to talk about them!
That’s certainly what my dad did with me at home. Thanks dad!
I dedicate this post to all those math teachers who went on a mathematical tangent this week. Keep up the great work!
Random Walks 
And your older brother learned trig functions in 6th grade by programming an Apple II with parametric equations y = a sin b t, x = c sin d t, creating Lissajou figures. He didn’t know about the right triangle definition or the Taylor series definition. He was able to treat them as fun black boxes.
Yes! Similarly, I remember you or Tim showing me how to get a particle to move around in a circle when I was programming in basic on the Apple II. I had no knowledge of parametric equations at all, and it was a black box to me too!
And the beautiful thing about computers with a Von Neumann architecture is that, for all the appearance of nonlinearity, it’s all a set of linear code/data under the hood. Something metaphysical about linear=nonlinear 😉