I’ve read the quote above somewhere a couple of weeks ago, but it escapes me where. Anyway, I gave it a lot of thought because as an instructor I’ve given a lot of the same advice to a lot of my students so I figured why not, let’s write a blog.

This particular blog will focus on math education so if you hate math, feel free to stop reading 🙂  In particular, it will focus on math prerequisites.

When I taught slightly higher level mathematics courses (like Calculus II and Differential Equations), I had a decent chunk of students who were just not mathematically prepared to be in those courses and yet they somehow made it in. So after a bit of analysis, I came up with a nice hierarchy of what you should know throughout your mathematical education.

Note: This goes through Differential Equations (typically one of the last courses required for a math minor) but a similar argument may be made for higher courses as well.

Have you ever played any of those online flash games where you complete stages and level up between them? And when you level up a whooole bunch and attempt the first stage again, it’s a piece of cake? Well, math works like that too. As you take math courses, you “level up” your mathematical knowledge, and earlier topics (should) become a piece of cake. In this case, the formula is very simple: You should be very familiar with the topics of a course two “levels” below the one you are currently attending. A typical math course progression in college looks like this:
Algebra
Trigonometry
Calculus I (Differential Calculus)
Calculus II (Integral Calculus)
Calculus III (Multivariate Calculus)
Differential Equations

[Note: I did not include precalculus, as, in my experience, it’s simply college algebra with a few chapters of trigonometry thrown in. Seriously, when I taught precalculus and college algebra at the same time, the author for both books was the same. She simply took the college algebra book she wrote, didn’t even change examples, and threw in four chapters of trig right in the middle of it and called it Precalculus. Great books though.]

So if you’re attempting, say, Calculus II, you should be great at Trigonometry. Every one of these courses uses some content of the course immediately preceding it, but it really requires mastery of the course two levels before it – Calculus I is very algebra heavy especially in things like optimization problems; the most difficult problems of Calculus II usually involve some sort of trigonometry; Calculus III is just Calculus I on steroids; Differential Equations involves a lot of integrating so you have to be familiar with techniques from Calculus II. If you can keep up your mathematical mastery to where it’s two levels below where you currently are on your mathematical journey, you will be successful.

Vazgen Zakaryan