So it was the Old Math Dog who pointed out that I never wrote a post explaining how I deal with the issue of kids not knowing basic algebra in calculus. I started this practice two years ago (when I also started standards based grading) and I have seen a remarkable difference in how my classes go from my life pre-bootcamps to my life post-bootcamps…
An issue in any calculus course — and I don’t care if you’re talking about non-AP Calculus or AP Calculus — is the student’s algebra skills. They might see and have no idea how to solve that. Or they might not know how to find . Or they might cancel out the -1s in to get . It depends on where they are coming from, but I can pretty much guarantee you that every calculus teacher says the same thing to their classes on the first day:
Calculus is easy. Algebra is hard.
In my first three years of teaching calculus, I started with how all the books started, and all my calculus teacher friends started: a precalculus review. Then we went into limits.
The problem with that is that we might review some basic trigonometry, and then we wouldn’t see it again for months. And by then, they had forgotten it. And who could blame them. The precalculus review unit at the beginning of the course wasn’t working.
As I transitioned into Standards Based Grading, I looked at everything I taught really closely, and I honed in on the particular skills/concepts I was going to be testing. And since I’d taught calculus for a number of years prior, I knew exactly where the algebra sticking points were. Thus was born The Algebra Bootcamp.
Before our first unit on limits, I carefully analyzed what things I needed students to know to understand limits to the depth I required. I then looked at all the skills and thought of all the algebraic things, and all the old concepts, they would need in order to understand limits. And from that, I crafted an algebra bootcamp, and I made SBG skills out of just those limited skills.
For example, here was our first bootcamp (which, admittedly, was longer than most of the others, because we were settling in and I was gauging where the kids were at):
and I did the same for other units… just the targeted prior knowledge that they tended to not know or struggle with…
Notice how they tend to be very concrete and specific? Like “rationalize the numerator” (because I knew we were going to be doing that when using the formal definition of the derivative) or “expand using the binomial theorem. Very specific things that they should know that they are going to be using in the following unit. It’s kind of funny because it is a hodgepodge of little (and often unconnected) things, and they have no idea why we’re doing a lot of what we’re doing (why are we rationalizing the numerator? why are we doing the binomial theorem?) and I don’t tell them. I say “it’s our bootcamp… once training is over you’ll see why these tools are useful.”
It is called “bootcamp” because I am not reteaching it from scratch. I’m reviewing it, and I go through things quickly. I only do a few of them in the first quarter and maybe the start of the second quarter. By that point, we’ve done what we needed to do, and they die off.
The reason that this has been so effective for me is because students aren’t having to relearn old topics/algebraic skills while concurrently learning the ideas of calculus. We review these very specific things beforehand so that when we approach the calculus topics, the focus is not on the algebraic manipulation or remembering how to find the trig values of special angles or what a piecewise function is… but on the larger picture…. the calculus.
Remember: calculus is easy, it’s the algebra which is hard.
So we took care of the algebra beforehand, so we can see how easy calculus is.
My kids in the past two years have made so many fewer mistakes, and we’ve been able to really delve into the concepts more, because I’m no longer fielding questions like “could you review how to do X?” Doing this has also forced me to think about what the purpose of calculus class is. The more I teach it, the more I take the algebraic stuff out and the more I put the conceptual stuff in. For example, I don’t use , , and in my course anymore , because I wasn’t trying to test them on their knowledge of trigonometry. Doing these bootcamps coupled with standards based grading has forced me to keep my eye on what I really care about. Students deeply understanding the fundamental concepts of calculus. And I think you can do that without knowing how to integrate just fine. 
 With the exception of for the derivative of .
 I teach a non-AP calculus, so I have this luxury. But it’s nice. Each year I strip more and more stuff off the course and add in more and more depth. And I am glad that I understand depth to mean something other than “more complicated algebra in the same old calculus problems.”