[Requesting Advice] Applying Instrumental Rationality to College Course Selection Dilemma
I'm faced with a dilemma and need a big dose of instrumental rationality. I'll describe the situation:
This fall, I'm entering my first semester of college. I'm aiming to graduate in 3-4 years with a Mathematics B.S. In order for my course progression to go smoothly, I need to take Calculus I Honors this fall and Calculus II in the spring. These two courses serve as a prerequisite bottleneck. They prevent me from taking higher level math courses.
My SAT scores have exempted me from all placement tests, including the math. But without taking a placement test, the highest any math SAT score can place me into is Pre-Calculus Honors, which is one level below what I want to take in the fall. The course progression goes Pre-Calculus Honors to Calc I Honors to Calc II Honors.
So in order to take Calc I Honors in the fall, I either need to:
(1) Score high enough on a College-Level Math placement test or
(2) Forgo the test and take Pre-Calc Honors for 9 weeks this summer
I've taken both pre-calculus and calculus in high school. I've also been studying precalculus material over the past few days, relearning a lot of what I've either forgotten or wasn't taught in class. If I decide to take the test, I'm pretty confident I'll place into Calculus I. I'd estimate that chance being within 0.8, plus or minus 0.1. If I pass the test, I'll save 9 weeks of studying in the summer and use them to prepare for classes I'll be taking in the fall. I'd also free me up to take another summer class worth 4 credits and fulfill a prerequisite.
But if I decide to forgo the test and take Precalc this summer, I'm also pretty confident I'll do very well in the class. I'd confidently wager above a 90%. The class would ensure I've got the material down better than the placement test and would also give me my first six credits.
The questions going through my mind right now include: How can I best decide between these two options? How can I compare the heterogeneous benefits/costs? Are there any other relevant factors that I'm leaving out?
Advice would be greatly appreciated.
Edit: Writing this post, as well as reading and responding to the comments, has clarified the situation for me. Unless there is something else important I've missed, I'll take the test, place into Calc I, spend the summer taking a different summer class and preparing for fall classes. Thanks to everyone who helped me out.
As someone who just finished my sophomore year as a math major, I think I can give some useful advice in the vale of tears that is a mathematics degree.
All in all, it comes down to how much your GPA matters to you versus how much math matters to you when choosing courses. Even if you are ridiculously smart, most of the stuff you see after calculus and linear algebra is going to be pretty damn hard, and in order to get something substantial out of those courses you'll have to spend a large amount of time staring at symbols.
So if you want to maintain a good GPA, limit your desire to speed ahead and focus on the recommended courses. You'll then have the time to be able to really understand the material and have good grades. Even if you were at the top of your class in high school, your GPA will benefit from understanding this. I would even recommend going slower than the pace set by the administrators. No matter how ready you think you are for a certain course, there will be a point where you have absolutely no clue what the fuck is going on. Trust me.
In my opinion, you will get as much out of doing this as you would if you sped ahead but kept the same work ethic. I use this heuristic: If I want to take another math course and have the same GPA and an increased net mathematical knowledge gain, I need to increase my work ethic by ten. If I'm missing a pre-requisite, I need to increase it by twenty. Grad courses are a hit or miss; sometimes they can be an easy, relaxed way to get into higher math, and sometimes they can be insanely hard.
Now, if you don't care about your GPA, then take as many math courses as you can. That's what I did. Worked my ass off for B's and C's. The only reason why it works for me (in terms of my level of satisfaction with my choices) is that I don't (and didn't) do much of anything other than math. So I was able to really delve into all of these topics and come out with internalized knowledge - but I had to sacrifice my ability to complete assignments on time and prepare adequately for exams. Had I focused on getting A's... I might've been able to do it, but it would be at the expense of optimal learning (not that I didn't try to get A's with my "internalized knowledge", I'm just really driving home the point that this shit is hard, especially in timed situations).
I guess what I'm getting at here is: don't overestimate yourself if you want to keep doing and loving math. Know your breaking point, or at least remember that you have one - you will hit it, and it will hurt. Even if you are not super into math and just want to use it another things, the core courses are still very hard and this advice is still valid. And if you do want to skip ahead and do as much as possible, think about how much harder you think you will have to work, and multiply that by ten. This is if you actually want to get anything out of these courses - I'm sure you can skip ahead and get A's, but you won't have gained much. Unless you're Gauss. (On that note, you will encounter a lot of this, even as an undergrad).