The mathematics of high school - explained

 Hi, my name is Garrett Zhou. I am currently a freshman at Durham Academy Upper School and enjoy playing soccer, studying, and generally maintaining discipline. I have been an AIME qualifier for those who know, and I consider myself a very strong student. 

The order of high school math can be hard to understand. Algebra, Geometry, Calculus, there is so much to unpack. Today, I'll explain the traditional math route that kids follow in High School, as well as what I took. At my school Durham Academy, traditionally kids took Algebra 1, Geometry, and Algebra 2 as required by our graduation requirements. Many kids went beyond this, though. There was the 'normal' group of kids, some of whom were pretty smart, who took Geometry in freshman year, then Algebra 2, then Precalculus, then Calculus AB. Then, there were the 'advanced' kids, who took Algebra 2 freshman year, typically Honors Precalculus sophomore year, Calculus AB or BC in junior year, and Multivariable Calculus in senior year. Then, you had the absolute nerds who took Honors Precalculus freshman year, Calculus BC sophomore year, Multi junior year, and took a free period in senior year. Then, there was me. I took Calculus BC as a freshman, along with AP Statistics (bad idea, this workload was pretty insane). I did very well in the class, getting a 98% average for the year, and (hopefully) a 5 on the AP exam. Yet, I am not the smartest person I know. One of my close personal friends took Multivariable Calculus in 8TH GRADE. I know, it's insane. However, what I am really here to talk about is how math can be taught in high school to lead more students to success. To me, math has come really easy, but I know that that isn't true for everyone. If more math teachers clearly explained every topic, it could make a huge difference. I know you may think this is a very low bar, but in my school, the number of teachers who had the label "doesn't actually teach" is crazy. It had to be half of the mathematics teachers in the school. There are many ways that math teachers could do this. One is by having more demonstrations of a topic. I vividly remember the Direct Comparison Test for convergence/divergence with a demonstration that my teacher gave my class: he put 2 airplanes, one he named a(n), and the other he named b(n). He said that if a(n) was larger than b(n), then if b(n) diverged, a(n) had to diverge because a(n) is larger than b(n). He then proceeded to show airplane b(n) flying to infinity, and because airplane a(n) is always higher than b(n), then a(n) had to fly to infinity as well. Teaching like this not only creates a more fun class environment where kids actually want to go to class, but it is also easier to remember. Because of this example, I always preferred the Direct Comparison Test to the Limit Comparison test because of the vivid image of airplanes in my mind. 

The actual math courses at your high school may differ from mine, such as having Math 1, Math2, etc, or maybe having very creative math classes (I have no idea what these would be). The main math sequence in high school will go: Algebra 1, Geometry, Algebra 2, Precalculus, Calculus (usually AB, sometimes BC), Multivariable Calculus, Linear Algebra, and if your school is very lucky, maybe you will have analysis or differential equations. There are also other side options of math like Statistics, Math Modeling, non-Euclidean geometry, etc. I will give you my opinion on courses that I have taken and learned about. 

Algebra 1 is very useful for not just Algebra 2 but also Precalculus and Calculus. It is relatively easy to study on your own and is decently useful in other classes that require math such as Chemistry or Physics. In terms of math competitions, it is a little too trivial, but you may find some problems that require it.

Geometry is not very useful beyond itself, maybe it has some uses in Multivariable Calculus, but is very practical in the real world. It is either really hard or pretty easy. It all depends if you can truly understand and visualize the concepts in your head. It may be used a little bit in Chemistry for molecules and in Physics, and it also has a heavy use in the AMC 10 and AMC 12.

Algebra 2 is a continuation of Algebra 1, and it is not very hard to learn by yourself. It is relatively useful in Precalculus and Calculus, but not really beyond that unless you go into abstract algebra. It is pretty heavily used in AMC tests, but is not very practical and used sparingly in Chemistry, Physics, or Biology.

Precalculus is a prerequisite to Calculus, and introduces heavy use of trig, complex numbers, and imaginary numbers. Not quite as easy to study on your own, it has very heavy use in AMC 12 and AIME. It is used in Calculus and Multi, but not much farther. It can be used for Chemistry and Physics, not used as much in biology.

Calculus is the epitome of high school classes. When little kids think of hard classes, they think of Calculus. For me, Calculus wasn't that hard. If you only want a good score on the AP exam, self studying is relatively easy because you have to get a 60% on the test to get a 5. Additionally, Calculus is very useful in very advanced Physics classes, which translate to Engineering and Economics. Math competitions that involve calculus are usually hard asf. Take the Putnam Test for example. The average scores is a 2/120. Crazy. 

I haven't taken beyond Calculus yet, so stay tuned for updates and thanks for reading!