As a high school student, how can I get ahead and teach myself math? [Context (Can be skipped)]
Hello, I am a grade 11 high school student in Canada. For most of my time as a student I have been pretty bad at math. It wasn't that I didn't like math or didn't have a natural aptitude for it, but rather had to do with a really bad education in elementary school that didn't get repaired until relatively recently.
Anyway, I decided to buckle down and fix the gaps in my knowledge during grade 10 which brought me from a 61% in grade 9 to a 75% in grade 10. I decided to go even further this year and pushed myself to a 92% in grade 11. Once again, I am looking to challenge myself.
[Question]
I want to have an advantage going into grade 12 by learning math from now until next fall. What subject should I learn and what resources do I need to learn it? I have gotten very good at self study, so that isn't an issue.
I have heard abstract algebra might be a good place to start. Is this true?
P.S I am in grade 11 and have not started any kind of calculus yet. My next course will be advanced functions and then calculus and vectors.
 A: There are several reasonable ways to approach this, but it really depends on what you are looking for. I'll list three options in order of increasing pragmatism.


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*The gutsy approach


The first option is my favorite: jump right into abstract algebra. For this, I would highly suggest using I. N. Herstein's Abstract Algebra (not to be confused with Topics in Algebra). If you search around, you can probably find a pdf online (there used to be one at least), or get a hard copy if possible.
This will likely not be all that helpful for preparing you for your classwork next year, but rather would be a longer-term investment. Having done this myself (slightly later in my school career), it was an exceedingly tough but exceedingly rewarding experience. It took me several read-throughs over more than a year to get an understanding of the content (for the most part). I should warn you that its a slow start - it took me probably two months to really get the first chapter. However, it speeds up from there.


*The balanced approach


Instead of abstract algebra, you might consider linear algebra first. It does not require calculus - actually, it might be quite helpful to do before calculus - but it does require a higher level of mathematical maturity than pre-calculus material. However, I personally think it is more accessible than calculus. Furthermore, it is a great introduction into higher mathematics that builds very naturally on content from pre-calculus algebra (in a way it is a multi-dimensional version of certain parts of pre-calculus algebra). Because of this, it might actually prove quite practical in preparing your for your classwork next year.
Really, linear algebra is the best preparation for multi-variable calculus (of which vector-calculus is a topic). One word of warning though - don't get misled by the terminology - the linear functions in linear algebra (really, linear transformations) are subtlety different than the linear functions you've likely seen before. Truthfully, things make sense in the end, but in the intermediate you might get quite mixed up.
Unfortunately, I can't really recommend a book for this. You may need to search around to find one at the right level. I would suggest something that focuses on vector geometry, since that makes the subject both more approachable and enjoyable; the elegant and practical way linear algebra combines geometry and algebra is really what makes it a spectacular topic to learn and to know.


*The pragmatic approach


The most practical approach might be to go straight to calculus. I'm not fond of this suggestion, since it's often given just because its the obvious solution (and not really the best). But there is a benefit to working on calculus that can't be achieved with the other approaches - you get the chance to go back and improve your foundations (i.e. review previous material) exactly when it's relevant. That is, you will be using knowledge from previous classes all the time in calculus. Every time you get to a new calculus topic, pause and figure out what background knowledge it assumes, and then go review that.
You will probably be surprised at how helpful this is. In my experience, every student has a harder time than necessary with calculus because of missing background knowledge. For most students, a weakness in algebra really holds them back in calculus. A while ago, when teaching someone calculus, I had the humorous experience of suddenly and unexpectedly understanding the purpose of a few pre-calculus topics - they happened to be essential for understanding a very particular calculus topic.
Put another way, learn calculus as a review tool. Rebuild your knowledge of previous material on top of an understanding of how it's used in calculus. This will give you a big head start on calculus while also strengthening your background. Plus it may give you a different perspective on the material, which is always helpful.
Again, I unfortunately don't have any book recommendations. For both calculus and linear algebra, you can find suggestions here on MSE. You might need to do more digging though to find the right book.
In addition, you may want to look at some videos on calculus and on linear algebra. There's two places in particular to look - on khan academy, and on the Youtube channel 3blue1brown. They are complementary to each other in style, but also could complement a good textbook too.
A: I would learn calculus next. It's what you are going to be learning in school and there are a ton of good resources out there to learn it via that you have more options than just buying a text book and going through it like you will for other subjects. The other benefit is that what you learn over the summer you will then be learning again in the fall at school so it will really cement in the knowledge, where if you learn something like linear algebra, you are just going to forget everything you learned unless you are using it for something (which you wont be).
Before you start trying to learn abstract algebra or anything more advanced, I would get some kind of book on logic and proofs, otherwise you are going to miss out on a lot of whats going on. I don't really know what is considered the gold standard, my proofs class in college used "How to prove it" by Daniel Velleman, and I thought it was fine.
A: One thing that I would strongly recommend if you really want to learn about advanced math is to learn about mathematical proofs. Understanding this is essential to understanding most advanced mathematics. In the higher level pure math courses, proofs are just a regular part of what you do. I would probably actually recommend this before doing abstract algebra. If you do this can actually get a better understanding of linear algebra as well (even though you can understand it without formally knowing proofs).
This is probably the most important thing for learning advanced mathematics.
I don't really know much about books for this, but one is How to Read and Do Proofs: An Introduction to Mathematical Thought Processes by Daniel Solow.
