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I’m a college student who just finished up the semester and has 4 months (and any free time during the following school year) to go all out and relearn all of the fundamentals of math up to but excluding calculus. I just placed a hefty order on Amazon, but I still feel confused about the content I chose. After tons of research, I’ve essentially narrowed down my list of resources in the following categories to the following books (they should be arriving by early-mid May).

**Geometry:**

Lang & Murrow - Geometry

Kiselev - Geometry Vol I & II

Moise - Elementary Geometry from an Advanced Standpoint

Gelfand - The Method of Coordinates

Gelfand - Trigonometry

**Algebra:**

Gelfand - Algebra

Gelfand - Functions and Graphs

Lang - Basic Mathematics (not sure if this is considered algebra or just thoroughly covers concepts that are needed for algebra)

**Logic & Proofs:**

Eccles - An Introduction to Mathematical Reasoning

Polya - How to Solve It

Velleman - How to Prove It

I chose these books because they were basically the most recommended for beginners in each topic. People said they were rigorous, meaty, and were great for adults who aren’t learning math for the first time. It really looks like they are, but I’m still a bit concerned.

I’d like to be able to work through the books I’ve ordered over the next year and spend my next summer trying to work through Spivak’s Calculus. The thing I’m kind of unsure of is how these books compare to other recommendations I’ve seen? For example, tons of people recommend Jacob’s geometry books, and comparing the page count alone, it looks like it’s a lot more content than the Kiselev books, yet it’s recommended at a lower frequency. Another example is comparing Gelfand’s Algebra, and Lang’s Basic Mathematics, to a more standard textbook like one from Blitzer or Jacob’s Elementary Algebra. Each of the latter books are many times the size of the former books combined, and even a comparison of the table of contents leaves a lot to be desired. Now I get that it’s an unfair comparison as there’s a ton of filler in the Blitzer books, but things like recursive formulas, logarithms, matrices, aren’t covered at all in Gelfand’s book, and Lang’s book although looks like it spends a few pages on each of those topics, there isn’t anything about recursion. All of those topics are covered at great length in Blitzer’s Algebra and Trigonometry 6th Edition. What I’m trying to get at is, should I have these other books as well to fill in these gaps with? I keep reading people saying that Gelfand and Lang are all you need to get ready for Spivak, but It seems like there’s a ton missing.

Another thing that’s puzzled me is how the books in my list compare to the Art of Problem Solving (AoPS) series. I’ve read they’re geared towards competition math, but they get great reviews. They’re regarded as a difficult set of beginner/intermediate level math books as well. How would someone who completes that curriculum compare to someone who goes through the list I’ve provided? Would one be better than the other at going further down the pure math path? Would the other have troubles adapting to competitive math later on? Is one type of student (competition vs pure math) generally regarded as a better mathematician than the other? I’ve noticed a lot of the AoPS books also have tons and tons of problems to solve. Would it be useful to just use them for the problems? Or will the things I learn not mesh well with them since they’re more focused on contests? I read through a short discussion regarding this on another site, not sure I'm allowed to post links so I won't, and it left me even more confused. There was barely any agreement throughout the whole discussion comparing the two approaches (AoPS vs Gelfand/Lang/Kiselev in this example). Two users went back and forth arguing over which is better than the other.

I guess I’m basically worried that I might not be learning all that I need to, or that things are explained in a different/better way elsewhere. I just want to make sure I’m using the best resources available to me. After all, if AoPS is the most popular curriculum for gifted students, they must be doing something right, right?

tl;dr I wanna spend my summer relearning math. I ordered some of the most recommended books for algebra/geometry but it looks like they don’t cover as much as other books. Is it just because I'm not knowledgeable enough yet that I'm not comparing books properly just by looking at their table of contents? Will the books in my list really help me learn as much as Jacobs, AoPS, and Blitzer considering they cover way more stuff?

Thanks for any guidance you may be able to provide!