I’m currently trying to teach myself mathematics from the ground up, so I’m in a similar situation as you. The biggest issue, as I see it, is attempting to forget everything I already “know” about math. Math curriculum at both the public high school and the state university I attended was generally bad; the focus was more on memorizing formulas and methods of solving prototypical problems than on honing one’s deductive reasoning skills, which if I’m not mistaken is the core of math as a field of inquiry.
So obviously textbooks are good place to start, but which ones don’t suck? Well, I can’t help you there, as I’m trying to figure this out myself, but I use a combination of recommendations from this page and looking at ratings on Amazon.
Here are the books I am currently reading, have read portions of, or are on my immediate to-read list, but take this with a huge grain of salt as I’m not a mathematician, only an aspiring student:
How to Prove It: A Structured Approach by Vellemen—Elementary proof strategies, is a good reference if you find yourself routinely unable to follow proofs
How to Solve It by Polya—Haven’t read it yet but it’s supposedly quite good.
Mathematics and Plausible Reasoning, Vol. I & II by Polya—Ditto.
Topics in Algebra by Herstein—I’m not very far into this, but it’s fairly cogent so far
Linear Algebra Done Right by Axler—Intuitive, determinant-free approach to linear algebra
Linear Algebra by Shilov—Rigorous, determinant-based approach to linear algebra. Virtually the opposite of Axler’s book, so I figure between these two books I’ll have a fairly good understanding once I finish.
Calculus by Spivak—Widely lauded. I’m only 6 chapters in, but I immensely enjoy this book so far. I took three semesters of calculus in college, but I didn’t intuitively understand the definition of a limit until I read this book.
I’m currently trying to teach myself mathematics from the ground up, so I’m in a similar situation as you. The biggest issue, as I see it, is attempting to forget everything I already “know” about math. Math curriculum at both the public high school and the state university I attended was generally bad; the focus was more on memorizing formulas and methods of solving prototypical problems than on honing one’s deductive reasoning skills, which if I’m not mistaken is the core of math as a field of inquiry.
So obviously textbooks are good place to start, but which ones don’t suck? Well, I can’t help you there, as I’m trying to figure this out myself, but I use a combination of recommendations from this page and looking at ratings on Amazon.
Here are the books I am currently reading, have read portions of, or are on my immediate to-read list, but take this with a huge grain of salt as I’m not a mathematician, only an aspiring student:
How to Prove It: A Structured Approach by Vellemen—Elementary proof strategies, is a good reference if you find yourself routinely unable to follow proofs
How to Solve It by Polya—Haven’t read it yet but it’s supposedly quite good.
Mathematics and Plausible Reasoning, Vol. I & II by Polya—Ditto.
Topics in Algebra by Herstein—I’m not very far into this, but it’s fairly cogent so far
Linear Algebra Done Right by Axler—Intuitive, determinant-free approach to linear algebra
Linear Algebra by Shilov—Rigorous, determinant-based approach to linear algebra. Virtually the opposite of Axler’s book, so I figure between these two books I’ll have a fairly good understanding once I finish.
Calculus by Spivak—Widely lauded. I’m only 6 chapters in, but I immensely enjoy this book so far. I took three semesters of calculus in college, but I didn’t intuitively understand the definition of a limit until I read this book.