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I was once in your shoes. Here are the books that helped me out of that spot. I recommend them in this order. 1/ Geometry and the Imagination. Yes, it's geomet

I was once in your shoes. Here are the books that helped me out of that spot. I recommend them in this order. 1/ Geometry and the Imagination. Yes, it's geometry. Yes, it's written in a somewhat older style. No, it doesn't do a lot of hand-holding. But if you can spend the time to really understand the dazzling intellectual fireworks going on there it will reward you with a very strong intuitive understanding of a lot of practical mathematics. The key goal of this book was to teach insight and it totally hits the mark. 2/ Contemporary Abstract Algebra by Gallian. This helped me learn more of the language of modern mathematics, so I could be more comfortable approaching recent developments in my field. It is an approachable, not-excessively-rigorous text and again, emphasizes intuition over deep rigor. Even if you don't really need abstract algebra, I'd recommend this book. 3/ Proofs From The Book. This is a brief survey of a selection of especially elegant and beautiful proofs. This is partly valuable for the immediate facts you'll learn-- I have surprised coworkers with a solution to a seemingly difficult problem that closely mirrors something in here more than once-- but is mostly about understanding mathematical standards of beauty and why proofs might be structured the way they are. 4/ TEA's problem solvers for whatever you actually need (calculus, etc). These books are just piles of worked out problems. Personally I found it useful to study several of the hard problems, understand the mechanics of solving them, and then go back to the easy ones and make sure I could work all the way through them without error. In my opinion, combining this practical and detail-oriented legwork with the above theoretical material helps avoid fooling yourself into thinking you can solve problems you really can't. I'll caution that in math there is no substitute for hard work. Where in programming being lazy is often a virtue, learning math was something that I really had to grind through even when I was enjoying it. It's just a guess, but I suspect the same might be true for you; best to expect it.

westoncb 8y ago
As a counterpoint, I have decent math chops and even do a lot of computational geometry professionally, and I really wanted to like "Geometry and the Imagination"—but for me I was till missing way too much information to figure out what he was saying half the time (much of because of the antiquated style), and after putting lots of work into some section I'd get to the end and be like, "this is kinda neat, but it's not rockin' my world or anything..." (to be fair, I did only make it halfway into the second chapter, on lattices iirc, and the later chapters did look more interesting). I would definitely not recommend it for someone working on high school level math. If I had made an attempt at that in high school, being told it was something I should be capable of then, I probably would have totally given up on any ideas I had about being able to do math well. If you look at a lot of reviews for math books, you'll find there is a major split between folks who, on one side, believe a math book should should be a sort of pure, elegant, perfect thing with barely anything in it but statements of definitions, theorems and proofs. It should be difficult above all. There should be no trace of how the contents within it came to be, just a pure presentation of some set of mathematical truths. The other side values pedagogy. The mathematical results in themselves are not considered sufficient on their own to be good teachers; instead, attention must be paid to the psychology of one's readers, and in consideration of it, the best route for making contact from the reader's knowledge to the author's must be taken. A typical trend is that the presentation includes context on why and how the results were developed. I've seen a similar split in CS, and I can say at least here, that I haven't seen any correlation between those insisting on doing everything the hard/austere way and doing interesting/technically impressive work. That said, I haven't personally done anything particularly impressive in mathematics, so I'll refrain from making too strong a comment about the necessity of any particular pedagogical style in that realm. But I would bet two things: 1) We lose many folks who could have become good mathematicians because they were freaked out by the sort of attitude and presentation making up the first half of the 'split' I describe above 2) It is probably significant that not all excellent mathematicians choose the trial by fire pedagogical approach in their own works.
- debatem1 8y ago
  Hmm. The books I put up here are frequently criticised for focusing on pedagogy at the expense of rigor. So I'm not sure what to say to that. Overall I agree that too much of mathematics is obfuscated. Whether that's because it makes the author sound smart or because excellent prose isn't a big timesaver for a reader who has to grapple with a complex proof anyway I don't know. Could be both. But I will reiterate is what I said above: deeply understanding an unfamiliar result or method often is irreducibly hard work that simply must be ground through. That's no reason to make simple things needlessly hard but when it's time to work through something tricky, well, people should expect to spend some time on it. Importantly, they should not feel stupid or bad at math when they have to do so, which I think is probably the thing that loses us the most mathematics besides obfuscated intro calculus.

westoncb 8y ago

As a counterpoint, I have decent math chops and even do a lot of computational geometry professionally, and I really wanted to like "Geometry and the Imagination"—but for me I was till missing way too much information to figure out what he was saying half the time (much of because of the antiquated style), and after putting lots of work into some section I'd get to the end and be like, "this is kinda neat, but it's not rockin' my world or anything..." (to be fair, I did only make it halfway into the second chapter, on lattices iirc, and the later chapters did look more interesting). I would definitely not recommend it for someone working on high school level math. If I had made an attempt at that in high school, being told it was something I should be capable of then, I probably would have totally given up on any ideas I had about being able to do math well. If you look at a lot of reviews for math books, you'll find there is a major split between folks who, on one side, believe a math book should should be a sort of pure, elegant, perfect thing with barely anything in it but statements of definitions, theorems and proofs. It should be difficult above all. There should be no trace of how the contents within it came to be, just a pure presentation of some set of mathematical truths. The other side values pedagogy. The mathematical results in themselves are not considered sufficient on their own to be good teachers; instead, attention must be paid to the psychology of one's readers, and in consideration of it, the best route for making contact from the reader's knowledge to the author's must be taken. A typical trend is that the presentation includes context on why and how the results were developed. I've seen a similar split in CS, and I can say at least here, that I haven't seen any correlation between those insisting on doing everything the hard/austere way and doing interesting/technically impressive work. That said, I haven't personally done anything particularly impressive in mathematics, so I'll refrain from making too strong a comment about the necessity of any particular pedagogical style in that realm. But I would bet two things: 1) We lose many folks who could have become good mathematicians because they were freaked out by the sort of attitude and presentation making up the first half of the 'split' I describe above 2) It is probably significant that not all excellent mathematicians choose the trial by fire pedagogical approach in their own works.

debatem1 8y ago
Hmm. The books I put up here are frequently criticised for focusing on pedagogy at the expense of rigor. So I'm not sure what to say to that. Overall I agree that too much of mathematics is obfuscated. Whether that's because it makes the author sound smart or because excellent prose isn't a big timesaver for a reader who has to grapple with a complex proof anyway I don't know. Could be both. But I will reiterate is what I said above: deeply understanding an unfamiliar result or method often is irreducibly hard work that simply must be ground through. That's no reason to make simple things needlessly hard but when it's time to work through something tricky, well, people should expect to spend some time on it. Importantly, they should not feel stupid or bad at math when they have to do so, which I think is probably the thing that loses us the most mathematics besides obfuscated intro calculus.