Learning Mathematics in Context

I once took a math course where the first homework assignment involved sending the professor an email that included what we wanted to learn in the course (this assignment was mostly for logistical reasons: professor's email now autocompletes, eliminating a trivial inconvenience of emailing him questions and such, professor has all our emails, etc). I had trouble answering the question, since I was after learning unknown unknowns, thereby making it difficult to express what exactly it was I was looking to learn. Most mathematicians I've talked to agree that, more or less, what is taught in secondary school under the heading of "math" is not math, and it certainly bears only a passing resemblance to what mathematicians actually do. You are certainly correct that the thing labelled in secondary schools as "math" is probably better learned differently, but insofar as you're looking to learn the thing that mathematicians refer to as "math" (and the fact you're looking at Spivak's Calculus indicates you, in fact, are), looking at how to better learn the thing secondary schools refer to as "math" isn't actually helpful. So, let's try to get a better idea of what mathematicians refer to as math and then see what we can do.

The two best pieces I've read that really delve into the gap between secondary school "math" and mathematician's "math" are Lockhart's Lament and Terry Tao's Three Levels of Rigour. The common thread between them is that secondary school "math" involves computation, whereas mathematician's "math" is about proof. For whatever reason, computation is taught with little motivation, largely analogously to the "intolerably boring" approach to language acquisition; proof, on the other hand, is mostly taught by proving a bunch of things which, unlike computation, typically takes some degree of creativity, meaning it can't be taught in a rote manner. In general, a student of mathematics learns proofs by coming to accept a small set of highly general proof strategies (to prove a theorem of the form "if P then Q", assume P and derive Q); they first practice them on the simplest problems available (usually set theory) and then on progressively more complex problems. To continue Lockhart's analogy to music, this is somewhat like learning how to read the relevant clef for your instrument and then playing progressively more difficult music, starting with scales. [1] There's some amount of symbol-pushing, but most of the time, there's insight to be gleaned from it (although, sometimes, you just have to say "this is the correct result because the algebra says so", but this isn't overly common).

Proofs themselves are interesting creatures. In most schools, there's a "transition course" that takes aspiring math majors who have heretofore only done computation and trains them to write proofs; any proofy math book written for any other course just assumes this knowledge but, in my experience (both personally and working with other students), trying to make sense of what's going on in these books without familiarity with what makes a proof valid or not just doesn't work; it's not entirely unlike trying to understand a book on arithmetic that just assumes you understand what the + and * symbols mean. This transition course more or less teaches you to speak and understand a funny language mathematicians use to communicate why mathematical propositions are correct; without taking the time to learn this funny language, you can't really understand why the proof of a theorem actually does show the theorem is correct, nor will you be able to glean any insight as to why, on an intuitive level, the theorem is true (this is why I doubt you'd have much success trying to read Spivak, absent a transition course). After the transition course, this funny language becomes second nature, it's clear that the proofs after theorem statements, indeed, prove the theorems they claim to prove, and it's often possible, with a bit of work [2], to get an intuitive appreciation for why the theorem is true.

To summarize: the math I think you're looking to learn is proofy, not computational, in nature. This type of math is inherently impossible to learn in a rote manner; instead, you get to spend hours and hours by yourself trying to prove propositions [3] which isn't dull, but may take some practice to appreciate (as noted below, if you're at the right level, this activity should be flow-inducing). The first step is to do a transition, which will teach you how to write proofs and discriminate between correct proofs from incorrect; there will probably some set theory.

So, you want to transition; what's the best way to do it?

Well, super ideally, the best way is to have an experienced teacher explain what's going on, connecting the intuitive with the rigorous, available to answer questions. For most things mathematical, assuming a good book exists, I think it can be learned entirely from a book, but this is an exception. That said, How to Prove It is highly rated, I had a good experience with it, and other's I've recommended it to have done well. If you do decide to take this approach and have questions, pm me your email address and I'll do what I can.

1. This analogy breaks down somewhat when you look at the arc musicians go through. The typical progression for musicians I know is (1) start playing in whatever grade the music program of the school I'm attending starts, (2) focus mainly on ensemble (band, orchestra) playing, (3) after a high (>90%) attrition rate, we're left with three groups: those who are in it for easy credit (orchestra doesn't have homework!); those who practice a little, but are too busy or not interested enough to make a consistent effort; and those who are really serious. By the time they reach high school, everyone in this third group has private instructors and, if they're really serious about getting good, goes back and spends a lot of times practicing scales. Even at the highest level, musicians review scales, often daily, because they're the most fundamental thing: I once had the opportunity to ask Gloria dePasquale what the best way to improve general ability, and she told me that there's 12 major scales and 36 minor scales and, IIRC, that she practices all of them every day. Getting back to math, there's a lot here that's not analogous to math. Most notably, there's no analogue to practicing scales, no fundamental-level thing that you can put large amounts of time into practicing and get general returns to mathematical ability: there's just proofs, and once you can tell a valid proof from an invalid proof, there's almost no value that comes from studying set theory proofs very closely. There's certainly an aesthetic sense that can be refined, but studying whatever proofs happen to be at to slightly above your current level is probably the most helpful (like in flow), if it's too easy, you're just bored and learn nothing (there's nothing there to learn), and if it's too hard, you get frustrated and still learn nothing (since you're unable to understand what's going on).)

2. "With a bit of work", used in a math text, means that a mathematically literate reader who has understood everything up until the phrase's invocation should be able to come up with the result themselves, that it will require no real new insight; "with a bit of work, it can be shown that, for every positive integer n, (1 + 1/n)^n < e < (1 + 1/n)^(n+1)". This does not preclude needing to do several pages of scratch work or spending a few minutes trying various approaches until you figure out one that works; the tendency is for understatement. Related, most math texts will often leave proofs that require no novel insights or weird tricks as exercises for the reader. In Linear Algebra Done Right, for instance, Axler will often state a theorem followed by "as you should verify", which should require some writing on the reader's part; he explicitly spells this out in the preface, but this is standard in every math text I've read (and I only bother reading the best ones). You cannot read mathematics like a novel; as Axler notes, it can often take over an hour to work through a single page of text.

3. Most math books present definitions, state theorems, and give proofs. In general, you definitely want to spend a bit of time pondering definitions; notice why they're correct/how the match your intuition, and seeing why other definitions weren't used. When you come to a theorem, you should always take a few minutes to try to prove it before reading the book's proof. If you succeed, you'll probably learn something about how to write proofs better by comparing what you have to what the book has, and if you fail, you'll be better acquainted with the problem and thus have more of an idea as to why the book's doing what it's doing; it's just an empirical result (which I read ages ago and cannot find) that you'll understand a theorem better by trying to prove it yourself, successful or not. It's also good practice. There's some room for Anki (I make cards for definitions—word on front, definition on back—and theorems—for which reviews consist of outlining enough of a proof that I'm confident I could write it out fully if I so desired to) but I spend the vast majority of my time trying to prove things.

What's your goal for which you want to learn math?

Most people are bad at understanding. As students they usually prefer to memorize things, because it is a strategy that works best in short term. When they grow up and become teachers, they recite things to students and expect them to memorize it.

In math, in addition to memorizing facts verbally, there is also a lot of procedural knowledge (solving equations). This is probably one of the reasons most people hate math. But even the procedural knowledge can be taught in the memorizing way; only the verbal memory is replaced by the muscle memory.

Understanding is a step yet beyond procedural knowledge. Most people don't get there; even most teachers don't.

And being able to explain stuff to beginners -- that's the ultimate art. It requires not only having a good understanding of the topic, but also being able to untangle it to a linear thread that can be gradually fed to a human and will allow them to build a proper model of the topic. This requires also an understanding of humans, and an understanding of understanding.

So why aren't most math textbooks better? I guess it's either because there are not enough good mathematicians who also happen to be good at explaining to beginners... or maybe the market for textbooks that teach understanding simply is not big enough.

If you want to learn a specific topic, maybe you could ask about it on LW.

Unfortunately there isn't a lot out there that is like what you are looking for. Here are some books that I've read that may come close.

I like the books and essays written by Steven Strogatz, a professor at Cornell University. He's written some things intended for the general public, including a pop science book called Sync and a series of essays in the New Yorker. He also writes journal articles and textbooks. He has a way with words, of being able to describe complicated mathematical concepts without equations. Here is his website: http://www.stevenstrogatz.com

However, the two books that most awoke my love of fractals and mathematics were:

Fractals: The Patterns of Chaos: Discovering a New Aesthetic of Art, Science, and Nature by John Briggs http://www.amazon.com/Fractals-Patterns-Discovering-Aesthetic-Touchstone/dp/0671742175/

An Eye For Fractals: A Graphic And Photographic Essay by Michael Mcguire http://www.amazon.com/Eye-Fractals-Graphic-Photographic-Nonlinearity/dp/0201554402/

They are introductions. They might not go into as much depth as you want. I'm not sure a single book would.

Life in Moving Fluids by Steve Vogel of Duke University is a mixture of biology, fluid dynamics and mathematics. It can be appreciated without background knowledge, but it skips over some explanations, so probably not all of it would be accessible. But it does delve into serious depths. The first edition has more explanations than the later edition. http://www.amazon.com/Life-Moving-Fluids-Princeton-Paperbacks/dp/0691026165/

Prime Mover: A Natural History of Muscle, by the same author Steven Vogel is easier to read, but has less math. http://www.amazon.com/Prime-Mover-Natural-History-Muscle/dp/0393021262/

Design in Nature: How the Constructal Law Governs Evolution in Biology, Physics, Technology, and Social Organizations by Adrian Bejan, also a professor at Duke University That's the introductory book, but if you want more detail, he has other books and scientific papers. I like his book Shape and Structure in Engineering and Nature, but it doesn't explain things very well, and a different book may be a better next step. http://www.amazon.com/Design-Nature-Constructal-Technology-Organizations/dp/0307744345/

Complex Adaptive Systems: An Introduction to comptutational models of social life (Princeton studies in complexity) by John H. Miller and Scott E. Page It was pretty readable to me, compared to textbooks in general.

Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics by John Derbyshire It's an interesting topic and the book is well-written, but somehow I didn't manage to finish it. So I don't recommend it as strongly as some of the others. http://www.amazon.com/Prime-Obsession-Bernhard-Greatest-Mathematics/dp/0452285259/

Structures: Or Why Things Don't Fall Down by J.E. Gordon A good introduction to concepts of tension and compression and how they are used in buildings. Uses some math. http://www.amazon.com/Structures-Things-Dont-Fall-Down/dp/0306812835/

What is Mathematics? by Courant and Robbins is a classic exploration that goes reasonably deep into most areas of math.

You could try Godel Escher Bach: An Eternal Golden Braid...

On a slightly meta-level, I only got over my bad attitude towards calculus by taking astronomy and physics courses in which it was used to figure out interesting stuff...

http://www.amazon.com/Calculus-Easy-Way-Series/dp/0812091418/ was pretty good, there are also others in the series. It teaches calculus with a sci-fi story where discovering new calculus ideas helps the protagonist. Basically HPMOR, except more directly didactical, on a specific subject, and with no real plot.

I'm convinced that humans must spike their blood sugar and/or pump their body full of stimulants such as caffeine in order to get past the natural tendency to find it unbearably dull to memorize words and syntax by rote and lifeless connection with the structures in their native language.

Just a comment: This is certainly not true for every human. Some people really enjoy that.

Learn about proofs and practice doing proof problems.

I recommend trying to solve problems. If you don't know how to solve a certain problem, join IRC, mailing list or some other math-related community. Look for humans rather than numbers. f(x)=y+5 are some dead characters that only make sense with some supplementary knowledge.

Books are good, but in the long run human communication is probably more important. You will eventually meet with a problem you can't solve or you might have questions or the answer might be unclear for some reason or another.

Perhaps a Mathematics for Philosophers book like this http://www.amazon.com/dp/1551119099 ?