One awesome thing about math is it is self organizing. Many topics are composed of many smaller topics. As an anecdote, we used to say people “actually learned” (high school) algebra in calculus 1 and trigonometry in calc 2/3. In those cases it was more that to solve those problems required using algebra and trig coherently.
So to learn those 1000 pages is to, to some extent, learn a core set of techniques across many different contexts. It compresses the required mental load. For the exceptions to that rule, well, you can skim over them and know enough about them to recognize their applications later.
Sure it does. If you really wanted to read and understand this book and dedicated time each day to understanding 3 pages worth of content, you could read the book in a year. Remembering 3 pages of content per day (especially in math where concepts build on each other) is really not hard.
> Remembering 3 pages of content per day (especially in math where concepts build on each other) is really not hard.
I disagree. Or rather, I think that's unsustainable. Any given three consecutive pages from Spivak's Calculus are probably doable on a daily basis. But is would be legitimately hard for most people to go through three pages of Rudin's Principles of Mathematical Analysis each day and consistently retain that information. Axler's Linear Algebra Done Right is very readable, but Halmos' Finite-Dimensional Vector Spaces will start getting just as dense as Rudin. These are difficult textbooks even when students are well-prepared for them with prerequisite courses. Terence Tao wrote two books to cover (with better exposition) what Rudin did in one. I think it would be pretty hard to read consistently three pages of Tao's Analysis I each day, before he even gets to limits.
I think you're underestimating the intellectual effort here. In my opinion, even if you're reading a math book targeted to your level, committing to reading and understanding three days of material each day would become exhausting. A typical semester is 15-16 weeks, with lectures 1 - 3 times a week, and most undergraduate courses do not actually work through the entirety of a 300 page textbook. Even at that slower pace it's not typical for most people to ace the course. If you read three pages a day and had a solid understanding of it, you'd be absolutely breezing through math courses.
In my experience students need to really step away from the material and let it percolate for a bit every so often in order to solidify their understanding. I really don't think you can partition the material into equal, bite-sized amounts each day. The learning progression doesn't tend to be that consistent or predictable.
If you assume that "run 200 miles" doesn't refer to a single run but rather to the capability of running 200 miles, the analogy works much better. If you stop training the ability to run 200 miles vanishes even more quickly than an equivalent feat of learning.
Check out “How to Read a Book” by Mortimer J. Adler. It seems like a pithy title, but he’s quite serious and has interesting things to say on the subject.
From the introduction to Shelah's Classification Theory and the number of nonisomorphic models (an extremely technical book with 700+ pages):
So we shall now explain how to read the book.
The right way is to put it on your desk in the day, below your pillow in the night,
devoting yourself to the reading,
and solving the exercises till you know it by heart.
I’ve read technical books like this for fun. Math is a little tricky because a lot of time you need to really spend time working out the algebra to get the concept, but sometimes you can still just sit down and read. And if it’s not a progression over a single subject you can just jump to a section you find interesting. Even if you don’t grok it 100% you’ll still have learned something by making the effort.
Nobody really does, IMO. You read it so you know what kinds of things are topics in the field, what the most important results are, and mostly to get a feeling for what people in the field focus on.
Then when someone mentions a term from the course, you know roughly what to expect and where to find details. Also, you know how big a bite you're taking. "Quantum Mechanics" might be several books. "Vector Calculus" maybe a few chapters. "De Morgan's laws" maybe a few pages.
It's just like any other math you do. Does anyone memorize all the trig relations, Laplace transforms, and geometric relations? Of course not. But you've seen at least one large example of every topic and you can reconstruct it from there.
I always wondered about how you could get your name on something as simple as the laws of binary combination, and I suppose there had to be some depth to it somewhere to merit this.
Pencil in hand. When you see an argument whose steps you don't follow, work it out. When you get to the problems, do at least some of them. Follow the assignments in the MIT course, and check your work against the solutions.
