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Comment by gmadsen | Hacker News Reader

This is just not true. Reading is necessary but hardly about as important. If you are lucky and get a good text, the exercises guide you to "invent" the important parts of the theory. Otherwise its just definitions and theorems, and its up to you to make your own examples to gain an intuition, which to be fair is closer to life outside of a classroom.

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The "if you are lucky and get a good text" is carrying a lot of weight here. My school identified certain young students as having high mathematical aptitude. While other students were learning the basics of addition and multiplication, the advanced students spent hours practicing multiplying nine digit numbers by hand.

The school was then always disappointed by by their performance in mathematics competitions. After all, the other teams were "wasting" their times unimportant reading about algebra, geometry, and combinatorics while our team was "practicing" math with hours of manual long division nightly.

The practice is certainly vital, but it's useless without a good text to guide you. Unless you're lucky and grab a good one on the first go, you'll need to read a few texts to find the good one.

I'd say that reading is as important to learning mathematics as breathing. You'll be a lousy mathematician if you spend all your time focused on your breathing, but you'll be worse one if you skip breathing entirely.

It really depends on the text, it doesn't have to be just a list of theorems and definitions.

Some intuition can be conveyed through text, or geometry and visualization. A good explanation can make things click.

I think you may be right about your own experience, that you get most of your intuition from exercises, but I'm confident this varies. Some people get much more than zilch from a good text, in addition to doing exercises.

The real world is harder than a classroom, but we don't have to make classroom learning as hard as research. It's okay to start with help and increase the difficulty gradually. You don't have to do everything on your own!

Could you elaborate on this please: "the exercises guide you to "invent" the important parts of the theory."

BTW: I think time not doing exercises is just as important; it's when your mind tries to piece together the data. Coincidentally(?) time resting, after physically exercising, is when your muscles strengthen.

Sure, texts like "linear algebra done right" or "Understanding Analysis" do a unusually good job of integrating large multi part examples where the reader works through them and proves the theorems themselves before they are explained in the book. The nominal case for most math texts is definition, lemmas, theorems, and sometimes they provide examples, and other times, readers are expected to make their own examples. For example, basic topology really only needs 10-20 pages to completely define, but to really understand why the axioms are chosen and the implications, you must work through examples, it is the only way. In fact there is a good book on topology specifically due to this " Counterexamples in Topology"

Thanks! I think I now recognize this from Spivak (Calculus), where much of the teaching is literally in the exercises. You are guided along, deriving/proving many things along the way, some incidental, some cumulative. (There's also important exposition in the exercises.)

A downside is you lose the thread if you skip exercises (e.g. do alternate ones) - the exercises are an integrated whole. But it's a lot to do all of them.

I hadn't gotten the impression that these helped show why exactly the axioms were choosen - though could well be there and I just didn't see it.

It depends a bit on what you call "reading" (i.e. whether you are reading passively or actively). If you stop and think through each proof yourself before reading the one in the book, that more-or-less turns the main content into a series of exercises.

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