Nobody teaches the same topic in the exactly the same way.

There is, for one, far more material in any given subject than an introductory course or textbook can (or should) cover, so the author/instructor must choose what to include.

Plus, the order of presentation matters. For example, here are two standard ways of introducing the real numbers:

#1 (Dedekind cut). Picture a square of side length 1. The length of the diagonal, √2, cannot be represented by a ratio of integers, so we need a new number system to represent it. These numbers "in between" rational numbers are called irrational numbers, and together they form the real numbers.

#2 (Cauchy completion). Non-repeating decimals, such as π ≈ 3.141592, cannot be represented by a ratio of integers. We call such decimal numbers irrational numbers. Any number representable by a (finite or infinite) decimal is called a real number.

You can deduce #2 from #1, and vice versa. It's entirely up to the author/instructor to decide which one to start with.

Lastly, there is always a better way to explain the same material.

Besides, books have different levels and audiences. I learned Linear Algebra from three books:

1) Gilbert Strang's "Introduction to Linear Algebra" was great because Gilbert goes straight to intuitions, the proofs are simple, most exercises have answers, but it does not cover advanced material. I used this book for self-teaching. You could probably learn from it with just high-school level maths. Good for engineers.

2) Hoffman and Kunze's "Linear Algebra" was given as a textbook for my first LA course. While it covered some topics that weren't found in many other textbooks and are not really "standard curricula" in many other universities for (jordan normal form, rational canonical form). I found it more similar to a reference than a textbook; it is intended for math majors. The proofs are imho a bit obtuse and it usually introduces topics without much justification. Determinants are introduced early.

3) Axler's "Linear Algebra Done Right" OTOH covered many of the topics in Hoffman&Kunze but the organization and the proofs were (imho) mucho more clear and motivated. Also intended for math majors. No determinants until the end.

I learned analysis from a self guided inquiry based learning packet: http://jiblm.org//downloads/jiblmjournal/V090212S/V090212S.p...

And the reals were introduced to me by the completeness axiom phrased like this: "If M is a point set and there is a point to the right of every point of M, then there is either a right-most point of M or a first point to the right of M."

Nice. I was thinking of real analysis texts as an example here too. Was going to say that not everyone can use Apostol, like Caltech, to teach calculus.

Well, there are books on subjects that are widely used in different countries, Sheldon&Ross for probability, Cormen for Algorithms, etc; Usually a professor choose his own way to teach things, with different example using maybe his own experience in the field, putting more focus into certain things, and students can just do better with a (usually free) reference of what is done in a particular class

We can argue if this method, where not every class on the same subject is equal, is good or bad, but till this is what is widely used, it's good for both students and professor to have a reference.

Maybe different learning styles? I spoke with one very young math prodigy a few years ago who told me that her secret to self teaching advanced material was that she tries several texts and ends up using the one that teaches the material in the way that she finds most intuitive.

People say that but usually the truth is the second or third book is the "best" (regardless of which book) but really it's because the material sinks in after multiple itersations, other worse because the material feels familiar even though the student hasn't actually learned to solve problems, they've just memorized bits.

So how would you solve that problem in a centralized system? I don't think "reread this section 2-3times" would necessary work.

Your comment reminds me of this Onion article :)

https://www.theonion.com/author-of-introduction-to-algebra-r...

I have a $250 textbook that not only answers your question, but also has some good insights on the topics referenced in the aforementioned pdf document.

Discrete Mathematics with Applications by Epp?

(I just finished Discrete I & II)

I bought an "only for non-US" version of this book... only to find out it was somehow illegal for me to own it. Doesn't really seem fair. Just because my country is "rich" doesn't mean that I am.

Yup.

Showing the proofs to the theorems taught in university-level math classes is expected. For starter, being that there can be many proofs to the same result you can't expect the teacher to show them all.

Opinions also differ on what is valuable and should be taught (you can't teach everything in a finite time), if the teacher can skip some partial result he doesn't find interesting he can opt for a different proof altogether.

Thus there'll always be some treasure left for the most ambitious student by tackling various such works.

Everyone needs to write his or her notes to understand the topic, to include something that he/she feels important. Some people stress the importance of logical flow and theory, whereas some people stress the importance of applications. I believe we all learn by organising knowledge in our own way. So there won't be a single definitive reference text or note.

I took this class, and although I can’t speak for Prof Aspnes, I always imagined he just preferred writing the notes himself (since he can edit them) and might want to publish them into an actual textbook some day.

He wrote similarly thorough notes for his Randomized Algorithms class.

Standard undergrad courses often have kind-of standard literature.

In Germany, for example, calculus courses will recommend Heuser. Linear algebra will recommend Beutelspacher. Every EE owns a copy of Tietze & Schenk.

Not sure if anybody else does it this way but sometimes I have to search for multiple sources or explanations of a given topic until I find one that "speaks to me".

Because most of the time it is the professor who prefers writing them. At least that is my case.

A single source is not better than many.

I see wikipedia as exactly this.

Wikipedia is terrible at teaching.

Because the material isn't quite the same, and because different people have different teaching and learning styles.

The material isn't quite the same because your ideas of what a first-year student needs to learn about probability may differ from someone else's. Not just because your and their prejudices are different: it'll be influenced by what else is in other courses. Imagine two universities with first-year probability courses for their mathematicians. One of them also has a statistics course that everyone takes, the other doesn't. They may make different choices about what goes in the probability course. And it'll be influenced by the students they get. Imagine one of the universities is a very prestigious highly-selective one that lots of good mathematicians apply to, and the other ... isn't. The first will likely want more material in their course.

Now, think again about those last two universities. As well as possibly presenting different material, they may well want it presented differently. Fancypants University will want extra-rigorous proofs, connections to other fields of mathematics (measure theory, probabilistic methods in combinatorics, ...), more challenging problems. Slowcoach Polytechnic will want a very clear and straightforward presentation that doesn't leave anything unexplained and doesn't distract students with irrelevancies. So even when they're presenting the same material, the approaches required will be different.

Different students will want different things, too. Besides the sort of differences in the previous paragraph (which of course apply between students as much as they do between universities), some students will be interested in the foundations of the subject (what is a probability, anyway?) and some won't; some will be interested in particular kinds of applications; some will like thinking visually and benefit from a lot of digrams, some will find them distracting; etc., etc., etc.

Now, with all that in view, imagine you're a professor who's teaching Probability 101 for the first time next year. No matter what, you're going to need to do a lot of thinking about what material you put in your lectures, how you divide it up, how you want to present it, etc. You're going to need some notes for your benefit. (The benefit comes both from making them and from having them.) The extra effort required to polish them up and make them usable by students too is actually pretty small. The fact that other people have presented other similar courses and made similar notes is neither here nor there. You can't just take their notes and avoid all the effort, because their course isn't identical to yours, and their students aren't identical to yours, and their style of presentation isn't identical to yours, and much of the benefit of making the notes comes from actually making them. So, you make your own notes, and then you might as well put them on the internet. And here we are.

This is probably an obvious statement, but if you write such a reference, then there will be one more reference, and someone else will have to write a new, more comprehensive reference based on all the existing ones, plus the new one you wrote. And so, on it goes.

To answer the question more seriously, there are a number of reasons for this: 1) terminology and notation changes over time, 2) there is not enough space in a standard textbook to include all material on any given subject in mathematics, even at an undergraduate level 3) there have been such standard textbook projects, e.g. the Bourbaki textbooks, but in order to cover everything in complete generality, they had to work at such a high level that the material is all but useless to a beginning student (many French professors may disagree with me on this point). 4) It is not uncommon for textbooks to start with different assumptions, to target the material for certain research applications, to give a different perspective on the material, to promote a favourite notation, to give new proofs or even new theorems, in lecture notes written by professors.

What this doesn't answer however, is why there is such a dearth of technological solutions to the problem of disseminating the 40 odd thousand pages of undergraduate mathematics taught around the world today. There are lots of videos at a very superficial level, very few in depth, and piles and piles of books and pdfs. There are virtually zero really well-produced, animated, scripted videos that cover mathematics in great depth.

The only reason I can come up with for not having such material is lack of time. Academics, especially professors, who would have to write and edit such material, would have to invest so much time, they would not have time to have families, or do research.

Teaching styles differ and student populations differ.

And re: in-depth, this is a long tail problem. The profit is in the mainstream, not in the tail. Producing really good in-depth work isn’t that rewarding because the market is small.

I'm still talking about undergraduate mathematics here. There's billions of dollars spent worldwide on providing exactly that. The issue is that a standard undergraduate course, especially in mathematics, is quite deep, relative to, for example, a YouTube "science" video.

On the other hand, technology offers new and interesting ways to disseminate this knowledge, but I see very little at the same depth as a standard undergraduate course, which is not just a video of someone doing the equivalent of a chalk-and-talk.

Since we’re discussing mathematics may I attempt to generalize your thought?

You can forget lack of time altogether and simply think of it as a lack of resources or money. The problem is to date, lack of an adequate self-sustaining model to provide these resources.

In this case by “model”, I mean either a business model where the work was incentivized by profit, or a well organized community that is able to motivate its members to collaborate and volunteer as has been shown to be possible in other areas.