# Future way of learning mathematics (towards graduate level)

Recently, there is an idea sparkling out from my mind due to the COVID-19 outbreak. In the past, we typically attend a course, complete homework and assignment, then finish the semester with a final exam. This takes around 4 courses per semester and 4 years for a whole degree. But I feel we can speed up this process with the aids of the online lecture, as provided from Quality of Videos Lectures and Lectures vs Textbooks. Scott Young had successfully done this by completing a 4-year computer science degree through his MIT Challenge. Thus, I believe online lectures are beneficial for the following reasons.

It helps people to accumulate the background to do research as fast as possible. As we know, mathematics is becoming more and more structured and complicated. To overcome this, we have to learn much faster and more effectively. Typically, online lectures are around 30 hours where people can watch all of them in a week (with note-taking and understanding). The best thing is we can always rewatch the part we don't understand and search for appropriate reference (using Google) to understand the idea better. Then we take another week to complete exercises and do final revision using Feynman techniques (teaching to yourself). We can complete a course ideally in two weeks.

Now let's see how many courses do we need minimally to be able to read research materials in mathematics. We need Analysis I, Analysis II, Linear Algebra, Abstract Algebra, Point Set Topology, Measure Theory, Complex Analysis, Functional Analysis, Commutative Algebra, Algebraic Topology, Algebraic Curves, Differential Geometry, Representation Theory... Let say around 20 courses to reach the level. Then we only need 40 weeks for the whole program. To be more relaxing, one year is sufficient for us to have the background to do research, of course with the aids of online lectures.

Here is my central idea: Self-studying by watching online lectures and then supplementing with exercises and textbook is much more effective than to read the textbook from pages to pages. It's like playing a new board game. Online lecturers (people who know board game) give the main idea of study (game) and then the students (players) do the exercises (play the games) after listening to the main rules. If there is any problem, they consult the reference book (player guide).

What do you guys think about this? Is learning online a better option for the future? I feel this is an interesting topic to discuss and I hope to get some opinions on this.

$$\textbf{Edit}:$$ I have to mention that doing this way only gives sufficient background to pursue research faster, but to consolidate the knowledge, the students have to read research papers, read more references, do more exercises on that particular research topic. For instance, people doing algebraic geometry should nevertheless do all the exercises in Hartshorne, read more advanced topics etc. When they face any difficulties, go back to the relevant prerequisites and consolidate them. (It should be easy to trace back since they have completed all courses)

• Who will be checking if the assignments are correct? – Joel Reyes Noche Jun 8 at 9:07
• Welcome to Mathematics Educators! Please note though that this is a Q&A website, not a discussion forum to get opinions, interesting though the topic is. – J W Jun 8 at 10:01
• Why not "reading lecture notes and then supplementing with exercises and textbook"? – Peter Saveliev Jun 8 at 11:02
• To me, the forced move to online education in the past few months has demonstrated how much more effective is face-to-face education. Absorbing a semester's worth of online lectures in a week or two will not lead to mastery of the topic. – Joseph O'Rourke Jun 8 at 13:03
• For me only a very small part of my learning occurred in lectures. Nearly all of my learning was during the (sometimes) hours and hours of trying to work a few of the more difficult problems, reading the text (probably over 70% of my time was spent doing this), and supplementary reading in books I checked out from the library. Indeed, in many cases I only attended a handful of lectures (often when the class met at a time that was not very convenient for me for various reasons), and in a nontrivial minority of these cases I don't think it had any effect on how much/fast I learned. – Dave L Renfro Jun 8 at 21:04

If you are pushing videos so hard, you haven't even read Scott's story carefully. He says a textbook with problems is far more important than online lectures (which are not that much needed).

https://www.scotthyoung.com/blog/2012/11/13/why-lectures/

As far as the "who will check the work" comment (below OP's question), the answer is to use books with answers for the drill so you can check your self for mistakes.

There: two changes that fundamentally reduce the demand for math educators. Now we won't have to pay them all double six figures. (Oops.) ;-)

• I mentioned that a course must be complemented with problems from a textbook. Let say you have two weeks to complete a course, you need need approx one week for the videos, note taking and understanding and another week for solving exercises and doing revisions. – Ling Min Hao Jun 9 at 4:55
• My main idea was, to self study, watching online lectures is far more efficient than reading the textbook from cover to cover. After we watched online lectures, we can directly go to problems and then read reference if we are stucked. – Ling Min Hao Jun 9 at 4:57
• I agree that most of the learning should be done with hands-on practice, but a minimum amount of explanation, Q&A and error correction is required. Otherwise you could just getting proficient at doing things wrong – David Jul 13 at 11:48
• @LingMinHao "My main idea was, to self study, watching online lectures is far more efficient than reading the textbook" — and the linked page directly contradicts it. Guest, thanks for the link, it agrees with my own experience. Although I find 3Blue1Brown's 10-minute videos more content-rich and very applicable in their use of visual presentation than a recording of a professor's talking head, I tend to forget the information I "learned" unless I do at least a couple of problems related to the information presented. – Rusty Core Jul 13 at 17:59
• As I said, I never said textbook is unimportant, but "solely" relying on textbook is inefficient because no people guide you with the correct way. (Especially for abstract courses). Nowadays, with online videos everywhere, you don't have to spend 4 years for a typical undergraduate curriculum. Instead, you could learn all of them in half of the time using the online lecture videos. 30-40 hours for a video set and directly go to problem set(which might take another 40-50 hours), then consult textbook when you get stucked. – Ling Min Hao Jul 13 at 20:27