Self studying math, how can I learn the most?

Question

I am currently studying Pre-Calculus on my own. I have a few texts I am working with but feel like I could learning a lot more than I am.

When people typically ask these kind of questions the common response for Pre-calculus->Calculus is to watch "Khan Academy" etc.

Is this really a wise approach, especially when confronting future topics? Wouldn't it better to develop more independent, self-sufficient learning strategies beforehand?

I always wonder how mathematicians teach themselves so much knowledge without resources like "Khan Academy", lectures or even teachers. People learned math before the internet, what was the strategy then?

Some other questions:

• Which is the better source to learn from, video lectures or reading the text? I feel like video lectures are too passive. Would time be better spent doing more exercises and less video lectures?

• If you watch a video lecture, should it be watched before reading the text or after?

• Math is quite concise as presented in textbooks, is summarizing the chapter as you read in a notebook effective or unnecessary?

• Should I completely read chapters before attempting exercises or try questions first and refer to the text as needed?

Final question: If a person was attempting to learn a topic in math from scratch with minimal knowledge on the content, what is an approach they could apply to learn it that would be both effective and efficient?

Thanks

Answer 1

• Video lectures are actually better for rapidly getting the intuition than studying directly from a textbook. And that's because the lecturer reduces the effort you have to make to understand, especially if his name is Sal.
• Yes, textbooks are superior to videos for genuinely learning the material.
• It depends on how you summarize your chapter. Try to look up for the Feynman technique of learning, I'm sure you'll appreciate it and it will help.
• Well, you can proceed as follows: ${\text{(1)}}$ Read first the chapter. $\text{(2)}$ Do some simple to medium exercises. ${\text{(3)}}$ Read the same chapter again, but this time more intensively and try to think about what you read a bit more. $\text{(4)}$ Do the hard exercises.

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If you are starting a topic from scratch with minimal knowledge on the content, what is an approach that is found to be effective and efficient?

Searching for the best textbook out there and doing the procedure I explained in the last sentences. And for your purpose, i.e. Pre-calculus the best book that I really recommend is this one: Precalculus with unit circle geometry - David Cohen. It discusses lot of aspects from the fundamental theorem of algebra to matrices that you can find in my answer here.

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After mastering Precalculus you may have a look at the other undergraduate topics.

Answer 2

All of your questions really come down to personal preference, all I can really see is that doing the exercises is the most important part. A good textbook is great to learn from, as is a good video lecture, but I think a live lecture is best because it allows you to interact with the teacher.

Should you read the chapter completely first, then do exercises or vice versa?

Most chapters are broken up into smaller parts, which often have their own exercise section, I generally quickly skim the chapter, then read the first part again and do the exercises that come with the first part, then proceed etc...

Answer 3

The problems are the key. In general texts are better than video since the problems are a designed part of the text. I don't know about Khan or some other tour de force video series. It is certainly possible to have a great video set with great accompanying problems. But I think much rarer.

My advice would be to go with a text. I find the ones from ~1900 to 1960 best. They are written more directly and with smaller words. They will generally have the answers (key requirement for self study). Will have a variety of problems from simple use the equation to harder. And will have sufficient volume of problems that you can do both drill and design self tests. The ideal text of course would be a programmed instruction text but this is rare.

Bulleted questions:

*"Would time be better spent doing more exercises and less video lectures?"

Yes.

*"Video before or after text".

In general, I would find a text that is the main thrust of your learning and not spend much time on videos unless for specific issues. (so...mostly "after").

*"summarize chapters".

I think systematic summarizing is overkill. Most importantly, work the example problems systematically (in same notebook as exercises as you read the text. Feel free to keep occasional notes, but don't rewrite each chapter. However, a great practice is to have section for Q&A. (end of notebook backwards works well.) Often you will find that as you work exercises or read further, you figure out the question, but for a few you may not, but then this gives you a very specific list to ask questions on SE or with a tutor or teacher or parent. (I think there will be few questions that really need to be escalated and many that you solve for yourself as you go. But just having a "parking lot" list like this keeps the mind at peace as you are learning, that you will get everythign sorted.) In any case the real proof of the pudding is in exercises and self tests, not in reading/parsing.

*"read first or do problems first?"

My advice is to read the section first. (Good books will not have long chapters, but short sections instead.) Read it fairly quickly but carefully. Sounds strange, but I mean not lackadaisically like a novel but in an engaged concentrating mode. But still with some speed. Not to skim or miss anything. But with a desire to be time efficient and get to problems. This will be a little bit of a strain until you get used it it.

After you finish the section, do the problems. Try to do this without looking at the text. Treat it as a practice test. If your book has a lot of problems, I would do the odds as first HW and save the evens for self testing as you progress (later tests for seeing if you remember.) Check your HW for answers. Any problems you missed, rework the entire problem from scratch (but not the problem set) even if it is a silly sign error or the like.

Answer 4

The strategy before the internet was to work with peers. The best tools for learning are other people. Whether you're having someone teach you something, or whether you're trying to teach them something, the act of cooperative practice is one of the most important elements of learning.

The notion of social relatedness as a tool for education is an old concept, and it has a formal psychological basis that has been extensively studied. Grad students get to share offices, university libraries have group study rooms, and high school teachers assign group projects. These are not new educational/study modalities.

The portrayal of the mathematician as an eccentric lone wolf doing work alone in his attic is a popular meme that persists; certainly, many great mathematicians were capable of solo work (or perhaps incapable of group work), but for the vast majority of working professionals, collaboration is a necessary part of their ongoing development.

Videos are great. Textbooks are great. But those things should be augmented by working with peers. It's the best way to learn.

Answer 5

I'm not an expert, but I'll offer an opinion. I think the most important thing is self-diagnostics. We don't all learn the same way, at the same speed, with the same methods. For me personally, I initially learn material to a point where I am able to complete medium-to-hard difficulty and do well on exams. After some time has passed and I have moved on in my education, I gain a firmer grasp on the material, I can see the big picture, and I go back for a refresher on old material. I'm able to relate my new knowledge to old lessons.

Short answer: Experiment, and find out what works for you.

Answer 6

(adapted from here)

Suggest trying to understand and unify along your study.

For example, how is this concept related to that, why is it needed, what problem it solves (or solved when it was introduced), any model where it is manifest, can it be done better (or worse)?

Can transform problems related to this concept to problems related to that concept? (see for example Isomorphism) Is there a unifying thread or more general concept where these 2 concepts are special/edge cases and so on.. (see for example Generalization)

Effectively (to use more mathematical terms) you build optimum descriptions (or compressions) of the concepts and framework studied.

(see for example, Kolmogorov complexity, Lempel-Ziv-Welch algorithm)

So reaching this point of understanding and unification, you have integrated a vast amount of knowledge in an optimum way (as mentioned above)

Answer 7

Mathematics is somewhat like playing an instrument or programming a computer. Musicians start their study with music in their heads they want to be able to play. Starting programmers may already see that killer game they want to program. Mathematicians want to solve problems or create new mathematics. That is where you start, by working on a problem: your problem. It is essential that you have one. Before anything I would read as much as I can on how to read mathematics, how to study mathematics and on how to solve problems. - About video lectures on internet I would like to say that they lack an essential quality: the possibility for the student to ask questions, you can't really interact and neither you are part of the group. I doubt if they really work.