Strategies for Effective Self-Study

Solution 1:

This was supposed to be a comment, but it is too long and there may be lessons in it from what I've experienced.

This is a great question (I would add a bounty from my own rep if this wasn't community wiki), and it is also a very personal issue that I struggle with myself. I have a similar style of study to the one you described when it comes to things I am really interested in, rather than things that happen to be part of the syllabus on an undergraduate/graduate course I am taking (those subjects falling into the latter category tend to all get the same treatment from me -- go to the lecture, absorb the main ideas, briefly look at my notes later to see they make sense, then ignore it until I need it for some problem set or an examination).

However, I find that your style of study (call it the hard grift method) means that I am often a little behind classes or lectures, despite the fact that I am trying to pursue a deeper understanding of the material I enjoy. Unfortunately, the style of exam questions means that this level of understanding rarely helps. One can ask oneself whether it is worth the trouble, and ultimately the answer to that question depends on what you want to get out of your learning.*

I am keenly aware of the approach of what Stefan Walter above calls the 'superficial' mathematician, which I do not see as at all disparaging (and for the record, I do not think he does either). I do not really believe in innate talent, but there are many mathematicians smarter than I who seem to pick up just as much knowledge as I might from the hard grift method, by instead coasting from an article to a textbook, to a set of exercises, to a pre-print, while making minimal notes and seemingly picking up the salient points naturally, and then having a fruitful discussion with others about their new findings almost immediately (call this the flowing method). From what I read of Terence Tao's blog this is the natural progression from an undergraduate mathematician to a so-called 'post-rigorous' mathematician.

The flowing method seems to reap more benefits, but it also doesn't seem to be a ticket you can buy. I have a few friends already on their PhDs (I am about to finish my humble MMath) and, without wanting to make this sound like a cop-out, their brains seem to work in a different way to mine. It may very well be the case that I am yet to make the transition because I have not yet put in the hours, but I believe that 'putting in the hours' boils down to passion. If you aren't passionate about what you are studying, you won't put in effective hours, and you won't make the transition to post-rigour.

(Aside: I would like to think that one day I might make that transition, but as it stands, I am not sure the life of a professional mathematician is for me!)

*To answer your question succinctly: you need to find out what it is you want out of your learning. If it is pure mastery, an effective method for you to try might be: stick to hard grift for a little while, but if you find you've reached a level where your intuition is guiding you more than rigour is, then stop and evaluate, and consider taking your learning to a higher level where the details of proofs are not the most important thing any more. In particular, re-read Terry Tao's post in the above link.

If, however, you enjoy learning for its own sake and want to pursue personal understanding (which I think the hard grift method is best suited for) then you should always keep this goal in mind. Personal understanding is more gratifying than pure mastery; it should be the goal of any true autodidact (see the last section of this great article by William Thurston).

Solution 2:

“He who seeks for methods without having a definite problem in mind seeks in the most part in vain.” -David Hilbert

Most of us don't have the power to learn everything in the world in sufficient depth to make meaningful progress. A way around this is to use a "lens"...learn what is needed to solve a specific relevant question in a desired field. Then keep doing this until you have your own personal toolbox and viewpoint on some specific area.

The pleasant surprise is: If you go deeply enough into any rich area of mathematics, you will end up learning the other areas in a very connected way. At least that is what the `unity of mathematics' tells us.

Best of luck in your studies!

Solution 3:

I like your approach in principle, but:

Not all theorems (or rather their proofs) are of the same difficulty level. So when you try to do some harder proof, you will often be forced to "peek" at the proof in the textbook. It could require a lot of discipline not to peek to often.

Also, in most cases knowing the proof of a theorem is not as important as knowing its statement and having a feeling how it is usually applied.

For these reasons I would prove only easy propositions and just read the proofs of the harder theorems, yet read them carefully. I would put the time you save by that into doing exercises. To this end, choose a textbook which has many exercises separated by small bits of text. This way you get a good balance between passive reading and working things out on your own. There are great differences between textbooks concerning the style of the exercises. Some authors just list things they want to mention and don't think too much about the solutions. What you want is books in which great care is taken to ensure a uniform difficulty level among the exercises.

Another aspect: Make sure you are really interested in the subject you are learning about. If you approach it with the mindset "It's boring, but I need it as a prerequisite for Cool Theory XY", then you are probably better off just having a go at Cool Theory XY and returning to the boring stuff later when you have gathered some motivation.

Finally: Although I like your strategy and do something similar myself, I have the impression that most mathematicians are more superficial, i.e. just reading through books and papers, yet arrive at an equally deep understanding in a significantly smaller amount of time. I would really like to hear the opinion of a research mathematician on this.

Solution 4:

I believe it is unlikely that you're able to prove a theorem pretty much on your own but unable to recall it well a month later: you will certainly be better able to recall it proceeding as you outline than if you had only read through a proof and applied it to, say, one example. We learn best by "doing", by "owning" the material we're studying, so to speak: making it "our own" by interacting with it, reconstructing it, proving conjectures, or finding counter-examples, etc. It will also generate questions and motivate you to pursue your studies at a deeper level.

Such an approach can seem overwhelming at times and involves a lot of solitary time without some sort of support/mentor available when you really feel stuck. That's where a site like this comes in. It gives you a community within which you can participate: both asking and answering questions. I know, for me at least, I often get a much deeper appreciation of a topic when attempting to answer a question than if I were to have just asked. So it's give and take, ask and answer. It all boils down to learning!

Solution 5:

I think that your approach is just fine. Just stay disciplined about it.