How to study?

Paul Halmos, I want to be a mathematician, pp.69

Eu poderia dizer que faço destas minhas palavras sobre como estudar. Ficam aí possíveis dicas (?), pelo menos pra divertir as idéias. Eu transcrevi do livro.

Here you sit, an undergraduate with a calculus book open before you, or a pre-thesis graduate student with one of those books whose first ten pages, at least, you would like to master, or a research mathematician (research or would-be) with an article fresh off the press – what do you do now? How do you study, how do you penetrate the darkness, how do you learn something? All I can tell you for sure is what I do, but do suspect that the same sort of thing works for everyone. It’s been said before and often, but it cannot be overemphasized: study actively. Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Another way I keep active as I read is by changing the notation; if there is nothing else I can do, I can at least change (improve?) the choice of letters. Some of my friends think that’s silly, but it works for me. When I reported on Chapter VII of Stone’s book (the chapter on multiplicity theory, a complicated subject) to a small seminar containing Ambrose and Doob, my listeners poked fun at me for having changed the letters, but I felt it help me to keep my eye on the ball as I was trying to organize and systematize the material. I feel that subtleties are less likely to escape me if I must concentrate on the bricks and mortar as well as gape admiringly at the architecture. I choose letters (and other symbols) that I prefer to the ones the author chose, and, more importantly, I choose the same ones throughout the subject, unifying the notations of the part of the literature that I am studying. Changing the notation is an attention-focusing device, like taking notes during lectures, but it’s something else too. It tends to show up the differences in the approaches of different authors, and it can therefore serve to point to something of mathematical depth that the more complacent reader would just nod at – yes, yes, this must be the same theorem I read in another book yesterday. I believe that changing the notation of every thing I read, to make it harmonious with my own, saves me time in the long run. If I can do it well, I don’t have to waste time fitting each new paper on the subject into the notational scheme of things; I have already thought that through and I can now go on to more important matters. Finally, a small point, but one with some psychological validity: as I keep changing the notation to my own, I get a feeling of being creative, tiny but non-zero – even before I understand what’s going on, and long before I can generalize it, improve it, or apply it, I am already active, I am doing something. Learning a language is different from learning a mathematical subject; in one the problem is to acquire a habit, in the other to understand a structure. The difference has some important implications. In learning a language from a textbook, you might as well go through the book as it stands and work all the exercises in it; what matters is to keep practicing the use of the language. If, however, you want to learn group theory, it is not a good idea to open a book on page 1 and read it, working all the problems in order, till you come to the last page. It’s a bad idea. The material is arranged in the book so that its linear reading is logically defensible, to be sure, but we readers are human, all different from one another and from the author, and each of us is likely to find something difficult that is easy for someone else. My advice is to read till you come to a definition new to you, and then stop and try to think of examples and non-examples, or till you come to a theorem new to you, and then stop and try to understand it and prove it for yourself – and, most important, when you come to an obstacle, a mysterious passage, an unsolvable problem, just skip it. Jump ahead, try the next problem, turn the page, go to the next chapter, or even abandon the book and start another one. Books may be linearly ordered, but our minds are not. Does your spouse believe everything you say? Mine doesn’t always accept my expert opinion. Once when my wife wanted to read some mathematics, I gave her the advice (about skipping) that I just expressed, and she looked skeptical. The next day, however, she showed me, pleased and excited, the preface of a pertinent book: “look, you were right, they say just what you said”. Sure enough, the preface said that a reader must not “expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often and argument will be clarified by a subsequent remark”. What my wife didn’t notice was that “they” were not exactly a source of independent confirmation. “They” were and editorial panel, with their names listed on the title page; I was a member of the panel, and the words “they” said were in fact written by me. As long as I am musing on how I learn mathematics through the eye, I might muse a moment on learning through the ear. Lecture courses are a standard way of learning something – one of the worst ways. Too passive, that’s the trouble. Standard recommendation: take notes. Counter argument: yes, to be sure, taking notes is an activity, and if you do it, you have something solid to refer back afterward, but you are likely to miss the delicate details of the presentation as well as the pig picture, the Gestalt – you are too busy scribbling to pay attention. Counter counter-argument: if you don’t take notes, you won’t remember what happened, in what order it came, and chances are, your attention will flag part of the time, you’ll day dream, and, who knows, you might even nod off. It’s all true, the arguments both for and against taking notes. My own solution is a compromise: I take very skimpy notes, and then, whenever possible, I transcribe them, in much greater detail, as soon afterward as possible. By very skimpy notes I mean something like one or two words a minute, plus possibly a crucial formula or two and a crucial picture or two – just enough to fix the order of events, and incidentally, to keep me awake and on my toes. By transcribe I mean in enough detail to show a friend who wasn’t there, with some hope that he’ll understand what he missed. One-shot lectures, such as colloquium talks, sometimes have a bad name, probably because they are often bad. Are they useful to a student, or, for that matter, to a grown-up who wants to learn something? The something might be specific (I need to know more about the relation between the topological and the holomorphic structures of Riemann surfaces) or only a vague but chronic pain in the conscience (my mathematical education is too narrow, I should know what other people are doing and why). My own answer is that colloquium talks, even some of the very bad ones, are of use to the would-be-learner (specific or vague), and I urge my students and colleagues to support them, to go to them. Why? Partly because mathematics is a unit, with all parts interlocking and influencing each other. Everything we learn changes everything we know and will help us later to learn more. A good colloquium talk, well motivated, with a sharp central topic, well organized, and clearly explained is obviously helpful – but even a bad one can be helpful. My favorite case in point is a bad talk I heard once on topology. I didn’t understand the definitions, the theorems, or the proofs – but I heard “stable homotopy groups” mentioned and, a few minutes later, “Bernoulli numbers”. I had only the dimmest idea of what stable homotopy groups were, and an equally dim one of Bernoulli numbers – but my knowledge grew and my ability to understand mathematics (and, in particular, future colloquium talks) became greater just by listening to that odd, surprising, shocking juxtaposition. (It has become commonplace to the experts since then). It’s worth 55 wasted minutes to learn, in 5 minutes, that the theory of the zeroes of meromorphic functions has a lot to do with the distribution of prime numbers. Seminars talks are thought to be different, but they are not really. The usual notion is that the audience in a seminar consists of experts who know everything that has been proved in the subject up to the day before yesterday and are just there to see the last filigree. That notion is false; a good seminar talk is good and a bad one is bad, the same as a colloquium talk. The idea that the speaker’s technical incomprehensibilities become conceptual insights if the occasion is called a seminar instead of a colloquium is just not realistic, just not true. The “experts” present, be they colleagues working on problems similar to the speaker’s (but remember: “similar” is almost never “identical”), or graduate students trying to work their way into the arcane technicalities, get lost and impatient and bored only 30 seconds than the so-called “general” audience at a colloquium. In my opinion good seminar talks and good colloquium talks are interchangeable – and even the bad ones are worth going to. The best kind of seminar has two members. It can last five minutes – a question, followed by a partial answer and a reference – or it can be a strong collaborative bond that lasts for decades, and it can be many things in between. I am very strongly in favor of personal exchanges in mathematics, and that’s one reason I am in favor of colloquia and seminars. The best seminar I ever belonged to consisted of Allen shields and me. We met one afternoon a week, for about two hours. We did not prepare for our meetings, and we certainly did not lecture at each other. We were interested in similar tings, we got along well, and each of us liked to explain his thoughts and found the other a sympathetic and intelligent listener. We would exchange the elementary puzzles we had heard during the week, the crazy questions we were asked in class, the half-baked problems that popped into our heads, the vague ideas for solving last week’s problems that occurred to us, the illuminating comments we heard at other seminars – we would shout excitedly, or stare together at the blackboard in bewildered silence – and, whatever we did, we both learned a lot from each other during the year the seminar lasted, and we both enjoyed it. We didn’t end up collaborating in the sense of publishing a joint paper as a result of our talks – but we didn’t care about that. Through our sessions we grew… wise? … well, wiser, perhaps.

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