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Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? When you learn mathematics, whether in books or in lectures, you generally only see the end product — very polished, clever and elegant presentations of a mathematical topic.
For instance, given a standard lemma in a subject, you can ask what happens if you delete a hypothesis, or attempt to strengthen the conclusion; if a simple result is usually proven by method X, you can ask whether it can be proven by method Y instead; the new proof may be less elegant than the original, or may not work at all, but in either case it tends to illuminate the relative power of methods X and Y, which can be useful when the time comes to prove less standard lemmas.
However, questions which do not immediately enhance the flow of the talk are probably best left to after the end of the talk. Comments feed for this article. When I study something myself either on my own or taking a reading course instead of learning it from a lecture course, I found I learn them a lot worse than I think I could have if I take a formal class. But I spend probably just the same amount of time. Do you have any advice on something that I should particularly pay attention to when I learn things myself? Terence Tao. Besides the advice already on these web pages, the one thing I can offer you is that when you are learning by yourself, it becomes very important to find ways to really test your knowledge of the subject, since you do not have homework, exams, or other feedback available.
But, as I already discuss in the above post, there are plenty of other usefully instructive tests you can make for yourself, for instance seeing whether you can somehow improve one of the lemmas in a text, or working through a special case of a theorem, etc. Unfortunately, I dont get at any result! I do not know what I will do and I do not have any precise brojet for my thesis. And yes, a reasonable amount […]. I have recently become obsessed with mathematics and proofs. I am already 24 years of age. Also this seems to terrible slow down the speed at which i can study.
I come from an engineering background but would like to do my graduate studies in maths. Would you still advice me to keep the same process going? I must say that my problem solving ability has improved a lot courtesy this process. Thanks a lot. I emphasize to them they are in my class to learn and not sit passively but participate actively. I think this is very important, especially at the undergraduate level. I invite my students to discuss topics we cover in the class, always critically questioning what we read in textbooks.
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I also learn from my students especially from their probing questions as much as they learn from me. I especially like students who ask the toughest questions!.
I have enjoyed being a college teacher for the past 20 years — even though the pay is poor. I have also learned a lot from my students. Essential Career Lessons. This is a really good post, thx. That would be great. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. It just happened to be a step process. If you never try because you are afraid of failure, you will never succeed. This can really help solidify the right way for you.
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Depression and problem solving in mathematics: the art of staying upbeat — Republic of Mathematics blog. Trivial reduction question is deep Nanoexplanations. That said, there are plenty of questions around the standard exercise of evaluating that are worthwhile.
Some examples: can the formula and its derivation be extended to the case where n is zero? What is the relationship between the sum and the integral? What can one say more generally about for k a natural number, integer, real, complex, etc. For instance, is it always a polynomial in n? Is there a way to assign a useful meaning to the infinite series , and how does this connect to the Riemann zeta function?
I discuss this latter point in this post. The quantity resembles a binomial coefficient — can one find a generalisation of the identity that emphasises this viewpoint? Related to this, is there a combinatorial proof of the identity?
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Can it be categorified? Is there a geometric proof? Does it extend to higher dimensions? What if one replaces by another arithmetic progression? What happens to all of the above questions if one replaces addition with multiplication? Or works in another group or ring? The standard proof of the identity uses induction; what happens if one somehow deletes the axiom of induction from the number system? Note that many of these questions are vague and open-ended, and thus quite distinct from the typical exercise one may find in textbooks.
Unlike such exercises, the point is not actually to find definitive answers to these questions, but rather to get your brain to start following original lines of inquiry, and to develop the type of mindset that is needed for genuine mathematical research. In many cases, the answers turn out to be degenerate or otherwise uninteresting, but the process of arriving at such an answer is often quite instructive. I love this answer. Many students actually ask a kind of questions that Pro. Tao mentions in the begining, but soon lost their interest in doing so because of formal education forcing them to find just the right answers.
This answer shows the very skill one should learn in his undergrads, but you are right, we do all get lost in the formalities of solving the problems, because that is the only way to get into exams, and then hopefully eventully graduate. On the other hand, this is also the very reason I have been losing passion and motivation for pure maths, because :.
No one gives you such good examples of genuine inquisitive thinking 2. The mindset that is valuable in research is actually counterproductive in an undergrads course for lack of time, and fast paced jumping from one chapter to the other. The fact that Professors are not accesible to students, as they need to go through tutors first, then assistant and only then Professor, should the former ones had failed to come with an answer to your question. Since tutors are students themselves, their interest is not in teaching but earning money, so usually one gets deflected reply to the question at hand.
We, the students, never see a proper proof to the homeworks we get, so for the most part we are left with huge gaps in understanding of the course. Since the time is always so limited, we almost never get to rethink an entire proof and thus try to find more elegant solutions, which would be a great way to keep high spirits and optimism.
My wish is to persue research in pure maths, but as I go on I only feel more unadequate to understand maths. This utter formalism, though so immensely neat and organized, is yet not the way that maths came about, which was very organic and natural in a sense, as ever more questions demanded answers.
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So I have to ask myself, what is my programme training me for? I am not sure if it helps to write my thoughts here, but I read your post and I certainly hear you. Spare time for the type of math you find interesting and get into it. You may want to visit a prof from the area of math you find interesting, and ask for accessible papers to read? If the prof is eager to give you a small research project and guide you through, it is even better.
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I think personal interactions with an active research mathematician will make you feel math is an organic product :. Am i correct?
I think that 1 is a good question. This is mainly because it leads you to look for different ways of proving the identity.