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237

In my view the central question that you should ask yourself is what is the end goal of your studies. As an example, American college life as depicted in film is hedonistic and certainly not centered on actual studies. Your example is the complete opposite - you describe yourself as an ascetic devoted to scholarship. Many people consider it important to ...


226

Karl Weierstrass was in his 40's when he got his PHD. There are a dozen other counterexamples, a number fairly recent. A good set of examples can be found in the thread on MO here: https://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners This myth of "science is a game for the young" is one of the falsest and most destructive canards ...


186

Let me tell you that the only thing that I have been doing for the last four years of my life is mathematics. I have enjoyed the experience thoroughly but I have also had points where I was somewhat unsure as to how to approach my learning. I think that there is no one rule that works for everyone; however, let me answer some of your questions. I hope that I ...


169

21 is not old at all. I personally know heaps of people my age (32) who started out at 18 as salesclarks/BA or BCom majors/lawyers/bookeepers etc and ended up having a PhD degree in some advanced math areas and landed a job in academia or industry. My personal case: I got a lousy BCom degree with little math at 22 and then worked in a primitive banking job. ...


165

Clearly the figure is a trapezoid because you can construct an infinite number of quadralaterals consistent with the given constraints so long as the vertical height $h$ obeys $0 < h \leq 9$ inches. Only one of those infinite number of figures is a square. I would email the above statement to the teacher... but that's up to you. As for the "politics" ...


164

Most people probably won't like this answer, but mathematics is a field where there's an unstable separation between those of genius caliber understanding and those who are just able to get by through dogged hard work. Way too many people want to do proofs and aesthetically pleasing artful mathematics for a career who are in the dogged-hard-work category. I ...


151

First, let me say that I'm impressed by your maturity and wisdom. It's not easy to recognize your own limitations, accept them, and adapt. Most people have to learn the hard way, by living through a few decades of struggle and frustration. Some people actually enjoy struggle and frustration, though. Your choice. I have been an "industrial" mathematician for ...


150

My advice would be: $\bullet $ Do many calculations $\bullet \bullet$ Ask yourself concrete questions whose answer is a number. $\bullet \bullet \bullet$ Learn a reasonable number of formulas by heart. (Yes, I know this is not fashionable advice!) $\bullet \bullet \bullet \bullet$ Beware the illusion that nice general theorems are the ultimate goal in ...


143

Of course, you are right. Send an email to the teacher with a concrete example, given that (s)he seems to be geometrically challenged. For instance, you could attach the following pictures with the email, which are both drawn to scale. You should also let him/her know that you need $5$ parameters to fix a quadrilateral uniquely. With just $4$ pieces of ...


111

I'm a little disappointed by the comments. Granted, it's hard to explain mathematics, but having the attitude that you're not even going to try is not doing mathematics PR any favors. We can't in good faith expect the public to fund our research if we're not even going to try to tell them what we're doing with their money. First, I hope you won't take this ...


88

Let me give you a personal story. As a young kid, I was always very strong in math but was pretty hampered by one of the worst educational environments in the USA. I ended up entering a magnet school for junior high and had to take a math placement exam to determine which of three math classes I would join: regular math, pre-algebra and algebra. I didn't do ...


86

At my university we use baby Rudin, Dummit and Foote, and Munkres in the first year analysis, algebra, and topology courses (respectively), which is the average schedule for an incoming grad student. You can also test out of these and move to the 2nd year if you happen to be good in one of those fields (or two, if you're really exceptional). My opinion is ...


85

FWIW, this question appears to come from a diagnostic test which can be perused at http://www.mathmatuch.com/presentations/diagnostic_test.pdf -- where the official answer is given as J (the square). So it's not just the teacher who is wrong. (Remark: I found the site by googling on "identify the figure shown" and "trapezoid" then looking for "76" and "J" ...


77

Absolutely! When I look back at some of my old papers, I find it hard to follow what's going on. The point about research is that you're pushing yourself and mathematics to its limits. It's impossible to maintain such a high standard over such a bredth of knowledge. PhD research requires independent and original thought. No-one cares if you can solve all ...


75

Sam sat with his eyes closed for several minutes, then said softly: "I have many names, and none of them matter." He opened his eyes slightly then, but he did not move his head. He looked upon nothing in particular. "Names are not important," he said. "To speak is to name names, but to speak is not important. A thing happens once that has never happened ...


66

I believe that I may be of some consolation. I had a very similar experience to you. I started doing "serious" math when I was a senior in high school. I thought I was very smart because I was studying what I thought was advanced analysis--baby Rudin. My ego took a hit when I reached college and realized that while I had a knack for analysis and point-set ...


65

In Israel kids are expected to serve in the army when they are 18, and they serve for three years (men do, women serve two years). After this period it is common to find yourself questioning what you should do with yourself and not many people have answers. Therefore it is common to take another two years to work and travel the world before settling down and ...


63

This is a great question and I think I may have responded to a similar one months ago on the companion site Math Overflow. I don't recall exactly what I said there,but to answer your question: Firstly, although I firmly believe mathematics has to be learned actively and looking up answers should be something you try to avoid, there has to be a practical ...


60

I'm convinced that I've proved a major conjecture. You are almost certainly mistaken. I say this on purely probabilistic grounds, so don't get upset $-$ even professional mathematicians are sometimes mistaken about their own 'proofs', and amateurs almost always. I suggest you tell us what this major conjecture is, and post a link to your proof (or just ...


60

Largely (very largely, so please take everything here with a grain of salt), there are two types of mathematical research, commonly referred to as 'theorem proving/problem solving' vs. 'theory building'. Typical characteristics of theorem proving/problem solving type research is to try and tackle a famous open problem, usually stated in the form of a ...


57

When I read a math book, and arrive at a proof I go through the following process: Try and prove the theorem without looking at the proof. Most of the time I fail. Read the proof and write it out by hand. The purpose of this step is to learn techniques which might be used later. Summarize the proof in one or two sentences. The summary is what you need to ...


57

Disclaimer: If you didn't consult your friends and/or parents on whether your situation is serious enough (i.e. requires specialist attention), then do it now. Major depressive disorder may have detrimental effects on your life and you should treat it accordingly. On the other hand, mild depression, melancholia, etc. may be temporary and may not require any ...


54

As a laymen myself, let me shed some light on this, and perhaps prevent this potentially awkward conversation from happening. The problem is you're trying to explain your research in a few sentences and induce wonder and so on. There is no solution. You'd have a better response if, instead of explaining what you're doing or how you're doing it, you told ...


51

I recently read an article on the 40 hour work week and I think it is somewhat related. The basic idea of it was that in the mid 20th century, they had a 40 hour work week and they had lots of research on it showing that it was optimal in many ways. That is, if you increased your work week from 40 hours to 60 hours, you wouldn't gain 50% extra productivity....


51

For better or worse, being unusually concerned about having one's precious ideas stolen is one of the classical hallmarks of a mathematical crank. Therefore if you get visibly protective, the loss of reputation you'll suffer simply because you sound like a crank is likely to outweigh, by far, the very minor risk that you lose the opportunity to earn some ...


51

Be honest with yourself about what you do and don't understand. Don't fall victim to "proof by intimidation," where someone attempts to shame you into saying you understand something by implying that you're dumb if you don't. Always ask questions until you really get it. Similarly, don't let yourself move on before you understand something fully; pretty much ...


50

Some very basic non mathematical advice and I'm sorry if I sound like your mother. If you feel like your memory is bad and you're not finding enough time to socialise, perhaps you're not finding enough time to eat well. Eating plenty of fresh fruit, fresh vegetables, fresh fish, olive oil and cereals will give your body the building blocks to do its best . ...


47

Regarding the "and have a healthy lifestyle" thing. Well, you also have to learn when to stop. Sometimes you figure out you don't understand something and you don't have time right then to understand it. Try to "box" that as much as possible. Figure out the general form of the kind of thing you don't understand. How does it function? What context is ...


45

It will be a very simple answer. If you like (not to say love) mathematics, don't ask this question. Just go and win ! Good luck and see you soon as a brilliant mathematician !


45

The point is that mathematically, you can't tell from the picture. It might be this: It is easy enough to describe a construction of this with compass and straightedge, so it is definitely a legitimate geometric figure by any reasonable definition. The same "diagnostic test" from which this came (thanks to Barry Cipra for finding it) has numerous other ...


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