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193

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: http://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 in ...


190

If one does not want to do something, then one can't do it. The question is how to get past any resistance and issues one has with, in this case, mathematics. From the conversation above, I can make a few recommendations. First, avoid asking simple questions for which the answer is obvious to you and should be obvious to your student. The reason is that it ...


145

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. ...


133

I am a high school teacher, so here are some comments specific to the situation that don't necessarily answer the question directly: One hour is around the maximum that a normal 13-year old student can concentrate on intense mathematical learning. You need to manage this by sticking to maximum one-hour sessions, maybe with a 10-minute break in the middle. ...


75

There is a continuum in the way one understands a theorem. At one end of the spectrum mathematicians just try to understand the statement and use it as a black box . At the other end they understand the theorem so well that they improve on it: this is called research. An important thing to keep in mind is that your attitude toward a result is not fixed ...


57

This is a hard question to answer, because the answer would depend a lot on personal aspects of your situation, but there are some general points of advice one can give: (1) Try different books. It may be that the particular textbook you are using doesn't click with you, but for classes like algebra and other pre-calc courses, and calculus itself, there ...


53

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 ...


52

Of course everybody has their own learning style. Here are some general suggestions. Find a teacher. It is hard to learn mathematics on your own until you have reached a certain level of mathematical sophistication; nobody is there to tell you what is important and what is unimportant. Take courses at a university; as Agusti Roig mentioned, video ...


50

First, let her calculate lots of easy exercises. There should be only about 3 types of exercises. She has to choose the correct algorithm. When she can calculate the trivial exercises without problems, move to the more complicated exercises. Add some abstraction, some real world examples. Add abstractions slowly. It's important letting her discover ...


49

This could be explained using algebraic transformation but i would rather show a very simple geometric proof for sum: 1 + 1/2 + 1/4 + ... = 2


48

If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people are genuinely rare. The other kinds of ...


48

Don't try to memorise the proofs: try to memorise the methods that are used in most analysis proofs. That way you only have to memorise a handful of methods instead of 30-50 proofs, and you can adapt them to prove things you have never seen before as well.


46

Alligator Eggs is a cool way to learn lambda calculus. Also learning functional programming languages like Scheme, Haskell etc. will be added fun.


42

This is a difficult question to answer, mainly because any advice must be very personal to be useful for you. I'll try anyway. Before learning mathematics, one has to learn how to learn mathematics. Concerning the contemporary school system, it simply does not do a good job at what it is supposed to do. The range of children who learn successfully from it, ...


42

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 ...


40

Riemannian geometry is a difficult subject to type in LaTeX. For example, there are concepts in Riemannian geometry which are best understood by pictures that would take no more than five seconds to draw by hand but much more time to incorporate into a LaTeX file. Also, the same can be said for long computations in Riemannian geometry (e.g., the computation ...


39

In my opinion there are a lot of things that cannot be imagined or understood. As John von Neumann said: "Young man, in mathematics you don't understand things: you just get used to them.".


38

I was recently in a similar situation. After finishing precalculus at my high school, when I was 15 I started taking calculus at my local university and studying higher mathematics on my own (out of the book "Modern Algebra: An Introduction" by John Durbin, which in retrospect seems laughably basic but at the time blew my mind). Three years later, I can say ...


38

I think all of us at some point will invoke theorems whose proofs we have forgotten. I would argue that memory is important for mathematics in the sense that it is important for practically every other field. Certainly having good memory will not hurt you and several mathematical giants were undoubted aided by their prodigous memories (notable examples ...


35

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 ...


35

To me, asking if you need to understand analysis is roughly* like asking "Is it necessary for one to understand how to operate a computer to pursue a career in mathematics?", in that the answer is technically no, but Everyone else does They'll assume that you do too There's no good reason not to know By not knowing, you are making things incredibly ...


34

I very often find some knowledge of category theory useful to understand things conceptually. A book that one could read before studying mathematics at the university is Lawvere's and Schanuel's Conceptual mathematics. This is an introduction to category theoretic ideas on a most elementary level. Edit: two days ago, there appeared a very ...


34

Luckily these days there is a beautiful text that teaches algebra and category theory at the same time: Aluffi - Chapter 0. It deserves to be more well-known. Besides the fact that it uses (basic) category language from the outset, it is very well-written. If I would ever teach an algebra course, this would probably the text I would use.


34

Jordan Maybe what you need is a coach or a mentor. If you don't believe you can do this yourself, you never will do it. That happens to most of us in Maths at some stage. But if you have someone telling you the answers or "how to do it", you'll never actually learn how to do it for yourself. I used to sit in front of problems until I could solve them (I ...


34

In no particular order: Algebraic number theory notes by Sharifi: http://math.arizona.edu/~sharifi/algnum.pdf Dalawat's first course in local arithmetic: http://arxiv.org/abs/0903.2615 Intro to top grps: http://www.mat.ucm.es/imi/documents/20062007_Dikran.pdf Representation theory resources: http://www.math.columbia.edu/~khovanov/resources/ Classical ...


33

This advice from a teacher I had, long, long ago has stuck with me all along. "There are three ways to react to any [troubling] situation: You can get frustrated... ARGH!!! You can feel intimidated... :-( or... You can be INSPIRED! In other words, try to catch yourself when you're feeling overwhelmed or frustrated; if you can reframe the situation ...


33

This is a recapitulation and extension of what we talked about in chat. Whatever you do, I recommend that you try a variety of areas in order to find out what you like best. Don’t feel obliged to stick to the most common ones, either; for instance, if you find that you’ve a taste for set theory, give it a try. My own interests are outside the ...


33

Of course you can make a career out of it! When I started reading your question I though you were around 50, but 22 is not old at all to go after a career in anything but sports. This kind of time don't affect your brains ability to think at all. The only thing is, if you have great ambitions, you are probably not gonna be able to win the fields medal ...


33

I'm not sure my personal experiences will be very helpful to a mere kid of 22, but here goes ... I made a complete mess of being an undergraduate when I was 18 (until 21), and followed a career for some decades before I finally got round to seeing if I was actually capable of doing maths at a more advanced level. I was almost 50 by the time I had published ...


33

Once you start doing "really abstract math", in my opinion, it is no longer possible OR useful to really imagine anything visually. You can ask yourself questions like: "How does a plane look like in $\mathbb{R}^{1000}$? But even this is a lot more "concrete" and "visualizable" than trying to imagine how, I don't know, the group ...



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