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174

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 ...


165

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 ...


124

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. ...


120

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. ...


62

There is a continuum in the way one understands a theorem. At one end 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 it: this is called research. An important thing to keep in mind is that your attitude toward a result is not fixed for ever: you may ...


48

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 ...


47

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 ...


43

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 ...


41

I can only speak from personal experience here, so don't take anything I say as a universal statement about doing math with iPads. Reading math on an iPad is great. I've been using my iPad to read during meals, at cafes, and in other places where it's more convenient because of its size than taking out my laptop, and it also has substantially better ...


35

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 ...


32

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 ...


32

You require two things: good stylus and good app. I also like to have stand but more about apps and accessories here. Before buying any Apple product, please, acknowledge the below limitations. Limitations Linux users should check whether they can do everything they need with Chromebook and Android phone because Apple discriminates you in many things ...


32

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 ...


32

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 ...


31

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


30

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 ...


30

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 ...


28

Many people get through Calculus by memorizing formulae. But you'll learn best by combining "conceptual understanding" with "procedural knowledge". To answer your question regarding how to develop the "problem-solving" creativity needed to manipulate expressions into forms you can apply theory: Answer: Practice! AND Effort (Perseverance)!, AND Time ...


27

First of all, thanks, Michael, for a great question! And framing it in terms of "imagine teaching mathematics to blind students..." helps us to recognize issues of math accessibility, in this case, accessibility to the visually impaired. You've helped me to educate myself, a bit: for example, whereas written mathematics depends extensively on "2-D" ...


27

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 ...


26

Your sister has developed a strategy that was successful to cope with a lot of typical problems given in school. So of course she is trying "more of the same" to solve any further problems. You are trying to force her to think in a very different way about these problems, often a way that will not instantly produce answers, but that is exactly what she is ...


25

This question partially belongs to the sister SE site: productivity.SE To fight the mental fatigue the following things will help: doing physical exercises, as they improve oxygen supply to the brain (e.g. walking, working out, etc) getting enough sleep keeping a healthy diet Essentially of all the above is to condition the brain to be in the best ...


24

There are perhaps 4 distinct stages in Mathematical understanding, and a lot of people never get past the first (and indeed may never need to). The stages are: Applying specific recipes to solve specific problems. Analysing problems to work out which are the right recipes to apply. Analysing recipes to understand why they give the right answer, and ...


24

It's hard to beat Feynman's Abacus story in Surely You're Joking, Mr. Feynman!. This excerpt copied from here which notes that the story is taking place in Brazil. A Japanese man came into the restaurant. I had seen him before, wandering around; he was trying to sell abacuses. He started to talk to the waiters, and challenged them: He said he could ...


24

I think the key to your problem is in your first paragraph. You say, "The correct answers came fast and intuitively. I never studied." This is the classic high school con that can lead one to doubt one's own abilities as soon as the going gets more challenging. No matter what your abilities, to do worthwhile work in mathematics you will need to study and ...


24

You can read every book ever written on chess, but if you never play you will still be, at best, a middling player. Even if you memorize every rule in every book on chess you still won't become a particularly good player. You must play! The same is true of math. You must solve problems! I was always a natural with math, and I almost always grasped ...


23

Oh my! $22$ years old? You are still very young, and certainly not too old to pursue any passion, math or otherwise! I've taken many (lengthy) breaks from math. But each time I've returned to serious and dedicated mathematical work (be it studying math, teaching math, or pursuing research), I've come at it from a fresh perspective. Sure, when I've ...


22

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 ...


22

I believe that all the above answers are quite splendiferous. (I do not use that adjective lightly. :)) I'd like to provide my own input, though it might be slightly Socratic: What do you care about? Ask yourself this and answer honestly. Does mathematics happen to be one of the things you care about? Or do you find mathematics to be a torturous beast that ...


22

Category theory and algebraic geometry. I spent a lot of time in undergrad studying things that were kinda nifty, but way too classical to be of any use/interest beyond "fun math". When I got to grad school, category theory was assumed and made some of my courses much harder than they should've been. In the words of Ravi Vakil, "algebraic geometry should ...



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