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Next year I'm going to start university (maths, obviously), and would like to ask you (I suppose you're mathematicians or, at least, people who studied maths somehow) some advice: what did you learn over the years that in your opinion could have been useful to you when you started? (e.g. note taking techniques: tablet or pieces of paper); things that I should know about maths (topics I'm supposed to know); things I should know about university maths; things I should know about how to study... etc. In a few words, anything that can help me to get the most out of my first year studing maths at university). Thank you for your help!

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closed as too broad by Potato, Did, hardmath, Pete L. Clark, T. Bongers Dec 21 '13 at 4:12

There are either too many possible answers, or good answers would be too long for this format. Please add details to narrow the answer set or to isolate an issue that can be answered in a few paragraphs.If this question can be reworded to fit the rules in the help center, please edit the question.

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Well, in my days there wasn't even a choice between tablet and paper :) –  Hagen von Eitzen Dec 16 '13 at 14:20
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Me too high five! study your books, listen to your teachers, upove this comment, and study some more. –  Bananarama Dec 16 '13 at 14:21
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Don't be dependent on the internet. It's tempting to just google the answers to questions but believe me it's really rewarding to be able to solve problems by yourself. :) –  chowching Dec 16 '13 at 14:57
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@user1729: That’s their problem, not yours. It’s even possible that some of them don’t really need the lecture. –  Brian M. Scott Dec 18 '13 at 19:50
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@user1729: As long as it’s silent, I don’t care: it’s their business, and in all honesty I consider your policy a bit unreasonable nowadays. // Some of the ones who don’t need the lecture simply aren’t there, unless you have some sort of attendance policy, which I dislike: at the university level I consider that the student’s business. Others are sitting there doing something else with half an eye/ear on what you’re doing, just in case. –  Brian M. Scott Dec 19 '13 at 9:50
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I could give you many advices but most of them would be quite generic. The most important one which I unfortunately realized only when it was a bit too late is that you have to study continuously. Maths is a skill - you have to work it, bit by bit, everyday. Don't expect superb results straight away - but they will come with persistence. Also try to keep your enthusiasm alive and make maths enjoyable, cultivate that feeling. But most importantly - study all the time, everyday. I am not saying 24/7, not at all, but just don't skip anything because 'you understand', or 'it's easy' or 'I'll do it tomorrow'. Be consistent and you will be rewarded big time.

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Agreed. "I'll do it tomorrow" is just another way to say "I'll never do it". –  Jean-Claude Arbaut Dec 16 '13 at 15:05
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+1 for study continuously. –  Student Dec 20 '13 at 23:24
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Keep asking questions, as others have said.

More specifically ... mathematicians have an annoying habit of presenting only the final "beautiful" results, without showing you all the ugly hard work and false starts that led up to them. As far as I can recall (it was 45 years ago) most things were presented this way. Everything looked like the magical products of transcendent minds. It was awe-inspiring, but it was also bewildering and discouraging.

Anyway, I wish I'd had the good sense to question all this magic. Question everything. Specifically ...

(1) Question the definitions: Why would you make a definition like this? What are some examples of things that have this property and things that don't? Why is it important to distinguish this class of objects? What happens if you weaken the definition? Where does the subsequent reasoning break down?

(2) Question the proof techniques: How did anyone think to use this line of reasoning in this proof? Are there some simple cases where the reasoning obviously works? Are other lines of reasoning possible? Are there analogies?

Don't let them fool you. Mathematics is not nearly so neat and tidy and magic as some teachers and textbook authors pretend. The good teachers will show you the dirt and mud, not just the pretty flowers that have grown in it. The not-so-good ones ... well, you might have to prod them a little, and they probably won't like it. Prod anyway. Good luck.

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Questioning definitions is very useful. Try weakening a definition and then see how close you can get to the proof of some theorem. If you can not get the whole theorem can you get something close. –  Jay Dec 16 '13 at 15:18
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Good teachers spend a great deal of class time trying to convey the intuitions underlying the results and their proofs — trying to make it look less magical, if you will. –  Brian M. Scott Dec 16 '13 at 19:22
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@Brian -- True. But it's useful to know how to deal with the not-so-good ones. I'd say they are fairly common. I attended two pretty good universities on two continents, and, in retrospect, I'd say my experiences were "mixed" (to put it politely). –  bubba Dec 18 '13 at 9:48
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When it comes to problem sets, be antisocial. Some people like to do homework together in teams. I don't think that's a good idea. Struggle with the problems yourself. Working with peers should be reserved for when you are stuck on a problem, or want to discuss a problem you have already solved.

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Make your notes beautiful: use lots of colours to highlight stuff, re-write good versions after a lecture, and liberally sprinkle with your own annotations. Love them dearly, for when exam time comes around you will be spending a lot of time with them.

You have to remember that your notes are your porthole into the course. They are also personal to you. I always found my own notes better than the lecture notes handed out. This was because I used three different colours, and annotated my notes heavily. (I used blue for definitions, statements and normal stuff, I used black for proofs and annotations, and I underlined stuff in red.)

Copying out your notes takes a lot of time, but is very useful. It helps you gauge the course better, giving you an understanding of the content. It also lets you pick up on things you don't understand. It forces you to read your notes in detail. Most importantly, though, it gives you a wonderful set of notes come exam time. If you have been doing it properly you will have annotated lots, and these annotations are useful in exam season. If you understood something in week 1 it is not unlikely that you will have forgotten 10 weeks later, but your annotations tell the story of your first understanding. Even if it turns out that you didn't understand it the first time, you will learn where you went wrong and so understand the concept better.

Oh, also, do lots of questions and the usual generic stuff that people say you should do...(but, of course, these generic things are tried and tested and work. Especially doing lots of questions).

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I'll get straight to the heart of the matter - Do the problem sheets, and every damned question on them!!

Never skip one 'because it's an easy one' or 'I know how to do that'; nor equally 'that looks too hard'. The 'easy' ones may surprise you (Im sure you'll have tried to evaluate a deceptive integral at some point) and the hard ones may be more accessible than you think (and you certainly wont find out without trying them).

Maths is a skill you learn and develop. Just as chess is a game where one's strategies are refined by playing many opponents of many styles, so is maths and its problem-solving/theorem-proving! Its is partly study, and partly experience. Reading a book or following a lecture is much easier than cultivating ability from experience, though.

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In addition to what others have suggested, here's something else to do:

Become skilled at, and use relentlessly, one or more computational tools to visualize and explore simple and complex problems in ways beyond what's in lecture, textbooks, and homework. Microsoft Excel and Mathematica, for example.

With luck, your university has a license for its students to use Mathematica, but if not, they should have something similar. At worst, struggle with a free tool like Sage or a programming language like R. Or learn some Python and see what you can do with libraries like pymath. Or learn the Racket language.

There are a million useful things you can do, and you can focus on what you enjoy most.

If you like integer stuff, like binomial coefficients, or Fibonacci numbers, compute hundreds of any of them, look at their remainders modulo various integers (color them with conditional formatting in Excel, even). Generate a huge Pascal's triangle, or a grid where row j is the a Fibonacci-like sequence with a $j$ in the recursion - maybe $f_n=f_{n-1}+f_{n-2}+j$. Where are the even ones? Compute ratios, differences, logs, anything you can think of, look for patterns, and make conjectures. If you find nothing interesting, try different formulas.

If you like calculus, create manipulators in Mathematica so you can see how graphs change if you vary one of the parameters. To choose an exploration at random: how does the graph of $(1+ax)^2/(2a+x)$ change when you vary the value of $a$? (For $a=0$, it's a hyperbola. When $a$ is very large, it's almost a parabola. In-between, what's going on?) Draw as many graphs as you can - polar graphs, 3D graphs, parametric graphs...

Reverse engineer arithmetic you see in or outside of class. If you encounter some calculations in chemistry or physics, figure out with an Excel spreadsheet how sensitive the results are to errors or rounding of the input values, then try to verify anything you see with some algebra.

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I've just started university and my advice is to found out what modules you are doing and see what is involved in them.

If you haven't studied further maths (like myself), read up on various topics that you haven't covered such as matrices, complex numbers etc. If you do, you won't be as far behind.

You're likely to be doing theoretical modules, which involve proofs etc., so you may want to have a read of how to read and understand proofs.

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Study hard basic logic (implication, equivalence, contraposition, negation, and so on...). Maybe you can survive at undergraduate level without it (especially in Calculus) but my claim is that if you understand it very well, you will see that many proofs are "mechanical" and become straightforward: one easy step after the other. It will help a lot for abstract topics like group theory or set theory.

Also, be over skeptical and never "accept" a proof. Question every detail of a proof (why is this number non zero? Why can I take the square root of this expression?...) until it becomes clear. My classmate at college never missed a gap in my proofs and I think it keeps me away from my natural tendancy to "intellectual lazyness". In a similar proof, don't accept the prof's proof as the best ones and question yourself about them. Why is it natural to use this method? Can I find a simpler/more natural way? In which kind of problems can I use the same trick?

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Lots of good advice here. I would add some more pieces...

Try and come top in every assessment and module. Strive to be the the best and aim for perfection bit don't fret. If you come up short. This involves working HARD.

If you like to party pick your spots and go all out. Work Hard, Work Often. Play Seldom, play hard. You might find in later years that you are too busy to party and you might regret not going out and experiencing that side of college earlier.

Read popular science maths and indulge in the lore of mathematics. Allow yourself to be inspired by the whole damn romanticism of it.

Most importantly be grateful for your gifts, enjoy them...and number one: have fun. Maths is supposed to be fun.

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Look to the future. I had no clue what I would do after school. If you want to be a professor one day, then it really helps getting a faculty person or two to be your friend so that you can do undergraduate research, get good letters of recommendation, and make contacts at possible graduate schools.

If not a professor, then you should take CS and/or statistics courses. There's no real jobs for those with a bachelors in pure math except NSA, and CS will help there too.

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I disagree with the last paragraph. There are no jobs which apply pure maths apart from those you mention, but the skills learned while studying maths are transferrable. Not doing CS or stats courses does not make you unemployable... –  user1729 Dec 19 '13 at 9:48
    
I applied to >40 non-teaching jobs such as the CIA, the Boy scouts, and banks after my PhD in pure math. I got several rejections saying I was overqualified, although several companies offered me a position because I programmed video games for a year (there's the CS). Only the postdocs expressed any interest, so that's what I did. –  Brian Rushton Dec 19 '13 at 13:43
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A sample population of one is not good statistics! Looking at a different population of one, I know a guy with a PhD in pure maths who walked into a job in a bank. I also know two pure maths PhDs who have become actuaries. A lot of my maths friends became accountants after their undergrads (and I think only one ever did any stats). –  user1729 Dec 19 '13 at 13:52
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Others have provided excellent answers here. I am going to say the most important topics you should know,

Set theory

and

Mathematical Logic

at basic level since they are the foundation of mathematics.

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This is incredibly subjective! Personally, I feel that the most important topics that the OP should study and geometry and group theory. But that is simply because I am a biased geometric group theorist. –  user1729 Dec 19 '13 at 9:37
    
@user1729 I actually study group theory, too. My reasoning is that you cannot be a good group theorist without knowing set theory. Remember the OP is just starting undergrad math. –  scaaahu Dec 19 '13 at 9:49
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Yes, you need set theory for group theory, but only in a very basic sense. What if they decided to study applied maths? Then understanding PDEs is important. Maths is a big subject, and at a basic level there are no topics which are generically more important than others. –  user1729 Dec 19 '13 at 10:03
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I still disagree though - just because they are the foundations of the subject does not mean you need to know them. I mean, I know next to nothing about mathematical logic, unless you mean proofs and stuff? –  user1729 Dec 19 '13 at 10:11
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Yes, but what do you mean by basic level? What would you consider to be included in a basic level of mathematical logic? –  user1729 Dec 19 '13 at 10:19
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