What things should one know in order to enjoy their undergraduate degree?

Question

From looking at undergraduate mathematics programmes it's quite apparent that mathematics degrees are demanding, one could even say the work load is grueling.

However I'm certain that there are things that one could do to prepare in advance for the rigours of such a degree. What i'd like to know is what foundations must be in place so that the experience of learning mathematics at university is an enjoyable one. Enjoyable in the sense that if you're exposed to a new topic you aren't floundering and you can dive straight in and enjoy the exposition and the process of learning, without having to go backwards plugging in numerous gaps and addressing other deficiencies in your knowledge.

I'm certain that a good grounding in pre-calculus mathematics and calculus are a prerequisite but aren't all that's needed.

What are the things that must one know in order to have a solid grounding in mathematics, with the aim of studying mathematics at a higher level?

Edit: Let's assume it's a quite a demanding degree programme: MIT, Harvard, Cambridge, etc.

Answer 1

Some skills you need to be familiar with:

1) Techniques of proofs: How should you think to prove a statement (induction, contrapositive, minimum counterexample etc), you can start with some proofs in number theory (a most excellent book):

2) Analysis: helps you make sense of numbers and continuity, you can then use sequences to decompose mathematical structures, baby Rudin is the most widely used introductory book in analysis.

3) Algebra (linear and abstract): Allows you to understand abstract mathematical structures. Hoffmann/Kunze for linear algebra.

Once you know have a basic understanding of these fields in mathematics you can be quite free to explore (Topology, Geometry, Number theory, etc)

Be curious! and Work hard! You'll find mathematics beautiful and rewarding!

Answer 2

The discipline of mathematics requires the following from you in order to excel in it:

1) Your love of the subject. 2) Your desire to do well in it. 3) Intellectual curiosity. 4) Willingness to work very hard. To excel in ANY discipline, you must work very hard. 5) Mental aptitude. This is the "luck component" of life. You have what you're born with as far as your maximum potential is concerned.

When asked why he didn't pursue a PhD in physics, my high school physics teacher responded in class, "Because you must have an uncontrollable desire to do physics."

If it were me, I would carefully examine all the branches of mathematics that interest you. Look at introductory texts for each of those branches. I strongly suspect that many people who end up with PhDs in mathematics have explored WAY beyond pre-calculus before they entered college. I would also investigate what courses are mandatory at your desired university and what courses are elective.

http://en.wikipedia.org/wiki/Areas_of_mathematics

Answer 3

I read this book from Lara Alcock and I think it answers your question: "How to Study as a Mathematics Major". I warmly suggest it!

http://ukcatalogue.oup.com/product/9780199661312.do