Mathematics major vs mathematics degree?

What is the essential core of an undergraduate mathematics degree? (Further details in first comment.)

• Here’s the thing: twenty years ago I enrolled on a combined degree in maths and computing. It wasn’t very well organised and neither was I. After a couple of years I transferred off to a BSc(Hons) in Mathematics Education. It’s served me well and I’ve been teaching maths at ages 11-16 for fifteen years since. I’m not bad at explaining the subject and it’s made me a Jedi at basic, secondary/high school level mental calculation. But I’ve always wanted to know more so, without any idea of what I was doing, over the years I’ve collected books: Fundamentals of Mathematical Analysis (Rod Haggarty) Introduction to Mathematical Analysis (Steven A. Douglass) Modern Abstract Algebra (Frank Ayres, Jr.) Discrete Mathematics (Kenneth A. Ross & Charles R.B. Wright) Algebraic Number Theory (I.N.Stewart & D.O.Tall) I’ve collected these because they seemed to embody that which has always felt just out of reach. At the moment I’ve gone back to Stroud’s Engineering Mathematics and I’ve determined to work through it until I’m once again adept at A-Level. Following that I plan to give myself up to five years’ part-time study to achieve knowledge equivalent to an honours degree in the subject. So here’s where you come in. As well as the above, what resources would you recommend to further my study and fulfill my ambition?

• Answer:

  To get a basic grip of various Mathematical concepts, in addition to the books you mentioned, you should go through: Introduction to Linear Algebra, Strang. Topology, Munkres. Introduction to Numerical Analysis, Hildebrand This is assuming that you are comfortable with "basic" advanced Calculus. If  this is your first look at rigorous mathematics, these books will easily keep you occupied for a few years. In fact, there is a good  chance that you may not even enjoy proof-based mathematics, so don't  think too far ahead. Start with Haggarty and see where you are placed when you are half way through. I would also advise against going through each book entirely. In most cases, the first six-seven chapters would be more than sufficient. It's better to figure out which area of Mathematics you are interesting before reading further.