Math route for someone of this background

Question

Apologies for a soft question. I do not have a lot of time because of my job (programmer) and my wife cannot work so I cannot quit. I also did not graduate from a very prestigious school with good grades. So grad school is out of the question for me.

I am 23 years old, I find that because I did not learn many topics well when I was younger, I have accumulated a lot of half baked knowledge about stuff. For example, I can spell out some theorems, results in fancy notation, but I don't really understand those things. So when I try to restart and re-educate myself I find that I have to unlearn a lot of gibberish that I put in my head just to pass through examinations.

I visited this site and saw some solutions people posted. I was so exhilerated after getting some of them that I wish I had the same understanding as the person who could think of such solutions. So in my private time, I have embarked on a project to learn math from scratch. I do not aspire to make meaningful contributions, because of the late start, but to discover mathematics and get some personal peace.

Before beginning, I decided that I am most interested in stuff like graph theory, and perhaps I would like to explore geometry as well (algebraic, differential)

So I planned a halfway route and here is my question... what is your advise for a person like me to learn mathematics and reach his goal (with literature recommendations possibly online)

1. Start from basics, do good problems in pre-calculus algebra (inequalities, permutations, combinations, sequences and series etc?) to get some mathematical thinking
2. Calculus from Rudin, to get rid of most of the algorithmic procedures in my minds (is this too tough for me)
3. Where do I go from here? Follow some college's standard curriculum? Is there a curriculum for undergrads designed to specialize in graph theory/topology?

Answer 1

I can speak from experience in saying that teaching oneself mathematics, while difficult, is possible. However, I cannot stress enough the importance of being able to talk with knowledgeable people in order to check one's understanding. I am fortunate enough to live in a university town with a mathematics faculty eager to talk math with anyone interested. This may well not be the case for you, in which case I suspect you will be making frequent use of this site.

As for your "curriculum", to some extend you need to find what works best for you. However, if you really are interested in learning higher math, I suggest you not spend much time memorizing formulas or doing algebraic busywork. Rather, try to reason your way through elementary problems in fields such as combinatorics or probability. Once you feel safe working with these, I suggest you either move on to basic calculus or an introductory text in abstract algebra. You didn't mention algebra in your list, but I really think that an introductory abstract algebra text will do the best job of introducing the fundamental concepts of abstract mathematics (which you will certainly need for graph theory or other branches of higher mathematics). Personally, I recommend "Modern Algebra: An Introduction" by John Durbin, which I found both intuitive and eye-opening. Of course, these recommendations are based off my own experiences and may or may not be well-suited to you.

Answer 2

First, I really admire your courage and perseverance.

Second, pick up a Schaum's outline on secondary school mathematics and make sure you understand every chapter. If you cannot answer a question by looking at it, then take out a pencil and paper and go to work on solving it. Make sure you understand the techniques involved and the reasoning behind each problem. Reread the chapter if you need to and make sure you check your answers with those in the book.

I suggest this approach rather than starting with a standard math text, because Schaum's outlines are specifically geared toward self-study. Remember, if your foundation is not strong, succeeding in mathematics --- even elementary mathematics --- will be nearly impossible.

If you have any math-related questions, don't be afraid to share them on Math.SE. Once you're done with a thorough evaluation of the fundamentals, then you're ready for something more advanced. In the meantime, go to work!