Expanding problem solving skill

I have a great passion for Math but my lack in problem solving skill always keeps me away from the "good stuff". I always wanted to be better at Math and one of the things I figured out was to keep practicing a specific set of problems until I understood the question. What techniques do you recommend, that implement to improve my problem solving skill and help me improve my Math?

EDIT: Are there any books you would recommend that can quickly help me implement your suggested technique?

Solution 1:

In addition to the other answers, perhaps you should look into these books and problem books (there are many).

Problem-Solving Strategies (Problem Books in Mathematics) [Paperback] Arthur Engel (Author)

How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library) [Paperback] G. Polya (Author)

How to Prove It: A Structured Approach [Paperback] Daniel J. Velleman (Author)

A Book of Abstract Algebra: Second Edition (Dover Books on Mathematics) [Paperback] Charles C Pinter (Author)

How to Think Like a Mathematician: A Companion to Undergraduate Mathematics [Paperback] Kevin Houston (Author)

The Nuts and Bolts of Proofs, Third Edition: An Introduction to Mathematical Proofs [Paperback] Antonella Cupillari (Author)

Mathematical Proofs: A Transition to Advanced Mathematics (2nd Edition) [Hardcover] Gary Chartrand (Author), Albert D. Polimeni (Author), Ping Zhang (Author)

How to Read and Do Proofs: An Introduction to Mathematical Thought Processes [Paperback] Daniel Solow (Author)

http://www.mathpropress.com/mathBooks/

http://www.springer.com/series/714?detailsPage=titles

HTH

Solution 2:

Find a forum to teach other people math. The process of explaining your thinking to other people and/or writing examples to frame notes on a subject will force you to think much harder about the structure. Personally, I understand much more once I write my own notes on a subject. Even if the notes are little more than what is usually found in standard books.

In retrospect, the job I had as an undergraduate tutor in math and physics was also very important to my education because it helped me understand just how little I really understood... despite the fact I could work the problems. Some people will tell you that if you can work the problems then you understand them, it's not true in my case.

Third, study the good book on the topic. I think this is one of the greatest aspects of this site. You can get a much better sense of what book you should be reading by asking a few pointed questions (or finding the thread where it's already been done).

Solution 3:

You mention that "one of the things I figured out was to keep practicing a specific set of problems until I understood the question." This is actually how East Asian mathematics education works.

From Leung (2006, p. 43):

The process of learning often starts with gaining competence in the procedure, and then through repeated practice, students gain understanding. Much of the mathematics in the school curriculum may need to be practiced without thorough understanding first. With a set of practicing exercises that vary systematically, repeated practice may become an important "route to understanding" [...].

Note that Western mathematics educators will frown at this. They believe that conceptual understanding should come before procedural skill, and not the other way.

I recommend that you read more about East Asian mathematics education using the keyphrase "repetition with variation."

Reference:

Leung, F. K. S. (2006). Mathematics education in East Asia and the West: Does culture matter? In F. K. S. Leung, K.-D. Graf, & F. J. Lopez-Real (Eds.), Mathematics education in different cultural traditions---A comparative study of East Asia and the West (Vol. 9, pp. 21-46). Springer. (The 13th ICMI Study)

Added:

The traditional Western idea of procedural instruction is sometimes called drill and skill (or drill and kill if you are against it). It's basically repetition.

But the East Asian idea of procedural instruction is called "repetition with variation." The large amount of problems solved provides the skill (procedural knowledge), but the slight differences between problems provides the understanding (conceptual knowledge). The variations allow the learner to distinguish what makes the problems different and what makes them similar.

Solution 4:

Aside from the said books, you can also enroll to coursera's free online course on Introduction to Mathematical Thinking by Keith Devlin. You can collaborate with other students and join study groups to help you think mathematically.

On the other hand, if you're looking for problems you can practice on, Project Euler is a great place to start.