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All throughout elementary school, middle school, high school and even in a large portion of college, math is all about solving problems using techniques taught in class.. You are taught the order of operations in elementary school; factoring techniques, graphing techniques, trigonometric proofs, and even some methods of solving for derivatives and integration in high school. The concentration is always on using theorems and techniques developed by other people. You are to assume that the theorems and techniques work.

I am currently a student of Abstract Algebra at Northern Illinois University. Abstract Algebra is the course that NIU uses to introduce its math majors to developing proofs. A while back got the results of our first exam. Many students who are now taking the class haven't ever done math proofs before (except for high school geometry, perhaps).

For those students who have previously excelled at solving math problems but are now struggling to complete proofs, there needs to come the realization that proving theorems does not need to be any different than solving a trigonometry problem, or a calculus problem.

At some point during a math student's career, they develop the ability to recognize problems, classify problems, and break the problems down into problems with which they are more familiar. This all comes with practice. My course in differential equations simply taught us some categories of differential equations and the step-by-step methods used to solve those equations.

Sure, there will be problems that can be solved multiple ways. Similarly, there will be thereoms that can be proved both directly and by contradiction. The key to proving theorems, however, is the same as solving any other math problems: recognition. With practice, a student will come to be able to apply different proving techniques to the current proof.

Perhaps I have an easier time proving things in this class because I've previously taken a class called Analysis. Analysis involved numerous proofs on homework assignments and exploration of many different proving techniques. But the lesson for me from that class is the same lesson for these students in Abstract Algebra.