Should I Bother with Studying Proofs?

A proof is an explanation of why a theorem is true. It is nothing more nor is it less. For a proof to be adequate it must adhere to strict rules of deductive logic. Often this means that proofs are long, tedious, and hard to follow. Students of mathematics know this without these notes having to tell them. Before every exam usually one student will ask what many others are thinking, “Will there be any proofs on the exam?”

In higher mathematics courses where the emphasis is on knowing the mathematical reasons behind a statement most, if not all, theorems are proved and students are expected to know how to prove theorems which are covered in class or in the text. You will find that in a course like Technical Calculus the emphasis is more to prepare you to learn how to use the tools of mathematics that will serve you well for the level that you are at. Consequently, not many theorems will be proved in a Technical Calculus course. However, whenever a theorem in the course is proved, as a student wishing to succeed in mathematics, you should ask yourself, “Why is the author or my instructor taking time to prove this particular theorem? Is it because the proof provides some interesting and useful insight into the mathematics we are studying?”

There are good reasons to consider as you decide whether to spend time studying proofs.

1. Proofs give an explanation for why a theorem is true.

2. Proofs help with understanding the interconnections of mathematical ideas and concepts.

3. Proofs are a valuable source of problem solving ideas and techniques. A theorem is sometimes like the shell of an oyster; it can conceal problem solving pearls.

4. Proofs can help you remember the statement of theorem.

5. Proofs can clarify the full extent of the limitations in the statement of a theorem. By studying the proof you may discover slightly different, less strict, conditions under which the conclusion of the theorem also applies.

Studying a proof can be hard work. But the effort can pay dividends in your understanding of mathematics.