Definitions, Definitions, Do We Need Them?

Definitions are useful for experts in a field. It is much faster to just refer to a concept by its name rather than explaining it every time.

Unfortunately, definitions are a terrible thing for students. Even more unfortunately, most books are written in the following style:

1. Here's a paragraph-long definition that makes no sense to you at this point.

2. Here's a proof of this thing you don't yet understand.

3. Oh, you want to practice? Here's an example problem and a solution. When you see a problem like this one, plug numbers into this formula in the bold box.

4. A bunch more problems to try plugging numbers into yourself.

5. The discussion on what this thing is actually useful for is outside the scope of this book. Read about that somewhere else.

The best teachers actually explain how the thing they are teaching works. They start with a limited concept the student can understand, and then refine it to make it more formal. Good textbooks do this too. The bad ones start with the definition in a box.

The problem is that by the time a professor writes a textbook, this stuff is second nature to them. The prof is an expert. So he/she starts with the definition, since that is how the professor commonly communicates this concept. The book then ends up looking like the author is trying to make him/herself look smart at the expense of the reader. Writing an accessible book is a rare talent.

Wow this turned into a long rant :)