"Originality is the essence of true scholarship. Creativity is the soul of the true scholar."
- Nnamdi Azikiwe
Don't get me wrong - mathematical theory, when written succinctly and rigidly, has its own austere beauty; however, it is the process of taking a statement and transforming it into your own words that can build your confidence and understanding of the subject while introducing your own beauty and originality.
Here is an example. The following is Rolle's Theorem as it is written in most calculus textbooks.
Let f be a function satisfying the following conditions.
- f is continuous on [a,b],
- f' exists on (a.b)
- f(a) = f(b)
Then there exists c in (a,b) such that f'(c) = 0.
Suppose that f is a differentiable function on some open interval (a,b). Then, of course, f must be continuous on [a,b]. Now, let's also say that f(a) and f(b) are anchored at the same height. Draw any curve connecting those two points (that passes the vertical line test) without lifting your pen. Make sure that your curve is smooth because we know that f is differentiable. Then you can easily see that, at some point, the curve has to have a tangent line with slope zero. That is, if the curve you drew started going up (increasing) initially, then there must be a moment where the curve goes from increasing to decreasing because it must eventually come back down to the original height by the time it gets to b!
Imagine a train track that starts at an elevation of 100 feet above sea level. You hop on and travel across a few states, going over mountains and descending into deep valleys during the entire trip. If your destination is also at an elevation of 100 feet above sea level, then there must have been some point between the start and the end when the train reached flat ground. That is, there must have been a moment where the train was neither climbing nor falling.
My Style: I expect my students to rewrite the concepts we cover in class in their own words. My exams almost always have questions requiring students to rephrase concepts in their own words or state how they would "describe this concept to a 12-year-old."
In this way, a major key to success in my courses is true understanding.