Creatively designing your own wording for a concept while making sure that it conforms to the spirit and structure of the original statement is an art requiring true mastery and understanding of the underlying material. It is a perfect marriage of critical thinking and artistic flare. This is, by far, why I have found mathematics to be a beautiful and exciting subject.

"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.

Theorem (Rolle's Theorem - standard)

Let

Then there exists

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.As most students in calculus find out early on, if a function is differentiable on an open interval (i.e.,

*f*' exists on (*a*,*b*)), then*f*must already be continuous on the corresponding closed interval (i.e.,*f*is continuous on [*a*,*b*]). Thus, there is some unnecessary wording in that official theorem. However, when putting something in my own words, I could honestly care less about reducing the complexity of the statement. My ultimate goal is to write an equivalent statement that makes sense to me! So here is my attempt.Theorem (Rolle's Theorem - own words for a calculus student)

Suppose that

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*!You can definitely see this is more wordy than the original statement; however, it has a

*lot*of theory and concepts built into it. This can also be rewritten to make it understandable for the general populace.Theorem (Rolle's Theorem - a good, basic understanding for non-calculus crowd)

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.

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.

During my years as a research mathematician, my colleagues and research professors praised my ability to truly understand the essence of a concept. I was always able to describe complex concepts in a way that made sense and brought a clearer understanding to the project. I use a lot of analogies when trying to understand something, putting things into a context I am familiar with and build from there.

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 istrue understanding.