Let me take a different approach: that of the liberal arts student. Since my interest vary I took sailing, theater arts, philosophy, etc and outside of school fencing related martial arts. Three segments stand out in my memory:
- 10 years ago when our Calculus teacher was writing the value of pi, he just kept going to 9th decimal places and we were all awed by the demonstration
- Our Linear Algebra professor used to quip: Parlez vous mathematique? and he even instilled the idea in me that: Mathematics is study of forms which will remain with me forever even though I could not complete the course (2x).
- The last is in our martial arts class related to stick fighting: It's not about just clicking sticks, but you must showcase yourself.
Now there have been some remarkable math personalities reading from biography like Tarski who would bring incredible energy to the classroom. I felt lack of it during the class.
Keeping these in mind and Howard Gardner's theory of multiple intelligence where we acquire learning in our unique way, I humbly point to this question in Math.SE which I opened under former account and admittedly the accepted bounty does not do justice to other answer, which may have been edited later.
As the thread would show, I as a student whose interests are bent more on humanities side, would like to take a still-life of a daily example, if I was the teacher, and break it down mathematically. How many sands are there in the universe? Why should 2+2 = 4? (and keep carrying on the conversation with the student). Anyone interested in knitting and crochet: how would you describe the concept of knot in mathematics? etc..
But at the end, it's about showcasing your art. Sometimes if a professor memorizes, mass amount of information and gives a dramatic showdown in class without looking at notes, it will be etched forever in student's mind.