Prelude: The Ways Mathematicians Think
Given a regular hexagon and a point in its plane: draw a straight line through the given point that divides the given hexagon into two parts of equal area. Please stop for a few moments, solve the problem, and think about the way you solved it.
Did you draw an actual diagram? Did you draw a mental diagram? Were you able to solve the problem without drawing one at all? If you had a diagram, do you find it hard to understand how others could proceed without one? Were you motivated at all by analytic considerations? If you were not, do you understand how others were? Did you use kinematic ideas, such as imagining the line to be moving or rotating about a point? If not, can you understand how others might have? In solving the problem for the hexagon, were you consciously motivated by the analogy to the circle? Now that the analogy has been mentioned, do you think you were unconsciously so motivated? If not, do you understand how others were? Did you have any other method not already mentioned? (For instance, some people, often chemists, say they thought of folding the plane.) Finally, did you try to prove your solution? Did you nevertheless “know” that you were right?
The diversity of problem-solving approaches and of levels of conviction, even for a simple problem like this one, makes clear that there are many kinds of successful mathematical thinking.