I am really not sure, what the message of this text on Scientific American is. It cannot be that math is always easy to grasp, since it isn’t. We have to admit the the formulation of Rolle’s theorem there is concealing the simple content of the theorem. And we have to admit that many a math talk just follows the same scheme throwing smoke grenades to hide the small impact of the presented results. But quite a few talks really contribute to human knowledge. And their math is often not easy to grasp.
What the article tells me is to be aware of the fact that doing math has three sides. I talked about that theory of mine on this blog before. Mathematics is (1) about visual content, (2) a mechanical craft, and (3) a challenge in logic. Rolle’s theorem demonstrates that very well. The visual content is obvious, as is the content of the main consequence of Rolle’s theorem, the mean value theorem. The craftsmanship is in the calculus we apply to find the zeros of the derivative to get the maximal value. The logic comes in, when we think about the requirements of the theorem, and why it must hold.
All in all, you need to connect the three areas of your brain, the visual, the mechanical and the reflection to get a complete mathematician. The visual alone will not do.