Herbert Simon is famous for having said that “The goal of science is to make the wonderful and the complex understandable and simple – but not less wonderful. (See H. Simon, Science of the Artificial, 1996).”In analogy to this quote, one might say that the aim of Mathematics Education is to design and create ways for making wonderful, sometimes complex mathematical results understandable and simple – but not less wonderful. Explaining ideas means decomposing them into understandable pieces and gluing them together with accessible arguments.
In a proper “waterproof” explanation, these arguments must be backed by logical rigor. Logical rigor is sometimes an enemy of beauty and even, at times, of intuition. Mathematics, no less than the natural sciences, is in itself a collection of wonders. It has grown at the interface between the human mind/brain and nature, enabling the mind both to model natural phenomena and to tame uncertainty. In science in general and mathematics in particular, there is a certain risk in decomposing concepts into their components and explaining processes formally because fascination may fade through the exercise of formal decomposition and explanation.
According to our version of Simon’s statement, the good mathematics instructor should teach and explain mathematical facts while keeping both motivation and wonder alive in his/her students’ minds. We cite a well-known mathematical example: It is a wonderful result of mathematics that there exist exactly five Platonic solids. Explaining this wonder requires a minimum of detailed formalism and going through this formalism demands special skills of a mathematics educator, if she wants to make the result clear and understandable without scratching its beauty.
(This is a slightly modified paragraph from “Simon’s legacies for Mathematics Education” in the Handbook of Bounded Rationaliy, by Martignon L., Laskey K, and Stenning, K. , to appear — published by Routledge).