My advice to teachers of mathematics: explain the intuition behind every theorem. Students can look up the statement of a theorem for themselves: you only add value if you give students a broader perspective by articulating the intuition that accompanies and underlies the formal details of a theorem.
This intuition is not merely a sign of the mathematical weakness of the human mind, but rather speaks to the essence of human understanding. From the intuition, an intelligent student can usually recreate the theorem, but the theorem does not readily lead to the intuition. This should suffice to show that the intuition is more powerful than the formal theorem.
History also reveals this pattern: the calculus, built upon the idea of limits, began with the intuition that limits existed. Substantial development took place in calculus before limits were formalized rigorously. Similarly, many of the counter-intuitive formulas in statistics are newcomers on the scene: the denominator of the sample variance formula was N for many decades before it became clear to statisticians that this formula led to a biased estimate of the population variance. Understanding the modern variance formula is impossible unless students are introduced to the concepts of expected value and bias: both of which are usually covered only cursorily — if at all — in introductory statistics courses. Students are therefore made to memorize formulas they cannot possibly understand — and which many of the great statisticians of history did not understand either. The subject of statistics, like much of mathematics, is made needlessly complicated by this insistence upon memorization rather than understanding.
Many teachers do not provide students with this perspective because they are unaware of it: they themselves simply memorized the formulas at the beginning and came — or did not come — over time to the intuition behind them. Historical ignorance pervades mathematics, in which historical development is too often felt to be irrelevant. This is doubly unfortunate: it obfuscates mathematical ideas that were inspired by intuitions that could be articulated clearly to students, and it makes the subject seem to have been developed more easily than it was. By concealing the difficulty of mathematical progress, mathematics is made to seem easier than it is — which has the effect of degrading the student who does not find mathematics as easy as it is made to seem.
But perhaps the worst consequence of this indifference to intuitions is that it makes mathematics seem like catechism class, when it should be an education in skepticism and rigor. Mathematics, like life, is a process, rather than a series of results. Any branch of mathematics could be taught to high school students to prepare them for advanced studies in mathematics: if the topic were covered rigorously, the students would be prepared to learn any other topic much more quickly. This is a consequence of the inescapable consilience of mathematics: the same core ideas occur in every branch of mathematics. It is the habit of mathematical thinking that is essential, not the results. Yet students rarely acquire this habit: the majority of students who succeed in their classes merely memorize the formulas that pass from the textbook to the blackboard to their notebooks without passing through either the instructor or the student’s mind.
Abstractions in the absence of strong examples are also pedagogically unsound: the human brain is far better at receiving clear examples and extracting generalizations from these examples than it is at deriving the implications of generalizations and applying these to new examples. The best textbooks in any field tend to be those that contain a small set of extremely clear examples that can be understood deeply and in their entirety. This is true not simply in mathematics, but also in programming and other fields. Kernighan and Richie’s “The C Programming Language” is such an amazing book because the reader is able to understand every single line of every example. Few other introductions to programming languages have been as clear as Kernighan and Richie’s because this principle has tended to be avoided. General rules must be learned in context, and the context must be comprehensible in its entirety.
In the end, these two traits are the essence of intuition: one understands something entirely and one can articulate a precise example of its application. If this were imparted to more students, a mathematically literature student body might emerge.