It might seem odd that someone who has come from a research organisation to one which is for education and research should ask a question such as the title! Some explanations are in order.

Mathematics is one of the oldest intellectual activities of mankind, so it is not surprising that the amount of mathematics that has already been done is enormous as compared with almost any other discipline. One consequence (that has not escaped notice!) is that people who prove theorems are often much older today than in earlier years.

Another important consequence is that for anyone active in mathematical research today, most mathematical learning has happened outside the classroom. Moreover, such non-classroom learning is far from linear. Monuments of mathematical beauty are built on wooden stilts; the latter are only turned into firm pillars when one finally writes down the fruits of one’s research.

The above paragraph is nothing new to working
mathematicians, but each incoming generation must learn it
anew. This is because courses and books in mathematics are
often structured in a linear way. There are clearly defined
prerequisites and everything new is either defined or proved in
strict deductive order. This serves the important purpose that
each fresh batch of students verifies the “grand edifice”.
However, it also leaves many a student with the false
impression that this is how mathematics is *done*. Even
worse, it may leave the impression that classroom and textbook
learning is what mathematics is about.

One should *certainly* strive to improve one’s
classroom skills, to write more readable textbooks and to
design better courses. However, one should never lose sight of
the wide open spaces where many new mathematicial objects are
built.