An important aspect of science is to look for patterns in the data and use that to grasp some simple underlying principles on the basis of which that data can be organised, studied, etc. Such a search for patterns is what mathematics (in its broadest form) is! Hence, it is no surprise that I would like to underline its importance.

Note that we are looking for *simple* principles. How
does one understand simplicity? Is "simplicity" in the eye of
the beholder? Indeed it often is! By the time we reach our
teens we are looking at the world through thick layers of
glasses of preconceived ideas --- some of them put there by our
teachers --- and sometimes simplicity involves removing some of
these filters/lenses.

The purpose of teaching is also simplification. Some may say that our purpose is to pass on the accumulated knowledge suitably distilled. However, "distillation" is insufficient to arrive at a learning time exponential lower than the time taken to collect the knowledge (note that what we teach in 12-15 years is based on 3000 years of data collection), unless this distillation involves simplification in a central way.

In other words, one of the functions of a teacher is to
*simplify* what the teacher already knows --- and one
measure of simplicity is that it should take the learner less
time than it took the teacher to learn the same thing!

The bottom line for teaching-researchers is this: Do not tell your students that they need to spend years to learn something since it took you that long --- instead, do some research and try to simplify the material! [*]

[*] | Students reading this should not automatically assume that this means that they can complain about long hours calculating and/or spent collecting data in a lab. There no short-cuts for acquiring skills! |