When I graduated with a bachelors degree in mathematics from one of
the most prestigious university mathematics programs in the world (Kings
College London) in 1968, I had acquired a set of skills that guaranteed
full employment, wherever I chose to go, for the then-foreseeable
future—a state of affairs that had been in existence ever since modern
mathematics began some three thousand years earlier. By the turn of the
new Millennium, however, just over thirty years later, those skills were
essentially worthless, having been very effectively outsourced to
machines that did it faster and more reliably, and were made widely
available with the onset of first desktop- and then cloud-computing. In a
single lifetime, I experienced first-hand a dramatic change in the
nature of mathematics and how it played a role in society.
The shift began with the
introduction of the electronic calculator in the 1960s, which rendered
obsolete the need for humans to master the ancient art of mental
arithmetical calculation. Over the succeeding decades, the scope of
algorithms developed to perform mathematical procedures steadily
expanded, culminating in the creation of desktop packages such as Mathematica and cloud-based systems such as Wolfram Alpha
that can execute pretty well any mathematical procedure,
solving—accurately and in a fraction of a second—any mathematical
problem formulated with sufficient precision (a bar that allows in all
the exam questions I and any other math student faced throughout our
entire school and university careers).
So what, then, remains in
mathematics that people need to master? The answer is, the set of skills
required to make effective use of those powerful new (procedural)
mathematical tools we can access from our smartphone. Whereas it used to
be the case that humans had to master the computational skills required
to carry out various mathematical procedures (adding and
multiplying numbers, inverting matrices, solving polynomial equations,
differentiating analytic functions, solving differential equations,
etc.), what is required today is a sufficiently deep understanding
of all those procedures, and the underlying concepts they are built on,
in order to know when, and how, to use those digitally-implemented
tools effectively, productively, and safely.