"Ingenuity", "far reaching vision", "unerring sense", "deep
curiosity" — and best of all,
"extraordinary creativity". These are some of the words that have been
used to praise this year's Fields medallists at the International
Congress of Mathematicians (ICM). These words aren't specific to maths. They
could be used to describe anyone whose work is about discovery and
beauty; writers, poets, or musicians for example. If there has
been one overarching theme at this ICM, it's just how creative a
subject mathematics is. To imagine what has never been imagined
before, spot new connections, look at things in a new light and find hidden patterns lies at the
heart of doing maths.

A great example is Fields medallist Manjul Bhargava, who in a flash
of inspiration discovered that a centuries' old maths problem
could be solved using a Rubik's cube, reducing long and tedious
calculations, quite literally, to child's play. But singling out
specific examples gives the wrong impression:
discovering the undiscovered and creating new ideas is central to the
work of every mathematician, whether they are presenting their work here at the ICM or working away at home.

What guides these creative minds is often a sense of beauty.
"There is an aesthetic side," says Phillip Griffiths,
winner of this year's Chern Medal. "You find out what the most harmonious
properties of a [mathematical] structure are, and then
you let those guide you." You might be wrong, of course,
things might be more complex, or perhaps even simpler, than you think,
but still, it's often aesthetic considerations that lead the
way.

The creative aspect of maths is one of mathematics' best kept
secrets. If you didn't have an inspiring teacher
at school, then you probably think that maths is all about repetition
with no room for imagination and exploration. "Many teachers still
teach that way; here is the problem, here is how you solve the
problem," says Martin Grötschel, Secretary of the International
Mathematical Union. "So what you do is learn recipes, how to solve a
quadratic equation, etc. But you are not taught why and how you get to this equation. The really successful countries, where
maths education has improved a lot, have taken the approach of teaching
students to figure out for themselves why things go this way or the
other way." (You can listen to our interview with Grötschel here.)

Fields medallist Bhargava suggested something similar to us: perhaps maths
students, rather than learning how to apply "finished" mathematical tools, should
be allowed to play with maths problems that have inspired past
masters, to develop their mathematical intuition and get a sense of
the excitement and adventure of doing mathematics (see our interview
with Bhargava to find out more) — the sense of discovering, and
shaping, something new.

A lot of research is currently going into finding out how to best
teach maths at school, part of it supported by the International
Mathematical Union. Luckily we know lots of teachers who
are passionate about creativity in maths. And we are currently involved in a project that
aims to foster this creativity, an effort that has resulted in the website Wild Maths. Equally luckily, we know lots of mathematicians who are
passionate about communicating the essence of maths to a wider
audience (Bhargava is one of them). So perhaps it won't be long until
mathematics' best kept — and most exciting — secret becomes public knowledge.