# The beauty of mathematics

## Creating an insight into 21st century maths

Imagine studying high school and first year university chemistry without learning about protons, electrons, neutrons or the periodic table; physics without ever discussing radiation, relativity or quantum mechanics; English literature without discussing Oscar Wilde, D H Lawrence or Virginia Woolf; information technology without mentioning the telephone, let alone the internet; biology without learning about evolution, DNA or the human genome project; music without learning about Schoenberg, Stravinsky or Rachmaninoff. In other words, imagine learning a subject and leaving out any developments since 1850 or even earlier. Yet this is what happens when students study mathematics. And there is good reason. Mathematics is like a giant scaffold. One needs to build the superstructure before ascending for the view. Although the calculus and algebra learnt in high school and early university is an essential part of this scaffolding and is fundamental for studying further mathematics, most of it was discovered in the 18th century or earlier.

Professor John Hutchinson of the Mathematical Sciences Institute is working towards addressing this issue with two unique approaches. Firstly, the creation of a series of professional development workshops for college and high school mathematics teachers and secondly by the introduction of the new ANU secondary college course for Years 11-12 students.

The goal of the professional development courses for college and high school mathematics teachers is to introduce some elements contemporary mainstream 20th and 21st century mathematics. Although the topics covered are not within the current maths curriculum, the aim is to give teachers an insight into current directions in mathematical research so that they are better able to give students a context for their current studies. The courses are open to mathematics teachers from across Australia. The ANU secondary college brings top year 11 and 12 maths students from across the ACT to the University for two hours a week for ¾ of the teaching year. The classes are taught by a small number of leading college and high school mathematics teachers, who have themselves been trained in the relevant disciplines.

In the college course and in the workshops we investigate some very exciting and useful concepts and get a feeling for “what mathematics is all about”. Topics includes RSA cryptography - based on number theory and fundamental to secure internet and banking transactions; infinity - a topic of philosophical ruminations for millennia and understandable only through mathematics; chaos, fractals and dynamical systems - very pretty mathematics extensively applied in fields as diverse as biology, statistical mechanics, diffusion in disordered media and image compression; and geometry and topology - visualising the fourth dimension, classifying surfaces up to topological equivalence (including those which most naturally sit in four dimensional space), and even briefly discussing what the journal Science described as the most significant scientific achievement of 2006, the solution of the Poincaré conjecture.

This mathematics is usually not seen until higher level courses in second or third year at University. Of course, we do not cover the mathematics in the same depth or generality. Instead, we proceed by studying carefully chosen parts and representative examples from major areas of mathematics, which illustrate important and general key concepts. The hope is that students and teachers will gain a real understanding and feeling for the beauty, utility and breadth of mathematics.

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