**Dr Peter Jenkins, Director of Mathematics**

Mathematics is undeniably useful. It is rightly referred to as both the queen and servant of the natural sciences, and its discoveries underpin the technological advancement of humanity. As a mathematics teacher, I often experience occasions that allow me to give students an understanding of the usefulness of mathematics in the ‘real world’, and I very much enjoy doing this. But when a student asks the question ‘When am I ever going to use this?’ (perhaps with the word ‘anyway’ appended) you can be pretty sure that no matter how good a response one gives, the student will be largely dissatisfied with it.

When students ask this question, they are typically not curious about applications of the mathematics currently being studied; they are frustrated. Mathematics is often hard, and it is perfectly understandable that people question whether it is worthwhile persevering with something hard when they cannot see the benefit. Providing a distant situation or occupation in which the learning could be personally useful is almost never a strong motivator to persevere. Furthermore, a significant quantity of mathematical knowledge learnt at school is rarely ‘used’ by anyone, but is rather prerequisite knowledge for future learning. This then raises the question of why learning higher-level school mathematics is worthwhile for a student who is not interested in pursuing a career in science or technology.

I have always believed that the primary goal of mathematics education is to teach students how to think. Learning mathematics is not so much about acquiring a body of mathematical knowledge, but rather developing a powerful set of cognitive tools that can be used in a wide variety of problems and situations. The actual mathematical content just happens to be the ideal vehicle to deliver such skills. To illustrate, consider one of the most challenging units for students in our Year 10 mathematics programme: Circle Geometry. Circles have an extremely rich mathematical structure; when combined with lines and resulting angles, many patterns emerge, some of which reveal deep truths about this structure, while others may simply be coincidences. This provides a perfect environment for students to practise identifying patterns, developing logically sound arguments that prove the truth or falsehood of a proposition, translating between symbolic and diagrammatic information, and making a creative selection and use of specific tools to solve unfamiliar problems. While the actual knowledge of many of the theorems and basic skills relating to circles may never be used again, the thinking skills developed by doing such mathematics will have future benefits across all fields of study.

It is also important for students to realise that, due to the rapid pace of technological change, the knowledge and skills required for various professions in ten years may be very different from those required today. Thus, students with a particular career path in mind may not appreciate how important particular thinking skills developed by doing higher level mathematics may be in their future careers.

Of course, I am not really doing complete justice to my field by reducing it solely to a sort of intellectual gymnasium, where students go to have their brains strengthened. The fact is that mathematics can be intrinsically interesting, irrespective of whether or not it is useful or good for us. Bertrand Russell, the famous English mathematician and philosopher, wrote that

Mathematics, rightly viewed, possesses not only truth, but supreme beauty — a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show (Russell, 1919, p. 60).

Most people do not see mathematics in this way. That is why Russell was wise to include the phrase ‘rightly viewed’. Helping students appreciate the aesthetic quality of mathematics is one of the most challenging tasks for the teacher, but it is the best way to foster interest and deep engagement. If a student enjoys and appreciates an abstract piece of mathematics simply because she sees it as interesting, it will not matter if the mathematics has no obvious real-life application or apparent future use to her.

Obviously, there is a significant difference between the aesthetic appeal of mathematics, and that of, say, music (which Russell refers to by comparison). The difference is that it almost always requires considerable intellectual effort in order to ‘hear’ the beautiful tones and harmonies of mathematics. But this effort does not have to be painful. Intense concentration and perseverance is required to play a video game, yet the experience can be extremely satisfying. I think it is important that students understand this. The fact that mathematics is often so challenging and frustrating is part of what makes studying it so intellectually engaging and fulfilling. The key is to embrace the challenge, and further, to understand that the time spent persevering with challenging tasks is always worthwhile. This calls to mind John F. Kennedy’s inspiring comment in 1962 about aiming to put a man on the moon by the end of that decade, not because it is an easy thing to do, but because it is a hard thing to do (Kennedy, 1962). Only by doing hard things do we experience personal growth and make the best of ourselves.

So, how should one respond to the question ‘When am I going to use this?’. I am afraid I am yet to develop the perfect response. I think the most important thing for teachers and parents is to continually help students to understand and appreciate the preceding ideas: that developing thinking skills is more important than the content, that so much of mathematics is inherently interesting and fascinating, and that the challenging nature of mathematics is part of what makes it so engaging and fulfilling. Then, if the question does crop up occasionally, a statement of reassurance that their effort is worthwhile will hopefully suffice.

**References**

Russell, B. (1919). The study of mathematics*. *In *Mysticism and logic and other **essays.* London: Longman.

Kennedy, J.F. (1962, September 12). *Address at Rice University on the Nation’s Space Program*. Retrieved from http://www.americanrhetoric.com/speeches/jfkriceuniversity.htm