# Old ideas from new directions

So much of the secondary and A-level mathematics curricula can be approached as if procedures are the way into mathematics, and all that is required to do well is to learn the procedures and spot where and when to apply them. By contrast, people who use mathematics work the opposite way round – curiosity about a phenomenon is the starting point and there is a subsequent need for procedures to make it accessible and usable.

So, how can the procedures that students are expected to acquire be approached through exploration and curiosity? One way that I have found effective is to approach an old idea from a new direction that focuses on a feature of the phenomenon. Here are two examples.

Suppose that measurement of angles in degrees has never been invented. ‘Angle’ can be thought of as a measure of turn, so what is the relationship between an arc of a unit circle and the angle at the centre? What happens to this relationship with a circle of a different radius?

This modelling approach to radian measure emphasises the necessary logic of radians. You do not need to refer to degrees at all, but can return to the elementary ideas of full-turn, half-turn, quarter-turn, and other fractions to build the relationship. With a bit of discussion, radian measure emerges as a mathematically logical way to deal with angles, whereas the more familiar degree measure was a human invention which is useful but arbitrary. Conversion between these measures is a linear relationship (why?), as is conversion between any measure and the notion of 100% that is required for drawing pie charts.  So the introduction to radians can be managed as an exploratory context for coordinating knowledge of circumference, fractions, ratio, and proportionality, with plenty of opportunity for modelling and conjecturing about relationships.

Note that in my diagram I have avoided using the usual orientation which has the turn starting as if from a horizontal axis. I find people are more creative if a diagram does not trigger conventional treatment.

Another example of ‘new direction’ that has been tried and tested over a number of years is the notion of ‘inter-rootal distance’ between the roots of a quadratic.

Using graph plotting software, find quadratic functions whose roots differ by 2.

This exploration works best with a whole class who frequently share their findings. Students have to know a little about quadratic functions to get started, and the work will progress differently according to their starting points.  Students who connect ‘roots’ to the formulation k(x – a)(x – b) will generate different examples to those whose general formulation is ax2+ bx + c You don’t need much more than this to get started. Transformation of functions by translation, reflection, and scaling all emerge in the process of trying to find all possible functions. Additionally, the idea of generalising the parameters crops up as a way to describe families of functions that have an ‘inter-rootal distance of 2’. Extending the exploration to other distances is an obvious way forward, followed by working backwards algebraically to find the inter-rootal distance for any quadratic – this is a bit tricky but worth doing because—hey presto!— it turns out to be square root of the discriminant.

Here are the beginnings of an example:

f(x) = x2 + 8x + 15
Sum of roots p + q = 8
Product of roots pq = 15

(p + q)2 – 4pq = (pq)2, which is the inter-rootal distance squared.

New directions

In each of these examples, ideas that are already familiar to students are approached from an unusual direction in order to trigger exploration and curiosity. This provides an alternative route to standard curriculum content and also provides contexts for using algebraic and proportional manipulations.

Anne Watson has two maths degrees and a DPhil in Mathematics Education, and is a Fellow of the Institute for Mathematics and its Applications.  Before this, she taught maths in challenging schools for thirteen years. She has published numerous books and articles for teachers, and has led seminars and run workshops on every continent.