One of the most important aims of Mathematics education in the current age, is to support exploration, investigation and inquiry as vital components in developing an in-depth understanding of Mathematics. The use of computers and advanced graphical calculators is very valuable in supporting this aim.
Of course there are many tools available for solving any mathematical problem, such as the traditional pencil and paper, concrete models, a ruler, a protractor and a calculator. However, we need to help our students become proficient users of computer packages such as the dynamical package, Geometer’s Sketchpad, or spreadsheets in Excel. Occasionally, we also use the visual programming language, Scratch.
In Years 7 and 8, we begin to introduce these computer packages to our students. They learn how to experiment, investigate and explore to enable them to make guesses or conjectures as they consider questions such as What is the angle sum in a triangle?, What are the optimal dimensions of a box made from a given amount of cardboard?, Which shapes will tessellate a plane? , What properties do star polygons have? or How do we minimise a cable length?
In the past, some of these problems required advanced Calculus methods to solve, however, now younger students can approach these with confidence and find approximate solutions using a computer.
As the students move further up the school, they become more independent in their investigation work and in Years 11 and 12, and many them use the skills they have learnt in their investigation and exploration work, which is now a mandatory component of their assessment in the both the IB and the HSC.
One of the most inspiring online presenters is “The Mathologer” (Associate Prof Burkard Polster, Monash University, Melbourne). In Year 8 a few years ago, one of his videos (https://www.youtube.com/watch?v=qhbuKbxJsk8) inspired a Geometer’s Sketchpad activity which combined the mathematics associated with straight lines, curves and the times table. It allowed us, as an extension activity, to discuss the idea of modulo or clock arithmetic, while creating interesting curves. They were constructed using straight lines based on the 2 times tables! Our resulting cardioid example is shown at right.
The Mathologer mentioned how curves such as this appear in the real world in many situations involving light rays.
Recently, it was wonderful to receive the following photograph from a Year 11 student. Audrey Au was cooking and noticed the kidney shaped curve as shown. This particular curve can be constructed by using the 3 times tables and Geometer Sketchpad just as the cardioid example was. Audrey’s curve is known as a nephroid. It is an example of a catacaustic curve created by rays of light emanating from the reflecting surface of the pot. Audrey had drawn a picture similar to the one above when she was a Year 8 student in 2015.
We hope that all of our students will notice the Mathematics in the world around them in examples such as this. Well done Audrey!
Robyn Gregory | Head of Mathematics