by Ed Staples
And yet, I have discovered through my own teaching that Mathematics can be much more than just learning algorithms. As we state in our preface, Mathematics is a way of thinking – logical, sequential and well defined. It is a celebration of culture and human endeavour – full of intrigue, inspiration, creativity and discovery. Mathematics is an art – rich in form, symmetry, and beauty, just as nature tends to present itself. Mathematics is an adventure – a journey into the unknown. With this in mind we attempted to produce a collection of supportive topic papers with five key pillars in mind.
2. An adventurous use of the Area of a Triangle rule to examine Kepler’s second law.
3. A description of Descartes Rule of Signs
4. The sketching of Rational Functions using form and symmetry.
5. The geometrical description of the Pythagorean Results.
6. Descartes geometrical solution to a quadratic equation.
7. A discussion of quarter-square tables that preceded logarithms.
8. A more general proof of the Fundamental Theorem of Integration
9. The solution in radicals to the cubic equation by the Italian abacists
10. A discussion of the ‘seconds’ pendulum and its relation to Simple Harmonic Motion.
We discuss the use of logarithms in the measurement of earthquake energy and noise. We show how the military have devised their own angle measure and how it is used to estimate distances. We show how a Ferris wheel can be modelled by a sine wave. We show how a pendulum was used to show that the earth is an oblate sphere. We derive and demonstrate the use of formulae for determining the distance to the horizon from a point above the ground. We take a look at Moore’s Law and the logistic equation as two growth phenomena. These are just a few of the applications we look at.
The idea to write a book originally came out of a discussion Erin, Paul and myself had about the future directions of mathematics teaching. Mathematics teachers are being challenged to think of new directions in the wake of the enormous impact of technology on learning. Mathematics cannot survive on the age-old algorithmic approach of the mid-20th century – we no longer exist in that world. Mathematics teaching has to change if it is to survive as a relevant and attractive curriculum offering. It is our hope that the some of the spirit of these whetstones rubs off on the reader– the narrative of human endeavour and a renewed emphasis on thinking, perspective and application. We hope that this work might offer a new way forward for the teaching and learning of this great subject.