by Ed Staples |

Aristotle School of Education |

Something occurred to me the other day as I reflected upon the different subject experiences of my schooling days. I recalled my English classes, where we studied the great writers and poets and their works. Science was full of inventors and philosophers who had pursued, discovered and formalized to laws the workings of the natural world. History was a fascinating journey of mankind’s conquests and defeats, mishaps and adventures. The language classes that I elected to take, not only included the language itself, but also explored the settings, the cultures and the people of the places where those languages were spoken. Each subject, as I remember, seemed to bring along with it a human story - each subject that is, except my experience in Mathematics.

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.

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.

Firstly, the papers had to have pedagogical fit – eighty four-page investigations supporting the material in the new National Curriculum. The

*whetstones*needed to be in a format so that photocopying for a class was possible. They needed, as their name suggests, to be thought of as a sharpening tool for the teacher – something they could pick up, read and digest in about 30 minutes.Secondly, the

*whetstones*needed to emphasize the narrative of human endeavour – the journey, the struggle and the triumph of great thinkers in their quest to solve real world problems. To name just a few: problems such as finding efficient methods for numerical calculation solved by Briggs with the invention of common logarithms; the solution to the cubic equation, which lead to Bombelli’s conception of an imaginary number; and the significant efforts of Hamilton that led to our modern understanding of vectors.Thirdly, we wanted the

*whetstones*to focus on the mathematical thinking that establishes certain results. How do we build new knowledge through creative combinations of the things we already know or assume to be true? This is what we do when we construct a proof. Can we harness our mathematical models to the problem of explaining real world phenomena? This is what we do when studying tides or the swing of a pendulum. What deeper realizations can we make from our discoveries? There are so many examples in the book, but one that I think about a lot is the Sine Rule. The length of any side of a triangle divided by the sine of its opposite angle is equal to the diameter of the circumscribing circle. I imagine the sine of the angle acting as an element of magnification, there to increase the opposing side length to that of the diameter, and of course, no increase is necessary for a right angle.Fourthly,

*whetstones*needed to offer**new perspectives**on the material, and to this end, I think we have gone some way to examine these things. Here are ten that immediately come to mind but there are many others in the book:*1.*The proof of the irrationality of the square root of 2 by the method of

*infinite descent.*

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.*

Finally the whetstones contain a myriad of applications so that students can sense a real-world connectedness to the material. We discuss for example the

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.

Ed Staples.

*Bankers rule of 72*, and the optimization of an investment under a fixed interest rate regime.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.

Ed Staples.