Magic Squares II

An article which gives an account of some properties of magic squares.

The Secret World of Codes and Code Breaking

When you think of spies and secret agents, you probably wouldn’t think of mathematics. Some of the most famous code breakers in history have been mathematicians.

History of Morse

This short article gives an outline of the origins of Morse code and its inventor and how the frequency of letters is reflected in the code they were given.

History of Trigonometry - Part 3

Stage: 3, 4 and 5

Published February 2010,February 2011.

Pedagogical Notes

Teaching Trigonometry: concepts and historical development.

There is no logical connection between the history of mathematics and the way in which a curriculum might be organised to support pupils' conceptual development today. History tells stories about the past. Our classrooms are in the present with all the different present contexts and contingencies, and our pupils are children of our culture today. That being said, we can learn certain things from history that may be used to provide ideas and contexts that can be adapted for use in the modern classroom to assist pupils' learning. But learning is not just about cognitive activity, learning also involves the affective aspects.

Motivations and conventions

Pupils often ask questions like where did this come from? Or, who invented algebra? Or, in the present example, why $360^\circ$ in a circle? These questions may require just a quick factual answer, but they might lead to a longer explanation or even an experiment or a demonstration. Very often, the origins of mathematical ideas come from outside mathematics; they arise from human problems that need solutions and thereby motivate the creation of some mathematics that at the time is quite new. A great deal of mathematics starts out as 'applied' mathematics, and the 'pure' mathematics of the classroom has too often been purged of its humanity. The answers to pupils' questions can explain conventions of naming or of notation, and offer the realisation that mathematics evolved slowly over many hundreds, even thousands, of years with contributions from a wide variety of people from many different cultures.

Models, analogies and activities

The simple observational equipment that early astronomers, star-gazers and shadow-measurers made and used are well within the reach of any pupil who is interested in observing the skies or checking the position of the Moon, stars or the shadow of the Sun**. Living in an urban environment, it is not easy to see stars near the horizon because of the glare from street lighting, and with our weather not always easy to see the sky at night at all! However, once we can recognise the Pole Star and realise that the other stars revolve round the Pole, we share one of the fundamental wonders of the universe with our ancestors. Exploring the language we use to describe these events helps us to realise that even today when we believe that Copernicus and Galileo showed us otherwise, we still say 'the sun rises'.

The planets and the moon revolve around us, the stars revolve around our solar system, the galaxies revolve in space. Rotation is a deeply embedded and fundamental aspect of our existence, or so it would seem.

Fundamental concepts

The summary below gives a brief list of the major events and developments in the evolution of the ideas that led to trigonometry as we find it in today's classroom. One thing we realise from even a casual encounter with the history of mathematics is that it took a long time for things to develop to the present, complex and sophisticated state we have now. Do we make too many assumptions about how quickly our pupils are expected to learn? The rotation of the heavens and the recognition that the seasons corresponded to special patterns of stars, led to the measurement of angle; the shadow stick measured the passage of the day, and contributed to the development of peg and cord geometry, the use of a right-angled triangle and the comparison of measurement led to the concept of ratio.

**[Make sure nobody looks directly at the sun without adequate protection]