Magazine article Mathematics Teaching


Magazine article Mathematics Teaching


Article excerpt

Jessica Steer, Maria Antioneta de Vila and James Eaton explore the teaching of trigonometry using a method developed by Jeremy Burke of Kings College.

A series of lessons was planned using an approach which looks at moving from a mathematical description of the topic, to a sequence plan, to a set of activities, which students can use to help them come to understand the topic. This is referred to as 'MNO' (Burke and Olley 2008). M is the map of the terrain to be covered, in this case the mathematical structure of the topic,, connections and potential problems that students might encounter. A consideration of the 'map' might be inaugurated with the question, 'what do we mean by...?' In this case, what do we mean by 'trigonometry'. N is the narrative, the sequencing of the lessons and activities - the story being told. O is 'orientation' the activities which will serve as strategies to facilitate students engagement with the topic.

The map

Trigonometry was initially studied a couple of millennia ago by the Babylonians, Greeks and Egyptians. They observed that in mathematically similar shapes, the ratios between pairs of sides were the same and could thus be used to find unknown lengths and angles of triangles. It was used mainly in the practical fields of astronomy and surveying. Today, students are first introduced to trigonometry in Years 9 and 10 where they are taught to understand and apply Pythagoras' Theorem and to use sine, cosine and tangent in right angled triangles when solving 2D problems. In Britain, study of trigonometry at KS 3 and KS4 focuses on real life examples, just as the Babylonians had done and pays very little attention to the more recent developments by Euler in analytical trigonometry.

It is not surprising then that students in Britain are taught to solve problems in trigonometry in very similar way to the ancient mathematicians. The most popular teaching method is known as the Ratio method. Using this method, trigonometric functions are defined as the ratios of the lengths of sides of right angled triangles. In order to find missing lengths or angles in right angled students, they merely need to identity the two sides and angle that interest them and apply the correct ratio to the problem. A common memory aid given to students to remember when to use each ratio is SOHCAHTOA where SOH means sin(x) = opposite/hypotenuse, CAH means cos(x) = adjacent/hypotenuse and TOA means tan(x) opposite/adjacent. Critics of this method have suggested that teachers jump in too early with the memory aid before properly deriving the formula? and this leads to a 'mystique about the subject' (Prichard, 1993). Furthermore, students have a poor grasp of what the trigonometric functions are, which means they are insufficiently prepared for the study of analytic trigonometry at A Level.

In the 17th and 18th centuries, Euler extended the subject to functional trigonometry. Some aspects of this work is studied by A Level students such as plotting the graphs of the trigonometric functions and using them to solve problems in simple harmonic motion. In other countries, there is a greater emphasis on this side of the topic from a much earlier stage. In countries where this is the case, teachers have found that a better method than the ratio method is the unit circle method, and some in Britain favour this method as it builds good foundations for A Level and degree studies. Using the unit circle method, sine, cosine and tangent are defined as functions of a real variable.

A study by Kendal and Stacey (1998) compared the successes of each of these methods. It was revealed that students who were taught the ratio method performed much better in the end of topic tests than those who were taught the unit circle method and in particular, the attainment gap was reduced in the ratio method. This was due to the fact that those using the ratio method were more successful at identifying which of the trigonometric functions were required in solving the question. …

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