5. april 2016

# The Trigonometric Functions - The transition from geometric tools to functions

Specialerapport. IND's studenterserie nr. 47. Matematik.

Lotte Nørtoft, februar 2016.

Vejleder: Carl Winsløw, Institut for Naturfagenes Didaktik.

Abstract

In the third year of high school Danish students are introduced to the trigonometric functions. Before, they have been working with sine and cosine as tools to investigate angles and other dimensions in (essentially) triangles, and this transition from geometric tool to function is often challenging. The aim of this thesis is to elucidate why the students have problems with the above transition. This is first done through a subject-matter didactic analysis where the complexity and diversity of the trigonometric functions are elaborated. Here it is suggested that, to hinder the problems concerning the transition, the teaching of trigonometric function could involve a simplified introduction to the natural parametrization of the unit circle. Thus the students would realize the ono-to-one correspondence between the arc length on the unit circle and the 𝑥-axis.

The subject-matter didactic analysis works, together with the theory of didactical situations, as a tool for analyzing concrete teaching episodes, observed at a Danish high school. Since the simplest and most common way to introduce sine and cosine as functions is as graphs, the analysis is focused on the students’ interactions with the graphic milieu.

Both the subject-matter didactic analysis and the observations showed that primarily two elements inhibit the students’ acknowledgement of sine and cosine as functions. Firstly a lack in the perception of functions, secondly that focus is not on the actual transition, neither in the official material, nor in the concrete teaching. Instead focus is on the possibilities the trigonometric functions gives us, illustrated though the sine curves.