Meanings shown by students and teachers in training on the sine and cosine of an angle

Abstract

Meaning and understanding are notions used in didactic to work on concepts, comprehension, curricular design, and knowledge assessment. This document aims to delve into the meaning of school mathematical concepts through their semantic analysis. This analysis is used to identify and establish the basic meaning of mathematical concepts and to value their understanding. We gathered the study data through a semantic questionnaire, and analised the responses using an established framework, included in the didactic analysis, that is developed along the dissertation. To illustrate the study, we have chosen the trigonometry relational system. Understanding the trigonometry relational system is one of high school mathematics most demanding topic. The angle, the unit circle, and the trigonometric functions are its foundational notions. Trigonometric contents meaning and their understanding involve these three concepts and their relationships. The study involves two stages. The aim of the first stage is to analyse the representations, concepts, notions, and the senses handled by secondary school students when describing the sine and cosine of an angle. This part of the report exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age on the several ways of expressing and interpreting the trigonometric notions aforementioned; it collects the variety of emergent notions and elements related to the trigonometric concepts involved when answering on the categories of meaning which have been asked for. From the analysis on the answers of this group of students emerges a categorization, whose relations are discussed and interpreted. The results show several types of representations and senses, some of which have already been recognized in previous studies, while some others are new. The subjects provide a diversity of meanings, interpreted and structured by semantic categories. These meanings underline different understandings of the sine and cosine, according to the inferred themes, such as length, ratio, angle and the calculation of a magnitude. Secondly, this research also aims to provide evidence on how pre-service mathematics teachers use trigonometric concepts, and to give examples and arguments that explain how they move between partial goniometry and partial analytic geometry systems; which structures and strategies use, and how they convert notions between representation systems. We characterize responses, as well as organized and interpreted the data obtained. The results indicate that participants’ meanings of the angle concept mediate their understanding of the conversions between the trigonometric representation systems involved. The scarcity of research related with school meaning of trigonometric contents provides an extra interest to the study.