Meaning and Understanding of School Mathematical Concepts by Secondary Students: The Study of Sine and Cosine

Abstract

Meaning and understanding are didactic notions appropriate to work on concept 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 a mathematical concept and to value its understanding. To illustrate the study, we have chosen the trigonometric notions of sine and cosine of an angle. The work exemplifies some findings of an exploratory study carried out with high school students between 16 and 17 years of age; 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. We gather the study data through a semantic questionnaire and analyze the responses using an established framework. 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.

Exploratory Case Study on Solving Word Problems Involving Triangles by Pre-Service Mathematics Teachers in a Regional University in Australia

“…Junior secondary students continue with learning Pythagorean theorem and basic trigonometry defined as ratios of sides and using these basic rules to solve problems associated with right triangles [3][4][5]. Senior secondary students progress further with trigonometric identities, functions and relationships, law of sines, law of cosines and law of tangent to solve oblique triangles in scientific and engineering problems [6][7][8][9]. As a result, triangles and their applications have been an important part in mathematics curriculums for pre-service mathematics teachers in tertiary institutions in the world [10][11][12][13].…”

An analysis of mathematics understanding of prospective student-teachers of mathematics

“…Mathematical understanding can be constructed within a suitable semantic framework to describe and interpret its meaning. The research of Martín-Fernández & Ruiz-Hidalgo (2019) shows that 88% of students have expressed their understanding of the concepts of sine and cosine based on the idea of coherent meaning with a good semantic framework.…”

“…This is in line with the study of Boaler et al (2016) that teaching and learning mathematics needs to explore mathematical concepts more visually. The study of Martín-Fernández & Ruiz-Hidalgo (2019) show that 88% of students have expressed their understanding of the concepts of sine and cosine based on the idea of coherent meaning with a good semantic framework.…”

Polya's Problem Solving Strategy in Trigonometry: An Analysis of Students' Difficulties in Problem Solving

“…Many studies [11,12] have reported that students understand sines and cosines in three ways namely (i) as the point-to-circle coordinates, (ii) as a coordinate of horizontal and vertical line, and (iii) as the ratio of the triangle sides. Trigonometric problem solving is strongly influenced by the stages chosen by students [13].…”