## High-school students' productive struggles during the simplification of trigonometrical expressions and the proving of trigonometrical identities

Doctoral Thesis

2023

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This study is an investigation into school students' productive struggles in the simplification of trigonometric expressions and proving of trigonometric identities. Although studies have been published on the teaching and learning of trigonometrical concepts in schools and teacher education, there is a lack of published research into students' productive struggles in the simplification of trigonometric expressions and proving of trigonometric identities. To fill this gap in the literature, this study used a sample of 16- and 17-year-old students at a rural high school in South Carolina in the United States of America to conduct a study on the use of productive struggles in the simplification of trigonometric expressions and the proving of trigonometric identities. The study used the Anthropological Theory of the Didactic by Chevallard (1992) as the main theoretical framework. However, this main framework was supported by other frameworks. The Anthropological Theory of the Didactic contends that mathematical activities such as simplifying trigonometric expressions and proving trigonometric identities must be interpreted as a human activity rather than seeing these mathematical activities as a language, the creation of concepts (for example, “simplification” or “proof”) or a cognitive process. A praxeology consists of two parts, namely (in Greek) the praxis, or “know how”, and the logos, or “know why”. The praxis is commonly known as the practical block, and the logos the theoretical block. This means that the Anthropological Theory of the Didactic can be used to describe how certain actions regarding the simplification of trigonometric expressions and proving of trigonometric identities take place, and why these actions take place. The exercises in the activities were obtained and adapted from the students' prescribed textbook. These activity questions were sequenced using the Development Cognitive Abilities Test (DCAT). The DCAT reflects Bloom's (1956) hierarchy of cognitive abilities. This means that the exercises were organised in three groups of increasing complexity, i.e., easy, medium, and difficult. The easy exercises related to the DCAT's Basic Cognitive Abilities, referred to as DCAT 1; the medium exercises related to Application Abilities, referred to as DCAT 2; and the difficult exercises related to Critical Thinking Abilities, referred to as DCAT 3. The data in this study consists of video recordings from classroom observations in real time transcribed verbatim, documentary analysis of students' assessments, and audio-recorded focus group interviews. The focus group interviews were also transcribed verbatim. Each transcription focused on a different aspect of the students' productive struggles in the simplification of trigonometric expression and the proving of trigonometric identities. Errors made by the students in written assessments were analysed using the Newman Error Analysis framework. By using Newman Error Analysis, this study could investigate and compare how the errors on the assessments were related to the students' struggles as observed during the teaching and learning of the activity questions. Due to Co-Vid 19 restrictions that resulted in logistical difficulties, only one class of 15 students participated in this study. After listening to the focus group recordings numerous times and reading the transcripts, common patterns were noted that had emerged, either from paraphrasing or from direct quotes. The primary research question is: What is the nature of the productive struggles experienced by high-school students during the simplification of trigonometric expressions and proving of trigonometric identities and how do these productive struggles influence the learning and teaching of trigonometry? The study findings were that the students struggled with “carrying out known mathematical processes” such as manipulating equations, knowing under what conditions cancellation of terms can be applied, adding and subtracting algebraic fractions involving trigonometric expressions, and factorisation of trigonometric expressions. In addition, there were misconceptions about the concept of “simplification”. Delayed impasse struggles occurred; this is when a student does not initially struggle to get started with a question, but the struggling becomes apparent as the student progresses with the question. The students committed fewer Newman errors in proving trigonometric identities than in the simplification of trigonometric expressions. Subsequently, students performed better at proving trigonometric identities than at simplifying trigonometric expressions. It could well be that through productive struggles, the students developed some of their own strategies from the simplification of trigonometric expressions. Alternatively, proving identities could be seen as “easier”, since the students already know what the “answer” should be. Nonetheless, students still struggled to carry out common mathematical processes such as factorisation and the manipulation of algebraic fractions. Regarding factorisation and manipulation of algebraic fractions, students compartmentalised knowledge. For example, most students knew how to factorise algebraic expressions, but failed to see the resemblance between algebraic expressions and trigonometric expressions (and consequently, how to factorise trigonometric expressions). Although there was a decrease in the number of Newman errors from the simplification of trigonometric expressions to the proving of trigonometric identities, there was an increase at the comprehension hierarchy, which may be attributed to the fact that the students might have struggled with the concept of “proof”. Additionally, students in this study struggled with the concept of “simpler”. Some students thought that the solution to a simplification question should be more complex than the original question. Nonetheless, with both the simplification of trigonometric expressions and the proving of trigonometric identities it remained a challenge for the students to apply prior knowledge in a new mathematical context such as trigonometry. The significance of the study's findings is that they suggest that teachers re-evaluate how to instruct known mathematical processes and procedures, so as not to compartmentalise mathematical knowledge. Productive struggles may not always produce correct answers; but given sufficient time and appropriate intervention by their teacher, students can build their own knowledge and become independent thinkers who can apply prior knowledge in new contexts such as the simplification of trigonometric expressions and the proving of trigonometric identities. In future research, productive struggles in the simplification of trigonometric expressions and the proving of trigonometric identities should be explored with a bigger, more diverse group of students, taught by more than one teacher at more than one school. In addition, to investigate the long-term effects of productive struggles a study lasting more than six months could be carried out.

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Sayster, A. 2023. High-school students' productive struggles during the simplification of trigonometrical expressions and the proving of trigonometrical identities. . ,Faculty of Science ,Department of Mathematics and Applied Mathematics. http://hdl.handle.net/11427/38539