Cognitive analysis of students’ learning of trigonometry in dynamic geometry environment : a teaching experiment

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2015
Şahin, Zülal
Trigonometry is a part of mathematics in which algebra and geometry converge. Dealing with trigonometric functions at secondary level is known as a difficult task because it requires to work with right triangles, the unit circle, and graphs of trigonometric functions simultaneously. For most students, this means excessive amount of formulas unless they can establish connections among different representational systems. There is a consensus in the literature that appropriate use of technology can be effective in helping students make such connections. Dynamic geometry environments can be a useful tool in teaching trigonometry due to their opportunities that enable to construct mathematical objects within different representational systems in a dynamically-and-linked way. The overarching purpose of this study was to design an instruction in dynamic geometry environment in order to support secondary students’ concept images on core trigonometric functions, i.e., sine and cosine, in different representations (i.e., symbolic, circular, and graphic). The instructional sequence was designed initially through inspiring from research literature on trigonometry, historical development of trigonometry, our exploratory teaching experience, and initial interview results. And then, design of the instruction was continued through revising as a result of the on-going prospective and retrospective cognitive analysis of the data that were collected during the 17-week teaching experiment from two pairs of secondary students separately. Students were encouraged to reason about dynamically-linked transformations of the core trigonometric functions within and between representational registers, as well as reasoning about dynamically-changed visual components referring to the core trigonometric functions. When compared with their initial serious recognition and discrimination troubles, as the study progressed, significant improvements were observed in students’ recognition and discrimination abilities within and between different representational registers. The cognitive analysis of the data revealed the importance of students’ constructions of well-defined concept definition images on foundational trigonometric concepts (i.e., angle, angle measure, trigonometric value, trigonometric functions, and periodicity) in order to recognize the same trigonometric object within different representational registers. The importance of the basic visual features’ discrimination in comprehension of trigonometry was also revealed in this study. When the basic visual features referring to trigonometric functions (i.e., radius of the circle, position of the center, position of the reference point on the circle referring to trigonometric value) were systematically varied in the (unit) circle register, and their dynamic-and-linked oppositions in the graphical register were constructed, the students developed significant understandings that enabled them to discriminate the basic form of sine and cosine functions from their general forms. Finally, the findings of the study revealed that the discrimination ability required to be reasoning about the new situations emerging as a consequence of the changed-visual features through focusing on trigonometrically relevant objects (e.g., reference point(s), reference right triangle, radius, displacement amount and direction) rather than detailed processes (e.g., ordinate of a point, procedural definition of sine or cosine).

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Citation Formats
Z. Şahin, “Cognitive analysis of students’ learning of trigonometry in dynamic geometry environment : a teaching experiment,” Ph.D. - Doctoral Program, Middle East Technical University, 2015.