A comparative study of traditional and transformational approaches to selected topics in trigonometry



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The primary purpose of this study was to compare selected topics in a transformational approach to trigonometry with their counterparts in the traditional approach to determine if there was a significant difference in the achievement of students using the two methods. Specifically, this study attempted to determine if one of these approaches would produce significantly higher achievement by students in the derivation of initial identities and in angle reduction problems. The sample population consisted of 130 students from six of seven classes of trigonometry at Clear Lake High School during the 1975-1976 school year. Clear Lake High School serves students in the Clear Lake area, an upper middle class suburb of Houston, Texas. The students were assigned to classes by their choice, without prior knowledge of which class would receive a particular treatment. The treatments were then randomly assigned to the classes. Classification by levels of ability was made on the basis of the Differential Aptitude Test, in each class. The transformational approach to trigonometry utilized circular functions based on rotations about the origin. The isomorphism between these rotations and the two-by-two matrices associated with these rotations was a key factor in this approach. A text written and piloted by the experimenter was the source material for the transformational approach. The traditional approach to trigonometry utilized circular functions based on the unit circle. Derivation of identities was based on the distance formula. No use was made of transformations or symmetry. State-adopted trigonometry texts were the source of material for the traditional approach. After treatment, students were administered an author-constructed examination on derivation of formulas and angle reduction, which served as the criterion measures. The students were also administered the Cooperative Mathematics Test for Trigonometry. It was found that students achieved better on derivation of initial identities using the transformational approach. The study also found that students achieved better on angle reduction problems using the traditional approach. With no significant difference in the overall achievement as measured on the Cooperative Mathematics Test for Trigonometry (Form A), this study also supports the feasibility of a transformational approach to trigonometry.