This thesis describes an investigation of the trigonometry schemas developed by a group of 16-18 year old English students during their study of A-level mathematics. It is concerned with identifying differences in the schemas of students who are successful with solving trigonometric problems to those who are less successful. The study is guided by the theoretical frameworks of mathematical schema development proposed by Sfard (1991) and Dubinsky’s (1991) APOS theory that describe how operational knowledge of one or more procedures develops into an understanding that is conceptual. The benefits of a conceptual understanding are greater flexibility in problem solving and greater cognitive economy. The study of trigonometry prior to starting the A-level course is predominantly concerned with problems relating to triangles either right-angled or scalene and it is during the Alevel course that trigonometry broadens into the study of the properties of function. Experience as a mathematics tutor suggests however that not all students finish the A-level course with a conceptual understanding of trigonometric functions that is a coherent entity. Some students have little more than a collection of arbitrary facts and procedures that they struggle to use cohesively. Traditionally trigonometry is taught by mediating the core ideas through a mixture of spatial-visual images and algebraic identities that together provide the basis for function properties and behaviour. This study examines student perceptions of these mediating representations through a phenomenological investigation based on concept maps, interviews, classroom observations and observed problem-solving by selected students. The results of the study suggest that different students focus upon different aspects of the mediating representations. Students schemas as evidenced by the concept maps varied between those that were predominantly composed of algebraic representations for instance formulae, to those that were composed of a mixture of algebraic and specific spatial visual representations such as graphs, the unit circle and special angles triangles, to those that portrayed trigonometry through a series of overlaid graphs that signified the essence of function behaviour. The students whose schemas included spatial-visual components were more successful in problem solving and assessments than those whose schemas were focused on algebraic aspects. The study also supported documented research by Gray, Pinto, Pinner & Tall (1999) that spatial-visual imagery has a qualitative aspect and by Delice & Monaghan (2005) that teaching style plays a considerable part in the students’ development of schema. A significant aspect to the development of a flexible schema is the teacher’s philosophy of trigonometry and approach to the construction of sub concepts. Finally the study considers the merits of the two main theoretical frameworks of mathematical development proposed by Sfard and Dubinsky’s APOS theory from a teaching perspective and concludes that the empirical findings of this study are better described by Sfard’s explanation of the dual nature of mathematical conceptions whereby a process schema has the potential to develop into a flexible, stable object conception through interiorisation, condensation and reification.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:524350 |
| Date | January 2009 |
| Creators | Challenger, Michele |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/1935/ |
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