Teaching Practicum

The purpose of the teaching practicum is to: serve as a historical document of student\u27s practicum; demonstrate the student\u27s understanding of how the actual courses they are involved in relate to and support the Curriculum Frameworks; demonstrate the student\u27s ability to develop classroom materials consistent with the Frameworks; provide the student the opportunity to assess his classes to determine the degree to which the Frameworks are being met; provide the student with opportunity to provide evidence of effective classroom management, promoting equity and meeting professional responsibilities; and require the student to reflect upon the connections between their experiences in the secondary education they are providing and the college education they are experiencing

A descriptive and evaluative bibliography of mathematics filmstrips.

Submitted by A.W. Clark and R.W. Allen for the degree of Master of Arts and by C.H. Gardner and R.F. Sweeney for the degree of Master of Education. Thesis (Ed.M.)--Boston UniversityThe purpose of this paper is to present in one volume (1) a bibliography of all mathematics filmstrips from those suitable for the first grade to those suitable for use in senior high school and college, (2) an accurate description of each filmstrip, and (3) unbiased evaluations of each filmstrip by qualified teachers invited to take part in the project. Concomitant problems. The foregoing three parts were the heart of the problem and the portion nearly completely solved. There were, however, concomitant problems which have been partially solved by this work. The first of these concerns the limited use of filmstrips by mathematics teachers. Undoubtedly many do not believe in using filmstrips in mathematics classes. Others have never given serious thought about the advisability of using filmstrips. In later sections of this chapter and throughout this work evidence is cited to support the contention that filmstrips should have serious consideration, and that they are useful in mathematics classes. The second concomitant problem concerns the revision of current filmstrips and production of new ones. The filmstrip producers were supplied, upon their request, with summaries of the evaluations. Summaries were supplied only at the producer's request; for unless they were interested enough to request the summaries, they probably would not be interested in changing or improving their filmstrips. Summary. The problem, then, had three major parts: listing , describing, and evaluating mathematics filmstrips, and two concomitant parts: arousing the mathematics teacher's interest in filmstrips, and encouraging producers to make better productions and necessary revisions in current productions. [TRUNCATED

From triangles to a concept: a phenomenographic study of A-level students' development of the concept of trigonometry

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

Music of the triangles: How students come to understand trigonometric identities and transformations

Trigonometry is an essential part of mathematics education (NCTM, 2000; NGA, 2010). Trigonometry is prevalent in studies of pure mathematics as well as physical applications. Trigonometric identities and transformations are particularly important. However, students and even teachers have struggled to articulate and justify trigonometric concepts (Moore, 2013; Tuna, 2013). Students have also struggled with identities and transformations in non-trigonometric contexts (Borba & Confrey, 1996; Tsai & Chang, 2009). This paper will describe a research project which articulates the critical stages through which students must pass to understand trigonometric identities and transformations. These critical stages were first hypothesized based on a review of the literature. Then undergraduate precalculus students were recruited to participate in a series of task-based interviews in order to examine the process by which students come to understand and justify trigonometric identities and transformations. The critical stages were revised based on the results of these interviews. Following the interviews, hypothesized lesson plans for the subjects were revised and implemented. The implementation of the lesson plans did not collect enough information to draw any conclusions, but the critical stages underscore the importance of students being able to move fluidly among representations