Beliefs and knowledge of middle grades teachers regarding functions
Wynne, Brian David
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The study of teaching mathematics is a complex endeavor. In fact, attempting to understand how teachers orchestrate mathematically sound, engaging, and meaningful lessons often generates more questions than answers. For many years, mathematics educators conducted quantitative studies—intended to show a correlation between the instructional decisions made by teachers and the number of content courses taken in college; however, no such correlation emerged. In the 1980s, researchers began to examine more affective or cognitive issues—such as attitudes, beliefs, or knowledge—using methods that were more qualitative in nature. The purpose of this study was to examine the beliefs that middle grades mathematics teachers hold about teaching functions, the knowledge that middle grades mathematics teachers have regarding functions, and the interplay between beliefs and knowledge during classroom instruction. This study is timely and appropriate given that there is not a large body of literature on middle grades mathematics education. Two middle grades mathematics teachers, Melodie and Rachel, taught Algebra I in different (yet similar) schools within the same district. The two teachers took part in completing an initial survey, three hour-long interviews, a card sorting activity that dealt with families of functions, selecting a favorite definition for the term function, and modeling a function using a Calculator-Based Ranger. Classroom observations were completed and artifacts were collected while each teacher provided instruction on quadratic functions. Data analysis was on-going throughout the study, and a theoretical framework was derived from literature pertaining to teacher beliefs, teacher knowledge, and teacher authority. Melodie’s belief that functions are the cornerstone of high school mathematics, coupled with her deep and flexible knowledge of mathematics, allowed her to teach procedures as well as to treat concepts. Her students were engaged in inquiry-based activities on a regular basis, and technology was a staple in her classroom. Rachel was a conveyor of direct instruction and believed that good mathematics teaching consisted of step-by-step instructions for her students to follow. She taught functions because they were part of her district’s Algebra I curriculum because her high school colleagues told her that functions were important in later mathematics courses.