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- All Subjects: Effective teaching
- Creators: Saldanha, Luis
Description
This study contributes to the ongoing discussion of Mathematical Knowledge for Teaching (MKT). It investigates the case of Rico, a high school mathematics teacher who had become known to his colleagues and his students as a superbly effective mathematics teacher. His students not only developed excellent mathematical skills, they also developed deep understanding of the mathematics they learned. Moreover, Rico redesigned his curricula and instruction completely so that they provided a means of support for his students to learn mathematics the way he intended. The purpose of this study was to understand the sources of Rico's effectiveness. The data for this study was generated in three phases. Phase I included videos of Rico's lessons during one semester of an Algebra II course, post-lesson reflections, and Rico's self-constructed instructional materials. An analysis of Phase I data led to Phase II, which consisted of eight extensive stimulated-reflection interviews with Rico. Phase III consisted of a conceptual analysis of the prior phases with the aim of creating models of Rico's mathematical conceptions, his conceptions of his students' mathematical understandings, and his images of instruction and instructional design. Findings revealed that Rico had developed profound personal understandings, grounded in quantitative reasoning, of the mathematics that he taught, and profound pedagogical understandings that supported these very same ways of thinking in his students. Rico's redesign was driven by three factors: (1) the particular way in which Rico himself understood the mathematics he taught, (2) his reflective awareness of those ways of thinking, and (3) his ability to envision what students might learn from different instructional approaches. Rico always considered what someone might already need to understand in order to understand "this" in the way he was thinking of it, and how understanding "this" might help students understand related ideas or methods. Rico's continual reflection on the mathematics he knew so as to make it more coherent, and his continual orientation to imagining how these meanings might work for students' learning, made Rico's mathematics become a mathematics of students--impacting how he assessed his practice and engaging him in a continual process of developing MKT.
ContributorsLage Ramírez, Ana Elisa (Author) / Thompson, Patrick W. (Thesis advisor) / Carlson, Marilyn P. (Committee member) / Castillo-Chavez, Carlos (Committee member) / Saldanha, Luis (Committee member) / Middleton, James A. (Committee member) / Arizona State University (Publisher)
Created2011
Description
Teachers must recognize the knowledge they possess as appropriate to employ in the process of achieving their goals and objectives in the context of practice. Such recognition is subject to a host of cognitive and affective processes that have thus far not been a central focus of research on teacher knowledge in mathematics education. To address this need, this dissertation study examined the role of a secondary mathematics teacher’s image of instructional constraints on his enacted subject matter knowledge. I collected data in three phases. First, I conducted a series of task-based clinical interviews that allowed me to construct a model of David’s mathematical knowledge of sine and cosine functions. Second, I conducted pre-lesson interviews, collected journal entries, and examined David’s instruction to characterize the mathematical knowledge he utilized in the context of designing and implementing lessons. Third, I conducted a series of semi-structured clinical interviews to identify the circumstances David appraised as constraints on his practice and to ascertain the role of these constraints on the quality of David’s enacted subject matter knowledge. My analysis revealed that although David possessed many productive ways of understanding that allowed him to engage students in meaningful learning experiences, I observed discrepancies between and within David’s mathematical knowledge and his enacted mathematical knowledge. These discrepancies were not occasioned by David’s active compensation for the circumstances and events he appraised as instructional constraints, but instead resulted from David possessing multiple schemes for particular ideas related to trigonometric functions, as well as from his unawareness of the mental actions and operations that comprised these often powerful but uncoordinated cognitive schemes. This lack of conscious awareness made David ill-equipped to define his instructional goals in terms of the mental activity in which he intended his students to engage, which further conditioned the circumstances and events he appraised as constraints on his practice. David’s image of instructional constraints therefore did not affect his enacted subject matter knowledge. Rather, characteristics of David’s subject matter knowledge, namely his uncoordinated cognitive schemes and his unawareness of the mental actions and operations that comprise them, affected his image of instructional constraints.
ContributorsTallman, Michael Anthony (Author) / Carlson, Marilyn P (Thesis advisor) / Thompson, Patrick W (Committee member) / Saldanha, Luis (Committee member) / Middleton, James (Committee member) / Harel, Guershon (Committee member) / Arizona State University (Publisher)
Created2015