Natasha M. Speer

Examining the Relationship Between Beliefs and Goals in Teacher Practice

Abstract This article presents a detailed analysis of how teacher beliefs interact with goals and influence the moment-to-moment actions of teaching. The beliefs, goals and instructional practice of two secondary mathematics teachers were examined as each conducted an algebra lesson. We discuss how specific beliefs organized to influence the selection and prioritization of goals that then influenced the actions of each teacher. Fine-grained analysis of classroom video and teacher interviews revealed that particular collections of beliefs become apparent when there is a shift in the teachers goals. Exploring the relationship between teacher beliefs and goals at this level of detail allows for the investigation of the mechanisms of the teaching process. Implications of this research for teacher education and professional development are also discussed.

Research on the Teaching and Learning of Calculus/Elementary Analysis

As noted in the introduction, there have been two very different traditions of research on calculus/introductory analysis. These traditions might almost be called ‘theory-driven,’ as reflected in section 2; and ‘practice-driven’ as described in section 3. Interestingly, there appears to be a move toward convergence of the two types. On the one hand, the theoretical work described in section 2 has given rise to some studies of ‘didactic engineering.’ On the other hand, now that various efforts at reform have been developed and stabilized, as described in section 3, such courses provide excellent sites for basic research. Ultimately, the field will make progress on effective teaching and learning only if it deals meaningfully with theoretical and pragmatic issues simultaneously. This paper reflects movement in that direction. All the articles cited — some of which focus on theoretical considerations, some on reform, and some on both theory and reform — are part of the foundations on which we build.

Calculus Students' Understanding of Volume.

Abstract Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students’ understanding of volume. This study investigated calculus students’ understanding of volume. Clinical interview transcripts and written responses to volume problems were analyzed. One finding is that some calculus students, when asked to find volume, find surface area instead and others blend volume and surface area elements. We found that some of these students believe adding the areas of an object’s faces measures three-dimensional space. Findings from interviews also revealed that understanding volume as an array of cubes is connected to successfully solving volume problems. This finding and others are compared to what has been documented for elementary school students. Implications for calculus teaching and learning are discussed.

Calculus Students' Understanding of Area and Volume Units.

Abstract Units of measure are critical in many scientific fields. While instructors often note that students struggle with units, little research has been conducted about the nature and extent of these difficulties or why they exist. We investigated calculus students’ unit use in area and volume computations. Seventy-three percent of students gave incorrect units for at least one task. The most common error was the misappropriation of length units in area and volume computations. Analyses of interview data indicate that some students think that the unit of the computation should be the same as the unit specified in the task statement. Findings also suggest that some students have difficulties correctly indicating the units for computations that involve the quantity π. We discuss students’ correct and incorrect use of unit in relation to their understanding of area and volume as arrays, as well as in terms of Sherin’s Symbolic Forms.