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Erna Yackel

A review of Teaching Problems and the Problems of Teaching by Magdalene Lampert(2001).

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Bridget Arvold, Pamela Turner and Thomas J. Cooney

Edited by Fernand J. Prevost

The visions of teaching set forth in the Professional Standards for Teaching Mathematics (NCTM 1991) do not, in themselves, ensure that we can successfully teach all students mathematics. To reach all students, we must understand how students think and then develop instruction compatible with their thinking. To begin to understand, we must observe, listen, and gather a variety of evidence of what and how students are learning. Although we might view a mathematical concept or algorithm as simple, it is “a mysterious, almost inexplicable phenomenon from the point of view of the outsider” (Davis and Hersh 1981, 43). Examining students' thinking through their interactions with mathematical tasks can help unlock a bit of the mystery. This process of analysis is the amalgamating item in the “Standards for Teaching Mathematics” section of the Professional Standards for Teaching Mathematics and emphasizes the need to listen carefully to our students.

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Sandy Spitzer and Christine Phelps-Gregory

For prospective teachers (PTs) to engage in lifelong systematic learning, they must be prepared to analyze teaching on the basis of its effects on student learning. We present the results of an intervention study aimed at developing PTs' ability to analyze a classroom video sample. The intervention used an online discussion board activity structured along three research-based dimensions, which allowed PTs to build their analysis skills outside of class time. Evidence for the effectiveness of this intervention includes findings that PTs engaged deeply with their peers' ideas, with many changing their mind about the lesson's success, and that PTs' final reflections showed increased attention to the mathematics of the learning goal. However, after the intervention, many PTs continued to take nonmathematical evidence as indicators of student learning. Implications illuminate key design features of interventions as well as the affordances and challenges of using online interactions for improving PTs' lesson analysis skills.

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Morton Friedman

Each teacher can improve his or her teaching by systematically analyzing it. Here is one way to analyze teaching.

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Christine Phelps-Gregory and Sandy M. Spitzer

One goal in teacher education is to prepare prospective teachers (PTs) for a career of systematic re_ ection and learning from their own teaching. One important skill involved in systematic re_ ection, which has received little research attention, is linking teaching actions with their outcomes on student learning; such links have been termed hypotheses. We developed an assessment task to investigate PTs' ability to create such hypotheses, prior to instruction. PTs (N = 16) each read a mathematics lesson transcript and then responded to four question prompts. The four prompts were designed to vary along research-based criteria to examine whether different contexts in_ uenced PTs' enactment of their hypothesizing skills. Results suggest that the assessment did capture PTs' hypothesizing ability and that there is room for teacher educators to help PTs develop better hypothesis skills. Additional analysis of the assessment task showed that the type of question prompt used had only minimal effect on PTs' responses.

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Amy Roth McDuffie, Mary Q. Foote, Corey Drake, Erin Turner, Julia Aguirre, Tonya Gau Bartell and Catherine Bolson

Mathematics teacher educators (MTEs) designed and studied a video analysis activity intended to support prospective teachers (PSTs) in learning to notice equitable instructional practices. PSTs from 4 sites (N = 73) engaged in the activity 4 to 5 times during the semester, using a set of 4 “lenses” to analyze teaching and learning as shown in videos. In an earlier analysis of this activity, we found that PSTs increased their depth and expanded their foci in noticing equitable instructional practices (Roth McDuf_ e et al., 2013). In this analysis, we shift the focus to our work as MTEs: We examine our decisions and moves in facilitating the video analysis activity with a focus on equity, and we discuss implications for other MTEs.

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Patty Anne Wagner, Ryan C. Smith, AnnaMarie Conner, Laura M. Singletary and Richard T. Francisco

As creating and critiquing arguments becomes more of a focus in mathematics classes, teachers need to develop their abilities to facilitate collective arguments. Many mathematics education researchers find Toulmin's (1958/2003) model of argumentation to be useful in analyzing arguments, raising the question of whether mathematics teachers would find it useful as well. We introduced the model to prospective secondary mathematics teachers and asked them to analyze arguments using it. We found that the prospective teachers developed an appropriate understanding of what collective argumentation looks like in the classroom, and the model provided them a lens for analyzing teaching practice. This suggests the use of Toulmin's model is a promising step in helping prospective teachers develop their conceptions of collective argumentation.

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Madhuvanti Anantharajan

van Es (2015) caution that noticing children’s thinking by itself does not ensure that teachers will “take up those ideas and use them as evidence to analyze teaching and learning or inform instructional responses” (p. 91). It is thus important for