Japanese and American Teachers' Evaluations of Videotaped Mathematics Lessons
Eiji Morita, Jennifer K. Jacobs
This article describes a novel assessment method used to examine Japanese and American teachers' ideas about what constitutes effective mathematics pedagogy. Forty American and 40 Japanese teachers independently evaluated either an American or Japanese mathematics lesson captured on videotape. Their comments were classified into over 1600 idea units, which were then sorted into a hierarchy of categories derived from the data. Next, the authors hypothesized underlying ideal instructional scripts that could explain the patterns of responses. Whereas the U.S. teachers were supportive of both traditional and nontraditional elementary school mathematics instruction and had different scripts for the two lessons, the Japanese teachers had only one ideal lesson script that was closely tied to typical Japanese mathematics instruction. The findings suggest that U.S. teachers may have more culturally sanctioned options for teaching mathematics; however, Japanese teachers may have a more detailed and widely shared theory about how to teach effectively.
Engaging Students in Proving: A Double Bind on the Teacher
Patricio G. Herbst
This article uses a classroom episode in which a teacher and her students undertake a task of proving a proposition about angles as a context for analyzing what is involved in the teacher's work of engaging students in producing a proof. The analysis invokes theoretical notions of didactical contract and double bind to uncover and explain conflicting demands that the practice of assigning two-column proofs imposes on high school teachers. Two aspects of the work of teaching--what teachers do to create a task in which students can produce a proof and what teachers do to get students to prove a proposition--are the focus of the analysis of the episode. This analysis supports the argument that the traditional custom of engaging students in doing formal, two-column proofs places contradictory demands on the teacher regarding how the ideas for a proof will be developed. Recognizing these contradictory demands clarifies why the teacher in the analyzed episode ends up suggesting the key ideas for the proof. The analysis, coupled with current recommendations about the role of proof in school mathematics, suggests that it is advantageous for teachers to avoid treating proof only as a formal process.<p> *In accordance with the policy of the JRME Editorial Panel regarding potential conflicts of interest involving the editor, the review and publication decision for this manuscript were handled by Jane Swafford, who was a member of the JRME Editorial Panel when the manuscript was originally submitted.
Abstraction in Expertise: A Study of Nurses' Conceptions of Concentration
Richard Noss, Celia Hoyles, Stefano Pozzi
Building on an ethnographic study of nurses' working practices on the ward (Hoyles, Noss, & Pozzi, 2001), we elaborate the notion of situated abstraction as an analytical tool for understanding nurses' conceptions of the intensive quantity of drug concentration. Data were gathered through interviews based on simulations of tasks observed to be problematic in the earlier study. The methodology was designed to explore nurses' conceptions in a more detailed way than is possible during in situ observations and to undertake a pointed examination of the degree of situatedness of nurses' knowledge and reasoning. Analysis of the data demonstrated that nurses' conceptions were abstracted within their practice when they were able to coordinate their mathematical knowledge with their professional expertise, yet their conceptions were situated, as evidenced by their difficulties in realizing this coordination in tasks too distant from their practice.