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July 1997, Volume 28, Issue 4


Teachers' Evolving Conceptions of One-to-One Tutoring: A Three-Tiered Teaching Experiment
Richard Lesh, Anthony E. Kelly
This article describes a three-tiered teaching experiment in which teachers were studied over a protracted period of time as they attempted to understand and improve their approaches to one-to-one tutoring. In a three-tiered teaching experiment model, emphasis is placed on (a) establishing a collaborative relationship between the research staff and the teachers, (b) careful choice of tasks for teachers and their students, (c) the development of "learning environments" for the study, and (d) the use of continuous and diverse dependent measures. The study documented the initial and revised strategies of teachers as they tutored children over a period of 10 weeks. Various indices of teacher change are reported, and some of the strengths and limitations of the methodology are discussed.

Designing Representations: Reasoning About Functions of Two Variables
Michal Yerushalmy
One major part of the effort to reform secondary school mathematics is the project of changing the goal of studying school algebra from the mastery of symbolic manipulations to the ability to reason algebraically. Another major component of these reform efforts is the creation of opportunities for students to communicate within and about mathematics. The ability to generalize, especially when the generalization requires a major breakthrough in habits of mind, is one indication of algebraic reasoning. In this article, I describe generalization activity as an opportunity to learn about seventh graders' understanding of functions. A group of students who had studied functions modeled a multivariable situation. Through individual and group work, they designed, described, and discussed various representations for functions of 2 variables. Their modeling efforts allowed them to analyze their understanding of representations of quantities, relationships among quantities, and relationships among the representations of quantities in both single- and multivariable functions.

Increased Knowledge in Geometry and Instructional Practice
Jane O. Swafford, Graham A. Jones, Carol A. Thornton
This study examined the effects on instruction of an intervention program designed to enhance teachers' knowledge of geometry and their knowledge of research on student cognition in geometry. Forty-nine middle-grade (4–8) teachers participated in a 4-week program consisting of a content course in geometry and a research seminar on van Hiele theory. The pretest and posttest results showed significant gains in content knowledge and in van Hiele level. The analysis of a lesson-plan task revealed a significant shift in goals and expectations to the next higher van Hiele level. Follow-up observations of 8 teachers found marked changes in what was taught, how it was taught, and the characteristics teachers displayed. Teachers attributed these changes to increased geometrical content knowledge and research-based knowledge of student cognition.

Families and Mathematics: A Study of Parent-Child Interactions
Ann Anderson
A group of 21 middle-class parents and their 4-year-old children participated in this study of mathematics and parent-child interactions. Each family chose 4 separate 15-minute periods over 2 days to share multilink blocks, a child's book, blank paper, and preschool worksheets with their child at home. Each 15-minute session was audiotaped, all artifacts generated during the sessions were kept, and each parent was interviewed. The results of the study indicate that a wide range of mathematics was evident, with counting being the most prevalent activity. All parents succeeded in injecting some mathematics in most sessions; some did so by explicitly setting mathematics as a goal, whereas others injected the mathematics as a means of carrying out, or as an aside to, the play activity they coconstructed. Questioning children's knowledge was the main way parents elicited mathematics, although some requested explanations, or clarifications, and some mediated patterns, relationships, and strategies.

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