Fairness of Dice: A Longitudinal Study of Students' Beliefs and Strategies for Making Judgments- FREE PREVIEW!
Jane M. Watson, Jonathan B. Mortiz
One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was "loaded") and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.
Low-Performing Students and Teaching Fractions for Understanding: An Interactional Analysis
Susan B. Empson
This article presents an analysis of two low-performing students' experiences in a firstgrade classroom oriented toward teaching mathematics for understanding. Combining constructs from interactional sociolinguistics and developmental task analysis, I investigate the nature of these students' participation in classroom discourse about fractions. Pre- and post-instruction interviews documenting learning and analysis of classroom interactions suggest mechanisms of that learning. I propose that three main factors account for these two students' success: use of tasks that elicited the students' prior understanding, creation of a variety of participant frameworks (Goffman, 1981) in which the students were treated as mathematically competent, and frequency of opportunities for identity-enhancing interactions.
A Survey Measuring Elementary Teachers' Implementation of Standards-Based Mathematics Teaching
John A. Ross, Douglas McDougall, Anne Hogaboam-Gray, Ann LeSage
Intensive case study is an expensive tool for measuring teachers' instructional practice. Previous research suggests that teacher self-report surveys provide a low-cost and relatively accurate picture of classroom practice. To examine the extent to which teachers implement mathematics education reform, we developed a 20-item survey based on nine dimensions of standards-based teaching. In this article, we provide evidence of the reliability (i.e., internal consistency) and validity of the instrument. The evidence consists of correlations of survey scores with a mandated performance assessment in Grade 6 mathematics, congruence with classroom observations of a small sample of teachers, and demonstrations that teachers who are similar in their claims about using a standards-based text series differ in how they use the text in ways predicted by the survey.