Joanne Rossi Becker

Formation of Pattern Generalization Involving Linear Figural Patterns Among Middle School Students: Results of a Three-Year Study

This chapter provides an empirical account of the formation of pattern generalization among a group of middle school students who participated in a three-year longitudinal study. Using pre-and post-interviews and videos of intervening teaching experiments, we document shifts in students’ ability to pattern generalize from figural to numeric and then back to figural, including how and why they occurred and consequences. The following six findings are discussed in some detail: development of constructive and deconstructive generalizations at the middle school level; operations needed in developing a pattern generalization; factors affecting students’ ability to develop constructive generalizations; emergence of classroom mathematical practices on pattern generalization; middle school students’ justification of constructive standard generalizations, and; their justification of constructive nonstandard generalizations and deconstructive generalizations. The longitudinal study also highlights the conceptual significance of multiplicative thinking in pattern generalization and the important role of sociocultural mediation in fostering growth in generalization practices.

Ethnomathematics in the Global Episteme: Quo Vadis?

This chapter discusses scholarly work in the field of ethnomathematics from three perspectives that seem to encompass much of the current work in the field: challenging Eurocentrism in mathematics; ethnomathematics praxis in the curriculum; and ethnomathematics as a field of research. We identify what we perceive to be strengths and weaknesses of these three perspectives for today’s learners who are faced with forces of a global nature. We propose a less traditional view of ethnomathematics that is compatible with postnational, global identities, and exemplify this approach through a professional development program in California. Finally, we raise several issues for future discussions relative to ethnomathematical theory and practice

Classroom Coaching as a Collaborative Activity in Professional Development

This chapter reports on a study of classroom coaching as a collaborative approach to professional development. Fourteen high school mathematics teachers were coached in 17 different classes for a combined total of 210 classroom visits in one academic year. Observations were combined with debriefing sessions after each visit and final interviews. The teachers had previously participated in one or two extensive professional development programmes, and the collaborative coaching was originally intended to help determine the impact of the professional development on classroom practices. The chapter discusses teacher behaviours in seven categories that were the focus of observations: equity; multiple representations; technology; student understanding; cooperative groups; alternative assessment; and innovative curriculum. However, the teachers reported that the coaching itself was a critical extension of the professional development. We concluded that classroom coaching, while expensive in resources, can be a crucial element of professional development, because it can help teachers both implement and sustain changes in classroom practice.

Commentary on the Chapter by Dowling and Burke, “Shall We Do Politics or Learn Some Maths Today? Representing and Interrogating Social Inequality”

This chapter posits the following conundrum: Is it possible to structure a mathematics lesson that integrates questioning of social inequality with worthwhile mathematical material? Dowling and Burke answer this question negatively. They conclude that a lesson must either privilege mathematics, providing all students with appropriate and rigorous mathematical content, or a lesson must privilege political motivations, providing for a full and extensive discussion of critical issues; according to the authors, it is not possible to accomplish both goals at the same time.