Eric J. Knuth

A Study of Whole Classroom Mathematical Discourse and Teacher Change

This article presents a comparison of the first 2 years of an experienced middle school mathematics teachers efforts to change her classroom practice as a result of an intervening professional development program. The teachers intention was for her teaching to better reflect her vision of reform-based mathematics instruction. We compared events from the 1st and 2nd years whole class discussions within a multilevel framework that considered the flow of information and the nature of peer- and teacher-directed scaffolding. Discourse analyses of classroom videos served both as an analytic tool for our study of whole classroom interactions, as well as a resource for promoting discussion and reflection during professional development meetings. The results show that there was little change in the teachers specific goals and beliefs in light of a self-evaluation of her Year 1 practices, but substantial changes in how she set out to enact those goals. In Year 2, the teacher maintained a central, social scaffolding role, but removed herself as the analytic center to invite greater student participation. Consequently, student-led discussion increased manifold, but lacked the mathematical precision offered previously by the teacher. The analyses lead to insights about how classroom interactions can be shaped by a teachers beliefs and interpretations of educational reform recommendations.

A CONCEPTUAL FRAMEWORK FOR LEARNING TO TEACH SECONDARY MATHEMATICS: A SITUATIVE PERSPECTIVE

This paper offers for discussion and critique a conceptual framework that applies a situative perspective on learning to the study of learning to teach mathematics. From this perspective, such learning occurs in many different situations -- mathematics and teacher preparation courses, pre-service field experiences, and schools of employment. By participating over time in these varied contexts, mathematics teachers refine their conceptions about their craft -- the big ideas of mathematics, mathematics-specific pedagogy, and sense of self as a mathematics teacher. This framework guides a research project that traces the learning trajectories of teachers from two reform-based teacher preparation programs into their early teaching careers. We provide two examples from this research to illustrate how this framework has helped us understand the process of learning to teach.

Teaching and Learning Proof Across the Grades : A K-16 Perspective

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proof

Middle School Students’ Understanding of Core Algebraic Concepts: Equivalence & Variable

Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.

How Teachers Link Ideas in Mathematics Instruction Using Speech and Gesture: A Corpus Analysis

This research investigated how teachers express links between ideas in speech, gestures, and other modalities in middle school mathematics instruction. We videotaped 18 lessons (3 from each of 6 teachers), and within each, we identified linking episodes—segments of discourse in which the teacher connected mathematical ideas. For each link, we identified the modalities teachers used to express linked ideas and coded whether the content was new or review. Teachers communicated most links multimodally, typically using speech and gestures. Teachers’ gestures included depictive gestures that simulated actions and perceptual states, and pointing gestures that grounded utterances in the physical environment. Compared to links about new material, teachers were less likely to express links about review material multimodally, especially when that material had been mentioned previously. Moreover, teachers gestured at a higher rate in links about new material. Gestures are an integral part of teachers’ communication during mathematics instruction.

Middle school students' production of mathematical justifications

Proof in advanced mathematics classes: semantic and syntactic reasoning in the representation system of proof

Preface to Part I

This chapter provides an overview of curricular part of the chapters in this volume. The authors focus on various aspects of curricula from different countries (including China, India, Japan Russia, Singapore, and the United States) in order to examine how curriculum might be designed and delivered to help students develop algebraic habits of mind. The chapters that comprise this part highlight various curricular approaches, experiences, and practices that illustrate ways in which students in earlier grades can be better prepared to think algebraically.

A Learning Progression for Elementary Students’ Functional Thinking

ABSTRACT In this article we advance characterizations of and supports for elementary students’ progress in generalizing and representing functional relationships as part of a comprehensive approach to early algebra. Our learning progressions approach to early algebra research involves the coordination of a curricular framework and progression, an instructional sequence, written assessments, and levels of sophistication describing students’ algebraic thinking. After detailing this approach, we focus on what we have learned about the development of students’ abilities to generalize and represent functional relationships in a grades 3–5 early algebra intervention by sharing the levels of responses we observed in students’ written work over time. We found that the sophistication of students’ responses increased over the course of the intervention from recursive patterning to correspondence and in some cases covariation relationships between variables. Students’ responses at times differed by the particular tasks that were posed. We discuss implications for research and practice.

Cycles of Generalizing Activities in the Classroom

This study considers classroom situations in which students and the teacher co-contribute to promoting generalization. It specifically focuses on the ways in which students and a teacher in one classroom engage in generalizing arithmetic. Generalized arithmetic is an important route into early algebra (Kaput in Algebra in the Early Grades. Routledge, New York, 2008); its potential as a way to deepen students’ understandings of concepts of school arithmetic makes it an important focus of early algebra research. In the analysis we identified generalizations around properties of arithmetic and the actions that promoted these types of generalizations, and then considered the relationship between these actions . Analysis revealed that generalizations became platforms for further generalization.

Implementing a Framework for Early Algebra

In this chapter , we discus s the algebra framework that guides our work and how this framework was enacted in the design of a curricular approach for systematically developing elementary-aged students’ algebraic thinking. We provide evidence that, using this approach, students in elementary grades can engage in sophisticated practices of algebraic thinking based on generalizing, representing, justifying, and reasoning with mathematical structure and relationships. Moreover, they can engage in these practices across a broad set of content areas involving generalized arithmetic; concepts associated with equivalence, expressions, equations, and inequalities; and functional thinking.