FACILITATING ENCOUNTERS AMONG TEACHERS WITH REPRESENTATIONS OF TEACHING: TWO REGISTERS

We report on the management of teacher study groups designed to elicit the practical rationality of mathematics teaching. These study groups appeal to practitioners as occasions for professional development yet are deliberately designed as research instruments, as catalysts of the often-implicit norms and dispositions that sustain mathematics teaching in secondary school. The dual purpose of these groups makes them amenable to tensions comparable to those that are endemic to teaching itself. Those tensions were handled by having two facilitators invested of different personas and agendas—a moderator and a researcher. Excerpts from these sessions are used to argue that these dual-facilitator sessions create a context for a conversation that can be described as interweaving two different registers.National Science Foundation, ESI-0353285http://deepblue.lib.umich.edu/bitstream/2027.42/64852/1/Nachlieli&Herbst-2007.pd

Evolving discourse of practices for quality teaching in secondary school mathematics

International audienceThis paper presents initial findings from the project PRAQTAL: PRActices for Quality Teaching. Our project focuses on conceptualizing and identifying teaching practices for quality teaching of secondary school mathematics and physics. This work is driven by the understanding that to document, conceptualize, analyze and promote quality teaching, we need to constitute a discourse which articulates the diverse practices of teachers, in their multiple resolutions, and link them to educational theories on one hand and specific instruments on the other. For this purpose, we adopted Commognition as a conceptual framework. Here, we present our working definition of a teaching practice and our criteria for quality teaching practices. We discuss the procedures for identifying and documenting practices and their representation and illustrate our arguments with an example from the project’s emerging database

STUDYING THE PRACTICAL RATIONALITY OF MATHEMATICS TEACHING: WHAT GOES INTO “INSTALLING” A THEOREM IN GEOMETRY?

This paper presents a way of studying the rationality that mathematics teachers utilize in managing the teaching of theorems in high school geometry. More generally, the study illustrates how to elicit the rationality that guides teachers in handling the demands of teaching practice. In particular, it illustrates how problematic classroom scenarios represented through animations of cartoon characters can facilitate thought experiments among groups of practitioners. Relying on video records from four study group sessions with experienced teachers of geometry, the study shows how these records can be parsed and inspected to identify categories of perception and appreciation with which experienced practitioners relate to an instance of an instructional situation. The study provides initial evidence that supports a theoretically-derived hypothesis, namely that teachers of geometry as a group recognize as normative the expectation that a teacher will sanction or endorse those propositions that are to be remembered as theorems for later use. In interacting with a story in which students had proven a proposition that the teacher had not identified as a theorem, the study also shows the kind of tactical resources that teachers of geometry could use to make it feasible for students to reuse such a proposition.The research reported in this article is supported by NSF, grant ESI-0353285 to Herbst and Chazan. Opinions expressed here are the sole responsibility of the authors and do not reflect the views of the Foundation.http://deepblue.lib.umich.edu/bitstream/2027.42/78023/1/ITH-PHTN&DC.pd

Ritual towards explorative classroom participation of pre-service elementary school mathematics teachers

International audienceIn this paper we follow preservice elementary school mathematics teachers learning processes in a course that was organized around problem solving and aimed at providing opportunities for students to participate exploratively. The goal of this study is to characterize pre-service teachers' participation on the "ritual-explorative" continuum, to understand better what opportunities for explorative participation are given and taken up by students. Findings show that the request to suggest various solution paths seems to help students focus on the explorative question of "where do I want to get at?" rather than at the ritual question of "how do I proceed?"

The discursive routine of personifying and its manifestation by two instructors

International audienceThis paper focuses on teaching processes in the mathematics classroom. We adopt Sfard's communicational framework to address the question: What were the teaching actions that seem to promote learning? To address the question, we analyzed four lessons taken from two courses about functions which were studied by elementary school prospective mathematics teachers. We identified a teaching mechanism that seems to relate teaching to students' learning. This mechanism includes flexible and dynamic transition between mathematizing (talk about mathematical objects), subjectifying (talk about the participants) and personifying (talk about human participants acting upon mathematical objects). We also identified several types of personifying talk

Opportunities for Adopting a Discourse of Explorations in a Professional Development Setting

International audienceWe examine the productiveness of discussions in a professional development (PD) towards explorative mathematics instruction (EMI). The artifacts used in the PD session under examination were a video-clip of a classroom discussion, together with the Quadrants coding scheme, a scheme originally developed for research purposes. We define productive discussions by relying on the linguistic tool of lexical chains that point to lexical cohesion. The analysis shows that the use of the coding scheme as an artifact, together with a video clip, offered increased opportunities for surfacing misalignments between the teachers' framing of learning and instruction and the PD leader's framing of it. These misaligned frames, which draw on the Acquisition Pedagogical Discourse, vs. Explorations Pedagogical Discourse, are usually difficult to surface

Opportunities for Adopting a Discourse of Explorations in a Professional Development Setting

International audienceWe examine the productiveness of discussions in a professional development (PD) towards explorative mathematics instruction (EMI). The artifacts used in the PD session under examination were a video-clip of a classroom discussion, together with the Quadrants coding scheme, a scheme originally developed for research purposes. We define productive discussions by relying on the linguistic tool of lexical chains that point to lexical cohesion. The analysis shows that the use of the coding scheme as an artifact, together with a video clip, offered increased opportunities for surfacing misalignments between the teachers' framing of learning and instruction and the PD leader's framing of it. These misaligned frames, which draw on the Acquisition Pedagogical Discourse, vs. Explorations Pedagogical Discourse, are usually difficult to surface

The Realization Tree Assessment tool: Assessing the exposure to mathematical objects during a lesson

International audienceWe present a new tool-Realization Tree Assessment (RTA) for assessing the mathematical quality of lessons and the ways in which the whole classroom discussion expose students to mathematical concepts. The tool, built upon the commognitive framework, depicts the different realizations of a mathematical object treated in a lesson, and then uses different shades to signify who articulated the realization-the teacher or the students. We exemplify the tool on two lessons implementing an identical Hexagon pattern generalization task. The RTA visualizes the manner in which one lesson gave students sufficient opportunities to "same" different algebraic expressions, while the other lesson did not. We show how this visual presentation of the mathematical ideas complements existing assessment tools, particularly, the Instructional Quality Assessment and Accountable Talk. We conclude by discussing the potential of the tool as an aid for lesson planning

Evolving discourse of practices for quality teaching in secondary school mathematics

International audienceThis paper presents initial findings from the project PRAQTAL: PRActices for Quality Teaching. Our project focuses on conceptualizing and identifying teaching practices for quality teaching of secondary school mathematics and physics. This work is driven by the understanding that to document, conceptualize, analyze and promote quality teaching, we need to constitute a discourse which articulates the diverse practices of teachers, in their multiple resolutions, and link them to educational theories on one hand and specific instruments on the other. For this purpose, we adopted Commognition as a conceptual framework. Here, we present our working definition of a teaching practice and our criteria for quality teaching practices. We discuss the procedures for identifying and documenting practices and their representation and illustrate our arguments with an example from the project’s emerging database