Susan D. Nickerson

The Need for Proof and Proving: Mathematical and Pedagogical Perspectives

This chapter first examines why mathematics educators need to teach proof, as reflected in the needs that propelled proof to develop historically. We analyse the interconnections between the functions of proof within the discipline of mathematics and the needs for proof. We then take a learner’s perspective and discuss learners’ difficulties in understanding and appreciating proof, as well as a number of intellectual needs that may drive learners to prove (for certitude, for causality, for quantification, for communication, and for structure and connection). We conclude by examining pedagogical issues involved in teachers’ attempts to foster necessity-based learning that motivates the need to prove, in particular the use of tasks and activities that elicit uncertainty, cognitive conflict and inquiry-based learning.

Hybrid lesson study: extending lesson study on-line

Purpose – Lesson study in the USA provides a space for teachers to discuss classroom teaching with a goal of improving student learning. In general, US teachers’ professional lives are constrained by a lack of time for planning and observation. Within a lesson study context, we designed and utilized a web site to support and extend collaboration among teachers of mathematics at 26 school sites across two school districts. The purpose of this paper is to describe the lesson study context and share results of the investigation and affordances of the web site. Design/methodology/approach – Through the use of a survey, discourse analysis, and grounded theory, we examined how teachers used the web site and how best to foster regular and inclusive participation. Findings – The results reveal the web site spans time and distance. A significant percentage of the teachers used the web site for accessing resources, while a smaller fraction of teachers used the web site to discuss pedagogy and students’ mathematical...

Challenges in Measuring Secondary Mathematics Teachers’ Professional Noticing of Students’ Mathematical Thinking

Our focus is on teachers’ professional noticing of children’s mathematical thinking which Jacobs et al. (Jacobs, Lamb, & Philipp, 2010) describe as a set of three interrelated skills: (1) attending to students’ strategies, (2) interpreting the students’ mathematical understandings, (3) deciding how to respond on the basis of students’ understandings. We focus on secondary teachers’ professional noticing of children’s mathematical thinking because we believe teachers with expertise in noticing children’s mathematical thinking are better poised to support their students’ mathematical performance and understanding. However, studying teacher noticing at the secondary level presents unique methodological challenges. We first consider methodological issues in measuring K–12 teachers’ professional noticing of children’s mathematical thinking, and then consider three methodological challenges that are particular to secondary mathematics classrooms. These issues center around the challenges of artifact selection, determining the relative sophistication of responses, and the lack of access to experts’ responses at the secondary level. Also, we consider the cultures of teaching in the elementary and secondary contexts.