Reflective Apprenticeship for Teaching and Learning Mathematical Proof

Daniel L Reinholz

doi:10.51355/jstem.2018.36

Abstract

This article explores teacher learning in a graduate-level analysis course for teachers. Drawing from the frameworks of extreme apprenticeship and Peer-Assisted Reflection (PAR), the course created authentic learning experiences for the teachers that served as models that they could use in their own classrooms. This paper describes how the teachers developed across the four dimensions of extreme apprenticeship. While this paper is grounded in mathematics, the extreme apprenticeship and PAR frameworks are cross-disciplinary, and thus there are implications for teaching and learning in all of the STEM disciplines.

Highlights

Who can be a mathematician? What is mathematics all about? How should mathematics be taught? We tend to answer these questions based on our experiences with our own mathematics teachers: what they looked like, what they asked us to do, and how they taught us
This paper addresses the following question: How were teachers in the course able to use PeerAssisted Reflection (PAR) to engage with the four components of extreme apprenticeship?
I focus on three teachers that decided to use PAR from the course in their own teaching

Summary

Introduction

Who can be a mathematician? What is mathematics all about? How should mathematics be taught? We tend to answer these questions based on our experiences with our own mathematics teachers: what they looked like, what they asked us to do, and how they taught us. Prospective teachers generally experience a variety of pedagogies in their mathematics education courses These particular activities often contrast their regular mathematics courses, which are primarily lecture-based. This can cause problems, because prospective teachers have limited opportunities to experience innovative pedagogies as mathematics learners themselves. PAR helps students develop stronger self-assessment skills as they develop objective lenses from critiquing the work of their peers (Black, Harrison, & Lee, 2003; Reinholz, 2015b). In this course, teachers used PAR to annotate one another’s proofs and provide constructive feedback. PAR is a cross-disciplinary technique, and has been used in other disciplines such as biology, engineering, and physics (e.g., Reinholz & Dounas-Frazer, 2016)

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