Designing Lessons for Mathematical Problem Solving: Researcher and Teacher Collaborative Inquiry

• In Event: Using Lesson Study to Build Mathematical Problem Solving in Three National Contexts

Abstract

Objectives
Our study researches the initiation, development and sustaining of lesson study communities in England with the objective of improving teaching and learning of problem solving process (OECD, 2013).

Theoretical Framework
Our analysis draws on Cultural Historical Activity Theory (CHAT) (Engeström, 2001) to understand boundaries, boundary crossing and boundary objects (Akkerman & Bakker, 2011) in lesson study that involves collaborative enterprise between multiple activity systems. This analysis, carried out in regular cycles, has highlighted the importance of boundary objects that support lesson study communities at the stages of planning and enactment of, followed by reflection on action in, the research lessons central to the lesson study cycle. The form of boundary objects is not left to chance: rather, careful design of such objects is found to facilitate professional learning. Our collaborative inquiry with teachers has therefore sought to design for expansive learning (Engeström, 2001). In this design we have drawn on the notion of artefacts becoming instrumental in their use (Trouche, 2014) and the theory of instrumental orchestration (Drijvers et al., 2010). This leads to boundary objects that help shape both the thinking of the user (the instrumentation process) and are themselves developed for use in specific situations (the instrumentalisation process).

Methods

Researchers and teachers have collaboratively engaged in action research investigating how to improve and adapt a lesson study methodology that originates in Japan. We focus at the classroom level on the lesson study cycle with teachers and educators working in each lesson study group on one research lesson per term focussed on problem solving process skills. Our detailed participatory research involves an ethnographic approach throughout the lesson study cycle with collaborative reflection on emergent needs during post-lesson discussions and at regular intervals teacher workshops. The work is ongoing and has lasted for three years to date.

Data sources
To inform case study development all meetings in lesson study cycles as well as teacher/researcher workshops are audio/video recorded. Photographs of the production of teacher and student outputs are captured. Teachers and educators are interviewed at key moments in the developmental cycle. Documentary evidence including lesson plans, materials and student productions are collected. This allows case studies of research lesson cycles to be developed.

Results
A number of artefacts have been designed as boundary objects. These take forms such as documentary templates (Figure 1) as well as a range of pedagogical practices that assist professional learning as well as classroom activity in problem solving lessons. Cycles of our participatory research show these being effectively used by both ongoing and newly formed lesson study communities.

Significance
The study provides insight into how new models of professional learning in problem solving adapted from the Japanese model of lesson study can be developed, scaled and made sustainable. A number of artefacts that become instrumental in their use have been shown to facilitate both teacher learning and teaching of problem solving.

Authors

• Geoff David Wake, The University of Nottingham

• Malcom B. Swan, University of Nottingham