Simon Goodchild

Mediating Mathematics Teaching Development and Pupils’ Mathematics Learning: The Life Cycle of a Task

A developmental research project in Norway, Learning Communities in Mathematics (LCM), a collaboration between university and schools, uses mathematical tasks as a basis for developing community in project workshops and for teachers’ design of tasks for classrooms. An aim in the project is that teachers and didacticians, through inquiry into design and use of tasks and reflection on and analysis of their use, will learn more about creating effective learning situations for pupils in mathematics. The processes involved are exemplified through an account of the design and use of the Mirror Task. An activity theory analysis traces the elements of learning of participants, teachers and didacticians, and highlights tensions, their nature and origins, in project activity and that of the established communities of school and university.

Theorising community of practice and community of inquiry in the context of teaching-learning mathematics at university

Introduction Biza, Jaworski, and Hemmi (2014) make a valuable contribution to theorising university mathematics teaching through the lens of community of practice theory (CPT) and by juxtaposing community of practice (CoP) and community of inquiry (CoI) within university settings. The paper has stimulated my thinking around the theoretical issues and I am pleased to have the opportunity to write this reaction. Ideally, this reaction would be just one link in a longer dialogue with the authors of the paper; appearing as it does as an afterword it could seem evaluative and judgmental – I do not want to engage in this type of critique. My purpose is to set out how the paper has provoked me to reflect on both development and application of theory. I do this by posing three questions: First, is it possible for an adapted community of practice theory (CPT) to offer all the constructs necessary to provide a theoretical underpinning of community of inquiry (CoI), without violating some of the a priori planks of CPT? Secondly, is it possible to create a genuine CoI with university students who are constrained in alignment to the teachinglearning practice chosen by their teacher within the context of strong institutional forces? Thirdly, does a community of inquiry theory (CIT) need to include constructs from cognitive and constructivist theories, requiring the swallowing of some paradigmatic camels? In the paper, Biza et al. first explain why they adopt sociocultural theory and more specifically CPT. From the introduction and second section it is evident that the authors intend the paper to focus on theory: “Our aim in this paper is to give more theoretical insight ... and to take this theorisation forward” (p. 161). They show how Wenger’s theorisation of CoP might be applied to university teaching and learning. The first of two Research Cases (Case 1) described in the second half of the paper demonstrates how this is achieved within an empirical study. Wenger’s mode of belonging, ‘alignment’, is specially scrutinised and questioned – Biza et al. explain that it is possible to be ‘critically aligned’ to practice, through adopting an inquiry stance. Many of the constructs or categories of Wenger’s theory are then re-examined and explained in the context of inquiry; hence the notion of CoI emerges. The second Research Case (Case 2) demonstrates how these adapted constructs are applied in developmental research. In

Commentary 1 on Reflections on Theories of Learning by Paul Ernest

Paul Ernest is an internationally recognised authority on the philosophy of social-constructivism particularly in the context of mathematics education. He has published widely on the issue, perhaps his two best known and widely cited works are ‘The Philosophy of Mathematics Education’ (Ernest 1991), and ‘Social Constructivism as a Philosophy of Mathematics’ (Ernest 1998). As one engages with this short paper it is evident that one is in the company of a ‘master’ of the topic. It is quite remarkable how within the space of about 4000 words he manages to produce an erudite and informative account of 4 related theories of learning, and outline some of their implications for teaching.