No abstract

The goal of this study is to characterise how pre-service teachers recognise students' reasoning, based in the unitising process, when students solve ratio comparison problems and, how they consider the unitising process as a Key Development Understanding to propose teaching decisions. Ninety-one preservice primary teachers interpreted primary school students' answers to a ratio comparison problem and proposed teaching decisions to improve students' reasoning. Findings indicate that pre-service teachers recognised features of the unitising process as a Key Development Understanding but had difficulties to propose teaching decisions addressed to improve the students' mathematical reasoning. These findings display the role played by specific mathematical content domain when pre-service teachers make teaching decisions considering the students' mathematical reasoning.

A problem identified in the literature around task design is the persistence of a gap between teacher intention and student activity. We show how principles designed around the making of distinctions and having an explicit language of mathematical thinking can eliminate the ''gap'' by guiding teacher planning, teacher actions in the classroom and student activity in the classroom. We show how our task design principles have developed during the time of our collaboration, over a period of 20 years, across several research projects. We argue for the importance, in task design, of an explicit theory of change and an explicit image of mathematical thinking, where the theory of change is applicable to researchers, teachers and students.

In this article, we present a re-framing of teacher development that derives from our convictions regarding the enactive approach to cognition and the biological basis of being. We firstly set out our enactivist stance and then distinguish our approach to teacher development from others in the mathematics education literature. We show how a way of working that develops expertise runs through all mathematics education courses at the University of Bristol, and distil key principles for running collaborative groups of teachers. We exemplify these principles further through analysis of one group that met over 2 years as part of a research project focused on the work of Gattegno. We provide evidence for the effectiveness of the group in terms of teacher development. We conclude by arguing that the way of working in this group cannot be separated from the history of interaction of participants.

We illustrate and exemplify how the idea of reflection is framed by the enactive concept of "deliberate analysis". In keeping with this frame, we do not attempt to define reflection but rather work on the question of "how do we do reflecting?" within such a frame. We set out our enactivist theoretical stance, in particular pointing to implications for how we can learn from experience and showing the role of "deliberate analysis". We then describe, drawing on education literature, what is generally seen as the purpose of reflection and review some existing conceptualizations in mathematics education, pointing out where we draw distinctions. To illustrate how we do reflecting, we offer excerpts from two lessons of an expert teacher and the writing of a prospective teacher. We exemplify how reflecting as deliberate analysis leads to a way of working with teachers supporting them in handling multiple views and ambiguity, their actions being contingent upon their students' actions in learning mathematics.