Alf T Coles

TEACHING STRATEGIES RELATED TO LISTENING AND HEARING IN TWO SECONDARY CLASSROOMS

The data for this study is taken from a research project [1] looking at year 7 students’ (aged 11–12) ‘need for algebra’ (Brown and Coles, 1999) in four teachers’ secondary classrooms. I introduce notions of evaluative, interpretive and transformative listening, (adapted from Davis, 1996), to analyse five transcripts taken from the lessons of two teachers on the project. The project design was informed by ideas of enactivist research (Varela, 1999, Reid, 1996, Brown and Coles, 1999, 2000). A significant change occurred in Teacher As classroom, as shown in three transcripts, and the listening of both students and teacher became transformative. There is evidence that specific teaching strategies were linked to this change and that once the change occurred the students started asking their own questions within the mathematics. The listening in Teacher Ds classroom was transformative from the start and there is evidence of the use of the same teaching strategies.

Planning for the unexpected in the mathematics classroom: an account of teacher and student change

This article focuses on ‘the unexpected’ in relation to teacher change. We take an enactivist approach to charting the parallel development of one teacher (Hannah) as she learnt to value and plan for unexpected events in her mathematics classroom, and one of her students as he learnt to notice and trust mathematical patterns. Shifts are juxtaposed not to suggest causality but to give a sense of the interplay over time of the teacher’s increasing focus on creative, co-produced, mathematical processes and all the unexpectedness that entails, and the student’s increasing sense of control over the subject. We analyse changes in Hannah’s teaching in relation to Gattegno’s notion of the subordination of teaching to learning and suggest that what changes for her was not centrally about new subject knowledge, but rather a new relationship to the unexpected (including, but not limited to, mathematics).

Returning to ordinality in early number sense: neurological, technological and pedagogical considerations

This chapter brings together recent research in neuroscience about the processing of number ability in the brain and new pedagogical approaches to the teaching and learning of number in order to highlight the significance roles of fingers and of ordinality in the development of early number sense. We use insights from these two domains to show how TouchCounts, a multitouch app designed for exploring counting and arithmetic, enables children to develop the symbol-symbol awareness that is characteristic of ordinality. We conclude by drawing out implications for further research making use of technology and neuroscience.

What is a Mathematical Concept

This presentation explores alternative approaches to the question: What is a mathematical concept? Philosophical and historical insights about the nature of mathematical concepts are discussed, with special attention to how concepts emerge and are established through particular mathematical practices. Such work shifts our attention to the material labour and onto generative nature of mathematical activity. New mathematical concepts emerge and old ones are creatively deformed when embodied practices redistribute what is considered sensible and perceptible. I discuss the pedagogical implications of this approach, and the important way such theoretical framing shifts our thinking about mathematics dis/ability. My aim is to rescue mathematical concepts from the staid curricular lists which entomb them, and to consider examples of how we might reanimate concepts in classroom activity.

SEEING PATTERNS: SOMATIC MARKERS IN TEACHERS’ DECISION-MAKING AND STUDENTS’ REASONING IN MATHEMATICS CLASSROOMS

Ideas of categorisation and pattern have been used in the past research of all three authors; on teachers’ complex decision-making (Brown and Coles, 2000) and students’ reasoning in mathematics classrooms (Reid, 1999, 2002). We illustrate these ideas with an analysis of two transcripts. As we reflect on how categories are formed it becomes clear that decision-making in the complex world of classrooms often takes place without time for reflection. Damasio (1996) develops what he calls his ‘somatic marker hypothesis’ to account for how people manage such decision-making. By ‘somatic marker’ Damasio means a bodily predisposition that informs decision-making. We then provide a second analysis of the two transcripts, illustrating how we currently observe the development of somatic markers through the language used and decisions made in classrooms.

Whole Number Thinking, Learning and Development: Neuro-cognitive, Cognitive and Developmental Approaches

This chapter focuses on the neuro-cognitive, cognitive and developmental analyses of whole number arithmetic (WNA) learning. It comprises five sections. The first section provides an overview of the working group discussion. Section 7.2 reviews neuro-cognitive perspectives of learning WNA but looks beyond these to explain the transcoding of numerals to number words. In the third section, children’s early mathematics-related competencies in reasoning about quantitative relations, patterns and structures are explored from new theoretical perspectives. Studies presented and discussed in working group 2 are presented in the following section as exemplars of intervention studies. The final section examines methodologies utilized in neuro-cognitive, cognitive and developmental analyses of children’s whole number learning. It discusses study designs and their potentialities and limitations for understanding how children develop competencies with whole numbers as well as task designs in cognitive neuroscience research pertinent to number learning. The chapter concludes with implications for further research and teaching practice.

Re-Thinking ‘Normal’ Development in the Early Learning of Number

In this article we suggest that, notwithstanding noted differences, one unmarked similarity across psychology and mathematics education is the continued dominance of the view that there is a ‘normal’ path of development. We focus particularly on the case of the early learning of number and point to evidence that puts into question the dominant narrative of how number sense develops through the concrete and the cardinal. Recent neuroscience findings have raised the potential significance of ordinal approaches to learning number, which in privileging the symbolic—and hence the abstract—reverse one aspect of the ‘normal’ development order. We draw on empirical evidence to suggest that what children can do, and in what order, is sensitive to, among other things, the curriculum approach—and also the tools they have at their disposition. We draw out implications from our work for curriculum organisation in the early years of schooling, to disrupt taken-for-granted paths.

Of Polyhedra and Pyjamas: Platonism and Induction in Meaning-Finitist Mathematics

This chapter considers the epistemology of mathematical concepts through classical Edinburgh School meaning finitism, a philosophical principle from the sociology of knowledge that is now several decades old. Meaning finitism is based on the idea that communities of people adaptively derive open-ended meanings and classifications from a finite basis of experience, choosing how to interpret the world by revising their past interpretations through social interactions. Using Barry Barnes’s metaphor for classification as hospitals issuing pyjamas alongside Imre Lakatos’s famous study of Euler’s theorem about polyhedra, I analyze the implications of this school of meaning finitism for the problems of epistemic induction (inferring about the future based on past experience) and Platonism (assuming the existence of ideal objects independent of concrete experiences) in the elaboration of complex mathematical concepts from simple models and examples.

The Role of the Facilitator in Using Video for the Professional Learning of Teachers of Mathematics

In this workshop we will address the following key questions: (1) how can and do facilitators guide work with mathematics teachers on video in a particular context?; (2) what are the principles, based upon research on teacher practice and teacher education, that guide our choices for teacher education and in particular our use of the video?; (3) what are the implications, for mathematics teacher learning, of different choices made by facilitators? The organisers will work actively with participants to demonstrate two ways of working (using the same video clip). We aim to share the detail of practice and how wider principles are enacted when using video. The first way of working we offer is based on principles derived from Jaworski (1990) and Coles (2013, 2014). The second way of working is based on principles derived from Horoks and Robert (2007), Chesné et al. (2009), Chappet-Pariès and Robert (2011), Robert and Vivier (2013). We are interested in learning from experiencing each others’ practice and hope that discussion grounded in the common experiences at the start of the workshop will be rich in connections. We will focus on similarities and differences in how work with video can be orchestrated (including the role of the mathematics) and work on how research could be taken forward into the role of the facilitator of the use of video.

Re-Looking at Teacher Discussions

Within enactivist methodology, there are two key ways to gain multiple perspectives on data. The first is to work on data with different people, the second is to consciously adopt a different theoretical frame for viewing. In this project, I engaged in both methods and report, in this chapter, on how re-viewing of data on teacher discussion served to trouble, or at least complexify, aspects of the role of the facilitator that I reported in Chapter 5. I begin by reporting on the outcomes of sharing two transcripts of lessons with the department of teachers and then report on what arose from my analysis of laughter during discussions of video extracts.