Introduction

Learning is a social and interactional endeavour, involving interactions between students and the teacher, between students themselves, and between all of these participants and the mathematics. By focusing on the process of learning itself in classroom interactions, the intersubjective negotations enabled by interactional structures and practices become visible. Detailed analysis of these interactional structures and practices, such as those offered in this book, can not only contribute to our understanding of the complex process of learning mathematics, but can also reveal opportunities to use and deviate from these structures.

Conversation Analysis

Conversation analysis offers an inductive approach to the analysis of classroom interaction. With its roots in ethnomethodology, conversation analysis is underpinned by some key principles that focus on how the learning of mathematics is made visible through teachers’ and students’ interactions. Using the tools developed by conversation analysts, the structures and patterns of interaction within mathematics classrooms can be described to reveal what it means to learn, and what it means to do, mathematics in school classrooms. This approach foregrounds what teachers and students themselves treat as learning and doing mathematics and reveals the multifaceted role of interaction in these processes.

Thinking, Understanding, and Knowing

Classrooms are all about thinking, knowing, and understanding. Epistemic issues are at the core of classroom interactions, yet teachers and students, as well as researchers, can treat knowing, thinking, and understanding in very different ways. Claims and demonstrations of knowing or understanding can achieve different actions in classroom interaction, which result in different meanings for what it means to know mathematics or understand mathematics in different classrooms. This negotiation of mathematical knowing or understanding is a theme of classroom interaction that needs further exploration.

Doing Mathematics

Learning mathematics is a way of acting. Mathematics is something that you do, not just something you know. Yet what it means to do mathematics can depend upon the norms and sociomathematical norms in each classroom. In turn, what it means to do mathematics in a classroom affects what it means to be a student or a teacher of mathematics. This doing of mathematics can be about communicating and problem solving, or it can be about remembering and knowing. Similarly, what it means to explain, argue, or justify varies between classrooms and the opportunities for students to do mathematics are constrained by what it means to do mathematics in each and every mathematics classroom.

Final Thoughts

Mathematics classroom interactions are both intricate and complex. They are contingent upon the contexts in which they occur but can both open up and close down opportunities to do mathematics in different ways. What it means to be successfully participating in the learning of mathematics can be challenged by considering how learning mathematics is contingently produced and negotiated. Conversation analytic approaches describe in detail the various, and often contrasting, paths that teachers and students follow as they interaction in mathematics classrooms. Yet there is so much more to discover about this process of mathematics learning as it happens in the classroom.

Trouble in Interaction

Making mistakes is part of learning mathematics but these mistakes can be handled in many different ways which in turn can lead to the process of making mistakes and learning from them very different. Using the conversation analytic idea of repair, and the associated preference organisation of repair, the implicit messages that the handling of mistakes can give is revealed. This structure around the preference organisation of repair is used by many teachers to encourage students to offer explanations and justifications that are a fundamental part of learning and doing mathematics.

Turn-taking

There is an almost universal pattern and structure to classroom interaction that both constrains what teachers and students can do, but also offers opportunities to use these structures to support the learning of mathematics. Turn taking structures that dominate learning interactions may control who can speak when and what can be said but turns can include a wide range of actions that constitute what it means to learn or do mathematics. It is through the deviations from these structures, such as when students speak ‘out of turn’, that these opportunities become clear. This reveals opportunities for teachers to work both within these structures, but also where deviations from these structures can also support learning. For example, by manipulating the wait time that can arise between turns in classroom interactions.