# Mathematical Discourse in Instruction (MDI) and its Mathematics Teaching Framework (MTF)

Opportunity to learn mathematics in school depends on skilful teaching, crucially the mathematical discourse in instruction (MDI) that evolves in a lesson, and over time across lessons. Of course, learners’ access to a set of resources – the means through which they can participate in mathematical discourse (i.e. learn) – goes beyond teaching. In WMCS schools, the quality of instruction is key, and mediated by the teacher.

MDI is constituted by two commonplaces of mathematics teaching: **exemplification** and accompanying **explanatory communication**. The quality of MDI is reflected in levels of coherence and connection within and between the example and explanation spaces teachers set up, and how these mediate (or connect learners to) mathematical object(s), particularly their **generality and structure**. The MDI analytic framework, operationalised for researching teaching, is described in Adler & Ronda (2015). The underlying assumption in WMCS is that MDI rests in the first instance on teachers’ mathematical knowledge for teaching, that is their subject matter and pedagogical content knowledge.

Through a range of qualitative studies we are investigating

- the relationship between
**exemplification and explanatory communication**in a lesson (Adler & Venkat, 2014; Venkat & Adler, 2012; Adler & Ronda, 2017a), and the development of these aspects of MDI through Lesson Study (Adler & Ronda, 2017b). - whether and how the MDI in our teachers’ classroom shifts over time, in what ways, and how MDI is related to what is made available to learn in school; (Adler, 2017; Ronda & Adler, 2019)
- the use of the MDI framework for identifying developmental processes of
**building generality**in and across lessons (Venkat & Adler, forthcoming, ERME book chapter); and for studying textbook lessons (Ronda & Adler, 2017) - whether and how teachers identify with and so learn from and with the
**ideational resources (ideas about mathematics, and ideas about mathematics teaching)**embedded in the MDI; (Ntow & Adler, 2019) - How teachers navigate between their learning related to MDI in their professional development work, and their practices in their school context (ongoing PhD studies)

The **Mathematics Teaching Framework (MTF)** is the teaching practice version of MDI, renamed and redescribed to suit teaching practices. It similarly focuses on key elements common to all teaching practices: identifying and articulating a lesson goal, designing and selecting examples sets, selecting representations, selecting and designing tasks, producing explanations and justifications, and building opportunities for meaningful learner participation in lessons. The MTF is explicitly mediated in the Transition Mathematics 1 (TM1) course for teachers, and is used to guide our **Lesson Study** work with teachers in schools.

Additional qualitative studies have investigated

- How the MTF, particularly
**exemplification**, is mediated in teacher education practice (Adler & Pournara, forthcoming) **Evolving example sets**through Lesson Study structured by the MTF (Adler & Alshwaikh, 2019)- Researchers and teachers’ collaboration in Lesson Study structured by the MTF (Alshwaikh & Adler, 2017)
- Pre-service teachers’ learning to teach mathematics in a community of practice using the MTF as a resource (ongoing PhD study).