The WMCS and its MDI framework
The WMCS project is a ten-year longitudinal research and PD project aimed at improving the quality of teaching, and, through this, learners’ access to mathematics in selected schools drawing students from low-income communities in one province in South Africa. The focus on teaching quality and access had its relevance in the South African education context, one which continues to be characterised by deep inequality, enduring high poverty levels, and related poor educational outcomes for students, particularly in low-income communities. The project began in 2010, as concern was growing nationally with the slow improvement, quantitatively and qualitatively, in educational outcomes, particularly in mathematics.
In 2010 we spent time in schools observing mathematics teaching and learning. At this time, of course, we were not newcomers to mathematics teaching and learning, nor to related research in schools. The description that follows reflects the deep ongoing interaction of the theoretical and empirical in our observations. As anticipated, expository (typically referred to as ‘traditional’) forms of teaching predominated. This was not surprising, nor are such practices unique to South Africa (Nachlieli &Tabach, 2019). At face value, the lessons observed across grades ranged from 30 to 45 minutes, and focused on a particular content. The topic was announced by the teacher, who proceeded with whole-class teaching to exemplify and explain some content, be it a mathematical idea, process and/or a procedure. This was followed by classwork and homework, typically on an example set from a textbook or other requisite materials. Learner participation in lessons was typically limited to individual or chorused responses to the teachers’ questions and copying of worked examples from the board.
Deeper analysis of the video recordings of these mathematics lessons showed, however, that they were characteristically incoherent.3 Students appeared to be ‘working’ (listening to the teacher, writing in their notebooks) and teachers were following a National Curriculum that recommended x number of weeks of instruction for a particular topic. In addition, there was a prescriptive Annual Teaching Plan that elaborated what was to be done in the x weeks in order to support curriculum coverage. Despite these resources, and bureaucratic monitoring, we struggled to interpret the intended mathematical message in a lesson. We puzzled over how specific mathematical goals prefigured lesson development activity for teachers.The mathematical object of learning was ‘out of focus’, despite an announced topic or content. We described this phenomenon as incoherence. There was no apparent mathematical story linking what learners were to know and be able to do. For example, in a four- part lesson ostensibly on multiplying algebraic expressions, different rules reliant on visible forms of the expressions were emphasized in the teacher’s talk in each part, fragmenting the story. The connecting thread of each being a product, and all linked to calculating products in some general way, was absent (Adler & Venkat, 2014). Additionally, there was widespread reading out of strings of symbols when teachers referred to mathematical objects and processes particularly in algebra lessons, with little attention to either naming or elaborating the operations and the objects being acted on. For example, a(b + c) = ab + ac was read out as “a bracket b plus c equals a b plus a c”.These observations reinforced the description of our initial observations in the project of mathematics lessons as incoherent. It was our sense that incoherence, fragmentation and disconnects within and between lesson parts reflected a contextual specificity that could account at least in part for enduring and pervasive poor mathematics attainment across the system.4
Initially, we worked with all secondary mathematics teachers in selected schools, but over time we focused in on Grades 8 and 9 and the teaching of core mathematics, particularly algebra and functions. The very low scores in the Grade 9 Annual National Assessments at that time were alarming, but not entirely surprising. Our own testing of learners revealed similar patterns (Pournara, Hodgen, Sanders, & Adler, 2016). Also, while most of the Grade 8 and 9 teachers we were working with were “qualified” this did not necessarily equate with them being “mathematically” prepared for teaching in these grades. Many had received poor-quality pre-service education (e.g. Welsch, 2002). In addition, there were ‘out of field’ teachers with some having being trained as primary teachers, and others qualified in different subjects. Underscored here are two dimensions of equity that interactively drove our work. First, and this is reflected in contexts of socioeconomic inequality elsewhere, is that learners in these schools had poor access to suitably qualified teachers. Second is that, at the level of teachers themselves, there was differential distribution of knowledge across teachers in the country.
Our major intervention, with our underlying equity agenda for teaching and learning, has been a mathematics for teaching course. Roughly two-thirds of the course provided opportunities for teachers to strengthen their mathematical knowledge and their relationship with mathematics in the first instance, and the remaining third focused on instructional practices. Called Transition Maths 1 (TM1), the course took place on the university campus, and stretched over sixteen days spread across the academic year. The course has run five times since 2012, with most of the 180 teachers who have completed it participating in the years 2016-2018. In the middle years of the project 2013—2016, we included a local version of collaborative school-based lesson study with some participating teachers. We focused on how their mathematical knowledge and instructional practices came to bear on planning, teaching and reflecting on specific lessons. I mention this here as the school-based work was critical in enabling us in the later years to refine how we worked with instructional practices in theTMl course.
The overall theoretical framing of our work is sociocultural, with the design of our interventions and related research drawing from key tenets of sociocultural theory. Mathematics is viewed as an interconnected network of scientific concepts. Mathematics teaching, therefore, is viewed as geared towards the mediation and appropriation of the increasingly sophisticated and increasingly general ways of thinking that constitute progression in the discipline (Vygotsky, 1978). Teaching as an activity is goal-directed and also always about something (Alexander, 2000). Bringing that something into focus with learners - its mediation - is the teacher’s work. We call this ‘something’ the object of learning5 - that which students are to know and be able to do. While in practical terms, it aligns with a lesson goal, it points intentionally to the mathematics that needs to be mediated and the related activity of the teacher. In line with previous research, mediational means, including language, are understood as cultural tools and/or resources in the practice of teaching (Adler, 2000,2001).
The committed research and development work in the project required that we elaborate what would count as improvement in the quality of mathematics teaching, and so access for learners, together with a trajectory towards such. Elsewhere, we have discussed our exploration of existing frameworks in the mathematics education literature and their limitations for our purposes (Venkat & Adler, 2020). At the surface level, we could describe almost all the lessons we observed as direct instruction, with a procedural and rule-bound approach to mathematics. But, in terms of opportunities made available to learn mathematics, the lessons were not all of the same quality. We needed a sociocultural framework that was sensitive to: (a) formalist (typically referred to as “traditional”) orientations to knowledge (Guthrie, 2011); and (b) expository orientations to instruction.The framework would then be more responsive to existing teaching practices, and disaggregating within and across. Additionally, such a framework needed to: (c) enable us to work with a developmental trajectory, guiding our intervention towards improving mathematics made available to learn in instruction.
The Mathematics Discourse in Instruction (MDI) framework we have developed is represented in Figure 5.1 and so named to foreground concern with the quality of mathematics made available to learn in school, and to highlight instructional practices that were key to this end.
In MDI, the key generative mechanisms for the work of teaching are exemplification, explanatory talk and learner participation (for details, see Adler & Ronda, 2015). What stands between (i.e. mediates) the object (here of learning) and the subject (the learner) are four key mediational means in mathematics classroom instruction - examples, tasks, names and legitimations - and, of course, how learners6 participate with these. While each of these four instructional, and ultimately
FIGURE 5.1 Constitutive elements of MDI (adapted from Adler & Ronda, 2015) cultural, tools has theoretical antecedents in the literature on research in mathematics education, their salience in the MDI is that they were resonant with on-the-ground practices, enabling us to use them separately and in combination to describe and/or work on the mathematics made available to learn in a mathematics lesson.
This framing has enabled us to describe and analyse what it is teachers do, and then to work with them developmentally on the mathematical quality of their teaching. Briefly, with a lesson as our unit of attention and analysis, we can produce a story of teaching that centres on identifying and/or articulating the object of learning, which, in any lesson, could be a concept, a procedure or mathematical practice, together with the relevant capability. Whatever this object and related capability, these need to be exemplified and explained. Exemplification will entail the selection of examples and associated tasks that can be used to bring the object of learning into focus with learners. Their elaboration unfolds through explanatory communication in the classroom, where language use includes both how concepts and procedures are named and how they are legitimated.
Elsewhere we have described how MDI operates as a boundary object in WMCS (Adler, 2017), moving flexibly across practices in the project (Adler & Ronda,2017a, 2017b) ‘bending’ to suit their purposes (Akkerman & Bakker, 2011). MDI is a living framework, with power lying in its iterative nature, moving between and supporting our research and development work and teachers’ teaching practices. We have operationalised it in our research so as to be able to describe and evaluate mathematics teaching (e.g. Adler & Ronda, 2015); it has framed the planning of professional development sessions in theTMl course (Pournara, Hodgen, Adler, & Pillay, 2015); and it was used as a tool for planning and reflection in our complementary lesson study work (Adler & Alshwaikh, 2019). We have also written specifically on the why, what and how of exemplification in our PD work (Adler & Pournara, 2020), and its take-up by teachers in their planning, teaching and reflection on lessons in lesson study activity.
I zoom in here onto the language-specific mediational means in the MDI, and so language-specific instructional practices. As I do so, it will become evident that our selections of and orientation to naming/word use (what is said and what is written in relation to the object of learning) and justifications of mathematics as specialised knowledge (what counts as mathematical) are empirically grounded in local teaching practices, as well as relevant research in the field of language and mathematics education.
