Example: Multilingual classrooms or second - language learners

Clearly, there are ditferences between people - but only some of these are named and spoken of as significant in relation to language in mathematics classrooms. One ditference of concern to many mathematics educators, including several of the authors in the current volume (see, in particular, Moschkovich (Chapter 4), Planas & Chronaki (Chapter 5), and Rangnes & Meaney (Chapter 12)), is difference in the ‘national’ or ‘home’ languages used by students and in classrooms.The range of linguistic resources available to an individual or group is a feature of the material world - but this material phenomenon may be spoken of in various ways.

Within the research domain concerned with the presence and use of more than one ‘national’ language, some researchers speak of bilingual, multilingual or monolingual students or classrooms. Others refer to ‘second-language learners’. While the first way of speaking orients us to think about the ways that multiple languages may operate together for individual students or within a classroom, the second way of speaking orients us to think about the need that such learners may have to acquire new language knowledge and skills. Such different orientations towards multilingualism have already been discussed within our field, contrasting deficit models of bilingualism with the idea of multiple languages as a resource for learning (e.g. Moschkovich, 2002; Planas & Setati-Phakeng, 2014) and raising issues about relationships between learning mathematics and acquiring a language that has political or economic power (Planas & Civil, 2013; Setati, 2008). Awareness of the consequences of using one term or another enables us to make an informed and explicit choice about which words to use.

It is important to note that speakers and writers do not generally make conscious and deliberate choices about which words to use. Most of the time we adopt the common ways of speaking within our social context without reflecting on our language practices. It is only when we are aware of alternatives and of the significance of their consequences that we take care in selecting our words. Researchers interested in language and mathematics education may have the awareness and freedom to choose deliberately whether to speak of multilingual classrooms or of second- language learners - though they may still not be conscious of how their choices may be interpreted by others. However, choices are not equally available for all or in all circumstances. In particular, it is worth considering how institutional practices constrain choice. Where official policy and curriculum documents refer to ‘second- language learners’, this is likely to be the way of speaking taken up by schools and teachers as they formulate their practices, making it more likely that they will orient their practice to expect such students to acquire the language of instruction rather than drawing on their existing language resources. More generally, if we researchers wish to communicate effectively with policy makers and practitioners, it can be a challenge to reconcile our messages, developed within the research domain, with the words of policy and practice. As soon as you change the words you change the ways in which the messages will be heard.

Communication as making functional choices

Thinking about choice brings me to the first set of theoretical and methodological resources I draw on to think about what happens in classrooms: Halliday’s social semiotics (Halliday, 1978), the associated tools of Systemic Functional Linguistics (SFL) and developments extending these ideas and analytical tools into the field of multimodal semiotics (Kress & van Leeuwen, 1996; O’Halloran, 2005). From this perspective, communicating using language or other semiotic systems involves selecting from a system of paradigmatic and syntagmatic options. That is, choosing (consciously or not) which words to use and how to connect them with other words in order to form coherent messages. The example above suggested how choices between words (multilingual classroom or second language learners) may affect the ways that teachers, policymakers and researchers orient their practices. Similarly, consider the possible effects of altering the ways such words are combined with others. For example, compare “I teach in a multilingual mathematics classroom” to “My mathematics classroom is full of second language learners”. Each of these statements could refer to the same material situation and might be construed as communicating a challenge for the teacher. However, the challenges are likely to be construed differently: attempting, in the first case, to operate effectively as a teacher within a complex environment or, in the second case, to accommodate alien or deficient students into a familiar setting.

A second important feature of social semiotics is that language and other modes of communication are seen as functioning in the world - not simply to convey ideas from a speaker to a listener but to play a constitutive role within social practices. Halliday’s categorisation of three metafunctions fulfilled by every utterance orients us to ask how classroom communications function ideationally to construe the objects and logic of mathematics, how they function interpersonally to construe social relationships within the classroom and relationships with mathematics and how they function textually as components of the social practice of the classroom (Halliday, 1978; Morgan, 2006).

When considering the ideational function as it is specialised to enable con- strual of mathematical objects, relationships and logic, Anna Sfard’s (2008) characterisation of mathematical discourse provides useful categories and tools that complement the more comprehensive and detailed grammar offered by SFL and multimodal social semiotics, highlighting the nature of mathematical objects and the forms of reasoning valued in academic mathematics. As has been argued elsewhere (Morgan & Sfard, 2016;Tabach & Nachlieli, 2011), fundamental assumptions of Sfard’s communicational theory and social semiotics are compatible, legitimising research that draws on both. Interestingly, Sfard’s theory also distinguishes between ‘mathematising’ and ‘subjectifying’ functions of discourse, paralleling the social semiotic distinction between ideational and interpersonal functions.