Subject-Specific Academic Language Versus Mathematical Discourse
The significance of language for the learning of mathematics has long been thematised in mathematics education research. Since Austin and Howson provided the first overview of the state of research in 1979, the field has become more differentiated. The present article will discuss one area of research emerging from this differentiation—multilingual contexts. This example shows how mathematics and language as a research field has developed from dichotomous approaches towards the idea that the language of mathematics is characterised differently in different cultural and group contexts, thus emphasising discursive aspects. This trend gives rise to the question of how the individual resources of participants can be acknowledged and exploited in groups with different abilities, while simultaneously providing the participants with the necessary linguistic support to participate in the linguistic discourse of that group.
KeywordsMathematical discourse Academic language Multilingual context Social interaction Interactionistic approaches of interpretive classroom research
1 The Development of a Linguistic Perspective on the Learning of Mathematics
“special vocabulary used to name mathematical objects and processes”
“the development of dense groups of words such as lowest common denominator”
“the transformation of processes into objects”.
According to Morgan, specialised domains of activity have their own specialised vocabularies and ways of speaking and writing. Pimm (1987) makes reference to the existence of a mathematical register in the English language (or any other language). Registers are specialized uses and meanings of a specific language for mathematical purposes (e.g., specialized meanings and purposes for vocabulary (words, phrases or expressions) as well as grammatical structures) that can be chosen by an individual to fit a situation or a context. Thus, a register is clearly different from a dialect, which is usually limited to a specific geographical region. Using or developing a register, according to Pimm, is not only a question of using technical terms; it is also about using certain phrases and characteristic modes of arguing (p. 76). In this context, Pimm draws on Halliday’s (1975, p. 65) definition of a register as “a set of meanings that is appropriate to a particular function of language, together with the words and structures which express these meanings” (Pimm, 1987, p. 75). Halliday says about the “mathematical register” that one can refer to it “in the sense of the meaning that belongs to the language of mathematics” (Pimm, 1987, p. 76). The mathematical register, like any other linguistic register, consists of words, phrases, and expressions borrowed from the English language (or any other language) and of terms that are solely created to describe something that only exists in mathematical contexts and has no meaning or different meaning outside of those. Pimm elaborates the creation of a mathematical register. For him, one important possibility is using the role of metaphors as a tool to create meaning. Pimm speaks of extra-mathematical metaphors and structural metaphors, seen as the two “main sources of metaphor” (Pimm, 1987, p. 93) to construe the mathematical register. The former are used to “explain or interpret mathematical ideas and processes in terms of real-world events […], e.g. a graph is a picture”, while the latter involve “a metaphoric extension of ideas from within mathematics itself” (Pimm, 1987, p. 95). In accordance with Hymes (1972), Pimm attaches a particular importance to “communicative competence” (p. 4) in mastering a language or language style. This is the ability to use language in social situations, i.e., in context. Like Halliday (1989), Pimm sees the learning of language as a process that depends on the language not being isolated from the context; indeed, it is always in context that the language must be invested with meaning. Only through learning language in context do learners become able to apply different linguistic styles appropriately in the respective situation: “Communicative competence, then, involves knowing how to use and comprehend styles of language appropriate to particular social circumstances” (Pimm, 1987, p. 4).
Although the dependence of mathematical language on respective contexts was thus postulated at an early stage, the contrasting notion of the universality of this language persisted for a long time—and unfortunately can still be seen in many studies today. However, comparison studies carried out in various cultures consistently concluded that the notion of mathematics as a universal language, a language that could be learnt regardless of cultural or native-language influence, was far less frequently appropriate than often being suggested. Recognition of the culture- and thus also language-dependent nature of mathematics, i.e., that native speakers of mathematics in fact speak this “language” differently in different cultures or groups, began to gain traction in the context of the increasing influence of the “ethnomathematical” stance (cf. D’Ambrosio, 1985). This approach emphasises that mathematics is sensitive to cultural idiosyncrasies, including those related to language (cf. Morgan, 2014), which has been confirmed in recent studies. This development occurred on a background of what Lerman (2000) calls the “social turn” of research on teaching and learning in mathematics education. The social turn, which appeared around the end of the 1980s, describes a development within mathematics education research where “the social origins of knowledge and consciousness” (p. 8) have increasingly been taken into consideration. This is not meant to imply that social factors have previously been ignored by other theories like Piaget’s theory of learning or radical constructivism, as they saw social interactions as stimuli for meaning-making within an individual. However, social activity was now increasingly seen as the source producing meaning, thinking, and reasoning. Lerman suggests three primary disciplines which contributed to the development of the social turn: anthropology, sociology, and cultural psychology. Essentially, the social turn no longer meant studying a person and his or her meaning-making separately, but taking into account the person’s actions within a social practice. This trend turned mathematics education into a mature field of study in which serious efforts are now being made towards the theorisation and problematisation of components, concepts, and methods, including language. Increased sensitivity to the role of the social environment within which mathematics education takes place has inevitably meant greater attention to language and all forms of communication within mathematical learning environments (cf. Morgan et al., 2014). Apart from the studies by D’Ambrosio, those by Bishop (1988), Cobb (1989), Lave and Wenger (1991) and Wertsch (1981) should be mentioned here.
In German-language research, the social turn was brought into clearer focus within the discussion through diverse studies using interactionist approaches of interpretive classroom research, for example those by Bauersfeld, Krummheuer and Voigt (see, among others, Bauersfeld, Krummheuer, & Voigt, 1988; Krummheuer, 1995). These studies explicitly renounce the previously dominant view that learning was merely an internal psychological phenomenon. The social turn and the inclusion of interactionist aspects of learning and teaching meant a shift(ing) of focus from the structure of objects to the structures of learning processes, and from the individual learner to the social interactions between learners (cf. Bauersfeld, 2000).
The transformed understanding of learning led to the development of theories that see meaning, thinking and reasoning as products of social activity. These are evidenced, for example, in the above-mentioned interactionist approaches of interpretive classroom research. Thus, Krummheuer (1992, 2011) argues that learners are involved in “collective argumentations” in the learning of mathematics in primary school (Krummheuer, 1992, p. 143) and it is through their increasingly autonomous participation that they learn mathematics (cf. Krummheuer & Brandt, 2001). Similar ideas are reflected in recent work by Sfard (2008). Underlining the significance of language for subject-specific learning, she writes that thinking is a form of communication and that learning mathematics means modifying and extending one’s discourse. So, following Krummheuer (2011), mathematics learning can be described as the “progress” of participation emerging from the coordination of interpretations in collective forms of argumentation. Sfard (2008, p. 92) suggests replacing the notion of “learning-as-acquisition” with that of “learning-as-participation” (p. 92). Lave and Wenger (1991, p. 35) conclude that the start of this kind of acquisition process can be understood as a “legitimate peripheral participation” (in German, “legitime periphere Partizipation”). A learning process can then be described on the interactional level as a path from legitimate peripheral participation towards being a “full participant” (p. 37). Based on the fundamental assumption of these approaches, i.e., that meaning is negotiated in interactions between several individuals and that social interaction is thus understood as constitutive of learning processes, language can no longer be only understood as the medium in which meaning is constructed; rather, speaking about mathematics in collective argumentations is in itself to be seen as the “doing” of mathematics and the development of meaning. Thus, language acquires a central significance, if not the central significance in the building of mathematical knowledge and the development of mathematical thought.
analysis of the development of students’ mathematical knowledge
understanding the shaping of mathematical activity
understanding processes of teaching and learning in relation to other social interactions
Concerning the development of students’ mathematical knowledge, past research analysed the language of students drawing a direct link to their mathematical knowledge, viewing language as an unambiguous medium for transmitting ideas. However, this conception of language has been challenged on the basis of new insights about language and communication, based for example on the semiotics of Peirce, Wittgenstein’s idea of language games, and post-structuralist theories, which object to the existence of a stable relationship between a word and its referent. These theories have been integrated and developed further within mathematics education research, thus adding to notions of how mathematical learning can be understood.
One example of the inclusion of semiotics into mathematics education research is Steinbring’s (2006) theoretical concept of the epistemological triangle, which can be used to describe the way mathematical knowledge is developed. This is done by focusing on the relationship between representations of mathematical concepts (e.g., symbols, words, etc.) and the concepts themselves, as well as on how students’ previous knowledge and experience (“reference context”) mediates these relationships. The second subfield, which focuses on the shaping of mathematical activity, follows a different approach when looking at how mathematical knowledge is developed. Here Vygotsky’s perception of verbal language and other semiotic systems as tools affecting human activities is used as a basis. An example of this is the communicational theory of Sfard (2008), which does not differentiate between communicating in mathematical forms and doing mathematics/thinking mathematically. The way learners engage in mathematical activities can, therefore, be described by characterising the nature of mathematical language in detail. Using tools like Systemic Functional Linguistics or other tools which enable tracing the development of ideas or meanings within language practices help understand how students obtain their conceptions of mathematics which are carried into adulthood (e.g., Chapman, 2003). These kinds of analyses can, for example, help identify how people may be empowered with mathematics through attention to language (cf. Wagner, 2007) or through particular language practices. This stresses the significance of language aspects for learning mathematics not only for young learners in school but also for adults later in life. The third subfield is concerned with understanding processes of teaching and learning in social interactions. Here, studies in mathematics education are connected to the general notion which sees learning as a social activity and meaning-making no longer as the activity of an individual but of an individual within a social environment. By using tools for analysing classroom interaction which often originate outside of mathematics education—for example, in ethnomethodology or linguistics—patterns within social interactions can be identified. These patterns—for example the funnelling pattern identified by Bauersfeld (1988)—have proven to be useful when working with teachers. In Germany, other recent studies in this field of interactionist approaches of interpretive classroom research in mathematics education include, for example, those by Brandt (2013), Fetzer and Tiedemann (2015), Krummheuer (2011, 2012), Meyer and Prediger (2011), Schütte and Krummheuer (2013) and Schütte (2009, 2014). Specific mathematical issues which emerge when analysing interaction within mathematics classrooms include socio-mathematical norms (Yackel & Cobb, 1996) and forms of interaction which are specific to mathematics, like argumentation (Krummheuer, 1998; Planas & Morera, 2011) or group problem solving.
Primarily, however, I will focus on the last of these subfields. Up-to-date international comparison studies are underway in this area concentrating on lower academic achievement and reduced educational opportunities among children with a migration background in many European countries; the present migration flows into Europe have also brought it into the focus of the current social and scientific discussion. The following will examine in more detail the research efforts on multilingual contexts and the developments in the field, in order to determine the tasks such research needs to address.
2 Multilingual Contexts—From Deficit to Resource
In their overview of the research field of mathematics and language, Austin and Howson (1979) also address multilingual contexts. They mention various studies from the end of the 1960s and the start of the 1970s, which present a rather fragmented picture. Alongside studies that address the topic in the context of cultural and linguistic diversity conditioned by migration, they also find studies that focus on learning among minorities in school systems that are characterised by majority groups. Furthermore, they dwell in particular on diverse studies that concentrate on learners who are taught in developing countries not in their native language but in the respective administrative languages, for example English. Their results appear just as diverse as the data they are based on. They point to positive effects of a bilingual education in developing countries in particular, though these results are partially contradicted by data gathered in other countries. Overall, a rather positive picture of learning under bilingual conditions is presented. Some studies show that bilingual children may well be in advantageous positions when compared to monolingual children (e.g., Gallop & Kirkman, 1972). The following decades, however, saw a change in this positive perspective on learning under conditions of cultural and linguistic diversity.
Around the end of the 1970s, an approach developed by Cummins in the context of English second-language acquisition began to gain influence in the teaching and learning research community in relation to learning under bi- or multilingual conditions. Cummins’ (1979, 2000) differentiation between “basic interpersonal communicative skills” (BICS) and “cognitive academic language proficiency” (CALP) had a fundamental influence on the consequent discussion in mathematics education. In Cummins’ work, “BICS” refers to fundamental communication and language abilities in everyday communication and interactions. These are situation-dependent and are understood in an informal context. Meanwhile, “CALP” represents special school-related cognitive language knowledge and abilities, which are relevant for example in subject-specific discourse in the classroom. An essential aspect of Cummins’ concept is the idea that children can quickly gain abilities in their second language which they can use in everyday situations, but need significantly longer to achieve the competences in the academic language of the classroom that are required for academic success. With this conceptualisation, the positive perspective on cultural and linguistic diversity appears to fall away, with the focus placed instead on the fact that some children learn academically relevant linguistic competences more slowly than others, or not at all. This suggests a closeness to Halliday’s (1975) register-based approach, mentioned above, which also influenced Pimm (1987). Pimm translates this approach more for mathematics in general, but in using the metaphor of the native speaker of mathematics he also distinguishes between natives and non-natives in terms of deficits. However, he emphasises that more than speaking like a mathematician, the point is to learn “to mean like a mathematician” (p. 207); this suggests a closeness to the above-mentioned approaches which consider the negotiation of meaning in interactions to be significant for the linguistic learning of mathematics. Cummins’ perspective, meanwhile, clearly focuses on a kind of “target register” which all children must learn to use in order to successfully participate in mathematics teaching. In the following years, this approach has influenced the emergence in the research field of a perspective focusing on children’s linguistic deficits above all, looking at the differences between children’s abilities and the demands of the target register. In Germany, a discussion developed around children’s abilities in an academic language (cf. the concept of “Bildungssprache”, Gogolin, 2006) in schools, seen as abilities which children need to master in order to be academically successful, resonating with Cummins’ concept. A definitive characteristic of this academic language is its conceptual written form, which means it shows high information density and context-independence, and fundamentally does not correlate with the features of the everyday oral communication engaged in by many pupils (cf. Gogolin, 2006). Various authors point out that this independence from the present situation—often described in terms of the ability to decontextualise—represents a fundamental characteristic of the school discourse based on academic language (cf. Bernstein, 1996; Cloran, 1999; Gellert, 2011).
Current approaches in international research on mathematics education criticise this narrow view of mathematics learning from a linguistic perspective. Aukerman (2007), for example, argues that the ability to decontextualise language, i.e., to separate language from its context, as emphasised by Cummins, needs to be re-thought. The CALPS concept is focused on leading children step by step from context-oriented language towards a language that is more context-independent. Thus, context is seen as a transient step towards an academic learning. But for Aukerman, it does not make sense to talk about a decontextualisation of language. A language will always remain incomprehensible to a child if he or she cannot find a meaningful context; this is always determined by what the child already knows and trusts. Therefore, she suggests the alternative concept of “re-contextualisation.” According to this, children acquire language that they need in order to carry out a range of academic and non-academic tasks. To do this, they must use the support of linguistic competencies they already have (including the non-academic competencies) and transform these competencies in new contexts. For Aukerman, the task of the teacher is therefore primarily to support children in their “recontextualisations,” acknowledging the children’s ways of thinking, and using this starting point to work together with the children to make new academic material meaningful and relevant to them.
In the German-language literature, Prediger (2002) uses the approaches of Bauersfeld (1988) and Maier and Voigt (1991) to consider communication processes in mathematics learning as intercultural communication. Her basic position is that all mathematical communication with learners is intercultural communication. The teacher functions as a representative of the mathematical culture while the learners apply their everyday culture—this suggests a closeness to Pimm’s (1987) notion of native speakers of mathematics. Prediger distances herself from a deficit-oriented perspective, emphasising the importance of acknowledging differences among learners without imposing values. Further similarities with Pimm can be found in the way she sees the central barriers to communication not only in individual problems or problems emerging through interactions, but above all in problems rooted in the subject-specific culture of mathematics (Prediger, 2002, p. 1).
Another link to interactionist approaches of interpretive classroom research can be observed here, which Prediger acknowledges. According to Krummheuer (1992), participants in classroom interaction interpret activities in extremely diverse ways, based on their different abilities and backgrounds. Following this interactionist perspective of mathematical learning, an individual context-dependent meaning is negotiated for the content of the respective situation in the interplay between connected oral contributions and accompanying actions. This is referred to as the “situation definition” (Krummheuer, 1992, p. 22). Situation definitions are constantly adapted and transformed with other participants in the interaction in “negotiations of meaning,” such that the process of the production of interpretations is never concluded. The production of “simple” situation definitions does not necessarily lead to new learning, however.
Standardised and routinised individual situation definitions are termed by Krummheuer (1992, p. 24) as “framings”, drawing on Goffman’s concept of “frame” (1974, p. 19). Pupils’ framings are often not in agreement with those of other pupils or the teacher (cf. Krummheuer, 1992). In this perspective, teaching can be understood in terms of the “juncture of framings from two different interactional practices, which itself becomes practice” (ibid. p. 64, translated by the author). Differences between framings can be explained in Pimm’s (1987) terms as differences between interactional practices of native and non-native speakers, and in Prediger’s (2002) as differences between those representing the subject-specific culture of mathematics and those representing the everyday culture. It is only the fundamental transformation or construction of framings that represents a learning process here, not the transformation of situation definitions (cf. also Schütte, 2009). According to Krummheuer, differences in framings between the participants, which obstruct the production of collective argumentations but at the same time represent the motor of learning, need to be increasingly coordinated by the individual with advanced skills in the interaction (usually the teacher).
With her “situated-sociocultural perspective”, Moschkovich (2002) also emphasises the significance of discourse in the learning of mathematics. The perspective completes a switch from a consideration of obstacles and deficits of learners to one of resources and competences of a diverse pupil population (cf. Planas & Civil, 2013, as well as the contrary results of Meyer & Prediger, 2011). In this perspective, the learning of mathematics always takes place in a public social and cultural context, and represents a discursive activity. However, there is not one correct mathematical discourse that needs to be achieved, contrary to what approaches based on the concept of register often suggest. Learners participate in mathematical discourses in different communities, using diverse resources from different registers in order to communicate successfully mathematically (Moschkovich, 2002). In contrast to register-based approaches, the concept of mathematical discourse makes clear that interactional or non-language aspects must take a central role in the understanding of the learning of mathematics.
In mathematical learning situations, it is perhaps not surprising that comprehension problems can emerge specifically among children with relatively unschooled or multilingual backgrounds as a result of migration, in classrooms where teaching is not accomplished in their native tongue and is directed by a native speaker of mathematics (Pimm, 1987), i.e., a representative of the mathematical culture (Prediger, 2002). In the context of the current discussion on children’s linguistic competences in school, this could be explained on the basis of children’s linguistic deficits. Many authors see a solution in training children’s linguistic competences in order to redress the presumed linguistic deficits in academic language (Cummins, 1979, 2000; Gogolin, 2006).
Potential misunderstandings in learning situations can also be explained by different interpretations of situations based on differing framings among participants (cf., Aukerman, 2007; Krummheuer, 1992; Moschkovich, 2002; Schütte, 2014), underlining the interactive aspect of doing mathematics. Significantly, however, according to Schütte (2014), framings of situations that differ from the framing of the teacher (or adult advanced in the interaction) can be reconstructed not only in children with presumed linguistic deficits, but also in children with monolingual and relatively schooled backgrounds. This enables us to hypothesise that the framings of children with clearly differing linguistic competencies can nevertheless be very close to each other, and that the framings of children with less linguistic deficits could develop just as large a difference to the framing of the teacher. This would resonate with Pimm (1987) and Prediger (2002) in suggesting that a perspective focusing exclusively on children’s possible linguistic deficits comes up short. It is certainly desirable for all participating children to be introduced to formal and subject-specific mathematical language aspects, and for the teacher to act explicitly as a linguistic role model. But even when children have a linguistic role model, they need a teacher who engages with their interpretations and tries to modulate the basic framings of all participants to enable “learning despite differences”.
Due to the increasing diversity of pupil populations, situations of multiple different interpretations in negotiations of meaning in the classroom will become more and more prevalent. In spite of this diversity, children’s basic framings, emerging from an everyday life that is at least partially shared, might have more in common with each other than with the subject-specific framing of the teacher based on his or her training in mathematical education—and they need to be appropriately coordinated. The goal should be firstly to accustom teachers to such diversity of interpretations of taught content, and, building on this, to develop their interpretive competency to recognise differences based on different framings, thematise them in the learning process, and thus produce the possibility of modulation. This is not to argue that teachers should not provide children with linguistic support. However, it seems that one future task of mathematics teaching will entail using children’s linguistic resources positively, for example allowing them to switch into their first language during group work, as well as providing them with opportunities to build linguistic competences in the principal teaching language.
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