Developing mathematical language

Vygotsky (1978) stressed the importance of language as a psychological and cultural tool. He argued that inter-psychological or social activity between people, mediated through language, can trigger intra-psychological activity in individual learners. These developments constitute a learner’s intellectual processes. Culture can contribute to mental development by providing tools to think with. From the resources available in learning contexts, learners can acquire the means and processes of their thinking: means to think, what to think, and how to think. This highlights the role of interpersonal communication in education. Several researchers (Mercer and Sams, 2006; Morgan, 2000) saw classroom discussions as a vital medium for developing effective learning. However, pupils cannot on their own, without the teachers’ support, either align their everyday language use to the vocabulary in mathematical discourse or acquire the desired communication skills: they need guidance which is unfortunately lacking. In a wider context, mathematical language and vocabulary need to attain an international status and commonality of terms criss-crossing physical borders of English-speaking countries. Sharing on the world stage will always struggle with true respect unless traditional or conservative values are both recognised and adhered to in one universal framework. With Latin, Greek and Arabic words underlining the origin of mathematical language, there is no better time than now to establish them as sacred in teaching fraternities across the English-speaking world before the prevalent use reverts to parochial dialect, limiting its ability to gain wide acceptance and thereby devaluing the quality of presentation and instruction. Evoking the notion of ‘verbal thought’ (Vygotsky, 1962), Pimm (1987) stated that

articulation assists reflection processes by affording better access to thought itself. The power in ‘talking out loud’ is that it requires expressing thoughts through the use of language,

including specialist mathematical terms in this context, which may easily go unnoticed when thinking to oneself. Externalising one’s thoughts can help to reshape thinking: it allows clarity or difficulty in one’s thinking to be exposed through clarity or difficulty in

expressions. Mathematical language conveys meaning in the words used (Durkin, 1991) as the speaker draws upon the knowledge derived from previous exchanges that have lain dormant in the memory. To Pimm (1987), talking out loud helps the speaker to organize their thoughts: it externalises ‘thinking’, rendering it public and readily accessible to both the speaker and the listeners. Promoting mathematical talk in classrooms can enable

externalisation of a speaker’s thoughts or ideas. Both the speaker and the listeners weigh what is being said against their own individual thoughts or ideas. Where there is agreement between these ideas, ‘internalisation’ occurs (Cheyne and Tarulli, 2005; Pimm, 1987; Vygotsky, 1987). In establishing classroom cultures where the pupils are required to take turns in speaking whilst sharing their thoughts or ideas on a mathematical problem, the teacher may focus the pupils’ attention on their language use.

The foundation of learning mathematics lies in solid development of mathematical language. Durkin (1991) underscored the crucial role of language in mathematical learning as:

“Mathematics education begins and proceeds in language, it advances and stumbles because of language, and its outcomes are often assessed in language” (p.1)

According to Morgan (2000), there is an emerging consensus regarding the vital importance of the role of language in mathematics education. Pupils’ mathematical language may be developed through active encouragement of verbal formulations of their mathematical

understandings. Durkin (1991) argued that some words used in mathematics may be endowed with alternative meanings that may be more familiar to pupils. Alexander (2008) emphasised the importance of encouraging pupils to talk through their mathematics in order to

consolidate their learning. Also, Cheyne and Tarulli (2005) regarded ‘pupil talk’ as crucial to learning processes. Pimm (1987) suggested that pupils need to be made aware of why they are being encouraged to talk. On further externalisation of pupils’ thoughts or ideas, either

through talk during discussions with ICT, with non-ICT resources, or ideas written on paper whilst they engage in pen-and-paper tasks, the learning may then be assessed (Black et al, 2003). Morgan (2000) considered whole-class discussions as teacher-guided, meaning- making experiences that offer interpersonal gateways for student to appropriate mathematical meanings for themselves. However, this can only happen if the material is suitable for

discussion, and the participants are used to discussing, with each valuing the other’s views. Lee (2006) explained that introducing discussion into mathematics classrooms can facilitate the ‘enculturation’ of pupils into the wider mathematics community; Noss and Hoyles concurred. Morgan (2000) regarded the growing acceptance of discussion in classrooms as a vital medium for enhanced mathematical learning that endorsed communication in current curricula. Alexander (2008) stressed that discussions encourage richer forms of

communication, which emphasise active use of language by learners.

Mercer and Sams (2006) presented empirical evidence from a study with Year 5 (age 9-10 years) pupils working on a computer-based activity, which focused on language as the tool of choice. This evidence supported the sociocultural conception of mathematics education as successful induction of learners into a community of practice (Lave and Wenger, 1991). It demonstrated the role of talk-based activities in enhancing individual pupils’ mathematical reasoning skills. Sutherland et al (2004) discussed the effect of integrating ICT within subject cultures. They stressed that, with or without ICT, the teacher’s role is important in

influencing the construction of new pupil knowledge. According to Vygotsky (1981, 1978), knowledge is first developed between the individual and the society through cultural tools, then shifts to the cognition of the individual; Valsiner and van der Veer (2005) and Lerman (2001) reiterated this view. The teacher’s role is pivotal in creating an enabling learning environment. Teachers guide pupils’ awareness and their use of mathematical language as one tool for reasoning with. Emphasis on oral proficiency may be a crucial avenue through which teachers can access the pupils’ conceptual understandings. Developing pupils’ mathematical ideas and language use is contingent upon mathematics educators providing suitable tools or resources, in well thought-out situations, to create environments for pupils to actively engage with mathematics.