Formal and informal mathematical discourses: Bakhtin and Vygotsky, dialogue and dialectic

Abstract

The importance of the role of language/discourse in the learning and teaching of mathematics is noted in many mathematics curricula and standards documents. In the research literature, this role has been widely theorised from a Vygotskian perspective. This perspective is limited by some of its underlying assumptions, including an instrumental and systemic view of language as tool and its basis in dialectic. In this paper, I propose a Bakhtinian, dialogic perspective as an alternative. I focus my discussion on the long-standing issue of the relationship between formal and informal mathematical language in the learning and teaching of mathematics. I illustrate this discussion with an examination of interaction in an elementary school mathematics classroom in Québec, Canada. Based on Bakhtin's ideas, I argue that mathematical meaning emerges through locally produced, situated dialogic relations between multiple discourses, voices and languages in mathematics classroom interaction. From this perspective, students do not follow a linear path from informal to formal mathematical discourse; rather, they work with the teacher to expand the repertoire of possible ways to make meaning in mathematics.

Stimulating students’ academic language. Opportunities in instructional methods in elementary school mathematics

“…By negotiating ideas, students expand their mathematical meaning and they will be equipped to think creatively to take mathematics further in a dialogic approach (Bakker et al, 2015;Barwell, 2016). Students need to learn specific AL features before they can join the whole class mathematical discourse in a proper way (Bailey, 2007;O'Malley & Chamot, 1994;Sfard, 2012;Stein et al, 2008).…”

“…According to Gibbons (2002) interaction is at the heart of the learning process and it is a significant factor in language development. However, the way in which interaction in the discourse during mathematics instruction takes place differs (Barwell, 2016;Gibbons, 2002;Niederdorfer & Kroon, 2014;Nijland, 2011). In mathematics lessons a global shift to goals around higher levels of mathematical proficiency and problem solving is taking place.…”

“…Traditionally, a dialectic approach, where students try to find the right answer and where speakers arrive at an intersubjectivity of shared truth, is used (Langer-Osuna & Avalos, 2015). Barwell (2016) argues that through the multiple discourses that take place in the dialogic approach, where students reach understanding instead of a right answer by negotiating ideas, mathematical meaning emerges. Besides expanding their mathematical meaning, students will be equipped to think creatively to take mathematics further in a dialogic approach (Bakker, Smit, & Wegerif, 2015).…”

Language as resource: a key notion for understanding the complexity of mathematics learning

“…From here, I have argued and exemplified that in language, there is always the potential for shared meaning as well as the potential for reconstructing texts, in the immediacy of the interaction and within a range of options given by the context of culture. The dialectics discussed are not exhaustive of the diverse dialectics in the functioning of language in communication (see, e.g., Barwell, 2016a), but they suffice to facilitate newer ways of thinking about language as resource. Language emerges as a shifting resource, mutually co-producing learning of school mathematics and of broader societal discourses and cultures.…”

“…All these communities of reference and their related languages work to provide social identity to learners in the processes of learning school mathematics and of acquiring school languages and societal ways that will make them people with particular cultural and social capitals. Research in sociolinguistics under the influence of Bakhtin (1981) has also preceded and prepared adoption in the field of a notion of social language (Barwell, 2014(Barwell, , 2016a. Social languages express the forms of language through which people and groups can be recognized and linked to specific worldviews in a given situation and context of culture.…”