Students’ Language Repertoires for Prediction

Abstract

The understanding of possibility, risk, and certainty, like the understanding of any mathematical idea, is mediated by language. Certain language repertoires are necessary to convey the ideas. At the same time, the language used to describe these ideas shapes the way people conceptualize them. This recursive nature of language compelled us to develop a research project to investigate children's language repertoires in relation to conjecture. Having noted similarities in the language of conjecture and of prediction, we structured the classroom activities and interviews in the project to prompt students to make predictions. In this paper, we focus on our research choices in relation to this endeavour. First, we describe choices we made to gain insight into children's language repertoires. Second, we use some of the data from the project to identify issues relating to interpreting data in the characteristically mathematical contexts of conjecture and prediction.Moving beyond our academic interest in mathematics education, we will argue that the issues we identify may be significant for understanding everyday experience. In particular, we will raise questions about the impact of mathematics class experiences that involve uncertainty on experience outside the classroom. We will also raise questions about the impact of intertextuality between uniquely mathematical ways of communicating about conjecture and everyday ways of interacting about authority.The investigation of conjectures (hypotheses) is one of the most important mathematical processes. Much mathematics teaching focuses on enabling students to perform particular mathematical procedures, such as adding fractions, factoring polynomials, and calculating probability. These skills appear as standards in curriculum documents and frameworks (e.g., CCSSO, 2010) that are used by curriculum planners and teachers. Research and professional literature, including curricula (e.g. New Brunswick Department of Education, 2010) and curriculum frameworks, point to the necessity of students learning these intended outcomes through the exploration of mathematical problems.When people explore a mathematical problem together, as with mathematical investigations in classrooms, it is necessary to have a way of suggesting an idea before knowing it is true. Rowland (2000) noted the centrality of such conjecture to mathematics, and coined this "space between what we believe and what we are willing to assert" (p. 142) as the Zone of Conjectural Neutrality (ZCN). Because of the recursive relationship between language and experience, the language resources available affect the possibilities for making conjectures.As our research exemplifies, the language of conjecture shares language that describes probability. Rowland's work refers often to the necessity of expressing uncertainty for conjecture, and he draws heavily on linguistics literature that describes the way people express uncertainty. Our research illustrates the complex relationship between probability, itself an important mathematical concept, and conjecture, which is at the heart of teaching for understanding.Our interest in language is not aimed to identify correct language. Rather it focuses on the language students use and asks what their language choices might tell us about the way they think about uncertainty. There is a range of English words that relate to uncertainty. Mathematics educators are likely to have particular ideas of what the words mean, which would differ from ideas of others. For example, in addition to everyday use of the word 'risk,' the concept has been studied in the fields of mathematics, psychology, business, and engineering. We find a general consensus that it references probability and uncertainty, especially as they relate to (perceived) consequences (e.g., Slovic, 2000). Our focus in this article is on the meaning and meaning-making we observe while students are confronted with uncertainty. …