This may be the author’s version of a work that was submitted/accepted for publication in the following source: English, Lyn & Watson, Jane M. (2018) Modelling with authentic data in sixth grade. ZDM - International Journal on Mathematics Education, 50(1-2), pp. 103- 115. This file was downloaded from: https://eprints.qut.edu.au/113835/ c Consult author(s) regarding copyright matters This work is covered by copyright. Unless the document is being made available under a Creative Commons Licence, you must assume that re-use is limited to personal use and that permission from the copyright owner must be obtained for all other uses. If the docu- ment is available under a Creative Commons License (or other specified license) then refer to the Licence for details of permitted re-use. It is a condition of access that users recog- nise and abide by the legal requirements associated with these rights. If you believe that this work infringes copyright please provide details by email to [email protected] Notice: Please note that this document may not be the Version of Record (i.e. published version) of the work. Author manuscript versions (as Sub- mitted for peer review or as Accepted for publication after peer review) can be identified by an absence of publisher branding and/or typeset appear- ance. If there is any doubt, please refer to the published source. https://doi.org/10.1007/s11858-017-0896-y 1 Title: Modelling with Authentic Data in Grade 6 First author: Lyn D. English Faculty of Education Queensland University of Technology Victoria Park Road Kelvin Grove Brisbane Queensland Australia, 4059 Email: [email protected] Phone: 617 31383329 Second author: Jane Watson University of Tasmania Faculty of Education Email: [email protected] Key words: modelling with data; data literacy; primary school; variation; uncertainty; informal inference; mathematisation; transnumeration 2 Modelling with Authentic Data in Sixth Grade Abstract This article explores 6th- grade students’ modelling with data in generating models for selecting an Australian swimming team for the (then) forthcoming 2016 Olympics, using data on swimmers’ times at various previous events. We propose a modelling framework comprising four components: working in shared problem spaces between mathematics and statistics; interpreting and reinterpreting problem contexts and questions; interpreting, organising and operating on data in model construction; and drawing informal inferences. In studying students’ model generation, consideration is given to how they interpreted, organised, and operated on the problem data in constructing and documenting their models, and how they engaged in informal inferential reasoning. Students’ responses included applying mathematical and statistical operations and reasoning to selected variables, identifying how variation and trends in swimmers’ performances inform model construction, recognising limitations in using only one performance variable, and acknowledging inform model construction, recognising limitations in using only one performance variable, and acknowledging uncertainty in model creation and model application due to chance variation. 1. Introduction and Background Statistical literacy is increasingly important in today’s society where data inform nearly all aspects of our lives. An ability to deal intelligently with such data is essential for a fulfilling and productive life in the community, school, workplace, and family (Engel, 2017; Gal, 2005). The notion of statistical literacy has been controversial for some time, with various definitions proposed, incorporating different aspects of informal and formal knowledge bases, dispositions, and attitudes (e.g., Gal, 2004; Gould, 2017; Watson, 2006). There has, however, been limited attention paid to primary school students’ statistical literacy, especially with 3 respect to how it might be developed in modelling with data. In this study, we consider statistical literacy to involve the ability to construct, interpret, reason, and communicate with data and data representations, make statistically sound decisions, and critically evaluate claims made in various contexts including the media. By undertaking their own investigations and generating different conclusions, primary school students can learn to make critical decisions with data, where variation and uncertainty are ever present. Yet these school students often do not receive the appropriate or adequate experiences that set them on the road to this statistical literacy. Developing statistical literacy takes a long time and must begin in the earliest years of schooling (English, 2014; Lehrer & Schauble, 2002; Watson, 2006). One approach to fostering this literacy with primary school students is through modelling with data. Although the term, modelling, has been used variously in the literature (e.g., Blum & Leiss, 2007; English, Arleback, & Mousoiludes, 2016; Gravemeijer, 1999; Kaiser, 2007; Lesh & Doerr, 2003), the linking of modelling with the development of statistical literacy has been limited especially in the primary school. As applied to the present study, modelling with data is a process of inquiry involving comprehensive statistical reasoning that draws upon mathematical and statistical concepts, contexts, and questions. The models produced should be supported by evidence and open to informal inferential thinking, with the latter including acknowledgement of uncertainty in model creations and applications arising from variation due to chance (Watson & English, 2015; Makar & Rubin, 2009). The aim of this article is to examine 6th- grade students’ generation of models for selecting Australian swimming teams for the (then forthcoming) 2016 Olympic Games, using data on swimmers’ competing times at various previous events. In studying students’ model generation, consideration is given to how they interpreted, organised, and operated on the 4 problem data in constructing their models, how they documented their models, and how they engaged in informal inferential reasoning, as previously defined. The problem was the final component of an activity implemented at the end of a three-year longitudinal study designed to develop 4th – 6th grade students’ statistical literacy, with a focus on informal inferential reasoning (Makar & Rubin, 2009). In reporting on students’ responses to the problem, we consider the following research questions: 1. How did students interpret, organise, and operate on the problem data in constructing their models? 2. What was the nature of the informal inferences students drew from their models? 2. Modelling with Data As used in this study, modelling with data differs from several other perspectives on modelling in that the focus is on developing both mathematical and statistical learning. As such, this modelling is an especially rich vehicle for developing foundational understandings that often are not addressed until the secondary school. An early introduction to modelling involving statistical learning is especially important in today’s society where studies of professional practice repeatedly highlight the complexity of the statistical world (e.g., Gould, 2017). Young learners need to be introduced to the rudiments of this complexity in ways that are pedagogically fruitful and tractable (Lehrer & English, in press; Lehrer & Romberg, 1996; Lehrer & Schauble, 2000). Modelling with data facilitates a balancing of this complexity and tractability but has been under-researched in the primary school years. The framework we address has four core components, namely, working in shared problem spaces (boundary interactions) between mathematics and statistics; interpreting and reinterpreting problem contexts and questions; interpreting, organising and operating on data in model construction; and drawing informal inferences. 5 2.1 Boundary Interactions in Modelling with Data As noted, our approach to modelling with data links children’s mathematical and statistical learning. Specifically, features of the statistical problem-solving framework of the Guidelines for Assessment and Instruction in Statistics Education (GAISE) Report (Franklin et al., 2007) are combined with aspects of research on the models and modelling approach (Lesh & Zawojewski, 2007; Doerr & English, 2003). A key feature of this combined perspective is the focus on both structure and variability (http://illuminations.nctm.org/Lesson.aspx?id=1189). As highlighted on the National Council of Teachers of Mathematics (NCTM) Illuminations website, “Statistical models extend mathematical models by describing variability around the structure” (p.11). As such, our framework supports the importance of “productive boundary interactions” or shared problem spaces between the two communities of practice in mathematics and statistics (Groth, 2015, p. 5). Two neglected problem spaces are variability and context, both of which can have different meanings in mathematics and statistics. Modelling with data provides one avenue for incorporating aspects of both meanings, in particular, by drawing on horizontal and vertical mathematisation (Freudenthal, 1991) and transnumeration (Wild & Pfannuch, 1999), as highlighted by Groth. Although (horizontal) mathematisation has frequently been identified as a key process in working modelling problems (Niss, 2010), “vertical mathematisation” has received less attention. Vertical mathematisation is applied when students look beyond the specific problem context to consider the general mathematical structure/s generated,