Dylan’s units coordinating across contexts
Introduction
Many elementary curricular standards now include an early focus on conceptions of fractions as measures (Lamon, 2007) alongside conceptual understanding of whole number arithmetic. For instance, the United States’ Common Core State Standards for School Mathematics includes both “represent and solve problems involving multiplication and division” and “understanding fractions as numbers [that can be plotted on a number line]” as third-grade objectives (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010, p. 21). Researchers have identified a shared cognitive necessity for understanding fractions and for conceptualizing multiplication (and division) – an ability to coordinate multiple levels of units (Hackenberg and Tillema, 2009, Steffe, 1992, Steffe and Olive, 2010).
Units coordination has emerged as an important construct for understanding mathematical thinking after elementary school as well. These domains include students’ writing of linear equations involving unknown quantities (Hackenberg and Lee, 2015, Olive and Çağalan, 2008), students’ ways of operating additively with signed quantities (such as integers; Ulrich, 2012), students’ combinatorial reasoning (Tillema, 2014), and teachers’ interpretations of fractions representations (Izsák, Jacobsen, de Arajuo, & Orhill, 2012). Meaningful attainment of middle and secondary school learning goals is likely to continue to present a challenge to teachers and students, as research suggests that many students enter sixth-grade yet to coordinate multiple levels of units (Boyce and Norton, 2016, Hackenberg, 2013, Hackenberg and Lee, 2015).
In this paper, we report on an investigation of how engagement in different mathematical situations might foster middle-grades students’ development of units coordination. More specifically, we consider an integrated development of whole number concepts and fractions concepts, with students’ development of structures that apply to both situations as an underlying objective. We focus our analysis on a single sixth-grade student, Dylan, who participated in paired-student constructivist teaching experiment (Steffe & Thompson, 2000). We analyze Dylan’s units coordination as he engaged with fractions and whole number tasks in discrete and continuous settings. We describe how Dylan progressed in his units coordination development even as he consistently approached fractions tasks by reasoning with whole number units instead of fractional units.
Section snippets
Theoretical and conceptual framing
We adopted a scheme theoretic perspective for our study (von Glasersfeld, 1995). In scheme theory, mental activity in service of a goal begins with the recognition of a salient situation; i.e., an individuals’ fitting of a situational goal to a scheme for which a sequence of available mental actions (operations) are expected to yield a satisfactory result (von Glasersfeld, 1995). Steffe (2001) describes units coordination as the “mental operation of distributing a composite unit across the
Purpose
In a previous teaching experiment (Boyce & Norton, 2014) focused on supporting sixth-grade students’ fraction scheme development, we noticed growth in one student’s units coordinating activity in the teaching session immediately following his successful coordination of (two levels of) units with whole numbers. He could confidently and correctly respond to fraction tasks that had engendered perturbation in previous sessions by coordinating two levels of fractional units. Though he did not
Methods
We report on the results of a constructivist teaching experiment (Steffe & Thompson, 2000) with a pair of sixth-grade students from the Southeastern United States. The two participants – Dylan and Maddie – were selected because they were available to meet after school and because they each were assessed as having two levels of (whole number) units interiorized during clinical interviews conducted by the authors as part of another study (Norton, Boyce, Phillips et al., 2015). Unlike Dylan,
Session 1: October 3, 2013
The first session began with a series of exploratory tasks with Cuisenaire rods (see Fig. 3). In this free play (Steffe, 1991), Dylan and Maddie suggested several relationships between the Cuisenaire rods’ sizes. Dylan reasoned that since five red rods fit into an orange rod and two yellow rods are the same length as five red rods, two yellow rods must fit into an orange rod. Dylan thus coordinated two levels of units: the orange as a unit of five red units.
Conclusions
In this paper, we have described results of a teaching experiment conducted with two sixth-grade students, Maddie and Dylan. The purpose of this study was to explore how varying the context of tasks involving composite units might support students’ development of units coordinating structures. The contexts included both whole numbers and fractions, and tasks were introduced in discrete and continuous settings with various manipulatives. We focused analysis on Dylan, who began the teaching
Acknowledgements
This work is supported by the National Science Foundation (NSF) under Grant No. DRL-1118571, the Institute for Society, Culture and Environment (ISCE) at Virginia Tech, and the Institute for Creativity, Arts, and Technology (ICAT) at Virginia Tech. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NSF, ISCE, or ICAT.
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