Co-construction of fractions schemes and units coordinating structures

Abstract

A growing body of research implicates students' ability to coordinate multiple levels of numerical units as an important aspect of their mathematical development. In this paper, we consider relationships between the ways students coordinate units with whole numbers (their multiplicative concepts) and the ways students coordinate units with fractions (their fractions schemes). Interviews with 50 sixth-grade students suggest commonality in the number of levels of units-within-units structures that students construct for the two contexts, consistent with Steffe's (2001) reorganization hypothesis. The results suggest that fractions may be a fertile domain for elementary and middle grades students to continue to develop whole number understandings.

The Connecting Quantities Process to Solve Fraction Mathematical Problems of Middle School Students

“…Therefore the bamboo would have a length of 6x (using coordination of addition or multiplication). The other type of unit coordination did not involve variables (Boyce & Norton, 2016, 2017Lobato & Siebert, 2002). In their research, Lobato & Siebert (2002) described an illustration from one of the participants (named Terry) in coordinating units from the results partitions.…”

“…Some research had evaluated the concept of fraction from the student-reasoning point of view (Baek et al, 2017;Boyce & Norton, 2016, 2017Hackenberg, 2007Hackenberg, , 2013Hackenberg et al, 2017;Hackenberg & Lee, 2015;Norton & Wilkins, 2009, 2010. Hackenberg (2007) studied unit coordination and fraction construction that required three-level unit interiorization.…”

Dylan’s units coordinating across contexts

“…These domains include students' writing of linear equations involving unknown quantities (Hackenberg & Lee, 2015;Olive & Çağalan, 2008), students' ways of operating additively with signed quantities (such as integers; Ulrich, 2012), students' combinatorial reasoning (Tillema, 2015), and teachers' interpretations of fractions representations (Izs k, 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 & Norton, 2016;Hackenberg, 2013;Hackenberg & Lee, 2015;Norton & Boyce, 2013).…”

“…The coordination of fractional units is fundamental to understanding fractions as measures, which is critical for students' development of proportional reasoning (Lamon, 2007), the rational number system (Norton & Wilkins, 2012) and the ability to reason with unknown quantities (Hackenberg & Lee, 2015). By the time they reach middle school, most students assimilate with the same number of levels of whole number units as levels of fractional units, but unfortunately many students remain at Stage 1, in which they require activity to coordinate even two levels of units (Boyce & Norton, 2016;Hackenberg, 2013;Hackenberg & Lee, 2015;Steffe, 2007). Some Stage 1 students' schemes for fractions situations instead involve partitioning without disembedding, resulting in a parts-within-wholes conception of a fraction as a ratio (Hackenberg, 2013).…”

“…Some Stage 1 students' schemes for fractions situations instead involve partitioning without disembedding, resulting in a parts-within-wholes conception of a fraction as a ratio (Hackenberg, 2013). Stage 2 students sometimes construct parts-out-of-wholes ratio conceptions that incorporate disembedding but not equi-partitioning (Boyce & Norton, 2016;Hackenberg, 2013;Norton & Wilkins, 2010). Though these students mentally -set apart‖ a part from a whole without modifying the whole, they unitize the part as a discrete unit of ‗1' prior to counting activity instead of a unitizing a size to be iterated.…”