Evolution of Unit Fraction Conceptions in Two Fifth-Graders with a Learning Disability: An Exploratory Study

ABSTRACT

The literature seems limited in what is known about conceptual processes that underlie evolution of students with learning disabilities (SLD) conceptions of fractions. This exploratory study examines how a foundational scheme of unit fractions (1/n) may evolve through the mathematical activity of two fifth grade girls. We analyze data segments from episodes conducted during a teaching experiment grounded in the activity of iterating estimates of one person's equal share. Our findings include four distinct conceptual stages: (1) No Conception of the Nature of Adjustment to the Magnitude of a Unit Fraction, (2) Evolving Anticipation of the Nature of Adjustment but not of its Relative Amount, (3) Anticipation of the Nature of Adjustment with an Evolving Partial Amount, and (4) a Dual Anticipation of the Nature and Amount of Adjustment. Findings demonstrate each girl was able to use her constructed scheme to successfully solve and reason about novel problems. We discuss the need for more research to confirm the findings from this study, while offering a conjecture of the possibilities for more SLDs to advance their conceptions of fractions in future interventions.

Notes

¹ von Glasersfeld‘s tripartite notion of scheme is a conceptual building block and an assimilatory structure consisting of: A situation (recognition template) that triggers one‘s goal in that situation, an activity executed to accomplish that goal, and a result (which may become predictable to the student).

² This postulation has not yet been studied empirically.

³ We do not claim here that the learning of SLDs and NAS is identical—only that we could build on research done with NAS to inform our work with SLDs.

⁴ A dual-discrepancy approach (Fuchs & Fuchs, 2006) is one of two predominant methods used in special education for diagnosing SLDs, where the difference between one’s score on an academic achievement test (here, mathematics) and her or his standard intelligence quotient (i.e., IQ score) is calculated. A student is labeled as LD in the tested content area if the discrepancy found (IQ > Math) exceeds one standard deviation. Due to privacy requirements, we do not have further information on the girls’ prior instructional opportunities, innate cognitive functioning (e.g., working memory), or special disabilities pertaining to fractions.

⁵ It should be noted that symbols for unit fractions (e.g., one-ninth or 1/9) were introduced during Episode 4 in the context of their activity of iterating a piece so-many-times (Tzur, 1999). For example, the researcher-teacher pointed out that one-fourth, or 1/4, is used to designate a unit that is iterated, and thus fit in the whole, exactly four times.

⁶ We use the term qualitative intuition in the sense proposed by Steffe and colleagues (2014).