A quantitative analysis of children's splitting operations and fraction schemes

Students’ partitive reasoning

“…The present study attempts to affirm and extend previous findings by addressing each of the hypotheses listed in Table 1. Norton and Wilkins (2009) found operational differences between students' partitive reasoning with unit fractions and non-unit fractions, despite finding no such differences in part-whole reasoning. We seek to corroborate that finding in addressing hypothesis 1.…”

“…Hackenberg (2007) has argued that "constructing a splitting operation is what allows students with partitive fraction schemes to reverse the operations of their partitive fraction schemes" (p. 46). Quantitative results from a preliminary study support this claim (Norton & Wilkins, 2009). Steffe (2010) further argues that splitting allows for multiplicative conceptions of fractions, whereas schemes that precede splitting, such as the PFS, are basically additive schemes.…”

“…Teaching experiments have produced numerous hypotheses regarding students' operational development of fractions knowledge, and initial attempts at quantitatively testing these hypotheses have largely affirmed those hypotheses (Norton & Wilkins, 2009;Wilkins & Norton, 2010). However, research on students' PFSs has yielded some conflicting hypotheses that do not seem to generalize as well across students or case studies.…”

“…We tested some of these hypotheses in a previous study (Norton & Wilkins, 2009), but results were tentative. The present study attempts to affirm and extend previous findings by addressing each of the hypotheses listed in Table 1.…”

“…We have previously documented operational differences between the PUFS and the PFS, as well as the hypothesis that the construction of the PUFS precedes that of the PFS (Norton & Wilkins, 2009;Wilkins & Norton, 2010). Based on this previous work, we again hypothesized this relationship (see Hypothesis 1).…”