Abstract
This study reports an analysis of how pre-service teachers (n=34) made sense of fraction division with remainders using pictorial modeling strategies, and how small-group and whole-class discussion helped them develop conceptual understanding. One and a half class sessions were video recorded, and 12 interviews were conducted. Results indicate that pre-service teachers can develop a conceptual understanding of fraction division with remainders using modeling strategies, and their understanding emerges in three levels: a) level one: ignoring the remainder or labeling it incorrectly; b) level two: interpreting the remainder in the original unit but not relating it to the new unit; and c) level three: interpreting the remainder both in the original unit and the new unit flexibly.
Highlights
The Common Core State Standards for Mathematics (CCSSM) state that students should be able to “... solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem” (National Governors Association Center for Best Practices & Council of Chief State School Officers (NGA & CCSSO), 2010, p. 42). Sharp and Adams (2002) reported that 5th grade students who participated in their study developed solutions that include pictorial methods and symbolic procedures such as repeated subtraction, but none invented the invert and multiply procedure
This study focuses on how pre-service teachers (PSTs) develop an understanding of the remainder in fraction division during class instruction that emphasizes the use of pictorial modeling strategies
Existing research revealed that PSTs have a weak conceptual understanding of fraction division (Ball, 1990; Li & Kulm, 2008; Ma, 2010; Nillas, 2003), only a few studies examined how they interpret the remainder in fraction division
Summary
The Common Core State Standards for Mathematics (CCSSM) state that students should be able to “... solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem” (National Governors Association Center for Best Practices & Council of Chief State School Officers (NGA & CCSSO), 2010, p. 42). Sharp and Adams (2002) reported that 5th grade students who participated in their study developed solutions that include pictorial methods and symbolic procedures such as repeated subtraction, but none invented the invert and multiply procedure. Teachers’ own conceptual understanding of mathematical ideas is a prerequisite to teaching students for understanding (Ball & Bass, 2000). It is important for pre-service teachers (PSTs) to make sense of both pictorial and symbolic representations for fraction division (Lubinski, Fox, & Thomason, 1998). Fraction division has been identified as one of the most difficult concepts in elementary mathematics (Elashhab, 1978; Warrington, 1997) Many students learn this concept through the “invert and multiply” procedure without making sense of why the procedure works (Hanselman, 1997). There is evidence that teaching rules to students, especially before students have developed a conceptual understanding of rational numbers, hinders sense making (National Research Council, 2001; Wearne & Kouba, 2000)
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