 Academic journal article International Electronic Journal of Elementary Education

# An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm

Academic journal article International Electronic Journal of Elementary Education

# An Alternative Route to Teaching Fraction Division: Abstraction of Common Denominator Algorithm

## Article excerpt

Introduction

Arithmetic operations, and teaching and learning of them have always been an interest for mathematics education community. In his historical analysis, Usiskin (2007) pointed out that operations (especially on fractions) still preserve its importance in school mathematics and they should be given enough emphasis. Division is one such operation that has taken considerable attention by many researchers. The attraction to this operation is partly because of its complexity. This complexity is caused by the fact that division requires a meaningful organization of a variety of interconnected relationships (Thompson, 1993). In other words, division can be considered as a relationship between three quantities (dividend, divisor, and quotient) and an invariant relationship exists among these three quantities (Post, Harel, Behr, & Lesh, 1991). Here, the invariant relationship is meant to describe the multiplicative relationship between divisor and dividend, divisor and quotient, and dividend and quotient. Abstractly thinking about these relationships among the quantities in a division situation is difficult even for most teachers (Simon, 1993), which is one of the reasons why division takes considerable attention by many researchers.

Division is a complex operation to conceptualize and treatment of it within fractional domain makes it even more complicated for learners (Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992; Ma, 1999; Sowder, 1995). The fact that division of fractions require conceptual proficiency in both division and fraction concepts (Armstrong & Bezuk, 1995) makes this area of mathematics problematic in the upper elementary and middle grades. One of the reasons for such problem is the fact that fractions, as part of the rational number set, itself has several different interpretations (Kieren, 1993) and division acting on that set makes this area more problematic. Therefore, division of/by fractions deserves a special attention in school mathematics.

Even though this topic deserves a special attention in school mathematics, research studies point out that teachers' understanding of this topic is not strong enough and they are not well-equipped to teach it conceptually. Teachers' understanding of division in fractional domain is closely associated with remembering a particular algorithm, invert and multiply algorithm (Ball, 1990), which is very poorly understood (Borko et al., 1992; Zembat, 2007) and dependent on rote memorization without conceptual basis (Li & Kulm, 2008; Simon, 1993). Teachers are not able to provide concrete examples or any rationale for invert and multiply algorithm (Ma, 1999). In fact, making sense of such an algorithm and conceptualizing it using the inverse relationship between multiplication and division is very difficult (Contreras, 1997; Tzur & Timmerman, 1997). In spite of this, a majority of teachers use it as a primary way to teach their students division of fractions (Ma, 1999). Most of the traditional mathematics textbooks make their introduction to division of fractions with this algorithm too. When explaining her previous experiences on teaching fraction division with the invert and multiply algorithm, a participant teacher from Sowder and her colleagues' (1998) study commented that

"[...] one of my students said, "why do you flip it and why are we multiplying? Th is is division." And she [referring to the student teacher] says "Because I just told you to do it." And I sat there and thought, "Boy that was a wonderful question, and that was a very common answer." And I don't know how I would [...] have to [...] think about it to give more concrete examples." (p. 46)

Teachers' lack of necessary mathematical background to delve into the rationale for algorithms such as invert and multiply algorithm (hereafter abbreviated as IMA) is one side of the issue whereas feasibility of this algorithm is another. From a curricular stand point, the traditional IMA for division of fractions provides few affordances for linking to a rich understanding of fractions. …

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