Two conceptions of fraction equivalence

Abstract

In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: proportional equivalence and unit equivalence. These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a) the distinction of equivalence gives a unified framework of equal fractions that has not previously been described in the literature; (b) a conceptual understanding of both fraction equivalences is integral to understanding rational numbers; and (c) knowledge of both conceptions of equivalence is important for developing a conceptual understanding of fraction arithmetic. Past research has largely overlooked the distinction between the two types of equivalence. However, this may provide an important foundation for central topics that build on equivalence, and a better understanding of these two types of equivalence may support a more flexible understanding of fractions. Last, we propose future directions for teaching equivalence in mathematics.

Learning trajectory for equivalent fraction learning: An insight

“…Fraction is classified as a topic with a formal concept and an advanced level (Wilkins & Norton, 2018). It is also a mathematical concept with diverse representations (Pedersen & Bjerre, 2021). These factors cause fractions a complicated topic for students (Siegler et al, 2013).…”

“…These factors cause fractions a complicated topic for students (Siegler et al, 2013). One of the sub-topics in fractions considered complex is equivalent fractions (Pedersen & Bjerre, 2021). Whereas the equivalent fraction is a basic fraction concept that contributes to the addition and subtraction of fractions, approaches to the division of fractions, and the relationship between fractions and decimals (Kara et al, 2018).…”

“…This condition drives students to understand equivalent fractions only as a procedure of multiplying or dividing the denominator and numerator by the same number (Wong, 2010). According to Pedersen and Bjerre (2021), to understand equivalent fractions, students can compare several different fractions, but the fractions refer to the same relative amount.…”

Early career teacher’s approach to fraction equivalence in Grade 4: A dialogic teaching perspective

“…Fractions is not only one of the most important topics in mathematics, but also one of the most multifaceted (Pedersen & Bjerre, 2021). An important part of this domain of mathematics is fraction equivalence.…”

“…Put another way, learners tend to engage in an overextension of natural number principles (Obersteiner, Alibali, & Marupidi, 2020). Pedersen and Bjerre (2021) attribute this common mistake to the fact that learners' first encounter with quantities involves natural numbers in which every number represents a unique quantity. As a consequence, Ni and Zhou (2005) point out, when learners subsequently encounter rational numbers, seemingly different numbers describe the same quantity in the fraction notation.…”