Substitution and sameness: Two components of a relational conception of the equals sign

Abstract

A sophisticated and flexible understanding of the equals sign is important for arithmetic competence and for learning further mathematics, particularly algebra. Research has identified two common conceptions held by children: the equals sign as an operator, and the equals sign as signalling the same value on both sides of the equation. We argue here that as well as these two conceptions, the notion of substitution is also an important part of a sophisticated understanding of mathematical equivalence. We provide evidence from a cross-cultural study in which English and Chinese children were asked to rate the "cleverness" of operational, sameness and substitutive definitions of the equals sign. A Principle Components Analysis revealed the substitutive items were distinct from the sameness items. Furthermore, Chinese children rated the substitutive items as 'very clever', whereas the English children rated them as 'not so clever', suggesting that the notion of substitution develops differently across the two countries. Implications for developmental models of children's understanding of equivalence are discussed.

Grade 9 Learners’ Structural and Operational Conceptions of the Equal Sign: A Case Study of a Secondary School in Soshanguve

“…Most research has documented that many elementary and middle school learners demonstrate inadequate understanding of the meaning of the equal sign, frequently viewing the symbol as an announcement of the result of an arithmetic operation rather than as a symbol of mathematical equivalence (Hohensee, 2017;Jones et al, 2012). Jones et al (2012) argue that learners should learn the meaning of the equal sign as being an equivalence relationship between two objects, such as numbers or expressions.…”

“…Jones et al (2012) argue that learners should learn the meaning of the equal sign as being an equivalence relationship between two objects, such as numbers or expressions. In this instance, the notion of the equal sign is correlated with an arithmetic competence, and assists learners to master algebra and other mathematical concepts (Matthews, Rittle-Johnson, Taylor, & McEldon, 2012).…”

“…In this instance, the notion of the equal sign is correlated with an arithmetic competence, and assists learners to master algebra and other mathematical concepts (Matthews, Rittle-Johnson, Taylor, & McEldon, 2012). Jones et al (2012); Li, Ding, Capraro, and Capraro (2008) and Matthews et al (2012) postulate that learners should have a flexible understanding of the equal sign for arithmetic competence and algebra. Essien and Setati (2006) concur that the equal sign is used to indicate the relationship between quantities or values in mathematics.…”

“…The equal sign has been defined and introduced differently in textbooks around the world (Jones et al, 2012). Studies conducted in the United States of America (USA) and China on teachers' guidebooks or textbooks were compared and it was found that, in the United States, the equal sign is rarely defined and is often used interchangeably with computations to obtain the answer (Li et al, 2008).…”

“…Much thought and research have gone into exploring learners' understanding of the equal sign in elementary and middle schools (Essien & Setati, 2006Hattikudur & Alibali, 2010Hohensee, 2017, Jones, Inglis, Gilmore, & Dowens, 2012Stephens, Knuth, Blanton, Isler, Gardiner, & Marum, 2013). Linchevski and Herscovics (1996) argue that teachers and learners should be made aware of the demarcation between arithmetic and algebra.…”