Teaching the substitutive conception of the equals sign

Abstract

A cumulative body of research has shown that children typically shift from an operational to a relational conception of the equals sign as they move through schooling. Jones (2008) argued that a truly relational conception of the equals sign comprises a substitutive component and a sameness component. Here we present two studies that build on this argument. The first investigated how the equals sign is typically presented to primary children in England, and we report that in the main an operational conception seems to be promoted. The second study measured the impact of a specially designed intervention on early secondary children's conceptions of the equals sign. Pre-and post-test data revealed that the intervention promoted substitutive and sameness components of symbolic equivalence. We consider the theoretical and pedagogical implications of the results.

The relationship between mental computation and relational thinking in the seventh grade

“…For example, without the explicit use of the equal sign, students could transform 49 + 51 into the equivalent expression 50 + 50, thereby making the expression more manageable to solve mentally. Despite the fact that the equal sign was not demonstrated while such mathematical transformations were shared and discussed during the intervention, the equal sign is implicit in this transformation, as the two expressions are equivalent and each can be substituted for the other [18]. In addition, students' scores on the relational thinking assessment improved immediately following the mental mathematics intervention.…”

“…When asked to think relationally about 99 × 3, for example, students can, and often do (e.g., [2]), use the distributive property by transforming 99 into (100 − 1), so the product can then be computed by subtracting 3 from 300 (i.e., 99 × 3 = (100 − 1) × 3 = 300 − 3 = 297). Such transformations rely on the notion of substituting 99 for (100 − 1), a key element to understanding the equal sign [18]. Thus, each transformation results in an expression that is mathematically equivalent to the first (in this case, (100 − 1) × 3 and 300 − 3 are both equivalent to 99 × 3).…”

“…In this way, a child performing 113 − 30 is essentially relying simultaneously on both notions of sameness and substitution: 113 − 30 is the same as and can be substituted for 113 − (10 + 10 + 10), which is the same as and can be substituted for 113 − 10 − 10 − 10. We argue, therefore, that choosing a mental computation strategy is an act of relational thinking because it relies on creating implicit transformations that rely squarely on both "sameness" and substitution principles of mathematical equivalence [18]. Once a child decides what strategy to use, he or she must then mentally perform the requisite arithmetical operations that follow from the selected strategy, which also rely on number knowledge, such as basic facts and place value [24,40].…”

Primary School Students' Understanding of Equation Structure and the Meaning of Equal Sign: A Chinese Sample Study

“…On the other hand, students' understanding of the equal sign is also influenced or even reinforced by the equation structure they experienced during primary school learning (Jones et al, 2013;McNeil et al, 2011). For example, equations with operations on both sides of the equal sign were found to be especially useful for the development of a relational understanding of the equal sign .…”

Polish teachers’ conceptions of and approaches to the teaching of linear equations to grade six students: an exploratory case study

“…This operational or procedural (Kieran, 1992) perspective frequently creates barriers when equations are introduced, not least because successful equation solving draws on a relational (Kieran, 1992) or structural understanding of the equals sign as an assertion of equality between two expressions (Alibali, Knuth, Hattikudur, McNeil and Stephens, 2007). While the transition from an operational to a relational 1 conception of the equals sign may be a function of learners' cognitive maturity (Alibali et al, 2007;Baroody and Ginsberg, 1983), focused interventions can facilitate this process (Saenz-Ludlow and Walgamuth, 1998), even after long exposure to operationally focused teaching (Jones, Inglis, Gilmore and Evans, 2013). In particular, even when textbooks encourage relational understanding, unless teachers explicitly draw attention to it, students may retain an operational perspective .…”