Context Variation and Syntax Nuances of the Equal Sign in Elementary School Mathematics

Chronoula Voutsina

doi:10.1007/s42330-019-00067-5

Abstract

Existing research suggests that young children can develop a partial understanding of the equal sign as an operator rather than as a relational symbol of equivalence. This partial understanding can be the result of overemphasis on canonical equation syntaxes of the type a + b = c in elementary school mathematics. This paper presents an examination of context and syntax nuances of relevant sections from the grade 1 Greek series of textbooks and workbooks. Using a conceptual framework of context variation, the analysis shows qualitative differences between equations of similar syntax and provides a nuanced determination of contextual and structural aspects of ‘variation’ in how the equal sign is presented in elementary mathematics. The paper proposes that since equations have context-specific meanings, context variations should constitute a separate element of analysis when investigating how the equal sign is presented. The implication for practice and future research is that nuanced considerations of equation syntax within varied contexts are needed for elaborating analyses of the equal sign presentation that move beyond dichotomized categorizations of canonical/non-canonical syntaxes.

Highlights

The equal sign, as a symbol that is associated with mathematical equivalence, has a pervasive role across all levels of mathematics (McNeil et al, 2019) and is used in equations as a relation between mathematical objects that are the same and interchangeable (Jones et al, 2012; McNeil, 2008)
The adopted definition is that canonical equations are equations of the syntactical form a + b = c where the arithmetical expression is on the left side of the equal sign and is followed by its outcome (McNeil et al, 2019), and the equal sign is in the second-to-last position (Jones & Pratt, 2006; Powell, 2015)
While the quantitative exploration of the occurrence of different equation syntaxes across the textbook series confirms the predominance of numerical canonical equations and operation-related contexts that has been found in previous textbook research (e.g., McNeil et al, 2006; Powell, 2012), qualitative analysis reveals syntax nuances and great within-context variation that warrant attention

Summary

Introduction

The equal sign, as a symbol that is associated with mathematical equivalence, has a pervasive role across all levels of mathematics (McNeil et al, 2019) and is used in equations as a relation between mathematical objects that are the same and interchangeable (Jones et al, 2012; McNeil, 2008). Seo and Ginsburg (2003), for instance, have argued that formal instruction introduces the equal sign predominantly within number sentences and equations of a ‘canonical’ form such as a + b = __, where the operation appears only on the left side of the equal sign. They point out that children have very few opportunities to see and use the equal sign in other, ‘non-canonical’ formats (e.g., 5 + 5 = 7 + __). When asked to make judgements about the correctness of equations, children who have a predominantly operator view of the equal sign as a symbol that denotes ‘find the answer’ or ‘do the operation’ only recognize canonical equations as correct and find it difficult to assign a meaning to non-canonical equations (e.g., Kieran, 1981; McNeil et al, 2011; Stephens et al, 2013)

Methods

Results

Conclusion