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

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 signaling the same value on both sides of the equation. We argue here that, in addition to 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 principal components analysis revealed that the substitutive items were distinct from the sameness items. Furthermore, Chinese children rated the substitutive items as 'sort of clever' or 'very clever', whereas 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.