A Substituting Meaning for the Equals Sign in Arithmetic Notating Tasks

We present 3 studies trialing arithmetic tasks that support both substitutive and basic relational meanings for the equals sign. The duality of meanings enabled children to engage meaningfully and purposefully with the structural properties of arithmetic statements in novel ways. In particular, they observed and exploited distinctive transformational effects when using different statement types to make substitutions of notation. We show how the initial version of the task emphasized the substitutive meaning but not the basic relational meaning, and an adapted version required both meanings of the equals sign to be flexibly coordinated. We found that some, but not all, children were successful at the adapted task and were able to connect making substitutions of notation with their existing mental calculation strategies.