More than the same: A substitutive conception of the equal sign is associated with better algebraic reasoning Author Accepted Manuscript

This study investigated conceptions of the equal sign, understanding of equivalence, and algebraic reasoning among students in Grades 3 and 6 in the United States. Participants (total N = 165) rated three possible definitions of the equal sign as “good” or “not good,” including definitions focusing on (a) sameness (“two amounts are the same”), (b) substitutability (“the two sides can be swapped”), and (c) the equal sign as an operation (“the total”). Participants also completed assessments of equivalence understanding and early algebraic reasoning. Nearly all students who endorsed the substitutive conception of the equal sign also endorsed the sameness conception. The reverse, however, was not the case, suggesting the sameness conception precedes the substitutive conception developmentally and may be a foundation upon which the substitutive conception builds. We used multiple regression to examine whether definition endorsements were associated with performance on the equivalence and algebra assessments. Participants who endorsed a substitutive definition performed better on both the equivalence and algebra assessments than those who did not. Further, participants who endorsed both the sameness and substitutive conceptions outperformed participants who endorsed only the sameness conception on the algebra assessment. The findings suggest that the substitutive conception of the equal sign is distinct—at least in students’ thinking—from the sameness conception, and that a substitutive conception is associated with stronger algebraic reasoning in the elementary and middle grades. These findings highlight the need for more robust and nuanced measures of the substitutive conception of the equal sign.