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The present correlational study examined third- and fourth-graders’ (N = 56) knowledge of mathematical equivalence after classroom instruction on the equal sign. Three distinct learning trajectories of student equivalence knowledge were compared: those who did not learn from instruction (Never Solvers), those whose performance improved after instruction (Learners), and those who had strong performance before instruction and maintained it throughout the study (Solvers). Learners and Solvers performed similarly on measures of equivalence knowledge after instruction. Both groups demonstrated high retention rates and defined the equal sign relationally, regardless of whether they had learned how to solve equivalence problems before or during instruction. Never Solvers had relatively weak arithmetical (nonsymbolic) equivalence knowledge and provided operational definitions of the equal sign after instruction.

An examination of third- and fourth-graders’ equivalence knowledge after classroom instruction