An Analysis of Third Grade Lessons Emphasizing a Relational Understanding of the Equal Sign Based on Sameness and Substitution

With the new standards for the “equal sign and equivalence relation” in grades 3 and 4 of the 2022 revised mathematics curriculum, there is a need to specifically explore and implement instructional practices that support a relational understanding of equality. In this study, we implemented three lessons emphasizing the relational understanding of the equal sign for third grade students and analyze the lessons. We developed and implemented teaching methods that emphasize both the sameness and the substitution meaning of the equal sign based on previous research. First, the teacher emphasized the meaning of substitution based on sameness, and students learned that one expression can be substituted for another expression based on the meaning that both sides of the equal sign are equal. Second, the teacher taught relational strategies by paying attention to the structure of equations as the object of reasoning, and students were able to use compensation and decomposition strategies to reason about equations. Finally, the teacher used relational strategies to work with equations with unknowns, such as creating different types of equations, and students were able to justify why the equations they created were true based on the meaning of the equal sign. Based on these findings, we discuss implications for instructional practices that can improve students' relational understanding of the equal sign.