Representational Implications for Understanding Equivalence

Teachers and researchers have long recognized that students tend to misunderstand the equal sign as an operator; that is, a signal for "doing something" rather than a relational symbol of equivalence or quantity sameness. Students' equal sign misconception has been researched for more than thirty years (Weaver, 1971, 1973) with little refinement in the theory. It was popularly believed that younger students were not developmentally ready to work variations of open numbers sentences, such as missing addend problems (Thompson & Babcock, 1978). In fact, misconceptions about the equal sign were identified in kindergarten students even before formal instruction (Falkner, Levi, & Carpenter, 1999). However, it is clear that with specific instructional guidance, elementary students can understand that the equal sign expresses a relation (Baroody & Ginsburg, 1983; Carpenter, Levi, & Farnsworth, 2000; Saenz-Ludlow & Walgamuth, 1998). In this article, the authors examine variables that could contribute to student's equivalence misconception and whether the equal sign misconception is still manifest in a U.S. sample and present in a Chinese sample. Findings indicate that misconceptions are still manifest in the U.S., and textbooks do little to mitigate the problem in the United States, while in China students are able to interpret the equal sign as a relational symbol of equivalence. The authors also found that the inclusion of multiple representations for equivalence in textbooks and guidebooks in China make a difference in assisting students to correctly interpret the equal sign.