Representational Implications for Understanding Equivalence

Inferiors revolt in order that they may be equal; equals revolt in order that they may be superior (Aristotle, 322BC). 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 (Behr, Erlwanger, & Nichols 1980; National Council of Teachers of Mathematics [NCTM], 2000; Sáenz-Ludlow & Walgamuth, 1998; Thompson & Babcock, 1978). 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 developmental ly 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 the equal sign expresses a relation (Baroody & Ginsburg, 1983; Carpenter, Levi, & Farnsworth, 2000; Saenz-Ludlow & Walgamuth, 1998). These previous studies did not employ random selection, examine the phenomenon internationally, or explore how materials used with elementary teachers prepare them to teach the equal sign. A major benefit of international comparisons is that cross-cultural comparisons lead to more explicit understanding of one's own implicit theories about how children learn mathematics (Stigler & Perry, 1988). We examined variables that could contribute to students' equivalence misconception and whether the 86 equal sign misconception was still manifest in a U.S. sample and present in a Chinese sample. First, six Ü. S. methods books were chosen and examined to determine what strategies were being presented to prepare U.S. elementary preservice teachers (PTs) to teach equivalence and the equal sign to their future students. Strategies ranged from nothing at all (Smith, 2001 ), to a single paragraph (Cathcart, Pothier, Vance, & Bezuk, 2006; Reys, Linquist, Lamdbin, Smith, & Suyday, 2004; Van de Walle, 2004), to an activity (Tucker, Singleton, & Weaver, 2006). Seemingly, the authors of these textbooks expect that PTs understand the issues related to the equal sign and the implications for their students. Both Reys et al. (2004) and Van de Walle (2004) alert PTs to the common misconception that the equal sign means "the answer is next." Both authors dutifully inform readers that using the calculator reinforces the equal sign misconception since the answer comes after the equal sign is pressed. To counteract this misconception, a balance scale can help students develop the correct conceptual understanding of equality and the equal sign (Reys et al., 2004). Van de Walle (2004) suggests that teachers should use the phrase "is the same as" (p. 139) instead of "equals" as students read number sentences.