Background
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). Falkner, Levi, and Carpenter (1999) reported in their investigation of the problem 8 + 4 = ? + 5 all 145 sixth-graders enrolled in the same school filled the box with 12 or 17. Apparently, students viewed the equal sign as a command to carry out the calculation such as a button on a calculator. Students may have had three different misconceptions: (1) thinking the answer comes next, that is, 8 + 4 = 12, ignoring the rest of the problem; (2) using all the numbers such as 8 + 4 + 5 = 17, arbitrarily restructuring the sentence; or (3) putting 12 in the box and extending the problem as 8 + 4 = 12 + 5 = 17 (Carpenter, Franke, & Levi, 2003). These misconceptions about the equal sign were common from grade one through six with findings indicating less than 10% of the students at each grade level answered this problem correctly (Carpenter et al., 2003).
Because the terms errors, bugs, and misconceptions are often used inconsistently or incorrectly in various mathematics education studies, it is necessary to delineate differences among them. Young and O’Shea (1981) provided clear interpretation of errors and faulty algorithms (bugs): “The ambiguity of problems highlights the need to distinguish carefully between, on the one hand, errors, i.e. actual wrongly answered problems, and on the other, faulty algorithms (or “bugs”), i.e., flaws in the program that generates the answers” (p. 156). On the other hand, misconceptions were students’ naïve explanations of concepts that were stable, robust, and resistant to instruction (Anderson & Smith, 1987). This view is consistent with that of Hammer (1996) who thought students’ misconceptions were strongly held, stable cognitive structures, affected by how students understand natural phenomena and scientific explanations, and must be overcome, avoided, or eliminated for students to achieve expert understanding. In this study, we use the term misconceptions similar to Capraro and Capraro (2005) who explained that regardless of whether the error is part of a developmental process that is, self-correcting as the student progresses through school or a more insidious pervasive type that is only corrected through direct intervention, the importance of the misconception is related to the immediate impact on understanding current content. Thus, we do not attempt to disentangle nuance differences among strength, stability, or persistence.