Given its important role in mathematics as well as its role as a gatekeeper to future educational and employment opportunities, algebra has become a focal point of both reform and research efforts in mathematics education. Understanding and using algebra is dependent on understanding a number of fundamental concepts, one of which is the concept of equality. This article focuses on middle school students' understanding of the equal sign and its relation to performance solving algebraic equations. The data indicate that many students lack a sophisticated understanding of the equal sign and that their understanding of the equal sign is associated with performance on equation-solving items. Moreover, the latter finding holds even when controlling for mathematics ability (as measured by standardized achievement test scores). Implications for instruction and curricular design are discussed.

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- Author or Editor: Martha W. Alibali x

### Eric J. Knuth, Ana C. Stephens, Nicole M. McNeil and Martha W. Alibali

### Eric J. Knuth, Martha W. Alibali, Shanta Hattikudur, Nicole M. McNeil and Ana C. Stephens

The equal sign is perhaps the most prevalent symbol in school mathematics, and developing an understanding of it has typically been considered mathematically straightforward. In fact, after its initial introduction during students' early elementary school education, little, if any, instructional time is explicitly spent on the concept in the later grades. Yet research suggests that many students at all grade levels have not developed adequate understandings of the meaning of the equal sign (Baroody and Ginsburg 1983; Behr, Erlwanger, and Nichols 1980; Falkner, Levi, and Carpenter 1999; Kieran 1981; Knuth et al. 2006). Such findings are troubling with respect to students' preparation for algebra, especially given Carpenter, Franke, and Levi's (2003) contention that a “limited conception of what the equal sign means is one of the major stumbling blocks in learning algebra. Virtually all manipulations on equations require understanding that the equal sign represents a relation” (p. 22).

### Caroline (Caro) Williams-Pierce, Elizabeth L. Pier, Candace Walkington, Rebecca Boncoddo, Virginia Clinton, Martha W. Alibali and Mitchell J. Nathan

In this Brief Report, we share the main findings from our line of research into embodied cognition and proof activities. First, attending to students' gestures during proving activities can reveal aspects of mathematical thinking not apparent in their speech, and analyzing gestures after proof production can contribute significantly to our understanding of students' proving practices, particularly when attending to dynamic gestures depicting relationships that are difficult to communicate verbally. Second, directing students to produce physical actions before asking them to construct a mathematical proof has the potential to influence their subsequent reasoning in useful ways, as long as the directed actions have a relationship with the proof content that is clearly meaningful to the students. We discuss implications for assessment practices and teacher education, and we suggest directions for future research into embodied mathematical proof practices.