Assessing knowledge of mathematical equivalence: A construct-modeling approach.

Abstract

Knowledge of mathematical equivalence, the principle that 2 sides of an equation represent the same value, is a foundational concept in algebra, and this knowledge develops throughout elementary and middle school. Using a construct-modeling approach, we developed an assessment of equivalence knowledge. Second through sixth graders (N ϭ 175) completed the assessment on 2 occasions, 2 weeks apart. Evidence supported the reliability and validity of the assessment along a number of dimensions, and the relative difficulty of items was consistent with the predictions from our construct map. By Grade 5, most students held a basic relational view of equivalence and were beginning to compare the 2 sides of an equation. This study provides insights into the order in which students typically learn different aspects of equivalence knowledge. It also illustrates a powerful but underutilized approach to measurement development that is particularly useful for developing measures meant to detect changes in knowledge over time or after intervention.

Substitution and sameness: Two components of a relational conception of the equals sign

“…Further work is required to establish the extent to which our measure corresponds to others based on definition generation, and equation analysis and solving (Rittle-Johnson et al, 2011 Level Description 4. Comparative relational Explicitly view "=" as a relation signalling the same value on each side and able to define it as such.…”

“…Their understanding of the meaning of the equals sign is correlated with arithmetic competence and success at further mathematics (Kieran, 1981;Li, Ding, Capraro, & Capraro, 2008;Rittle-Johnson, Matthews, Taylor, & McEldoon, 2011). However, many young children, in Western countries at least, view the equals sign not as expressing a relation but as an operation meaning "work out the answer" (Baroody & Ginsburg, 1983;Behr, Erlwanger, & Nichols, 1976;Kieran, 1981;Knuth, Stephens, et al, 2006;RittleJohnson & Alibali, 1999).…”

“…Over the past three and a half decades researchers have presented increasingly detailed models of development from operational to relational conceptions (Baroody & Ginsburg, 1983;Behr et al, 1976;Carpenter, Franke, & Levi, 2003;Rittle-Johnson et al, 2011). The most recent model, by Rittle-Johnson et al, comprises four levels from operational through to relational, as shown in Table 1.…”

“…To explore this hypothesis we adapted the Conceptions of the Equals sign (CES) instrument from RittleJohnson & Alibali (1999), which has been demonstrated to identify children at different levels of development McNeil & Alibali, 2005;Rittle-Johnson et al, 2011). Our adapted CES included substitutive-relational items along with samenessrelational and operational items.…”

Inducing mental set constrains procedural flexibility and conceptual understanding in mathematics

“…The mathematical equivalence assessment was adapted from previous research (DeCaro & Rittle-Johnson, 2012;Matthews, Rittle-Johnson, McEldoon, & Taylor, 2012;Rittle-Johnson, Taylor, Matthews, & McEldoon, 2011). The assessment included four subscales (see Table 2): Conceptual Knowledge items evaluated both individuals' understanding of the meaning of the equal sign as a relational symbol and the structure of equations.…”