Benefits of Practicing 4 = 2 + 2: Nontraditional Problem Formats Facilitate Children’s Understanding of Mathematical Equivalence Nicole M. McNeil, Emily R. Fyfe, Lori A. Petersen, and April E. Dunwiddie University of Notre Dame Heather Brletic-Shipley Pleasant Union Elementary School This study examined whether practice with arithmetic problems presented in a nontraditional problem format improves understanding of mathematical equivalence. Children (M age = 8;0; N = 90) were randomly assigned to practice addition in one of three conditions: (a) traditional, in which problems were presented in the traditional ‘‘operations on left side’’ format (e.g., 9 + 8 = 17); (b) nontraditional, in which problems were presented in a nontraditional format (e.g., 17 = 9 + 8); or (c) no extra practice. Children developed a better understanding of mathematical equivalence after receiving nontraditional practice than after receiving tradi- tional practice or no extra practice. Results suggest that minor differences in early input can yield substantial differences in children’s understanding of fundamental concepts. Decades of research in cognitive development and mathematics education have shown that children struggle to understand mathematical equivalence, particularly in symbolic form (e.g., Alibali, 1999; Baroody & Ginsburg, 1983; Behr, Erlwanger, & Nichols, 1980; McNeil, 2008; Renwick, 1932). Mathe- matical equivalence is the relation between two quan- tities that are interchangeable (Kieran, 1981), and its symbolic form specifies that the two sides of a math- ematical equation are equal and interchangeable. Mathematical equivalence is arguably one of the most important concepts for developing young chil- dren’s algebraic thinking (Falkner, Levi, & Carpen- ter, 1999; Knuth, Stephens, McNeil, & Alibali, 2006). Difficulties with mathematical equivalence are most apparent when children are asked to solve equations that have operations on both sides of the equal sign (e.g., 3 + 7 + 5 = 3 + __), henceforth referred to as ‘‘mathematical equivalence prob- lems.’’ Although mathematical equivalence prob- lems are not typically included in traditional K–8 curricula (McNeil et al., 2006; Seo & Ginsburg, 2003), most people are shocked to discover that chil- dren (ages 7–11) solve the problems incorrectly. Across nine studies, McNeil (2005) found that the vast majority of children (about 82%) did not suc- ceed on the problems. Children’s difficulties with mathematical equivalence have been shown to be robust and long term, persisting among some mid- dle school, high school, and even college students (Knuth et al., 2006; McNeil & Alibali, 2005a; Renwick, 1932). This is cause for concern because individuals who do not develop a correct under- standing of mathematical equivalence will have difficulties advancing in mathematics and science. Although there is growing evidence that children have difficulties with mathematical equivalence, the mechanisms underlying these difficulties and the eventual emergence of correct understanding are less clear. A central goal of research in cognitive development is to characterize how knowledge is constructed over time. To achieve this goal, we need to move beyond simply assessing children’s successes and failures on mathematical tasks toward providing detailed accounts of why mathe- matical equivalence is particularly difficult to learn and how such difficulties are ultimately overcome (cf. Siegler, 2000). This research was supported by the Institute of Education Sci- ences, U.S. Department of Education through Grant R305B070297 to McNeil. The opinions expressed are those of the authors and do not represent views of the Institute or the U.S. Department of Education. We thank Martha Alibali for feedback on the method and for comments on a previous version of this paper. Thanks also to Matt Gibson, Krysten Williams, Noelle Crooks, and Jenny Heil for help with coding and tutoring. This research would not have been possible without the support of many administrators, teachers, parents, and students in the greater South Bend area. Correspondence concerning this article should be addressed to Nicole M. McNeil, Department of Psychology, University of Notre Dame, 118 Haggar Hall, Notre Dame, IN 46556. Electronic mail may be sent to nmcneil@nd.edu. Child Development, September ⁄ October 2011, Volume 82, Number 5, Pages 1620–1633 Ó 2011 The Authors Child Development Ó 2011 Society for Research in Child Development, Inc. All rights reserved. 0009-3920/2011/8205-0021 DOI: 10.1111/j.1467-8624.2011.01622.x