Contents lists available at ScienceDirect Contemporary Educational Psychology journal homepage: www.elsevier.com/locate/cedpsych Number lines, but not area models, support children’s accuracy and conceptual models of fraction division Pooja G. Sidney a, ⁎ , Clarissa A. Thompson b , Ferdinand D. Rivera c a Department of Psychology, University of Kentucky, United States b Department of Psychological Sciences, Kent State University, United States c Department of Mathematics and Statistics, San Jose State University, United States ARTICLE INFO Keywords: Fractions Diagrams Problem solving Mathematics cognition Number lines ABSTRACT The Common Core State Standards in Mathematics recommends that children should use visual models to re- present fraction operations, such as fraction division. However, there is little experimental research on which visual models are the most effective for helping children to accurately solve and conceptualize these operations. In the current study, 123 fifth and sixth grade students solved fraction division problems in one of four visual model conditions: number lines, circular area models, rectangular area models, or no visual model at all. Children who solved the problems accompanied by a number line were more accurate and showed evidence of consistently producing sound conceptual models across the majority of problems than did children who com- pleted problems with either area model or no visual model at all. These findings are particularly striking given that children have experienced partitioning area models into equal shares as early as first grade, thus circles and rectangles were likely familiar to children. The number line advantage may stem from the fact that they afford the ability to represent both operand magnitudes in relation to one another and relative to a common endpoint. Future work should investigate the optimal order that instructors should introduce various visual models to promote children’s representational fluency across number lines and area models. 1. Introduction Reasoning about fraction operations is a critical aspect in the de- velopment of children’s deep understanding of mathematics. The National Mathematics Advisory Panel (NMAP, 2008) considers under- standing fractions to be foundational for algebra (p. xviii). Empirically, children’s understanding of fractions predicts later mathematics achievement and success in algebra (e.g., Bailey, Hoard, Nugent, & Geary, 2012; Siegler et al., 2012). However, despite the importance of children’s understanding of fraction operations, this facet of early and middle mathematics is notoriously difficult for children (e.g., Mack, 1990, 1995, 2001; Siegler, Thompson, & Schneider, 2011) and adults (e.g., Ball, 1990; Luo, Lo, & Leu, 2011; Ma, 1999). Even though fraction learning begins early in first grade, many students continue to struggle to accurately represent and perform fraction operations (Lortie- Forgues, Tian, & Siegler, 2015; Sidney & Alibali, 2015, 2017; Siegler & Pyke, 2013; Siegler et al., 2011). One common way of supporting children’s understanding of chal- lenging fraction concepts is by using visual models, and other external representations, during instruction and problem-solving activities. Reflecting this common practice, the IES Practice Guide for Developing Effective Fraction Instruction for Kindergarten through 8th Grade (Siegler, Carpenter, Fennell, Geary, Lewis, Okamoto, & Wray, 2010) directly recommends that instructors use visual models to engage stu- dents in sense-making activities and to ground their understanding of fraction concepts and procedures. According to the Common Core Standards Writing Team (2013), fractions should be first introduced with visual models in first and second grade under the Geometry strand (e.g., partitioning circles and rectangles into two, three, and four equal shares; 1.G.A.3; 2.G.A.3). The number line is introduced in third grade, when students use partitioning to place fractions on the line. Fourth and fifth graders should learn about multiplication (4.NF.B.4, 5.NF.B.4, 5.NF.B.5, 5.NF.B.6) and division (5.NF.B.3, 5.NF.B.7) with fractions with reference to visual models. Many research-based and empirically- tested effective fraction interventions, such as the Rational Number Project Curriculum (Cramer, Behr, Post, & Lesh, 1997; Cramer, Post, & del Mas, 2002) and several others (e.g., Fazio, Kennedy, & Siegler, 2016; Fuchs et al., 2013; Kellman et al., 2008; Moss & Case, 1999; Rau, Aleven, Rummel, & Pardos, 2014), include visual models as key com- ponents. As this body of research and practice recommendations https://doi.org/10.1016/j.cedpsych.2019.03.011 ⁎ Corresponding author at: Department of Psychology, University of Kentucky, 012E Kastle Hall, Lexington, KY 40506, United States. E-mail address: pooja.sidney@uky.edu (P.G. Sidney). Contemporary Educational Psychology 58 (2019) 288–298 Available online 26 March 2019 0361-476X/ © 2019 Elsevier Inc. All rights reserved. T