Conceptual explanations and understanding fraction comparisons Emma H. Geller a, * , Ji Y. Son b , James W. Stigler a a University of California, Los Angeles, Department of Psychology,1285 Franz Hall, Box 951563, Portola Plaza Building, Los Angeles, CA, 90095, USA b California State University, Los Angeles, Department of Psychology, KH C-3104, 5151 State University Dr., Los Angeles, CA, 90032, USA article info Article history: Received 2 September 2016 Received in revised form 1 March 2017 Accepted 21 May 2017 Available online xxx Keywords: Fractions Explanation Understanding Conceptual abstract Explanations are used as indicators of understanding in mathematics, and conceptual explanations are often taken to signal deeper understanding of a domain than more superficial explanations. However, students who are able to produce a conceptual explanation in one problem or context may not be able to extend that understanding more generally. In this study we challenge the notion that conceptual ex- planations indicate general understanding by showing that e although conceptual explanations are strongly associated with correct answers e they are not employed equally across different contexts, and the highest performing students tend to use more general explanations, which may or may not be conceptual. Overall, our results suggest that explanations of fraction magnitudes follow a learning tra- jectory reflected in students’ accuracy and explanations: weak students focus on concrete, non- conceptual features, stronger students use concepts to explain their answers, and the highest per- formers tend to use general (but not necessarily conceptual) rules. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Fractions are notoriously difficult for students to understand. Although the topic is introduced as early as 3rd grade, many stu- dents graduate high school and enter college with only a superficial and fragile understanding of rational numbers (Stigler, Givvin, & Thompson, 2010). One hypothesized reason for this fragility is that students have memorized procedural rules and techniques for dealing with fractions without developing a corresponding con- ceptual understanding of fraction magnitudes, which makes many operational rules appear meaningless (Siegler & Pyke, 2014; Stigler et al., 2010). When students believe that mathematical rules are arbitrary, they lose the motivation to believe that those procedures should make sense, which, in turn, hinders their ability to evaluate outcomes of procedures to decide whether they are reasonable or not (Hiebert & Lefevre, 1986; Siegler & Pyke, 2014). Developing conceptual knowledge has recently become the focus of education reforms, such as the Common Core, which point to poor perfor- mance on standardized tests as evidence that students do not un- derstand the mathematical procedures they use to solve problems. If students had a deeper conceptual understanding of mathematics, the argument goes, then they would not make such errors (Jordan et al., 2013). Indeed, there is a large literature showing that con- ceptual understanding is critical to flexible thinking in mathe- matics (Gray & Tall, 1994; Niemi, 1996; Rittle-Johnson & Alibali, 1999; Rittle-Johnson & Siegler, 1998; Rittle-Johnson & Star, 2009), whereas procedural knowledge alone may not lead to accurate and flexible performance. Recent work with community college students has demon- strated this procedural fragility in a number of mathematical do- mains, particularly with fractions (Givvin, Stigler, & Thompson, 2011; Stigler et al., 2010). In this study, students were asked to indicate which of two fractions e a/5 and a/8 e was larger, and to explain their answers (Stigler et al., 2010). Performance on the task was dismally poor, with students on average doing no better than they would by simply guessing. But students’ explanations revealed an interesting caveat; although very few students produced con- ceptual explanations for their answers, all of those who did chose the correct answer. So, for example, a student who could explain that if the same quantity (a) were divided into 5 pieces, the pieces would be larger than if it were divided into 8 pieces, would instantly see that a/5 is the larger fraction. One might naturally conclude from this that students who provide conceptual explanations have a deeper understanding of fractions than students who don't. While the gist of this conclusion is not controversial, per se, the specific claim is not supported with * Corresponding author. Present address: University of California, San Diego, Department of Psychology, 9500 Gilman Drive, La Jolla, CA, 92093, USA. E-mail addresses: emma.h.geller@gmail.com (E.H. Geller), jiyunson@gmail.com (J.Y. Son), stigler@ucla.edu (J.W. Stigler). Contents lists available at ScienceDirect Learning and Instruction journal homepage: www.elsevier.com/locate/learninstruc http://dx.doi.org/10.1016/j.learninstruc.2017.05.006 0959-4752/© 2017 Elsevier Ltd. All rights reserved. Learning and Instruction xxx (2017) 1e8 Please cite this article in press as: Geller, E. H., et al., Conceptual explanations and understanding fraction comparisons, Learning and Instruction (2017), http://dx.doi.org/10.1016/j.learninstruc.2017.05.006