Benefits of “concreteness fading” for children's mathematics understanding * Emily R. Fyfe a, * , Nicole M. McNeil b , Stephanie Borjas b a Department of Psychology and Human Development, Vanderbilt University, USA b Department of Psychology, University of Notre Dame, USA article info Article history: Received 29 April 2014 Received in revised form 6 October 2014 Accepted 23 October 2014 Available online Keywords: Math equivalence Concrete manipulatives Learning and transfer Concreteness fading abstract Children often struggle to gain understanding from instruction on a procedure, particularly when it is taught in the context of abstract mathematical symbols. We tested whether a “concreteness fading” technique, which begins with concrete materials and fades to abstract symbols, can help children extend their knowledge beyond a simple instructed procedure. In Experiment 1, children with low prior knowledge received instruction in one of four conditions: (a) concrete, (b) abstract, (c) concreteness fading, or (d) concreteness introduction. Experiment 2 was designed to rule out an alternative hypothesis that concreteness fading works merely by “warming up” children for abstract instruction. Experiment 3 tested whether the benefits of concreteness fading extend to children with high prior knowledge. In all three experiments, children in the concreteness fading condition exhibited better transfer than children in the other conditions. Children's understanding benefits when problems are presented with concrete materials that are faded into abstract representations. © 2014 Elsevier Ltd. All rights reserved. 1. Introduction When we teach children a procedure for solving a mathematics problem, we not only want them to learn the procedure and apply it correctly, but also want them to understand why the procedure works. Indeed, a key question in the development of children's mathematical thinking is how we can help children gain under- standing of underlying concepts from the procedures they are taught, so they can transfer those procedures beyond the specific, instructed context. Unfortunately, children struggle to gain con- ceptual understanding from a procedure, especially when it is taught in the context of abstract mathematics symbols (e.g., McNeil & Alibali, 2000; Rittle-Johnson & Alibali, 1999). Because under- standing abstract symbols and manipulating them in meaningful ways are critical aspects of learning mathematics, the use of ab- stract symbols during instruction cannot and should not be avoided altogether. However, relatively minor changes to when and how abstract symbols are introduced during instruction may improve children's ability to extend their knowledge beyond the instructed procedure. In the present study, we tested one hypothesized method for helping children extend their mathematical knowledge beyond a simple, instructed procedure: beginning with concrete examples and then explicitly fading to the abstract symbols. This “concrete- ness fading” technique is hypothesized to facilitate conceptual understanding by fostering knowledge that is both grounded in meaningful concrete contexts, and also generalized in a way that promotes transfer (e.g., Fyfe, McNeil, Son, & Goldstone, 2014; Goldstone & Son, 2005). Students spend a lot of time learning and practicing mathe- matical procedures. For example, in representative eighth-grade math classrooms, students spent approximately two-thirds of in- dividual work time solving problems using an instructed procedure (Hiebert et al., 2003). Unfortunately, students typically just memorize the procedure and rotely apply it as instructed. This leads to misunderstandings and failure to transfer the procedure appropriately. Indeed, children rarely benefit from procedural * This work was based, in part, on a senior honors thesis conducted at the Uni- versity of Notre Dame by Emily R. Fyfe under the supervision of Nicole M. McNeil. The work was supported by the University of Notre Dame through a grant from the Center for Undergraduate Scholarly Engagement to Fyfe and summer fellowships to Fyfe and Borjas from the Undergraduate Research Opportunity Program of the Institute of Scholarship in the Liberal Arts. It was also supported by U.S. Department of Education, Institute of Education Sciences 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. The authors would like to thank April Dunwiddie for her help with data collection and scheduling as well as Vladimir Sloutsky for his insightful comments that inspired Experiment 2. * Corresponding author. 230 Appleton Place, Peabody #552, Vanderbilt Univer- sity, Nashville, TN 37203, USA. Tel.: þ1 615 343 7149. E-mail address: emily.r.fyfe@vanderbilt.edu (E.R. Fyfe). Contents lists available at ScienceDirect Learning and Instruction journal homepage: www.elsevier.com/locate/learninstruc http://dx.doi.org/10.1016/j.learninstruc.2014.10.004 0959-4752/© 2014 Elsevier Ltd. All rights reserved. Learning and Instruction 35 (2015) 104e120