2-082 - External Representations in Mathematical Thinking and Learning

Session Type: Paper Symposium

Integrative Statement

In line with Common Core State Standards (2010), teachers often use external representations, such as physical manipulatives and diagrams, to instantiate mathematical relationships. Indeed, such representations of key concepts can facilitate children’s conceptual understanding. For example, external representations sometimes help learners construct accurate mental models of symbolic problems (Cromley et al., 2013; McNeil, Uttal, Jarvin, & Sternberg, 2009). However, children’s learning with external representations does not always extend to similar tasks when those representations are not present (Kaminski, Sloutsky, & Heckler, 2008). Furthermore, mathematical concepts can be instantiated in many ways, and potential benefits of an external representation may depend on its alignment with children’s internal representations of mathematical concepts.

In this symposium, we present evidence that the quality of learners’ thinking with external representations depends on which external representations are used and how learners use them. Paper 1 demonstrates that when children understand the connections between physical manipulatives and the mathematical symbols they are meant to represent, their learning from a lesson using manipulatives transfers to tasks without them. Paper 2 demonstrates the promise of connecting symbolic fractions with nonsymbolic ratio magnitude representations that align with learners’ internal ratio representations. Paper 3 demonstrates that number lines uniquely afford more consistently accurate fraction division reasoning than other external representations. The discussant will consider ways to conceptualize the range of instructional and learner differences presented, focusing on mechanistic accounts that can explain why these differences lead to various outcomes, as well as the implications of these findings for mathematics teachers.

Sub Unit

• Cognitive Processes

Chair

• Pooja Gupta Sidney, University of Kentucky

Discussant

• Kelly S. Mix, University of Maryland

Individual Presentations

• Connecting Manipulatives and Symbols Promotes Mathematics Learning - Presenting Author: Emily R. Fyfe, Indiana University; Non-Presenting Author: Andrea Marquardt Donovan, University of Wisconsin-Madison

• Perceptual Routes to Rational Number - Presenting Author: Percival Matthews, University of Wisconsin-Madison; Non-Presenting Author: John Binzak, University of Wisconsin-Madison; Non-Presenting Author: Priya Kalra, University of Wisconsin-Madison; Non-Presenting Author: Yunji Park, University of Wisconsin-Madison; Non-Presenting Author: Edward Hubbard, University of Wisconsin-Madison

• Number Lines, but not Area Diagrams, Support Children’s Fraction Division Problem Solving - Presenting Author: Pooja Gupta Sidney, University of Kentucky; Non-Presenting Author: Clarissa A Thompson, Kent State University; Non-Presenting Author: Ferdinand Rivera, San Jose State University