The Process of Probability Problem Solving: Use of External Visual Representations

Abstract

We investigate the role of external inscriptions, particularly those of a spatial or visual nature, in the solution of probability word problems. We define a taxonomy of external visual representations used in probability problem solving that includes pictures, spatial reorganization of the given information, outcome listings, contingency tables, Venn diagrams, trees, and novel graphical representations. We also propose a process model for probability problem solving (PPS) and use it as a framework to better understand how and why external visual representations are used. In a study of 34 novice probability problem solvers, participants worked to solve six probability word problems covering six probability subtopics. Both written and verbal structured interview protocols were analyzed to investigate when and how external visual representations are spontaneously used by problem solvers. Analyses of the coded transcripts showed that participants' probability problem-solving efforts move through the stages of PPS in a sequential but not always linear manner, sometimes exhibiting iterated attempts to represent the problem mathematically and to find a solution strategy. Results showed that use of specific external visual representations was associated with specific probability topics, and that certain choices of representation are associated with higher rates of solution success. These findings suggest that an external visual representation can facilitate probability problem solving, but only when an appropriate representation is chosen. Finally, we present evidence to show that external visual representations are usually created and first used during the stages of representing the problem mathematically and finding a solution strategy. However, pictures are often created during the initial stage of problem text understanding, and tables are sometimes created during computation of the solution.

Support for this project was provided by the National Science Foundation under Grant No. SBE-0350288, Science of Learning Centers. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. Portions of this work were submitted by the first author in partial fulfillment of the requirement for the Ph.D. degree at Teachers College, Columbia University; some results were also reported at the 2006 annual meeting of AERA. We thank Herb Ginsburg, John Black, and Barbara Tversky for useful comments on prior drafts of this manuscript.