Visual Support for Instructional Analogy: Context Matters

Visual Support for Instructional Analogy: Context Matters Kreshnik Nasi Begolli (kbegolli@uci.edu) School of Education, 3200 Education Irvine, CA 92697 USA Lindsey Engle Richland (lrichland@uchicago.edu) Department of Comparative Human Development, 5730 S. Woodlawn Ave. Chicago, IL 60637 Abstract Instructional analogies can overload children’s executive function and working memory resources (see Richland, Morrison & Holyoak, 2006), though structure-mapping lies at the core of recommended pedagogy in mathematics instruction (National Mathematics Panel, 2008; NRC, 2001). Videotaped mathematics instruction was manipulated to test the role of visual representations in instructional analogy. Pretest, posttest, and delayed posttest measures assessed 11- 13 year old children’s learning from one of three versions of the same lesson in which three solution strategies (one a misconception) were compared. Analogs were either a) Not Visible (NV) - presented only orally, b) Partially Visible (PV) – only the most recent solution was visible, or 3) All Visible (AV) - all solutions were visible throughout the instruction. Overall, AV students experienced greater learning gains in procedural knowledge, procedural flexibility, and conceptual/ schematic knowledge compared to PV students. These results persist after one-week delay. Apart from procedural knowledge, the same trend is evident when comparing AV students’ to NV students’ immediate learning gains. Overall, visual representations of analogs within an instructional analogy appear to support schema formation only when they are all visible simultaneously and throughout structure- mapping. Showing students visual representations of analogs but not enabling them to be simultaneously visible led to the lowest performance overall, suggesting this may lead to more object-level encoding than schema formation. Keywords: analogy; comparison; mathematics education; video stimulus; misconception; executive function. Comparing different student solutions to a single instructional problem is a key recommended pedagogical tool in mathematics, however the cognitive underpinnings of successfully completing this task are complex. Students must represent the multiple solutions as relational systems, align and map these systems to each other, and draw inferences based on the alignments (and misalignments) for successful schema formation (see Gentner, 1983; Gick & Holyoak, 1983; Richland, Zur & Holyoak, 2007). Orchestrating classroom lessons in which learners successfully accomplish relational structure mapping is not straightforward, particularly because opportunities for learning through structure mapping often fail in laboratory contexts (e.g., Gick and Holyoak, 1983; Ross, 1989). Specifically, reasoners regularly fail to notice the utility of aligning and mapping two or more available relational structures. The low success rate with which participants notice and use relational structure mapping, or analogy, within laboratory studies to solve problems may in part reflect limitations in the working memory system (see Waltz, Lau, Grewal & Holyoak, 2000). Working memory is required to relationally represent systems of objects, in this case steps to solution strategies, to re-represent these systems of relations so that their structures can align and map together, to identify meaningful similarities and differences, and to derive conceptual/ schematic inferences from this structure- mapping exercise to better inform future problem solving (see Morrison, Krawczyk, Holyoak et al 2004). The current study tests the role of visual representations of the source and target analogs within an opportunity for structure-mapping. The manipulation assesses whether 1) making source and target analogs visual (versus oral) increases the likelihood that participants will notice and successfully benefit from structure mapping opportunities, and 2) whether the visual representations must be visible simultaneously during structure-mapping in order to increase the likelihood of future success in problem solving and schema formation. The former is likely to increase the salience of the relational structure of each representation, while the latter is likely to reduce the working memory load and executive function resources necessary for participants to engage in structure-mapping and inference processes. These are research questions with high ecological validity. A cross-cultural study of 8 th grade mathematics instruction revealed that comparing verbal and visual structured representations is a common practice in U.S. mathematics classrooms as well as in higher achieving regions (Hong Kong and Japan), but that U.S. teachers are less likely to make visual representations visible during a structure-mapping episode than the teachers in higher achieving countries (Richland, Zur & Holyoak, 2007). Thus findings from this experiment will yield both theoretical insight into the resource load necessary for complex structure mapping and schema formation, and practice relevant implications for everyday mathematics teachers. Because the study takes ecological validity and the complexity of everyday classrooms as serious constraints, a novel methodology was used to derive rigorous, experimental data that incorporates the complexities of situated cognition. Specifically, the stimuli for the experiment derive from videotapes of a public school