Looking at Versus Looking Through: Designing for Problem Solving and Conceptual Understanding

• In Event: 29.027 - Designing Digital Environments to Support Mathematical and Scientific Reasoning: Theoretical and Disciplinary Perspectives

Abstract

A persistent challenge of designing for complex problem solving involves balancing between support for problem solving activity and support for underlying conceptual understanding; that is, understanding not just how to execute a procedure, but also what a procedure might tell you and why it might be useful in a particular situation (Gresalfi, 2015; Gresalfi & Barnes, 2015; Santos-Trigo, 2007; Schoenfeld, 2007).
This paper reports findings from two rounds of design and implementation, focusing in particular on the design of a ratio tool intended to support students’ conceptual understanding of proportion, and the influence of this tool on student learning about ratio and their overall problem solving activity. We studied mathematical reasoning and problem solving in the context of a virtual educational game called Boone’s Meadow, based on a project-based mathematics activity from the Adventures of Jasper Woodbury (Bransford, Zech, Schwartz, Barron, & Vye, 2000; Pellegrino et al., 1992).
Data come from two years of implementation in the same 7th grade teacher’s classroom (Ms. Lynn) at a diverse middle-school (92% free and reduced lunch, 30% English language learners) in a southeastern U.S. city. The students generally performed at least two grades below level; in both years students struggled with fundamental understanding of multiplication and proportion. . We collected and analyzed pre and post tests, videos from two thirds of focus groups (six pairs), and transcripts of the teacher’s talk about ratio.
Although a simple t-test revealed that all students learned in both years (t=6.00, p<.001); however, a multiple regression revealed that students in year 1 scored significantly higher than students in year 2 on the posttest (t= -2.31, p <.05). This was surprising, as we observed higher levels of engagement in year 2. To better understand why the ratio tool might have contributed to lower posttest scores in year 2, we examined videos of student and teacher interactions, focusing specifically on times when the ratio tool was the object of activity. These videos were coded to determine whether students were reasoning with ratio in a separate or coordinated way. Seeing ratios as a single coordinated unit rather than two separate units has been established as a core concept for building proportional reasoning (Lobato, Ellis, & Charles, 2010). Students did not seem to use the tool in a coordinated manner. Instead, many students cued into separate patterns within the tool. Rather than building the conceptual understanding of proportion, the tool became the object to look at and get answers, not learn with (Greeno & Hall 1997). However, student problem solving also changed over the two years, with students including significantly more problem elements, mathematical justification, and more precision in their final solutions in Year 2 than in Year 1.
These findings suggest that having a straightforward way to find an answer to the ratio problems might have prevented robust learning of ratio, but did allow students to engage more deeply with problem solving, suggesting that more investigation into the interrelations between understanding and problem solving is necessary.

Authors

• Melissa Sommefeld Gresalfi, Vanderbilt University

• Isaac Thomas Nichols, Vanderbilt University - Peabody College

• Panchompoo Wisittanawat, Vanderbilt University