Examples and Mathematics: How to Design Physics Materials for Learning and Transfer

• In Event: Excellence in Education Research: Early-Career Scholars and Their Work

Abstract

Using equations to represent functional relations among variables is a key skill in mathematics. For example, high schoolers should “Create equations in two or more variables to represent relationships between quantities” (CCSS.MATH.CONTENT.HSA.CED.A.2). Representing relations using equations is also crucial in scientific thinking and argumentation. The Next Generation Science Standards call upon students to use mathematics to model scientific phenomena, such as momentum (HS PS2-2). How can educators design materials to support students in representing relations mathematically? Furthermore, how can instruction promote transfer of these ideas to new topics? I suggest two possible ways: selecting a comprehensive set of examples and providing a task orientation.

I draw on ideas of information sufficiency from Perceptual Learning (Gibson & Gibson, 1959) to systematically design contrasting cases, instructional materials for learning general explanations in science. During these inductive activities, students are asked to notice the invariant relationship that underlies each example. Contrasting cases have been developed for many topics, such as statistical variation (Schwartz & Martin, 2004) and density (Schwartz, Chase, Oppezzo, & Chin, 2011).

In the present study, 108 community college student participants used contrasting cases for learning two-factor physics problems, where two variables are multiplicatively combined to create an intensive quantity. Students worked with contrasting cases showing inelastic collisions problems on the initial task. A balance scale contrasting cases activity served as a Preparation for Future Learning (PFL) transfer assessment (cf. Bransford & Schwartz, 1999).

I created two sets of contrasting cases for the collisions task (Figure 1). The ME materials isolated the main effects of each variable (mass and velocity) and did not include any examples where mass and velocity "trade off”. In the ME+I condition, the contrasting cases showed the main effects of each variable (mass and velocity) and the interaction of the two (mass x velocity). Furthermore, students’ task orientation was manipulated between-subjects: half of the students receiving each set of cases were prompted to “use math” in their solutions to the collisions contrasting cases task. In summary, four versions of the collisions task were compared between-subjects using a 2 (Materials: main effects vs. main effects + interactions) x 2 (Prompt: “use math” vs. no prompt). To assess learning, I recorded students’ solution strategies and measured their knowledge with a posttest asking them to predict the results of new collisions.

I examined how students’ experiences on the collisions task influenced future learning on the balance scale task. All participants received the same materials for this task: a series of cases that included main effects and interactions cases. No explicit prompts about using math were provided. Again, students’ solutions and performance on a posttest were noted.

The choice of materials impacted students’ learning and transfer. Students who received ME+I materials were more likely to generate multiplication solutions on the collisions and balance scale tasks, and, as a result, performed higher on the posttests. Most students who learned from ME materials on the collisions task generated qualitative solutions and negatively transferred, producing qualitative solutions on the balance scale as well. Providing simplified materials to make it easier for students in the short run may have negative consequences in the long run.


Though less strong, an effect of the “use math” prompt was observed. The prompt was especially effective in conjunction with the ME materials – it influenced more than half of those students to create mathematical solutions for collisions. Some students found a multiplication solution and transferred the solution to the balance scale task. Others found an additive solution, which could not directly transfer to the balance scale. Some of these students abandoned the addition solution but transferred the general strategy of mathematizing, moving to multiplication. However, others did not recover and failed to find a multiplication solution on the second task. The prompt was less effective than the selection of materials.


These findings have direct implications for the selection of examples and the design of instructional prompts in educational settings. I see parallels between the participants who created strategies to be transferred within these studies and students who leave school with knowledge that is soon-to-be-insufficient in broader contexts. Future work must help students with insufficient, but correct-at-the-time understanding learn to learn from more complex information in the future.

Author

• Nicole Rose Hallinen, Temple University