Learning From Worked Examples: Conceptually Rich Explanations Predict Conceptual Gains

• In Event: Poster Session 09
  In Poster Session: PS 09 Topic - Cognitive Processes

Abstract

Teachers often use worked examples as a way to help students learn to solve mathematics problems. Asking students to explain such examples has been shown to lead to improvements in problem solving, as well as gains in conceptual understanding (Booth et al., 2013; Sweller & Cooper, 1985). Some work further suggests that explaining worked examples may be particularly beneficial for students with low prior knowledge (Kalyuga et al., 2001). In this research, we investigated the effectiveness of explaining worked examples for students learning to solve linear equations in algebra, with a focus on their conceptual understanding of the procedures—that is, their knowledge of the rationale for each of the steps in the procedure (Crooks & Alibali, 2014). We asked whether the quality of students’ explanations was associated with learning, and whether this relation depended on their prior knowledge.

Participants were 41 middle school students (M age = 12.82 years, SD = 0.73, 44% female, 76% white). All students completed a pretest, a brief computer-based lesson about linear equations, three worked examples with explanations, and a posttest. Half of the students were randomly assigned to an experimental condition in which they also completed warm-up items and received a version of the lesson that contained visual representations. The pretest included seven items and was designed to assess prior knowledge of algebraic concepts. After completing the pretest, students were asked to explain the steps in the worked examples. To elicit explanations for each step, participants were asked, “What did the student do to get from Step X to Step Y and why?” (Figure 1). Explanations were coded for whether students mentioned five key concepts: equality, inverse operations, do the same thing to both sides, isolate the variable, and preserve the solution. Finally, students completed a seven-item posttest similar to the pretest.

On average, students produced three conceptual explanations (M = 3.22, SD = 2.34) across all three worked examples. In a regression analysis with concept mentions, pretest score, their two-way interaction, and condition predicting posttest score, there was a significant positive effect of pretest score, F(1, 36) = 33.56, b = 0.67, p < 0.001 and a nonsignificant effect of concept mentions, F(1, 36) = 1.13, b = 0.10, p = 0.295. However, these effects were qualified by a significant concept mentions-by-pretest interaction, F(1, 36) = 8.96, b = -0.12, p = 0.005 (Figure 2). The effect of concept mentions was significantly greater for students with low prior knowledge than those with high prior knowledge. Condition did not influence posttest performance, F(1, 36) = 0.50, b = 0.29, p = 0.485.

Our results suggest that expressing concepts when explaining worked examples may help to activate and strengthen conceptual knowledge, particularly for learners with low prior knowledge. Therefore, prompting explanations of worked examples may promote gains in conceptual understanding for struggling learners. In future work, we plan to investigate what kinds of examples elicit more conceptually-rich responses and greater learning, both in general and for learners with low prior knowledge.

Authors

• Nicholas Allan Vest, University of Wisconsin - Madison
  Presenting Author

• Elena Silla, University of Wisconsin - Madison
  Non-Presenting Author

• Anna Bartel, University of Wisconsin-Madison
  Non-Presenting Author

• Tomo Nagashima, Carnegie Mellon University
  Non-Presenting Author

• Vincent Aleven, Carnegie Mellon University
  Non-Presenting Author

• Martha Wagner Alibali, University of Wisconsin - Madison
  Non-Presenting Author