Comparing and Explaining Examples of Multiple Strategies to Promote Algebra Learning: Instructional Features That Predict Learning

• In Event: Using Examples to Help Students Learn Mathematics: How, When, and With Whom to Use Worked Examples

Abstract

The purpose of this study was to explore variability in students’ algebra learning and instructional features that predict learning when using an example-based instructional intervention. The intervention focused on comparing and explaining examples of multiple solution strategies, a recommended instructional method (NCTM, 2014). However, textbook analyses and classroom observations suggest that this instructional method is not frequently or optimally used by many teachers (Richland, Holyoak, & Stigler, 2004). Thus, we have created a set of instructional materials and associated professional development supports called Comparison and Explanation of Multiple Strategies (CEMS) for Algebra I instruction based on evidence from cognitive psychology and mathematics education (e.g., Begolli & Richland, 2015; Rittle-Johnson & Star, 2007).

Method. We focused on 7 ninth-grade teachers from 3 schools and their 361 students who used our CEMs approach as part of their instruction during a full-year Algebra I course. Data included a researcher-developed algebra assessment (with questions on conceptual knowledge, procedural knowledge and procedural flexibility), coding of videos of classroom lessons, and teacher logs. Latent transition analysis (LTA) was used to identify student knowledge profiles on the algebra assessment at the beginning of the school year and profile transition from the beginning to the end of the school year. Then, we explored variability between teachers in their students’ initial knowledge profile and profile transitions and evaluated if 3 instructional features (see Table 1) predicted this variability.

Results. Three student knowledge profiles were identified in the LTA: students with a low, medium and high level of knowledge. There was large variability among teachers in their students’ initial knowledge level as well as in the probability to transition to a higher knowledge profile on the end-of-year assessment (see Figure 1). We explored instructional features that could explain this variability. The higher teachers’ use of our materials and the more teachers facilitated high-quality student interactions, the more likely their students were to have a higher knowledge profile at the beginning of the school year (Likelihood Ratio χ2 (2) = 17.08, p < .001 and χ2 (2) = 21.93, p < .001, respectively) and to transition to a higher-knowledge profile at the end of the school year (χ2 (2) = 6.20, p = .045 and χ2 (2) = 18.77, p < .001, respectively).
Significance. Greater use of our CEMS approach was related to greater knowledge gains, providing preliminary support for the effectiveness of the approach, albeit with a small number of teachers. Greater support for high-quality student interaction was also associated with greater knowledge gains, highlighting the importance of students explaining ideas with classmates. However, some teachers struggled to implement our approach and some students did not learn much of our target content, especially in classrooms with many students with low initial knowledge, suggesting that our CEMS approach and teacher PD was not sufficiently powerful to aid learning by all students. The current findings highlight the potential of evidence-based instructional approaches for improving student learning, as well as persistent gaps in improving teaching quality and student learning broadly.

Authors

• Bethany Rittle-Johnson, Vanderbilt University

• Marian Hickendorff

• Jon R. Star, Harvard University

• Kelley Durkin, Vanderbilt University

• Abbey M. Loehr, Washington University in St. Louis