Problem-solving before instruction for learning linear algebra in university mathematics

Abstract Problem-solving before instruction has been shown to be a more effective learning design than traditional tell-and-practice for several mathematical concepts at the secondary school level. In particular, the more a problem-solving before instruction design follows the productive failure principles, such as comparing and contrasting student-generated solutions, the higher the effect on students’ conceptual understanding and transfer. University mathematics education poses several inherent constraints that complicate the implementation of these principles. In the present study, we implemented a problem-solving before instruction design in a university linear algebra course adhering to the productive failure principles as closely as possible. Participation in the preparatory problems was voluntary. We investigated the effect on students’ learning over four one-year iterations in a design-based research approach. Compared to the baseline (aggregate of cohorts prior to the intervention), we observed a significant increase in final exam performance for all four cohorts with effect sizes between Cohen’s d = 0.28 and d = 0.59. For students who agreed to further analyses, our results show that up to 16% of the variance in students’ performance can be explained by variance in their participation in the problem-solving before instruction design. As our design did not include a control group, we refrain from conclusions regarding any design components that might have caused these effects. However, these results are promising, given that our implementation involved only minor changes to the original course structure and required little extra time for students.