Productive Failure in Learning the Concept of Variance

In a study with 140, ninth-grade mathematics students on learning the concept of variance, students experienced either direct instruction (DI) or productive failure (PF), wherein they were first asked to generate a quantitative index for variance without any guidance before receiving direct instruction on the concept. Whereas DI students relied only on the canonical formulation of variance taught to them, PF students generated a diversity of representations and formulations for variance but were ultimately unsuccessful in developing the canonical formulation. On the posttest however, PF students performed on par with DI students on procedural fluency, and significantly outperformed them on data analysis, conceptual insight, and transfer items. These results challenge the claim that there is little efficacy in having learners solve problems targeting concepts that are novel to them, and that direct instruction alone is the most effective approach for teaching novel concepts to learners.