[The Role of Task Prompts in Fostering Strategy Flexibility, Procedural, and Conceptual Knowledge when Solving Quadratic Equations].

In secondary mathematics education, students are expected to apply procedures not only correctly but also flexibly, for example when solving quadratic equations. Tasks that deliberately stimulate cognitive and metacognitive processes support the development of such flexibility. The aim of this paper is to investigate the relationship between students' learning gains and the activation potential of the tasks provided. To this end, prompts included in tasks on solving quadratic equations based on a sample of 39 classes. In parallel, the learning gains of N = 739 students in these classes were measured over the course of an instructional unit on rule-based solving of quadratic equations.Linear regression analyses showed that (a) the development of strategy flexibility and procedural knowledge depends on the number of tasks that prompt students to compare worked-out solutions; (b) the development of strategy flexibility and conceptual knowledge depends on the number of tasks with metacognitively activating prompts; and (c) a high number of tasks with prompts that prescribe a specific procedure tend to hinder the development of strategy flexibility. These findings indicate that the activation potential of task prompts is a significant predictor of learning gains in solving quadratic equations.