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Discovery learning has a long tradition in educational psychology and in mathematics education. While these two research strands are often only weakly connected, we demonstrate that it is fruitful to draw on the theories and findings from both perspectives. We base the design of our study on the instructional model “productive failure”, which is investigated in educational psychology, and for which one can find analogous models in mathematics education. Productive failure combines a phase of divergent discovery (exploration phase) with a phase of convergent organization encompassing structured instructional elements (instruction phase). Our research questions target both phases: Regarding the exploration phase, we investigated students’ strategies when they attempt to solve fraction problems prior to instruction. Regarding the instruction phase, we tested whether a guided elaboration on typical errors had an impact on students’ understanding. The study included two learning units. In both units, students of all experimental conditions first engaged in an identical exploration activity and then worked on elaboration tasks that introduced: a) only correct solutions; b) correct and erroneous solution attempts; or c) correct and erroneous solution attempts with prompts to compare these solutions. Our qualitative categorization of students’ solutions showed that the identified errors match the solution attempts used for the instruction phase. Posttest results indicate that including erroneous solution attempts in the instruction phase can be beneficial for learning units in which most students fail to come up with a correct solution themselves, but only if students are prompted to compare the erroneous solution attempts with correct solutions.

To progress from intuitive ideas to deep conceptual understanding, students need to become aware of gaps in their ideas. Attempting to solve problems prior to instruction may lead to a global awareness of knowledge gaps (i.e., awareness without being able to identify which specific component is lacking). These gaps may subsequently be specified by comparing students' solutions to the canonical solution. In our first experiment, the teacher highlighted specific gaps by comparing typical student solutions to the canonical solution before or after problem solving. The second experiment varied the factors form of instruction (with or without student solutions) and timing of instruction (problem-solving prior to or after instruction). Problem-solving prior to instruction triggered a global awareness of knowledge gaps. While this was beneficial for learning when combined with instruction with student solutions, our results indicate that comparing student solutions during instruction to specify the gaps is the most relevant factor.

Multiple studies have shown benefits of problem-solving prior to instruction (cf. Productive Failure, Invention) in comparison to direct instruction. However, students’ solutions prior to instruction are usually erroneous or incomplete. In analogy to guided discovery learning, it might therefore be fruitful to lead students towards the discovery of the canonical solution. In two quasi-experimental studies with 104 students and 175 students, respectively, we compared three conditions: problem-solving prior to instruction, guided problem-solving prior to instruction in which students were led towards the discovery of relevant solution components, and direct instruction. We replicated the beneficial effects of problem-solving prior to instruction in comparison to direct instruction on posttest items testing for conceptual knowledge. Our process analysis further revealed that guidance helped students to invent better solutions. However, the solution quality did not correlate with the posttest results in the guided condition, indicating that leading students towards the solution does not additionally promote learning. This interpretation is supported by the finding that the two conditions with problem-solving prior to instruction did not differ significantly at posttest. The second study replicated these findings with a greater sample size. The results indicate that different mechanisms underlie guided discovery learning and problem-solving prior to instruction: In guided discovery learning, the discovery of an underlying model is inherent to the method. In contrast, the effectiveness of problem-solving prior to instruction does not depend on students’ discovery of the canonical solution, but on the cognitive processes related to problem-solving, which prepare students for a deeper understanding during subsequent instruction.

Affective states play a significant role in students’ learning behaviour. Positive affective states can enhance learning, whilst negative affective states can inhibit it. This paper describes a Wizard-of-Oz study which investigates whether the way feedback is presented should change according to the affective state of a student, in order to encourage affect change if that state is negative. We presented high-interruptive feedback in the form of pop-up windows in which messages were immediately viewable; or low-interruptive feedback, a glow-\ud ing light bulb which students needed to click in order to access the messages. Our results show that when students are confused or frustrated high-interruptive feedback is more effective, but when students are enjoying their activity, there is no difference. Based on the results, we present guidelines for adaptively tailoring the presentation of feedback based on students’ affective states when interacting with learning environments.

Recent studies have shown benefits of problem-solving prior to instruction (cf. productive failure) for learning. These findings seem to contradict well-established assumptions of cognitive load theory. However, there are two possible mechanisms in line with cognitive load theory that may explain these beneficial effects: the activation of prior knowledge and intuitive ideas during the problem-solving phase to generate solution approaches and the focusing of attention on relevant components of the canonical solution by comparing and contrasting typical student solutions to the canonical solution during the instruction phase. It is unclear whether the reported benefits originate from the activation of prior knowledge and intuitive ideas during the problem-solving phase or from the specific form of instruction in which student solutions are compared and contrasted to the canonical solution. To investigate this question, we compared three conditions in a quasi-experimental study: standard instruction prior to problem-solving (I − PS), instruction in which typical student solutions are contrasted to the canonical solution prior to problem-solving (Icontrast − PS), and problem-solving prior to instruction in which typical student solutions are contrasted to the canonical solution (PS − Icontrast). I − PS was outperformed by the other two conditions on conceptual knowledge. This finding suggests that student solutions are fruitful learning resources. We argue that the comparison of student solutions and the canonical solution focuses attention on the relevant components of the solution, which leads to deeper processing. Indeed, our cognitive load measures suggest that comparing and contrasting typical student solutions during instruction is germane for learning.