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To explain successful subject matter learning with digital tools, the specification of mediating cognitive processes is crucial for any empirical investigation. We introduce a cognitive process framework for the mechanisms of learning with digital tools (CoDiL) that combines core ideas from the psychology of instruction (utilization-of-learning-opportunity framework), cognitive psychology (knowledge-learning-instruction framework), and domain-specific research on learning and instruction. This synthesizing framework can be used to theoretically ground, firstly, the design of digital tools for learning, and secondly, the empirical analysis of students' learning activities in digitally enriched educational settings via the analysis of specific student-tool interactions.

PS-I learning approaches with an initial problem-solving phase (PS) followed by an instruction phase (I) can be optimised by providing adaptive (individualised) instruction that builds on students' problem-solving products. A computer-based implementation of both phases offers the possibility of adaptivity at the transition from the problem-solving to the instruction phase. However, it also poses two challenges for the problem-solving phase, which should allow for (1) complex problem-solving processes (process validity); and (2) accurate diagnosis of problem-solving products. Increased openness of the learning environment can increase the process validity and decrease diagnostic accuracy and vice-versa (complexity-accuracy trade-off). Using two variants of a computer-based learning environment for fraction comparison with different extent of openness (O+ and O-), we investigate whether both requirements (complex problem-solving processes and accurate diagnosis of the problem-solving products) can be met simultaneously. To check process validity (complexity of problem-solving processes), we compare the problem-solving products with those from previous research on an analogous paper-based version. Our results show that valid and comparable problem-solving processes can be achieved in both computer-based learning environments (O+ and O-). Moreover, only the learning environment O- enables an accurate computer-based interpretation of students' products and thus resolving the complexity-accuracy trade-off.

Productive Failure (PF) facilitates students' conceptual knowledge by delaying instruction until after problem solving. While PF is well investigated in middle and high school students, little is known about its effectiveness in younger students. Studies in younger samples, which implemented delayed instruction designs similar to those used in PF studies, showed mixed results. However, these studies did not implement two core design components of PF: (1) contrasting and comparing student-generated solutions and the canonical solution during the instructional phase (contrasting activity), and (2) students' collaboration in small groups during the initial problem-solving phase. Both components can be expected to contribute to the effectiveness of PF. In a quasi-experimental study with 228 fifth graders, we implemented the first component (contrasting activity) with all students to establish whether under this condition, problem solving prior to instruction would be more effective for younger students' conceptual knowledge acquisition than direct instruction (i.e., problem solving after instruction). Further, we experimentally tested the effect of the second component (collaborative vs. individual problem solving) on students' conceptual knowledge and the number of solution ideas generated during initial problem solving. We found no empirical support for either of our hypotheses. To explore the extent to which students' collaboration actually achieved its potential and relates to students' conceptual knowledge and solution ideas in PF, we conducted analyses of collaborative processes. Our study adds to the mixed results regarding the superiority of problem solving prior to instruction for young students, thus opening the discussion about age-related prerequisites as boundary conditions for PF.