Both processes are said to be more beneficial than active e. The mode of engagement that the ICAP framework terms interactive synthesizes beneficial collaborative processes that have been described by a large body of previous research. However, we found no clear superiority of interactive processes in our study.

During the PF phase, the cumulative contributions of individual students, rather than the co-constructed ideas, determined how many solution ideas a dyad noted down. Regarding the individual posttest score, the relative numbers of interactive, constructive, and active utterances all showed a positive relationship. Against this background, it seems that the most relevant factor for learning is content-related talk and not the social mode in which it was generated. This major effect might have overshadowed the differential effects of the different types of utterances i. Second, the overt learning processes might not always reveal the underlying cognitive learning processes, especially when differentiating between constructive and interactive processes.

Even though the theoretical and coded differentiation between interactive and constructive activities seems clear, the point when a constructive contributions becomes an interactive contribution might not always be visible. This unclear relation between the overt learning activity and the cognitive learning processes may explain the missing superiority of interactive utterances in our study.

To shed more light on this relation in particular and on the role of collaboration in PF, more in-depths analyses are needed.

However, these boundary conditions need to be investigated and validated in future studies. A total of German fifth graders aged 10—12 years from nine classes from three different schools participated in the study. Classes approximately 22—30 students per class were randomly assigned to conditions i. As in previous PF studies 3 , 7 and in line with the German core curriculum, 28 students did not have any formal knowledge about the target domain.

Students acquired conceptual knowledge about fraction equivalence comprising three interconnected target concepts. The first concept relation required students to understand the relation between the size and the number of fractional parts within a given fraction i. The second concept division concerned the division of fractions. It required students to understand to divide, for example, a circle representation through the central point and to fairly distribute these parts among persons. The third concept comparison focused on the comparison across two given fractions for identifying fraction equivalence.

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To be able to compare two fractions with different denominators and numerators, students needed to understand the necessity to first find a common denominator prior to comparing. According to the German standards of the core curriculum, arguing with mathematical concepts is a key competence that students should have acquired at the latest by the end of grade 4. Equivalent fraction problem used in both Productive Failure conditions. In the direct instruction conditions, students had to solve an isomorphic problem differing only in the numbers being used in problem-solving step 1 and 2 i.

As students had no formal target knowledge, PF students encountered a rather complex problem-solving situation, which is an important and suitable design component for PF 6 as described in the Introduction. Further, and also in line with these design recommendations, the problem permitted students to develop multiple solution ideas. For example, they could try to find a solution by calculating with actual fractions e. Students in the PF conditions tried to solve the problem shown in Fig.

To prevent frustration while working on a challenging problem, the instructor provided PF students with motivational prompts e. In the DI condition, the problem shown in Fig. Thus, DI students received an isomorphic problem with different numbers for each problem-solving step i.

As the problem-solving phase in DI only occurred after the instruction, students were not expected to generate multiple solution ideas. The instruction was constant across all four experimental conditions. It used the problem employed in the problem-solving phase of the PF conditions, and included comparing and contrasting between student-generated solution attempts and the canonical solution. A table with more details on the comparing and contrasting activity is presented in the supplementary material.

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We employed typical student solutions taken from two previous studies 36 rather than the solutions of the students in this study. This enabled us to keep the instruction constant and it prevented a variety of different and unpredictable solutions across experimental conditions, which would have led to unwanted variance. The instruction also allowed students to practice the solution steps required to solve the types of problem the PF students already had encountered during the initial problem-solving phase of PF or likewise the DI students still had to encounter during the subsequent problem-solving phase of DI.

During the intervention lessons, all students received a to min introduction about the study background. Afterwards, students in the PF conditions started with the problem-solving phase and then received instruction. Students in the DI conditions first received instruction and then engaged in the problem-solving phase. In line with previous PF studies, 7 the pretest did not measure the targeted knowledge which students could not be assumed to already possess , but rather prerequisite mathematical knowledge.

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This ensured that students were not prompted to generate solution ideas prior to receiving instruction. There were a total of nine questions, from which students could receive a maximum of 9 points. Therefore, it included non-trivial tasks that required students to solve story problems of similar complexity to the problem, which they had encountered during the learning phases and was limited in time to one regular min school lesson.

The posttest comprised six items: one multiple-choice item including a drawing task, and five story problems with subtasks that required explanation-based answers. We rewarded the canonically correct explanation with 2 points, the visual-based explanation with 1 point, and any other or no explanation with 0 points. For the multiple-choice item, students were rewarded with 1 point for the correct answer and an additional point if they were able to correctly visualize their answer.

The maximum total possible score was Given the heterogeneous nature i. To investigate whether the number of solution ideas differs across both PF conditions cf. As a coding example, drawing the four salami pizzas and dividing them into halves was coded as one solution idea for problem-solving step 1, and again drawing the four salami pizzas and this time dividing them into quarters was coded as another solution idea for problem-solving step 1. We summed up the number of solution ideas across all problem-solving steps into one sum score.

Two coders were intensively trained to assess the number of solution ideas. Disagreements were resolved by discussion. We developed and iteratively improved a coding scheme by following a top-down and bottom-up approach: 37 using the bottom-up approach, the coding scheme builds on verbal process data from a previous study 36 in which dyads tried to solve the very same equivalent fraction problem prior to receiving instruction see Fig.

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From the top-down perspective, the coding scheme is rooted in the ICAP framework 23 , 24 and broader literature about collaborative communication processes. To avoid ambiguity of interpretation and thus to increase objectivity, we coded on the utterance level rather than on an episode level. The final coding scheme differentiates between six different types of collaborative utterances: interactive, constructive, active, coordinative, back channeling, and off-task.

While interactive, constructive, and active utterances are derived from the ICAP framework, coordinative, back channeling, and off-task utterances are taken from the broader literature on collaborative learning. Table 4 presents the conditions that needed to be fulfilled to code a single utterance as interactive, constructive, active, coordinative, back channeling, or off-task. Overall, we used a conservative coding approach, choosing the lower code in cases of ambiguity i.

Two coders were intensively trained on the coding scheme. Inconsistencies were resolved by discussion. To assess the number of learning opportunities students encountered during the problem-solving phase, we used the absolute frequency i. To examine how efficient students used their collaborative discourse time, we referred to the relative frequencies i. To investigate the relation between the number of solution ideas and the absolute frequency of different kinds of utterances, we referred to the absolute number of the respective kind of utterances a dyad made.

Note that the methods were performed in accordance with relevant regulations and guidelines and approved by members of the Institute of Educational Research. Further information on experimental design is available in the Nature Research Reporting Summary linked to this article. Due to legal data privacy issues, the data to support the findings of this study are not publicly available, as individual students might be identifiable to insiders.

The anonymized datasets may be provided by the corresponding author on reasonable request. Kapur, M. Productive Failure in mathematical problem solving. A further study of Productive Failure in mathematical problem solving. Productive Failure in learning the concept of variance. Productive Failure in learning math.

Designing for Productive Failure. Loibl, K. Schwartz, D. Inventing to prepare for future learning: the hidden efficiency of encouraging original student production in statistics instruction.

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## Representation and Productive Ambiguity in Mathematics and the Sciences by Emily Grosholz

Practicing versus inventing with contrasting cases: the effects of telling first on learning and transfer. Belenky, D. Motivation and transfer: the role of mastery-approach goals in preparation for future learning. Rittle-Johnson, B. Conceptual and procedural knowledge of mathematics: does one lead to the other? Towards a theory of when and how problem solving followed by instruction supports learning. DeCaro, M. Exploring mathematics problems prepares children to learn from instruction.

Child Psychol. Fyfe, E.

An alternative time for telling: when conceptual instruction prior to problem solving improves mathematical knowledge. Loehr, A. Wait for it… Delaying instruction improves mathematics problem solving.