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Educational technologies in mathematics typically focus on fostering either procedural knowledge by means of structured tasks or, less often, conceptual knowledge by means of exploratory tasks. However, both types of knowledge are needed for complete domain knowledge that persists over time and supports subsequent learning. We investigated in two quasi-experimental studies whether a combination of an exploratory learning environment, providing exploratory tasks, and an intelligent tutoring system, providing structured tasks, fosters procedural and conceptual knowledge more than the intelligent tutoring system alone. Participants were 121 students from the UK (aged 8–10 years old) and 151 students from Germany (aged 10–12 years old) who were studying equivalent fractions. Results confirmed that students learning with a combination of exploratory and structured tasks gained more conceptual knowledge and equal procedural knowledge compared to students learning with structured tasks only. This supports the use of different but complementary educational technologies, interleaving exploratory and structured tasks, to achieve a “combination effect” that fosters robust fractions knowledge.

Accurate diagnostic judgments from teachers are important in education, e.g., for providing feedback to students that supports learning. Explaining diagnostic judgment requires understanding of the components of cognitive processes underlying judgment formation. Many aspects ascribed to these components are specific to a particular domain, like physical education. In the physical education domain, research on judgment accuracy and cognitive processes of judgment formation is scarce, and not substantially driven and reflected by theory. In this paper, we provide a theoretical framework on cognitive processes of judgment formation for physical education, with the aim to support guiding and pooling of future research activities in this domain. In addition, this framework serves as a heuristic example to other domains which may want to adopt it according to their needs.

Productive Failure (PF) is an instructional design that implements a problem-solving phase which aims at preparing students for learning from a subsequent instruction. PF has been shown to facilitate students’ conceptual knowledge acquisition in the mathematical domain. Collaboration has been described as a vital design component of PF, but studies that have investigated the role of collaboration in PF empirically so far, were not able to confirm the necessity of collaboration in PF. However, these studies have diverged significantly from prior traditional PF studies and design criteria. Therefore, the role of collaboration in PF remains unclear. In an experimental study that is based on the traditional design of PF, we compared a collaborative and an individual problem-solving setting. It was hypothesized that collaboration facilitates the beneficial preparatory mechanisms of the PF problem-solving phase: prior knowledge activation, awareness of knowledge gaps, and recognition of deep features. In a mediation analysis, the effects of collaborative and individual problem solving on conceptual knowledge acquisition as mediated through the preparatory mechanisms were tested. In contrast to the hypotheses, no mediations or differences between conditions were found. Thus, collaboration does not hold a major preparatory function in itself for the design of PF.

A question that many parents face when their child learns motor skills like riding a bicycle concerns the temporal order of instruction and self-determined practice. In physical education, a systematic evaluation of this temporal order on learning outcomes is so far missing. In the present study, ninety 7th graders learned javelin throwing. They either received instruction on the technical movement features of javelin throwing before self-determined practice (I-PS), or they started with self-determined practice and received the instruction afterwards (PS-I). We measured changes in the throwing distance and technical features of the throws. Students with PS-I, but not students with I-PS, significantly increased javelin throwing distance and improved technical features of their throws. We discuss potential underlying mechanisms of this effect, in particular the role of awareness of knowledge gaps through initial practice that could be elaborated on during subsequent instruction, and the suppression of implicit learning processes with instruction.

Diagnostics is an essential part of teachers’ profession. We investigated judgement accuracy and cognitive processes of judgment formation in physical-education teachers who observed exercises in gymnastics, and compared teachers with gymnastics trainers as a reference group. Judgment accuracy was lower in teachers compared to trainers, and agreement on the ratings was lower among teachers than among trainers. Agreement about the temporal structuring of the exercises was lower among teachers than trainers. We discuss these findings in context of the task to perform accurate judgments, and discuss the relevance of movement phases, their features, and their functionality for teacher training.

ZusammenfassungDer Forschungsstand zu diagnostischen Kompetenzen von Lehrkräften – als wesentliche Voraussetzung für Feedback oder adaptives Unterrichten – wird zurzeit als unbefriedigend angesehen, da kaum überzeugende Theorien über die kognitiven Prozesse bei der Genese diagnostischer Urteile bestehen. An dieser Stelle setzt das Rahmenmodell DiaCoM (Explaining Teachers’ Diagnostic Judgements by Cognitive Modeling) an. Das DiaCoM-Rahmenmodell bietet eine theoretische Basis für Forschungsansätze, die diagnostische Urteile von Lehrkräften als Informationsverarbeitungsprozesse erklären wollen. Es konzeptualisiert diagnostisches Urteilen in Bildungskontexten als kognitive Prozesse einer Lehrkraft über Schülerinnen und Schüler (z. B. deren Fähigkeit) oder über Anforderungen (z. B. Aufgabenschwierigkeiten) auf der Grundlage der Informationen, die explizit oder implizit in einer diagnostischen Situation bestehen. Es bezieht sich auf Theorien der kognitiven Informationsverarbeitung und erfordert eine Spezifikation von vier Komponenten: die Personencharakteristika, die Situationscharakteristika, das diagnostische Denken als Informationsverarbeitung und schließlich das Diagnoseverhalten. Der Beitrag stellt dar, wie das DiaCoM-Rahmenmodell als forschungsheuristisches Modell eingesetzt werden kann, um Erklärungswissen zur Genese diagnostischer Urteile zu generieren: Durch Spezifikation der informationsverarbeitenden Prozesse können theoretische Voraussagen darüber getroffen werden, welche Personen- und Situationscharakteristika zu welchem diagnostischen Verhalten führen. Diese Annahmen sind dann einer experimentellen Prüfung durch systematische Variation der Situation oder der Personen (z. B. durch Instruktion) zugänglich.

ZusammenfassungBei der Auswahl von Aufgaben für einen adaptiven Unterricht schätzen Lehrkräfte deren fachliche Anforderungen ein. Im Rahmen eines Modells der Informationsverarbeitung wird angenommen, dass bei solchen diagnostischen Urteilen über Aufgaben auf Basis fachdidaktischen Wissens schwierigkeitsgenerierende Merkmale identifiziert und diese hinsichtlich ihres Einflusses auf die Aufgabenschwierigkeit gewichtet werden. Dabei erfolgt die Verarbeitung von Oberflächen- und Tiefenmerkmalen unterschiedlich schnell und ist daher abhängig von der verfügbaren Zeit. Anliegen der Studie ist es, diese Annahmen über die kognitiven Prozesse bei der Urteilsbildung zu prüfen, indem Aufgabenmerkmale systematisch variiert und fachdidaktisches Wissen als Personenmerkmal sowie Zeitdruck als Situationsmerkmal experimentell variiert werden. Zur Prüfung der Modellannahmen werden bei Lehramtsstudierenden (N = 175) zwei Bedingungen verglichen: Einer Experimentalgruppe wird spezifisches fachdidaktisches Wissen über schwierigkeitsgenerierende Aufgabenmerkmale vermittelt. Ihre aufgabendiagnostischen Urteile werden durch paarweise Schwierigkeitsvergleiche erfasst und mit einer Kontrollgruppe verglichen – jeweils mit hoher und mit geringer Zeitrestriktion. Es zeigt sich, dass fachdidaktisches Wissen dazu führt, dass schwierigkeitsgenerierende Aufgabenmerkmale signifikant besser identifiziert und gewichtet werden, Zeitdruck hingegen hat signifikant negative Auswirkungen auf die Urteilsgüte. Die beschriebene Merkmalsvariation schlägt sich demnach hypothesenkonform in der Urteilsakkuratheit nieder, wobei die Prozesse der Identifizierung und der Gewichtung von schwierigkeitsgenerierenden Aufgabenmerkmalen unterschieden werden. Diese Ergebnisse erlauben Rückschlüsse auf die Bedeutung von spezifischem fachdidaktischem Wissen für diagnostische Urteile und geben damit Impulse für die Lehrkräfteaus- und -fortbildung. Die Unterscheidung der kognitiven Prozesse beim Einschätzen von Aufgabenschwierigkeit mit und ohne Zeitdruck legt nahe, dass sich Lehrkräfte hierüber während der Unterrichtsplanung (ohne Zeitdruck) und im Unterrichtsgeschehen (mit Zeitdruck) bewusst sein sollten, um reflektiert damit umgehen zu können.

Assessing students’ learning processes and products is considered a core requirement of effective teaching. As such, it is an object of research in several disciplines and research areas. To structure the various corresponding research perspectives and provide a broader, yet still systematic view of the field, we propose an overarching framework that allows for systematizing foci of interest, goals, methodologies, and theoretical premises as four integral aspects of conducting research in this area. We demonstrate the benefits of the overarching framework by using it as a tool to analyze and systematize previous research from four different research perspectives. Based on this framework, we discuss the strengths and limitations of existing studies and, in particular, highlight theoretical premises that are rarely explicitly addressed but become more obvious by using the overarching framework. In addition, we provide directions for future research by drawing connections within and between research perspectives. Our analyses illustrate how the overarching framework can provide a foundation for research synthesis and inform future studies.

Research on conceptual change, learning from errors, and ‘productive failure’ suggests that elaborating on errors during instruction is crucial for learning in settings which put problem solving before instruction. To reveal the role of elaboration and comparison of errors, we investigated the effect of prompting students to engage in such cognitive processes: Students in all conditions first worked on an identical problem-solving activity (targeting the comparison of fractions) without prior instruction about the targeted concepts and procedures. In the subsequent instruction phase, students in the different conditions received 1) only correct solutions, 2) correct and erroneous examples, or 3) additional comparison prompts. Students who were prompted to compare erroneous solution attempts to correct solutions significantly outperformed their peers at posttest. Elaborating on errors seemed to mediate this effect. In accordance with the theoretical assumptions, the difference between conditions was only significant for students whose initial solution resembled the erroneous examples.

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., Productive Failure: young students don’t learn productively with Productive Failure The beneficial effect of Productive Failure does not usually transfer to students who are younger than the typical age group. Productive Failure proposes that students gain more understanding if they are allowed to collaboratively struggle to solve problems before they are explicitly taught the solution. To investigate whether also younger students benefit from Productive Failure, Claudia Mazziotti and colleagues investigated whether problem-solving prior to as compared to after instruction leads to more understanding. Furthermore, they examined what role students’ collaboration played by comparing collaborative problem-solving with individual problem-solving. They found no advantage of Productive Failure for this age group but found first hints for the benefits of small groups’ collaboration. These data suggest that the extent to which students are capable to deal with the initial struggle and to collaborate might display boundary conditions for Productive Failure.

A diagnostic judgment of a teacher can be seen as an inference from manifest observable evidence on a student’s behavior to his or her latent traits. This can be described by a Bayesian model of inference: The teacher starts from a set of assumptions on the student (hypotheses), with subjective probabilities for each hypothesis (priors). Subsequently, he or she uses observed evidence (students’ responses to tasks) and knowledge on conditional probabilities of this evidence (likelihoods) to revise these assumptions. Many systematic deviations from this model (biases, e.g., base-rate neglect, inverse fallacy) are reported in the literature on Bayesian reasoning. In a teacher’s situation, the information (hypotheses, priors, likelihoods) is usually not explicitly represented numerically (as in most research on Bayesian reasoning) but only by qualitative estimations in the mind of the teacher. In our study, we ask to which extent individuals (approximately) apply a rational Bayesian strategy or resort to other biased strategies of processing information for their diagnostic judgments. We explicitly pose this question with respect to nonnumerical settings. To investigate this question, we developed a scenario that visually displays all relevant information (hypotheses, priors, likelihoods) in a graphically displayed hypothesis space (called “hypothegon”)–without recurring to numerical representations or mathematical procedures. Forty-two preservice teachers were asked to judge the plausibility of different misconceptions of six students based on their responses to decimal comparison tasks (e.g., 3.39 > 3.4). Applying a Bayesian classification procedure, we identified three updating strategies: a Bayesian update strategy (BUS, processing all probabilities), a combined evidence strategy (CES, ignoring the prior probabilities but including all likelihoods), and a single evidence strategy (SES, only using the likelihood of the most probable hypothesis). In study 1, an instruction on the relevance of using all probabilities (priors and likelihoods) only weakly increased the processing of more information. In study 2, we found strong evidence that a visual explication of the prior–likelihood interaction led to an increase in processing the interaction of all relevant information. These results show that the phenomena found in general research on Bayesian reasoning in numerical settings extend to diagnostic judgments in nonnumerical settings.

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.

Recently, there has been a growing interest in learning approaches that combine two phases: an initial problem-solving phase followed by an instruction phase (PS-I). Two often cited examples of instructional approaches following the PS-I scheme include Productive Failure and Invention. Despite the growing interest in PS-I approaches, to the best of our knowledge, there has not yet been a comprehensive attempt to summarize the features that define PS-I and to explain the patterns of results. Therefore, the first goal of this paper is to map the landscape of different PS-I implementations, to identify commonalities and differences in designs, and to associate the identified design features with patterns in the learning outcomes. The review shows that PS-I fosters learning only if specific design features (namely contrasting cases or building instruction on student solutions) are implemented. The second goal is to identify a set of interconnected cognitive mechanisms that may account for these outcomes. Empirical evidence from PS-I literature is associated with these mechanisms and supports an initial theory of PS-I. Finally, positive and negative effects of PS-I are explained using the suggested mechanisms.

Research on diagnostic competencies of teachers nowadays raises the question which person or situational characteristics moderate judgement accuracy. Beside this correlational approach, a stronger interest in understanding the cognitive processes involved in the genesis of diagnostic judgements has emerged. To address the theoretical gap regarding cognitive processes underlying diagnostic judgements, we propose a framework, called DiaCoM (Explaining Teachers’ Diagnostic Judgements by Cognitive Modeling). It aims at supporting (existing or envisioned) research that strives to test cognitively oriented explanations for processes and products of diagnostic judgements of teachers.

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.

We describe a Wizard-of-Oz study that investigates the impact of different types of feedback on students’ affective states. Our results indicate the importance of matching carefully the affective state with appropriate feedback in order to help students transition into more positive states. For example when students were confused affect boosts and specific instruction seem to be effective in helping students to be in flow again. We discuss this and other effective ways to and implications for the development of our system and the field in general.