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To adapt teaching to the prerequisites of students, teachers have various options at their disposal to gather and process information as the basis to form a judgment, such as carrying out tests, talking to and observing the behavior of students, or administering tasks. The complexity of such a judgment arises from the multitude of observations and their different possible explanations. This complexity might be reduced when teachers focus on one hypothesis instead of considering multiple hypotheses, interpret information in a confirmatory way, and not collect diagnostically relevant information. However, in this way, they run the risk of undesirable biased judgments. It therefore seems important to improve diagnostic judgments by selecting and processing information in a more reflective way. Research indicates that if information on a student is not easily available but restricted (e.g., by time pressure, difficult access to the student or high effort), a teacher who wants to make a careful decision is forced to rely on more reflective processes in the selection of tasks and in the interpretation of solutions. The present experimental study therefore investigates how the restricted availability of information in a specific diagnostic situation—when diagnostically inexperienced prospective mathematics teachers determine misconceptions in decimal fractions—influences the underlying cognitive processes. We assume that restricting the availability of information on student behavior augments the attentional focus and therefore reduces cognitive biases. Such more reflective processing can be observed by an increased time spent per piece of information, which should lead to the processing of relevant information and further increase judgment accuracy. To investigate these hypotheses, prospective teachers without prior knowledge in diagnosing misconceptions (N = 81) were asked to diagnose misconceptions on decimal fractions of virtual students by collecting information on students’ solutions. Data concerning the effects of restricting the availability of information on teachers’ cognitive processes were analyzed. The results show that with restricted information, participants indeed select a greater proportion of diagnostically relevant tasks, which positively influences judgment accuracy. These results are discussed with respect to their significance for framing teacher training and for further research.

Tasks in which learners are asked to compare two data sets using box plots and decide which distribution contains more observations above a given threshold have already been investigated in research. There are indications that these tasks are solved schema-based and that different (correct and erroneous) schemas are used depending on the arrangement of the quartiles around the threshold. Erroneous schemas can cause systematic errors and are often based on typical misconceptions. For example, if learners did not complete the conceptual change and assume that in box plots – like in most other statistical representations (e.g., bar or circle diagrams) - more (box) area also represents more observations, they decide the task according to which box plot shows more box area above the threshold. However, this can lead to incorrect answers, as the box area does not represent frequency but the range of the middle half of the data (interquartile range) and thus a measure of variability. So far, these schema-based reasoning processes have mainly been investigated via differences in solution rates of congruent and incongruent items. The present study investigates whether eye-tracking data can help to better understand which information is processed in the different schemas. Our research interest is based on hypotheses specifying which box plot components are significantly involved in the different schemas. We assume that the gaze patterns of learners using different schemas differ both regarding the number and duration of fixations on the relevant box plot components (areas of interest) and in terms of the number of transitions between them. We asked N = 14 participants to solve congruent and incongruent items and simultaneously collected eye movement data. In the analysis, we first used the solution rates to assign the schemas most likely used. Subsequently, the eye-tracking data were analyzed regarding differences in line with our hypotheses. We found hypothesis-compliant effects in all schemas regarding the number of fixations and transitions, but not regarding fixation duration. These results not only validate the schemas identified in previous studies, but also indicate that the schemas differ primarily in terms of which quartile is focused.

Trotz vielfältiger Bemühungen, Problemlösen als prozessbezogene Kompetenz nachhaltig in den Mathematikunterricht zu integrieren, sind problemlösende Elemente im deutschen Mathematikunterricht nach wie vor unterrepräsentiert. Eine Möglichkeit, dem entgegenzuwirken, stellen Fortbildungsangebote für Mathematiklehrkräfte dar. Wie ein solches Fortbildungsangebot zum Problemlösen in Zeiten der Digitalisierung gestaltet werden kann, wird im folgenden Beitrag dargestellt. Ein besonderer Fokus liegt dabei auf der Gestaltung eines digitalen Angebots, das von den Teilnehmenden flexibel und adaptiv genutzt werden kann. Basierend auf Rückmeldungen von 166 Lehrkräften aus drei Durchführungen der Online-Fortbildung konnten das Angebot optimiert und Implikationen für zukünftige Konzeptionen vergleichbarer Fortbildungsangebote (Online-Fortbildung mit synchronen und asynchronen Elementen) herausgearbeitet werden.

IntroductionDiagnostics is an essential part of teachers’ profession. We investigated judgment accuracy and cognitive processes underlying judgment formation in physical education teachers who observed tasks in gymnastics, and compared teachers with gymnastics trainers as a reference group.MethodsTeachers and trainers judged performance of prepuberal students in gymnastics, namely students exercising squat vault, underswing, and handstand. To investigate cognitive processes of judgment formation, participants were asked to structure the movements via event segmentation as well as to explain their judgments. All teachers and trainers had experience in working with prepuberal children similar to those they observed in this experiment, and the teachers completed a gymnastics class during their studies.ResultsJudgment accuracy (with reference to judgments made by expert trainers) was found to be significantly lower in teachers compared to trainers (p < 0.001). Moreover, agreement on the ratings among teachers was lower than among trainers. Agreement about the temporal structuring of the tasks from event segmentation was lower among teachers than among trainers (p < 0.05). When explaining their ratings, trainers referred more often than teachers to kinematic features of the task that were relevant to the judgments.DiscussionWe discuss these findings in context of the teachers’ task to perform accurate judgments. For suggestions on teacher training, we particularly emphasize the relevance of implementing knowledge about kinematic features of the tasks and student errors into real-life scenarios resembling the complex skill of making accurate judgments in the physical education classroom.

The selection of tasks based on the evaluation of task features can be considered a core practice of teaching and a relevant component of teaching quality. This is typically part of teachers' preparation for their classroom teaching, which prompts the following question: What are the characteristics of the tasks that teachers use when selecting tasks for differentiated teaching? To answer this question, we analyzed systematic differences in the focus of 78 in-service high school and lower secondary school teachers during the evaluation of task features. The teachers had to select eight tasks about the practice of fractions with respect to their differentiation potential--operationalizing their adaptive teaching competence from a mathematics educational perspective. To analyze the differences, we performed a cluster analysis of the task features that the teachers drew upon. Three groups of teachers could be identified with variations in their focus on directly or indirectly relevant, domain-specific or domain-general task features. Taking into account such variations may explain differences in teaching quality and student outcomes and may be relevant when designing teacher professional development programs.

Self-efficacy is an important predictor of learning and achievement. By definition, self-efficacy requires a task-specific assessment, in which students are asked to evaluate whether they can solve concrete tasks. An underlying assumption in previous research into such assessments was that self-efficacy is a one-dimensional construct. However, empirical evidence for this assumption is lacking, and research on students’ performance suggests that it depends on various task characteristics (e.g., the representational format). The present study explores the potential multi-dimensionality of self-efficacy in the topic of linear functions. More specifically, we investigate how three task characteristics – (1) the representational format, (2) embedding in a real-life context, or (3) the required operation – are related to students’ self-efficacy. We asked 8th and 9th graders (N = 376) to evaluate their self-efficacy on specific linear function tasks which systematically varied along the three dimensions of task characteristics. Using confirmatory factor analysis, we found that a two-dimensional model which includes the task characteristic of real-life context (i.e., with vs. without a real-life context) fitted the data better than other two-dimensional models or a one-dimensional model. These results suggest that self-efficacy with linear functions is empirically separable with respect to tasks with vs. without a real-life context. This means that in their self-evaluation of linear function tasks students particularly rely on whether or not the linear function task is embedded in a real-life context. This study highlights the fact that even within a specific content domain students’ self-efficacy can be considered a multi-dimensional construct.

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.

Ein entscheidender Faktor für erfolgreiches Unterrichten und hohen Lernzuwachs ist die Anpassung des Unterrichts an die Lernvoraussetzungen der Schülerinnen und Schüler. Daher werden im Rahmen des adaptiven Unterrichtens unter anderem Aufgaben ausgewählt, welche den individuellen Voraussetzungen der Lerngruppe entsprechen. Die von den Lehrkräften hierfür benötigen Fähigkeiten beinhalten die Diagnose der Lernvoraussetzungen und der in den Aufgaben enthaltenen Anforderungen. Die Lehrkräfteausbildung kann den Grundstein für solche diagnostischen Fähigkeiten legen, aber auch bei erfahrenen Lehrkräften gilt es diese Kompetenzfacette in Lehrerfortbildungen weiterzuentwickeln. Die hier vorgestellte Studie beschreibt eine Maßnahme für die Förderung der diagnostischen Kompetenz im Bereich der Bruchrechnung bei Lehrpersonen aus der Praxis. Dabei wurden die teilnehmenden Personen im Bereich des fachdidaktischen Wissens hinsichtlich schwierigkeitsgenerierender Aufgabenmerkmale bei Textaufgaben mit konkreten Beispielen aus der Bruchrechnung fortgebildet. Die Kurzintervention verzeichnet große Effekte für die Experimentalgruppe im Vergleich zu einer Warte-Kontrollgruppe und kann somit als Hinweis für den Einfluss von Wissensfacetten bei Aus- und Fortbildungsmaßnahmen im Bereich der diagnostischen Kompetenz dienen. Die Urteilsakkuratheit der Einschätzung von Aufgabenschwierigkeit zeigt sich bei erfahrenen Lehrkräften stabil gegen den Einfluss von Zeitrestriktion. Dieses Ergebnis unterstreicht die Auswirkungen von Berufserfahrung auf Denkprozesse und wird als Facette professioneller Kompetenz gedeutet.

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.

Teachers' ability to accurately judge difficulties of mathematical tasks is an essential aspect of their diagnostic competencies. Although research has suggested that pedagogical content knowledge (PCK) is positively correlated with the accuracy of diagnostic judgments, experimental studies have not been conducted to investigate how PCK affects perception and interpretation of relevant task characteristics. In an intervention study with a control group, 49 prospective mathematics teachers judged the difficulty of 20 tasks involving functions and graphs while an eye tracker tracked their eye movements. Some of the tasks included characteristics well known to be difficult for students. Participants' domain-specific PCK of typical student errors was manipulated through a three-hour intervention, during which they learned about the most common student errors in function and graph problems. We found that the process of perception (relative fixation duration on the relevant area in the tasks) was related to judgment accuracy. Pre-post comparisons revealed an effect of the intervention not only on participants' domain-specific PCK of typical student errors but also on their perception and interpretation processes. This result suggests that domain-specific PCK of typical student errors allowed participants to focus more efficiently on relevant task characteristics when judging mathematical task difficulties. Our study contributes to our understanding of how professional knowledge makes teachers' judgment processes of mathematical tasks more efficient.

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.