IntroductionA considerable number of students enter secondary school with fundamental gaps in basic mathematical skills, particularly in their understanding of operations related to multiplication and division. The conceptual understanding of operations - operation sense - is the ability to relate situations (e.g., word problems) to mathematical-symbolic notations (e.g., calculations, equations) and vice versa and it is crucial for further learning in mathematics. In order to provide adaptive support for operation sense, the understanding or the lack thereof must be systematically assessed. Therefore, a sensitive assessment instrument specifically tailored to the operations of multiplication and division is needed, which can be used for a focused diagnosis and for evaluation of specific interventions in this domain. Thus, we developed an instrument for assessing multiplicative operation sense, which represents theoretically grounded levels of understanding operations in multiplicative situations.MethodsThe test was empirically validated in a pilot study (N = 66). Item responses were analyzed using general linear mixed models to investigate differences in solution rates across levels and to estimate the proportion of variance in item difficulty explained by the theoretically derived levels. Based on the pilot results, three items were revised and additional shortcomings of the study design were addressed in a main study (N = 464).ResultsIn the pilot study, general linear mixed models showed that - as expected - the estimated solution rate decreases with increasing level and that 86% of the variance in item difficulty can be explained by the four theoretically derived levels of multiplicative operation sense. The main study showed 94% explained variance and significant mean differences between all levels.DiscussionThese findings support the validity of the instrument for assessing multiplicative operation sense and its usefulness for both research and practice.

Bayesian reasoning requires the processing of data in probabilistic situations to revise risk estimations. Research has shown that this is difficult when data are presented as single-event probabilities; the multiplicative combination of priors and likelihoods often is disregarded, resulting in erroneous strategies such as prior neglect or averaging. Proportions (relative frequencies) are computationally equivalent to probabilities: They also require a multiplicative combination. However, proportions are connected to natural mental representations (ratio sense). More specifically, mental representations of nested proportions (e.g., 70% of 20%) allow for mental operations that correspond to multiplicative combinations. In three experimental studies, we avoided numerical calculations and focused on the conceptual understanding underlying Bayesian reasoning by administering tasks with bar chart representations without numbers. In all studies, we compared two conditions: The tasks were verbally framed either in terms of proportions or in terms of single-event probabilities. The studies revealed that the framing had no substantial effect on whether participants combined priors and likelihoods or neglected part of the information. However, the findings supported our hypothesis that the framing impacts how the information is combined. In line with our hypothesis, proportions increased the correct Bayesian judgment and reduced an incorrect averaging strategy-a strategy for combining information that was predominant with single-event probabilities. Thus, proportions appear as a natural view on combining information in Bayesian situations. (PsycInfo Database Record (c) 2026 APA, all rights reserved).

The Knowledge-Learning-Instruction (KLI) framework is a cognitive framework describing the causal interplay between knowledge, learning, and instruction. It has an explicit cross-domain scope and has had considerable impact in the learning sciences. Despite this, to the best of our knowledge, it has not yet been applied in research on instructed second language (L2) learning. This is notable because, although L2 research is sometimes siloed within the learning sciences, it has undergone a cognitive turn emphasizing domain-general learning mechanisms. We propose that a KLI-informed cognitive view on instructed L2 learning helps to systematically integrate the field into the learning sciences, encompassing not only domain-general learning mechanisms but also domain-general perspectives on knowledge and instruction. Importantly, this integration may generate novel, testable hypotheses in L2 research, grounded in an interdisciplinary understanding of knowledge, learning, and instruction. We present a cognitive KLI-based perspective on instructed L2 learning and discuss challenges of this perspective, including the fundamentally integrative nature of language knowledge, the inherently expressive character of language (verbal, written, signed), and the limited relevance of understanding and sense-making processes in instructed L2 learning. Moreover, we provide an illustrative KLI-informed analysis of two empirical L2 studies. Based on these analyses, we discuss novel, testable hypotheses generated through the KLI lens that offer interdisciplinary approaches to known issues in L2 pedagogy, such as integrating multiple induction- and refinement-enhancing instructional principles to mitigate individual differences in L2 classrooms.

Designing adaptive online teacher professional development (oTPD) requires an understanding of how both teacher characteristics and the nature of learning opportunities shape teacher engagement. While indicators of behavioral engagement (e.g., time spent) have received some attention, cognitive engagement-how deeply teachers process and reflect on learning opportunities-remains underexplored. This study investigates both behavioral and cognitive engagement across three types of learning opportunities (theoretical input, planning material, representations of practice) among N = 125 in-service teachers. Results indicate that aspects of behavioral engagement were only weakly related to cognitive engagement, and for some types of learning opportunities, not at all. In addition to prior knowledge, individual beliefs and perceived time resources emerged as relevant predictors of engagement. These findings highlight the importance of considering teachers' cognitive engagement more systematically in oTPD design and research.

Behavioral data analysis in physical education has traditionally relied on manual coding, a process that is time consuming, resource intensive, and prone to observer bias. To address these limitations, researchers and practitioners increasingly turn to machine learning methods to support the analysis of behavioral data. This paper outlines how two machine learning approaches, Computer Vision and Natural Language Processing, can be applied in physical education. Computer Vision enables the detection and tracking of students, identification of body landmarks and equipment, and assessment of movements as proxies of learning, useful in both classroom research and teacher training. Natural Language Processing methods are applied to transcribe, segment, and classify teacher utterances, providing scalable insights into verbal behavior. These methods enhance efficiency, analytical depth, and scalability while reducing the limitations of manual coding. Future progress will depend on improving model performance and building domain-specific data sets to fully unlock the potential of machine learning in naturalistic physical education settings.

Inquiry-based learning has been shown to foster conceptual understanding through cycles of hypothesis generation and testing, making it particularly relevant for conceptual change when prior knowledge conflicts with new concepts. While inquiry-based learning has been extensively applied in science education, its use in mathematics is still developing. For example, it is often connected to exploratory tasks or problem-solving phases prior to instruction (PS-I). The present study aims to address this gap by investigating whether digital inquiry, including explicit cycles of experimentation, can foster conceptual change in the transition from natural to rational numbers. Using the Scientific Discovery as Dual Search (SDDS) model, we designed a digital learning environment that enables students to generate hypotheses, create visual representations with dynamic fraction bars, test their hypotheses through experimentation, and revise their reasoning based on feedback. We examined the impact of prompts that required the use of digital tools for generating and manipulating dynamic fraction bars, as well as providing empirical feedback on students’ understanding of fractions in two contexts: basketball and color mixing. In an experiment with 231 fifth graders, we found a significant indirect effect of prompts to use the digital tools on the post-test, mediated by the quality of an external visual representation generated with dynamic fraction bars and verbal reasoning. Additionally, there was a positive and significant indirect effect of context on the post-test, favoring the basketball context, mediated by the quality of verbal reasoning. However, no effect of empirical feedback was found. The findings of this study suggest that using dynamic fraction bars and familiar contexts leads to more elaborated learning activities and helps students to shift from natural number concepts to understanding fractions.

Background Conceptual change emerges from cognitive conflicts between new information and prior understanding. Confronting learners with their incorrect solutions and asking them to compare these to correct solutions can facilitate successful acquisition of new concepts. This learning process of committing errors and comparing solutions occurs, for instance, in problem-solving prior to instruction (PS-I). However, it is not clear how well this comparison has to be aligned to students’ individual errors. Aims We examine if conceptual change is more effectively supported by comparing a correct solution to an individually aligned incorrect solution rather than to a typical incorrect solution example. Sample Participants were 184 sixth graders. Methods In a computer-based learning environment, learners solved a fraction problem. They then received: 1) the correct solution only (control condition), 2) a correct and an individually aligned incorrect solution (alignment condition), or 3) a correct and non-aligned incorrect solution (contra-alignment condition). Results Unlike in previous studies, using both correct and incorrect solutions (i.e., regardless of the alignment) did not outperform the correct solution only approach. However, learners in the alignment condition had the highest learning outcomes, supporting conceptual change theories. Intermediate knowledge after the problem-solving phase (visible in students’ solutions) significantly influenced learning outcomes. Prior knowledge predicted learning outcomes only in the alignment and the control condition, a trend not observed in the contra-alignment condition. Conclusions This study highlights that comparison processes can support conceptual change when tailored to individual misconceptions.

Students often encounter difficulties when learning and processing fractions. In fraction comparison tasks, they tend to rely on a mixture of successful strategies, e.g., fraction magnitude processing or benchmarking, and erroneous strategies, e.g., natural number-based reasoning or gap thinking. Reinhold et al. (2023) used a theory-driven approach to classify students into distinct profiles depending on their strategy choice based on their performance on a comparison task with 24 single-digit fractions. The authors identified single strategy profiles (e.g., typical natural number bias) and composite profiles (e.g., benchmarking or typical bias). The current study aimed to replicate this study (RO1) and extend its design by incorporating multi-digit fractions (i.e., the denominator has at least two digits; RO2) and additional biased comparison strategies, specifically gap thinking (RO3). A set of 101 fraction comparison tasks was administered to 285 fifth- and sixth-grade students in Flanders, Belgium, controlling for benchmarking to 1/2, numerical distance, natural number-based reasoning and gap thinking. A Bayesian classification approach based on students’ performance (accuracy, reaction time, and individual distance effect) replicated the distinct profiles identified by Reinhold et al. (2023), not only for single- (RO1) but also for multi-digit fractions (RO2). Furthermore, we identified additional single strategy and composite profiles (e.g., applying benchmarking where possible, otherwise gap thinking), both based on accuracy and reaction time data (RO3). This study enhances our understanding of individual differences in fraction processing and emphasizes the importance of controlling for gap thinking.

Research indicates that generating multiple solutions during a problem-solving phase before subsequent instruction (problem-solving prior to instruction) has a beneficial effect on the acquisition of conceptual understanding. This effect can be supported by visualizations that enable the use of multiple solution strategies through interactive manipulation. In a digital learning environment for fraction comparison tasks using dynamic fraction bars (i.e., fraction bars that can be interactively manipulated), we investigated how solution strategies adopted by students predicted their learning outcomes in relation to the basic (non-symbolic, representational) fraction concept and how learning in a dynamic learning environment compared to learning in a static environment (i.e., fraction bars without interactive manipulation) leads to a better basic fraction concept. The experiment included 92 fifth graders. We found a descriptive but nonsignificant trend of students in the dynamic condition performing better on a post-task than those in the static condition. However, there was a substantial aptitude–treatment interaction. In the static environment, students’ prior mathematical conceptual understanding of basic arithmetic operations of addition, subtraction, division, and multiplication of natural numbers had a significant effect on the learning outcomes, whereas in the dynamic learning environment, students’ prior mathematical knowledge did not influence their learning outcomes. Moreover, the quality of the solution strategies employed in the dynamic digital learning environment predicted the learning outcomes in the post-task. This finding shows that students who use the dynamic fraction bars effectively achieve a better learning outcome.

Akkurate Diagnosen sind von zentraler Bedeutung für die Förderung von Schüler*innen im Unterricht. In Bezug zum Bewegungslernen im Sportunterricht erfordert dies von Lehrpersonen, Bewegungsmerkmale akkurat zu beurteilen. Hierfür sind sowohl Fachwissen über relevante Bewegungsmerkmale und deren korrekte Interpretation als auch praktische Erfahrungen in der Beobachtung und Bewertung der Schüler*innen unerlässlich. Diese Arbeit widmet sich der Stärkung der diagnostischen Kompetenzen von Sportlehrkräften mittels einer digital gestützten Fortbildung auf Basis des Four-Component-Instructional-Design (4C/ID)-Modells von Van Merriënboer und Kollegen (1992; 1997). Die Fortbildung hat zum Ziel, Sportlehrkräfte dabei zu unterstützen, Bewegungsmerkmale akkurat zu diagnostizieren. Dabei wird der Erwerb von fachdidaktischem Wissen in der praktischen Anwendung gefördert. Im Rahmen der Fortbildung werden den Sportlehrkräften Aufgaben zur Analyse von Bewegungen anhand von Videovignetten gestellt, wobei der Fokus auf der Identifikation und Bewertung von Bewegungsmerkmalen beim Schlagwurf von Grundschülerinnen und Grundschülern liegt. Entsprechend der verschiedenen Komponenten des 4C/ID-Modells werden die Sportlehrkräfte mit authentischen Lernaufgaben konfrontiert und in der praktischen Umsetzung der Diagnose durch Teilaufgaben sowie durch die Bereitstellung von Fachwissen unterstützt. In einer Pilotstudie mit 8 Referendar*innen wurden erste Hinweise auf die positive Wirkung der Fortbildung gefunden, nämlich darauf, dass Bewegungsmerkmale anschließend akkurater diagnostiziert werden können. Es werden Möglichkeiten aufgezeigt, wie die Fortbildung in den verschiedenen Phasen der Lehrkräfteausbildung implementiert werden kann.

In the boxplot, the box always represents – regardless of its area – the middle half of the data and thus a measure of variability (interquartile range). However, when students first learn about boxplots, they are usual already familiar with other forms of statistical representations (e.g., bar or circle graphs) in which a larger area represents a higher frequency of observations. If students erroneously apply this well-established area-represents-frequency schema to boxplots, it results in a systematic error which we describe as the consequence of an incomplete conceptual change. We empirically validated difficulty-generating characteristics that allow the differentiation between item types with varying complexity (item level) and aimed to identify profiles (person level) that differ depending on which schema was used in which item type. For this purpose, we conducted two cross-sectional studies with N = 100 university students (study 1) and N = 297 participants who finished secondary school or higher (study 2) and used generalized linear mixed models (item level) and k-means clustering with predefined cluster centers (person level) to test our hypotheses. We could replicate the systematic error that was described in previous research and found new difficulty-generating characteristics in boxplot items. Our results support the notion of different profiles potentially emerging based on varying degrees of conceptual change. From an instructional perspective, information about individual progress in conceptual change could be considered for tailoring individualized interventions.

Der Beitrag betrachtet drei Typen der Evidenzorientierung beim mikroadaptiven Unterrichten: Theorieorientierung, Produktnutzung und Methodenadaption. Beim Diagnostizieren und Fördern als Kern mikroadaptiven Unterrichtens kann die Lehrkraft auf evidenzbasierte Theorien und Prinzipien zurückgreifen, dabei evaluierte Produkte und Materialien einsetzen und sich bei der Beobachtung und der Interpretation des Handelns der Schülerinnen und Schüler an wissenschaftlichen Methoden orientieren. Anhand konkreter Beispiele aus den Fächern Mathematik, Sport und Naturwissenschaften wird aufgezeigt, wie diese Evidenzbasierungstypen die Qualität der Diagnose und der Unterrichtsanpassung verbessern können.

In this paper, we draw on the distinction of states and traits to investigate the influence of humor and awareness prompts on solving word problems with a reflective component. Two studies with n = 144 and n = 401 mathematics pre-service teachers were conducted, using the three-item Cognitive Reflection test and a ten-item test on mathematical critical thinking, respectively. In both studies, students seeing jokes or awareness prompts showed significantly better results than students solving routine tasks before working on the test items. Drawing on the mindset theory by Gollwitzer, we explain these results with deliberate and implemental mindsets being induced by humoristic and awareness prompts prompts or routine tasks, respectively.

Wenn Lehrkräfte Mathematik unterrichten, sind sie gefordert, die Inhalte an die Voraussetzungen der Lernenden anzupassen. Grundlegend hierzu ist diagnostische Kompetenz, also die Fähigkeit, die Leistung von Lernenden oder die Qualität des Lernmaterials nach fachdidaktischen Kriterien angemessen zu beurteilen. In diesem Zusammenhang spielen Aufgaben eine bedeutende Rolle. Welche Aufgabe wann und vor allem für wen geeignet ist, darüber muss die Lehrkraft gute Entscheidungen treffen können. Ferner sollte sie fehlerhafte Schüler*innenlösungen schnell einordnen und geeignete Hilfestellung anbieten können. Wie insbesondere die Fähigkeit der Aufgabeneinschätzung als Facette diagnostischer Kompetenz geschult werden kann, soll in diesem Artikel verdeutlicht werden. Dazu haben wir auf Grundlage aktueller Forschungsbefunde zum Einfluss von fachdidaktischem Wissen auf die Kompetenz der diagnostischen Aufgabeneinschätzung im Bereich Funktionen eine Fortbildung konzipiert, die den Prinzipien des Instruktionsmodells 4-Component-Instructional-Design (4C/ID) folgt. Das Modell formuliert auf der Basis lernpsychologischer Befunde Prinzipien, wie Lernsituationen für komplexe Fähigkeiten gestaltet sein sollten. Die hier beschriebene Fortbildung ist eine Weiterentwicklung eines Hochschulseminars mit ähnlichem Inhalt. Im Rahmen einer wissenschaftlichen Evaluation dieses Seminars konnten bedeutende Lerneffekte auf die Fähigkeiten zur effizienten Aufgabeneinschätzung der Studierenden gezeigt werden. Ziel der hier vorgestellten Fortbildung ist es, die komplexe diagnostische Fähigkeit der Einschätzung von Aufgaben und Schüler*innenlösungen zu unterstützen. Dieser Beitrag zeigt exemplarisch, wie eine Fortbildung nach dem 4C/ID-Modell gestaltet werden kann, um komplexe Kompetenzen von Lehrkräften in einer kurzen Fortbildungsmaßnahme zu fördern.

The four-component instructional design (4C/ID) model is designed to support complex learning by facilitating the transfer of theoretical knowledge into practice. This study presents a systematic research synthesis on the implementation of the 4C/ID model in training programs with a special focus on teachers. Specifically, we investigate how the four components and 10 steps of 4C/ID are applied in training programs, for which professional fields and real-life tasks it is used, the concreteness of the instructional design, and the effects on learning. A special focus is on the model's implementation in teacher education. A systematic database search following PRISMA guidelines yielded 55 relevant studies, which were systematically coded and analyzed. Surprisingly, we found only a few papers of n = 11 on in-service training with 4C/ID for teachers. Our findings indicate that many studies referencing the 4C/ID model lack detailed descriptions of its implementation. To complement our qualitative synthesis, we conducted a quantitative subanalysis, applying a multilevel meta-analysis to estimate the overall effect size of training programs based on pre-post comparisons. The results yielded a moderate positive effect (g = 0.76, SE = 0.31, p = 0.014), suggesting that the 4C/ID model has a meaningful impact on learning outcomes. These findings underscore the need for more structured reporting and further research on the implementation of 4C/ID, particularly in teacher education.

Dieser Beitrag zeigt auf, wie Lehrkräfte fachdidaktisches Wissen nutzen können, um Fehlkonzepte von Schüler*innen zutreffend zu diagnostizieren – insbesondere in Situationen, in denen ein identifiziertes Fehlermuster in Aufgabenbearbeitungen durch unterschiedliche Fehlkonzepte erklärt werden kann. Im Fokus stehen also mehrdeutige Diagnosesituationen. Ausgehend von einer idealtypischen Diagnose von Fehlkonzepten in mehrdeutigen Diagnosesituationen bei Aufgabenbearbeitungen zum Größenvergleich von Brüchen und zur Einschätzung von Wahrscheinlichkeiten werden zunächst die Anforderungen an eine Lehrkraft und die zu deren Umsetzung notwendigen diagnostischen Kompetenzen explizit gemacht. Anschließend wird ein entwickeltes Fortbildungsmodul vorgestellt, welches zum Ziel hat, diese Kompetenzen praxisbezogen durch geeignete vignettenbasierte Lernaufgaben systematisch aufzubauen. Die Wirksamkeit des entwickelten Fortbildungsmoduls wurde in einer experimentellen Studie mit 40 Lehrkräften empirisch überprüft. Die Ergebnisse dieser Studie bestätigen die positive Wirkung des Fortbildungsmoduls (im Vergleich zu einer Kontrollgruppe) auf die diagnostische Kompetenz der Lehrkräfte, Fehlkonzepte in mehrdeutigen Diagnosesituationen zutreffend zu diagnostizieren.

Im Beitrag wird ein Fortbildungskonzept nach dem 4C/ID-Modell zur Förderung diagnostischer Kompetenzen bei Lehrkräften mit Hilfe eines digitalen Diagnosetools vorgestellt. Dabei stehen authentische Lernaufgaben im Mittelpunkt, die nach dem 4C/ID-Modell den Wissenstransfer begünstigen sollen. Im Rahmen der Fortbildung arbeiten die Lehrkräfte in einer digital gestützten Lernumgebung, in der die Auswahl, Bewertung und das Validieren relevanter diagnostischer Informationen für das Begründen diagnostischer Urteile gefördert werden. Zudem sind weitere Komponenten des 4C/ID-Modells (wie die unterstüt-zenden Informationen) in das digitale Tool integriert und Prozeduralisierungshilfen werden flankierend durch die Fortbildner*innen bereitgestellt. Analysen deuten darauf hin, dass die Lernumgebung zum diagnostischen Begründen anregt. Diskutiert wird, inwieweit das simulationsbasierte Lernen nach dem 4C/ID-Modell einen aussichtsreichen, noch weiter auszubauenden Ansatz für die Lehrkräftefortbildung darstellt, insbesondere auch im Hinblick auf den Transfer von Fortbildungsinhalten in die Schulpraxis.

Seit Jahrzehnten findet man im deutschsprachigen Raum begründete Praxisvorschläge für Übungsformate im Mathematikunterricht, ebenfalls werden in der empirischen Lehr-Lernforschung Instruktionsmodelle des Übens untersucht. Während die empirischen Studien meist ihren Fokus auf das Konsolidieren von Faktenwissen und Fertigkeiten richten, ist den Praxisvorschlägen eher eine breitere Sicht auf Übungsziele zu eigen: Das Konsolidieren bezieht sich dann sowohl auf Fertigkeiten als auch auf begriffliches Verständnis, oft auch beides im Zusammenhang. Der vorliegende Beitrag diskutiert die verschiedenen Forschungsstände, sowie die möglichen Konsequenzen für die Praxis des Mathematikunterrichts und für wünschenswerte weitere Forschung zum Üben. Für die Gestaltung von Übungsphasen braucht es zunächst eine fachliche Spezifizierung der Übungsziele (vor allem mathematische Begriffe und mathematische Verfahren und Strategien) und nachfolgend eine Identifikation der zugehörigen psychologischen Wissensarten (Faktenwissen, Fertigkeiten und Verständnis) und der Funktion des Übens (Sichern, Flexibilisieren oder Vertiefen). Diese Festlegungen erlauben dann die Auswahl und Ausgestaltung geeigneter Übungsformate, von denen die folgenden fünf Typen – jeweils mit ihrer theoretischen Fundierung, empirischer Evidenz und Überlegungen zur praktischen Umsetzung – vorgestellt werden: (1) Abrufübungen, (2) verteiltes Üben (vor allem bei deklarativem Faktenwissen oder einfachen Fertigkeiten), (3) verschachteltes Üben mit der impliziten Möglichkeit für Vergleiche (insbesondere bei leicht zu verwechselnden Fakten oder Fertigkeiten) oder mit expliziten Vergleichsaufforderungen (auch für komplexere Wissensarten wie prozedurale Flexibilität oder Verständnis), (4) elaboratives Üben (mit einem Fokus auf Verständnis) sowie (5) kombinierendes Üben als „intelligente“ Verbindung mehrerer Übungsziele. Während die ersten drei Übungsformate systematisch beforscht sind, findet man die beiden letzten eher unter dem Sammelbegriff „produktives Üben“ in lerntheoretisch begründbaren, aber wenig beforschten Praxisvorschlägen.