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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.
Der 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.
Research on productive failure suggests that attempting to solve a problem prior to instruction facilitates conceptual understanding compared to receiving instruction prior to problem solving. The assumptions are that during the problem-solving phase, students activate their prior knowledge, become aware of their knowledge gaps, and discover deep features of the target content, which prepares them to better process the subsequent instruction. Unclear is whether this effect results from merely changing the order of the learning phases (i.e., instruction or problem solving first) or from additional features, such as presenting problem-solving material in the form of cases that differ in one feature at a time. Contrasting such cases may highlight the deep features and provide grounded feedback to students’ problem-solving attempts. In addition, the effect of the order of instruction and problem solving on procedural fluency is still unclear. The present experiment (N = 181, mean age = 14.53) investigated in a 2 × 2 design the effects of order (instruction or problem solving first) and of contrasting cases in the problem-solving material (yes/no) on conceptual understanding and procedural fluency. Additionally, the quality and quantity of students’ solution attempts from the problem-solving phase were coded. Regarding the learning outcomes, the ANOVA results suggest that for procedural fluency instruction prior to problem solving was more beneficial than problem solving prior to instruction. Merely delaying instruction did not increase conceptual understanding. The contrasting cases did not affect the quality of solution attempts, nor the posttest results. As expected, students who received instruction first generated fewer, but higher-quality solution attempts.
Bei 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.
Dass bei pädagogischen Entscheidungen die Heterogenität der Lernenden berücksichtigt werden muss,ist keine neue Erkenntnis. Viele unterschiedliche Strategien des Differenzierens werden schon seit Jahrzehnten diskutiert, in der täglichen Praxis umgesetzt und auch empirisch untersucht. Ein Überblick über wichtige Ansätze.
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.
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.
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.
We use numbers and fractions every day, for example when we are doing our shopping or baking a cake. But mathematics is, of course, much more: it is the language of science, or, to use Galileo's words, “the book of Nature is written in mathematical language” (Galileo, 1623) and some mathematical competencies beyond basic arithmetic are required in most professions. Basic mathematics, i.e., elementary arithmetic, elementary geometry and some elements of calculus, is taught in school, not just for everyday life, but as a tool for many different professions. In school, however, mathematics is either “loved” or “hated”, as Hersh and John-Steiner masterfully describe in their book “Loving and Hating Mathematics” (Hersh and John-Steiner, 2010). Research in mathematics education has definitely contributed to reducing school students' hatred of mathematics and this reduction may be seen as one of its many goals.
In contrast with mathematics, the field of mathematics education is strongly interdisciplinary; the closest field to influence it directly is psychology. In fact, mathematics education is consistently shaped by both behavioral and cognitive perspectives, since so many factors—the power of visualizations, the effect of representation formats, but also factors like gender, self-efficacy, etc.—influence and sometimes determine students' performance.
Our aim for this Research Topic and for the collection of papers we are now publishing has thus been to illustrate the relevance of such various psychological perspectives for mathematics education using the contributions of colleagues from around the world. All the contributions we have collected address these interdisciplinary perspectives explicitly or implicitly.