In der vorliegenden Studie werden die Präferenz und die Performanz von Schüler:innen der Sekundarstufe bei der Nutzung der beiden schulüblichen Visualisierungen Baumdiagramm und Vierfeldertafel sowie einer rein textuellen Darstellung zur Lösung Bayesianischer Aufgaben untersucht. Bayesianisches Denken ist ein zentraler Bestandteil der Risikokompetenz (Risk Literacy) und essenziell für fundierte Entscheidungen, etwa in Fragen der Rechtsprechung oder bei der Beurteilung von Gesundheitsfragen. Visualisierungen können dabei helfen, die Problemzusammenhänge zu verdeutlichen. Jedoch ist derzeit noch unklar, welche Visualisierungen Problemlöser:innen von sich aus bevorzugen. In der vorliegenden Studie wurden zur Beantwortung dieser Frage mittels Eyetracking und Kurzinterviews die Präferenzen hinsichtlich je zwei gleichzeitig präsentierten Darstellungsarten von insgesamt 36 Schüler:innen der Klassenstufen 5/6 und 10 in einem dreifaktoriellen experimentellen Design untersucht. Dabei wurden empirisch die Präferenzvergleiche a) Baumdiagramm vs. Vierfeldertafel, b) Baumdiagramm vs. Text, und c) Vierfeldertafel vs. Text implementiert. Die Ergebnisse zeigen, dass das Baumdiagramm insgesamt gegenüber der Vierfeldertafel präferiert wird. Die rein textuelle Darstellung der Informationen wird nur sehr vereinzelt einer der beiden Visualisierungen vorgezogen. Die Präferenz, aber auch die Performanz werden zudem von der Klassenstufe beeinflusst. So zeigt sich zwar das Baumdiagramm in beiden Klassenstufen als bevorzugte Visualisierung, in Klassenstufe 10 wird jedoch individuell auch häufig die Vierfeldertafel präferiert, während in Klassenstufe 5/6 gelegentlich auch die rein textuelle Präsentation der statistischen Informationen präferiert wird.
In addition to basic calculation skills, mathematical competencies that focus on understanding relationships and structures are also gaining importance in primary school education. Functional relationships are a key component of this. Currently, in Germany only proportional relationships are routinely addressed. The project presented in this paper investigates a possible approach to non-proportional linear functions via figural growing patterns. The approach used here is to directly prompt the children to colour an additive constant in the pattern in order to make the rate of change visible at the same time. Although colouring approaches are worked on in previous studies, the colouring of a constant approach is novel and, moreover, primary school children have not been engaged with it before now. The results of an exploratory study with over 200 third and fourth graders indicate that the approach can be fruitful and that highly significant correlations and medium size effects between colouring and functional thinking strategies can be revealed. However, the children’s functional thinking strategies demonstrate that the project approach of colouring a constant does not necessarily result in explicit functional thinking. It may therefore be worthwhile to pursue the approach in more depth in follow-up studies.
The concepts of probability and risk are closely related, yet they stand in a complex relationship that poses particular challenges from a didactic perspective. While both objectivist and subjectivist axiomatic theories exist for probability, no comparable theoretical consensus has been established for the concept of risk. Risk is primarily associated with individual decisions, consequences, and evaluations, whereas probability, as a property of repeatable situations, can also be applied to singular cases. The assessment of consequences proves to be substantially more subjective than the measurement of probabilities, and measures of risk derive their meaning only through comparisons among alternative courses of action. This paper aims to provide a foundational clarification of the concepts of Risk Literacy and Risk Competence. To this end, a hermeneutic line of argument is developed that relates statistical and probabilistic concepts to intuition as a guiding factor in human cognition. Three central dimensions are identified: statistical literacy, probability literacy, and intuitive thinking. These dimensions are not understood as additive components but as interwoven perspectives on risk competence. The paper is explicitly intended as an exploratory conceptual positioning rather than as a reductive competence model. Drawing on paradigmatic contexts, risk competence is presented as a multidimensional construct that extends beyond the mere understanding of probabilities. In this way, a theoretical framework is developed that prepares the ground for a didactic engagement with risk without simplifying its complexity or prematurely operationalising it. It is argued that Risk Literacy and, building upon it, Risk Competence emerge from the interplay of probabilistic conceptions, statistical interpretation, and intuitive patterns of thinking. The respective context plays a decisive role, influencing decision-making in risk-laden situations to varying degrees depending on intuitive judgments.
Die Begriffe Wahrscheinlichkeit und Risiko sind eng miteinander verknüpft, stehen jedoch in einem spannungsreichen Verhältnis, das aus didaktischer Sicht besondere Herausforderungen mit sich bringt. Während für Wahrscheinlichkeit sowohl objektivistische als auch subjektivistische axiomatische Theorien vorliegen, existiert für den Risikobegriff kein vergleichbarer theoretischer Konsens. Risiko ist primär mit Einzelentscheidungen, Konsequenzen und Bewertungen verbunden, während Wahrscheinlichkeit als Eigenschaft wiederholbarer Situationen auch auf Einzelfälle bezogen werden kann. Die Einschätzung von Folgen erweist sich dabei als wesentlich subjektiver als die Messung von Wahrscheinlichkeiten, und Risikomaße erhalten ihren Sinn erst im Vergleich alternativer Entscheidungsoptionen. Der Beitrag verfolgt das Ziel, die Begriffe Risk Literacy und Risikokompetenz grundlagenorientiert zu klären. Dies geschieht mittels einer hermeneutischen Argumentation, die statistische und probabilistische Konzepte mit Intuitionen als erkenntnisleitendem Faktor in Beziehung setzt. Dabei werden drei zentrale Dimensionen herausgearbeitet: statistische Literalität, probabilistische Literalität und intuitive Psychologie. Diese werden nicht additiv verstanden, sondern als miteinander verschränkte Perspektiven auf Risikokompetenz. Der Beitrag versteht sich ausdrücklich als explorative Verortung und nicht als reduktives Kompetenzmodell. Anhand paradigmatischer Kontexte wird Risikokompetenz als mehrdimensionales Konstrukt sichtbar gemacht, das über das bloße Verstehen von Wahrscheinlichkeiten hinausgeht. Auf diese Weise wird ein theoretischer Orientierungsrahmen entwickelt, der die didaktische Auseinandersetzung mit Risiko vorbereitet, ohne die Komplexität des Gegenstands zu simplifizieren oder vorschnell zu operationalisieren. Im Beitrag wird gezeigt, dass Risk Literacy und darauf aufbauend Risikokompetenz auf dem Zusammenspiel von probabilistischen Vorstellungen, statistischer Einordnung und intuitiven Deutungsmustern basieren. Dabei spielt der jeweilige Kontext eine entscheidende Rolle, der die Entscheidungsfindung in risikobehafteten Situationen mehr oder weniger stark – abhängig von der intuitiven Einschätzung – beeinflusst.
Risk is seen as a relevant topic in school across all disciplines. However, there is a need for research in mathematics education, especially regarding the design of curriculum-embedded instruction units. The Risk-Design project addresses this gap. It is a design-based research project aiming to develop a teaching unit to promote the handling of risk in stochastics lessons. In this article we present the analysis of the first of eight lessons. This lesson introduces expected value in the context of multiple coin tosses. It investigates decision-making processes under uncertainty and learners' reasoning processes that accompany these decision-making processes. It combines two theoretical approaches. A literacy concept of risk and the inferential view of conceptualization embedded in the social practice of giving and asking for reasons. These approaches are used for the lesson design and its empirical investigation by using videography. The data analyses follow a reconstructive approach aiming to characterize learners' decision-making and reasoning practices. The main results are the identification of teaching phenomena regarding decision-making processes, a reasoning model with four modes of informal stochastic reasoning, and the finding that risk competency can already be activated in the initial lesson on the topic of risk. In the practice, these results can make teachers aware of the design of lessons on the topic of risk. In particular, the model has diagnostic potential and may reveal developmental nuances of reasoning in other decision-making situations. Theoretically, the model points to different reasoning qualities extending previous research on informal stochastical reasoning.
In primary arithmetic lessons, manipulatives should be used to support the development of number concepts, number and task relationships, and concepts of operations. For this development of concepts to succeed, manipulatives must be specifically selected, introduced and supervised by the teacher. It is assumed, that teachers' implicit beliefs, among other factors, guide their actions. This study focusses on the beliefs of primary school teachers regarding the use of manipulatives in primary arithmetic lessons. Episodic interviews were conducted with twelve primary school teachers and analysed using the documentary method to gain insights into explicit and implicit beliefs. Four sensegenetic typologies could be reconstructed, each with two types of beliefs, with one type having beliefs close to didactical standards and the other having beliefs (rather) far from didactical standards. We further found that inconsistencies arise between explicit and implicit beliefs. Four profiles of teachers were identified across the typologies: a didactically close and a didactically far profile as well as two mixed profiles. Overall, our findings point to a discrepancy between the findings of mathematics education research and teachers' beliefs, from which conclusions can be drawn for the initial and further training of teachers.
Risk literacy includes the ability to accurately verbalize statistic information, such as proportions and probabilities. Research on conditional probabilities and Bayesian tasks has shown that using visualizations and so-called "natural frequencies" (e.g., "80 out of 100 people") instead of probabilities (expressed as percentages) can reduce errors in determining probabilities. However, little is known about the reverse process-the verbalization of information presented in visualizations-although these verbalizations are also considered essential for the development of conceptual knowledge. Therefore, this study investigated how well students succeed in verbalizing proportions and natural frequencies presented in visualizations. A paper-and-pencil test was conducted with 138 secondary school students (grade 9), in which participants were asked to describe the meaning of the statistic information presented in probabilistic visualizations as detailed as possible. The study focuses on the influence of the visualization (tree diagram vs. net diagram) and the information format of the visualization (proportion in percentages vs. natural frequencies) on the accurate verbalization of different types of relationships (joint information vs. conditional information). The results show, among other things, that information expressed in natural frequencies is verbalized significantly better compared to proportions in percentages, and that joint information expressed in percentages is better verbalized using net diagrams than using tree diagrams. Furthermore, the findings provide insights into typical (incorrect) verbalizations by students and could serve as a basis for developing language-sensitive instruction on proportions and probabilities in school-level stochastics classes.
In times of crisis, when people are faced with an ongoing series of unstable situations characterized by uncertainty, they lack the qualitative statistical evidence needed to help them deal with risks. Therefore, critically understanding and dealing with uncertain events in a risk-literate manner cannot be based on the comprehension of evidence and the calculation of risks. However, individuals who are risk literate may still succeed in uncertain situations. We hypothesize that two facets of risk literacy—numeracy and graph literacy—can facilitate learning about decision cues and beneficial strategies in situations where qualitative evidence is lacking. Our aim is to indicate how risk literates use beneficial rules of thumb as strategies for overcoming everyday life challenges. We conducted an online survey with a sample of adults from Germany (N = 632) to investigate whether self-reported sensitive use of beneficial rules of thumb in everyday life (as opposed to those with unclear benefits) is related to numeracy, graph literacy and risk literacy, as measured by a composite outcome. A cluster analysis was performed to explore different demographic profiles of sensitive use. There was no evidence of an association between numeracy, graph literacy and risk literacy and a sensitive use of beneficial rules of thumb. Rules of thumb were more likely to be reported as being used with increasing age. Although facets of risk literacy theoretically help one to learn about decision cues and beneficial strategies, they are not necessarily related to more sensitive use of heuristic decision-making strategies in situations of uncertainty. It seems likely that a relevant group of low-literate people will practice heuristic decision-making under uncertainty to their benefit. We discuss implications for teaching decision-making under uncertainty.
Mathematical problem solving is a highly complex competence that needs to be supported throughout the learning process. This study examines the role of problem-solving heuristics among 73 sixth-grade students in a German secondary school (Gesamtschule). Specifically, we focus on the empirical relationship between the occurrence of heuristics, their suitability, and application, as well as their contribution to problem-solving success. To this end, we developed a heuristic training that introduced strategy keys (heuristic aid cards) accompanied by explanatory videos. By leveraging variations in the successful application of the heuristics between students and across multiple sessions (73 students in 4 training situations and 3 problems in a final test), we discover from regression models with student and task-fixed effects that a successful use of heuristics (i.e., an heuristic occurred, that heuristic had a high suitability regarding the problem, and it was applied correctly) leads to more successful mathematical problem solving. Regarding the underlying mechanism, we find that all dimensions of heuristic use (occurrence, suitability, and application of heuristics) are positively related to problem-solving success. When comparing the predictive capacity of all three aspects, we identified the correct application of heuristics as particularly conducive for better problem-solving success. In comparing the group of students with high to those with low initial math competence, we find no significant differences in the relationship between heuristic use and problem-solving success across achievement groups. Our results emphasize the crucial role of heuristic application in mathematical problem solving, which generally benefits students regardless of their overall math competence.
Learning difficulties with respect to division have been documented and discussed during the last three decades in the German as well as in the international literature. There have been several proposals for changing traditional teaching approaches. At the same time various international studies indicate that young children successfully solve context-related partitive and quotitive division problems even before they start school. However, frequently these studies were conducted many years ago and/or have various limitations. The study presented in this paper seeks to examine the early division skills of a total of 537 first and second graders working on paper-pencil tests with quotitive (n(A) = 291) and partitive (n(V) = 246) division problems with and without remainder before division was formally introduced in their classrooms. The tasks were set in familiar contexts for young children and presented as drawings combined with verbal instructions. The results demonstrate that the children who took part in the study were highly successful in solving the tasks irrespective of the underlying basic concept, i.e. quotioning or partitioning. However, children's drawings turned out to be rather diverse and they differ significantly in terms of quotioning and partitioning. Implications for classroom practice and further research are discussed in conclusion.
Das Thema Risiko wird interdisziplinär als relevantes schulisches Thema angesehen. In der Mathematikdidaktik besteht jedoch Forschungs- und Entwicklungsbedarf, insbesondere in Hinblick auf curricular eingebettete Unterrichtseinheiten. Das Projekt Risk-Design greift dieses Desiderat auf. Es ist ein Design-Based Research Projekt mit dem Ziel, eine Unterrichtseinheit zu entwickeln, die den Umgang mit Risiko im Stochastikunterricht fördern soll. Im vorliegenden Beitrag stellen wir die Untersuchung der ersten von insgesamt acht Unterrichtsstunden vor. In diesem Unterricht geht es um erste Erfahrungen mit dem Erwartungswert im Kontext eines mehrfachen Münzwurfs. Beforscht werden Entscheidungsprozesse unter Unsicherheit sowie die Begründungsprozesse der Lernenden, die diese Entscheidungsprozesse begleiten. Zwei theoretische Ansätze werden dazu kombiniert. Ein „Literacy-Konzept“ zu Risiko und das inferentialistische Verständnis von Begriffsbildung, das sich in der sozialen Praxis des Verlangens und Lieferns von Gründen zeigt. Diese Ansätze werden für das Unterrichtsdesign und seine empirische Untersuchung mittels Videographie genutzt. Die Datenanalysen erfolgen rekonstruktiv mit dem Ziel, die Entscheidungs- und Begründungspraxis der Lernenden bei Entscheidungen unter Unsicherheit zu charakterisieren. Zentrale Ergebnisse sind die Identifizierung von Unterrichtsphänomenen zu den Entscheidungsprozessen, ein Begründungsmodell mit vier Modi informell-stochastischen Begründens, sowie der Befund, dass bereits in der Einführungsstunde zum Thema Risiko Kompetenzfacetten zu Risiko aktiviert werden können. Die Relevanz der Ergebnisse geht theoretisch und praktisch über die aktuelle Erkenntnissituation hinaus. Praktisch können diese Ergebnisse Lehrkräfte für die Gestaltung von Unterricht zum Thema Risiko sensibilisieren, insbesondere hat das Begründungsmodell diagnostisches Potenzial und kann Entwicklungsnuancen von Begründungen auch in anderen Entscheidungssituationen erschließen. Theoretisch weist das Modell auf unterschiedliche Begründungsqualitäten hin, die die bisherige Forschung zum informell-stochastischen Begründen erweitern.
Risk literacy is increasingly important in today’s society. It involves the adequate understanding of risks for informed decision making. Probability is central for defining risks and thus, an adequate understanding of probabilities (in the mathematical world) is crucial for risk literacy. Further, risk literacy is characterised by a switch between the mathematical world and the risks’ context. As language is used for communicating in both (mathematical world and risks’ context), language reception and production may be particularly relevant for risk literacy. Therefore, we study receptive and productive tasks in risk literacy. Thereby, we focus on typical Bayesian reasoning tasks including conditional probabilities. We vary the verbal expression of conditional probabilities by manipulating the syntactic structure (conditional clause vs. noun phrase) and the sequence of condition and conditional (condition in the beginning vs. end). We measure the ability to determine conditional probabilities based on these varying given verbal expressions (i.e., receptive tasks) and study students’ verbal expression of conditional probabilities (i.e., productive tasks). Moreover, we study the influence of language proficiency on receptive and productive tasks. According to our study with N = 124 secondary-school students, differences in the sequence of condition and conditional are more influential than the syntactic structure in receptive tasks, implying no inherent difficulty in specific linguistic expressions. Further, performance in receptive tasks only partly predicts performance in productive tasks, suggesting a need for explicit training of verbalising probabilities. Finally, language proficiency predicts performance in both tasks.
Risk is an integral part of modern societies and therefore school education, and in particular mathematics education, should develop students’ risk literacy. Students’ risk literacy is part of the interdisciplinary research project siMINT (Understanding complex STEM topics: Using simulations to promote competences for the 21st century). One first challenge in this project was to find a common ground for the inconsistently conceptualised terms of risk and risk literacy. Consequently, the aim of this paper is to contribute to the conceptualisation of risk and risk literacy. The first part of this paper provides a literature review on these areas, resulting in a working conceptualisation of the two concepts. We further carried out two consecutive steps of a Delphi study with N = 15 and N = 12 experts respectively, to develop a common conceptualisation based on the plurality of different dimensions and elements of risk and risk literacy. Based on the results of the first step of the Delphi study, we modified our working conceptualisation, using a common model of risk and risk literacy with different dimensions. In the second step of the Delphi study, the experts comment on (i) the dimensions and (ii) the elements of risk and risk literacy in these dimensions. The results showed that the experts are able to locate their individual definitions of risk in the model with three dimensions (relation between risk/uncertainty, connotation, and mathematical object) that we developed and to formulate reasons for their allocation. Furthermore, the results revealed that risk literacy consists of subsets of mathematical and non-mathematical elements that the experts rate differently with regard to their importance. In general, the results contribute to clarifying the constructs of risk and risk literacy as a basis for developing approaches to improve risk literacy.
Schwierigkeiten im Kontext des Lehrens und Lernen der Division in der Grundschule wie auch der Sekundarstufe I werden bereits seit über 30 Jahren in (inter-)nationaler Literatur dargestellt und diskutiert. Entsprechend wurden immer wieder auch Änderungen im Bereich der Vermittlung der Division gefordert. Demgegenüber deuten Studien deuten darauf hin, dass Kinder bereits vor Schulanfang erfolgreich Divisionsaufgaben zu den Grundvorstellungen Aufteilen und Verteilen lösen. Diese Studien liegen jedoch bereits viele Jahre zurück und/oder weisen verschiedene methodische Limitationen auf. Die vorliegende Studie untersucht Kompetenzen von insgesamt 537 Erst- und Zweitklässler*innen mithilfe von Paper-Pencil-Tests zum Aufteilen (nA = 291) sowie Verteilen (nV = 246). Die ausgewählten Aufgaben mit und ohne Rest greifen Kontexte auf, mit denen junge Kinder vertraut sind. Sie wurden den Kindern bildlich präsentiert, die Instruktionen erfolgten mündlich durch die Lehrkräfte. Die Ergebnisse weisen darauf hin, dass die an der Studie teilnehmenden Kinder bereits über nennenswerte Kompetenzen in Bezug auf Aufteilen und Verteilen verfügen, bevor die Division formal im Unterricht eingeführt und behandelt wurde. Die von den Kindern im Rahmen ihres Lösungsprozesses oder zur Dokumentation ihrer Lösungen angefertigten Zeichnungen sind insgesamt heterogen und unterscheiden sich deutlich in Abhängigkeit der zugrunde liegenden Aufteil- und Verteilsituationen. Abschließend werden Implikationen sowohl für die Unterrichtspraxis als auch für die weitere Forschung diskutiert.
The interaction of early childhood teachers and children is crucial for the development of mathematical skills. Designing mathematically stimulating interactions is a complex task that should be given greater emphasis in the training and professional development of early childhood educators. To develop targeted qualification measures, it is necessary to understand which characteristics of the educator, the child, and the situation are associated with high-quality mathematics-specific interactions. Empirical evidence on this topic is still limited.This exploratory study examined potential associations between the quality of mathematics-specific interactions of educators (N = 38) and various individual characteristics of both educators and children. Furthermore, it investigated whether interaction quality differs across two games addressing different mathematical content areas. The results showed that both the child's performance in the game and the educator's perception of the child's ability were significant predictors of interaction quality in one of the two games. There was no significant correlation between the interaction qualities of the two games; however, they also did not differ significantly. These findings are discussed with regard to practical implications for educator training as well as methodological limitations.
Die Interaktion zwischen Fachkraft und Kind ist entscheidend für die Entwicklung mathematischer Fähigkeiten. Mathematisch anregende Interaktionen zu gestalten ist eine komplexe Aufgabe, die in der Aus- und Fortbildung frühpädagogischer Fachkräfte verstärkt berücksichtigt werden sollte. Um gezielte Qualifizierungsmaßnahmen zu entwickeln, ist es erforderlich zu verstehen, mit welchen Merkmalen der Fachkraft, des Kindes und der Situation eine hohe mathematikspezifische Interaktionsqualität zusammenhängt. Bisher liegen hierzu jedoch nur wenige empirische Befunde vor. In dieser explorativen Studie wurden mögliche Zusammenhänge zwischen der Qualität mathematikspezifischer Interaktionen von Fachkräften (N = 38) und verschiedenen Personenmerkmalen von Fachkraft und Kind untersucht. Zudem wurde geprüft, ob sich die Interaktionsqualität in zwei Spielen, die unterschiedliche mathematische Inhaltsbereiche adressieren, unterscheidet. Die Ergebnisse zeigten, dass sowohl die Spielleistung des Kindes als auch das Maß, wie die Fachkraft die Fähigkeit des Kindes einschätzt, signifikante Prädiktoren der Interaktionsqualität in einem der beiden Spiele darstellten. Zwischen den Interaktionsqualitäten der beiden Spiele bestand keine signifikante Korrelation, sie unterschieden sich jedoch auch nicht signifikant. Diese Ergebnisse werden im Hinblick auf praktische Implikationen für die Fachkräftequalifizierung sowie methodische Limitationen diskutiert.
In Germany, not only the basic conditions for teaching mathematics, but also its content and technical requirements have changed significantly since the political change in 1989-1990. These curriculum and teaching changes took place-at least in part-very differently in the eastern and western German states. The more recent changes, which include the introduction of a central Abitur and the establishment of a common pool of tasks for the Abitur examination, aim to standardize the requirements and make final grades comparable. Nevertheless, a large number of empirical studies show that learners in upper secondary school still show deficits in the area of scientific propaedeutics and that their performance varies greatly from federal state to federal state. Since scientific propaedeutics education should focus on teaching various forms of mathematical argumentation, the question arises to what extent the central Abitur examination faces learners in federal states with different educational policy backgrounds with uniform requirements or to what extent the integration of pool tasks into the Abitur examination contributes to standardizing those requirements.To answer this question, this paper analyzes Abitur examination tasks from 1981-2023 from the perspective of justification. For this purpose, the federal states of Thuringia and Bavaria were selected, both are states with a permanent centrally organized, structurally similar Abitur examination, but with a different education policy background. On the one hand, the findings show fluctuations over time, that can be linked to educational policy decisions in the federal states under investigation. On the other hand, there is a clear shift in emphasis from formal, deductive proof to lower levels of argumentation in both the regions examined. Overall, it can be stated that the requirements in central Abitur examinations cannot be assessed without considering the respective educational policy situation and are not per se the same in two different federal states. The study also showed that the use of the common task pool in the federal states under investigation is attended by the quantitative and qualitative alignment of requirements related to argumentation.
This paper investigates students' use of language when dealing with negative numbers. On the one hand, theoretical and methodological considerations are made as to what 'students' use of language' actually consists of and how a methodical approach to it can be found. On the other hand, these considerations are used to gain initial empirical insights into the students' use of language when dealing with negative numbers. For this purpose, lexical means used by 7th grade students when explaining the concept of negative numbers in writing are collected and analysed in three steps: descriptive, didactic-normative and explanatory. Possible indications of students' individual mental representations will be discussed for three linguistic phenomena that could be observed in students' language.
The graph-as-picture error is prevalent when students are asked to match contextual graphs and realistic images. Previous research has not yet sufficiently revealed how and why this error emerges in students’ approaches. This paper presents an exploratory empirical study analyzing students’ approaches to matching a contextual graph and a realistic image in the racing car context, focusing on the characteristics of approaches that lead to the graph-as-picture error and those that do not. Using eye movement data and verbal utterances from 19 ninth graders, we traced students’ approaches. We analyzed the characteristics of students’ approaches and developed a model comprising five characteristic approach components that describe how students connect graph, speed, track features, and realistic images. Applying this model to individual students’ approaches, we found that when the graph-as-picture error did not emerge, students engaged in a three-step process: They (1) extracted relevant information about the speed from the graph, (2) translated what this information means for the real-world situation (track features) and (3) linked this back to the realistic images by establishing the connection between the shape of the realistic image and the real-world situation (track features). In contrast, the graph-as-picture error tended to emerge when students made an error in one of these steps or directly merged the graph and realistic image based on visual similarity. Our findings suggest that the graph-as-picture error should be considered in a differentiated, process-oriented way rather than viewed as a simple error category to facilitate tailored support for students.
In secondary mathematics education, students are expected to apply procedures not only correctly but also flexibly, for example when solving quadratic equations. Tasks that deliberately stimulate cognitive and metacognitive processes support the development of such flexibility. The aim of this paper is to investigate the relationship between students' learning gains and the activation potential of the tasks provided. To this end, prompts included in tasks on solving quadratic equations based on a sample of 39 classes. In parallel, the learning gains of N = 739 students in these classes were measured over the course of an instructional unit on rule-based solving of quadratic equations.Linear regression analyses showed that (a) the development of strategy flexibility and procedural knowledge depends on the number of tasks that prompt students to compare worked-out solutions; (b) the development of strategy flexibility and conceptual knowledge depends on the number of tasks with metacognitively activating prompts; and (c) a high number of tasks with prompts that prescribe a specific procedure tend to hinder the development of strategy flexibility. These findings indicate that the activation potential of task prompts is a significant predictor of learning gains in solving quadratic equations.
