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.

Bayesian reasoning has been intensively investigated, due to its high relevance. Prior research has focused on conventional Bayesian reasoning, i.e. the reasoning behind the calculation with probabilities. We extend this focus by studying covariational reasoning as part of extended Bayesian reasoning. We analyze whether covariational reasoning varies with regard to four different visualizations known to facilitate calculation, i.e. tree diagram, double tree, unit square and 2 x 2-table (RQ1). Furthermore, we study how people's strategy in calculation tasks affects their covariational reasoning in covariation tasks (RQ2). N = 221 undergraduate students participate in the experimental study. The results show significant differences in the covariational reasoning between visualizations: covariational reasoning is best with 2 x 2 tables and worst with tree diagrams. Also, the calculation strategy significantly influences covariational reasoning. Hence, the results provide insights about the support of visualizations for covariational reasoning and have implications on the teaching of extended Bayesian reasoning.

Background: Bayesian reasoning is understood as the updating of hypotheses based on new evidence (e.g., the likelihood of an infection based on medical test results). As experts and students alike often struggle with Bayesian reasoning, previous research has emphasised the importance of identifying supportive strategies for instruction. Aims: This study examines the learning of Bayesian reasoning by comparing five experimental conditions: two "level-2" training courses (double tree and unit square, each based on natural frequencies), two "level-1" training courses (natural frequencies only and a school-specific visualisation "probability tree"), and a "level-0" control group (no training course). Ultimately, the aim is to enable experts to make the right decision in high-stake situations. Sample: N = 515 students (in law or medicine) Method: In a pre-post-follow-up training study, participants' judgments regarding Bayesian reasoning were investigated in five experimental conditions. Furthermore, prior mathematical achievement was used for predicting Bayesian reasoning skills with a linear mixed model. Results: All training courses increase Bayesian reasoning, yet learning with the double tree shows best results. Interactions with prior mathematical achievement generally imply that students with higher prior mathematical achievement learn more, yet with notable differences: instruction with the unit square is better suited for high achievers than for low achievers, while the double tree training course is the only one equally suited to all levels of prior mathematical achievement. Conclusion: The best learning of Bayesian reasoning occurs with strategies not yet commonly used in school.

Previous studies on Bayesian situations, in which probabilistic information is used to update the probability of a hypothesis, have often focused on the calculation of a posterior probability. We argue that for an in-depth understanding of Bayesian situations, it is (apart from mere calculation) also necessary to be able to evaluate the effect of changes of parameters in the Bayesian situation and the consequences, e.g., for the posterior probability. Thus, by understanding Bayes’ formula as a function, the concept of covariation is introduced as an extension of conventional Bayesian reasoning, and covariational reasoning in Bayesian situations is studied. Prospective teachers ( N =173) for primary ( N =112) and secondary ( N =61) school from two German universities participated in the study and reasoned about covariation in Bayesian situations. In a mixed-methods approach, firstly, the elaborateness of prospective teachers’ covariational reasoning is assessed by analysing the arguments qualitatively, using an adaption of the Structure of Observed Learning Outcome (SOLO) taxonomy. Secondly, the influence of possibly supportive variables on covariational reasoning is analysed quantitatively by checking whether (i) the changed parameter in the Bayesian situation (false-positive rate, true-positive rate or base rate), (ii) the visualisation depicting the Bayesian situation (double-tree vs. unit square) or (iii) the calculation (correct or incorrect) influences the SOLO level. The results show that among these three variables, only the changed parameter seems to influence the covariational reasoning. Implications are discussed.

International literature is increasingly disclosing the relevance of cultural aspects within the processes of teaching and learning mathematics. Knowledge is inextricably linked to the activities in which the subjects engage, and must be considered in relation to the socio-cultural context wherein the activity takes place. Literature reveals the relationship between the background culture (e.g. language, nationality, etc.) of prospective elementary teachers and their beliefs about mathematics and its teaching. In this paper, we define culture with reference to the wider discussion from cross-cultural psychology literature about cultural values, and we investigate if and how differences in individuals' cultural values are related to prospective teachers' beliefs about being successful in mathematics and in its teaching. We adopt a questionnaire from the studies by Schwartz to measure participants' values. We assign each prospective teacher of our sample to a cluster of beliefs and we analyse how the beliefs of prospective teachers are related to their values. Results show that cultural values and beliefs about mathematics are related, while this is not the case for beliefs about the teaching of this subject.

Previous research on Bayesian reasoning has typically investigated people’s ability to assess a posterior probability (i.e., a positive predictive value) based on prior knowledge (i.e., base rate, true-positive rate, and false-positive rate). In this article, we systematically examine the extent to which people understand the effects of changes in the three input probabilities on the positive predictive value, that is, covariational reasoning . In this regard, two different operationalizations for measuring covariational reasoning (i.e., by single-choice vs. slider format) are investigated in an empirical study with N = 229 university students. In addition, we aim to answer the question wheter a skill in “conventional” Bayesian reasoning is a prerequisite for covariational reasoning.

Although research in mathematics education has yielded a great number of results on mathematics teachers’ beliefs in recent decades, it is still an open question as to whether these teachers’ beliefs are stable or not. However, since mathematics teachers’ beliefs are perceived to be the default of their classroom practices, it is important to gain insight into these beliefs and into their development. In this paper, we contribute to this field of research by providing an approach designed to analyse a change in teachers’ belief systems. We study a sample of 20 prospective teachers during their first practical experience. We analyse the structure of teachers’ belief systems with interviews and a questionnaire set twice within one year. To analyse development of these teachers’ belief systems, we assigned the young teachers of the first and second survey to clusters that were developed in a former study and re-analysed for our purposes here. Our results show considerable changes in mathematics teachers’ central beliefs during their first practical experience. However, a general pattern of belief changes is not found.

Research on fostering teachers’ diagnostic competence and thinking has become increasingly important. To this end, research has already identified several aspects of effective fostering of teachers’ diagnostic competence. One of the aspects is assignment of the role as a teacher in interventions but, so far, assignment of the role of student has hardly been considered. Based on a model of the diagnostic thinking process, this paper operationalizes the role of the student by solving specific tasks and the role of the teacher by analyzing student solutions. Furthermore, based on previous research, it is assumed that assigning both roles is effective in promoting diagnostic competence. The following research addresses the development of 137 prospective teachers’ diagnostic thinking in an experimental pre-post-test study with four treatment conditions, which vary prospective teachers’ working with tasks and students’ solutions to those tasks. The quantitative results show that a treatment integrating focus on tasks and students’ solutions is equally as effective as a treatment focusing solely on students’ solutions, and also that a treatment focusing solely on tasks has no effect.

Diagnostic competence is a significant part of teachers' professional competencies that influences the quality of teaching and students' learning. Therefore, developing teachers' diagnostic competence has become a relevant topic of educational research. In this regard, we present a quasi-experimental study that investigates the effect of an intervention on the promotion of diagnostic competence in prospective elementary teachers. The study follows a pre-post design. It comprises a treatment group and a control group (N = 75). The intervention takes place as part of a university seminar and includes components that are known to be effective for promoting diagnostic competence. We measure prospective teachers' diagnostic competence regarding so-called epistemic activities when assessing students' solutions to problem-oriented tasks. The results reveal our treatment to be effective, particularly regarding generating and reasoning hypotheses about students' abilities. With the specific focus on epistemic activities in the diagnostic process, our research contributes to developing and measuring diagnostic competence in a multi-perspective way.

Zusammenfassung Diagnostische Kompetenz ist eine zentrale Komponente der professionellen Kompetenzen von Lehrkräften, die die Qualität von Unterricht und somit das Lernen von Schülerinnen und Schülern beeinflusst. Aufgrund dieser zentralen Bedeutung rückt die systematische Schulung der diagnostischen Kompetenz zunehmend in den Forschungsfokus. Diese Arbeit zielt auf die Erforschung der Förderung diagnostischer Kompetenz im Bereich der Mathematik ab. Wir stellen eine quasi-experimentelle Studie mit Treatment- und Kontrollgruppe vor, die den Effekt einer Intervention bei Studierenden des Grundschullehramts ( n = 74) untersucht. Die Intervention erfolgt im Rahmen eines Seminars, das die als effektiv bekannten Bestandteile der Förderung diagnostischer Kompetenz aufgreift. Diagnostische Kompetenz modellieren wir als Fähigkeit, Lernprodukte von Schülerinnen adäquat und multiperspektivisch beurteilen zu können und messen deren Entwicklung in einem Pre-Post-Design. Die Ergebnisse zeigen, dass unsere Schulung insbesondere auf einen Teil sogenannter epistemischen Aktivitäten wirkt: die Entwicklung und Stützung von Hypothesen zu Fähigkeiten von Schülerinnen und Schülern. Hier unterscheidet sich die Treatmentgruppe im Post-Test signifikant von der Kontrollgruppe. Die Arbeit leistet insgesamt einen Beitrag zur Entwicklung und mehrperspektivischen Messung diagnostischer Kompetenz bezogen auf epistemische Aktivitäten im diagnostischen Prozess und der multiperspektivischen Beurteilung der Lernprodukte von Schülerinnen und Schülern zu offenen Lernangebote zur Arithmetik.

Questions involving Bayesian Reasoning often arise in events of everyday life, such as assessing the results of a breathalyser test or a medical diagnostic test. Bayesian Reasoning is perceived to be difficult, but visualisations are known to support it. However, prior research on visualisations for Bayesian Reasoning has only rarely addressed the issue on how to design such visualisations in the most effective way according to research on multimedia learning. In this article, we present a concise overview on subject-didactical considerations, together with the most fundamental research of both Bayesian Reasoning and multimedia learning. Building on these aspects, we provide a step-by-step development of the design of visualisations which support Bayesian problems, particularly for so-called double-trees and unit squares.

Bayesian Reasoning is both a fundamental idea of probability and a key model in applied sciences for evaluating situations of uncertainty. Bayesian Reasoning may be defined as the dealing with, and understanding of, Bayesian situations. This includes various aspects such as calculating a conditional probability (performance), assessing the effects of changes to the parameters of a formula on the result (covariation) and adequately interpreting and explaining the results of a formula (communication). Bayesian Reasoning is crucial in several non-mathematical disciplines such as medicine and law. However, even experts from these domains struggle to reason in a Bayesian manner. Therefore, it is desirable to develop a training course for this specific audience regarding the different aspects of Bayesian Reasoning. In this paper, we present an evidence-based development of such training courses by considering relevant prior research on successful strategies for Bayesian Reasoning (e.g., natural frequencies and adequate visualizations) and on the 4C/ID model as a promising instructional approach. The results of a formative evaluation are described, which show that students from the target audience (i.e., medicine or law) increased their Bayesian Reasoning skills and found taking part in the training courses to be relevant and fruitful for their professional expertise.

Patients need to be informed correctly and comprehensibly about the implications of their medical test results. Reasoning in such situations, where, for example, a medical test result is used to make inferences on a particular disease, is called Bayesian reasoning. Prior research mostly concentrated on the ability to correctly calculate risks in Bayesian situations (so-called performance) and repeatedly demonstrated that performance is very low—even among medical experts. The need to also study communication within Bayesian situations has been brought forward. Here, we broaden the focus of Bayesian reasoning and present first insights into a study where medical students participated in a training course on the aspect of performance and show that this already improves the ability to judge doctor–patient communication within Bayesian situations.

There have been intensive research efforts to improve Bayesian reasoning over the last 25 years. Much of this research focuses solely on improving performance on Bayesian tasks. In addition to performance, however, it is also important to establish an understanding of the effect on the positive predictive value when parameters of Bayesian formula are changed. We call this ability “covariation” in Bayesian tasks. To this end, training courses were developed to support understanding of covariation, based on strategies that have been proven helpful by previous studies concerning performance, using: (a) natural frequencies and (b) visualisations, i.e., double trees and unit squares. Results of a comparative study in a pre-, post-, and follow-up test design show that the developed training courses can improve understanding of covariation.

Background Functional thinking is characterized as a specific way of thinking in relationships, dependencies, and changes. Hence, beyond mathematics, it is also crucial for other (STEM) disciplines as well as for everyday situations. In particular, dealing with different representations of functions and changing between them are core function-related competencies, which are correspondingly needed for the formation of appropriate concepts and flexible problem-solving in various situations. Therefore, this study investigated students’ ( N = 856) competencies related to representational changes of elementary functions and, in particular, assessed which changes are especially easy or difficult for students. Moreover, possible school track and gender differences were investigated by performing DIF analyses within the framework of Rasch modeling. The data were gathered using a paper–pencil test administered after the students had completed the teaching unit on linear functions in their mathematics lessons. Results Altogether, students were found to have limited competencies related to representational changes of elementary functions. There was no clear pattern regarding the types of representational change that were difficult or easy for them. Moreover, girls performed better on purely mathematical tasks, whereas boys did better at a complex modeling and problem-solving task. Classes from the academic track produced better results in tasks with a situational context compared to their peers from non-academic tracks, who performed relatively strongly on purely mathematical tasks. Conclusions These findings imply that various representations and representational changes should be included in lessons on functions to support students in building a rich concept of function and flexible problem-solving skills, thus fulfilling curricular requirements and responding to didactical considerations. In particular, the teaching of functions should be more balanced by mixing tasks with and without a situational context and the corresponding representational changes. These findings should motivate teachers, in particular those teaching non-academic tracks, to give a more prominent role to situational contexts in their lessons on functions in order to foster their students’ learning and build a bridge between mathematics and real-world situations.

Prospective teachers often perceive a “double discontinuity” between school mathematics and university mathematics. The first discontinuity can be described as the belief that there is no coherence between school mathematics and university mathematics, which forms part of the notoriously problematic transition from school to university. The second discontinuity can be described as a belief about the lack of relevance of university mathematics for the later professional practice of prospective teachers. Beliefs about coherence and relevance have been known to impact prospective mathematics teachers’ interests and academic success. In this paper, we discuss an intervention involving 72 prospective secondary school mathematics teachers, aimed at influencing their beliefs about coherence and relevance. For this, we refer to the construct of beliefs as the main part of our theoretical framework, as well as the sub-constructs of beliefs regarding coherence and relevance. We then describe an intervention implemented within the first two years of mathematics courses, involving so-called “teacher-oriented” tasks that aim to trigger reflection on the benefit of university mathematics for teaching mathematics in school. The effect of the intervention was measured with a pretest-posttest experimental design using a questionnaire concerning teachers’ beliefs about coherence and relevance. Our results show that the prospective teachers’ beliefs about coherence and relevance generally decrease during the semester. However, statistically significant differences between the treatment group and a control group were found, especially regarding their beliefs about relevance.