Theresa Büchter

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.

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.

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.