DFG-Project: TrainBayesSchool

Project leaders: Andreas Eichler (University of Kassel), Stefan Krauss (University of Regensburg), Karin Binder (Paderborn University), Markus Vogel (Heidelberg University of Education)

Being able to evaluate hypotheses in uncertain situations by using Bayes' formula is an essential part of probabilistic thinking, which should already be promoted in school. Therefore, approaches to understanding Bayesian situations (i.e., situations in which Bayes' formula can be applied) have been discussed in mathematics education for decades. Beyond school, Bayes' formula is central to probability theory and highly relevant in many professions such as medicine or law as well as in everyday life. However, numerous findings of psychologically oriented research, but hardly ever directly related to learning in school, show that laymen as well as experts often fail to apply Bayes' formula adequately. Based on these findings, strategies have been developed in psychology and mathematics education to increase competence in Bayesian situations. These include natural frequencies as a format of statistical information, their visualization, and training courses, although school-based instruction has hardly been studied. In the previous DFG project ("Training Study on Bayesian Reasoning"), the strategies were combined to derive conditions for success in promoting Bayesian reasoning among the professional groups of medicine and law, for which Bayesian situations are highly relevant. The present follow-up application aims at adapting the strategies of a psychologically oriented training developed and systematically combined in the previous DFG project for the learning at school of students of grade 11, taking over the controlled instruction from the previous project and integrating it into the learning at school. As an innovation, the aspects "Calculation" (calculation of a positive-predictive value), the situation-appropriate "Communication" as well as the "Covariation" (effect of parameter changes on the positive-predictive value), which were focused on in the previous DFG project, will be investigated as extended Bayesian thinking for learning in school. In school, Calculation and Covariation relate to functional thinking and Communication to the interpretation and communication of a model. For comparing the different instructional approaches, two optimal training courses with natural frequencies will also be implemented with the in school more commonly used probabilities. Additionally, two other school-specific curricular approaches will be designed. The students are taught in class and assigned to the different conditions, whereby the instructional phases are strongly controlled, but the project as a whole progresses from the strongly controlled experimental design to the weaker controlled, but more ecologically valid learning in school.  

Analysis of erroneous strategies in Bayesian Reasoning

Bayes‘ formula is a fundamental model for estimating risks. It is of essence in probability theory and in various professions such as medicine and law. Yet, multiple research results from cognitive psychology and mathematics education suggest that learners as well as experts struggle immensely when judging Bayesian situations (we thereby understand situations in which Bayes’ formula can be applied). Based on the relevance of adequately judging Bayesian situations, much empirical research is carried out in order to detect influences on the performance of participants in studies about Bayesian situations. Two strategies are particularly relevant for increasing performance, first the format of the statistical information in a Bayesian situation in so-called natural frequencies (e.g., “80 out of 100 people” instead of probabilities such as “80%”) and second the visualization of the statistical information. So far, it has been shown that depending on the support the performance of Bayesian situation varies between 5% without any support, 25% if the statistical information is given in natural frequencies and 60-75% for comprehensive support in form of a visualization in combination with natural frequencies. Despite all different strategies, which may be used in Bayesian situations, a considerable proportion of erroneous strategies is still observed in all samples. Yet, so far studies on identifying the erroneous strategies are rare. However, empirical results on typical erroneous strategies are central as they are part of every learning process. Moreover, to know about erroneous strategies is important for regulating the learning process and therefore for successful learning. After all, it can also be useful to know about erroneous strategies in the sense of “error-knowledge”. The few contributions on erroneous strategies in Bayesian situations have shown that different typical strategies are used by the participants of the studies. However, some of the studies find partially contrasting results. The differences may be based on different contexts of the Bayesian situation, the format of the statistical information, the visualization or the question type. Therefore, it is a central interest of this project to systematize and broaden the knowledge on erroneous strategies in Bayesian situations. For that, we plan to systematically vary the format of the statistical information (natural frequencies vs. probabilities), the kind of visualization and the question type for the first time. The results should contribute to the understanding of the construct of Bayesian Reasoning, which goes beyond simply measuring the performance of participants.

Being able to evaluate hypotheses in uncertain situations by using Bayes' formula is an essential part of probabilistic thinking, which should already be promoted in school. Therefore, approaches to understanding Bayesian situations (i.e., situations in which Bayes' formula can be applied) have been discussed in mathematics education for decades. Beyond school, Bayes' formula is central to probability theory and highly relevant in many professions such as medicine or law as well as in everyday life. However, numerous findings of psychologically oriented research, but hardly ever directly related to learning in school, show that laymen as well as experts often fail to apply Bayes' formula adequately. Based on these findings, strategies have been developed in psychology and mathematics education to increase competence in Bayesian situations. These include natural frequencies as a format of statistical information, their visualization, and training courses, although school-based instruction has hardly been studied. In the previous DFG project ("Training Study on Bayesian Reasoning"), the strategies were combined to derive conditions for success in promoting Bayesian reasoning among the professional groups of medicine and law, for which Bayesian situations are highly relevant. The present follow-up application aims at adapting the strategies of a psychologically oriented training developed and systematically combined in the previous DFG project for the learning at school of students of grade 11, taking over the controlled instruction from the previous project and integrating it into the learning at school. As an innovation, the aspects "Calculation" (calculation of a positive-predictive value), the situation-appropriate "Communication" as well as the "Covariation" (effect of parameter changes on the positive-predictive value), which were focused on in the previous DFG project, will be investigated as extended Bayesian thinking for learning in school. In school, Calculation and Covariation relate to functional thinking and Communication to the interpretation and communication of a model. For comparing the different instructional approaches, two optimal training courses with natural frequencies will also be implemented with the in school more commonly used probabilities. Additionally, two other school-specific curricular approaches will be designed. The students are taught in class and assigned to the different conditions, whereby the instructional phases are strongly controlled, but the project as a whole progresses from the strongly controlled experimental design to the weaker controlled, but more ecologically valid learning in school.