Bruckmaier, Georg
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Didaktische Kompetenzen von Mathematiklehrkräften. Weiterführende Analysen aus der COACTIV-Studie
An eye-tracking study of statistical reasoning with tree diagrams and 2 x 2 tables
Changing the information format from probabilities into frequencies as well as employing appropriate visualizations such as tree diagrams or 2 × 2 tables are important tools that can facilitate people’s statistical reasoning. Previous studies have shown that despite their widespread use in statistical textbooks, both of those visualization types are only of restricted help when they are provided with probabilities, but that they can foster insight when presented with frequencies instead. In the present study, we attempt to replicate this effect and also examine, by the method of eye tracking, why probabilistic 2 × 2 tables and tree diagrams do not facilitate reasoning with regard to Bayesian inferences (i.e., determining what errors occur and whether they can be explained by scan paths), and why the same visualizations are of great help to an individual when they are combined with frequencies. All ten inferences of N = 24 participants were based solely on tree diagrams or 2 × 2 tables that presented either the famous “mammography context” or an “economics context” (without additional textual wording). We first asked participants for marginal, conjoint, and (non-inverted) conditional probabilities (or frequencies), followed by related Bayesian tasks. While solution rates were higher for natural frequency questions as compared to probability versions, eye-tracking analyses indeed yielded noticeable differences regarding eye movements between correct and incorrect solutions. For instance, heat maps (aggregated scan paths) of distinct results differed remarkably, thereby making correct and faulty strategies visible in the line of theoretical classifications. Moreover, the inherent structure of 2 × 2 tables seems to help participants avoid certain Bayesian mistakes (e.g., “Fisherian” error) while tree diagrams seem to help steer them away from others (e.g., “joint occurrence”). We will discuss resulting educational consequences at the end of the paper.
Visualizing the Bayesian 2-test case: The effect of tree diagrams on medical decision making
In medicine, diagnoses based on medical test results are probabilistic by nature. Unfortunately, cognitive illusions regarding the statistical meaning of test results are well documented among patients, medical students, and even physicians. There are two effective strategies that can foster insight into what is known as Bayesian reasoning situations: (1) translating the statistical information on the prevalence of a disease and the sensitivity and the false-alarm rate of a specific test for that disease from probabilities into natural frequencies, and (2) illustrating the statistical information with tree diagrams, for instance, or with other pictorial representation. So far, such strategies have only been empirically tested in combination for “1-test cases”, where one binary hypothesis (“disease” vs. “no disease”) has to be diagnosed based on one binary test result (“positive” vs. “negative”). However, in reality, often more than one medical test is conducted to derive a diagnosis. In two studies, we examined a total of 388 medical students from the University of Regensburg (Germany) with medical “2-test scenarios”. Each student had to work on two problems: diagnosing breast cancer with mammography and sonography test results, and diagnosing HIV infection with the ELISA and Western Blot tests. In Study 1 (N = 190 participants), we systematically varied the presentation of statistical information (“only textual information” vs. “only tree diagram” vs. “text and tree diagram in combination”), whereas in Study 2 (N = 198 participants), we varied the kinds of tree diagrams (“complete tree” vs. “highlighted tree” vs. “pruned tree”). All versions were implemented in probability format (including probability trees) and in natural frequency format (including frequency trees). We found that natural frequency trees, especially when the question-related branches were highlighted, improved performance, but that none of the corresponding probabilistic visualizations did.
Prediction of elementary mathematics grades by cognitive abilities
In the present study, the relationship between the mathematics grade and the three basic cognitive abilities (inhibition, working memory, and reasoning) was analyzed regarding possible alterations during elementary school. In a sample of N = 244 children, the mathematics grade was best predicted by working memory performance in the second grade and by reasoning in the third and fourth grades. Differentiation of these abilities during elementary school was considered as a cause for this pattern but discarded after the analysis of structural equation models. Thus, with respect to output-orientated curricula, scholastic standards, and a large inter-individual heterogeneity of students, it is implied for teachers to account for different cognitive strengths and weaknesses of their students, using adequate tasks and teaching strategies like self-differentiating tasks and adaptive explorative learning.