@article{59437, author = {{Steib, Nicole and Büchter, Theresa and Eichler, Andreas and Binder, Karin and Krauss, Stefan and Böcherer-Linder, Katharina and Vogel, Markus and Hilbert, Sven}}, issn = {{0959-4752}}, journal = {{Learning and Instruction}}, publisher = {{Elsevier BV}}, title = {{{How to teach Bayesian reasoning: An empirical study comparing four different probability training courses}}}, doi = {{10.1016/j.learninstruc.2024.102032}}, volume = {{95}}, year = {{2024}}, } @inproceedings{59986, author = {{Garnelo Abellanas, Irene and Liebendörfer, Michael}}, booktitle = {{Beiträge zum Mathematikunterricht 2024}}, title = {{{Herausforderungen beim Einsatz interaktiver Theorembeweiser in der Hochschullehre}}}, year = {{2024}}, } @inproceedings{59990, author = {{Eichler, Andreas and Floren, Henrik and Garnelo Abellanas, Irene and Liebendörfer, Michael and Müller, Raphael and Schürmann, Mirko and Speer, Annabelle}}, booktitle = {{INDRUM2024 PROCEEDINGS Fifth conference of the International Network for Didactic Research in University Mathematics}}, issn = {{2496-1027 (online)}}, location = {{Barcelona}}, publisher = {{Escola Univerist`aria Salesiana de Sarri`a – Univ. Aut`onoma de Barcelona and INDRUM}}, title = {{{Digital STACK tasks and exam results}}}, doi = {{https://theses.hal.science/INDRUM2024/hal-04944194v1}}, year = {{2024}}, } @article{59436, abstract = {{BackgroundCommunicating well with patients is a competence central to everyday clinical practice, and communicating statistical information, especially in Bayesian reasoning tasks, can be challenging. In Bayesian reasoning tasks, information can be communicated in two different ways (which we calldirections of information): The direction ofBayesian information(e.g., proportion of people tested positive among those with the disease) and the direction ofdiagnostic information(e.g., the proportion of people having the disease among those tested positive). The purpose of this study was to analyze the impact of both the direction of the information presented and whether a visualization (frequency net) is presented with it on patient’s ability to quantify a positive predictive value.Material and methods109 participants completed four different medical cases (2⨯2⨯4 design) that were presented in a video; a physician communicated frequencies using different directions of information (Bayesian information vs. diagnostic information). In half of the cases for each direction, participants were given a frequency net. After watching the video, participants stated a positive predictive value. Accuracy and speed of response were analyzed.ResultsCommunicating with Bayesian information led to participant performance of only 10% (without frequency net) and 37% (with frequency net) accuracy. The tasks communicated with diagnostic information but without a frequency net were correctly solved by 72% of participants, but accuracy rate decreased to 61% when participants were given a frequency net. Participants with correct responses in the Bayesian information version without visualization took longest to complete the tasks (median of 106 seconds; median of 13.5, 14.0, and 14.5 seconds in other versions).DiscussionCommunicating with diagnostic information rather than Bayesian information helps patients to understand specific information better and more quickly. Patients’ understanding of the relevance of test results is strongly dependent on the way the information is presented.}}, author = {{Brose, Sarah Frederike and Binder, Karin and Fischer, Martin R. and Reincke, Martin and Braun, Leah T. and Schmidmaier, Ralf}}, issn = {{1932-6203}}, journal = {{PLOS ONE}}, number = {{6}}, publisher = {{Public Library of Science (PLoS)}}, title = {{{Bayesian versus diagnostic information in physician-patient communication: Effects of direction of statistical information and presentation of visualization}}}, doi = {{10.1371/journal.pone.0283947}}, volume = {{18}}, year = {{2023}}, } @phdthesis{59988, author = {{Müller, Raphael}}, title = {{{On the asymptotics of wildly ramified local function field extensions}}}, year = {{2023}}, } @inproceedings{59984, author = {{Garnelo Abellanas, Irene and Liebendörfer, Michael}}, booktitle = {{Thirteenth Congress of the European Society for Research in Mathematics (CERME13)}}, location = {{Budapest}}, title = {{{Introducing mathematics undergraduate students to the theorem prover lean}}}, year = {{2023}}, } @article{59434, abstract = {{ Background. Medical students often have problems with Bayesian reasoning situations. Representing statistical information as natural frequencies (instead of probabilities) and visualizing them (e.g., with double-trees or net diagrams) leads to higher accuracy in solving these tasks. However, double-trees and net diagrams (which already contain the correct solution of the task, so that the solution could be read of the diagrams) have not yet been studied in medical education. This study examined the influence of information format (probabilities v. frequencies) and visualization (double-tree v. net diagram) on the accuracy and speed of Bayesian judgments. Methods. A total of 142 medical students at different university medical schools (Munich, Kiel, Goettingen, Erlangen, Nuremberg, Berlin, Regensburg) in Germany predicted posterior probabilities in 4 different medical Bayesian reasoning tasks, resulting in a 3-factorial 2 × 2 × 4 design. The diagnostic efficiency for the different versions was represented as the median time divided by the percentage of correct inferences. Results. Frequency visualizations led to a significantly higher accuracy and faster judgments than did probability visualizations. Participants solved 80% of the tasks correctly in the frequency double-tree and the frequency net diagram. Visualizations with probabilities also led to relatively high performance rates: 73% in the probability double-tree and 70% in the probability net diagram. The median time for a correct inference was fastest with the frequency double tree (2:08 min) followed by the frequency net diagram and the probability double-tree (both 2:26 min) and probability net diagram (2:33 min). The type of visualization did not result in a significant difference. Discussion. Frequency double-trees and frequency net diagrams help answer Bayesian tasks more accurately and also more quickly than the respective probability visualizations. Surprisingly, the effect of information format (probabilities v. frequencies) on performance was higher in previous studies: medical students seem also quite capable of identifying the correct solution to the Bayesian task, among other probabilities in the probability visualizations. Highlights Frequency double-trees and frequency nets help answer Bayesian tasks not only more accurately but also more quickly than the respective probability visualizations. In double-trees and net diagrams, the effect of the information format (probabilities v. natural frequencies) on performance is remarkably lower in this high-performing sample than that shown in previous studies. }}, author = {{Kunzelmann, Alexandra K. and Binder, Karin and Fischer, Martin R. and Reincke, Martin and Braun, Leah T. and Schmidmaier, Ralf}}, issn = {{2381-4683}}, journal = {{MDM Policy & Practice}}, number = {{1}}, publisher = {{SAGE Publications}}, title = {{{Improving Diagnostic Efficiency with Frequency Double-Trees and Frequency Nets in Bayesian Reasoning}}}, doi = {{10.1177/23814683221086623}}, volume = {{7}}, year = {{2022}}, } @article{59432, abstract = {{ZusammenfassungIn stochastischen Situationen mit zwei dichotomen Merkmalen erlauben weder die schulüblichen Baumdiagramme noch Vierfeldertafeln die simultane Darstellung sämtlicher in der Situation möglicher Wahrscheinlichkeiten. Das im vorliegenden Beitrag vorgestellte Netz hat die Kapazität, alle vier möglichen Randwahrscheinlichkeiten, alle vier Schnittwahrscheinlichkeiten sowie alle acht bedingten Wahrscheinlichkeiten gleichzeitig darzustellen. Darüber hinaus ist – aufgrund der Knoten-Ast-Struktur des Netzes – die simultane Darstellung von Wahrscheinlichkeiten und absoluten Häufigkeiten mit dieser Visualisierung ebenfalls möglich. Bei der sukzessiven Erweiterung des typischen Baumdiagramms zunächst zum Doppelbaum und schließlich zum Netz sinkt der Inferenzgrad (d. h. weniger kognitive Schritte sind erforderlich) z. B. für Fragen nach bedingten Wahrscheinlichkeiten, aber gleichzeitig steigt die Komplexität der Darstellung und somit die extrinsische kognitive Belastung. Im vorliegenden Artikel erfolgt zunächst ein theoretischer Vergleich dieser Knoten-Ast-Strukturen. Eine anschließende Studie illustriert, dass sich die sukzessive Erweiterung bereits vollständig ausgefüllter Diagramme positiv auf die Performanz von N = 269 Schülerinnen und Schülern auswirkt. Obwohl Häufigkeitsdoppelbäume und Häufigkeitsnetze den Schülerinnen und Schülern gänzlich unbekannt waren, unterstützten diese Visualisierungen die Schülerinnen und Schüler bei der Bearbeitung der Aufgaben am meisten.}}, author = {{Binder, Karin and Steib, Nicole and Krauss, Stefan}}, issn = {{0173-5322}}, journal = {{Journal für Mathematik-Didaktik}}, number = {{2}}, pages = {{471--503}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Von Baumdiagrammen über Doppelbäume zu Häufigkeitsnetzen – kognitive Überlastung oder didaktische Unterstützung? Moving from tree diagrams to double trees to net diagrams—cognitively overwhelming or educationally supportive?}}}, doi = {{10.1007/s13138-022-00215-9}}, volume = {{44}}, year = {{2022}}, } @article{59429, abstract = {{In the present paper we empirically investigate the psychometric properties of some of the most famous statistical and logical cognitive illusions from the “heuristics and biases” research program by Daniel Kahneman and Amos Tversky, who nearly 50 years ago introduced fascinating brain teasers such as the famous Linda problem, the Wason card selection task, and so-called Bayesian reasoning problems (e.g., the mammography task). In the meantime, a great number of articles has been published that empirically examine single cognitive illusions, theoretically explaining people’s faulty thinking, or proposing and experimentally implementing measures to foster insight and to make these problems accessible to the human mind. Yet these problems have thus far usually been empirically analyzed on an individual-item level only (e.g., by experimentally comparing participants’ performance on various versions of one of these problems). In this paper, by contrast, we examine these illusions as a group and look at the ability to solve them as a psychological construct. Based on an sample ofN= 2,643 Luxembourgian school students of age 16–18 we investigate the internal psychometric structure of these illusions (i.e., Are they substantially correlated? Do they form a reflexive or a formative construct?), their connection to related constructs (e.g., Are they distinguishable from intelligence or mathematical competence in a confirmatory factor analysis?), and the question of which of a person’s abilities can predict the correct solution of these brain teasers (by means of a regression analysis).}}, author = {{Bruckmaier, Georg and Krauss, Stefan and Binder, Karin and Hilbert, Sven and Brunner, Martin}}, issn = {{1664-1078}}, journal = {{Frontiers in Psychology}}, publisher = {{Frontiers Media SA}}, title = {{{Tversky and Kahneman’s Cognitive Illusions: Who Can Solve Them, and Why?}}}, doi = {{10.3389/fpsyg.2021.584689}}, volume = {{12}}, year = {{2021}}, } @article{59439, abstract = {{AbstractWhen physicians are asked to determine the positive predictive value from the a priori probability of a disease and the sensitivity and false positive rate of a medical test (Bayesian reasoning), it often comes to misjudgments with serious consequences. In daily clinical practice, however, it is not only important that doctors receive a tool with which they cancorrectlyjudge—thespeedof these judgments is also a crucial factor. In this study, we analyzed accuracy and efficiency in medical Bayesian inferences. In an empirical study we varied information format (probabilities vs. natural frequencies) and visualization (text only vs. tree only) for four contexts. 111 medical students participated in this study by working on four Bayesian tasks with common medical problems. The correctness of their answers was coded and the time spent on task was recorded. The median time for a correct Bayesian inference is fastest in the version with a frequency tree (2:55 min) compared to the version with a probability tree (5:47 min) or to the text only versions based on natural frequencies (4:13 min) or probabilities (9:59 min).The scorediagnostic efficiency(calculated by: median time divided by percentage of correct inferences) is best in the version with a frequency tree (4:53 min). Frequency trees allow more accurateandfaster judgments. Improving correctness and efficiency in Bayesian tasks might help to decrease overdiagnosis in daily clinical practice, which on the one hand cause cost and on the other hand might endanger patients’ safety.}}, author = {{Binder, Karin and Krauss, Stefan and Schmidmaier, Ralf and Braun, Leah T.}}, issn = {{1382-4996}}, journal = {{Advances in Health Sciences Education}}, number = {{3}}, pages = {{847--863}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Natural frequency trees improve diagnostic efficiency in Bayesian reasoning}}}, doi = {{10.1007/s10459-020-10025-8}}, volume = {{26}}, year = {{2021}}, } @article{59433, abstract = {{AbstractWhen physicians are asked to determine the positive predictive value from the a priori probability of a disease and the sensitivity and false positive rate of a medical test (Bayesian reasoning), it often comes to misjudgments with serious consequences. In daily clinical practice, however, it is not only important that doctors receive a tool with which they cancorrectlyjudge—thespeedof these judgments is also a crucial factor. In this study, we analyzed accuracy and efficiency in medical Bayesian inferences. In an empirical study we varied information format (probabilities vs. natural frequencies) and visualization (text only vs. tree only) for four contexts. 111 medical students participated in this study by working on four Bayesian tasks with common medical problems. The correctness of their answers was coded and the time spent on task was recorded. The median time for a correct Bayesian inference is fastest in the version with a frequency tree (2:55 min) compared to the version with a probability tree (5:47 min) or to the text only versions based on natural frequencies (4:13 min) or probabilities (9:59 min).The scorediagnostic efficiency(calculated by: median time divided by percentage of correct inferences) is best in the version with a frequency tree (4:53 min). Frequency trees allow more accurateandfaster judgments. Improving correctness and efficiency in Bayesian tasks might help to decrease overdiagnosis in daily clinical practice, which on the one hand cause cost and on the other hand might endanger patients’ safety.}}, author = {{Binder, Karin and Krauss, Stefan and Schmidmaier, Ralf and Braun, Leah T.}}, issn = {{1382-4996}}, journal = {{Advances in Health Sciences Education}}, number = {{3}}, pages = {{847--863}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Natural frequency trees improve diagnostic efficiency in Bayesian reasoning}}}, doi = {{10.1007/s10459-020-10025-8}}, volume = {{26}}, year = {{2021}}, } @article{59423, author = {{Krauss, S. and Bruckmaier, G. and Lindl, A. and Hilbert, S. and Binder, Karin and Steib, N. and Blum, W.}}, issn = {{1863-9690}}, journal = {{ZDM}}, number = {{2}}, pages = {{311--327}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Competence as a continuum in the COACTIV study: the “cascade model”}}}, doi = {{10.1007/s11858-020-01151-z}}, volume = {{52}}, year = {{2020}}, } @article{59431, author = {{Krauss, Stefan and Weber, Patrick and Binder, Karin and Bruckmaier, Georg}}, issn = {{0173-5322}}, journal = {{Journal für Mathematik-Didaktik}}, number = {{2}}, pages = {{485--521}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Natürliche Häufigkeiten als numerische Darstellungsart von Anteilen und Unsicherheit – Forschungsdesiderate und einige Antworten Natural Frequencies as Numerical Representation of Proportions and Uncertainty—Research Desiderata and Some Answers}}}, doi = {{10.1007/s13138-019-00156-w}}, volume = {{41}}, year = {{2020}}, } @article{59424, author = {{Binder, Karin and Krauss, Stefan and Wiesner, Patrick}}, issn = {{1664-1078}}, journal = {{Frontiers in Psychology}}, publisher = {{Frontiers Media SA}}, title = {{{A New Visualization for Probabilistic Situations Containing Two Binary Events: The Frequency Net}}}, doi = {{10.3389/fpsyg.2020.00750}}, volume = {{11}}, year = {{2020}}, } @article{59989, author = {{Müller, Raphael and Klüners, Jürgen}}, journal = {{Journal of Number Theory}}, title = {{{The conductor density of local function fields with abelian Galois group}}}, volume = {{212}}, year = {{2020}}, } @article{59428, author = {{Bruckmaier, Georg and Binder, Karin and Krauss, Stefan and Kufner, Han-Min}}, issn = {{1664-1078}}, journal = {{Frontiers in Psychology}}, publisher = {{Frontiers Media SA}}, title = {{{An Eye-Tracking Study of Statistical Reasoning With Tree Diagrams and 2 × 2 Tables}}}, doi = {{10.3389/fpsyg.2019.00632}}, volume = {{10}}, year = {{2019}}, } @article{59425, author = {{Weber, Patrick and Binder, Karin and Krauss, Stefan}}, issn = {{1664-1078}}, journal = {{Frontiers in Psychology}}, publisher = {{Frontiers Media SA}}, title = {{{Why Can Only 24% Solve Bayesian Reasoning Problems in Natural Frequencies: Frequency Phobia in Spite of Probability Blindness}}}, doi = {{10.3389/fpsyg.2018.01833}}, volume = {{9}}, year = {{2018}}, } @article{59426, author = {{Binder, Karin and Krauss, Stefan and Bruckmaier, Georg and Marienhagen, Jörg}}, issn = {{1932-6203}}, journal = {{PLOS ONE}}, number = {{3}}, publisher = {{Public Library of Science (PLoS)}}, title = {{{Visualizing the Bayesian 2-test case: The effect of tree diagrams on medical decision making}}}, doi = {{10.1371/journal.pone.0195029}}, volume = {{13}}, year = {{2018}}, } @article{59430, author = {{Hilbert, Sven and Bruckmaier, Georg and Binder, Karin and Krauss, Stefan and Bühner, Markus}}, issn = {{0256-2928}}, journal = {{European Journal of Psychology of Education}}, number = {{3}}, pages = {{665--683}}, publisher = {{Springer Science and Business Media LLC}}, title = {{{Prediction of elementary mathematics grades by cognitive abilities}}}, doi = {{10.1007/s10212-018-0394-9}}, volume = {{34}}, year = {{2018}}, } @inbook{59427, author = {{Binder, Karin and Krauss, Stefan and Hilbert, Sven and Brunner, Martin and Anders, Yvonne and Kunter, Mareike}}, booktitle = {{Diagnostic Competence of Mathematics Teachers}}, isbn = {{9783319663258}}, publisher = {{Springer International Publishing}}, title = {{{Diagnostic Skills of Mathematics Teachers in the COACTIV Study}}}, doi = {{10.1007/978-3-319-66327-2_2}}, year = {{2017}}, }