Stefan Krauss- Professor at University of Regensburg
Stefan Krauss
- Professor at University of Regensburg
About
212
Publications
142,817
Reads
How we measure 'reads'
A 'read' is counted each time someone views a publication summary (such as the title, abstract, and list of authors), clicks on a figure, or views or downloads the full-text. Learn more
9,555
Citations
Introduction
Current institution
Publications
Publications (212)
Representing statistical information in terms of natural frequencies rather than probabilities improves performance in Bayesian inference tasks. This beneficial effect of natural frequencies has been demonstrated in a variety of applied domains such as medicine, law, and education. Yet all the research and applications so far have been limited to s...
In their research articles, scholars often use 2 × 2 tables or tree diagrams including natural frequencies in order to illustrate Bayesian reasoning situations to their peers. Interestingly, the effect of these visualizations on participants’ performance has not been tested empirically so far (apart from explicit training studies). In the present a...
In both the United States and Europe, concerns have been raised about whether preservice and in-service training succeeds in equipping teachers with the professional knowledge they need to deliver consistently high-quality instruction. This article investigates the significance of teachers’ content knowledge and pedagogical content knowledge for hi...
Research interest in the professional knowledge of mathematics teachers has grown considerably in recent years. In the COACTIV
project, tests of secondary mathematics teachers’ pedagogical content knowledge (PCK) and content knowledge (CK) were developed
and implemented in a sample of teachers whose classes participated in the PISA 2003/04 longitud...
„20 % von 50 %“ kann entweder als 40 % oder als 10 % interpretiert werden, da die Präposition „von“ je nach Sachsituation als Multiplikation oder Division gedeutet werden muss. Wir begründen beide Interpretationen, die wir mit „Multiplikations-von“ und „Divisions-von“ bezeichnen, mithilfe der Grundgleichung der Prozentrechnung und zeigen, dass eine...
It has been established that, in Bayesian tasks, performance and typical errors in reading information from filled visualizations depend both on the type of the provided visualization and information format. However, apart from reading visualizations, students should also be able to create visualizations on their own and successfully use them as he...
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...
Das Projekt SelVi ist ein durch Drittmittel gefördertes Kooperationsprojekt
der Universität Regensburg, in dessen Rahmen jede der 12 beteiligten Fakultäten eine prototypische Vorlesung fokussiert. Ziel des Projekts ist die
Verankerung digitaler Lehre auf qualitativ hohem Niveau in allen Studiengängen. Dabei haben die digitalen Semester der letzten...
Natural frequencies are known to improve performance in Bayesian reasoning. However, their impact in situations with two binary events has not yet been completely examined, as most researchers in the last 30 years focused only on conditional probabilities. Nevertheless, situations with two binary events consist of 16 elementary probabilities and so...
Das Lösen Bayesianischer Aufgaben erfordert das anspruchsvolle Umgehen mit bedingten Wahrscheinlichkeiten, was häufig mit Schwierigkeiten und fehlerhaften Schlussfolgerungen verbunden ist. Es gibt zahlreiche empirisch gestützte didaktische Ansatzpunkte, wie Bayesianisches Denken gefördert werden kann, bspw. indem die statistischen Informationen im...
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 c...
Das berühmte Würfelproblem des Chevalier de Méré (siehe z. B. Kütting & Sauer, 2011) verdeutlicht die zentrale Bedeutung von Ergebnismenge und zugeordneten Wahrscheinlichkeiten bei der Modellierung von mehrstufigen Zufallsexperimenten. Doch schon bei einstufigen Zufallsexperimenten kann die Festlegung einer Ergebnismenge problematisch sein, wie im...
Der Mathematikunterricht bietet Lerngelegenheiten, sich systematisch mit mathematischen Inhalten und Konzepten auseinanderzusetzen, die eigenen Kompetenzen zu erweitern sowie Freude und Interesse an Mathematik zu entwickeln. In Deutschland berichten die Fünfzehnjährigen, dass sie in ihrem Mathematikunterricht wenig Störungen erleben. Gleichzeitig f...
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...
Der vorliegende Band vereinigt erstmals die vielfältigen und einflussreichsten empirischen Untersuchungsansätze zum professionellen Wissen von Mathematiklehrkräften der beiden letzten Jahrzehnte. Er ermöglicht allen Leserinnen und Lesern unabhängig von ihren Vorkenntnissen einen raschen Einstieg in dieses zentrale Thema fachdidaktischer Forschung u...
Calculation in a so-called Bayesian situation is known to be challenging. According to a meta-analysis the performance is only 4% if the relevant information is given in form of probabilities. Still, previous research has identified helpful strategies for improving Bayesian reasoning (i.e., so-called natural frequencies and visualizations). Yet, it...
There are many ways in expressing relative frequencies with numerical values: 1) percentages 25%, 2) decimal fractions 0.25, 3) common fractions 1/4, 4) natural frequencies "1 out of 4", 5) notations like "every fourth", or 6) odds "1 to 3". The latter one plays an important role in communicating changes of risk, e.g., in medical research results i...
Stochastik in der Schule 43(1)
In empirical educational research, it is of great interest to predict student performance. In contrast to other disciplines, however, machine learning methods for identifying promising predictors are not yet widely used. We will use a machine learning approach to study the effect of learning strategies and cooperative behaviors of German mathematic...
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 Ba...
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 a...
Zusammenfassung
In 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 vie...
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...
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...
Inzidenzen, R-Wert, Anzahl von Hospitalisierungen, Corona-Ampel, Wirksamkeit der
Impfstoffe – beim Thema COVID-19 wird mit Zahlen und Statistiken geradezu um sich geworfen. Doch was bedeuten all diese Zahlen im Detail?
Die polarisierende Debatte um Vorzüge, Wirksamkeiten und Nebenwirkungen von Impfstoffen macht deutlich, dass eine angemessene Inter...
mathematik lehren, 230, 22-25
Machine learning (ML) provides a powerful framework for the analysis of high‐dimensional datasets by modelling complex relationships, often encountered in modern data with many variables, cases and potentially non‐linear effects. The impact of ML methods on research and practical applications in the educational sciences is still limited, but contin...
This article presents a seminar concept designed to promote students’ ability to explain as an aspect of their professional competence. The seminars were carried out as part of the project FALKE at the University of Regensburg. The concept includes: discussing criteria of good explanations, preparing an explanation, being filmed while explaining, r...
Abstract
When 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 to...
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 select...
Classical statistical methods are limited in the analysis of highdimensional datasets. Machine learning (ML) provides a powerful framework for prediction by using complex relationships, often encountered in modern data with a large number of variables, cases and potentially non-linear effects. ML has turned into one of the most influential analytic...
Classical statistical methods are limited in the analysis of highdimensional datasets. Machine learning (ML) provides a powerful framework for prediction by using complex relationships, often encountered in modern data with a large number of variables, cases and potentially non-linear effects. ML has turned into one of the most influential analytic...
This article details how the FALKE research project (Fachspezifische Lehrerkompetenzen im Erklären; Engl.: subject-specific teacher competency in explaining) integrates 14 heterogeneous disciplines in order to empirically examine the didactic quality of teacher explanations in eleven school subjects by bringing together trans-, multi-, and interdis...
Das Forschungsprojekt FALKE-digital untersucht in sechs Unterrichtsfächern die Wirk-samkeit von Erklärvideos in einem Flipped-Classroom-Setting. Die Forschungsfrage ist, unter welchen Bedingungen die Methode Flipped Classroom unter Verwendung von Erklärvideos den Unterricht sinnvoll ergänzen kann und hinsichtlich des Lernzuwach-ses von Schüler*inne...
Das „Handbuch Lehrerinnen- und Lehrerbildung“ bietet aus fachlicher, fachdidaktischer, bildungswissenschaftlicher und schulpraktischer Perspektive einen umfassenden forschungsbasierten Überblick zu allen Bereichen der Lehrerinnen- und Lehrerbildung. Das von 174 Autorinnen und Autoren erarbeitete Orientierungswissen wird in 107 Beiträgen präsentiert...
Wason cards were originally designed for an experiment evaluating subjects' reasoning abilities. Our claim is that they may be used as an educational tool in the classroom and their didactical value is enormous. Fourth graders can be trained in logical reasoning by playing not only with four Wason cards, as in the original experiment, but with samp...
A central task of educational research is to examine common issues of teaching and learning in all subjects taught at school. At the same time, the focus is on identifying and investigating unique subject-specific aspects on the one hand and transdisciplinary, generalizable effects on the other. This poses various methodological challenges for educ...
In: W. Riemer und H.-S. Siller (Hrsg.): Mathematik lehren, 220
In teaching statistics in secondary schools and at university, two visualizations are primarily used when situations with two dichotomous characteristics are represented: 2 × 2 tables and tree diagrams. Both visualizations can be depicted either with probabilities or with frequencies. Visualizations with frequencies have been shown to help students...
Two different tools for assessing pedagogical content knowledge (PCK) of mathematics teachers used in the framework of the COACTIV study are systematically compared in this paper, namely the paper-and-pencil test consisting of items on the three facets knowledge of explaining and representation, knowledge of student thinking and typical mistakes, a...
The article describes materials and activities which can be used with great success to foster first robust intuitions about Stochastics in general.
Das aus der Kognitionspsychologie stammende Konzept der sogenannten natürlichen Häufigkeiten wird seit etwa 20 Jahren auch in der Mathematikdidaktik diskutiert. Im vorliegenden Beitrag soll illustriert werden, dass trotz der mittlerweile enormen Fülle an empirischen Studien noch zahlreiche fachdidaktische Fragestellungen unbeantwortet sind. So ist...
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 ar...
Mit der Lehrplanreform zur Einführung des achtjährigen Gymnasiums in Bayern ab 2004 verschwanden Inhalte wie Grundlagen der Akustik, Stimmungen und Tonsysteme aus den schulischen Curricula und in der Folge auch aus dem Pflichtprogramm der Lehramtsstudiengänge Musik. Anders im Bereich der Mathematik und der Mathematikdidaktik: Eine ganze Reihe aktue...
Die vorliegende Studie untersucht statistisches Schlussfolgern bei Bayesianischen Aufgaben. Den Studienteilnehmer/inne/n wurden hierzu insgesamt 20 Aufgaben präsentiert, wobei die statistischen Informationen in Baumdiagrammen oder Vierfeldertafeln gegeben waren. Indem die Blickbewegungen der Probanden bei der Aufgabenbearbeitung aufgezeichnet werde...
Bei Definition, Förderung und Erwerb von Expertise spielt Problemlösen eine zentrale Rolle. Expertise kann als eine Art wissensbasierten Problemlösens angesehen werden. In Bezug auf Schülerinnen und Schüler misst PISA (Programme for International Student Assessment) Problemlösefähigkeit als eigenständige Kompetenz und betrachtet diese als fächerübe...
Die neue Zahl ist durch Multiplikation der zweiziffrigen Zahl mit 1001 entstanden. In unserem Beispiel gilt 23023 = 23 • 1001. Die Zahl 1001 ist aber durch 7 teilbar (1001 = 7 • 143), somit ist (nach B1) auch die neue Zahl durch 7 teilbar. Zusätzlich gilt wegen 1001 = 7 • 11 • 13, dass alle Zahlen dieser Form nicht nur durch 7, sondern auch durch 1...
Es wird die Formel zur Bestimmung von einzelnen Folgengliedern arithmetischer Folgen verwendet.
Wenn jemand die Frage stellt ” 1, 3, 6, 10, 15, . . . und was kommt dann?“, setzt er sich mit Zahlenfolgen auseinander. Auch in Intelligenz- oder Einstellungstests wird man oft aufgefordert, bestimmte Muster in solchen Folgen zu erkennen und diese fortzusetzen. Dies gilt offenbar als eine grundlegende mathematische Fähigkeit.
In der Natur ist die Zahl 7 nur selten zu finden, aber im Volkstum begegnet sie uns häufig: Die sieben Weisen, die sieben Weltwunder, die sieben Zwerge, die sieben Todsünden, die sieben Tage der Schöpfung und danach die sieben Wochentage.
Wie lassen sich Spiele und Puzzles mathematisch analysieren?
Wie kann man unendliche Strukturen zutreffend beschreiben?
Wie kann man Nachrichten gut verschlüsseln?
Achtzehn ausgewählte mathematische Themen mit Aufgaben und Lösungen laden zum Entdecken und Knobeln ein und bieten Einblicke in die faszinierende Welt der Mathematik – von A wie Aussagen...
For more than 20 years, research has proven the beneficial effect of natural frequencies when it comes to solving Bayesian reasoning tasks (Gigerenzer and Hoffrage, 1995). In a recent meta-analysis, McDowell and Jacobs (2017) showed that presenting a task in natural frequency format increases performance rates to 24% compared to only 4% when the sa...
In der COACTIV‐Studie 2003/04 wurden Mathematiklehrkräfte deutscher Sekundarschulen im Verbund mit ihren PISA‐Klassen umfangreich befragt und getestet. Im vorliegenden Beitrag fokussieren wir auf verschiedene Aspekte des Modellierens, die im Rahmen dieser Studie aus verschiedenen Perspektiven (Lehrkräfte, Schüler, externe Rater) beurteilt wurden. E...
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...
Objectives:
Pyrrolizidine alkaloids (PA) exist ubiquitously in our environment. More than 6000 plants, about 3% of the world’s flowering plants, are known to synthesize PA. As a consequence, many herbal ingredients, including St. John’s wort (SJW), are contaminated with PA that can possess acute and subchronic toxic effects as well as mutagenic an...
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 situatio...
Conditional independence.
(DOCX)
In the present chapter we introduce theoretical and empirical approaches to the construct of diagnostic competence within the COACTIV research program. We report eight conceptualizations and operationalizations of diagnostics skills and add in addition three constructs that seem to be close to diagnostic skills. Correlational analyses reveal only m...
Humans are exposed to pyrrolizidine alkaloids (PA) through different sources, mainly from contaminated foodstuff. Teas and herbal infusions (T&HI) can be contaminated by PA producing weed. PA can possess toxic, mutagenic, genotoxic, and carcinogenic properties. Thus, possible health risks for the general population are under debate. There is a stro...
This paper provides a meta-analytical synopsis of central results of the FALKO-project in five sections: In the first section, the different domain-specific conceptualizations and operationalizations of content knowledge (CK) and pedagogical content knowledge (PCK) are summarized and critically compared with regard to recurrent challenges of test c...
Poster presented at the 13th International Congress on Mathematical Education (ICME), Hamburg
