Bruckmaier, Georg
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- PublikationEine Frage der Taktik. Fairness beim Spiel „Schere – Stein – Papier“(Friedrich, 08.02.2022) Binder, Karin; Bruckmaier, Georg [in: Mathematik lehren]Anhand des Spiels Schere – Stein – Papier lässt sich das Thema Fairness im Unterricht spielerisch behandeln, die zugehörigen mathematischen Zusammenhänge geeignet visualisieren und psychologische Aspekte des Spiels Spielstrategien) thematisieren. Wir werden hier, ausgehend von der Grundform des Spiels, drei Varianten betrachten: unterschiedlich viele Symbole für einzelne Spieler (Variante 1), zusätzliche Symbole (Variante 2) und unterschiedlich hoher Gewinn je Spielausgang (Variante 3).01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationCorona-Impfungen. Was 95% Wirksamkeit bedeuten – und was nicht(Verlag Klaus Seeberger, 2022) Ollesch, Julia; Bruckmaier, Georg; Vogel, Markus; Krauss, Stefan [in: Stochastik in der Schule]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 die Vorzüge, Wirksamkeiten und Nebenwirkungen von Impfstoffen zeigt deutlich, dass eine eindeutige Interpretation der Daten nicht immer ganz einfach ist. Dies zeigt sich nicht zuletzt darin, dass es einem Großteil der Bevölkerung schwerfällt, die Statistiken zur Wirksamkeit angemessen zu deuten. In diesem Artikel wollen wir Berechnungen zur Wirksamkeit der Impfstoffe gegen COVID-19 und die damit verbundenen Probleme bzw. Fehlinterpretationen genauer betrachten und diesbezügliche Anregungen für den Mathematikunterricht präsentieren.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationHow do self‐efficacy and self‐concept impact mathematical achievement? The case of mathematical modelling(Wiley, 06.07.2021) Holenstein, Mathias; Bruckmaier, Georg; Grob, Alexander [in: British Journal of Educational Psychology]Background. According to the self-enhancement perspective, self-efficacy and self concept are shaped by prior achievement and have a crucial impact on future development. Their role in improving performance on challenging tasks, such as mathematical modelling (i.e., solving realistic problems mathematically), has barely been studied. Aims. We investigated patterns of self-efficacy and self-concept and their predictive effects on mathematical modelling while taking into account school grades as measure of prior achievement and reasoning to reveal cognitive and motivational effects on achievement. Sample. N = 279 secondary students in Grade 8 or 9 from 16 classes and 6 schools participated in the study. Method. The multi-informant design consisted of teachers’ reports of school grades, students’ reports of self-efficacy and self-concept (questionnaire-based), and assessment of students’ reasoning and mathematical modelling. Results. Using random-intercept models, we found that the predictive effect of self efficacy on mathematical modelling withstood taking the school-classroom-related nested structure into account, whereas self-concept lost its predictive value. Further, self efficacy fully mediated the effect of school grades on mathematical modelling. Conclusions. In line with the self-enhancement perspective on self-efficacy, our findings highlight the strength of motivational effects on mathematical modelling. When we take the nested structure into account, our results indicate an impact of school grades via self efficacy on mathematical modelling independent of students’ cognitive level or classroom. Given the diverse challenges such complex tasks present, important pedagogical and didactical recommendations, such as targeting the enhancement of students’ self-efficacy by teachers and educational decision makers, can be drawn.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationTversky and Kahneman’s cognitive illusions. Who can solve them, and why?(Frontiers Research Foundation, 12.04.2021) Bruckmaier, Georg; Krauss, Stefan; Binder, Karin; Hilbert, Sven; Brunner, Martin [in: Frontiers in Psychology]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 of = 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).01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationTransfer effects of mathematical literacy: an integrative longitudinal study(Springer, 07.08.2020) Holenstein, Mathias; Bruckmaier, Georg; Grob, Alexander [in: European Journal of Psychology of Education]Mathematical literacy (ML) is considered central to the application of mathematical knowledge in everyday life and thus is found in many comparative international educational standards. However, there exists barely any evidence about predictors and outcomes of ML having a lasting effect on achievement in non-mathematical domains. We drew on a large longitudinal sample of N= 4001 secondary school students in Grades 5 to 9 and tested for effects of ML on later academic achievement. We took prior achievement in different domains (information and communication technology literacy, scientific literacy, reading comprehension, and listening comprehension), socioeconomic status, and gender into account and investigated predictive effects of math grade, mathematical self-concept, reasoning, and prior achievement on ML. Using structural equation models, we found support for the importance of integrating multiple predictors and revealed a transfer effect of ML on achievement in different school domains. The findings highlight the importance of ML for school curricula and lasting educational decisions.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationCompetence as a continuum in the COACTIV study: the “cascade model”(Springer, 11.04.2020) Krauss, Stefan; Bruckmaier, Georg; Lindl, Alfred; Hilbert, Sven; Binder, Karin; Steib, Nicole; Blum, Werner [in: ZDM]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, and knowledge of the potential of mathematical tasks, and the video vignettes instrument that examines teachers’ proposed continuations for presented lesson video clips specific to their subject-related and methodological competence aspects. Initially, both COACTIV PCK assessment tools are systematically contrasted for the first time with respect to their predictive validity for instructional quality (N = 163 German secondary mathematics teachers) as well as student learning gains (N = 3806 PISA students from 169 different classes) by means of path models showing that PCK, when assessed by the paper-and-pencil method, can better predict instructional quality than the video vignettes instrument can. Next, we theoretically propose the cascade model as capable of integrating pertinent theories on teacher competence and instructional quality. This model implies five ‘columns’ that are ordered according to a sequential causal chain (teacher disposition → situation-specific skills → observable teaching behavior → student mediation → learning gains). Finally, we specify four out of the five ‘columns’ of this cascade model, based empirically on the COACTIV data.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationNatürliche Häufigkeiten als numerische Darstellungsart von Anteilen und Unsicherheit – Forschungsdesiderate und einige Antworten(Springer, 28.01.2020) Krauss, Stefan; Weber, Patrick; Binder, Karin; Bruckmaier, Georg [in: Journal für Mathematik-Didaktik]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 die Ersetzung von Wahrscheinlichkeiten (wie z. B. „80 %“) durch zwei absolute Häufigkeiten in der Form von natürlichen Häufigkeiten (z. B. „4 von 5“) zwar als verständnisfördernd anerkannt, es ist aber noch unklar, wie genau sich natürliche Häufigkeiten definieren lassen, welche Eigenschaften entsprechende Verknüpfungen haben, aber auch, welche Grundvorstellungen für den verständnisfördernden Effekt verantwortlich sein könnten. Ein drängendes Desiderat ist darüber hinaus, dass natürliche Häufigkeiten bislang zwar im Zusammenhang mit Bayesianischen Aufgabenstellungen diskutiert werden (d. h. beim Thema Wahrscheinlichkeit), aber noch nicht im Hinblick auf ihr tatsächliches Vorkommen in der Welt (d. h., beim Thema Daten). Obschon aktuelle Strömungen in der Didaktik der Stochastik nahelegen, dass gerade eine Analyse der Darstellungsformate statistischer Informationen, denen wir in der Welt begegnen, überhaupt erst die Voraussetzung ist, um Schülerinnen und Schüler im Sinne einer statistical literacy adäquat auf eine reflektierte Teilnahme an unserer Informationsgesellschaft vorzubereiten, geschieht dies im Zusammenhang mit Daten bislang meist mit einem Fokus auf graphische Darstellungen. Im vorliegenden Artikel (a) analysieren wir numerische Darstellungen von Anteilen und Wahrscheinlichkeiten in Alltagskommunikation und Medien, (b) vergleichen diese mit entsprechenden Darstellungen im schulischen Stochastikunterricht und (c) machen konstruktive Vorschläge, wie die hierbei zu Tage tretende Diskrepanz zwischen (a) und (b) im Stochastikunterricht adressiert werden könnte. Der Schwerpunkt liegt dabei auf dem Konzept der natürlichen Häufigkeiten.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationLösungsvorschläge zu Thema 17(Springer Spektrum, 2019) Bruckmaier, Georg; Löh, Clara; Kilibertus, Niki; Krauss, Stefan [in: Quod erat knobelandum. Themen, Aufgaben und Lösungen des Schülerzirkels Mathematik der Universität Regensburg]04A - Beitrag Sammelband
- PublikationAn eye-tracking study of statistical reasoning with tree diagrams and 2 x 2 tables(Frontiers, 2019) Bruckmaier, Georg; Binder, Karin; Krauss, Stefan; Kufner, Han-Min [in: Frontiers in Psychology]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.01A - Beitrag in wissenschaftlicher Zeitschrift
- PublikationDidaktische Kompetenzen von Mathematiklehrkräften. Weiterführende Analysen aus der COACTIV-Studie(Springer Spektrum, 2019) Bruckmaier, Georg02 - Monographie