Measuring people's covariational reasoning in Bayesian situations.

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 the three input probabilities on the positive predictive value, that is, covariational reasoning. In this regard, two different operationalizations for measuring covariational reasoning (i.e., by single-choice vs. slider format) are investigated in an empirical study with N = 229 university students. In addition, we aim to answer the question wheter a skill in "conventional" Bayesian reasoning is a prerequisite for covariational reasoning.

Fostering diagnostic competence in prospective mathematics elementary teachers through epistemic activities.

Research on fostering teachers' diagnostic competence and thinking has become increasingly important. To this end, research has already identified several aspects of effective fostering of teachers' diagnostic competence. One of the aspects is assignment of the role as a teacher in interventions but, so far, assignment of the role of student has hardly been considered. Based on a model of the diagnostic thinking process, this paper operationalizes the role of the student by solving specific tasks and the role of the teacher by analyzing student solutions. Furthermore, based on previous research, it is assumed that assigning both roles is effective in promoting diagnostic competence. The following research addresses the development of 137 prospective teachers' diagnostic thinking in an experimental pre-post-test study with four treatment conditions, which vary prospective teachers' working with tasks and students' solutions to those tasks. The quantitative results show that a treatment integrating focus on tasks and students' solutions is equally as effective as a treatment focusing solely on students' solutions, and also that a treatment focusing solely on tasks has no effect.

Different Visualizations Cause Different Strategies When Dealing With Bayesian Situations.

People often struggle with Bayesian reasoning. However, previous research showed that people's performance (and rationality) can be supported by the way the statistical information is represented. First, research showed that using natural frequencies instead of probabilities as the format of statistical information significantly increases people's performance in Bayesian situations. Second, research also revealed that people's performance increases through using visualization. We have built our paper on existing research in this field. Our main aim was to analyze people's strategies in Bayesian situations that are erroneous even though statistical information is represented as natural frequencies and visualizations. In particular, we compared two pairs of visualization with similar numerical information (tree diagram vs. unit square, and double-tree diagram vs. 2 × 2-table) concerning their impact on people's erroneous strategies in Bayesian situations. For this aim, we conducted an experiment with 540 university students. The students were randomly assigned to four conditions defined by the four different visualizations of statistical information. The students were asked to indicate a fraction in response to four Bayesian situations. We documented the numerator and denominator of the students' responses representing a basic set and a subset in a Bayesian situation. Our results showed that people's erroneous strategies are highly dependent on visualization. A central finding was that the visualization's characteristic of making the nested-sets structure of a Bayesian situation transparent has a facilitating effect on people's Bayesian reasoning. For example, compared to the unit square, a tree diagram does not explicitly visualize the set-subset relations that are relevant in a Bayesian situation. Accordingly, compared to a unit square, a tree diagram partly hinders people in finding the correct denominator in a Bayesian situation, and, in particular, triggers selecting a wrong numerator. By analyzing people's erroneous strategies in Bayesian situations, we contribute to investigating approaches to facilitate Bayesian reasoning and to further develop the teaching of Bayesian reasoning.

How to Improve Performance in Bayesian Inference Tasks: A Comparison of Five Visualizations.

Bayes' formula is a fundamental statistical method for inference judgments in uncertain situations used by both laymen and professionals. However, since people often fail in situations where Bayes' formula can be applied, how to improve their performance in Bayesian situations is a crucial question. We based our research on a widely accepted beneficial strategy in Bayesian situations, representing the statistical information in the form of natural frequencies. In addition to this numerical format, we used five visualizations: a 2 × 2-table, a unit square, an icon array, a tree diagram, and a double-tree diagram. In an experiment with 688 undergraduate students, we empirically investigated the effectiveness of three graphical properties of visualizations: area-proportionality, use of discrete and countable statistical entities, and graphical transparency of the nested-sets structure. We found no additional beneficial effect of area proportionality. In contrast, the representation of discrete objects seems to be beneficial. Furthermore, our results show a strong facilitating effect of making the nested-sets structure of a Bayesian situation graphically transparent. Our results contribute to answering the questions of how and why a visualization could facilitate judgment and decision making in situations of uncertainty.

Utility indifference pricing of insurance catastrophe derivatives.

We propose a model for an insurance loss index and the claims process of a single insurance company holding a fraction of the total number of contracts that captures both ordinary losses and losses due to catastrophes. In this model we price a catastrophe derivative by the method of utility indifference pricing. The associated stochastic optimization problem is treated by techniques for piecewise deterministic Markov processes. A numerical study illustrates our results.

The Impact of Visualizing Nested Sets. An Empirical Study on Tree Diagrams and Unit Squares.

It is an ongoing debate, what properties of visualizations increase people's performance when solving Bayesian reasoning tasks. In the discussion of the properties of two visualizations, i.e., the tree diagram and the unit square, we emphasize how both visualizations make relevant subset relations transparent. Actually, the unit square with natural frequencies reveals the subset relation that is essential for the Bayes' rule in a numerical and geometrical way whereas the tree diagram with natural frequencies does it only in a numerical way. Accordingly, in a first experiment with 148 university students, the unit square outperformed the tree diagram when referring to the students' ability to quantify the subset relation that must be applied in Bayes' rule. As hypothesized, in a second experiment with 143 students, the unit square was significantly more effective when the students' performance in tasks based on Bayes' rule was regarded. Our results could inform the debate referring to Bayesian reasoning since we found that the graphical transparency of nested sets could explain these visualizations' effect.

Integrated learning: ways of fostering the applicability of teachers' pedagogical and psychological knowledge.

In teacher education, general pedagogical and psychological knowledge (PPK) is often taught separately from the teaching subject itself, potentially leading to inert knowledge. In an experimental study with 69 mathematics student teachers, we tested the benefits of fostering the integration of pedagogical content knowledge (PCK) and general PPK with respect to knowledge application. Integration was fostered either by integrating the contents or by prompting the learners to integrate separately taught knowledge. Fostering integration, as compared to a separate presentation without integration help, led to more applicable PPK and greater simultaneous application of PPK and PCK. The advantages of fostering knowledge integration were not moderated by the student teachers' prior knowledge or working memory capacity. A disadvantage of integrating different knowledge types referred to increased learning times.

Integrating pedagogical content knowledge and pedagogical/psychological knowledge in mathematics.

In teacher education at universities, general pedagogical and psychological principles are often treated separately from subject matter knowledge and therefore run the risk of not being applied in the teaching subject. In an experimental study (N = 60 mathematics student teachers) we investigated the effects of providing aspects of general pedagogical/psychological knowledge (PPK) and pedagogical content knowledge (PCK) in an integrated or separated way. In both conditions ("integrated" vs. "separated"), participants individually worked on computer-based learning environments addressing the same topic: use and handling of multiple external representations, a central issue in mathematics. We experimentally varied whether PPK aspects and PCK aspects were treated integrated or apart from one another. As expected, the integrated condition led to greater application of pedagogical/psychological aspects and an increase in applying both knowledge types simultaneously compared to the separated condition. Overall, our findings indicate beneficial effects of an integrated design in teacher education.

Real-time observation of molecular motion on a surface.

The laser-induced movement of CO molecules over a platinum surface was followed in real time by means of ultrafast vibrational spectroscopy. Because the CO molecules bound on different surface sites exhibit different C-O stretch vibrational frequencies, the site-to-site hopping, triggered by excitation with a laser pulse, can be determined from subpicosecond changes in the vibrational spectra. The unexpectedly fast motion--characterized by a 500-femtosecond time constant--reveals that a rotational motion of the CO molecules, rather than pure translation, is required for this diffusion process. This conclusion is corroborated by density functional theory calculations.

Adsorption and dissociation of NO on stepped Pt (533).

We present an experimental and theoretical investigation of the adsorption, desorption, and dissociation of NO on the stepped Pt (533) surface. By combining temperature programmed desorption and reflection absorption infrared spectroscopy, information about the adsorption sites at different temperatures is obtained. Surprisingly, metastable adsorption structures of NO can be produced through variation of the dosing temperature. We also show that part of the NO molecules adsorbed on the step sites dissociates around 450 K. After dissociation the N atoms can desorb either by combining with an O fragment, or with another N atom, resulting in NO and N(2). The N(2) production can be enhanced by coadsorbing CO on the surface: CO scavenges the oxygen atom, thereby suppressing associative recombinative desorption of N and O atoms. Density functional theory calculations are used to reveal the adsorption energies and vibrational frequencies of adsorbed NO as well as barriers for dissociation of NO and for diffusion of N atoms. The combined experimental results and theoretical calculations reveal that dissociation of NO is the rate limiting step in the formation of N(2).