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.
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.
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.
In classrooms today, teachers are asked to support their teaching with digital tools. For this purpose, teachers require not only technological knowledge but also corresponding beliefs about the ...advantages of digital tools. The development of those beliefs should already be embedded in the university education of teachers. To this end, we developed a university seminar aimed at fostering prospective teachers’ confidence in the utility of digital tools, using the digital tool STACK as an example. The seminar is based on learning mathematics with the digital tool STACK, independently designing digital tasks with said tool, and finally, reflecting on a teaching experiment with school students using STACK. To make the development of prospective teachers’ beliefs visible throughout the seminar, we worked with different qualitative methods. The results of this case study show that there are four developmental phases of prospective teachers’ beliefs which include an initial situation, a purely positive phase, a disillusionment, and a phase of differentiated beliefs. It becomes apparent that it is possible to develop prospective teachers’ beliefs about digital tools in a positive way.
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.
Background
Functional thinking is characterized as a specific way of thinking in relationships, dependencies, and changes. Hence, beyond mathematics, it is also crucial for other (STEM) disciplines ...as well as for everyday situations. In particular, dealing with different representations of functions and changing between them are core function-related competencies, which are correspondingly needed for the formation of appropriate concepts and flexible problem-solving in various situations. Therefore, this study investigated students’ (
N
= 856) competencies related to representational changes of elementary functions and, in particular, assessed which changes are especially easy or difficult for students. Moreover, possible school track and gender differences were investigated by performing DIF analyses within the framework of Rasch modeling. The data were gathered using a paper–pencil test administered after the students had completed the teaching unit on linear functions in their mathematics lessons.
Results
Altogether, students were found to have limited competencies related to representational changes of elementary functions. There was no clear pattern regarding the types of representational change that were difficult or easy for them. Moreover, girls performed better on purely mathematical tasks, whereas boys did better at a complex modeling and problem-solving task. Classes from the academic track produced better results in tasks with a situational context compared to their peers from non-academic tracks, who performed relatively strongly on purely mathematical tasks.
Conclusions
These findings imply that various representations and representational changes should be included in lessons on functions to support students in building a rich concept of function and flexible problem-solving skills, thus fulfilling curricular requirements and responding to didactical considerations. In particular, the teaching of functions should be more balanced by mixing tasks with and without a situational context and the corresponding representational changes. These findings should motivate teachers, in particular those teaching non-academic tracks, to give a more prominent role to situational contexts in their lessons on functions in order to foster their students’ learning and build a bridge between mathematics and real-world situations.
Density functional theory calculations have been used to investigate the adsorption of carbon monoxide on the (1
1
1) surface of nickel. For various surface coverages between
Θ=1/12 and 1 ML with CO ...occupying different adsorption sites geometric structure, adsorption energy, CO stretch frequency (and intensities) and O1s and C1s binding energies have been calculated. By comparison to experimental data from the literature, the site occupation with coverage could be traced: at low coverage (
Θ<0.5 ML) CO occupies mainly bridge sites, accompanied by small patches of hollow (for
Θ<0.2 ML) and on-top (for
Θ>0.2 ML) adsorbed species. The relatively high-dynamic dipole moment of the linear bond species (especially at low coverage) leads to a distinct signal even for very small amounts. At
Θ=0.5 the well established
c(2×4) structure with a mixed occupation of fcc and hcp adsorbed molecules is formed. In this structure strong dipole interaction between the adsorbates shift the CO stretch frequency to a value that is traditionally assigned to bridging species. For even higher coverage more complicated structures within
7
×
7
(
Θ=4/7 ML) and
c(2
3
×4)
rect (
Θ=5/8 ML) cells are discussed. A further fundamental observation of this study is a very pronounced coverage dependence of the dynamic dipole moment for the CO stretch frequency and hence of the corresponding peak intensities in vibrational spectroscopy.
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8.
Real-Time Observation of Molecular Motion on a Surface Backus, Ellen H. G; Eichler, Andreas; Kleyn, Aart W ...
Science (American Association for the Advancement of Science),
12/2005, Volume:
310, Issue:
5755
Journal Article
Peer reviewed
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.
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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 of changes to the parameters of a formula on the result (covariation) and adequately interpreting and explaining the results of a formula (communication). Bayesian Reasoning is crucial in several non-mathematical disciplines such as medicine and law. However, even experts from these domains struggle to reason in a Bayesian manner. Therefore, it is desirable to develop a training course for this specific audience regarding the different aspects of Bayesian Reasoning. In this paper, we present an evidence-based development of such training courses by considering relevant prior research on successful strategies for Bayesian Reasoning (e.g., natural frequencies and adequate visualizations) and on the 4C/ID model as a promising instructional approach. The results of a formative evaluation are described, which show that students from the target audience (i.e., medicine or law) increased their Bayesian Reasoning skills and found taking part in the training courses to be relevant and fruitful for their professional expertise.
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