Andreas Eichler

Statistics Teachers and Classroom Practices

Why do statistics teachers teach certain topics, how do they teach these topics, and to what extent does the teaching affect students’ learning? In this chapter, a theoretical framework combining a curriculum model with the construct of beliefs will be provided to analyse previous research concerning teachers’ instructional planning, their classroom practices, and the impact of these practices on their students’ learning. Each section includes a brief discussion of research results referring to mathematics education in general and statistics education in particular, and exemplifying results from research that address the three questions posed.

Three Approaches for Modelling Situations with Randomness

Three different approaches to the concept of probability dominate the teaching of stochastics: the classical, the frequentistic and the subjectivistic approach. Compared with each other they provide considerably different possibilities to interpret situations with randomness. With regard to teaching probability, it is useful to clarify interrelations and differences between these three approaches. Thus, students’ probabilistic reasoning in specific random situations could be characterized, classified and finally, understood in more detail. In this chapter, we propose examples that potentially illustrate both, interrelations and differences of the three approaches to probability mentioned above. Thereby, we strictly focus on an educational perspective.

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.

CO adsorption on a Cu(211) surface : First-principle calculation and STM study

Chemisorption of CO on the stepped Cu(211) surface is studied within ab-initio density functional theory (DFT) and scanning tunneling microscopy (STM) imaging as well as manipulation experiments. Theoretically we focus on the experimentally observed ordered (2x1) and (3x1) CO-phases at coverages 1/3, 1/2 and 2/3 monolayer (ML). To obtain also information for isolated CO molecules found randomly distributed at low coverages, we also performed calculations for a hypothetical (3x1) phase with 1/3 ML. The adsorption geometry, the stretching frequencies, the work functions and adsorption energies of the CO molecules in the different phases are presented and compared to experimental data. Initially and up to a coverage of 1/2 ML CO adsorbs upright on the on-top sites at step edge atoms. Determining the most favorable adsorption geometry for the 2/3 ML ordered phase turned out to be nontrivial both from the experimental and the theoretical point of view. Experimentally, both top-bridge and top-top configurations were reported, whereby only the top-top arrangement was firmly established. The calculated adsorption energies and the stretching frequencies favor the top-bridge configuration. The possible existence of both configurations at 2/3 ML is critically discussed on the basis of the presently accessible experimental and theoretical data. In addition, we present observations of STM manipulation experiments and corresponding theoretical results, which show that CO adsorbed on-top of a single Cu-adatom, which is manipulated to a location close to the lower step edge, is stronger bound than CO on-top of a step edge atom.

Calorimetric evidence for two magnetic phases in CePb3 under pressure

Low-temperature specific heat measurements on the heavy-fermion antiferromagnet CePb3 have been performed under pressures up to 1.3 GPa. From the shift of the anomaly atTN the magnetic phase diagram is constructed. The change of the shape of the anomaly, when the pressure exceeds 0.7 GPa, clearly indicates the crossover from one magnetic structure to another as had already been concluded from formern-scattering results. The Kondo temperature does not show a peculiar variation with pressure.

Specific heat of the superconducting high pressure phase Ga II

The high pressure modification gallium II, which becomes superconducting near 6.4 K, has for the first time been investigated calorimetrically at about 35 kbar in the temperature range of liquid helium. From the data a Debye temperatureΘD = 178K and an electronic specific-heat coefficient γ=1.75 mJ/mole K2 are obtained. These values are in good qualitative accord with the increased superconductingTc with respect to that of Ga I.

Ground state of CeCu4M (M Al, Ga) under pressure

Abstract The continuous substitution of one Cu atom by Ga or Al in antiferromagnetically ordered CeCu 5 leads to a suppression of magnetism and a strengthening of the Kondo effect. CeCu 4 Al and CeCu 4 Ga are non-magnetic and non-superconducting heavy fermion compounds lying near to the border of magnetic order. CeCu 4 Ga shows distinct maxima in the susceptibility, specific heat and resistivity at about 1 K commonly considered as an indication of the evolution of a coherent state. In CeCu 4 Al, on the other hand, the corresponding features are not as clearcut, making an interpretation in terms of coherence difficult. We have studied the influence of pressure on the ground state using specific heat and magnetoresistance measurements. Our recent data show unusual small pressure dependences in contrast to other Ce-based heavy fermion compounds.

Specific heat ofd-hcp and fcc lanthanum under high pressure

Specific heat measurements have been performed under pressures up to 2.5 GPa on single phased-hcp lanthanum and a mixed phase sample yielding also results on fcc La. An evaluation of the pressure dependences of the Debye temperatureΘD and the electronic specific heat coefficient γ has been possible and shows distinct differences between the two phases. While for thed-hcp phase marked variations with pressure have been found, there is hardly no change seen in fcc La. Ind-hcp La the most remarkable results are an initial increase of γ, which cannot be explained by the increase of the electron-phonon coupling parameter λ alone and, therefore, indicates an increase of the band structure density of states, as well as a strong softening ofΘD at higher pressures. These results are discussed in the context of the existing knowledge on lanthanum.

Secondary Teachers’ Beliefs on Modelling in Geometry and Stochastics

This chapter presents two combined qualitative studies on secondary teachers’ beliefs on modelling in geometry and stochastics. The teachers’ views on modelling, which are described in detail, differ considerably in both parts of mathematics from a pragmatic approach to modelling. In case of elementary geometry, a conflict with a traditional view on geometry is detected and elucidated. In case of stochastics, the need for data and real situations are revealed as controversial. The chapter ends with the invitation to analyse the parts of factual school mathematics including teachers’ beliefs more specifically, that is, to compare applied-oriented aims with other didactical requests, and to design tasks which are supposed to be a response to the teachers’ hesitations on modelling analysed before.

Breakdown of ferromagnetism in CePd2Ga3 under high pressure

Abstract Heavy fermion systems in the hexagonal PrNi2Al3 structure show a manifold of phenomena including superconductivity and different types of magnetic order. In antiferromagnetic CePd2Al3 (TN = 2.8 K) it has been found out by several groups that the magnetic properties depend sensitively on small changes in stoichiometry, lattice order and stress. The related compound CePd2Ga3, which has a slightly smaller lattice cell volume, but a larger c/a ratio orders ferromagnetically already near 6 K. We have investigated the stability of the ferromagnetic state up to high pressures by measurements of the magnetization in a diamond anvil cell. After an incipient slope of dTc/dp ≅ −0.32 K/GPa, a dramatic decrease of the Curie temperature Tc close to 2.4 GPa has been found. In recent resistivity data from the literature, on the other hand, signs of an ordering transition have been observed up to 3.5 GPa. The implications from this discrepancy on the magnetic phase diagram are discussed, and an explanation in terms of anisotropic exchange interactions is suggested.