Einflüsse lehrergeführter Schüler-Interviews auf Schülereinschätzung und Unterrichtsunterstützung im Mathematikunterricht der Grundschule

In Folge der Ergebnisse der PISA und IGLU Studien ist das Thema Diagnose und individuelle Förderung in die öffentliche Diskussion geraten. Vor diesem Hintergrund richtet sich im Herbst 2002 die Aufmerksamkeit der Arbeitsgruppe Wollring auf ein mathematikdidaktisches empirisches Forschungsprojekt in Australien: Early Numeracy Research Project (ENRP) (Clarke et al. 2002). Eine Besonderheit dieses Projektes besteht in der Eins-zu-eins-Situation zwischen Lehrer und Schüler bei einem Interview über Mathematik. Dieses Projekt bildet den Ausgangspunkt der vorliegenden Arbeit. Im ersten Kapitel wird das australische Projekt sowie seine Umsetzung in Deutschland vorgestellt. Ziel des Projektes ist es, die individuellen mathematischen Performanzen von Grund-schulkindern mit Hilfe eines Interviews in einer Eins-zu-eins-Situation des Schülers mit dem Lehrer (Schüler-Interview) zu erheben und damit mathematikdidaktische Orientierungshilfen für den Unterricht zu liefern. Das Schüler-Interview bestimmt den Lernstandort eines Kindes, der als Ausgangspunkt für eine Diagnose dienen kann. Daher werden unterschiedlichen Sichtweisen der Disziplinen – Psychologie, Medizin, Pädagogik, Sonderpädagogik und Fachdidaktik – in Hinblick auf den Begriff „Diagnose“ diskutiert. Die Durchführung von Schüler-Interviews kann neben ihrem diagnostischen Wert auch eine Bedeutung für die Professionalisierung von Lehrern einnehmen, da sie die Lehrer herausfordert, sich mit den Denk- und Lösungswege von Kindern aller Leistungsniveaus intensiv auseinanderzusetzen. In einer Studie von Steinberg et al. (2004, p. 238) wird deutlich, dass dieses Wissen des Lehrers sowohl als ein Index der Veränderung als auch als ein Mechanismus zur Veränderung des Unterrichts dient. In dieser Arbeit werden über den Zeitraum eines Jahres der Umgang der Lehrer mit dem Führen von Schüler-Interviews und den von ihnen daraus gewonnenen Erkenntnissen ausgewertet. Dabei werden mit den Lehrern nach einem halben und nach einem Jahr Erprobung mehrerer von ihnen selbst geführter Schüler-Interviews je ein Interview mit der Forscherin geführt, um herauszufinden, in welchen verschiedenen Bereichen das Führen von Schüler-Interviews den einzelnen Lehrern Unterstützung bietet. Die erhobenen Daten werden qualitativ mit Hilfe der Grounded Theory ausgewertet. Im empirischen Teil der Arbeit werden drei, der am Projekt beteiligten, Lehrerinnen in Form von Fallstudien vorgestellt und ausgewertet. Bei der Lehrerin, die Mathematik nicht als Fach studiert hat, besteht vor allem ein eigener Lernzuwachs in der Sicht auf Mathematik. Zu Beginn der Untersuchung hatte sie laut ihrer eigenen Aussagen eine eher ergebnisorientierte Sicht auf die Mathematik. Die Aussagen der drei Lehrerinnen beruhen auf einzelnen Schülern und ihren Besonderheiten. Im Laufe der Studie verallgemeinern sie ihre Erkenntnisse und beginnen Konsequenzen für ihren Unterricht aus den Schüler-Interviews zu folgern, wie sie in den abschließenden Interviews berichten. Das Schüler-Interview scheint dem Lehrer einen geschützten Raum zu bieten, um die Reflexion über die mathematischen Performanzen seiner Schüler und seinen eigenen Unterricht anzuregen, ohne ihn bloßzustellen und ohne ihm Vorschriften zu machen. Nach der einjährigen Erprobung von Schüler-Interviews betonen alle drei Lehrerinnen größeren Wert auf prozessorientiertes Mathematiklernen zu legen. Sie berichten, dass sie die Performanzen der Kinder stärker kompetenzorientiert wahrnehmen. Jedoch haben sie Schwierigkeiten, die für sich selbst gewonnene Transparenz über die mathematischen Performanzen des interviewten Kindes, den Schülern mitzuteilen und ihnen ermutigende Rückmeldungen zu geben. Außerdem können die Lehrer die problematischen mathematischen Bereiche der Schüler zwar beschreiben, sehen sich laut ihrer eigenen Aussage aber nicht in der Lage mit den Schülern daran zu arbeiten und sie angemessen zu för-dern. Selbst nach den ausführlichen Analysen der ausgewählten Lehrerinnen bleibt unklar, ob und in welcher Weise sie die Erkenntnisse aus dem Führen der Schüler-Interviews für ihren Unterricht nutzen. Laut der Aussage zweier beteiligter Lehrerinnen sollten Lehrer offen und interessiert sein und sich bereitwillig mit ihren eigenen Kompetenzen auseinandersetzen, damit das Führen von Schüler-Interviews für die Lehrer selbst und für die Schüler einen besonderen Nutzen besitzt. Um diese Auseinandersetzung stärker anzuregen und zu vermeiden, dass sich im Schüler-Interview mit dem Kind nicht die Einstellungen des Lehrers gegenüber den Leistungen des Schülers widerspiegeln, könnten sie vor Beginn des Führens von Schüler-Interviews verstärkt in der Ausbildung ihrer Interviewkompetenzen unterstützt und geschult werden. Obwohl sich die Lehrer zuerst Freiräume schaffen mussten, in denen sie trotz ihres Zeitmangels Schüler interviewen konnten, bietet das Führen von Schüler-Interviews die Chance, den Ist-Zustand der Schülerperformanzen in den mathematischen Bereichen Zahlen, Größen und Raum zu erfassen.

Numerical Simulation of Flows at Low Mach Numbers with Heat Sources

This work is concerned with finite volume methods for flows at low mach numbers which are under buoyancy and heat sources. As a particular application, fires in car tunnels will be considered. To extend the scheme for compressible flow into the low Mach number regime, a preconditioning technique is used and a stability result on this is proven. The source terms for gravity and heat are incorporated using operator splitting and the resulting method is analyzed.

Double-autoionization decay of resonantly excited single-electron states

Perturbation theory in the lowest non-vanishing order in interelectron interaction has been applied to the theoretical investigation of double-ionization decays of resonantly excited single-electron states. The formulae for the transition probabilities were derived in the LS coupling scheme, and the orbital angular momentum and spin selection rules were obtained. In addition to the formulae, which are exact in this order, three approximate expressions, which correspond to illustrative model mechanisms of the transition, were derived as limiting cases of the exact ones. Numerical results were obtained for the decay of the resonantly excited Kr 1 3d^{-1}5p[^1P] state which demonstrated quite clearly the important role of the interelectron interaction in double-ionization processes. On the other hand, the results obtained show that low-energy electrons can appear in the photoelectron spectrum below the ionization threshold of the 3d shell. As a function of the photon frequency, the yield of these low-energy electrons is strongly amplified by the resonant transition of the 3d electron to 5p (or to other discrete levels), acting as an intermediate state, when the photon frequency approaches that of the transition.

Photoionization of Kr near 4s threshold: II. Intermediate-coupling theory

The Kr 4s-electron photoionization cross section as a function of the exciting-photon energy in the range between 30 eV and 90 eV was calculated using the configuration interaction (CI) technique in intermediate coupling. In the calculations the 4p spin-orbital interaction and corrections due to higher orders of perturbation theory (the so-called Coulomb interaction correlational decrease) were considered. Energies of Kr II states were calculated and agree with spectroscopic data within less than 10 meV. For some of the Kr II states new assignments were suggested on the basis of the largest component among the calculated CI wavefunctions.

Dirac-Fock-Slater calculations for the elements Z = 100, fermium, to Z = 173

Listed here for the elements Z = 100, fermium, to Z = 173 are energy eigenvalues and total energies found from relativistic Dirac-Fock-Slater calculations. The effect of high ionization on the energy eigenvalues is presented for two exarnples. The use of these tables in connection with the energy levels of superheavy elements and molecular orbital (MO) x-ray transitions in superheavy quasiatoms, is discussed. In addition, abrief comparison between the results of the Dirac-Fock-Slater and Dirac-Fock calculations is given.

Relativistic effects in physics and chemistry of element 105. [Part] II.

Relativistic self-consistent charge Dirac-Slater discrete variational method calculations have been done for the series of molecules MBr_5, where M = Nb, Ta, Pa, and element 105, Ha. The electronic structure data show that the trends within the group 5 pentabromides resemble those for the corresponding pentaclorides with the latter being more ionic. Estimation of the volatility of group 5 bromides has been done on the basis of the molecular orbital calculations. According to the results of the theoretical interpretation HaBr_5 seems to be more volatile than NbBr_5 and TaBr_5.

Reduced L_1 level width and Coster-Kronig yields by relaxation and continuum interactions in atomic zinc

Calculations of the level width \gamma( L_1) and the f_12 and f_13 Coster-Kronig yields for atomic zinc have been performed with Dirac-Fock wave functions. For \gamma(L_1), a large deviation between theory and evaluated data exists. We include the incomplete orthogonality of the electron orbitals as well as the interchannel interaction of the decaying states. Orbital relaxation reduces the total rates in all groups of the electron-emission spectrum by about 10-20 %. Different, however, is the effect of the continuum interaction. The L_1-L_23X Coster-Kronig part of the spectrum is definitely reduced in its intensity, whereas the MM and MN spectra are slightly enhanced. This results in a reduction of Coster-Kronig yields, where for medium and heavy elements considerable discrepancies have been found in comparison to relativistic theory. Briefly, we discuss the consequences of our calculations for heavier elements.

Tomographic imaging of molecular orbitals in length and velocity form

Recently Itatani et al. [Nature 432, 876 (2004)] introduced the new concept of molecular orbital tomography, where high harmonic generation (HHG) is used to image electronic wave functions. We describe an alternative reconstruction form, using momentum instead of dipole matrix elements for the electron recombination step in HHG. We show that using this velocity-form reconstruction, one obtains better results than using the original length-form reconstruction. We provide numerical evidence for our claim that one has to resort to extremely short pulses to perform the reconstruction for an orbital with arbitrary symmetry. The numerical evidence is based on the exact solution of the time-dependent Schrödinger equation for 2D model systems to simulate the experiment. Furthermore we show that in the case of cylindrically symmetric orbitals, such as the N2 orbital that was reconstructed in the original work, one can obtain the full 3D wave function and not only a 2D projection of it. Vor kurzem führten Itatani et al. [Nature 432, 876 (2004)] das Konzept der Molelkülorbital-Tomographie ein. Hierbei wird die Erzeugung hoher Harmonischer verwendet, um Bilder von elektronischen Wellenfunktionen zu gewinnen. Wir beschreiben eine alternative Form der Rekonstruktion, die auf Impuls- statt Dipol-Matrixelementen für den Rekombinationsschritt bei der Erzeugung der Harmonischen basiert. Wir zeigen, dass diese "Geschwindigkeitsform" der Rekonstruktion bessere Ergebnisse als die ursprüngliche "Längenform" liefert. Wir zeigen numerische Beweise für unsere Behauptung, dass man zu extrem kurzen Laserpulsen gehen muss, um Orbitale mit beliebiger Symmetrie zu rekonstruieren. Diese Ergebnisse basieren auf der exakten Lösung der zeitabhängigen Schrödingergleichung für 2D-Modellsysteme. Wir zeigen ferner, dass für zylindersymmetrische Orbitale wie das N2-Orbital, welches in der oben zitierten Arbeit rekonstruiert wurde, das volle 3D-Orbital rekonstruiert werden kann, nicht nur seine 2D-Projektion.

Relativistic prediction of the ground state of atomic Lawrencium

In contradiction to the prediction of the Periodic Table but in agreement with earlier suggestions by Brewer and Mann, the ground state configuration of atomic Lawrencium (Z = 103) will not be 7s^2 6d^2 D_3/2 but 7s^2 7p ^2p_1/2. The reason for this deviation from normal trends across the Periodic Table are strong relativistic effects on the outermost 7P_l/2 orbital. Multicontiguration Dirac-Fock calculations are reported for Lawrencium and analogous lighter atoms. These calculations include contributions from magnetic and retardation interactions and an estimation of quantum electrodynamic corrections.

Interchannel interactions and relaxation in the 2p auger spectra of Mg-like ions

Relativistic Auger rates for the 2p spectra of Mg-like ions have been calculated in the atomic range 13 < Z < 36. We used the multiconfiguration Dirac-Fock method but beyond a simple frozen-orbital approach we include also relaxation for the bound electrons and the interchannel interaction between the continuum states. Both effects may alter the individual transition rates remarkably. This is analysed for a few selected states within the isoelectronic sequence. Weak transitions within the 2p spectra can be changed by an order of magnitude because of the continuum coupling. The influence of both effects for higher-Z ions is reduced but still remain visible.

Non-relativistic and relativistic molecular calculations for the chalcogen hexafluorides: SF_6, SeF_6, TeF_6, PoF_6

Non-relativistic Hartree-Fock-Slater and relativistic Dirac-Slater self-consistent orbital models are applied for the analysis of the electronic structure of the chalcogen hexafluorides: SF_6, SeF_6, TeF_6 and PoF_6. The molecular eigenfunctions and eigenvalues are generated using the discrete variational method (DVM) with numerical basis functions. The results obtained for SF_6 are compared with other ab initio calculations. Information about relativistic level shifts and spin-orbit splitting has been obtained by comparison between the non-relativistic and relativistic results.

Molecular Imaging using Strong-Field Processes

Die Arbeit befasst sich mit dem Zusammenhang zwischen einfachen Molekülen und deren Verhalten in starken, kurzen Laserfeldern. Einerseits wird versucht, strukturelle Daten des Moleküls in den Elektronen- und Photonenspektren wiederzuerkennen. Andererseits geht es darum, ein Bild der elektronischen Wellenfunktion aus den spektralen Daten abzuleiten.

Many-body theory of laser-induced ultrafast demagnetization and angular momentum transfer in ferromagnetic transition metals

An electronic theory is developed, which describes the ultrafast demagnetization in itinerant ferromagnets following the absorption of a femtosecond laser pulse. The present work intends to elucidate the microscopic physics of this ultrafast phenomenon by identifying its fundamental mechanisms. In particular, it aims to reveal the nature of the involved spin excitations and angular-momentum transfer between spin and lattice, which are still subjects of intensive debate. In the first preliminary part of the thesis the initial stage of the laser-induced demagnetization process is considered. In this stage the electronic system is highly excited by spin-conserving elementary excitations involved in the laser-pulse absorption, while the spin or magnon degrees of freedom remain very weakly excited. The role of electron-hole excitations on the stability of the magnetic order of one- and two-dimensional 3d transition metals (TMs) is investigated by using ab initio density-functional theory. The results show that the local magnetic moments are remarkably stable even at very high levels of local energy density and, therefore, indicate that these moments preserve their identity throughout the entire demagnetization process. In the second main part of the thesis a many-body theory is proposed, which takes into account these local magnetic moments and the local character of the involved spin excitations such as spin fluctuations from the very beginning. In this approach the relevant valence 3d and 4p electrons are described in terms of a multiband model Hamiltonian which includes Coulomb interactions, interatomic hybridizations, spin-orbit interactions, as well as the coupling to the time-dependent laser field on the same footing. An exact numerical time evolution is performed for small ferromagnetic TM clusters. The dynamical simulations show that after ultra-short laser pulse absorption the magnetization of these clusters decreases on a time scale of hundred femtoseconds. In particular, the results reproduce the experimentally observed laser-induced demagnetization in ferromagnets and demonstrate that this effect can be explained in terms of the following purely electronic non-adiabatic mechanism: First, on a time scale of 10–100 fs after laser excitation the spin-orbit coupling yields local angular-momentum transfer between the spins and the electron orbits, while subsequently the orbital angular momentum is very rapidly quenched in the lattice on the time scale of one femtosecond due to interatomic electron hoppings. In combination, these two processes result in a demagnetization within hundred or a few hundred femtoseconds after laser-pulse absorption.

Magnetic interactions between transition metal impurities and clusters mediated by low-dimensional metallic hosts: A first principles theoretical investigation

The magnetic properties and interactions between transition metal (TM) impurities and clusters in low-dimensional metallic hosts are studied using a first principles theoretical method. In the first part of this work, the effect of magnetic order in 3d-5d systems is addressed from the perspective of its influence on the enhancement of the magnetic anisotropy energy (MAE). In the second part, the possibility of using external electric fields (EFs) to control the magnetic properties and interactions between nanoparticles deposited at noble metal surfaces is investigated. The influence of 3d composition and magnetic order on the spin polarization of the substrate and its consequences on the MAE are analyzed for the case of 3d impurities in one- and two-dimensional polarizable hosts. It is shown that the MAE and easy- axis of monoatomic free standing 3d-Pt wires is mainly determined by the atomic spin-orbit (SO) coupling contributions. The competition between ferromagnetic (FM) and antiferromagnetic (AF) order in FePtn wires is studied in detail for n=1-4 as a function of the relative position between Fe atoms. Our results show an oscillatory behavior of the magnetic polarization of Pt atoms as a function of their distance from the magnetic impurities, which can be correlated to a long-ranged magnetic coupling of the Fe atoms. Exceptionally large variations of the induced spin and orbital moments at the Pt atoms are found as a function of concentration and magnetic order. Along with a violation of the third Hund’s rule at the Fe sites, these variations result in a non trivial behavior of the MAE. In the case of TM impurities and dimers at the Cu(111), the effects of surface charging and applied EFs on the magnetic properties and substrate-mediated magnetic interactions have been investigated. The modifications of the surface electronic structure, impurity local moments and magnetic exchange coupling as a result of the EF-induced metallic screening and charge rearrangements are analysed. In a first study, the properties of surface substitutional Co and Fe impurities are investigated as a function of the external charge per surface atom q. At large inter-impurity distances the effective magnetic exchange coupling ∆E between impurities shows RKKY-like oscillations as a function of the distance which are not significantly affected by the considered values of q. For distances r < 10 Å, important modifications in the magnitude of ∆E, involving changes from FM to AF coupling, are found depending non-monotonously on the value and polarity of q. The interaction energies are analysed from a local perspective. In a second study, the interplay between external EF effects, internal magnetic order and substrate-mediated magnetic coupling has been investigated for Mn dimers on Cu(111). Our calculations show that EF (∼ 1eV/Å) can induce a switching from AF to FM ground-state magnetic order within single Mn dimers. The relative coupling between a pair of dimers also shows RKKY-like oscillations as a function of the inter-dimer distance. Their effective magnetic exchange interaction is found to depend significantly on the magnetic order within the Mn dimers and on their relative orientation on the surface. The dependence of the substrate-mediated interaction on the magnetic state of the dimers is qualitatively explained in terms of the differences in the scattering of surface electrons. At short inter-dimer distances, the ground-state configuration is determined by an interplay between exchange interactions and EF effects. These results demonstrate that external surface charging and applied EFs offer remarkable possibilities of manipulating the sign and strength of the magnetic coupling of surface supported nanoparticles.