The presented study was motivated by the dynamic phenomena observed in basic catalytic surface reactions, especially by bi- and tristability, and the interactions between these stable states. In this regard, three reaction-diffusion models were developed and examined using bifurcation theory and numerical simulations. A first model was designed to extend the bistable CO oxidation on Ir(111) to include hydrogen and its oxidation. The differential equation system was analyzed within the framework of bifurcation theory, revealing three branches of stable solutions. One state is characterized by high formation rates (upper rate state, UR), while the other two branches display low formation rates (lower rate (LR) \& very low rate (VLR) states). The overlapping branches form the shape of a `swallowtail', the curve of saddle-node bifurcations forming two cusps. Increasing the temperature leads to a subsequent unfolding and hence decreases the system complexity. A series of numerical simulations representing possible experiments was conducted to illustrate the experimental accessibility (or the lack) of said states. Relaxation experiments show partially long decay times. Quasistatic scanning illustrates the existence of all three states within the tristable regime and their respective conversion once crossing a fold. A first attempt regarding the state dominance in reaction-diffusion fronts was done. While UR seems to dominate in 1D, a 2D time-evolution shows that LR invades the interphase between UR and VLR. Subsequently, a generic monospecies mock model was used for the comprehensive study of reaction-diffusion fronts. A quintic polynomial as reaction term was chosen, derived by the sixth-order potential associated with the `butterfly bifurcation'. This ensures up to three stable solutions($u_{0}$,$u_{1}$,$u_{2}$), depending on the four-dimensional parameter space. The model was explored extensively, identifying regions with similar behaviors. A term for the front velocity connecting two stable states was derived, depending only on the relative difference of the states' potential wells. Equipotential curves were found, where the front velocity vanishes of a given front. Numerical simulations on a two-dimensional, finite disk using a triangulated mesh supported these findings. Additionally, the front-splitting instability was observed for certain parameters. The front solution $u_{02}$ becomes unstable and divides into $u_{01}$ and $u_{12}$, exhibiting different front velocities. A good estimate for the limit of the front splitting region was given and tested using time evolutions. Finally, the established mock model was modified from continuous to discrete space, utilizing a simple domain in 1D and three different lattices in 2D (square, hexagonal, triangular). For low diffusivities or large distances between coupling nodes, fronts can become pinned, if the parameters are within range of the equipotential lines. This phenomenon is known as propagation failure and its extent in parameter space was explored in 1D. In 2D, an estimate was given for remarkable front orientations respective to the lattice using a pseudo-2D approximation. Near the pinning region, front velocities differ significantly from the continuous expectation as the shape of the curve potential becomes significant. Factors that decide the size and shape of the pinning regions are the coupling strength, the lattice, the front orientation relative to the lattice, and the front itself. The bifurcation diagram shows a snaking curve in the pinning region, each alternating branch representing a stable or unstable frozen front solution. Numerical simulations supported the observations concerning propagation failure and lattice dependence. Furthermore, the influence of front orientation on the front velocity was explored in greater detail, showing that fronts with certain lattice-dependent orientations are more or less prone to propagation failure. This leads to the possibility of pattern formation, reflecting the lattice geometry. An attempt to quantify the front movement depending on angular front orientation has shown reasonable results and good agreement with the pseudo-2D approach.

Studien zur Wirksamkeit der Ausbildung von Lehrerinnen und Lehrern sind in der Regel fragebogenbasiert. In der vorliegenden Untersuchung wird ein anderer Ansatz gewählt, um die Kompetenzen angehender Mathematiklehrkräfte zu erheben und zu evaluieren. Die Studie untersucht anhand der unterrichtlichen Performanz, welche Kompetenzen Anwärterinnen und Anwärter in ihrer praktischen Ausbildung erworben haben. Der Analyse liegen 137 anonymisierte Niederschriften zu benote-ten Lehrproben von Anwärterinnen und Anwärtern im Fach Mathematik zugrunde. Diese bilden die Ausbildungsjahrgänge von 2004 bis 2011 ab. Die Anwärterinnen und Anwärter werden unterrichtspraktisch anhand der Niederschriften in Anlehnung an eine qualitative Inhaltsanalyse auf ihre Kompetenzausprägungen hin getestet. Dies erfolgt schwerpunktmäßig in den Bereichen Fachdidaktik, Fachmethodik, Classroom-Management und Gesprächsführung. Im Verlauf der Untersuchung wird deutlich, dass die Teilneh-mergruppe vor allem Stärken im methodischen Bereich hat. Hier können durchgehend gute Kompetenzausprägungen festgestellt werden. Über den gesamten Erhebungszeitraum zeigen sich hingegen in den Bereichen Classroom-Management und Gesprächsführung deutliche Defizite. Die fachdidaktischen Kompetenzen der angehenden Lehrkräfte entwickeln sich negativ: Ist zu Beginn der Erhebung noch etwa die Hälfte der Anwärterinnen und Anwärter im Bereich der Fachdidaktik zufriedenstellend einzustufen, gilt das im letzten Erhebungsjahrgang nur noch für knapp ein Fünftel der Probandinnen und Probanden. Es kann gezeigt werden, dass eine gute Notenfestsetzung eng mit den er-forderlichen Kompetenzen in der Fachdidaktik zusammenhängt. Die vorliegende Studie bestätigt im Bereich der Fachdidaktik die Ergebnisse jener Studien, die auf Fragebögen basieren. Es konnte zudem ein statistischer Zusammenhang zwischen den fachdidaktischen Fähigkeiten und den Steuerungskompetenzen hergestellt werden.