Methodological reflections on a three-step-design combining observation,stimulated recall and interview

In this paper a tripartite qualitative design combining abservation, stimulated recall and interview is presented and discussed. This three-step-design makes it possible to get insight into the interaction of internal and external processes when solving mathematical tasks. The data analysis depends on the research question and the methodological approach. In the light of two research projects in mathematics education two different methods of data analysis are presented and methodologically reflected.     

Particle Characterization by Projected Area Determination

The projected areas of non-spherical particles do not represent an unambiguous particle characteristic. Depending on the orientation towards a constant observational direction, different projected areas result. The spectrum of all projected area values of a particle, if determined representatively, gives the probability with which a certain value is obtained by a single measurement. In this work, the frequency distributions of different examples of test objects were both calculated and measured. The objects were a cube, a rectangular parallelepiped and also three model agglomerates consisting of spheres of the same size. Instead of just one projected area, during each measuring procedure three projected areas from three orthogonal directions can be obtained. A mean value is then calculated to reduce the ambiguity of the particle characteristic and enhance the resolution. A suitable measurement set-up is introduced. The results of calculation and measurement are compared for observation from just one direction and also simultaneous observation from three directions. The frequency distributions of the equivalent diameters of the particle projected areas show a characteristic trend of the total curve with remarkable properties. The simultaneous measurement of three values from mutually orthogonal directions and their mean value calculation result in a much narrower distribution. In this case, a non-sphericity factor can additionally be calculated, whose frequency distribution contains information in a characteristic manner about the degree to which the particle shape differs from a sphere.     

Runge–Kutta Methods for Itô Stochastic Differential Equations with Scalar Noise

A general class of stochastic Runge–Kutta methods for Itô stochastic differential equation systems w.r.t. a one-dimensional Wiener process is introduced. The colored rooted tree analysis is applied to derive conditions for the coefficients of the stochastic Runge–Kutta method assuring convergence in the weak sense with a prescribed order. Some coefficients for new stochastic Runge–Kutta schemes of order two are calculated explicitly and a simulation study reveals their good performance.     

Minimum boundary touching tilings of polyominoes

We study the problem of tiling a polyomino P with as few squares as possible such that every square in the tiling has a non‐empty intersection with the boundary of P . Our main result is an algorithm which given a simply connected polyomino P computes such a tiling of P . We indicate how one can improve the running time of this algorithm for the more restricted row‐column‐convex polyominoes. Finally we show that a related decision problem is in NP for rectangular polyominoes. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)