Springer Science and Business Media LLC

The ability to easily and flexibly operate with mathematical pattern and structure, in short, having a structure sense is important for mathematics learning. This assumption of a relation between early structure sense and mathematical competences is supported by a longitudinal study with 74 children from kindergarten to second grade. Quantitative analyses reveal a significant, moderate correlation between the two variables at the entry to formal schooling. Furthermore, early structure sense is a predictor for mathematical, in particular, arithmetical competences at the end of grade 2. Its predictive power is almost as good as the explanation of variance by early mathematical abilities. Low and high achieving children’s competences regarding pattern and structure at the beginning of grade 1 are then comparatively analysed in order to evaluate the range of school starters’ early structure sense. The results suggest overall high pre-instructional competences which, however, vary strongly between the mathematical high and low achievers. Cognitive milestones for the development of a sound early structure sense are identified based on the analyses.

Daily lesson planning deviates — particularly with experienced teachers — considerably from the patterns taught in teacher education. Empirical studies in various countries show a relatively homogenous picture of the process which emerges from professional experience. It is the conflict between lack of time and great complexity of the aspects which require consideration in planning which causes this pattern of lesson preparing to emerge. The paper summarizes present empirical results and discusses potential conclusions for lessons planning. Further is discussed which results apply to mathematics teachers in particular and which have a more general relevance.

What additional value for mathematical coursework can new technology achieve? This question is to be answered here by the specific example ofthe tangent problem in differential calculus. The article indicates detailed possibilities for the use of this technology in differential calculus and shows its additional value for coursework. Already known methods such as graphie differentiation and the secant method gain a new quality by the use of modem tools. The internal numerical methods of these tools are compared with the known exact methods, thereby demonstrating the limits and the potential of these tools. Both the limits and the benefits must be known to permit constructive and critical work with new technology.