Mathematical modelling in mathematics education and instruction

This paper aims at giving a concise survey of the present state-of-the-art of mathematical modelling in mathematics education and instruction. It will consist of four parts. In part 1, some basic concepts relevant to the topic will be clarified and, in particular, mathematical modelling will be defined in a broad, comprehensive sense. Part 2 will review arguments for the inclusion of modelling in mathematics teaching at schools and universities, and identify certain schools of thought within mathematics education. Part 3 will describe the role of modelling in present mathematics curricula and in everyday teaching practice. Some obstacles for mathematical modelling in the classroom will be analysed, as well as the opportunities and risks of computer usage. In part 4, selected materials and resources for teaching mathematical modelling, developed in the last few years in America, Australia and Europe, will be presented. The examples will demonstrate many promising directions of development.

Design, implementation and application of a generic framework for integrated regional land-use modeling

Land use is a crucial link between human activities and the natural environment and one of the main driving forces of global environmental change. Large parts of the terrestrial land surface are used for agriculture, forestry, settlements and infrastructure. Given the importance of land use, it is essential to understand the multitude of influential factors and resulting land use patterns. An essential methodology to study and quantify such interactions is provided by the adoption of land-use models. By the application of land-use models, it is possible to analyze the complex structure of linkages and feedbacks and to also determine the relevance of driving forces. Modeling land use and land use changes has a long-term tradition. In particular on the regional scale, a variety of models for different regions and research questions has been created. Modeling capabilities grow with steady advances in computer technology, which on the one hand are driven by increasing computing power on the other hand by new methods in software development, e.g. object- and component-oriented architectures. In this thesis, SITE (Simulation of Terrestrial Environments), a novel framework for integrated regional sland-use modeling, will be introduced and discussed. Particular features of SITE are the notably extended capability to integrate models and the strict separation of application and implementation. These features enable efficient development, test and usage of integrated land-use models. On its system side, SITE provides generic data structures (grid, grid cells, attributes etc.) and takes over the responsibility for their administration. By means of a scripting language (Python) that has been extended by language features specific for land-use modeling, these data structures can be utilized and manipulated by modeling applications. The scripting language interpreter is embedded in SITE. The integration of sub models can be achieved via the scripting language or by usage of a generic interface provided by SITE. Furthermore, functionalities important for land-use modeling like model calibration, model tests and analysis support of simulation results have been integrated into the generic framework. During the implementation of SITE, specific emphasis was laid on expandability, maintainability and usability. Along with the modeling framework a land use model for the analysis of the stability of tropical rainforest margins was developed in the context of the collaborative research project STORMA (SFB 552). In a research area in Central Sulawesi, Indonesia, socio-environmental impacts of land-use changes were examined. SITE was used to simulate land-use dynamics in the historical period of 1981 to 2002. Analogous to that, a scenario that did not consider migration in the population dynamics, was analyzed. For the calculation of crop yields and trace gas emissions, the DAYCENT agro-ecosystem model was integrated. In this case study, it could be shown that land-use changes in the Indonesian research area could mainly be characterized by the expansion of agricultural areas at the expense of natural forest. For this reason, the situation had to be interpreted as unsustainable even though increased agricultural use implied economic improvements and higher farmers' incomes. Due to the importance of model calibration, it was explicitly addressed in the SITE architecture through the introduction of a specific component. The calibration functionality can be used by all SITE applications and enables largely automated model calibration. Calibration in SITE is understood as a process that finds an optimal or at least adequate solution for a set of arbitrarily selectable model parameters with respect to an objective function. In SITE, an objective function typically is a map comparison algorithm capable of comparing a simulation result to a reference map. Several map optimization and map comparison methodologies are available and can be combined. The STORMA land-use model was calibrated using a genetic algorithm for optimization and the figure of merit map comparison measure as objective function. The time period for the calibration ranged from 1981 to 2002. For this period, respective reference land-use maps were compiled. It could be shown, that an efficient automated model calibration with SITE is possible. Nevertheless, the selection of the calibration parameters required detailed knowledge about the underlying land-use model and cannot be automated. In another case study decreases in crop yields and resulting losses in income from coffee cultivation were analyzed and quantified under the assumption of four different deforestation scenarios. For this task, an empirical model, describing the dependence of bee pollination and resulting coffee fruit set from the distance to the closest natural forest, was integrated. Land-use simulations showed, that depending on the magnitude and location of ongoing forest conversion, pollination services are expected to decline continuously. This results in a reduction of coffee yields of up to 18% and a loss of net revenues per hectare of up to 14%. However, the study also showed that ecological and economic values can be preserved if patches of natural vegetation are conservated in the agricultural landscape. -----------------------------------------------------------------------

Statistical evaluation of texture analysis from the biocrystallization method

The consumers are becoming more concerned about food quality, especially regarding how, when and where the foods are produced (Haglund et al., 1999; Kahl et al., 2004; Alföldi, et al., 2006). Therefore, during recent years there has been a growing interest in the methods for food quality assessment, especially in the picture-development methods as a complement to traditional chemical analysis of single compounds (Kahl et al., 2006). The biocrystallization as one of the picture-developing method is based on the crystallographic phenomenon that when crystallizing aqueous solutions of dihydrate CuCl2 with adding of organic solutions, originating, e.g., from crop samples, biocrystallograms are generated with reproducible crystal patterns (Kleber & Steinike-Hartung, 1959). Its output is a crystal pattern on glass plates from which different variables (numbers) can be calculated by using image analysis. However, there is a lack of a standardized evaluation method to quantify the morphological features of the biocrystallogram image. Therefore, the main sakes of this research are (1) to optimize an existing statistical model in order to describe all the effects that contribute to the experiment, (2) to investigate the effect of image parameters on the texture analysis of the biocrystallogram images, i.e., region of interest (ROI), color transformation and histogram matching on samples from the project 020E170/F financed by the Federal Ministry of Food, Agriculture and Consumer Protection(BMELV).The samples are wheat and carrots from controlled field and farm trials, (3) to consider the strongest effect of texture parameter with the visual evaluation criteria that have been developed by a group of researcher (University of Kassel, Germany; Louis Bolk Institute (LBI), Netherlands and Biodynamic Research Association Denmark (BRAD), Denmark) in order to clarify how the relation of the texture parameter and visual characteristics on an image is. The refined statistical model was accomplished by using a lme model with repeated measurements via crossed effects, programmed in R (version 2.1.0). The validity of the F and P values is checked against the SAS program. While getting from the ANOVA the same F values, the P values are bigger in R because of the more conservative approach. The refined model is calculating more significant P values. The optimization of the image analysis is dealing with the following parameters: ROI(Region of Interest which is the area around the geometrical center), color transformation (calculation of the 1 dimensional gray level value out of the three dimensional color information of the scanned picture, which is necessary for the texture analysis), histogram matching (normalization of the histogram of the picture to enhance the contrast and to minimize the errors from lighting conditions). The samples were wheat from DOC trial with 4 field replicates for the years 2003 and 2005, “market samples”(organic and conventional neighbors with the same variety) for 2004 and 2005, carrot where the samples were obtained from the University of Kassel (2 varieties, 2 nitrogen treatments) for the years 2004, 2005, 2006 and “market samples” of carrot for the years 2004 and 2005. The criterion for the optimization was repeatability of the differentiation of the samples over the different harvest(years). For different samples different ROIs were found, which reflect the different pictures. The best color transformation that shows efficiently differentiation is relied on gray scale, i.e., equal color transformation. The second dimension of the color transformation only appeared in some years for the effect of color wavelength(hue) for carrot treated with different nitrate fertilizer levels. The best histogram matching is the Gaussian distribution. The approach was to find a connection between the variables from textural image analysis with the different visual criteria. The relation between the texture parameters and visual evaluation criteria was limited to the carrot samples, especially, as it could be well differentiated by the texture analysis. It was possible to connect groups of variables of the texture analysis with groups of criteria from the visual evaluation. These selected variables were able to differentiate the samples but not able to classify the samples according to the treatment. Contrarily, in case of visual criteria which describe the picture as a whole there is a classification in 80% of the sample cases possible. Herewith, it clearly can find the limits of the single variable approach of the image analysis (texture analysis).

Ethnomethodologie als handlungstheoretische Grundlage einer Didaktik der Kommunikation

Our focus is the question as to how communication in school and instruction can be organized in an open-ended yet methodical fashion. Accordingly we proceed from the thesis that teaching methodology must be in accordance with the methodology od everyday activities so that school instruction can build on the structure of reflexive communication. We therefore adopt ethnomethodology as a theory of everyday activities in order to establish a repertoire of teaching methodologies. Taking an ethnomethodological perspective we analyze two examples of pupil aggression from the aspect of the reflexivity of accounts and develop suggestions for openended teaching methodologies.