Mathematical algorithms play an important role in our highly technologized daily life and show that mathematics is more then just formulas and exhausting theory. Mathematics also means experimenting and gaining knowledge by trial and error. The numerical experiment is essential for the development of mathematical algorithms: does the algorithm really calculate the desired solution? How fast is the algorithm? What are the limits of algorithms? Only by experimentation can the mathematical-algorithmic theory be proven to be right or new approaches be found. Figuratively speaking computer experiments for an algorithmic mathematician are comparable to the experimental research of natural scientists in the laboratory.
Together with students mathematicians from the research project EMMA developed two topics of research from the field of algorithmic mathematics. The focus of this cooperation was on computer experimentation which was to a great extent carried out by the students. The experiment formed an important bridge between the students and the researchers involved. The students were able to directly participate in the research activities and provided a valuable contribution. The experiments enabled the researchers to impart important aspects of mathematical algorithmic to the students and allowed a profitable exchange for both sides. In the first research topic, assigned to numerics as a field of applied mathematics, numerical solution schemes for variational inequalities, which occur for example in mechanical contact systems, were developed and analyzed. The students were asked to implement and analyze different algorithms for solving the arising systems as independently as possible and to compare them with respect to their properties, dependency of different parameters and efficiency.
The second research topic was set in the field of discrete mathematics and addressed elliptic curves with high rank and Diophantine tuples. Elliptic curves are generalizations of conic sections, i.e. ellipses, parabolas and hyperbola; the rank is a measured value, which counts the number of infinite spirals (in its arithmetic) of an elliptic curve. In collaboration with the students new rank records were established and curves with the highest possible rank elliptic were constructed. Diophantine tuples played a role here, as the starting point for these new constructions. Applications of both research topics can be found for example in crash simulations and data encryption.
The optional course University Mathematics at the affiliated school HTL Braunau as well as the extra course Secret Messages at the Akademisches Gymnasium Salzburg provided the theoretical and practical basic knowledge for the research topics and established a concrete framework for the contact with the students. During the course of the project the basic tuition evolved into working in small groups in which the algorithms were be implemented and analyzed. The experiments and studies as well as the graphical processing were carried out under direct supervision by the involved researchers. The results were subsequently presented and discussed. The preparation of diploma theses in the field of the research topics were another important part of the cooperation with the students.
In addition to the optional course University Mathematics and the extra course Secret Messages further activities within the EMMA project were provided to ensure an intense cooperation between the students and the researchers. These included in particular the workshops at the Bundesinstitut für Erwachsenenbildung in Strobl and in the Nesin Mathematics Village in S¸ erince, Turkey. The workshops took place over several days and intensified the cooperation with the students. There were also further activities such as a programming day, a science day as well as the successful participation at the Science Slam. The compulsory optional subject Computer-oriented applications at Salzburg University meant that teacher trainees of mathematics were also involved.
The EMMA project was also supported by the school of education so that in addition to the research topics, investigations into the effectiveness of teaching principles and concepts, as well as teaching methodology and the evaluation of technical content from the point of view of the school in questions of mathematical algorithmic were carried out. In conclusion it can be ascertained that while the opportunity of experimentation initially presented the students with an unusual situation; the positive experiences which were gained, clearly changed the students perspective of the mathematics class during the EMMA project. Instead of the tenacious manual process of algorithms (for instance the determining of extremes etc.) the students were able to first implement methods themselves and then compare them by means of various criteria. The screening of the available tools in the form of different methods as well as the evaluation of the experiments by means of various criteria such as efficiency, precision and stability were new experiences for the students and encouraged a critical analysis with the given properties of the task requirements and the return dates for example in the form of computing time and the quality of the solution.
The obtained knowledge will directly influence the further planning and structuring of the optional course University Mathematics and mathematic orientated extra courses, as well as the contents and methods of the didactical modules of the new teacher training for the school subjects mathematics, informatics and informatics management. There is also the aim of creating a guideline of how to include mathematical algorithms in school teaching.