Fostering Mathematical Competences by Preparing for a Mathematical Competition

Abstract

Mathematical competences are important for mastering the problems that are encoun-tered in a modern society that values knowledge. Such competences are relevant not only for mastering the mathematical problems encountered in school but also for managing everyday life. In practice, mathematical competences are required for finding solutions to society’s major problems (e.g., the prediction of global warming). Mathematical competences are thereby assumed to be individual cognitive abilities and skills as well as the outcomes of learning processes. An individual is ascribed with sophisticated mathematical competences if he or she is able to come up with new mathematical problems by applying previously existing mathematical competences meaningfully. Therewith, fostering mathematical competences is of major importance. Based on a cognitive-socio-constructive understanding of learning in mathematics, students need learn-ing possibilities that lock in their individual potential. Several mechanisms and factors have been shown to drive the acquisition of mathematical competences. To foster mathematical competences, challenging learning opportunities are necessary. Especially for students who are already able to solve curriculum-based tasks. One extracurricular enrichment approach that has been suggested to challenge students are (domain-specific, mathematical) academic competitions. But, to ensure that these students will be able to master the challenging prob-lems they will face in the competition, they must prepare appropriately to solve such prob-lems. Therefore, and to protect them from negative experiences such as failure, corresponding training programs have been suggested and implemented in practice. Such training programs prepare students to participate in a specific academic competition. Paper 1 reviews the appropriateness of academic competitions by summarizing the roles ascribed to academic competitions with regard to the promotion of gifted students. Using the example of the Mathematical Olympiad for elementary school students, a training program that considers the strengths and weaknesses of mathematically gifted elementary school stu-dents is introduced. The training was aimed at enhancing the performance in the Mathemati-cal Olympiad as well as (process-based) mathematical competences. The effectiveness of this particular training was examined in two empirical studies: In Paper 2, a quasi-experimental pre- and posttest design was used to investigate the effects of the training. Dependent variables were success in the Mathematical Olympiad, mathematical competences, and the motivation to do mathematics (i.e., math self-concept and value beliefs for mathematics). A total of 201 third- and fourth-grade students participated in this study. Positive effects were found for third and fourth graders’ performance in the Mathematical Olympiad, their mathematical competences, and the task-specific interest in mathematics of fourth-grade students. In Paper 3, the effects of a training that was aimed at fostering process-based mathe-matical competences on cognitive factors were investigated in detail. Dependent variables were success in the Mathematical Olympiad, content- and process-based mathematical com-petences, as well as domain-general cognitive abilities. Results of a randomized controlled field trial with 97 students indicated significant effects of the training on process-based com-petences but also transfer effects on domain-general abilities. In summary, this dissertation provides evidence for the positive influences of a training for an academic competition in mathematics on students’ performance in the competition and, additionally, their mathematical competences. Based on the results of the studies, questions for further educational research with regard to trainings and academic competitions can be deduced. The findings suggest that the effectiveness of separate core components should be investigated more detailed. Further, some implications for educational practice are summa-rized.