Abstract:

The ability to defend and to reason own attitudes and opinions is an inherent part of everyday life. According to long-term objectives in education, correct and proper argumentation and information processing is increasingly required. Mathematics and science are based on valid relations and laws and their characteristic feature is deductive construction of knowledge. The mathematical thinking, the reasoning and the usage of arguments are skills necessary for the construction of mathematical proofs as one of the higher skill showing the level of mathematical literacy. Mathematical proof problems are not common in school mathematics, but necessarily part in the deeper understanding of mathematical principles, especially in connection with the open-ended problem solving.

This paper is aimed on the analysis of authentic students’ solutions of open-ended problems represented the mathematical proof. Problems were selected from Mathematical B-day contest assignment. Students’ performance in solutions was assessed by the rubrics created for observing chosen attributes: manifested mathematical competencies, progress of mathematical modelling, level of mathematical argumentation, mathematical writing skills and creativity. Mentioned attributes also described the didactical variables for analysing solutions of open-ended problems as mathematical proofs. We described the relations, which influenced the mathematical level of solution, by statistical implicative analysis based on comparison of 50 authentic students’ solutions.

Found relations were useful in identifying competencies for finding and formulating the mathematical proofs. By defined competencies, the rubric for assessing the mathematical proving processes can be created. They can be also used for assessing of students’ wider mathematical thinking skills.

Keywords:

Mathematical proof, competency, open-ended problem.