Abstract
The transition between school- and university-level mathematics is a challenging task for students, which includes the need to change their views of mathematics. In particular, this task encompasses coping with the discrepancy between a mostly concrete and application-oriented school perspective on the subject and the abstract and comprehensive university perspective. Preservice teachers are especially concerned as they might see their university studies more or less as an interruption between the two ways of working at school, namely as a student and as a teacher. In a small study, we addressed the question how teacher students change their view on mathematical definitions during the first months at the university. We will present results and discuss them with respect to the transformation of knowledge.
Die Mathematiker sind eine Art Franzosen: Redet man zu ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anders. (Mathematicians are like a sort of Frenchmen: if you talk to them, they translate it into their own language, and then it is immediately something quite different). Johann Wolfgang von Goethe (1749–1832).
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Notes
- 1.
Elementary Mathematics from an Advanced Standpoint: Arithmetic, Algebra, Analysis.
- 2.
Baumert et al. (2010) used, e.g., the task “Is \( 2^{1024}-1 \) a prime number?” in their study. Solving this task presupposes the concept of prime number (approx. grade 6 in German curricula) and the formula \( x^{2n}- 1 = (x^n + 1)(x^n - 1) \) (approx. grade 7 in German curricula).
- 3.
German preservice teacher students who will teach at the upper secondary level are supposed to choose two major subjects during their BA and MA studies as well as elementary courses in pedagogy and psychology.
- 4.
This means a course at a level somewhat below Walter Rudin’s “Principles of Mathematical Analysis” (1976).
- 5.
The notation “y ≤ Y” was introduced in the course to express that a real number y is a lower bound of a set Y of reals.
- 6.
A real number s is the supremum or least upper bound of a set X of reals, if it is the smallest real number s such that \(X\leq s.\).
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Deiser, O., Reiss, K. (2014). Knowledge Transformation Between Secondary School and University Mathematics. In: Rezat, S., Hattermann, M., Peter-Koop, A. (eds) Transformation - A Fundamental Idea of Mathematics Education. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3489-4_3
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