Didactical implications of using various methods to evaluate zeta(2).

International audienceThere are mathematical problems, which could be solved by several techniques that belong to different areas of mathematics and vary in their requirements for prerequisite knowledge or level of sophistication. One example is given in this paper. It involves summation of an infinite series that uses either calculus or advanced (complex or Fourier) analysis. A survey that I conducted with students who recently graduated with a major in mathematics reveals that students were more likely to be familiar with some advanced methods than with relatively elementary views. Similarly, instructors' survey showed that calculus based methods were less popular than some advanced methods despite the fact that instructors found the former being appropriate for inclusion in corresponding courses. I argue that this situation in undergraduate mathematics curriculum needs a more careful consideration. </p

Bounds for algorithms in differential algebra

We consider the Rosenfeld-Groebner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld-Groebner algorithm, in which for every intermediate polynomial system F, the bound M(F) is less than or equal to (n-1)!M(G), where G is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. We also give an algorithm for converting a characteristic decomposition of a radical differential ideal from one ranking into another. This algorithm performs all differentiations in the beginning and then uses a purely algebraic decomposition algorithm.Comment: 40 page

The promise of interconnecting problems for enriching students’ experiences in mathematics

The interconnecting problem approach suggests that often one and the same mathematical problem can be used to teach various mathematical topics at different grade levels. How is this approach useful for the development of mathematical ability and the enrichment of mathematical experiences of all students including the gifted ones? What are the benefits for teachers’ and what would teachers need to implement this approach? What directions would further research on these issues take? The paper discusses these and closely related questions. I propose that a long-term study of a progression of mathematical ideas revolved around one interconnecting problem is useful for developing a perception of mathematics as a connected subject for all learners. Having a natural appreciation for linking learned material, mathematically-able students exposed to this approach could develop more comprehensive thinking, applicable in many other problem solving situations, such as multiple-solution tasks. Because the problem’s solutions vary in levels of difficulty, as well as conceptual richness, the approach allows teachers to form a strategic vision through a systematic review of various mathematical topics in connection with one problem. General pedagogical ideas outlined in this paper are supported by discussions of concrete mathematical examples and classroom applications. While individual successful practices of using this approach are known to be taking place, the need for more data collection and interpretation is highlighted