Open Mathematical Education Notes

1-30

Authors: Bernd Steinbach and Christian Posthoff

It will be shown that many finite combinatorial problems can be solved by using Boolean equations. Therefore, it is necessary to teach how to transform the different problems into this area in a correct way. We demonstrate the required modeling steps in the first part using very different interesting examples. It is no longer necessary to develop different solution methods for different tasks; Boolean equations can uniquely be used. Ternary vectors and ternary arrays are powerful data structures that help to extenuate the complexity. Naturally, problems of a reasonable size cannot be solved by hand, therefore existing software systems must be used. The application of such a software system can require a huge amount of details. We show how a simplified environment of the software supports the teaching process because the key issues can be explored on a reasonable level of abstraction. Both the XBOOLEMonitor and a SAT-solver will be used. It must be learned how to use these systems, i.e., the structure and the working of the systems must be well understood. It can, however, be that the complexity of the problem is beyond the size of the existing systems. Then we have the last level that requires an excellent knowledge of Mathematics and excellent programming skills. We take as an example the rectangle-free coloring of grids using four colors. The solution of these last problems of the highest level show that only the unified efforts of Mathematics and Computer Science can solve the problems of the highest complexity. It is possible to solve problems that could not be solved before in a constructive way. You do not only know whether solutions exist or not, it is possible to build a very large number of solutions or even all the solutions. Therefore, the education of Mathematicians, Computer Scientists and Engineers must take this necessary cooperation into consideration. To be a specialist in one of these areas is not enough!

47-54

Authors: Jelena Kurtuma and Daniel A. Romano

Our intention in this paper is to open a discussion with different people, who have influence on principle-philosophical attitudes on mathematics teacher education policy, about social and political aspects of mathematics education in the Republic of Srpska (an entity of Bosnia and Herzegovina) school systems.

65-86

Authors: Sergei Abramovich

The paper introduces the computational experiment approach to school mathematics curriculum by investigation a variety of mathematical models that were typically considered advanced in the pre-computer age. This approach makes it possible to connect sophisticated mathematical context and the modern day teaching practice. The talk will demonstrate how mathematical experimentation in the technological paradigm creates conditions for collateral learning to occur including the development of skills important for engineering applications of mathematics.

9-18

Authors: Sergei Abramovich

This note explores the efficacy and drawbacks of moving from visual to symbolic in the teaching of school mathematics. It stems from the author’s experience working with prospective teachers in different mathematical education courses. In particular, the note demonstrates how the appropriate use of computing technology can overcome possible pitfalls associated with an uncritical use of computer-generated images of mathematical concepts. The importance of conceptual perspective on visual representations and the teacher’s role in helping students to develop such perspective are emphasized.

17-29

Authors: Sergei Abramovich and Gennady A. Leonov

This note demonstrates how, using commonly available computer applications, allows for the discovery of patterns that structure oscillations of generalized golden ratios formed by the roots of the so-called Fibonacci-like polynomials. It is shown how the universality of oscillations associated with the smallest root, regardless of the length of the string of the ratios, can be established and formulated in combinatorial terms. In addition, the use of circular diagrams informed by computational experiments in the case of the strings of odd number lengths made it possible to recognize patterns associated with the largest root.

23-37

Authors: Jaska Poranen and Pentti Haukkanen

The purpose of this article is to present a connection between school and university mathematics. We examine infinite periodic (non-pre-periodic) decimal expansions of rational numbers with mathematical rigor by using the university-level Number Theory and Group Theory.We will find there connections to various concepts and results concerning the multiplicative group modulo m. We illustrate and concretize our considerations, and propose some preliminary task ideas for all school levels.We also describe how the didactic mathematics programs are organized at the University of Tampere, Finland. We present the mathematical background needed in this subject in Appendix section at the end of the paper.

17-26

Authors: Daniel A. Romano and Milovan Vinčić

The European Society for Research in Mathematics Education was founded in 1998. Since then, not a single institution or an individual from the areas of the former Yugoslavia have, yet joined the society or participated in any of its activities. This article has been written for the mathematics community of Bosnia and Herzegovina, and particularly for the Ministries of Education and Science, because of the need to begin the process of joining the European Society for Research in Mathematics Education. In particular, it is important that mathematicians become involved with research on mathematics education. In this article, the authors try to explain a number of problems related to mathematics education. Thus, we will consider the need for a change in the goals of mathematics education within the school system. On the other hand, we will explore the opinion that the Education ministry should consider the possibility of introducing different ways of teaching mathematics in primary and secondary schools, in order to follow, at least partially, the important reforms which are already taking place in this field.