Fifth Central- and Eastern European Conference
on Computer Algebra- and Dynamic Geometry Systems
in Mathematics Education

A course of discrete mathematics calls for different accentuation when taught for different future specialists. A lot of ingenuity is needed to create a successful course for would-be teachers of mathematics. The basic topics of our course includes set theory, mathematical logic, combinatorics, and graph theory in teaching which we make a good use of GeoGebra tools. A wide range of instruments (such as Spreadsheet and Algebra view, some categories of commands and tools, etc) allow a teacher to expand the range of tasks, increase their complexity, foster a research component, create dynamic demonstrations to explore mathematical objects with multiple representations.

We offer the collection of interactive dynamic demonstrations designed in GeoGebra in support of discrete mathematics course for future teachers of mathematics. The collection focuses on four types of demonstrations: teaching and learning basic concepts, stages of theorem proving, implementation of well-known algorithms from graph theory, the real-life problem-solving situations. One of the course assignments includes a student's project on creating a dynamic demonstration. The best projects formed the supplement of the collection.

The result of the experimental implementation (2010-2014) of the collection of interactive dynamic demonstrations will be presented.

We can consider two basic views, when using mathematical software in the teaching of the mathematical subjects.

First: How to learn to use specific software for the specific tasks, e. g., software Statistica for the subjects of Applied statistics, probability and mathematical statistics, or financial mathematics.

Second: How to learn to use the software that is available to us to solve specific mathematical problems from different areas of mathematics and applied mathematics.

My article describes the practical use of the software in the teaching of the mathematical subjects and my experience with its use by the students and the pedagogues.

In our computer laboratory our students use the following software: MS Office (MS Excel), LibreOffice (LO Calc), MATLAB 2010b, Octave, wxMaxima, and LaTeX.

We use this software in the following mathematical subjects: Operational Analysis, Linear and Quadratic Programming, Numerical Mathematics, Applied Statistics, Queuing Theory and Fundamentals of the LaTeX.

Eight years of experience with the usage of these programs have shown us, how we can improve the teaching process of the mathematical subjects at the technical universities with freely available software.

It is usually assumed that a computer algebra system (CAS) offers the answer that is expected by the student or the teacher. However, some answers can be somewhat unexpected (but not necessarily wrong). Such answers could induce rich mathematical discussion. Then again, the unexpected answers could only be obstacles to learning without any productive discussion.

The CADGME paper analyzes pair discussions of first-year students who participated in a course in elementary mathematics. The pairs had worksheets with trigonometric equations and questions. In the beginning, the students solved the equation (correctly or not) on paper without the help of a CAS. Then they solved the same equation with a particular CAS (that gave a different answer from the answer expected by the students). The given questions guided them to analyze the differences, equivalence and correctness of their own answers and CAS answers.

Their discussions were audio-recorded and the paper is based both on the audio-records and on the students’ worksheets. In most cases the students engaged in active discussions about the tasks. The discussions varied in dynamics, productivity, mathematical depth, etc. This paper highlights and discusses some of the patterns that emerged from the discussions.

Dynamic Algebra has been implemented in EpsilonWriter as a way for doing calculations by direct manipulation of algebraic objects mouse, especially step-by-step calculations with explanations. A first group of functionalities is currently available (http://epsilonwriter.com). It contains equivalent drag&drop, e.g., a drag&drop of 3x onto 2x in 2x-4=3x provides –x-4=0, a drag&drop of 3x inside the parentheses in 3x(2x+3) provides 3x²+9x. Factoring out from a sum, simplifying fractions, applying substitutions to expressions, multiplying an equation by an expression, side-by-side adding equations can also be performed by drag&drop. It is particularly easy to solve simultaneous equations. Each calculation step is explained in a way chosen by the user.

These functionalities also contain schemas for solving quadratic equations with the discriminant and cubic equations with Cardan’s formulas, see demos at: http://www.epsilonwriter.com/en/top10demos/

New functionalities have been recently implemented and will be available soon. They appear as proposals displayed in a balloon when the mouse flies over an operator or when an expression is selected. This is the case for “Expand and simplify”, “Complete the square”, and “Factor with A²-B²”. A rich set of functionalities concerns functions with: definition conditions, calculations of limits, calculations of derivatives, and drawing tables of signs of derivatives and of variations of functions. Last, a lot of approximate values of several functions, from values of the variable expressed as a sequence, can be obtain in a table for helping understanding the behavior of functions.

These tools are currently used to design learning units in the MCSquared project (http://mc2-project.eu/). In the first phase, the domain concerns functions. In the second phase, it will be enlarged in particular to algebra.