Volume XV , issue 1 ( 2012 ) | back |

An exploration of students' conceptual knowledge built in a first ordinary differential equations course (Part I) | 1$-$20 |

**Abstract**

This study aims to analyze and document the types of knowledge that university students exhibit to deal with fundamental issues that they had studied in a first ordinary differential equation course. Questions that helped us structure the research included: How do students interpret and deal with the concept of solution to an Ordinary Differential Equation (ODE)? To which extent do students use mathematical concepts they have previously studied to answer basic questions related to ODEs? And, to what extent do the students' answers privilege the use of certain type of representation to explore and examine issues related to ODEs? Results indicate that, in general, students choose one of two methods to verify whether a function represents a solution to a given ODE: a substitution method or by solving directly the given equation. It was observed that they do not rely on concepts associated with the meaning of derivative to make sense and deal with situations that involve basic ODEs' ideas; rather, they tend to reduce their knowledge of ODEs to the search for an algorithm (analytical approach) to solve particular groups of equations. In addition, there is evidence that students do not use graphic representations to explore meanings and mathematical relations and they experience difficulties to move back and forth from one type of representation to another.

**Keywords:** Learning of Mathematics; the concept of ordinary differential equation; representations.

An approach to incorporate dynamic geometry systems in secondary school---model with module | 21$-$31 |

**Abstract**

The paper presents the key elements of a model for in-service teachers training. An educational module about parabola as geometry object is the illustrative part of the model. The inclusion of dynamic geometry system applets allows some properties of the parabola to be established by examining dynamic constructions. Starting with the focus-directrix definition, the reflective property of the parabola has been proven and the equation of the parabola is worked out. A comprehensive comment on doubling-the-cube problem is given. Some directions for further development of the topic are pointed out.

**Keywords:** Dynamic geometry; parabola.

On the existence of $\lim_{x\to x_0}f(g(x))$ | 33$-$42 |

**Abstract**

In this article we discuss limits of composite functions in the general setting of topological spaces. We include here some of its technical applications.

**Keywords:** Limits; composite functions; topological spaces.

Factors that influence students to do mathematics | 43$-$54 |

**Abstract**

The aim of this investigation was to study the factors that influence students to do mathematics in a level higher than the usual level arising from the usual syllabus. The sample was 339 students who participated at 25th National Mathematical Olympiad ``Archimedes'' in March 2008. They completed a questionnaire designed to measure the factors that influenced them to do mathematics. The results of this investigation show that the more important factors affecting students to do mathematics are: Mathematical competitions, their fathers, books, their school teachers, the publications of the Hellenic Mathematical Society and their mothers.

**Keywords:** Mathematics competitions; factors of motivation.

The fundamental theorem on symmetric polynomials | 55$-$59 |

**Abstract**

In this work we are going to extend the proof of Newton's theorem of symmetric polynomials, by considering any monomial order $>$ on polynomials in $n$ variables $x_{1},x_{2},\dots,x_{n}$ over a field $k$, where the original proof is based on the graded lexicographic order. We will introduce some basic definitions and propositions to support the extended proof.

**Keywords:** Symmetric polynomials.

Zoran Pop-Stojanović (Obituary) | 61$-$62 |

**Abstract**

In this work we are going to extend the proof of Newton's theorem of symmetric polynomials, by considering any monomial order $>$ on polynomials in $n$ variables $x_{1},x_{2},\dots,x_{n}$ over a field $k$, where the original proof is based on the graded lexicographic order. We will introduce some basic definitions and propositions to support the extended proof.

**Keywords:** Symmetric polynomials.