Volume XIII , issue 2 ( 2010 ) | back |

Local rate of change: a Socratic experience in van Hiele's model framework | 63$-$92 |

**Abstract**

Our aim is to provide an educative experience concerning the study of rate of change, to be implemented prior to its formal treatment in the classroom. By means of a Socratic dialog and the manipulation of a computer generated visualization tool, we shall provoke answers to questions such as How much?, How steep? and How fast? by identifying all key ingredients necessary to provide a sound concept image of the concept of derivative of a function at a point. Our study is framed in van Hiele's educational model methodology. This first of two articles deals with the interview and the list of descriptors which allow the detection of the levels of reasoning the model postulates. In a forthcoming article we shall concern ourselves with the analysis of all students' responses, cognitive obstacles encountered and tutorial actions needed to overcome them.

**Keywords:** Model; change; rate of change; visualization; approximation; local rate of change.

Zur Integration der Hyperbelfunktion | 93$-$104 |

**Abstract**

This is about several ways how to integrate the hyperbola function (not using the chain rule). Some ways may be not so common. This article is not about new mathematical results, but focusses on several ways while always keeping in mind the students' perspective. From a purely mathematical point of view, the article could be read as proposing several alternatives on how to define $\exp(x)$ or $\ln(x)$, but this would not be the author's intention. He argues from the students' view, so this article is about several alternatives on how to obtain the area under the hyperbola function. At the end, the relation of $\ln 2$ to the harmonic series will be discussed.

**Keywords:** Hyperbola; integration.

Phases of mathematical modelling and competence of high school students | |

**Abstract**

Competence of mathematical modelling are these days an important part in person's competence. This paper deals with student's problems with mathematical modelling as a part of problem solving. Three phases of mathematical modelling are suggested and some types of models are described in this paper. Presented results are based on a pedagogical survey which is characterized here too.

**Keywords:** 105--112

The influence of the Geometer's Sketchpad on the geometry achievement of Greek school students | 113$-$124 |

**Abstract**

The aim of this study is to investigate the question: ``Does the use of a Geometer's Sketchpad help secondary students to improve their geometry performance?'' The respondents were students enrolled in a high school (grade~7) in a northern suburb of Athens. The experimental group consisted of approximately 40 students, who spent at least one hour per week doing computer explorations for the first six weeks of the last semester. There were 39 students in the control group, which was not exposed to the computer explorations. Students in both groups were pre-tested and post-tested for their geometry performance. The results of the study indicated that the use of technology is needed for students to make significant progress in geometry.

**Keywords:** Proof; geometry; van Hiele model; Geometer's Sketchpad; educational software.