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

This paper presents a research work where has as objective to check if the proposal of an algorithm converted into a computer program can help high school students in the learning of the zeros of the 2nd degree polynomial function. The research was conducted in two stages. The first stage was with a 1st year high school student in order to verify if the activities were appropriate and, in the second part, we have selected four participants for the second stage. After the analysis of the first stage development, the activities were improved to the second one, composed of three activities, among which the software Visualg 2.0. The APOS theory by Ed Dubinsky, theoretical support of the research, presents the action levels, process, object and scheme, that allow the verification of the individual`s capacity to develop actions over an object and think about its properties. The research participants had improvements in their learning, because besides developing a computer program to determine the zeros of 2nd degree polynomial function, they have started to elaborate other functions previewing their possible solutions, presenting all the levels of the APOS theory. As research methodology we have adopted the Design Experiments. We have justified its use, for adjustments could be done during the work development. Analyzing the activities which were done we have concluded that the students have achieved a satisfactory learning level over the object of study.

Key words: 2nd degree polynomial function, APOS theory, Algorithm.

The mathematics in technical problems can be discovered by computer-aided experiments. Examples are presented from four different courses in the areas of statics, elasticity, finite elements and partial differential equations. It will be reported on the implementation within the curriculum at Beuth University of Applied Sciences Berlin, the classroom experiments and the teacher’s role.

Current trends in research on the impact of technologies in mathematics education emphasize their increased role in supporting students' conceptual understanding in comparison with numerous previous studies about technology contribution in procedural understanding. This talk exemplifies the role of Dynamic Geometry Systems utilizing students' conceptual understanding of dot product of vectors in the transition between upper high school and university education. Students' conceptual understanding is identified as constituting a structured network of: concept definitions and concept images (Tall & Vinner, 1981) of dot product of vectors developed by students; three modes of description and thinking (Hillel, 2000; Sierpinska, 2000) of dot product of vectors: arithmetic, geometric and axiomatic-structural; and concept's applications in problem solving situations. Authentic video recordings and students' written works serve as two collected data sets for qualitative analysis of students' interactions in the designed Dynamic Geometry Environment, within the framework of instrumental genesis (Drijvers et al., 2010). The study is part of a larger design-based research (The Design-Based Research Collective, 2003) undergoing seven phases in a cyclic manner, ending with evaluation and dissemination of created teaching and learning materials as visual dynamic applets and worksheets.

What students describe when they explore computerbased multiple representations of functions does not necessarily reflect how, or whether at all, they understand. In fact, the case of one student explaining his observations while exploring properties of a multiple representation environment suggests that such observations - even if correctly stated - could be based on a superficial perception of the properties only. A qualitative content analysis of further interviews results in a three level model of reasoning with multiple representation learning environment. The theoretical base of this analysis is formed by cognitive theories that describe the learning process in mathematics as a process of abstracting from superficial aspects of representations to structural coherencies between them.