Using GeoGebra to study complex functions

The aim of this workshop to present some of the strategies studied to use GeoGebra in the analysis of complex functions. The proposed tasks focus on complex analysis topics target for students of the 1st year of higher education, which can be easily adapted to pre-university students. In the first part of this workshop we will illustrate how to use the two graphical windows of GeoGebra to represent complex functions of complex variable. The second part will present the use of the dynamic color Geogebra in order to obtain Coloring domains that correspond to the graphic representation of complex functions. Finally, we will use the threedimensional graphics window in GeoGebra to study the component functions of a complex function. During the workshop will be provided scripts orientation of the different tasks proposed to be held on computers with Geogebra version 5.0 or high.

European student mobility and intercultural learning at a portuguese university

The purpose of this study is to understand, impact and evaluate the development of intercultural communicative competencies among European credit-seeking exchange students and other sojourners through purposeful intercultural pedagogy. This pedagogy encompasses intentional intercultural- educational approaches which aim to support and enhance sojourners’ intercultural learning throughout the study abroad cycle (pre-departure, in-country and reentry phases). To test and validate these pedagogies a 20-hour intervention was designed and implemented among two cohorts of 31 sojourners during the in-country phase of their sojourn in Portugal. The process to develop and validate the intercultural intervention was driven by a mixed-methods methodology which combined quantitative and qualitative data to triangulate, complement and expand research results from a pragmatic stance. The mixed methods research design adopted is multi-phased and encompasses a multi-case study and an evaluative component. The multi-case component is embodied by sojourner cohorts: (1) the primary case study involves 19 incoming students at the University of Aveiro (Portugal) as participants in the European exchange program Campus Europae; (2) the second case study comprises three incoming Erasmus students and nine highly skilled immigrants at the same university. All 31 sojourners attended two intermediate Portuguese as Foreign Language classrooms where the intervention was employed. Data collection was extensive and involved collecting, analyzing and mixing quantitative and qualitative strands across four research phases. These phases refer to the: (1) development, (2) implementation and (3) evaluation of the intervention, as well as to (4) a stakeholder analysis of the external value of the intervention and of the Campus Europae program. Data collection instruments included pre and posttest questionnaires and semi-structured interviews. Results confirm the intercultural effectiveness of the intervention and the positive impact upon research participants’ intercultural gains. This impact was, however, greater in case study 2. Among explanatory variables, three stand out: (1) participant meaning-making abilities, (2) host language proficiency and related variables, and (3) type of sojourn or exchange programs. Implications for further research highlight the need to systematize purposeful intercultural pedagogy in sojourner populations in general, and in European credit student mobility in particular. In the latter case, these pedagogies should be part of the design and delivery of credit-bearing exchange programs in pre- departure, in-country and re-entry phases. Implications for practice point to the urge to improve intercultural practices in: macro (higher education institutions), mezzo (exchange programs) and micro (sojourner language classrooms) contexts where this research took place, and wider social scenarios they represent.

Cassini states for black hole binaries

Cassini states correspond to the equilibria of the spin axis of a body when its orbit is perturbed. They were initially described for planetary satellites, but the spin axes of black hole binaries also present this kind of equilibria. In previous works, Cassini states were reported as spin-orbit resonances, but actually the spin of black hole binaries is in circulation and there is no resonant motion. Here we provide a general description of the spin dynamics of black hole binary systems based on a Hamiltonian formalism. In absence of dissipation, the problem is integrable and it is easy to identify all possible trajectories for the spin for a given value of the total angular momentum. As the system collapses due to radiation reaction, the Cassini states are shifted to different positions, which modifies the dynamics around them. This is why the final spin distribution may differ from the initial one. Our method provides a simple way of predicting the distribution of the spin of black hole binaries at the end of the inspiral phase.