Applications and modelling in mathematics teaching

The aim of this paper is a comprehensive presentation of some important basic and general aspects of the topic applications and modelling, with emphasis on the secondary school level. Owing to the review character of this paper, some overlap with the survey paper Blum and Niss (1989) for ICME-6 in Budapest is inevitable. The paper will consist of three parts. In part 1, I shall try to clarify some basic concepts and remind the reader of a few application and modelling examples suitable for teaching. In part 2, I shall formulate some general aims for mathematics instruction and, on that basis, summarise the most important arguments for and against applications and modelling in mathematics teaching. Finally, in part 3, I shall discuss some relevant instructional aspects resulting from the considerations in part 2.

Knowledge Building in Landscape Architecture

The objective of this study was to develop an internet-based seminar framework applicable for landscape architecture education. This process was accompanied by various aims. The basic expectation was to keep the main characteristics of landscape architecture education also in the online format. On top of that, four further objectives were anticipated: (1) training of competences for virtual team work, (2) fostering intercultural competence, (3) creation of equal opportunities for education through internet-based open access and (4) synergy effects and learning processes across institutional boundaries. This work started with the hypothesis that these four expected advantages would compensate for additional organisational efforts caused by the online delivery of the seminars and thus lead to a sustainable integration of this new learning mode into landscape architecture curricula. This rationale was followed by a presentation of four areas of knowledge to which the seminar development was directly related (1) landscape architecture as a subject and its pedagogy, (2) general learning theories, (3) developments in the ICT sector and (4) wider societal driving forces such as global citizenship and the increase of open educational resources. The research design took the shape of a pedagogical action research cycle. This approach was constructive: The author herself is teaching international landscape architecture students so that the model could directly be applied in practice. Seven online seminars were implemented in the period from 2008 to 2013 and this experience represents the core of this study. The seminars were conducted with varying themes while its pedagogy, organisation and the technological tools remained widely identical. The research design is further based on three levels of observation: (1) the seminar design on the basis of theory and methods from the learning sciences, in particular educational constructivism, (2) the seminar evaluation and (3) the evaluation of the seminars’ long term impact. The seminar model itself basically consists of four elements: (1) the taxonomy of learning objectives, (2) ICT tools and their application and pedagogy, (3) process models and (4) the case study framework. The seminar framework was followed by the presentation of the evaluation findings. The major findings of this study can be summed up as follows: Implementing online seminars across educational and national boundaries was possible both in term of organisation and technology. In particular, a high level of cultural diversity among the seminar participants has definitively been achieved. However, there were also obvious obstacles. These were primarily competing study commitments and incompatible schedules among the students attending from different academic programmes, partly even in different time zones. Both factors had negative impact on the individual and working group performances. With respect to the technical framework it can be concluded that the majority of the participants were able to use the tools either directly without any problem or after overcoming some smaller problems. Also the seminar wiki was intensively used for completing the seminar assignments. However, too less truly collaborative text production was observed which could be improved by changing the requirements for the collaborative task. Two different process models have been applied for guiding the collaboration of the small groups and both were in general successful. However, it needs to be said that even if the students were able to follow the collaborative task and to co-construct and compare case studies, most of them were not able to synthesize the knowledge they had compiled. This means that the area of consideration often remained on the level of the case and further reflections, generalisations and critique were largely missing. This shows that the seminar model needs to find better ways for triggering knowledge building and critical reflection. It was also suggested to have a more differentiated group building strategy in future seminars. A comparison of pre- and post seminar concept maps showed that an increase of factual and conceptual knowledge on the individual level was widely recognizable. Also the evaluation of the case studies (the major seminar output) revealed that the students have undergone developments of both the factual and the conceptual knowledge domain. Also their self-assessment with respect to individual learning development showed that the highest consensus was achieved in the field of subject-specific knowledge. The participants were much more doubtful with regard to the progress of generic competences such as analysis, communication and organisation. However, 50% of the participants confirmed that they perceived individual development on all competence areas the survey had asked for. Have the additional four targets been met? Concerning the competences for working in a virtual team it can be concluded that the vast majority was able to use the internet-based tools and to work with them in a target-oriented way. However, there were obvious differences regarding the intensity and activity of participation, both because of external and personal factors. A very positive aspect is the achievement of a high cultural diversity supporting the participants’ intercultural competence. Learning from group members was obviously a success factor for the working groups. Regarding the possibilities for better accessibility of educational opportunities it became clear that a significant number of participants were not able to go abroad during their studies because of financial or personal reasons. They confirmed that the online seminar was to some extent a compensation for not having been abroad for studying. Inter-institutional learning and synergy was achieved in so far that many teachers from different countries contributed with individual lectures. However, those teachers hardly ever followed more than one session. Therefore, the learning effect remained largely within the seminar learning group. Looking back at the research design it can be said that the pedagogical action research cycle was an appropriate and valuable approach allowing for strong interaction between theory and practice. However, some more external evaluation from peers in particular regarding the participants’ products would have been valuable.

Methods of generating carrier-envelope-phase stabilized, intense, few-cycle laser pulses

Ultrafast laser pulses have become an integral part of the toolbox of countless laboratories doing physics, chemistry, and biological research. The work presented here is motivated by a section in the ever-growing, interdisciplinary research towards understanding the fundamental workings of light-matter interactions. Specifically, attosecond pulses can be useful tools to obtain the desired insight. However access to, and the utility of, such pulses is dependent on the generation of intense, few-cycle, carrier-envelope-phase stabilized laser pulses. The presented work can be thought of as a sort of roadmap towards the latter. From the oscillator which provides the broadband seed to amplification methods, the integral pieces necessary for the generation of attosecond pulses are discussed. A range of topics from the fundamentals to design challenges is presented, outfitting the way towards the practical implementation of an intense few-cycle carrier-envelope-phase stabilized laser source.