A quantitative approach to evolution of music and philosophy

The development of new statistical and computational methods is increasingly making it possible to bridge the gap between hard sciences and humanities. In this study, we propose an approach based on a quantitative evaluation of attributes of objects in fields of humanities, from which concepts such as dialectics and opposition are formally defined mathematically. As case studies, we analyzed the temporal evolution of classical music and philosophy by obtaining data for 8 features characterizing the corresponding fields for 7 well-known composers and philosophers, which were treated with multivariate statistics and pattern recognition methods. A bootstrap method was applied to avoid statistical bias caused by the small sample data set, with which hundreds of artificial composers and philosophers were generated, influenced by the 7 names originally chosen. Upon defining indices for opposition, skewness and counter-dialectics, we confirmed the intuitive analysis of historians in that classical music evolved according to a master apprentice tradition, while in philosophy changes were driven by opposition. Though these case studies were meant only to show the possibility of treating phenomena in humanities quantitatively, including a quantitative measure of concepts such as dialectics and opposition, the results are encouraging for further application of the approach presented here to many other areas, since it is entirely generic.

Competitive artistic gymnastics and coaches' philosophy

The current study is a piece from the original project entitled "Diagnosis of the Artistic Gymnastics Development Program in Brazil". Among others issues, the coaching philosophy of coaches who are developing gymnasts that may be representative of the national team was a matter of analysis. We interviewed 46 coaches from 29 sports institutions in Brazil. As data collection we used a semi-structured interview and for data treatment we adopted the content analysis method of Bardin (2010). We have found out that there is an inconsistent philosophical basis, and in many institutions this has been mixed up with the objectives of the training program or just doesn't exist. This fact needs consideration and criticism, but we know that the coaching philosophy is not developed during the coaching education courses.

CLIL materials in Secondary Education: focusing on the language of instruction in the subject area of Mathematics

[EN]Applying a CLIL methodological approach marks a shift in emphasis from language learning based on linguistic form and grammatical progression to a more ‘language acquisition’ one which takes account language functions. In this article we will study the elements of the “language of instruction” of the area of Maths in Secondary Education, by focusing on the analysis of the communicative functions, and the lexical and the cultural items present in the textbook in use. Our aim is to present the CLIL teacher with the linguistic and didactic implications that he or she should take into consideration when implementing the bilingual syllabuses with their students. In order to do that, we will present our conclusions emphasizing the need for coordination in different content areas, linguistic and communicative contents, between the foreign language teacher and the CLIL subject one.

Friedrich Riesz' Beiträge zur Herausbildung des modernen mathematischen Konzepts abstrakter Räume

Der ungarische Mathematiker Friedrich Riesz studierte und forschte in den mathematischen Milieus von Budapest, Göttingen und Paris. Die vorliegende Arbeit möchte zeigen, daß die Beiträge von Riesz zur Herausbildung eines abstrakten Raumbegriffs durch eine Verknüpfung von Entwicklungen aus allen drei mathematischen Kulturen ermöglicht wurden, in denen er sich bewegt hat. Die Arbeit konzentriert sich dabei auf den von Riesz 1906 veröffentlichten Text „Die Genesis des Raumbegriffs". Sowohl für seine Fragestellungen als auch für seinen methodischen Zugang fand Riesz vor allem in Frankreich und Göttingen Anregungen: Henri Poincarés Beiträge zur Raumdiskussion, Maurice Fréchets Ansätze einer abstrakten Punktmengenlehre, David Hilberts Charakterisierung der Stetigkeit des geometrischen Raumes. Diese Impulse aufgreifend suchte Riesz ein Konzept zu schaffen, das die Forderungen von Poincaré, Hilbert und Fréchet gleichermaßen erfüllte. So schlug Riesz einen allgemeinen Begriff des mathematischen Kontinuums vor, dem sich Fréchets Konzept der L-Klasse, Hilberts Mannigfaltigkeitsbegriff und Poincarés erfahrungsgemäße Vorstellung der Stetigkeit des ‚wirklichen' Raumes unterordnen ließen. Für die Durchführung seines Projekts wandte Riesz mengentheoretische und axiomatische Methoden an, die er der Analysis in Frankreich und der Geometrie bei Hilbert entnommen hatte. Riesz' aufnahmebereite Haltung spielte dabei eine zentrale Rolle. Diese Haltung kann wiederum als ein Element der ungarischen mathematischen Kultur gedeutet werden, welche sich damals ihrerseits stark an den Entwicklungen in Frankreich und Deutschland orientierte. Darüber hinaus enthält Riesz’ Arbeit Ansätze einer konstruktiven Mengenlehre, die auf René Baire zurückzuführen sind. Aus diesen unerwarteten Ergebnissen ergibt sich die Aufgabe, den Bezug von Riesz’ und Baires Ideen zur späteren intuitionistischen Mengenlehre von L.E.J. Brouwer und Hermann Weyl weiter zu erforschen.

Intuizione e visualizzazione in matematica con particolare riferimento a Felix Klein

Questo lavoro trae spunto da un rinnovato interesse per l’«intuizione» e il «pensiero visivo» in matematica, e intende offrire un contributo alla discussione contemporanea su tali questioni attraverso lo studio del caso storico di Felix Klein. Dopo una breve ricognizione di alcuni dei saggi più significativi al riguardo, provenienti sia dalla filosofia della matematica, sia dalla pedagogia, dalle neuroscienze e dalle scienze cognitive, l’attenzione si concentra sulla concezione epistemologia di Klein, con particolare riferimento al suo uso del concetto di ‘intuizione’. Dai suoi lavori e dalla sua riflessione critica si ricavano non solo considerazioni illuminanti sulla fecondità di un approccio «visivo», ma argomenti convincenti a sostegno del ruolo cruciale dell’intuizione in matematica.

How teachers think about the role of digital technologies in student assessment in mathematics

This study concerns teachers’ use of digital technologies in student assessment, and how the learning that is developed through the use of technology in mathematics can be evaluated. Nowadays math teachers use digital technologies in their teaching, but not in student assessment. The activities carried out with technology are seen as ‘extra-curricular’ (by both teachers and students), thus students do not learn what they can do in mathematics with digital technologies. I was interested in knowing the reasons teachers do not use digital technology to assess students’ competencies, and what they would need to be able to design innovative and appropriate tasks to assess students’ learning through digital technology. This dissertation is built on two main components: teachers and task design. I analyze teachers’ practices involving digital technologies with Ruthven’s Structuring Features of Classroom Practice, and what relation these practices have to the types of assessment they use. I study the kinds of assessment tasks teachers design with a DGE (Dynamic Geometry Environment), using Laborde’s categorization of DGE tasks. I consider the competencies teachers aim to assess with these tasks, and how their goals relate to the learning outcomes of the curriculum. This study also develops new directions in finding how to design suitable tasks for student mathematical assessment in a DGE, and it is driven by the desire to know what kinds of questions teachers might be more interested in using. I investigate the kinds of technology-based assessment tasks teachers value, and the type of feedback they give to students. Finally, I point out that the curriculum should include a range of mathematical and technological competencies that involve the use of digital technologies in mathematics, and I evaluate the possibility to take advantage of technology feedback to allow students to continue learning while they are taking a test.