This research studies the interaction among the following contexts: natural language, geometrical language and musical language and it can provide new instruments to accord didactical situations and for a deeper understanding of communication processes. It springs from the consideration that the geometrical transformations are usually used in the compositional processes and the “role of the music to learn geometrical transformations” is actually a new study. In the field of the theory of situations by G. Brousseau (1986) we can assume to be in front of a learning teaching-situation including non-teaching situation as the teacher of musical instruments, while transmitting the knowledge of musical language (theoretical-practical) didn’t have the intention to transmit the geometrical transformation.

One charter school’s path to tracking and pushing for privileged passage is examined. The school as it increased in size began to track students first by grade level and then by ability. Realizing that moving mathematics out of the main school program compromised the teaching ideals of the school and potentially student learning. The school has embarked on a program to create a place-based, integrated curriculum developed around mathematics so that mathematics can be reintroduced to multi-age classrooms. Examining the data in terms of trust, size and the behaviors of administrators, teachers and parents at this school in this process are highlighted.

Cryptography fascinates people of all generations and is increasingly presented as an example for the relevance and application of the mathematical sciences. Indeed, many principles of modern cryptography can be described at a secondary school level. In this context, the mathematical background is often only sparingly shown. In the worst case, giving mathematics this character of a tool reduces the application of mathematical insights to the message ”cryptography contains math”. This paper examines the question as to what else cryptography can offer to mathematics education. Using the RSA cryptosystem and related content, specific mathematical competencies are highlighted that complement standard teaching, can be taught with cryptography as an example, and extend and deepen key mathematical concepts.

In this paper we examine an Australian project in which we seek to develop leadership capacity in staff and students throughout the country, such that they may contribute to and lead others to contribute to the development and sharing of learning support resources for mathematics and statistics across disciplines and universities. One of the tangible outputs is a set of video based learning support resources that can be embedded in subjects across disciplines and shared across institutions. However the guiding aim is to develop leadership capacity, in its simplest form leading others to lead others to contribute to the project. Leadership may also be developed and exercised across different aspects of the project whether it be mapping needs, drawing together disciplines groups, finding ways to recognise and reward those engaged in the process, developing resources and the associated skills, ensuring copyright adherence, creating learning designs for optimal use of resources, evaluating the impact on student outcomes, peer review and the dissemination of findings.

Virtual manipulatives are employed by both preservice and inservice teachers to enhance the instructional effectiveness of physical manipulatives and related tools by addressing limitations of access, cost, and adaptability. While research into the use of emerging technologies continues, there are several variables to consider when measuring the effects of virtual manipulative use. Research design, sampling characteristics, and the type of manipulative used may influence achievement. Variables that may influence the effectiveness of virtual manipulatives include: previous experience with computers, grade level, mathematical topic, treatment length, student attitudes toward mathematics, and computer-to-student ratio.

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Die vorliegende Arbeit beschäftigt sich mit der Bures-Geometrie auf Zustandsräumen über von-Neumann-Algebren. Diese basiert auf jenem Abstandsbegriff für normale Zustände, der von Bures im Jahre 1969 eingeführt wurde. Eng damit verbunden ist der Begriff der algebraischen Übergangswahrscheinlichkeit, der von Uhlmann 1976 vorgeschlagen wurde. An einem Beispiel wird gezeigt, dass man den Bures-Abstand unter Umständen nicht implementieren kann, wenn man einen der implementierenden Vektoren vorgeben will. Im weiteren wird der vom Bures-Abstand induzierte Paralleltransport von Vektoren entlang Loops von normalen Zuständen untersucht. Um die Holonomiegruppe im unendlichdimensionalen Fall zu untersuchen, werden Sätze über Produkte positiver Operatoren hergeleitet. Diese Sätze, die durchaus auch von eigenständigem Interesse sein könnten, werden mit Ergebnissen aus der Literatur verglichen. Schließlich wird der Bures-Abstand unter infinitesimalem Blickwinkel betrachtet. Die so entstehenden Bures-geodätischen Bögen werden untersucht. Speziell wird gefragt, ob gewisse Strata stets geodätisch konvex sind, also als Beispiel für Umgebungen dienen können. Um diese Frage am Ende negativ zu beantworten, werden mehrere Sätze über Sakaische Radon-Nikodym-Operatoren hergeleitet, die auch ohne Bezug zur Bures-Geometrie interessant sein könnten. Das entscheidende Gegenbeispiel nutzt Gohbergs Ergebnis zum Spektrum bestimmter Toeplitzoperatoren aus. Ein Nebeneffekt des beschriebenen Verfahrens ist, dass es auch zur Konstruktion von Operatoren mit hinreichend nichttrivialem Spektrum benutzt werden kann. / The present paper deals with Bures' geometry in the state space over von-Neumann algebras. This geometry is based on the distance introduced by Bures in 1969. Closely related with it is the concept of algebraic transition probability as proposed by Uhlmann in 1976. It is shown by an example that there are cases where one can not implement Bures' distance if one of the implementing vectors is given. In the following, the parallel transport of vectors along loops of normal states, which is induced by Bures' distance, is examined. In order to investigate the holonomy group in the infinite-dimensional case, theorems on products of positive operators are derived. These theorems, which could be of interest on their own, are compared with the literature. Finally, Bures' distance is examined infinitesimally. The thus arising Bures-geodesic arcs are investigated. Especially, it is asked whether certain strata are geodesically convex and can therefore serve as examples of neighbourhoods. In order to finally give a negative answer, several theorems on Sakai's Radon-Nikodym operators, which could also be of interest without a connection to Bures' geometry, are derived. The critical counterexample exploits Gohberg's result on the spectrum of certain Toeplitz operators. A by-product of the described procedure is that it can be used to construct operators which have a sufficiently non-trivial spectrum.