Using Records of Practice to Bridge from Teachers’ Mathematical Problem Solving to Classroom Practice

It is often the case that high quality mathematics education professional development involves enhancing both teachers’ content knowledge and their pedagogical skills, specifically for teaching mathematics. However, when teachers are immersed in their own learning of mathematics, they are often unaware of the facilitators’ instructional decisions and moves that influence their own learning outcomes, as well as how these might apply to their future teaching. Thus, while the teachers can appreciate their new understanding of content, they may not have added significantly to their understanding of the instructor’s pedagogical moves that facilitated their growth. As a result, teachers may leave even high quality professional development without assurance that they will be able to adjust their teaching in ways that support their own students’ meaningful learning of mathematics. In this work we describe one way in which professional development can both enhance teachers’ subject matter knowledge and help to transform these new understandings into pedagogical content knowledge; the mathematics content sessions provide the platform for reflection on pedagogy. To facilitate this reflection, a “record of practice” is created by facilitators, and thereafter utilized for participants and facilitators to identify and analyze critical moments in the mathematics content session. This paper offers two specific examples of records of practice and how they were used, as well as teachers’ reactions and insights. It also discusses various formats of records of practice, the logistics of developing them, and ends with the potential benefits of using records of practice in professional development for teachers

New types of bialgebras arising from the Hopf equation

New types of bialgebras arising from the Hopf equation (pentagonal equation) are introduced and studied. They will play from the Hopf equation the same role as the co-quasitriangular do from the quantum Yang Baxter equation.Comment: Latex2e, Comm Algebra, in pres

Toward an ecological aesthetics: music as emergence

In this article we intend to suggest some ecological based principles to support the possibility of develop an ecological aesthetics. We consider that an ecological aesthetics is founded in concepts as “direct perception”, “acquisition of affordances and invariants”, “embodied embedded perception” and so on. Here we will purpose that can be possible explain especially soundscape music perception in terms of direct perception, working with perception of first hand (in a Gibsonian sense). We will present notions as embedded sound, detection of sonic affordances and invariants, and at the end we purpose an experience with perception/action paradigm to make soundscape music as emergence of a self-organized system

The Hopf modules category and the Hopf equation

We study the Hopf equation which is equivalent to the pentagonal equation, from operator algebras. A FRT type theorem is given and new types of quantum groups are constructed. The key role is played now by the classical Hopf modules category. As an application, a five dimensional noncommutative noncocommutative bialgebra is given.Comment: 30 pages, Letax2e, Comm. Algebra in pres