Automorphisms of free products and their application to multivariable dynamics

We examine the completely isometric automorphisms of a free product of noncommutative disc algebras. It will be established that such an automorphism is given simply by a completely isometric automorphism of each component of the free product and a permutation of the components. This mirrors a similar fact in topology concerning biholomorphic automorphisms of product spaces with nice boundaries due to Rudin, Ligocka and Tsyganov. This paper is also a study of multivariable dynamical systems by their semicrossed product algebras. A new form of dynamical system conjugacy is introduced and is shown to completely characterize the semicrossed product algebra. This is proven by using the rigidity of free product automorphisms established in the first part of the paper. Lastly, a representation theory is developed to determine when the semicrossed product algebra and the tensor algebra of a dynamical system are completely isometrically isomorphic.Comment: 21 pages, final version, accepted to Indiana U. J. of Mat

An Analysis of Cooperating Teacher Feedback: A Qualitative Inquiry

Cooperating teachers are a key component to the success of student teaching internships, serving an integral part in “raising” a teacher. To effectively facilitate the student teaching internship, teacher preparation programs must identify cooperating teachers who align philosophically with the pedagogical training delivered by university programs (Korthagen & Kessels, 1999; Tom, 1997), specifically, cooperating teachers who can reinforce the theoretical framework underpinning the professional coursework pre-service teachers experience in university teacher preparation programs. This qualitative study sought to better understand the feedback provided to future school-based agricultural education (SBAE) teachers during their student teaching experience. Through initial and secondary coding, the research team identified themes among the feedback provided to student teachers by their cooperating teachers. The study revealed cooperating teacher feedback reflects the pedagogical training provided via the teacher preparation program, specifically, around effective teaching behaviors. Understanding the cooperating teacher feedback provides insight for teacher preparation programs