Abstract. Whittaker modules have been well studied in the setting of complex semisimple Lie algebras. Their definition can easily be generalized to certain other Lie algebras with triangular decomposition, including the Virasoro algebra. We define Whittaker modules for the Virasoro algebra and obtain analogues to several results from the classical setting, including a classification of simple Whittaker modules by central characters and composition series for general Whittaker modules.

Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed, and asked to describe how they used their textbook. The results indicate that students tend to use examples, instead of the expository text, to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself, as well as students' beliefs about reading and the nature of mathematics. Although many instructors may not clearly convey how they want their students to use the textbook, students do report using it more productively when they believe they are asked to do so. This suggests that instructors should carefully choose text materials to promote mathematical reasoning, and actively encourage their students to read the text in a way that supports the development of this reasoning.

Textbooks have the potential to be powerful tools to help students develop an understanding of mathematics. However, many students are unable to use their textbooks effectively as learning tools. This paper presents a framework that can be used to analyze factors that impact the ways students read textbooks. It adapts ideas from reader-oriented theory and applies them to the domain of mathematics textbooks. In reader-oriented theory, the reader is viewed as actively constructing meaning from a text through the reading process; this endeavor is shaped and constrained by the intentions of the author, the beliefs of the reader, and the qualities the text requires the reader to possess. This paper also discusses how reading mathematics textbooks is further constrained by the authority and closed structure of these textbooks. After describing the framework, the paper discusses recommendations for future avenues of research and pedagogy, highlighting the importance of teachers' roles in mediating their students' use of textbooks.

Researchers have increasingly focused on how gestures in mathematics aid in thinking and communication. This paper builds on Arzarello's (2006) idea of a semiotic bundle and several frameworks for describing individual gestures and applies these ideas to a case study of an instructor's gestures in an undergraduate abstract algebra class. We describe the role that the semiotic bundle plays in shaping the potential meanings of gestures; the ways gestural sets create complex relationships between gestures; and the role played by polysemy and abstraction. These results highlight the complex ways in which mathematical meaningsboth specific and general-are expressed in gesture, and to highlight the integrated nature of elements of the semiotic bundle.

Following analogous constructions for Lie algebras, we define Whittaker modules and Whittaker categories for finite-dimensional simple Lie superalgebras. Results include a decomposition of Whittaker categories for a Lie superalgebra according to the action of an appropriate sub-superalgebra; and, for basic classical Lie superalgebras of type I, a description of the strongly typical simple Whittaker modules.