The Mathematics of Beauty and the Beauty of Mathematics

Designed for all audiences Whether subjective or objective, ephemeral or eternal, arousing the senses or charming the intellect, the definition of beauty has forever challenged artists and philosophers alike. This engaging and highly visual presentation invites you to ponder the meanings of beauty, examine the mathematics behind the beauty of things and enjoy aspects of mathematics that delight students, teachers, mathematicians, and all lovers of mathematics. This lecture is designed in three parts: Part 1: A general discussion on ―beauty.‖ How artists, philosophers, and writers have tackled the problem of defining and expressing beauty. Part 2: An exploration of mathematics through the beauty of nature and human creations in our western culture. Monica Neagoy will attempt to explore the math hidden behind art such as sculpture, music and theatre, as well as the wonders of nature. Part 3: An exploration of what can be considered as ―beautiful‖ in mathematics. How can we explain reactions such as, ―That’s wonderful!‖, ―That’s marvelous‖ or ―That’s beautiful!‖ when talking about mathematics? Monica Neagoy will explain the passion that professors, students and math lovers have for the subject

TEACHERS' PEDACOGICAL CONTENT KNOWLEDGE OF RECURSION

"Pedagogical Content Knowledge" (PCK) consists of topic-level knowledge of learners, of learning, and of the most useful forms of representation of ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations --in a word, the ways of representing and formulating the subject that makes it comprehensible to others" (Shulman, 1986). Recursion is a process that permeates many aspects of the real world-both natural and man-made. In discrete mathernatics, recursion is a powerful idea, a problem solving strategy that enables us to describe or predict future results as a function of past results. The purpose of this study was to explore the nature of high school teachers' PCK of recursion prior to, and as a result of, their participation in a carefully designed summer institute that focused on the important emerging concept of discrete dynamical systems. The study also explored how teachers plan to use this knowledge in teaching recursion. The framework for studying teachers' PCK was one inspired by Shulman's model ( 1987), but modified in its connectedness among components and its dynamics of change. The in-service program that served this study was the 1991 Summer Institute in Mathematics Modeling with Discrete Mathematics, (SIMM) offered at Georgetown University and partially funded by NSF. Forty high school math teachers from Washington metropolitan area schools, who attended the SIMM were the subjects of this research. The instruments that helped assess the nature and growth of teachers' PCK as a result of the SIMM intervention were: A personal data questionnaire, a pretest, and a post-test; follow-up, one-on-one interviews were conducted with a random sample of nine teachers. The test results and interview transcripts were analyzed in terms of teachers' subject matter and pedagogical knowledge (knowledge of teaching and learning) of recursion: For that purpose, this study developed an original model of six categories of knowledge for each domain. Overall, teachers' PCK of recursion, as exhibited by their performance on the totality of the test items, grew as a result of the in-service intervention. The only category in which teachers' knowledge showed no growth was Student Errors