Game Theory, defined in the broadest sense, can be used to model many real world scenarios of decision making in situations involving conflict and cooperation. Further, mastering the basic concepts and tools of game theory require only an understanding of basic algebra, probability, and formal reasoning. These two features of Game Theory make it an ideal path to developing habits of quantitative literacy among our students. This audience participation mini-course develops some of the material used by the presenters in their general education courses on Game Theory and encourages participants to develop their own, similar, courses.

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Catalan numbers are both fascinating and ubiquitous. They pop up in quite unexpected places, such as triangulations of convex polygons, correctly parenthesized algebraic expressions, rooted trees, binary trees, full binary trees, trivalent binary trees, lattice-walking, Bertrand’s ballot problem, abstract algebra, linear algebra, chess, and the World Series, to name a few. Beginning with a brief history of Catalan numbers, this minicourse presents numerous examples from different areas. We will develop a number of combinatorial formulas for computing them, investigate their parity and their primality-link to Mersenne numbers, and present the various ways they can be extracted from Pascal’s triangle and several Pascal-like triangles. As a bonus, we will investigate tribinomial coefficients and extract Catalan numbers from them.

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Many colleges and universities have a gateway course to help mathematics students make the transition to more theoretical courses, with a goal of helping students learn how to understand and construct proofs. The organizer of this course, guided by 6 years of videotaping his students doing their homework for a proof-based course, will lead participants in an exploration of effective approaches to teaching “proof.” We will discuss appropriate types of problems, the wording of problems, effective hints and prompts, and a variety of pedagogical approaches. Suggestions and questions from participants will be encouraged. 

Mathematical circles are modeled after those in Eastern Europe and are as successful here as they have been there. Circles bring mathematicians into direct contact with middle or high school students who work together on problems that require deep thinking rather than rote solutions. Running a math circle is a rewarding activity but also presents many challenges. In this mini-course participants will be involved in two abbreviated math circles led by the organizers trying their hand at developing math circles of their own. In addition, there will be discussions of the history, goals, and achievements of existing math circles, as well as suggestions on how to start and maintain new circles.

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The execution of the simplest line drawings in perspective can pose math problems that challenge the brightest of students. Nevertheless, the solutions are pleasingly easy to use and remember. The other side of the coin—viewing a work in perspective from the correct viewpoint—poses similarly interesting problems. When applied to viewing real artwork (or posters), these techniques lead to an astonishing experience of depth and realism that leaves a lasting impression on viewers. This Minicourse conveys these techniques through hands-on activities, which the facilitators have taught to over 140 instructors in faculty development workshops. No artistic experience is required.

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It is election year! But will we elect whom we really want? What raises a concern is that, by now, most of us know that voting rules can cause unexpected outcomes and delicious paradoxes. The standard plurality ranking, for instance, can be Alice > Barb > Connie even though the “vote for two” outcome is precisely the opposite. The mathematical issues - which constitute the theme of this course - are to identify everything that can possibly happen and explain why they occur, to describe how to construct any number of illustrating examples, to identify the “best” voting rule, and to learn how to convert some of these recent research results into rich course offerings for our undergraduates – particularly for Fall term of this coming year.

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