3:00 – 5:30 Thursday June 26, 2014
Jennifer Halfpap, University of Montana
EXPLORING, CONJECTURING, AND PROVING: USING PYTHON IN TRANSITION AND ANALYSIS COURSES.
Transition to proof courses and analysis courses are notoriously hard for students. Part of the reason is that students see a proof as an end in itself rather than as the last step in a process that begins with an interesting question, proceeds through exploration and conjecture, and only then produces a proof as a way of communicating the explanation to the community. Students thus often approach a proofs course as yet one more template-matching course in which they learn to classify problems as “induction problems”, “divisibility problems”, or “delta-epsilon problems.”
It is, of course, hard to teach students to explore, conjecture, and prove (rather than just to prove) because it requires us to pose interesting open-ended problems that are nonetheless accessible to students. In this workshop, we discuss these sorts of problems and illustrate how teaching students basic Python programming may give them some of the tools they need to explore and conjecture before trying to prove.
Jennifer Quinn, University of Washington Tacoma
COMBINATORIALLY THINKING: CONNECTING DIGRAPHS AND DETERMINANTS
Most mathematicians appreciate clever combinatorial proofs. But faced with an identity—in particular a determinantal identity— can you create one? This workshop will provide you with some useful combinatorial interpretations, lots of examples, and the challenge of finding your own combinatorial proofs. We will work to connect digraphs and determinants using two approaches:
Given a “pretty” matrix, can we design a (possibly weighted) digraph that clearly visualizes its determinant?
Given a “nice” directed graph, can we find an associated matrix and its determinant?
Previous knowledge of determinants is an advantage but not a necessity. This will be a hands-on session, so bring your colored pencils, your creativity, and be prepared to explore the mathematical connections.