Capstone and One-Semester Research Projects for a Variety of Students Panel presentation Mary Shepherd, Moderator Sr. Barbara Reynolds Steve Morics William Fenton January 8, 2008

Goals Create a true research experience for math majors of varying ability and within a small time frame (one semester). Recognize and find appropriate problems, match problems to students and mentor these young researchers from initial problem selection to final product (paper and/or presentation).

Questions What are some resources for good but “small” research questions for either one semester projects or the weaker students? How do we mentor these students successfully in the short time frame of a single semester? How can we move these students from a “research paper” type paper approach to a “research” approach? By that I mean move them from just looking up different resources on a subject and trying to put it together into a paper to actually experiencing a true research experience.

The absent presenter Bill Fenton Dept. of Mathematics Bellarmine University Louisville, KY

Capstone course Readings in Mathematics Asks the students to write three small papers in the first half of their final semester, Follow these with a larger final project that takes up the second half of the semester.

Syllabus description “In this paper you are to explore in depth a topic in mathematics. This should not be something from a previous course, though a previous course may suggest a good topic to explore. You are expected to seek out references and to learn about this topic. But the paper is to be more than a synthesis of what you find in your references. You should go beyond that, to contribute something of yourself and show that you have deeply understood the topic. What are some ways you could demonstrate this deep understanding? It will vary considerably, depending on your topic and on yourself … Finding a suitable topic is the first, and perhaps the hardest, part of the paper.”

Assignment goals To learn something in depth that is not part of regular course work; To show that they can be an independent learner of mathematics; and To demonstrate through written and oral presentations that they have gained a deep understanding of the topic.

URL for assignments & materials ATH450/asp

Choosing a topic—opening questions What was your favorite course offered from the Mathematics Department? Why was it your favorite? What was your favorite non-mathematics course? Why was it your favorite? What connections did this course have to mathematics? What topics in mathematics do you find interesting? What are your career goals? What are some of your interests outside mathematics, and what connections do they have to mathematics? Do you have any ideas about what you might like to do for your final paper? If so, what are they?

Some topics chosen Bayes’ Theorem and subjective probability to settle lawsuits explained the Traveling Salesman Problem in detail and talked about the history of attempts to solve it. (weaker student) Mathematics behind Benford’s Law and how it is used in detecting accounting fraud Student on the baseball team wrote about the physics of baseball amateur juggler wrote about theorems on juggling patterns

More topics Tennis player attempted a game-theoretic analysis of serve-&-volley strategies Baseball player wrote a linear programming program that found a more efficient travel schedule for the baseball teams in our university’s athletic conference Used methods of mathematical geography to analyze the highway system in Indiana and the possible effects of building I-69 Applied the critical path method from O.R. to analyze the operations of a local swim club Critical comparison of retirement plan proposals from his internship with a local actuarial firm

Suggestions: Start early Try to set clear expectations Have regularly scheduled meetings with the student Set deadlines Finish each meeting with a clear plan for the next meeting Require a first draft of the paper in advance of the final deadline

Items requiring caution Do not do the project for the student. Once a topic is chosen, stay focused on the objective. Be realistic in the expectations. Pay attention to the critical path.

Summary Some projects have required more work from me than others, and some have produced better results than others. However, I believe that every student has benefited from the experience—not necessarily in the mathematical knowledge they gained, rather in the confidence that they can work independently in mathematics. This is worth the work. –Bill Fenton

Steve Morics University of Redlands Redlands, CA

Sources - REU’s Cascades of Period-Doubling Bifurcations and The Cascade Theorem

Sources-Student Interests Chemistry: Group Theory and Physical Chemistry Env. Studies: Allocation of Colorado River Water Economics: Game Theory and Transition Costs Music: Fretting a Guitar

Sources-Faculty Interests Fair Division: Ramsey Partitions Coding Theory: NTRU Cyryptography Juggling: Site-swapping and possible patterns Music: Hexachord Theorem

Sources-Education Track Penrose Tilings Mathematics of the Incas and Mayas Understanding Infinity Victorian Women’s Mathematics Origami Constructions

Sources-Summary Most every project generated by a combination of faculty and student interests Very few, if any, started life as a “back of the journal” problem or project suggestion in a textbook Full department commitment pays off!

Redlands Capstone One Coordinator Anywhere from 4 to 20 students Every faculty member serves as advisor on one or two projects Two weeks spent hunting up a problem

Student Expectations Significant written product Significant mathematical component Not covered in a regular class More than an article review 30-minute presentation More for honors

Sr. Barbara Reynolds, SDS Cardinal-Stritch University Milwaukee, WI

Rubric and Feedback for Seminar Project Preliminaries (10 points) Format (10 points) Writing Style (15 points) Computer Science/Mathematics Content (50 points) Synthesis and integration (15 points)

Preliminaries (10 points) Project proposal/revised project proposal submitted and approved. Rough draft submitted for review.

Format (professional presentation) (10 points) Overall, your project gives a good professional impression.

Writing style (spelling, grammar, etc) (15 points) Are you using correct spelling and appropriate grammatical structures throughout your project?

Computer Science/Mathematical Content (50 points) Your project demonstrates knowledge of fundamental concepts from your mathematics or computer science major. Your project demonstrates an ability to apply concepts from your major in new problem-solving settings and/or to extend your knowledge base. Your project demonstrates higher-level reasoning and analysis. The computer science and/or mathematics content presented in your project is correct. Your project showcases knowledge and skills appropriate for a student who is completing an undergraduate major in computer science or mathematics.

Synthesis and integration (15 points) Your project demonstrates synthesis and integration of skills developed through your undergraduate major. Your project demonstrates an ability to do original research in your major. That is, you demonstrate that you can develop and verify new ideas, not merely search the literature for results that others have developed. Your project includes a complete bibliography, and you have sited your sources correctly.

Questions from audience?

Suggestions for sources of topics

General discussion

The End Thank you: Mary Shepherd Steve Morics Barbara Reynolds