Capstone and One-Semester Research Projects for a Variety of Students

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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

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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).

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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.

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The absent presenter

• Bill Fenton
• Dept. of Mathematics
• Bellarmine University
• Louisville, KY 40205
• wfenton_at_bellarmine.edu

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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.

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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.

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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.

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URL for assignments materials

• http//www.bellarmine.edu/faculty/fenton/450/MATH4
  50/asp

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Choosing a topicopening 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?

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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 Benfords 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

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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
  universitys 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

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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

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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.

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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
  experiencenot 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

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Steve Morics

• University of Redlands
• Redlands, CA
• Steven_morics_at_redlands.edu

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Sources - REUs

• Cascades of Period-Doubling Bifurcations and The
  Cascade Theorem

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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

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Sources-Faculty Interests

• Fair Division Ramsey Partitions
• Coding Theory NTRU Cyryptography
• Juggling Site-swapping and possible patterns
• Music Hexachord Theorem

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Sources-Education Track

• Penrose Tilings
• Mathematics of the Incas and Mayas
• Understanding Infinity
• Victorian Womens Mathematics
• Origami Constructions

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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!

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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

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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

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Sr. Barbara Reynolds, SDS

• Cardinal-Stritch University
• Milwaukee, WI
• breynolds_at_stritch.edu

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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)

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Preliminaries (10 points)

• Project proposal/revised project proposal
  submitted and approved.
• Rough draft submitted for review.

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Format (professional presentation) (10 points)

• Overall, your project gives a good professional
  impression.

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Writing style (spelling, grammar, etc) (15 points)

• Are you using correct spelling and appropriate
  grammatical structures throughout your project?

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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.

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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.

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Questions from audience?
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Suggestions for sources of topics
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General discussion
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The End

• Thank you
• Mary Shepherd
• Steve Morics
• Barbara Reynolds