The purpose of the senior capstone is to engage each senior in the exploration and communication of mathematical ideas beyond material covered in courses taken, or to be taken, by the student.

Basic Components

  1. The student independently studies a topic of interest.
  2. The student writes a paper on the topic.
  3. the student takes a standardized test, the ETS Major Field Exam in mathematics.

Requirements and Important Dates

1. Students must choose a topic.

A student may approach faculty members for suggestions for topics, or browse for one independently. The shelves of the Finkbeiner Reading Room and the stacks in the library are obvious places to look for ideas. The student's topic should be of suitable difficulty - not too easy and not too hard. If in doubt, the student should consult with faculty members about this. Once the student has a topic and a few print resources (books or articles) in mind, they should approach one or more faculty members to consult on the suitability of the proposed topic and to discuss ideas for the proposal.

Students should have topic ideas and some resources in hand no later than the first week of the Fall semester. (Classes begin Monday, August 31.) We encourage students to begin thinking about possible topics in the spring of their junior year.

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Writing a good proposal requires a reasonably good understanding of the major ideas and technical material involved in the topic.  The student needs to understand the general motivation behind the topic, and needs to identify the end-goal(s) for the senior capstone.  The student needs to make sure they have the necessary background and appropriate resources for completing the topic.  The student needs to be fairly sure the project is doable in the given timeline.  All of this requires a substantial amount of reading and background work prior to writing the proposal. 

The student should not attempt to write the proposal without studying their major resources enough to get a good sense of the topic, its difficulty, and feasibility.  Therefore the preliminary work needs to begin several weeks in advance of this first deadline.  In addition to extensive reading, the student should be consulting with one or more faculty members who will be able to give guidance on end goals and feasibility.  It will ordinarily take several discussions with faculty members to really narrow down the goals of the project.  Even though the proposal itself is fairly short, the amount of work that goes into it is much greater.  Do not underestimate the importance of the preparation step or the amount of work that goes into it.  This is a big part of the overall senior capstone project and a big deadline.  As with all other steps in the process, the student may fail the senior capstone by turning in “unacceptable work” at the proposal stage.

The proposal must include:

  • a description of the topic: this is more than the mere title of the topic. The description should include motivation and describe the basic ideas underlying the topic.  It should also state specific end-goals and the nature of the mathematical material that is needed and will be covered. The description should also explain how the topic fits the students’ background and interests in mathematics. The student should also be careful to choose a topic that has substantial content and that is also doable in the timeline given for the senior capstone.  
  • a bibliography with annotations saying generally what is in each source and how the student expects the source to contribute to the eventual paper, (these annotations need not be extensive; keep them general and short, but writing them will require the student to have done some reading in the source.)
  • a list of courses the student has had that will support his or her study of the topic, and
  • A summary of the discussions the student has had with faculty members concerning the topic.

Deadline for the Proposal:  Monday, September 14, 2020
Departmental Response by:  Wednesday, September 23, 2020

Upon acceptance of the student's proposal, the department will appoint a first and second reader for the students’ senior capstone. The first reader will serve as primary advisor for the project, but the student may work with either reader at various times during the process.  

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By the first full week in October, student should have a detailed outline of the paper and a reasonable understanding of the ideas (including proofs of major theorems) that are going to be included in the paper. This outline should be shared with the student's adviser, and a meeting scheduled to discuss the outline and the progress of the paper generally. This meeting should take place before October reading days. The advisor should be given a week's notice on a request for a meeting, and should be given the outline at least a day in advance of the meeting.

Timeframe for Meeting:  Wednesday, October 7 - Tuesday, October 13, 2020, or before.

At the advisor's discretion, further meetings, outlines, or drafts may be required.

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The paper should include all major definitions, examples, theorems (along with complete proofs), and any other critical content. Please submit two hard copies to the department chair before the end of business hours on the due date. (The department administrative assistant or any math faculty member may also receive the paper and pass it on to the chair.)

Deadline for Draft:  Friday, October 23, 2020
Departmental Response by:  Monday, November 9, 2020

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General instructions:

  • The paper should not exceed 20 pages in LaTeX 12-point article format (or comparable font/spacing). Exceptions for longer papers may be requested from the Department at any time between the submission of the outline and submission of the initial complete draft in October.
  • The paper should be self-contained. The necessary background, basic definitions, and useful examples must be explicitly set out in the paper so that the reader can understand it without consulting other sources. The paper may, however, assume the general background covered in the core courses in mathematics and (if necessary) the material covered in other math courses the student has taken at Kenyon.
  • The paper should be written so that it is understandable to fellow math majors; thus we strongly encourage seniors to read and comment on each others' papers before turning them in to the department.
  • Diagrams and calculations that are central to the understanding of the paper should be included in the body of the paper, not relegated to an appendix at the end. It may be appropriate to put as appendices ancillary materials that fill gaps in the development, if their absence from the body of the paper does not interrupt the flow of the reading.
  • The student will be expected to work out the details that are not explicitly set out in the references, so as to put together a complete picture of the topic.
  • The student will be expected to coordinate and systematize material from various sources into a coherent whole. this includes standardizing the notation, which often varies from one source to another.
  • The paper will be graded on the correctness, completeness, and clarity of the writing. As part of this, of course, the student is expected to use correct English and mathematical grammar throughout.
  • Papers must be typed. Use of the LaTeX typesetting system is strongly encouraged, but is not required. Acceptable alternatives include Microsoft Word (with Equation Editor and, if appropriate, MathType), OpenOffice (with appropriate mathematical add-ons), and Maple.

Deadline for Paper: Wednesday, December 2, 2020, by 4 p.m.
Departmental Response by: After the results of the ETS test are known

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  • To test students' general mastery of undergraduate mathematics.
  • To give graduates an additional and widely acknowledged level of certification for their mastery of mathematics, and
  • To help assess the program in mathematics at Kenyon.

Test Date: Beginning of Spring Semester; Date TBA when we know more.

Unless they have a class conflict, students must take the test on this date. The results of the test can be expected to be received from the ETS a few weeks after the test dates.

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The department will evaluate each senior capstone (the result being either distinction, pass or failure) by considering both the independent study/paper and the performance on the ETS exam.

Each paper project will be rated on the following criteria, and each will be evaluated as exemplary, acceptable, marginally acceptable, or unacceptable.

    • Meeting Deadlines
    • General Acceptability of the Work Submitted at Each Stage
    • The Paper, which itself will be rated on the following, before getting an overall score:
      • Accessibility
      • Mathematical/Statistical Depth
      • Mathematical/Statistical Correctness
      • Evidence of Understanding
      • Originality and Independence
      • Organization
      • Style (Citation, Grammar, etc.)

The ETS Major Field test will also be evaluated as outstanding (80th percentile or above, nationally), acceptable (50th-79th percentile), marginal (35th-49th percentile) or unacceptable (below 35th percentile).

An evaluation of "unacceptable" on any component of the senior capstone may be considered grounds for failure, although a particularly strong paper may be used to offset a performance below the 35th percentile on the ETS exam at the discretion of the department. Unacceptable work at any stage of the paper project may be considered immediate grounds for failure.

To achieve distinction on the senior capstone, both the paper and the test must be examples of high quality work, typically meaning that each broad component of evaluation was considered “exemplary.”

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College policy provides each student who fails a senior capstone the first time with a second chance. If the senior capstone is evaluated as a failure, the form of the second chance will depend on which component(s) was/were rated unacceptable.

If the paper project was evaluated as unacceptable (either at a preliminary stage or in the final project), the student will be required to write a paper on a topic chosen by the department. The final version of this second paper will be submitted to the department during the next to the last week of classes in the spring semester. Other (preliminary) deadlines will be imposed by the department in consultation with the student.

If the score on the ETS exam was unacceptable (below 35th percentile) and the paper was not judged strong enough to offset this low performance, the student will retake the ETS exam by the end of the 6th week of the spring semester. If on the second attempt at the ETS exam, the student scores in the 35th percentile or above, then the exam component of the senior capstone will be considered passing. If the student does not achieve the 35th percentile on the second attempt at the ETS exam, it is still possible to pass the senior capstone overall, by writing a second-chance paper considered strong enough to offset the low ETS score, in the judgement of the department.

Even if the fall paper was at a passing level, the department urges any student who had an unacceptable first attempt at the ETS exam to immediately begin work on a second-chance paper, obtaining a topic from the department and submitting a formal proposal. If the second attempt at the ETS is successful, then work on this second-chance paper may be stopped (unless it was required because of a failing paper in the fall semester). But it would be nearly impossible to write a second-chance paper strong enough to offset a low ETS score starting only after a second attempt at the ETS exam.

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