A full-year senior capstone course has evolved at a small, private women's liberal arts college in the Midwest to become the principal tool for assessing the major. Within this two-semester seminar each student has to develop an independent study project, known as the comprehensive project. Preliminary work on the project begins in the first semester and oral and written presentations of the completed project are given in the second semester.
Background and Purpose
Saint Mary's College is a catholic, liberal arts college for women with about 1500 students. We typically have ten to a dozen mathematics majors graduating each year; all of the mathematics and computer science courses are taught within the department, which consists of eight full-time faculty and two or three adjuncts teaching specialized courses. Since the 1930s the college has required that each student pass a comprehensive examination in her major. The requirement was rewritten in the early 1970s to give each department the freedom to determine the most appropriate form for examining its students. At that time the mathematics department replaced the "test" format with a requirement for an independent study project, which has come to be known as the comprehensive project.
A capstone seminar was developed to provide a framework for student work on the comprehensive project and to foster student independence in learning and skill in discussing mathematics. The spur for developing a formal program for overall assessment of student learning was more bureaucratic. In 1996 the college had to present to North Central, as part of the reaccreditation process, a plan for assessing student learning. The mathematics department had responsibility for developing the plan for assessment in the major and built its plan around the existing senior capstone course and the project.
In developing the assessment program, we had to articulate our goals for the major in terms of student achievement rather than department action.
Goals for a Mathematics Major
As a result of study in mathematics, a graduate of Saint Mary's College with a major in Mathematics will have met the following goals.
The senior seminar and senior comprehensive project, as assessment tools, are intended to provide both development and information on goals 1, 3, and 4. Some of the work in the first semester of the seminar touches on goal 5, and individual projects may get into actual problem-solving of the sort envisioned in goal 2. In practice, the senior seminar, like all of our courses, was designed to foster development toward these goals, as well as to assess their achievement.
In addition to the assessment of student learning described here, the department evaluates its programs through a biennial program in which recent graduates are invited back to campus for a day-long discussion with current students, through contact with alumnae, through contacts with interviewers who come to campus, and by using the information in the college's surveys of recent graduates.
The assessment method described here consists of (1) a full-year course meeting twice a week, called the Senior Seminar, and (2) an independent study project culminating in a formal oral and written presentation and response to faculty questions, known as the Senior Comprehensive Project. Roughly speaking, the first semester seminar begins the experience of extended independent learning and presentation and provides a vehicle for organizing the students' search for a topic and advisor. The second semester seminar is built around students' presentation of preliminary findings while they are working on their individual projects.
During the first semester of the Senior Seminar, all students work on a common topic and from a common text. They are required to learn and present material on the level of a junior-senior mathematics course. The way this usually works is that the seminar director selects a text involving material not already covered by the students in other courses and determines a course schedule based on one-week sections of the material. Each section of material is assigned to a team of two students which is responsible, with advice from the seminar director, for learning the material, presenting the material to the class and assigning and correcting written problems based on the material. One semester generally allows time for two rounds of presentations, with the teams changed between first and second rounds, and for several problem presentation days in which individual students are assigned to present solutions of some of the more difficult problems. In weeks in which they do not present, students are responsible for reading the material, asking questions of the presenters, critiquing the presentations, and working the problems. Each student is required to have an advisor and topic for her senior comprehensive project by mid-October. The writing assignment for the fall seminar often requires an introduction to the topic of the comprehensive project. The grade in the first semester of the seminar is based on demonstrated knowledge of the material, the presentations, participation (including asking questions, completing the critiques, etc.), and the required paper.
During the second semester, each student is working on her own senior comprehensive project. She works with the advisor in learning the material and preparing a paper and a formal public presentation based on her work. In the seminar, she presents two fifty-minute preliminary talks on her topic and is expected to follow other student talks in enough detail to ask questions and to critique the presentations. The seminar talks are expected to contain sufficient information for her classmates to understand her formal presentation. The grade in the second semester of the seminar is based on demonstrated learning of the topic, presentation of the topic, participation, and work with the comprehensive advisor.
The comprehensive project must develop material that is new to the student. This may be an extension of an area previously studied or a completely new area for the student and may be in any area of mathematics, including applications and computer science. The comprehensive advisor and the student work out a project that is of sufficient depth and breadth and that can be completed in the time available. The progress is monitored by both the advisor, who meets regularly with the student, and the seminar director, who observes the preliminary talks. The student provides a formal written presentation of her work, with a summary of the material covered in the seminar talks and a more detailed treatment of the final material. Once the paper is written, it is submitted to a committee of three faculty members the advisor, the seminar director and one other who read the paper and prepare questions related to the material. The formal presentation is made before an audience consisting of this committee, the seniors in the seminar, and any others who may wish to attend. It consists of a forty-minute presentation on the final material in her work and twenty minutes of response to questions from the faculty committee. The formal paper, the oral presentation, and the response to questions constitute the senior comprehensive examination required by the college. The paper for the senior comprehensive project also serves as the final submission for the student's advanced writing portfolio.
Grading of the senior comprehensive projects occurs in two stages. The committee determines whether the student passes or fails; there is usually some reworking of the paper needed, but except in extreme cases this not an impediment to success. After all the presentations have been completed, the department faculty meets to discuss all of the comprehensives and to determine which, if any, are to be awarded honors. The discussion is based on five criteria:
This discussion also serves as an overview of the achievement of the students in the senior class, and the seminar director writes a report to the department based on the discussion.
Since the seminar and comprehensive project have been in place longer than our formal assessment program, most of the information has been gained and used in planning in an informal and implicit fashion. There are, however, a few results that can be stated directly.
The first is that taking courses does not prepare students to explain what they know to people who do not already know it that is, to anyone other than teachers or other students who have seen the material. There are really two different but related issues here. The first is the identification of assumptions and background for a particular piece of work understanding where it fits into a larger picture. The second is the need to move from the "show that I know it" mindset of passing a test to the "giving you an understanding of something I know and you don't" mindset. There is a stage of development here that does not occur automatically and it seems to be very closely related to learning to work independently and to critique the results. We do see growth here during the senior year.
A related fact is that students find it difficult to work with, and especially to combine, different presentations of the same idea. Searching for examples and dealing with varying notation in different sources is a major difficulty for many. Seeing old ideas in a new form and a new setting is often a challenge, especially to a student who is working at the limit of her experience. The failure to recognize ideas from previous courses, and a consequent inability to use them in new settings, is often another form of this problem. The seminar also provides a sometimes discouraging reminder that there are always topics from previous study that have been forgotten or were never really understood.
Discussions with alumnae and interviewers have strongly supported our belief that several aspects of the seminar and comprehensive project are very important for those graduates a majority, in our case who do not go into teaching or into explicitly mathematics-related fields. The important aspects seem to be the independent work on a long-term project and the organization and presentation of results. The practice in independent learning and interpretation has been equally important to those who have gone on to graduate study in mathematics or mathematics-related fields.
Use of Findings
The department's experience with the seminar and the comprehensive project has affected all planning and discussion for the last twenty years. One major result occurred at the level of the college curriculum when the department strongly supported the introduction of the "advanced writing" requirement, to be fulfilled in the student's major. Our experience in asking students to explain mathematics made it clear that we needed to develop this skill, and we had already begun introducing writing requirements within courses when this college-wide framework came under discussion. Currently the second year writing requirement in mathematics focuses on exposition and the third year adds technical writing, but we may find that some other approach works better.
The fall semester of the seminar has changed often, usually in small ways, from experience with the second semester and the comprehensive project. In order to make students more conscious of the decisions made in preparing a presentation, we introduced a student critique form, which must be filled out by each student for each speaker. We are considering having each student fill out such a form for herself, as well, to foster more reflection on the process. When the seminar was put in place, students worked on separate topics in both semesters shorter topics in the fall, longer in the spring. We have found that the process of explanation takes more effort and learning, and that working on a common topic in the fall allows for more mutual support and cooperation. By encouraging more student questions of the presenters it also improves the feedback on the presentations and student awareness of the process of presentation. The use of teams of students arose for the same reason. The use of written exercises puts more pressure on the "audience" to really work at learning the material, and somewhat reduces the tendency to let the presenters get by with "good enough."
Our experience with the seminar and comprehensive has also contributed to an increase in assignments requiring longer explanations in earlier mathematics courses and an increase in requirements for in-class presentations by students in earlier years.
Alumnae who have gone to graduate programs and those who have gone to work report that the experience has been valuable in preparing them to deal with the work they have to do. A part of this is certainly the confidence they develop from having already worked on extended projects and reported on their work. We have also gained information on some things that work and some that do not work in developing the skills.
A major advantage to this program as an assessment tool is that it is a required part of the educational program of the department, not an added testing requirement. The seminar involves graded academic credit and the whole program is approached as a learning tool, so that problems of student participation are minimized. It is a great help that seniors in other majors are also involved in meeting the comprehensive requirement in ways tailored to these majors, even though the mathematics comprehensive is generally recognized as one of the most demanding.
There are two major drawbacks to this program use of student time and use of faculty time. Since the seminar involves two semesters, the number of other courses taken by a student is reduced, though the fall seminar does, in fact, deal with standard content in a fairly standard way. The comprehensive project involves a great deal of time, and a student who has thought in terms of "a two-hour course" for the spring semester may find herself overextended. The program also uses considerably more faculty time than would appear on official records. The seminar director is credited with two semester hours of teaching in each semester, but the faculty who serve as project advisors receive no official teaching load credit. During the first semester, the seminar director spends a great deal of time in one-on-one or one-on-two discussion with the students as they first learn the material and then begin thinking about how to present it; this takes far more time than preparing lectures and exercises on the material. In addition, the seminar director provides written critiques to the students, incorporating the information from the student critiques. In the senior comprehensive phase, each faculty member is advising one or two students on projects which extend over several months and require planning of presentations and writing. Naturally, weaker students require a great deal of direction and advice, but faculty find that they are more likely to plan more ambitious projects with stronger students, so that these also require a lot of time. There is a positive side to this effort in the opportunity for faculty to explore new areas in their fields or in related fields. The seminar director has the unique opportunity of reading all the projects and seeing new areas in many fields of mathematics.
In working with a program such as this, it is essential to have requirements explicit and clear from the beginning. There is always time pressure on students, and for many the long-term nature of the project is a new experience. Without deadlines and dates the weaker students, especially, may get so far behind that they can catch up only with heroic effort and at the cost of work in other courses. Grading criteria need to be as explicit as possible, because there are many opportunities for misunderstandings and incorrect expectations in a mathematics course that is very different from students' previous experience.
The involvement of all faculty in the senior
comprehensive projects means that the program cannot work
without full faculty support. Not only is there a serious time
commitment, but it is necessary that all faculty members
keep track of deadlines, formats for papers, etc., and be willing
to adjust schedules to serve on the committees for other
students. Working out the inevitable differences of
opinion about interpretation of requirements before they
become critical falls mainly to the seminar director, but would
not be possible if other faculty were not interested in the
program as a whole.