# Mathematics Capstone Course - 1987

**The Capstone Course and Degree Cognate Proposals in 1987**

One of the main issues when Grand Valley reorganized in 1983 by merging the cluster colleges into one college was the structure of the general education program. The College of Arts and Sciences, William James College, and Kirkhof College each had distinctive approaches to general education. The compromise that was made was that for the time being, the general education program would essentially be that of the College of Arts and Sciences but that a committee would be formed to develop a new general education program for Grand Valley State College. It took some time for a new program to be developed and approved, but a new general education program and new graduation requirements were instituted for the 1987 – 88 academic year.

Two features of these new general education and graduation requirements that affected the mathematics major (as well as other majors) were:

- A new definition of the B.A. or B.S. degree cognate; and
- The requirement of a capstone course in the major.

** B.A. or B.S. Cognate Proposal**

Following is a description of these requirements.

The B.A. degree requires a third-semester proficiency in a foreign language of the student’s choice. (This was the same as before.)

The B.S. degree requires a three-semester sequence of courses that emphasize either natural science or social science methodology as prescribed by the major department. In addition, at least one of the courses had to be a prerequisite for one of the other courses, at least one course had to be from outside the discipline, and at least one of the courses had to meet the requirements of the Mathematical Sciences category in the General Education Program.

This new requirement had little impact on the mathematics major since the major already required courses that satisfied the B.S. degree cognate requirement. All mathematics majors were required to complete PHY 230 – Principles of Physics I and either CS 151 – Introduction to Programming or CS 152 – Programming in FORTRAN. So the B.S. cognate for the mathematics major was listed as completion of the first two courses in the calculus sequence and PHY 230. So students seeking a B.A. in mathematics also completed the B.S. degree cognate and still had to acquire a third-semester proficiency in a foreign language.

**The Capstone Course Proposal**

According to the catalog:

Each major must include a senior-level capstone course aimed at providing the student with a broad and comprehensive perspective on the fundamental assumptions, issues, and problems of the field.

With the exception of the statistics emphasis, implementing a capstone course for the mathematics major caused some problems. For the statistics emphasis, it was clear that MTH 415 – Mathematical Statistics II would be an appropriate capstone course. However, in order to include a capstone course in the mathematics major for the other emphases, the department had to use an existing course as the capstone course or develop a new course. If a new course was developed, it would either have to replace an existing course in the major, reduce the number of electives in the major, increase the number of credits required for the major, or restructure the major. The first three options were not appealing to the department, but the faculty did not feel it was ready to redesign the major in order to have a new capstone course. In addition, there was no agreement at this time as to what a capstone in mathematics should be. Some options considered were a history of mathematics course, a philosophy of mathematics course, or a mathematical modeling course, but the department did not feel it was ready at this time to propose a new capstone course and restructure the major.

Consequently, the department decided to use existing courses as options for a capstone course even though most faculty members felt that none of the courses in the major actually fit the definition of a capstone course. Proposals for capstone courses required approval of the General Education Committee and the College Curriculum Committee, and the department decided to submit the following two courses for approval as capstone courses in the mathematics major.

- MTH 415 – Mathematical Statistics II
- MTH 420 – Abstract Algebra

Following is part of the justification for these capstone courses in the proposal submitted to the General Education Committee in January 1986.

A capstone course in mathematics should be a course that requires the student to be involved with the theoretical development of a significant area of mathematics. The course should study an area of mathematics as a branch of learning, a discipline, rather than as a set of problem solving techniques. The theory should be developed from first principles, and general principles should take precedence over problem solving. Even with this emphasis on theory, the course should be structured so that students will see how an understanding of the theory can provide insights into the proper use of certain problem solving techniques or how the theory can be used to solve problems that they could not solve with other techniques. The theory developed within the course should also provide insights into some of the fundamental concepts of mathematics.

For students with a statistics emphasis, the course that satisfies these requirements is MTH 415, Mathematical Statistics II. (This is actually the second course in a two course sequence. MTH 315, Mathematical Statistics I, is a prerequisite for MTH 415.) In lower level statistics courses, the emphasis is on using statistical techniques, but in these two courses, the emphasis is the development of the theory of statistical inference from first principles. Students study the nature of statistical inference based on the probability distributions underlying the techniques of statistical inference. In this course, the students not only have to be able to use statistics, but with an understanding of the nature of statistical inference, will be required to decide what is the appropriate statistical method to use in real world situations.

For other mathematics majors, this statistics course would not be an appropriate capstone course since they would not have taken enough statistics to reach this level without sacrificing other upper division courses that are more appropriate for their emphasis. For these students, the course that we are proposing as a capstone course is MTH 420, Abstract Algebra. This is not a course in the symbol manipulation of elementary algebra but is a study of algebra as a broad branch of mathematics that is capable of revealing general principles which apply equally to all known and all possible algebras.

Historically, abstract algebra has its roots in the study of such fundamental mathematical questions as the solvability of equations and the study of symmetry. As taught in the undergraduate curriculum, an abstract algebra course deals with the development of algebraic structures from first principles (the axioms of the structures). As such, the students work with the development of a unified and coherent body of knowledge where logic and proof are central to this development. Although students study the concept of proof in other courses, this may be their only course where proving fundamental principles is of paramount importance. The course will deal with fundamental algebraic structures and then show how these structures provide insight into the nature of our standard number systems. Even though the emphasis of this course is on development, of the algebraic structures, students should not leave such a course with the impression that these concepts are of interest only to mathematicians. As mathematicians, they must be concerned with these concepts and an insistence on rigorously proving results, but these students should also gain some appreciation of the power and usefulness of the concepts in applied areas. As a result, applications are studied in the course. Applications are selected from such topics as combinatorics, cryptography, symmetry, crystallography, error-correcting codes, and geometric constructions.

Neither course was at first approved as a capstone course. The General Education Committee tabled the proposal for MTH 420. The legitimate concerns centered around the narrowness of MTH 420 as a capstone course. The College Committee tabled the proposal for MTH 415 on similar grounds. Both committees sought further clarification from the department. It is interesting to note that the memo from the College Curriculum Committee stated that:

Some members wondered whether MTH 415 is sufficiently advanced to be a senior-level course. It was observed that topics covered are those frequently covered in courses below the 400-level. That objection, too, should be met in your justification.

To address these concerns of the two committees, Don Vander Jagt (as chair of the department) and Ted Sundstrom (as chair of the department curriculum committee) drafted two memos: one to the College Curriculum Committee to address their concerns about MTH 415; and one to the General Education Committee to address their concerns about MTH 420. For MTH 415, the central argument was that other courses in the Statistics Emphasis were not theoretical or abstract and did not deal with the fundamental issue of the development of statistical theory from a mathematical standpoint. For MTH 420, the argument the course was chosen not as a content capstone but as a capstone that deals with the spirit and methods of modern mathematics and the fundamental issues of abstraction, axiomatization, and proof. In addition, it was argued that in this sense, abstract algebra is not narrow since the basic ideas and ways of thought of abstract algebra permeate most areas of mathematics.

MTH 415 and MTH 420 were eventually approved as capstone courses for the mathematics major. The mathematics faculty then started an on-again, off-again discussion for the next several years regarding a capstone course and the restructuring of the major. In the fall of 1997, the Department of Mathematics proposed a new capstone course and a restructuring of the major, which was approved for the 1998 – 99 academic year.