The CUPM Curriculum Guide 2004

The CUPM Curriculum Guide 2004

Introduction

Tertiary level mathematics departments at colleges and universities in the United States serve a variety of communities at the undergraduate level: students whose major field of study is mathematics and who take “mathematics major courses,” students who are required to include certain mathematics courses to complement their studies in other disciplines and who take mathematics “service courses,” and students whose major programs do not require the study of mathematics but who take mathematics courses to satisfy university “general education” requirements. The sets of courses for the three groups of students overlap to a certain extent. For example, the same calculus course is often taken both by mathematics majors and by majors in other disciplines.

History of CUPM

The Committee on the Undergraduate Program in Mathematics (CUPM) was founded by the Mathematical Association of America in 1953. Itoriginally viewed its task as developing lists of topics for mathematics courses and recommendations for course sequences for the mathematics major. The first CUPM recommendations for the major were published in 1963. The program for the first two years included calculus and analytic geometry, some multivariable calculus, series, probability or more multivariable calculus, differential equations, and linear algebra. In the third and fourth years, students were to select from various more advanced mathematics courses, including probability, statistical theory, numerical analysis, and applied mathematics but not including modern algebra. Soon after the publication of the program a need for revision was recognized, and a tradition of regular review and revision was established.

During the 1970’s and 1980’sCUPM came to advocategreater flexibility in the nature of mathematics major courses than were contained in the original recommendations (e.g., addition of statistics and actuarial science “tracks”), curricular response to the broadening of mathematics beyond its classical boundaries (for instance, inclusion of probability and statistics and modeling in the major for all students), greater concern for and attention to “average students,” greater concern for and attention to service courses, incorporation of technology into mathematics courses, and, finally, specific pedagogical innovations such as the use of interactive teaching styles, asking students to explain ideas both in writing and in speaking, and giving students experience working on team projects. The recommendations triedto preserve the essence of the “traditional major” by, for example, including some proof-based courses and an “in-depth experience” (year-long sequence) in a relatively advanced topic. Many of these recommendations, however, were not widely publicized. Although they reflected trends and interests among some mathematics faculty members, a majority remained largely unaware of them.

Issues That Led to the Recommendations in the CUPM Curriculum Guide 2004

Relative Decline in the Number of Mathematics Majors: For mathematics departments a significant phenomenon of the last twenty yearshas been the relative decline in the number of mathematics majors. From 1985-2000 there was a 28% increase in bachelor’s degrees and a 20% increase in science and technology degrees, but thenumber of mathematics degrees remained essentially constant. Moreover, while it might appear that the increase in the number of science and technology degrees would result in increased enrollment in advanced math courses, exactly the opposite occurred. Enrollment in upper-division mathematics courses declined significantly from 1985-2000.

A primary reason for the declinemay have been increasedoptions for mathematically talented students. Because other scientific disciplines have developed mathematically oriented sub-specialties, students who might once have electeda mathematics degree may have chosen majors in computer science or software engineering, economics (econometrics), psychology (statistics), biology (bioinformatics), physics (theoretical or computational), or engineering (especially electrical and computer engineering).

Another possible reason for the decline in the number of mathematics majors is that the great increase in the number of college graduates in the United States may have led to increased competition for jobs with a consequent greater value placed on degrees that appear to provide skills perceived to give immediateentrée to post-graduate employment.

Drift of Mathematics Courses Away from Mathematics Departments: Another trend over the past several decades has been a drift of mathematics courses away from mathematics departments. Many elementary statistics courses are now housed in psychology, economics, sociology, biological sciences, business, computer science, and nursing departments. Some physics departments teach their own differential equations courses. Coverage of multivariable calculus and linear algebra has been incorporated into some engineering courses, business mathematics is sometimes taught by faculty in schools of business, mathematics for the life sciences in biology departments, discrete mathematics in computer science and electrical engineering departments, and dynamical systems and modeling in some engineering departments. While there has been great growth in quantitative literacy and quantitative reasoning programs around the United States, many are run by special committees or programs located outside mathematics departments. A relatively new trend is the increasing numbers of mathematical skills courses that are being offered through distance learning, sometimes run by adjunct faculty outside mathematics departments.

Notwithstanding this drift of courses, enrollments in many mathematics departments are at record highs. The reason appears to be tremendous growth in the number of students taking courses below the level of calculus. For instance, according to a study by the Conference Board of the Mathematical Sciences (CBMS)of students taking college or university mathematics in the fall of 2000,1,979,000 (62%) were incourses classified as “remedial” or “introductory” (with names likeelementary algebra, intermediate algebra, college algebra, precalculus, finite mathematics, contemporary mathematics, and quantitative reasoning), whereas only 676,000 (21%) were in calculus I, II, or III, 264,000 (8%) were in elementary statistics, and 287,000 (9%) were in all other undergraduate mathematics courses.

Influence of New Technologies: Another significant impetus for change in the past twenty years is the influence of new technologies. These have affected both the nature of the mathematics that is used for applications and the range of possible ways mathematics courses can be taught.

The fact that many real-world problems are not solvable by classical methods combined with good techniques for computer approximation of solutions has ledfaculty in partner disciplines to place reduced importance on traditional material and greater importance on understanding the concepts needed to be able to use technology effectively. It has also led to increased use of discrete mathematical methods. Moreover, the development of computer science as a mature discipline has resulted in pressures from faculty in that area to provide a course in discrete mathematics at the first-year level.

The use of technology in mathematics education has provided both challenges and opportunities. On the one hand, powerful graphical and other statistical tools for data analysis encourage direct exploration and analysis of raw data, and graphing calculators and computer software have the capability of bringing mathematical concepts to life for students. Many faculty members have developed ingenious ways to take advantage of technology to promote students understanding and extend their mathematical power. On the other hand, an unthinking use of technology can impede understanding, as students too often demonstrate. Finding ways to ensure that students learn to use technology intelligently is a major curricular challenge at all levels of the educational system.

Experience with Calculus Reform and Other Reform Efforts: Over 25 years of experience with calculus and other mathematical reform efforts have given many faculty members experience and insight into the issues involved both in using technology and in trying to lead students to a deeper understanding of mathematical concepts. Development of the discipline of research in undergraduate mathematics education has also improved understanding of the difficulties students encounter in learning mathematics and ways that these might be overcome.

Description of the CUPM Curriculum Guide 2004

The CUPM Curriculum Guide 2004(hereinafter referred to as the Guide) is divided into several sections. Part I contains six overarching recommendations for departments, programs, and all courses, and Part II elaborates the recommendations for the various student populations enrolled in mathematics courses: students taking general education or introductory courses in the mathematical sciences, students majoring in partner disciplines, students majoring in the mathematical sciences, and mathematical sciences majors with specific career goals (majors preparing to be secondary school (9-12) teachers, majors preparing for the nonacademic workforce, and majors preparing for post-baccalaureate study in the mathematical sciences and allied disciplines). Thefocus of all the recommendations is on the activities in which departments and faculty need to engage to achieve the goal of providing the best possible mathematical education for the variety of students they serve.

The six overarching recommendations are as follows:

Recommendation 1 Understand the student population and evaluate courses and programs: Mathematical sciences departments should

Understand the strengths, weaknesses, career plans, fields of study, and aspirations of the students enrolled in mathematics courses;
Determine the extent to which the goals of courses and programs offered are aligned with the needs of students as well as the extent to which these goals are achieved;
Continually strengthen courses and programs to better align with student needs, and assess the effectiveness of such efforts.

Recommendation 2 Develop mathematical thinking and communication skills: Every course should incorporate activities that will help all students progress in developing analytical, critical reasoning, problem-solving, and communication skills and acquiring mathematical habits of mind. More specifically, these activities should be designed to advance and measure students’ progress in learning to

State problems carefully, modify problems when necessary to make them tractable, articulate assumptions, appreciate the value of precise definition, reason logically to conclusions, and interpret results intelligently;
Approach problem solving with a willingness to try multiple approaches, persist in the face of difficulties, assess the correctness of solutions, explore examples, pose questions, and devise and test conjectures;
Read mathematics with understanding and communicate mathematical ideas with clarity and coherence through writing and speaking.

Recommendation 3 Communicate the breadth and interconnections of the mathematical sciences: Every course should strive to

Present key ideas and concepts from a variety of perspectives;
Employ a broad range of examples and applications to motivate and illustrate the material;
Promote awareness of connections to other subjects (both in and out of the mathematical sciences) and strengthen each student’s ability to apply the course material to these subjects;
Introduce contemporary topics from the mathematical sciences and their applications, and enhance student perceptions of the vitality and importance of mathematics in the modern world.

Recommendation 4 Promote interdisciplinary cooperation: Mathematical sciences departments should encourage and support faculty collaboration with colleagues from other departments to modify and develop mathematics courses, create joint or cooperative majors, devise undergraduate research projects, and possibly team-teach courses or units within courses.

Recommendation 5 Use computer technology to support problem solving and to promote understanding: At every level of the curriculum, some courses should incorporate activities that will help all students progress in learning to use technology

Appropriately and effectively as a tool for solving problems;
As an aid to understanding mathematical ideas.

Recommendation 6 Provide faculty support for curricular and instructional improvement: Mathematical sciences departments and institutional administrators should encourage, support and reward faculty efforts to improve the efficacy of teaching and strengthen curricula.

Perhaps the most innovative part of the Guide is a compilation of resources designed to help departments implement and improve practices to satisfy the recommendations. It is organized and numbered the same way as the recommendations in Parts I and II and contains descriptions of courses, programs, curricular materials, and other items that illustrate the feasibility of the recommendations in each subsection of the Guide. This part of the Guide is being published on the Internet as a hyper-linked web document called Illustrative Resources for CUPM Guide 2004. Pointers to additional resources, such as websites and publications, are included, with live links where appropriate, and it is planned to update the Illustrative Resources on a regular basis.

The full text of the CUPM Curriculum Guide 2004and the Illustrative Resourcesare available at

Susanna S. Epp

Member, Writing Group for CUPM Curriculum Guide 2004

Department of Mathematical Sciences

DePaulUniversity

2320 N. Kenmore

Chicago, IL60614

USA