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Introduction and Table Of Contents

Revised September 2007

This online document describes a variety of experiences and resources associated with Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide 2004, and it follows the organization in Parts I and II of the Guide. As a major part of the development of this document, CUPM made a broad request for reports on experiences from individuals who had implemented various ideas discussed in earlier drafts of CUPM Guide 2004, including specific requests to departments with large numbers of majors and/or recent NSF or FIPSE awards for curricular projects. (See Appendix 1 of CUPM Guide 2004 for additional details on the gathering of information.) There was a large response to the CUPM request, and thanks are due to the many mathematics faculty who gave generously of their time and wisdom. The contributed examples, experiences and resources were instrumental in developing this section, and they provided evidence that the CUPM recommendations are indeed feasible. References to many of these contributions are made in this document.

The experiences described here and the associated resources are far from exhaustive. They were not endorsed by CUPM, nor is there an implication that they represent better practice than other implementations not included here. Rather they are offered as experiences and ideas that appear interesting and may serve as a starting point for those considering enhancement of components of their department’s programs.

CUPM invites all members of the mathematical sciences community to report on additional resources they have successfully used or developed to work toward the recommendations in this guide. Please click here to obtain a submission form.

Revised versions of ILLUSTRATIVE RESOURCES for CUPM Guide 2004 (known as CUPM-IR) will replace this version approximately once a year on the MAA website at

Most entries in CUPM-IR include a web address; it may link to a site containing further material or it may provide a means of contacting a person or department for additional information. An annotated bibliography with additional reference information is at the end of the document. The links in these resources were all operational as of mid-August 2007. In case of difficulty using one, there are several possible reasons, each of which has a potential remedy.

1. Sometimes the server is simply busy or being worked on for an hour or a day or so. In this case the remedy is simply to try again later.

2. Sometimes the URL for a given link has changed, usually because the website for the supporting institution has been restructured. One possible remedy is to google the name of the item (program, article title, course name and/or number of the syllabus, etc.), possibly together with the name of the institution, and try to identify the new URL from the listing that appears. Another remedy is to go to the website of the department connected with the item and search for the item among the department’s webpages. Yet another remedy is to do an ’Advanced Searchâ? on the name of the item but restrict the search to the domain name of the institution in which the item is likely to be located (e.g.,

3. Sometimes the item is no longer available through a current URL. In this case, it is often possible to find the original webpage by using the WayBackMachine: Insert the URL for the address given in the Illustrative Resources, and from the listing of dates that appears click on various ones to find the desired item.


Part I: General Recommendations

1: Understand the student population and evaluate courses and programs
Placement Exams
Mathematical Autobiography
Redesigning Courses and Programs in Response to Information about Students
Assessing Programs, Courses, and Blocks of Courses
External Support for Assessing Undergraduate Mathematics
Assessment Tools for the Classroom
Gathering Information about Students and Alumni to Improve Programs

2: Develop mathematical thinking and communications skills
Discovery Learning/Inquiry-Based Learning/Problem-Based Learning
Research on Reasoning and Problem Solving
Activities to Help Students Learn to Reason and Work Logically to Conclusions
Strategies for Problem Solving
Motivating Student Reading
Specific Techniques to Improve Students’ Ability to Read Mathematical Writing
Use of Mathematical Language in the Classroom
Mathematical Writing Assignments
Assessing Students’ Skills in Writing Mathematics
Additional Resources

3: Communicate the breadth and interconnections
Key Ideas and Concepts from Varied Perspectives
Promote Awareness of Connections between Mathematics and Other Subjects
Introduce Contemporary Topics
Enhance Perception of Vitality and Importance of Mathematics
Additional Resources

4: Promote Interdisciplinary Cooperation
Connecting with other Disciplines within a Mathematics Course
Interdisciplinary Courses
Interdisciplinary-Project-Based Curriculum
Interdisciplinary Programs
Additional Resources

5: Use computer technology to support problem solving and understanding
Tools for Visualization and for Promoting Understanding
Technology Throughout The Curriculum
Additional Resources

6: Provide faculty support for curricular and instructional improvement
Teaching and Learning
Faculty and Professional Development Programs
Practices at Specific Institutions
Additional Resources

II. Additional Recommendations Concerning ...

A: Students taking general education or introductory
collegiate courses in the mathematical sciences

A.1: Offer suitable courses
General Introductory Courses
Precalculus ’ New Approaches
Integrating Precalculus and Calculus
Introductory Statistics ’ New Approaches
(See A.2 for College Algebra ’ New Approaches)
Engaging Students in Project Work
Quantitative Literacy
Developing Mathematical and Quantitative Literacy Across the Curriculum
Offering Choices to Satisfy a General Mathematics Requirement
Examples of Introductory Course Syllabi
Support for Faculty Teaching Developmental Mathematics

A.2: Examine the Effectiveness of College Algebra
Refocusing College Algebra
College Algebra ’ New Approaches

A.3: Ensure the effectiveness of introductory courses
College Algebra ’ New Approaches
Precalculus ’ New Approaches
Integrating Precalculus and Calculus

B: Students majoring in partner disciplines&
prospective teachers

B.1: Promote interdisciplinary collaboration
Strengthening Mathematics Courses to Support Future STEM Study
Learning Communities
(See C.5 for Interdisciplinary Majors)

B.2: Develop mathematical thinking and communication
Research on Teaching and Learning
Improving Students’ Abilities to Think about and Do Mathematics
Writing in Introductory and Service Courses
Additional Resources

B.3: Critically Examine Course Prerequisites
Including 3-Dimensional Topics in the First Year.

B.4: Pre-service elementary (K-4) and middle-school (5-8) teachers
Guidance to Colleges and Universities
Programs for Elementary Teachers
Programs for Middle School Teachers
Programs for All Grade Levels
Research to Practice ’ Elementary and Middle School Teachers
Programs for Mathematicians Teaching Future Teachers
Use of Video in Teacher Preparation
Additional Resources

C: Students majoring in the mathematical sciences

Examples of Effective Majors
Descriptions of Some Programs at Schools with a Large Number of Mathematics Majors

C.1: Develop mathematical thinking and communication skills
Research on Reasoning and Proof
Inquiry-Guided, Problem-Oriented Learning
Classroom Practice: Writing, Reading, and Exploring Proofs
Reasoning with Data: Probability and Statistics
Reading, Writing, and Speaking Mathematics
Evaluating Oral Presentations

C.2: Develop skill with a variety of technological tools
Resources for the Use of Technology
Using a Computer Language

C.3: Provide a broad view of the mathematical sciences
A General Resource
Discrete Mathematics and Data Analysis
Geometry and Geometric Thinking
(See Section C.1 for Statistics and Probability and Data Analysis)
Linkages ’ Algebra and Discrete Mathematics
Linkages -- Algebra and Geometry
Linkages ’ Number Theory and Geometry
Linkages ’ Complex Variables and Geometry
Linkages ’ Probability and Analysis
Powerful Applications and Contemporary Questions
Breadth of Mathematics and Connectedness to other Disciplines
Broader and More Flexible Major

C.4: Require study in depth
Pairs of Courses
Capstone Courses and Projects

C.5: Create interdisciplinary majors
Joint Majors
Tracks Within the Major

C.6: Encourage and nurture mathematical sciences majors
General References
Designing Introductory Courses To Be Effective and Engaging
Encouraging Prospective Majors
Providing Career Information
Mentoring and Advising Mathematics Majors
Co-Curricular Activities for Mathematics Majors

D: Mathematical sciences majors with specific career goals

D.1: Majors preparing to be secondary (9-12) school teachers
Connecting Students’ Learning to their Future Teaching
Capstone Courses for Secondary Teachers
Programs for Mathematicians Teaching Future Teachers
Collaboration with Local School Districts

D.2: Majors preparing for the nonacademic workforce
Upper-Level Statistics
Skills Needed for Industry
Advising and Mentoring for the Nonacademic Workforce
Internships and Summer Research
Professional Master’s Degree
Additional Resources

D.3: Majors preparing for post-baccalaureate study in the
mathematical sciences and allied disciplines

Internships and Summer Research
Special Programs for Graduate School Preparation
Mentoring and Supporting Students from Under-represented Groups
Advising Mathematics Students
Program Examples