**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 **www.maa.org/cupm/**.

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., umich.edu).

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: **www.webarchive.org**. 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.

**CONTENTS OF ILLUSTRATIVE RESOURCES**

**Part I: General Recommendations**

**1: Understand the student population and evaluate courses and programs**

Placement Exams

Mathematical Autobiography

Advising

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**

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

Geometry

History

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

**ANNOTATED BIBLIOGRAPHY**