I want to take advantage of the new blog format of this column to generate a discussion that I hope will help us identify the greatest problems facing mathematics departments. The intent is to assist the MAA’s Committee on the Undergraduate Program in Mathematics (CUPM) in determining the kind of information and advice, especially with regard to the major in the mathematical sciences, that would be most helpful.

As I’ve travelled the country for the MAA over the past few years, I have seen that the greatest problem facing most departments is simply lack of resources. These include the inability to replace tenured faculty who have retired or left, the need to rely on adjunct and part-time faculty to meet expanding student demand for mathematics classes, and inadequate facilities. The MAA will not be able to help directly with any of these. It can help by providing access to what is known about the effects of large classes and part-time faculty on mathematics instruction and by illustrating best practices that have enabled departments to do more with less. It also can help indirectly by assisting departments in improving the major while attracting more students. I have observed in many settings that, especially in these times of tight budgets, what resources are available tend to flow toward those departments that the administration sees as having a healthy and successful majors’ program that is clearly tied to the core mission of the college or university.

Ten years ago, CUPM was engaged in preparing the *CUPM Curriculum Guide 2004*. That *Guide* looked broadly at the role of mathematics departments and made a number of recommendations. I began this column in February, 2005 for the purpose of drawing attention to and further explicating these recommendations. It is now time to think about the next iteration of the *Guide*. A writing group, chaired by Martha Siegel and including Carol Schumacher, James Sellers, Michael Starbird, Alan Tucker, Betsy Yanik, and myself, has begun the task of identifying the issues and problems that the new *Guide* should address. This time, the focus will return to the mathematics major.

We have collected suggestions from many people about the issues and questions for this iteration of the *Guide*. I am interested in your reactions: What have we missed? Where should our priorities lie as we prepare the next *Guide*? What is going to be most useful and how can it be made useful to more people? I do not promise that the next issue of the *Guide* will address all of the important issues. Some of them would take us too far afield. For some, the relevant information does not exist or is simply too difficult to obtain. Even so, knowing that something is a burning issue for many departments will help the MAA decide future areas of focus and may help in getting funding to seek out the needed information.

I encourage you to respond via a comment on this blog or by sending me an email, bressoud@macalester.edu. I am also interested in what effect, if any, the *CUPM Curriculum Guide 2004* had for your department. Has your department used the *Guide*? If so, what in it or about it was helpful? What did you find inadequate or lacking?

As you think about the issues listed below, it would be helpful for us to know what you and your department would find most useful, such as

a. Examples of models that work (with evidence of why they are believed to work),

b. Examples of what different departments are attempting or experimenting with,

c. Descriptions of current common practice organized according to size and mission of the department,

d. Specific recommendations with evidence for the recommendation.

**Potential Issues and Questions Relevant to the Major**

1. **Core courses.** What are/should be the core courses that every math major takes? What is commonly required that is not essential for all majors? How many credit hours of mathematics should a math major be required to take?

2. **Curricular goals.** What are/should be the goals of the courses in the core curriculum?

3. **Tracks and programs.** What tracks or programs should or could be made available to students? How does the size and mission of the institution affect this choice? What courses belong in, say, a track in financial mathematics or in biological mathematics? How does a department go about building an interdisciplinary track or program?

4. **Double majors, concentrators, and minors.** What can/should departments do to attract students to a minor or concentration in the mathematical sciences, or to pick up a second major that is in the mathematical sciences?

5. **Career opportunities.** What are the options for students who graduate with a major in mathematics? How can these be communicated more effectively to our students? How can they be used to help shape a student’s course through the major?

6. **Preparation for upper division courses.** What are examples of the variety of ways in which departments prepare their majors for proof-based upper division courses? What strategies are successful, and for whom are they successful?

7. **Preparation of future teachers.** What mathematics should future teachers of mathematics know (elementary, middle-school, or secondary teachers)? For prospective secondary teachers, should their preparation differ from that of other mathematics majors? (Note: An update to the MAA/AMS publication *The Mathematical Education of Teachers*, with guidelines that address these questions, is currently underway.)

8. **Preparation for work or graduate study.** What do employers of mathematicians in business, industry, or government (BIG) need their employees to have learned as undergraduates? Which courses are going to be most useful for students heading into a career in BIG and why? What do graduate programs in the mathematical sciences need their students to have learned as undergraduates? Which courses are going to be most useful for students heading into graduate studies and why?

9. **Statistics and computer science.** What statistics and what computer science should be required of all mathematics majors? What about those majors headed toward specific careers? What software should all majors be able to use (Excel, LaTeX, MatLab, Mathematica/Maple, R)? What are the different ways in which familiarity with these packages can be incorporated into the mathematics major? Should operations research be required of all majors or of all majors in particular tracks or programs?

10. **Reading and communication.** What is being done to improve students’ ability to read, write, and orally communicate mathematics? What should be done? What expectations should we have of all mathematics majors and how can we know whether they have these skills?

11. **Proof.** What is being done across the curriculum to improve student ability to read, understand, and construct proofs? What expectations should we have of all mathematics majors regarding the ability to construct proofs and how can we determine whether they have these skills?

12. **Internships.** What are examples of successful internship programs? How were they developed and how are they run? What expectations for internships should there be for all departments?

13. **Undergraduate research.** What does undergraduate research mean and how is it interpreted at various institutions? Should there be guidelines for what it means and how it is implemented? What are examples of programs that successfully incorporate undergraduate research?

14. **Capstones.** What are the various capstone experiences that departments have in place for mathematics majors? Which are the most successful programs, and for whom are they successful?

While the next iteration of the *Guide* will be concerned with issues of the major, this seems an opportune time to identify other issues, some of which—such as attracting talented students to courses that could lead them toward a major in mathematics—are tangentially relevant. While we do not expect that the general issues listed below will be treated in the next *Guide*, CUPM still wants to know which of these are of concern so that it can help coordinate and publicize the work of MAA on these issues.

**General Issues**

15. **Placement programs.** What’s available? What works? Who does it work for? How do departments identify and properly place students with uneven preparation?

16. **Students requiring developmental mathematics.** What are the successful programs in developmental mathematics? If developmental mathematics is currently outside the department’s purview, should the department become involved in developmental education?

17. **Students with weak or uneven preparation.** How do departments meet the needs of students who are almost ready for calculus or may even enter with credit for some calculus, but who still have significant gaps in their mathematical preparation for college?

18. **Talented entering students.** How do departments attract talented students to their courses and keep them engaged in mathematics?

19. **Calculus instruction.** What kinds of calculus courses are now being offered? What should be offered? What works for which groups of students?

20. **Articulation with two-year colleges.** How can/should this be made as seamless as possible? What are the issues?

21. **Working within state requirements and articulation agreements.** How can a department be creative within imposed strictures?

22. **Online courses.** Should they be encouraged or discouraged? Should there be guidelines? When are they appropriate?

23. **Workforce. **Who is teaching which courses (full-time vs. part-time, tenured vs. adjunct vs. graduate student)? Does it make a difference? What can be done to assist instructors to improve the learning that takes place in their classes?

24. **Pedagogy.** What pedagogical approaches are most effective under which circumstances? What professional development should be available for college faculty, and who is responsible for supplying it?

25. **Technology.** How and when is it used most effectively? Is it ever essential, and if so when?

26. **Departmental self-assessment.** What are the strategies being used by departments to assess their effectiveness? What do we know about which strategies are most effective, and for whom?

The length and breadth of suggested "issues" attests to the many challenges facing math departments today and in the near future. There is a larger issue of whether (more likely when and how) the structure of higher education will change from its presencial, discipline-centered organization that "certifies" broad general knowledge to a virtual, working-group structure that "certifies" specific skills and knowledge. The extent to which the traditional university organization, and consequently the mathematics bachelor's degree, meets societal expectations and demands will likely affect the growth and development of alternatives.

ReplyDeleteHow math programs address two issues regarding student ability to "prove" mathematical statements may have a large effect on the number and types of students who succeed in these programs. The following suggests some questions within the questions.

1) Preparation for upper division courses. What are examples of the variety of ways in which departments prepare their majors for proof-based upper division courses? What strategies are successful, and for whom are they successful?

Additional questions: What does it mean to be prepared for proof-based upper division courses? Does it mean able to construct correct original proofs? Does it mean ready to learn to construct correct proofs with limited guidance? Are the strategies under consideration limited to experiences prior to upper division courses?

2) Proof. What is being done across the curriculum to improve student ability to read, understand, and construct proofs? What expectations should we have of all mathematics majors regarding the ability to construct proofs and how can we determine whether they have these skills?

Additional questions: How common are cross-curricular skill development efforts? How common are curricular requirements that funnel students through one "learning proofs" course? Who should participate in the discussion of expectations for math major skills in constructing proofs? Should the discussion include professional mathematicians such as actuaries, employers of professional mathematicians such as NSA, computer scientists, statisticians, physicists, engineers, or math teachers as well as research mathematicians?