The recommendations in this section refer to all major programs in the mathematical sciences, including programs in mathematics, applied mathematics, and various tracks within the mathematical sciences such as statistics or operations research. Also included are programs designed for prospective mathematics teachers, whether they are “mathematics” or “mathematics education” programs. These recommendations also provide a basis for discussion with colleagues in other departments about possible joint majors with applied science, economics or life sciences.

Examples of Effective Majors

The case studies project that resulted in Models that Work (Tucker, 1995) included site visits to ten mathematics departments with undergraduate programs that are considered effective in one or more of the following categories:
   * attracting and preparing large numbers of mathematics majors
   * preparing students to pursue advanced study in mathematics
   * preparing future school mathematics teachers
   * attracting and preparing underrepresented groups in mathematics.

Mathematics departments at the following schools were at the heart of this project: Lebanon Valley College, Miami University of Ohio, Mount Holyoke College , Saint Olaf College, Southern University, Spelman College, State University of New York – Potsdam, University of Chicago , University of Michigan, and University of New Hampshire.

A book on the Potsdam program is Mathematics Education at Its Best: The Potsdam Model by D. K. Datta, Kingston, RI: Rhode Island Desktop Enterprises, 1993. Articles about the program are “The Basis for the Success of the Potsdam Program” by Rick Luttmann, “On Attracting and Retaining Mathematics Majors – Don’t Cancel the Human Factor” by Armond Spencer, Notices of the AMS 1995, “A Humanistic Academic Environment for Learning Undergraduate Mathematics” by Clarence Stephens, who founded the program, “Thoughts on power and pedagogy” by P. Rogers (which appeared in Leone Burton, Ed., Gender and Mathematics: An International Perspective (pp. 38-45), London: Cassell, 1992),  and “A modern fairy tale?” by J. Poland, Amer. Math. Monthly, 94(3), 291–295, 1987. Additional publications about the SUNY Potsdam experience are available through a link on the mathematics department website.

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

Montclair State University (public coeducational; 11,000 undergraduates). The Department of Mathematical Sciences offers programs leading to the bachelor’s degrees in mathematics, mathematics with a concentration in applied mathematics, mathematics with certification as a teacher of mathematics, physics, and physics with certification as a teacher of physics. The department also offers minors in mathematics and in physics and honors programs in mathematics and physics for qualified students. The applied mathematics concentration has two tracks: (1) statistics and (2) discrete applied mathematics and operations research. The programs introduce central ideas in a variety of areas in mathematics and physics, and are intended to develop problem-solving ability by teaching students to combine critical thinking with rigorous reasoning. All majors require single and multivariable calculus, linear algebra and probability. The mathematics major adds to that core advanced calculus, algebra, and electives; courses in introductory physics and computer science are required as well. The applied mathematics major adds to the core modeling, algebra, two computer science courses, and either discrete mathematics and operations research or statistics. The mathematics education major adds geometry and algebra and also requires physics and computer science. The probability course includes discrete and continuous.  There is also an advanced course in combinatorics and graph theory.

Spelman College (Private, independent, liberal arts, historically black college for women; 2,000 undergraduates). At Spelman the “primary goal” of the mathematics major is “to teach students to think logically and critically.” Both the B.A. and the B.S. versions of the major require single and multivariable calculus, linear algebra plus applications, a bridge course, algebra, real analysis, a second semester of either algebra or analysis, 3 advanced electives, and a senior seminar. The B.S. requires 8 more credits of advanced electives. Discrete mathematics is offered with a calculus and computing prerequisite. There is no requirement of discrete mathematics, probability, or statistics. Through choices of electives, students can emphasize operations research, computation, statistics, actuarial science, business administration or preparation for teaching. With Bryn Mawr, Spelman runs the EDGE Program (Enhancing Diversity in Graduate Education) for graduating seniors or students entering graduate school.

State University of New York (SUNY) - Fredonia (Public, 4900 undergraduates).  The department offers several undergraduate program options in mathematics: liberal arts mathematics, applied mathematics: economics option, applied mathematics: statistics/operations research, mathematics: adolescent education, mathematics: middle childhood education, and an honors program.  All options require calculus and linear algebra and one laboratory science course. The liberal arts option requires programming, discrete mathematics, differential equations, analysis, algebra, probability and statistics, a senior seminar and two electives.  The applied mathematics: economics option omits the algebra and analysis courses and adds mathematical modeling, financial mathematics, a mathematics or statistics elective, and 7 economics electives. The applied mathematics: statistics/operations research option omits financial mathematics and adds one more statistics course, 2 more mathematics or statistics courses, and a minor in a field that uses statistics or operations research.  The mathematics: adolescent education option is similar to the liberal arts option, except that it does not include differential equations and adds geometry, history of mathematics and a mathematics reading and writing course.  See B.4 for information about the mathematics: middle childhood education option.

State University of New York (SUNY) - Stony Brook (Public, research university; 14,200 undergraduates). SUNY-Stony Brook offers majors in mathematics and in applied mathematics and statistics, with the applied mathematics and statistics major offering tracks in actuarial science or applied environmental sciences. Two distinct departments in two separate colleges support these programs, the Department of Mathematics in the College of Arts and Sciences  and the Department of Applied Mathematics and Statistics in the College of Engineering and Applied Sciences. The major program in mathematics is broadly based, and contains courses that feature the history of mathematics and the use of computers in mathematics as well as the standard undergraduate courses in analysis, geometry and algebra and a set of high-level seminars for advanced students. It is very flexible and may be combined with other majors, such as physics, economics, biochemistry, computer science or applied mathematics. Double major and major/minor combinations are designed to give a solid background for a student who is interested in graduate school either in another discipline or in mathematics itself. Stony Brook also offers a program, open to both mathematics and applied mathematics and statistics majors, which prepares future teachers of high school mathematics. Students graduate from the program with provisional certification to teach mathematics, grades 7-12, in New York State. The applied mathematics and statistics (AMS) department encourages its students to have a broad exposure to many types of mathematical reasoning and to its diverse roles in the social and natural sciences. The department was cited in Towards Excellence as having a popular major oriented toward the “decision sciences” side of applied mathematics. Almost all electives in this program are in probability and statistics or operations research. During their first two years, students considering an AMS major are encouraged to take a required calculus sequence, two semesters of physics, an appropriate-level computer science or computer science for engineers course; one other computer course (because competence in computer programming is deemed essential for many professional careers), and some economics. At the end of the sophomore year or the beginning of the junior year, students begin taking upper division AMS courses, usually starting with finite mathematical structures or probability and statistics. At the same time, they are strongly encouraged to continue taking pure mathematics and computer science courses and mathematically oriented courses in other departments.

University of California at Los Angeles (Urban public research university; 25,300 undergraduates). The department at UCLA offers many kinds of majors: mathematics (recommended for students planning graduate school), applied mathematics, mathematics of computation, and “general” (recommended for prospective high school teachers). It also offers joint majors as described in Part 2, Section C.5. The mathematics major requires “preparation” consisting of single and multivariable calculus, series and differential equations, an introduction to linear algebra, and an introduction to programming. The upper-division requirements are linear algebra, algebra (2 quarters), analysis (2 quarters), complex analysis, differential geometry, and electives. The applied mathematics major has the same preparation, followed by linear algebra, analysis (two quarters), modeling, and two courses chosen from numerical methods, probability/statistics, differential equations, and electives. Prospective teachers are required to take algebra, geometry, probability or statistics, applied mathematics, with recommended electives in the history of mathematics, number theory, and statistics.  Both analysis and algebra are available with an applied emphasis. 

University of Chicago (Private, coeducational; 3500 full-time undergraduates). The University of Chicago mathematics department was cited in Towards Excellence and Models that Work as being clearly committed to excellence in undergraduate mathematics education. The department offers five bachelor’s degree programs leading to a B.A. or B.S. in mathematics, B.S. in applied mathematics, B.S. mathematics with specialization in computer science, and B.S. mathematics with specialization in economics. The department requires all majors to complete both a yearlong sequence in calculus (or to demonstrate equivalent competence on the calculus placement test), year-long (three-quarter) sequence in analysis, and two quarters of a sequence in algebra. Candidates for the B.S. degree in mathematics must take a three-quarter sequence in algebra. The remaining mathematics courses needed in the mathematics concentration programs (three for the B.A., two for the B.S.) are selected from an extensive list of over 35 courses. Mathematics B.S. candidates are further required to select a minor field, which consists of an additional three-course sequence, outside the mathematics department but within the division of physical sciences. Candidates for the B.S. in applied mathematics all take prescribed courses in numerical analysis, algebra, complex variables, ordinary differential equations, and partial differential equations. In addition, candidates are required to select a minor field, which consists of a three-course sequence that is outside the mathematics department but within the division of physical sciences. The programs of “with specialization in” are versions of the B.S. in mathematics and have specific mathematics and disciplinary requirements. (See the descriptions in Part 2. Section C.5.)

University of Michigan (Public research university; 24,500 undergraduates). The department offers several majors:  “pure mathematics,” “mathematical sciences,” actuarial mathematics, and a version of the major for teaching certification. All majors require a core consisting of calculus, linear algebra, and differential equations. The pure mathematics major requires an additional 9 courses: 4 basic, 4 elective and 1 cognate. The 4 basic courses consist of 1 each from the following groups: differential equations, algebra, analysis, geometry/topology. There is great freedom on the choice of electives, but the choice must be “coherent”. The cognate course can be anything involving significant use of mathematics at the intermediate level or higher. There is no discrete or probability/statistics requirement for the pure mathematics major. The mathematical sciences major also requires an additional 9 courses: 4 basic courses chosen from differential equations, discrete mathematics and algebra, analysis and probability, and 5 electives chosen to fit one of 9 options: discrete and algorithmic methods, numerical and applied analysis, operations research and modeling, probabilistic methods, mathematical economics, control systems, or finance and risk management. Prospective teachers take the core plus 1 course from each of discrete mathematics/modern algebra, geometry, probability, and secondary mathematics and one additional mathematics course. They must also demonstrate competence with a computer programming language or high-level mathematics software and take a series of education courses.

University of Rochester (Private, coeducational, nonsectarian; 4440 full time undergraduates). The department offers programs in mathematics, mathematics/statistics, applied mathematics, and mathematics education (for prospective high school teachers). Each major has a “preparation,” 3 core course sequence, and 5 advanced electives. The preparation typically includes single and multivariable calculus and a course combining linear algebra and differential equations. The core for mathematics majors requires linear algebra, algebra or topology, and one of a long and diverse list of options. The electives must include 3 upper-level mathematics courses plus two more that can either be mathematics courses or courses in other departments with significant mathematical content. The core for applied mathematics is linear algebra, probability, real and complex analysis. Mathematics education majors take a core of probability, linear algebra and algebra. All majors must satisfy an upper-level writing requirement by taking two ‘W’ mathematics courses or by taking one ‘W’ course plus a 2 credit ‘W’ supplement to another mathematics course. The University of Rochester, which has over 5% of students majoring in mathematics, was the focus of the article “College math on the rebound?” by Mark Clayton, Christian Science Monitor, August 13, 2002).

Vanderbilt University (Independent, private, coeducational; 6200 undergraduates). The Department of Mathematics offers an undergraduate major with several types of emphasis. Students planning to continue in graduate study may choose to emphasize pure mathematics. Students with other interests emphasize applied mathematics, statistics, or preparation for teaching. A solid background in mathematics is advertised as providing an excellent foundation for several professions—many students go on to professional studies in law, medicine, or business. The department offers two kinds of concentration. Program I is intended for most mathematics majors in the College of Arts and Science and requires a minimum of 32 semester hours in the department. Program II is intended for students in the School of Engineering who elect a second major in mathematics, but it is also available for other students. Program II requires a minimum of 29 semester hours in the department in addition to 6 semester hours outside the department. Both programs require a calculus sequence, linear algebra, differential equations, and 4, from a list of 20, mathematics courses. Students have choices for the remaining requirements. Vanderbilt attracts a significant number of engineers who double major in engineering and mathematics. Counting double majors, Vanderbilt claims to have the highest percentage of its bachelor’s degrees awarded to mathematics majors of any U.S. doctoral university.

Williams College (Private, coeducational, liberal arts; 2000 full time undergraduates). In 2000, 8% of Williams College graduates were mathematics majors. The department (which includes both mathematics and statistics) offers a single, flexible major with the goal of developing “problem solving ability by teaching students to combine creative thinking with rigorous reasoning.” It requires single and multivariable calculus; one course chosen from applied mathematics, discrete mathematics or statistics; 3 core courses: linear algebra, real analysis or applied real analysis, algebra; 3 advanced courses, one of which must be numbered at the 400 level and taken in the senior year; and participation in the weekly colloquium in which senior majors present talks on subjects of their choosing. Although there are no formal tracks, the department offers suggestions for different interests. For example, students interested in applied mathematics and the physical sciences are advised to take differential equations and vector calculus, discrete mathematics, applied real analysis, and groups and characters. The discrete mathematics is offered with a calculus prerequisite.

York College, City University of New York (CUNY) (Public, urban, 3600 undergraduates).  The major in mathematics at York College is very flexible.  It requires single and multivariable calculus, differential equations, linear algebra, modern algebra, a one-credit Seminar in Contemporary Mathematics, a mathematics elective, and 3 courses in an area of specialty.  In consultation with advisors, students can tailor their programs for particular career paths, including actuarial science.

Courses designed for mathematical sciences majors should ensure that students:

• Progress from a procedural/computational understanding of mathematics to a broad understanding encompassing logical reasoning, generalization, abstraction and formal proof;
• Gain experience in careful analysis of data;
• Become able to convey their mathematical knowledge in a variety of settings, both orally and in writing.
  
  

Research on Reasoning and Proof

In “Secondary School Mathematics Teachers' Conceptions of Proof” (Journal for Research in Mathematics Education, 33 (5), 379-405, 2002, available through ProQuest), Eric J. Knuth concluded that although teachers recognize a variety of roles of proof, they lack the view that proof is an important tool for learning mathematics, hold limited views regarding the nature of proof, and demonstrate inadequate understandings of what constitutes proof. Knuth called for changes in undergraduate mathematics courses, as well as further research on required levels of understanding of proof and mathematical reasoning for secondary school teachers. Knuth concluded that “the responsibility for enhancing teachers' conceptions of proof lies with both mathematicians and mathematics educators, the parties who are chiefly responsible for the nature of teachers' experiences with proof.”

In “Making the transition to formal proof” (Educational Studies in Mathematics 27: 249-266, 1994, available through JSTOR), Robert Moore found that students in a transition-to-higher-mathematics course “did not know how to use definitions to obtain the overall structure of proofs,” “were unable to understand and use mathematical language and notation,” and “did not know how to begin proofs.”  He noted that “several students in the transition course had previously taken upper-level courses requiring proofs.  All of them said they had relied on memorizing proofs because they had not understood what a proof is nor how to write one.”  Analysis of data collected from a previous group theory course indicated that “the students appeared to be overwhelmed by the necessity of grappling with difficult group theory concepts, problem solving, abstraction, and generalization while learning what a proof is and how to write one.  A transition course on mathematical language and proof would have reduced their cognitive load in subsequent upper-level courses while also preparing them for the formal mathematical approach used in those courses.”

Annie and John Selden also studied students in a transition-to-higher-mathematics course and found that even third-or fourth-year university students specializing in mathematics or secondary mathematics education had great difficulty translating informal mathematical statements into the formal versions that would help them determine the truth or falsity of the statements.  In “Unpacking the logic of mathematical statements” (Educational Studies in Mathematics, 29; 123-151, 1995, available through JSTOR) the Seldens define “the term validation to describe the process an individual carries out to determine whether a proof is correct and actually proves the particular theorem it claims to prove.”  They state that “[t]his process involves much more than just passive reading – it is often quite complicated and includes making affirming assertions, asking and answering numerous questions of oneself, and perhaps even constructing subproofs.”  To help students learn to validate a proof, they suggest presenting “theorems and definitions both in a more informal way and in a more formal way” in order to “accommodate the needs both for intuitive understanding and for the careful validation of proofs or even validation of less formal arguments.”  They also suggest that “it might be useful to offer university students some explicit instruction or advice on validation, an area currently more or less neglected.”

Another study by the Seldens addressed “The Role of Example in Learning Mathematics.” They wrote, “Examining examples and non-examples can help students understand definitions. ... When we teach linear algebra and introduce the concept of subspace, we often provide examples and non-examples for students. We may point out that the polynomials of degree less than or equal to two form a subspace of the space of all polynomials, whereas the polynomials of degree two do not. Is the provision of such examples always desirable? Would it perhaps be better to ask undergraduate students to provide their own examples and non-examples? Would they be able to? Given a false conjecture, would students be able to come up with counterexamples?” The “sampler” includes several studies aiming to shed light on these questions: “Successful Math Majors Generate Their Own Examples,” “Being Asked for Examples Can Be Disconcerting,” “Generating Counterexamples That Are Explanatory, Coda,” and "If I Don't Know What It Says, How Can I Find an Example of It?"

The Seldens also explored the way in which mathematics majors read and reflect on student-generated arguments purported to be proofs of a single theorem. They found that undergraduates tend to focus on surface features of arguments, and that these students have little ability to determine whether a particular argument constitutes a proof. The article “Validation of Proofs Considered as Texts: Can Undergraduates Tell Whether an Argument Proves a Theorem” (Selden, A. & J. Selden, Journal for Research in Mathematics Education, 34 (1), 4-36. Reston, VA: National Council of Teachers of Mathematics, 2003, available through ProQuest) includes details of the study, “relates the mathematics research community’s views of proofs and their validations to ideas from reading comprehension and literary theory,” and points to implications for teaching.

In “The Role of Logic in Teaching Proof” (American Mathematical Monthly (110)10, 886-899) Susanna S. Epp proposes two hypotheses to explain some of the reasons why so many students have difficulty with proof and disproof: differences between mathematical language and the language of everyday discourse, and the kinds of shortcuts and simplifications that have been part of students' previous mathematical instruction.  The article describes research about whether instruction can help students develop formal reasoning skills and suggests that such instruction can be successful when done with appropriate parallel development of transfer skills.

To prepare “Group Dynamics in Learning to Prove Theorems,” Connie Campbell, Georgia S. Miller, and G. Joseph Wimbish silently videotaped students working in groups on set problems and found that “[m]any groups took far longer to formulate a proof than we expected. We looked on, unable to comment or give help, as groups struggled with a proof, pursued several incorrect paths, but were finally successful. This prompted the authors to wonder how often we preempt our students from making self discoveries. In several cases we were convinced that a group was destined for failure, but were proven wrong as they finally came around to a valid argument. Had the environment been different, and we been able to interact with the students, we most certainly would have offered input, potentially sending the incorrect message to these students that they could not have developed a correct solution independently.”

In “Students’ Proof Schemes: Results from Exploratory Studies,” Guershon Harel and Larry Sowder addressed questions revolving around the development of college students’ proof understanding, production, and appreciation (PUPA) (Harel and Sowder, 1998). In this first of a series of planned reports, these researchers focused on various students’ schemes of mathematical proof. They have developed three categories of proof schemes – each with several subcategories: External Conviction Proof Schemes, Empirical Proof Schemes, and Analytical Proof Schemes. None of the schemes are mutually exclusive and students can operate within several schemes simultaneously. Research continues to examine and refine these schemes, as well as to document students’ progress in developing a conception of proof, offer developmental models of the concept of proof based on educational research, and offer principles for instructional treatments that will facilitate proof understanding, production, and appreciation.

Additional information about research on reasoning and proof is in Section 1, Part 2.

Inquiry-Guided, Problem-Oriented Learning

Instructors at North Caroline State University use Inquiry-Guided Learning (IGL) to teach Foundations of Advanced Mathematics, Abstract Algebra, and Introduction to Analysis). The primary aim of the IGL method is to promote habits of inquiry through guided and increasingly independent investigation of questions and problems for which there is no single answer while requiring students to take responsibility for their own work through weekly graded homework and conventional closed book exams. Foundations of Advanced Mathematics is geared toward mathematics majors but taken by majors in many disciplines. The course focuses on critical thinking (reading and analyzing mathematical arguments and writing mathematical proofs) and content (logic and the language of mathematics, methods of proof, theories of sets, relations and functions). Michael Shearer and Jo-Ann Cohen are working on integrating the IGL method into other courses. To make a classroom conducive to group work, they recommend having plenty of blackboard or whiteboard space and round tables if possible.

One type of inquiry-guided learning is the method of R.L. Moore, which is discussed in Part 1, Section 2.  The following are some articles recounting experiences using either the Moore method or a “modified Moore method” in upper-level courses for mathematics majors: “My Experiences with the Various ‘Texas Styles’ of Teaching” by Jack Brown, Auburn University, “The Texas Method and the Small Group Discovery Method” by Jerome Dancis and Neil Davidson, University of Maryland,  “The Moore Method” by F. Burton Jones, University of Colorado, and “Comments on Moore-Method Teaching” by  Mike Reed, Duke University.

At Harvey Mudd College, there are two versions of the Putnam Seminar. One is suitable for college underclassmen and the other is for more advanced students. All participants are encouraged to try out for the Putnam Exam team. In addition, every mathematics major is required to take a full year of the Mathematics Clinic or the Senior Thesis. In the Mathematics Clinic “teams employ mathematical modeling, statistical analysis, and a whole host of formidable numerical approaches to concentrate on unsolved problems for industry and government.” The Senior Thesis “offers the student, guided by the faculty advisor, a chance to experience a taste of the life of a professional research mathematician” doing work that “is largely independent with guidance from the research advisor.”

In Texas A & M University’s Putnam Challenge course students deal with a variety of mathematical problems to develop problem-solving techniques and prepare to take the Putnam exam.  See Doug Hensley’s fall 2004 syllabus.  Jonathan Duncan at Walla Walla College occasionally offers a course with a similar focus. 
 

The website The Art of Problem Solving is intended for faculty preparing students for pre-college mathematics competitions but contains many resources useful at the college level as well. The MathPro Press website contains links to thousands of online problems among other resources.

Amy Cohen cited Rutgers University’s Introduction to Mathematical Reasoning course, which is required as a prerequisite for Advanced Calculus and Abstract Algebra, as addressing “not only how to write proofs but how to investigate a conjecture to find proofs or counterexamples. Depending on the instructor, there is some group work and workshop write-up and even sometimes presentations in class where the students attempt to understand and critique other students’ efforts.” In 1994-95 Rutgers University added workshop meetings to Advanced Calculus (Math 311) and Introduction to Abstract Algebra (Math 351).  (See the listing of Rutgers math courses and Stephen Greenfield’s homepage and click on the course numbers.) All mathematics majors must pass one of these courses or an upper-level linear algebra course, and all candidates for secondary school teacher certification must pass the algebra course.  Cohen wrote:
      “The introduction of workshops was intended primarily to enhance both the subject matter mastery and the mathematical communication skills of the majors – and secondarily to increase the first-time pass rates in courses that had acquired a discouraging reputation of requiring two efforts to pass.
     “In advanced calculus (where we prove the theorems of calculus I and II) and in introduction to abstract algebra, we have two lectures a week and one workshop. Faculty members have realized that the one ‘extra’ contact period saves them many hours a week in extra-help outside office hours. Ideally, workshop write-ups are critiqued and returned for revision before grading. Explicitly making students responsible for correcting their own errors seems to be the most effective method.
     “The workshops present problems to be addressed by students in groups of size 2-4.  Some problems explore the subject matter of past lectures; others, accessible examples relevant to upcoming lectures.  The students work under the supervision of the faculty member teaching the course.  The supervision does not consist of showing the students elegant solutions – but rather of prodding with questions and comments which are intended to get the students to see ways to proceed on their own.  At the end of the period, the faculty member assigns one of the day's problems to be written up in good expository style.  Each student must produce his/her own write-up. These write-ups are turned in for correction and grading.  In some cases faculty provide a critique of the first draft, and require a revision before assigning a grade....
    “The initial faculty reaction was uniform dismay at just how inarticulate and confused students could be when working on the workshop problems.  Coaching students in workshop was very different from grading homework or tests. The initial student reaction was mixed.  Many students were angry to discover only as juniors or seniors that doing mathematics involved reasoning and explaining as well as formal computations following text-book templates. Faculty had to confront the awkward fact that effective teaching requires more than cogent lectures. But faculty also found that the investment of time in workshops paid off with a decrease in demand for extra help on homework outside ordinarily scheduled office hours....
     “Since 1995, Rutgers has seen a change in student culture – most clearly among those students who serve as peer mentors in calculus. Group study and intellectual interaction among the majors seems more common.  The students preparing to be high school teachers seem particularly pleased with the new format. They feel that they are understanding more about mathematics, and that they are seeing class-room activities consistent with the NCTM standards.  The best students have benefited the most.  Many C and C+ students continue to resist the idea that mathematics is more than rote computation.  The weakest students seem to be hit hardest by the demand for coherent explanation.”

 

Additional information about Inquiry-Guided, Problem-Oriented Learning is in Part 1, Section 2.

 

Classroom Practice: Writing, Reading, and Exploring Proofs

Robert Rogers, State University of New York Fredonia, provides students with a simulation on the blackboard of what a mathematician might do on scratch paper while developing a proof. Since starting this practice, he has refined his method and published a description and some reflections in “Using the Blackboard as Scratch Paper” (2002).

Virginia (Jimmy) Buchanan of Hiram College reports that she begins a class period by randomly assigning homework problems to students for them to present to the class.  Students come to class early to see which problem has been assigned to them and to negotiate and trade problems, if desired, with their classmates. Once class begins, each student is responsible for explaining his or her problem to the class by writing a proof, solution, or construction on the board and giving a verbal explanation. The student then answers questions posed by classmates and by the instructor. If errors occur, the class as a group works to resolve the problem and complete the solution.

David Gibson, Murray State University, used task-based interviews with students to research “Students’ Use of Diagrams to Develop Proof in an Introductory Analysis Course,” (in Schoenfeld, A., Kaput, J. & E. Dubinsky (Eds.), Research in Collegiate Mathematics Education III (284-307), 1998) These indicated that students used diagrams to understand information, judge truthfulness, discover ideas, and express ideas in writing. Gibson reported that diagrams helped students to complete subtasks that they could not complete with verbal or symbolic representation alone.

Susanna Epp, DePaul University, assembled suggestions for teaching proof from a variety of sources: 
   * Include some logic puzzles to stimulate students' interest and help them develop a sense for the flow of a deductive argument.
   * Discuss a few basic aspects of logic explicitly,  e.g., nonequivalence of a conditional statement and its converse, equivalence of a conditional statement and its contrapositive, interpretation of multiply-quantified statements, negations of quantified statements.
   * Emphasize definitions. Much of the inner structure of mathematical proofs is determined by the meanings of the terms in the hypothesis and the conclusion. Both the “if” and the “only if” parts of definitions are used in most proofs. Logically equivalent versions of definitions may be obvious to instructors but many students have little feeling for them. Students may benefit from practice recognizing and translating among various formal and informal versions of definitions. It may also be helpful to ask them to think about what it means for an object not to satisfy a definition and to give them a broad range of instances of a definition to prevent their forming an overly narrow concept of a term (e.g., identifying a function with a formula).
   * Distribute a proof-writing guideline and provide model answers to some of the proofs assigned as homework. (Many students have difficulty believing that instructors really want them to write proofs in complete, coherent sentences.) It may also be helpful to have students complete a few fill-in-the-blank proofs before they start writing their own.
   * Develop proofs with the class as a committee of the whole, allowing each person to give only one step, and discussing the motivation for each step and its role in the proof.
   * Suggest that students read their proofs out loud to test whether they are written in coherent sentences.
   * Have students present proofs from homework assignments at the blackboard.  If the proofs are good, the other students see that the demands made by their instructor can actually be met by one of their own kind. If the proofs contain mistakes or sections that are not well expressed, an instructor can involve the rest of the class in finding ways to improve it. A ploy is to ask students to imagine they are a research team for a large company and that if they can collectively come up with a perfect answer they will get to share a sizeable bonus. After the class has finished its critique and some changes have been recorded, the instructor can take a turn, using the opportunity both to comment on significant errors that have gone undetected and also to show students the kinds of things the instructor will be looking for when grading students' work.
   * Discuss the kinds of errors often made in writing proofs and assigning homework problems of the find-the-mistake or “proofs to grade” variety.
   * Make sure to give each student detailed feedback about the work they have done on each type of proof for which they will be responsible.
   * Have students submit one or two drafts of their solutions to a few selected problems and make suggestions for improvement on each draft.
   * Have students work to develop some proofs in pairs or in groups of three or four.
   * Ask students to identify the crux, or essential idea, of a proof.
   * Spend some class time discussing the psychological aspects of the process of mathematical discovery. (Students are often very grateful for this discussion.)

James Sandefur, Georgetown University, gave a talk “Writing Proofs: How Do We Teach Students What Is Second Nature To Us?,” with a link to a more detailed paper, in the 2004 MAA session “Getting Students To Discuss And Write About Mathematics,” which contains many other excellent discussions about the subject. 

Moira McDermott, Gustavus Adolphus College, emphasizes reading, writing, and proofs in her relation-based structures course. The following are excerpts from the syllabus for this course. (McDermott credits Barbara Kaiser and John Holte for some of the ideas):
SYLLABUS EXCERPTS:
Class time will be a mixture of lectures, discussions, problem solving and presentation of proofs. At various times you will be asked to present problems, reflect on the reading and generate questions for your classmates. It is essential that you come to class prepared to do the day's work. In particular, you should read the text and attempt homework before coming to class. Class meetings are not intended to be a complete encapsulation of the course material. You will be responsible for learning some of the material on your own.
Homework: I encourage you to work with other students on the homework provided that you do so in such a way that every one in your group learns the material. The most effective way to do this is to first discuss each problem as a group and then have each person work on the problem individually. When you're done (or stuck) compare your work and discuss it. Remember that doing the homework is how you learn the material and that you are not allowed to work cooperatively on tests.  If you do work with other students on the homework, I would like you to follow these guidelines:
* Each person should write up the answers independently.
* Each person should be able to work each one of the problems independently.
* Each person gives credit to the others who helped.
Writing: MCS236 is a ``W'' course. Writing assignments will take several forms. First, homework assignments will often require written proofs. These proofs will be checked for logical and grammatical accuracy, as well as for style and exposition. It is important to be able to express your mathematical thoughts in writing, using clear, well-organized paragraphs comprised of English sentences. This means more than separating your equations with a few well placed “Thus it follows that...” or “Plugging (a) into (b) shows that ...” During the course we will work on writing mathematical prose effectively and clearly. In addition, you will be compiling a proof portfolio containing one example of each type of proof discussed. You will also be expected to write one expository paper. Details of the paper, including deadlines for drafts and revisions, will be described later in the course.
Homework Guidelines  (excerpts)
* For problems that don't involve proofs, you should show enough work so that any student in the class can follow your solution. Just writing the answer is never enough.
* Proofs should be written in complete English sentences. Proofread what you have written to make sure it makes sense.
* Don't try to fake a proof. Instead, acknowledge the gap in your proof. Better yet, come talk with me beforehand and see if I can help you close the gap.
Proof Portfolio
* You may hand in up to three versions of each proof. If you decide to rewrite a proof, you should hand in previous (graded) versions. You can modify a homework problem, provided the original grade on it was no more than 7. In this case, you should hand in your original solution as well as your rewrites.
* Each proof is graded on: mathematical content (80%) and does the proof technique fit the problem (20%).
 

Reasoning with Data: Probability and Statistics

Gustavus Adolphus College offers an introduction to statistics course, designed primarily for mathematics and science majors. It uses Introduction to the Basic Practice of Statistics by David S. Moore and George P. McCabe and includes supplementary material to make connections to calculus through the topics of the normal distribution, least-squares regression and probability. Mathematics majors typically take the course in their sophomore year, often when they are also taking a more proof-oriented course such as linear algebra or theory of calculus. Since introducing the course, the department has seen an increase in the number of students who have selected the upper-level probability and mathematical statistics sequence to satisfy their depth requirement.

Allan Rossman and Beth Chance (Cal Poly-San Luis Obispo) have developed a calculus-based introduction to probability and statistics. The course is intended for mathematics, statistics, computer science, economics, and engineering majors and attempts to combine data analysis, an emphasis on concepts, exploration of the mathematical underpinnings of the subject, and active learning.  It includes a variety of specific data analytic techniques, such as exploratory data analysis, confidence intervals, tests of significance, t-tests and intervals, regression analysis, contingency table analysis, analysis of variance, along with a broad introduction to fundamental statistical ideas, such as variability, randomness, distribution, association, transformation, resistance, sampling, experimentation, confidence, significance, power, and model.

Duke University offers a calculus-based course to introduce the concepts, theories, and methods of statistical modeling and inference for mathematics majors. Its goal is to explore the foundations of scientific reasoning and inference and arouse curiosity through applications in medicine, genetics, policy, astronomy, physics, economics, finance, and education, among others. Students also learn to use statistical computing software.

The course based on Stat Labs: Mathematical Statistics Through Applications emphasizes reasoning and is suitable for mathematics majors. It was developed by Deborah Nolan and Terry Speed, University of California at Berkeley and is described in Part I, Recommendation 4 of the CUPM Curriculum Guide 2004.

Reading, Writing, and Speaking Mathematics

Several people have written guides for college students about writing mathematics:
   * A Guide to Writing Mathematics by Dr. Kevin Lee, Purdue University Calumet: This is an attractively written article with good advice.
   * A Guide to Writing in Mathematics Classes by Dr. Annalisa Crannell, Franklin & Marshall College: This guide has been widely used in classes throughout the country. Among other things, it contains a checklist for students to use as they write answers to problems.
   * How To Write a Solution by Richard Rusczyk & Mathew Crawford: This article (actually written for advanced high school students preparing for mathematical competitions) gives examples of bad and good ways to write solutions to challenging mathematical problems. Reading through the two versions gives a vivid sense for why the good solution is better than the bad one. The authors have tried to make the presentation amusing as well as helpful. (Table of Contents: Have a Plan; Readers Are Not Interpreters; U s e   S p a c e;  sdrawkcaB knihT, Write Forwards; Name Your Characters; A Picture Is Worth a Thousand Words; Solution Readers, not Mindreaders; Follow the Lemmas; Clear Casework; Proofreed; Bookends)
   * A Short Guide to Writing Mathematics (a guide for undergraduates) by Stephen B Maurer, Professor of Mathematics, Swarthmore College:  This link describes a book about mathematical writing that can be downloaded from Professor Maurer. It also contains links to a few freely available sections of the book: Advice on Note Taking, Common Work Errors in Writing Mathematics, and a complete table of contents for the book.
   * Writing a Research Paper in Mathematics by Ashley Reiter, Mathematics Instructor, Maine School of Science & Mathematics: This article contains advice about writing a research paper in mathematics. 

Fernando Guvea, Colby College, has developed a self-evaluation checklist for students to use before handing in a mathematical paper.

Mount Holyoke College) requires all mathematics majors to take the sophomore-level Laboratory in Mathematical Experimentation, a course in which students typically write six 10-page reports on mathematical experiments.  The link contains a description of the course and the table of contents and first two chapters of a book that was developed for it.

Six week-long writing assignments are a main learning and assessment tool in some versions of the sophomore-level linear algebra course at New Mexico State University. Writing is also emphasized in some versions of the linear algebra course at Illinois State University.  Descriptions of both are in Part 2, Section B2.

In the fall semester at Macalester College mathematics seniors attend sessions to prepare for the capstone seminar in which they receive general advice about how to write their senior project report. Each student also has a faculty advisor to help guide his or her work. According to David Bressoud, seminar meetings “address issues such as using information resources, tips on TeX or other mathematical typesetting, and how to use figures and diagrams effectively. In the meantime, students are working individually with their advisers.” Students receive feedback on their first draft and produce a second, which is reviewed with one more chances for revision before the final submission date. In exit interviews with Macalester students, “all have praised the capstone seminar as a positive experience, several have described it as the most important part of their undergraduate careers, and many have said how important it was during employment interviews.”

At the University of Redlands, mathematics students take a senior research seminar during which they write a formal research proposal, at least three preliminary research reports of 3-5 pages each, an abstract for a formal research presentation, and at least two 20- to 30-page drafts of their research paper.

Carlton College requires every major in the fall or winter of the senior year to “give a public presentation on an assigned mathematical topic. He/she will have a total of five weeks to prepare the talk. The student will give a private version of the talk at the three week point to a committee of two faculty members. This presentation gives the student a chance to rehearse the talk and to receive feedback and suggestions from the committee. ...  Each major must attend twelve other comps talks during his or her junior and senior years.”   The Oral Comps website contains an extensive set of preparation material, much of which was adapted from “Giving Oral Presentations in Mathematics,” by Deborah S. Franzblau (PRIMUS, March 1992, Vol. II, no. 1), “How to talk mathematics” by Paul R. Halmos (Notices Amer. Math. Soc., 21(3):155–158, 1974) and Handbook of Writing for the Mathematical Sciences by Nicholas J. Higham, SIAM, Philadelphia, PA, 1993, Second edition 1998.)

Additional information about creating and assessing writing assignments is in Part 1, Section 2.

Evaluating Oral Presentations

According to Pam Crawford, speaking in the 2003 MAA session “Helping Students Give Effective Mathematics Presentations,” mathematics majors at Jacksonville University satisfy the university requirement of a speaking-intensive course in their major by taking a history of mathematics capstone course in which they make two general-audience presentations (10 minutes each) and also report orally on a term paper  (30 to 45 minutes). Because the presentations are most students’ first experience of speaking about mathematics for a lengthy period of time, the course distributes handouts with advice. One is “Giving Oral Presentations in Mathematics,” by Deborah S. Franzblau (PRIMUS, March 1992, Vol. II, no. 1).
       Crawford stated that one of the most effective methods for helping students internalize presentation advice is to make them evaluators of other speakers.  For classroom presentations, the students in the class as well as the instructor fill out evaluation forms. Students are also required to evaluate at least three of the speakers in the Science & Engineering Lecture Series, a weekly luncheon seminar with faculty presentations.
       The evaluation forms used by the department were designed in consultation with a speech professor and are available at Crawford’s website.  They include rating the presentation in various categories as well as giving an overall rating for a talk, offering an opinion as to what the speaker did most effectively, and providing a suggestion for how the presentation could have been improved. When students make their in-class presentations, the evaluation scores they receive for each category are averaged and reported to them privately. They also receive anonymous comments from classmates regarding the strong and weak points of their presentations. The average-overall-presentation-rating determines the student's grade for the presentation, and students meet individually with the instructor to discuss their own view of the presentation and their response to the comments and suggestions for improvement. Crawford reported a noticeable improvement in quality of presentations over the course of a semester and stated that several former mathematics majors have returned to campus to tell the department that employers were impressed with the quality of their presentation skills.

Oral presentations are also required in a number of the courses at Keene State College, where faculty identified students’ ability to communicate mathematics effectively through oral presentations as an important learning goal. For example, students in an introductory statistics course make brief but formal presentations on group projects, and students in most upper-level mathematics courses make longer presentations of their project work. The presentations by upper-level students are sometimes made not only before the students’ peers, but also before the mathematics faculty as part of a weekly seminar program. Additionally, students have made presentations outside the department at a college-wide Academic Excellence Conference, MAA Northeastern Section regional meetings, and the Hudson River Undergraduate Mathematics Conference.  For further information, including the guidelines that are used for making presentations and grading rubrics, see the assessment study by Richard Jardine and Vincent Ferlini. 

Additional information about developing mathematical thinking and communication skills is in Part 1, Section 2.

 

All  majors should  have experiences with a variety of technological tools, such as computer algebra systems, visualization software, statistical packages, and computer programming languages.

See Part 1 Section 5, for general information about technology resources and examples of how technology is being used in undergraduate mathematics courses, especially in lower-division courses.

Resources for the Use of Technology

 

The Electronic Proceedings of the International Conference on Technology in Collegiate Mathematic (EPICTCM) contains articles about a wide variety of topics related to the use of technology in the classroom, a number of which concern upper-level courses. The articles are indexed by year, by author, and by keyword. In particular, links from the main page lead to various mathematical topics and also to topics identified by type of software. For instance, the following are links to articles from the EPICTCM about the use of CAS packages in mathematics courses:
   * MATHEMATICA in calculus, number theory, numerical methods and other courses;
   * Maple in calculus, group theory, analysis, linear algebra, complex variables, and other courses
   * MATLAB in calculus, linear algebra, and other courses. 

Another general source of information is The Math Forum Internet Mathematics Library. The Library contains links to a very large number of technology resources for topics in modern algebra , real analysis, complex analysis,  geometry, topology, number theory, discrete mathematics, probability, statistics, numerical analysis, dynamical systems, history and biography, logic and foundations, and others.

A number of software programs are widely used in geometry courses for prospective teachers: the Geometer's Sketchpad, Cabri-3D, Cinderella  and GeoGebra (which is free). Flash applications illustrating straightedge-and-compass constructions of a perpendicular bisector, parallelogram, rectangle, rhombus, square, circumcircle, and incircle  are available on the website of R. Dassonval. The text is in French but the animations can be viewed simply by clicking on the start arrows.

The Journal of Online Mathematics and its Applications (JOMA) contains articles, modules, mathlets, and reviews, many of which involve uses of technology in mathematics education. For example, the article Technology in the Upper-Level Curriculum by Ellen J. Maycock, DePauw University, describes her use of technology in several upper-level mathematics courses: abstract algebra, real analysis, and geometry, and gives references for each. In her conclusion Maycock states: “Instead of working through one example with paper and pencil in the course of an hour, the student can generate six or eight with the computer – and with the dynamic geometry software, thousands. Patterns can emerge from the examples. Students are much more able to see the concepts behind the formalism and the theory. . . Primarily, however, the lab experiences changed the dynamics of the courses. A carefully constructed syllabus became a hindrance for each course – the unpredictability of the lab experience meant that I had to be prepared to discard my lesson plans for the day and respond to their comments and questions. I had to ask myself what my basic goal was in each class and be flexible about whether a list of theorems could be covered – I refocused on a sparse collection of fundamental concepts in each course. Students felt empowered by their own discoveries, and they began to provide at least as much energy to the classroom as I did.”

The MAA Online website contains a review by Andrew B. Perry, Springfield College, of the book Innovations in Teaching Abstract Algebra, ed. by Allen C. Hibbard and Ellen J. Maycock (MAA Notes, volume 60, Mathematical Association of America, 2002). Perry writes: "Many of the papers in this volume describe the author's experience teaching with a particular software package, with six packages represented in at least one paper: Finite Group Behavior (FGB), ISETL, GAP, MATLAB, Maple, and Mathematica. There is enough description of each software for an instructor to get a sense of whether he or she might profitably include the software in his or her course. The editors have thoughtfully included information on obtaining any of the software described in the book.” A website has been created with abstracts of all the articles in the book and sources for all materials, software, and websites referenced in the book’s articles.

Another review in MAA Online, by Mihaela Poplicher, University of Cincinnati, of the book Multimedia Tools for Communicating Mathematics: Compression, Simplification, and Multiresolusion, edited by Jonathan Borwein, Maria H. Morales, Konrad Polthier, and José F. Rodrigues, contains several links to papers illustrating the use of technology in geometry, linear algebra, topology, graph theory, and the history of mathematics.

A PowerPoint presentation by Matthias Kawski, Arizona State University, discusses the use of technology as a tool to visualize mathematical concepts. Kawski's webpage provides information, including technology links, for integrating technology into over 18 courses, including various levels of calculus, differential equations, linear algebra, analysis, control theory, and other advanced mathematics.

A variety of statistical packages are used in upper-level undergraduate courses. They may be incorporated into a year-long course in probability and statistics, or offered as a laboratory accompaniment to such a course, or given as a stand-alone course in statistical software. SAS, Minitab, SPSS, or BMDP are widely used commercial programs, and the statistical software package R, which is open-source freeware, appears to be gaining broader acceptance.  The archive of the online Journal of Statistics Education is searchable and contains all articles published since 1993. The UCLA Stat Computing Portal contains links about statistical computing using SAS, Stata , SPSS , and  S-Plus, and R, as well as other statistics resources. The Electronic Proceedings of the International Conference on Technology in Collegiate Mathematic (EPICTCM) also has links to articles about the use of the statistical package Minitab.

Gordon Royle, University of Western Australia, compiled a Combinatorial Catalogue, which gives detailed information about specific graphs, geometries, designs, and groups.

John Stembridge, University of Michigan, constructed a home page consisting of Maple Programs, which provide an environment for computations involving symmetric functions, partially ordered sets, root systems, finite Coxeter groups, and related structures.

Daniel R. Grayson, University of Illinois at Urbana-Champagne and Michael E. Stillman, Cornell University, developed Macaulay 2, a package that supports work in algebraic geometry and commutative algebra.

Using a Computer Language

The Electronic Proceedings of the International Conference on Technology in Collegiate Mathematic (EPICTCM) contains a number of articles that link programming to the teaching of mathematics.  Examples include “Implicit Differentiation on the TI-92+ Calculator as an Illustration of Some Powerful Programming Features,” “Using Microsoft COM for Complex Variables,” and “Using Visual BASIC to Create a Graphical User Interface for Matlab.”

Ed Dubinsky and others have produced versions of courses from calculus and discrete mathematics to abstract algebra that use programming in the free software language ISETL to try to lead students to understand sophisticated mathematical concepts. Resources include the book Learning Abstract Algebra with ISETL by Ed Dubinsky and Uri Leron (Springer-Verlag, 1994) and the article Discovering Abstract Algebra with ISETL by Ruth I. Berger in the volume Innovations in Teaching Abstract Algebra, ed. by Allen C. Hibbard and Ellen J. Maycock (MAA Notes, volume 60, Mathematical Association of America, 2002).

Many colleges and universities require mathematics majors to take a course in a specific programming language or in general principles of computer science. At present, the most frequently used languages used in such courses are Java and C++, although Pascal and Fortran meet the requirements at some institutions. 

All majors should have significant experience working with ideas representing the breadth of the mathematical sciences.  In particular students should see a number of contrasting but complementary points of view:

• Continuous and discrete,
• Algebraic and geometric,
• Deterministic and stochastic,
• Theoretical and applied.

Majors should understand that mathematics is an engaging field, rich in beauty, with powerful applications to other subjects, and contemporary open questions.

A General Resource

The site www.geometry.net/math.html has a large number of links to mathematics resources. Major headings are Mathematics Sites, Mathematics Books and Pure Mathematics. Under each heading are twelve to twenty additional subheading links to again a large number of index links. For example under Mathematics Sites and then Pure and Applied Mathematics, there are 96 subject indexes from Abstract Algebra to Wavelets. Under Abstract Algebra there are 119 links to various books, publications, course offerings and information web sites. The book entries are linked to a commercial provider, but the other resource links connect to institutions and individual sites. Within each index there is a search option.

The website Innovative Mathematics Majors in Small/Medium Departments contains summaries of the talks given at the session of the same name at the 2005 MathFest.

The Massachusetts Institute of Technology MITOpenCourseWare is a free publication of course materials used at MIT. Materials in the mathematics section include syllabi, lecture notes, problems sets, exams, etc., for a large variety of courses.

Discrete Mathematics and Data Analysis

The United States Military Academy has devised a curriculum in which all students, including mathematics majors, take data analysis and discrete mathematics in the first two years.

At California Polytechnic University at San Luis Obispo, the Mathematics B.S. Curriculum requires Methods of Proof in Mathematics, Combinatorics, and either two courses in statistics or the combination of one in statistics and one in probability.

 

At the State University of New York at Oswego, the mathematics major requires a course in discrete mathematics and a course in statistics.

The Smith College A Guide to Mathematics at Smith recommends discrete mathematics as a first mathematics course for students who have taken a year of calculus in high school, and the discrete mathematics course is one of the three courses listed as an “entryway requirement” for the major. The other courses are linear algebra and multivariable calculus. An alternative suggested first mathematics course for students entering with a year of calculus is introduction to probability and statistics. While not helping to meet the entryway requirement, this course would count as part of a mathematics major for a student who concentrates in statistics.

At Oberlin College, the Handbook for Mathematics Majors recommends that mathematics majors use the discrete mathematics course as a bridge between the two-semester introductory calculus sequence and the courses in multivariable calculus and linear algebra. All three courses are required for both the pure mathematics and the applied mathematics concentrations.

At St. Cloud State University mathematics majors are required to take two courses in discrete mathematics. The first consists of logic, proof, mathematical induction, finite and infinite sets, relations, functions, introduction to number theory, and the second includes basic counting techniques, permutations and combinations with and without repetitions, binomial and multinomial coefficients, inclusion-exclusion, pigeonhole principle, recurrence relations, generating functions, complexity of algorithms, introduction to graph theory.

The Laboratory in Mathematical Experimentation at Mount Holyoke College (see also the listing under Part 2, Section C2) is used as a bridge course for all majors. It consists of a half -dozen two-week explorations chosen from among more than sixteen.  The choice of projects varies from year to year and is drawn from algebra, analysis, discrete mathematics, geometry, and statistics. Often the choice includes one on graph-coloring and one on randomized response surveys, so both the discrete and stochastic are represented. Introductory discrete mathematics and statistics courses are available to majors, but relatively few take them. Most majors who enroll in a data-based statistics course take Applied Regression, for which the prerequisite is either linear algebra or an introductory statistics course.

At Albion College, all mathematics majors are required to take discrete structures.  Pure mathematics majors must take either probability and statistics or mathematical modeling, applied mathematics majors choose two courses from a list of twelve, of which two are in statistics and one is in mathematical modeling, and students interested in teacher certification choose two courses out of three, one of which is mathematical statistics and another of which is mathematical modeling.

Geometry and Geometric Thinking

The Geometry in Action, developed and maintained by David Epstein, University of California Irvine, offers information and links for a large number of sites involving real-world applications of ideas from discrete and computational geometry. While most of the site is devoted to applications, some general mathematical techniques are also included. The major topic categories are Geometric References and Techniques, Design and Manufacturing, Graphics and Visualization, Information Systems, Medicine and Biology, Physical Sciences, Robotics, and Other Applications, with each category consisting of approximately ten subcategories. For items not fitting into one of the categories, Epstein has a Geometry Junkyard site, which contains usenet clippings, web pointers, lecture notes, research excerpts, papers, abstracts, programs, problems, and other material related to discrete and computational geometry.

The Creative Visualization Labs session at the 2003 Joint Mathematics Meetings, organized by Cathy Gorini, Sarah Greenwald, and Mary Platt, invited papers describing a complete lab or series of labs using computers, technology, dynamic software and/or manipulatives aimed at increasing visualization skills. Ten of the thirteen papers that were presented are posted on the website, and all thirteen have abstracts and contact information. Sample titles include: Computer Activities for College Geometry; Walking, Folding, and Computing to Visualize Geometric Concepts; Making the Transition from Euclidean to Non-Euclidean Geometry Through Exploration; and the Spherical Geometry Project.

Geometry in the Undergraduate Syllabus contains a 1993 report by a working group in Europe that considered the role of geometry in the undergraduate curriculum.  The sections of the report are Introduction, Geometry for its own sake, Geometry and Algebra, Exploring Geometry, Geometry, Logic and Language, and Geometry and Planetary Motion.

The website of Joe Malkevitch, City University of New York, contains many links having to do with mathematics education.. In particular, it has a link to his article Geometry in Utopia, which links to a set of problems and also contains a bibliography of geometry resources organized into categories. Malkevitch’s website also links to his Mathematical Tidbits page, which contains links to short sets of notes he has used for talks and classroom presentations.

Several recent textbooks offer materials for integrating geometry and geometric thinking into the curriculum. The Geometry of Spacetime: An Introduction to Special and General Relativity by James Callahan uses the framework of spacetime geometry to explore Einstein’s special and general relativity theorems. The book includes:  relativity before 1905, special relativity—kinematics, special relativity—kinetics, arbitrary frames, surfaces and curvatures, intrinsic geometry, general relativity, and consequences. Background needed is linear algebra, multivariable calculus, and familiarity with the physical applications of calculus.

The book Geometry by David A. Brannan, Matthew F. Esplen, and Jeremy J. Gray, The Open University, United Kingdom, adopts the Klein approach, viewing a geometry as a set with an associated group of natural transformations. One then studies those properties left invariant by the transformation groups. The book assumes basic knowledge of group theory and linear algebra. An example of a syllabus for a course using the book is by Zheng-Chao Han, Rutgers University.

 The book Continuous Symmetry: From Euclid to Einstein by Roger Howe, Yale University, and William Barker, Bowdoin College, is currently available in manuscript form and will be published in two volumes. The first volume focuses on a coordinate-free approach to two- and three-dimensional Euclidean geometry based on symmetry transformations. The second introduces coordinates and ends with the study of the geometries of space-time. The over-riding principle of the two volumes is that Felix Klein's approach to geometry, known as the Erlangen Program, should be taken seriously: geometry is the study of invariants under symmetry groups, and different geometries are obtained by varying the choice of symmetry groups. The manuscript has been used at Wesleyan and Yale Universities and at Bowdoin, Colby, and Rockhurst College, among others.

Links to additional resources for geometry include David Royster’s  Hyperbolic Geometry Links page, Michael Reid's Polyomino page, Torsten Sillke’s Tiling and Packing results, John C. Polking’s The Geometry of the Sphere page,  and The Geometry Center’s Tessellation Resources and  Geometric dissections on the web pages.

Examples of geometric thinking and visualization outside geometry courses include the Bridge Project at Oregon State University, which encourages geometric visualization as a problem-solving technique in vector calculus.

See also the resources listed under visualization in Part 1, Section 5 and the geometry resources in Part 2, Section D.1. 

Statistics and Probability and Data Analysis

See Part 2, Section C.1, Reasoning with Data: Probability and Statistics

Linkages – Algebra and Discrete Mathematics

Algebra and discrete mathematics encompass theoretical and applied aspects of mathematics that are foundational for matrix analysis, modern algebra, number theory, combinatorics, and graph theory.  They have significant impact on applications arising in statistics (linear models, experimental designs), probability (random models), operations research (mathematical programming, network analysis), communication engineering (coding theory, cryptography), and computer science (analysis of algorithms, nonnumerical computing). One result is the possibility to make connections among these mathematics topics in undergraduate courses. Some books that can be used as resource material are From Error-Correcting Codes through Sphere Packing to Simple Groups, by Thomas M. Thompson, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture by David Bressoud , Algebra and Tiling by Sherman Stein and Sandor Szabo, Identification Numbers and Check Digits Schemes by  Joseph Kirtland, Introduction to the Theory of Error-Correcting Codes by Vera Pless, A First Course in Coding Theory by  Raymond Hill, and Elements of Algebraic Coding Theory by L. R. Vermani.  

Examples of courses that link algebra and probability are Discrete Markov Chain Monte Carlo by George Cobb, Mount Holyoke College, which involves a mix of elementary graph theory, probability, and linear algebra, and Stochastic Processes courses, such as those at University of California Berkeley and Arizona State University. Additional information about George Cobb’s course is contained in the article An Application of Markov Chain Monte Carlo to Community Ecology by George Cobb and Yung-Pin Chen.

Another way to encourage students to become aware of connections among mathematical topics is through a regular schedule of seminar or math club talks. For instance, each semester, the Clemson University mathematics department sponsors an Algebra and Discrete Mathematics Seminar to which undergraduate students are invited. In Tulane University’ Senior Seminar students are required to attend talks in the Student Seminar of the department and to prepare a talk to deliver there themselves. One purpose of the talks is to broaden students’ experience as mathematics majors and to make them aware of parts of mathematics that they may not have been exposed to in their classes. Another goal of the Senior Seminar is to improve their oral and written expression about mathematics. A large number of the articles recommended to students as bases for their presentations are on topics that link algebra and discrete mathematics.

Linkages – Algebra and Geometry

The emergence of computational methods in algebraic geometry led to interactions with a number of other areas, such as combinatorics, optimization, statistics, and splines. The 1998 workshop: Algorithmic Algebra and Geometry: Summer Program for Graduate Students of MSRI Sponsoring Institutions sought to introduce participants to these ideas and topics. Ideals, Varieties and Algorithms (Cox, Little and O’Shea, 1997), an undergraduate text with a focus on computational methods, was the prerequisite for the main lectures. The book discusses systems of polynomial equations ("ideals"), their solutions ("varieties"), and how these objects can be manipulated ("algorithms"). This workshop also introduced the use of specialized computer algebra systems such as Macaulay, Macaulay2, and GAP, which are aimed at these problem domains.  This text has also been used in the Seminar in Mathematics (Elementary): Algebraic Geometry from an Algorithmic Point of View at the State University of New York – Stony Brook and in the Groebner Bases course taught by Rekha R. Thomas at the University of Washington .

Field Theory and its Classical Problems by Charles Hadlock (1978) begins with the geometric construction problems of antiquity, continues through the constructibility of regular n-gons and the properties of roots of unity, and then moves on to the solvability of polynomial equations by radicals, and. beyond. The logical pathway is historic, but the terminology is consistent with modern treatments. No previous knowledge of groups, fields, or abstract algebra is assumed. Notable topics treated along this route include the transcendence of e and of pi, cyclotomic polynomials, polynomials over the integers, Hilbert's, irreducibility theorem, and many other gems in classical mathematics. Historical and bibliographical notes are provided, as are complete solutions to all problems.

The MathWorld site provides some historical background to geometric construction and algebra. The site links the basic terms used in this description to definitions and explanations.

Linkages – Number Theory and Geometry

One part of the 2001Park City Mathematics Institute (PCMI) was a course on the Euclidean algorithm and its applications to algebra and the theory of numbers. A complete set of course notes, including a section on curve fitting, can be downloaded from the website.

The Geometry of Numbers by C.D. Olds, Anneli Lax and Giuliana Davidoff presents a self-contained introduction to the geometry of numbers, which begins with lattice points on lines, circles and inside simple polygons in the plane and gradually leads up to the theorems of Minkowski and others who succeeded him.
 
The Geometry Junkyard site (described earlier) includes a section called Lattice Theory and Geometry of Numbers, which provides many references and resources addressing connections between number theory and geometry.

The part of David Rusin’s Mathematical Atlas site (described earlier) that is devoted to number theory contains information and web links for connections between number theory and many other branches of mathematics.

Linkages – Complex Variables and Geometry

With computer graphics programs, one can now visually explore the geometry of the complex plane and various mappings of one or more complex variables. A website from The Geometry Center displays several such images, and the Living Mathematics Project, hosted at SunSITE University of British Columbia, has an applet that allows one to experiment with the behavior of certain functions of a complex variable. Other applets on the site encourage exploration of flows of vector fields, Bessel functions, and Fourier series, among other things.

While somewhat advanced to use as a primary text for undergraduates, Complex Analysis: The Geometric Viewpoint by Steven G. Krantz is frequently cited as supplementary reading for undergraduate courses. Krantz explains the role of Hermitian metrics and of curvature in understanding the Schwarz lemma, normal families, Picard's theorems, conformal mappings, and many other topics. 

Linkages – Probability and Analysis

At Duke University, Greg Lawler’s Real Analysis course supplements the primary text with Measure Theory and Probability by Malcolm Adams and Victor Guillemin. and Probability and Measure by Patrick Billingsley. The prerequisites for Lawler’s Stochastic Processes course are a calculus-based undergraduate probability course and a course in linear algebra.

Additional resources about connections between probability and analysis are on Dave Rusin’s Mathematical Atlas pages on Probability Theory and Stochastic Processes.

Powerful Applications and Contemporary Questions

A number of areas of mathematics have led to important contemporary applications. These include, among many others, cryptography as an application of number theory, error-correcting codes as an application of algebra, computer graphics as an application of linear algebra, projective geometry, and splines, and robotics as an application of computational algebraic geometry.

Allen Broughton’s Mathematics of Image Processing course at Rose-Hulman Institute of Technology covers the mathematical basis of many of the ideas behind image processing such as filtering, filter banks, the discrete Fourier and cosine transforms and the discrete wavelet transform. The theory is balanced by concrete applications to various image processing problems with a special emphasis on image compression.

Breadth of Mathematics and Connectedness to other Disciplines

The department of mathematics at Hiram College requires a Correlative Experience of all mathematics majors, which is intended to emphasize the applications of mathematics and its connectedness to other disciplines.  The experience must involve significant use of mathematics but must take place outside the mathematics department’s course offerings.  It may take a variety of forms and may be tailored to meet the goals of the individual student.  Examples of ways in which a student may satisfy the requirement include approved coursework in a field outside mathematics, an approved summer research experience, an approved internship, or two years’ participation in the Mathematical Contest in Modeling together with public presentations of solutions.  

 

Victor Katz, University of the District of Columbia, and Karen Dee Michalowicz, the Langly School, edited Historical Modules for the Teaching and Learning of Mathematics, which are available on CD. The modules are collections of lesson materials designed to demonstrate the use of the history of mathematics in the teaching of mathematics. They are intended for use in both college mathematics courses, especially those for prospective teachers, and the K-12 classroom.

Broader and More Flexible Major

At St. Olaf College students who major in mathematics are encouraged to structure their program to be broad and flexible, arranging their programs through individual contracts that include both course work and independent mathematical activity. Contracts normally include seven courses beyond linear algebra, balanced so as to represent analytic, axiomatic, and applied mathematics. Independent activities include tutoring, attending colloquia, grading papers, computer consulting, internships, and undergraduate research. 

The Mathematical Sciences program at the University of Michigan is designed to provide broad training in basic mathematics together with some specialization in one of eight areas of application. Options include discrete and algorithmic methods, numerical and applied analysis, operations research and modeling, probabilistic methods, mathematical economics, control systems, mathematics of finance and risk management, mathematical physics, and mathematical biology.

There are four undergraduate programs in mathematics at the Massachusetts Institute of Technology. The first three lead to the degree Bachelor of Science in Mathematics (General Mathematics, Theoretical Mathematics, and Applied Mathematics), and the fourth to the degree Bachelor of Science in Mathematics with Computer Science.  There is also an option that allows students, with the help of their advisors, to design their own programs. This self-designed option is particularly popular with students who plan to combine their mathematical studies with an in-depth exploration of another field, such as economics, physics, or business.  In 2002, 76 students completed the BS in Mathematics and 3 the BS in Mathematics and Computer Science. The General Mathematics option requires only differential equations plus eight 12-unit electives, at least six at an advanced level.  Gilbert Strang observed that while his colleagues knew how many students chose the major, it has little knowledge of why students chose (or didn't choose) to study mathematics. As an experiment, he sent five questions [see below] to the MIT math majors and got 50 answers that same evening. In reporting on his experiment he wrote:
     I can summarize some of the answers.  I found them thought-provoking -- just giving the students a chance to express their thoughts is good for them and good for the faculty.  Here are three ideas that came through in the replies:
    (1)  Having flexible requirements is extremely attractive.  MIT has a lot of double majors because we don't require a specific list of courses (as engineering departments tend to do).  Mathematics is an ideal subject for a less rigid curriculum, especially as a very high proportion of majors go on to work or to graduate school in other fields.
    (2) Statistics is a key course particularly for students who are thinking ahead to their future career.  (By far the most frequent reply to Question 2 was "I just like mathematics" but a significant number want to know how mathematics is applied.  That group mentioned statistics.)
    (3)  Find a way to introduce ‘real mathematics’ or ‘cool mathematics’ early enough to balance the effect of the large calculus courses.  Make it known that mathematics is different, and make it easy for prospective majors to speak with faculty.  ‘Get people hooked....’
    QUESTIONS:
    (1) When did you first know that you wanted to major in mathematics?  (age 12, before college, freshman year, sophomore, junior, senior)
    (2) Which TWO of these were most important in your decision to major in math?
    (a) You just like mathematics.  (b) You like the freedom in the mathematics major to take courses in other fields. (c) You see mathematics as good preparation for your (different) career. (d)  Particular math course(s) (which?) (e) Particular people (teachers, family, advisers, ...)
    (3) Do you plan to go to graduate school, in mathematics or another subject?  Do you have a specific career in mind, and did that enter heavily into your choice of major?
    (4) FOR SENIORS: Did you apply to graduate schools in mathematics (how many?)
    (5) What could mathematics departments do to encourage students to major in mathematics?  (This is an open-ended question about mathematics here and everywhere.)

 

Mathematical sciences departments should require all majors to:

• Study a single area in depth, drawing on ideas and tools from previous coursework and making connections, by completing two related courses or a year-long sequence at the upper level;
• Work on a senior-level project that requires them to analyze and create mathematical arguments and leads to a written and oral report.

Pairs of Courses

See ideas for linked courses in Part 2, Section C.3.

Capstone Courses and Projects

The University of Redlands Department of Mathematics has required a Senior Research Seminar of its mathematics majors for over 40 years. The major assignment for the course is a (usually expository) research project in an area of mathematics selected by the student. Primary emphasis is placed on improving students' independent study, problem solving, research, reading, writing and oral presentation skills. An unwritten objective is to provide peer support for students as they prepare to make the transition to graduate school and/or a career. The seminar homepage has links to a great deal of information on project guidelines, recent topics, etc.

The Macalester College mathematics and computer science degree requirements include a capstone project involving a written report and an oral presentation. All presentations are given on a single capstone day during which regular mathematics and computer science classes are canceled. The students in those classes are required to attend at least one of the capstone presentations. Because of canceled classes and required attendance, the seniors speak to large audiences. A capstone seminar is designed to prepare seniors to communicate their results effectively. Detailed information about the seminar is now on a restricted website, but a previous site lists examples of past capstone projects, and another has links to information on expectations, topics, etc.

The Readings in Mathematics course serves as the capstone experience at Bellarmine University. The required text is a volume of What’s Happening in the Mathematical Sciences, but students are expected to do library research as well. Three short papers and one long paper are required. In the first paper two students work together to summarize a chapter from the text and present the topic orally to the class. In the second each student is to reflect on the mathematics courses taken at Bellarmine and how they have addressed the five goals of the Mathematics Department. In the third paper each student is to write a short history of a topic in mathematics and also present the topic to the class. In the final paper, 10-15 pages long, the student is to explore in depth a topic in mathematics and present the topic orally to an audience of senior mathematics majors.

The Hiram College mathematics department requires each major to complete a capstone project in a senior seminar. Students are expected to undertake significant independent learning in an area not included in the standard undergraduate curriculum; gain experience in locating, reading, and interpreting mathematical literature; and gain experience in communicating mathematical ideas with clarity and precision, both orally and in writing. Students select a topic and begin research in the Junior Seminar, at the end of which each student submits a project proposal that must include at least five annotated references. Students complete their projects during the Senior Seminar. They turn in three preliminary drafts of their paper and have several practice oral presentations before submitting the final paper and making a public oral presentation at the end of the spring semester. For more information, contact the chair, Virginia (Jimmy) Buchanan.

At Portland State University, the faculty in the department of mathematics and statistics used a student survey to assist in designing a senior-level capstone experience for their students. A report discussed students’ responses to the survey questions. One consequence of the process was the decision to make the course satisfy the university’s capstone requirement, the courses for which are listed under “university studies” rather than mathematics and involve both working on a community project and linking study in the major to students’ broader programs. Thus, for example, the decision was made to include presentations to inner city high school students as part of the course requirement.

The Senior Exercise at Kenyon College is designed “to engage each senior in the exploration and communication of mathematical ideas beyond material covered in courses taken, or to be taken, by the student.” It includes two components: (1) students take the Major Field Test; (2) they study independently and write a paper on a topic of their choice. The website gives detailed information to the student on expectations at each step of the project.

Some schools that have instituted senior level capstone courses are using the courses to help assess and improve their programs. For example, see “An Assessment Program Built Around a Capstone Course,” by Charles Peltier, St. Mary’s College. The webpage for St. Mary’s Senior Comprehensive Project contains information about the capstone course, called Proseminar, and the more general project of which it is a part.  Another example illustrating how a capstone course may be used to assess the effectiveness of the mathematics major is “Using a Capstone Course to Assess a Variety of Skills,” by Deborah A. Frantz, Kutztown University.

One resource for capstone course project ideas is the Math Pages website. It contains links to articles on a variety of topics, with each topic heading linking to as many as 50 or more articles. To give a sense of the website, a couple of titles are given with each of the following topic headings: combinatorics and graph theory (The Four Color Problem and The Dartboard Sequence), geometry (Sphere Packing in Curved 3D Space and Heron's Formula and Brahmagupta's Generalization), probability and statistics (The Gambler's Ruin and Biased and Anti-Biased Variance Estimates), number theory (Fermat's Last Theorem for Cubes and  Quadratic Congruences), set theory and foundations (Fractal Logic and Reconstructing Brouwer), calculus and differential equations (Curvature: Intrinsic and Extrinsic and Series Solutions of the Wave Equation), and history (Zeno and the Paradox of Motion and Legendre's Prime Number Conjecture). In addition, the website has articles on relativity, physics, music, animated (Java) illustrations, and a list of quotations. 

Mathematicians should collaborate with colleagues in other disciplines to create tracks within the major or joint majors that cross disciplinary lines.

Joint Majors

UCLA has a joint major in mathematics and applied science, which is intended for students who are interested in mathematics but also have a substantial interest in the applications of mathematics to other areas. Options include actuarial mathematics, management and accounting, medicine and life sciences, history of science and an individual plan subject to approval by the undergraduate vice-chair. UCLA also has a joint major in mathematics and economics. Courses are quarter courses, and the department majors require a list of preparatory course plus 13 upper-division courses.

Brown University has several interdepartmental concentrations for undergraduate students in the department of mathematics and the division of applied mathematics. The AMS book Towards Excellence (p. 143) states: “Current standard concentrations include: mathematics-computer science, mathematics-economics, mathematics-physics, applied math-biology, applied math-computer science, applied math-economics … and applied math-psychology … Some of these programs are small and geared to preparation for advanced study. Others are large (mathematics-economics and applied math-economics) and have gained a reputation as excellent preparation for careers in business.”  Additional information for the applied mathematics program combinations is given in the Applied Mathematics Guide to Undergraduate Programs.

The University of Washington offers an interdepartmental major in applied and computational mathematical sciences. The AMS book Towards Excellence (p. 143) states: “The Departments of Applied Mathematics, Statistics and Computer Science recently worked together to create a new interdisciplinary undergraduate degree program … called the Applied and Computational Sciences (ACMS) degree program. … The program seeks to prepare its students to pursue a variety of positions in industry after graduation or to go on to graduate or professional school in many fields.”

The Interdisciplinary Mathematics Program at the University of New Hampshire allows students to complete a joint major in mathematics and one of five other disciplines, including computer science, economics, electrical science, physics, and statistics. The program is designed to prepare students for employment in various areas of applied mathematics, as well as for graduate work in these fields. Each interdisciplinary major option consists of ten mathematics courses and at least six courses in the discipline of the option. The statistics option requires eight courses in mathematics in addition to the core requirements.

The mathematics program in the Department of Mathematics and Computer Science at Ithaca College offers bachelor's degrees in mathematics-computer science, mathematics-computer science (teacher education), mathematics-physics, and mathematics-economics. The mathematics-economics degree, for example, requires 27-28 credits of mathematics/computer science and 27 credits of economics.

At Indiana University the Interdepartmental Major in Mathematics and Economics is designed to enable students to model economic questions mathematically, and to analyze and solve those models. Students take 7 courses in mathematics, 7 in economics, and at least 1 in statistics.

As Simmons College the Major in Financial Mathematics is offered jointly by the mathematics and economics departments. It is intended to serve students who are interested in applying the principles of mathematical and economic analysis in the financial services industry.

Harvey Mudd College (HMC) offers two joint majors that involve mathematics. The mathematics and computer science departments together administer a joint major in computer science and mathematics. Its purpose is to provide an integrated program of study for students who are interested in the interdisciplinary connections between computer science and mathematics. Depending on how electives are selected, the program would position joint majors for graduate study in either computer science or mathematics, or to enter the work force. The mathematics and biology departments together administer a mathematical biology major. This major prepares students for graduate study in either biology or applied mathematics or for employment in industry. HMC's technical core provides mathematical biology majors with a strong multidisciplinary foundation, and the college offers many opportunities for students to engage in interdisciplinary research in biomathematics and quantitative biology. The major is sufficiently flexible to allow students to concentrate in a particular area of mathematical biology. Every HMC student, regardless of major, takes the equivalent of four semesters of mathematics (calculus, multivariable calculus, linear algebra, and differential equations) as part of the general core curriculum. Fifteen courses beyond the HMC core are required for the joint major.

DePaul University has a Joint Mathematics-Computer Science major, which is intended to enable students to develop the necessary background to be able to work in areas that depend on knowledge from both fields. It is designed to prepare the student for graduate study in various areas of computer science such  as theoretical computer science, graphics, and computational methods and in areas in applied mathematics such as numerical analysis or discrete mathematics. It is also expected to be good preparation for the more intellectually demanding jobs in computer software development.

Rutgers University has a BioMathematics Interdisciplinary Major in which students do about half their coursework in biology and half in mathematics.  Mathematics requirements include single and multivariable calculus, introductory linear algebra, introduction to differential equations, probability and mathematical statistics, differential equations in biology, discrete and probabilistic models in biology, and one additional elective. Another joint major is Statistics/Mathematics to provide a stronger preparation for graduate study in statistics.

The program offerings at Utah State University include several “composite majors,” which allow students to choose from mathematics and statistics, mathematics and statistics education, mathematics and computer science, mathematics and electrical engineering, and mathematics and physics. Utah State also offers a minor in bio-mathematics.

Other schools with joint majors in mathematics and computer science include Drew University, New York University, University of Oregon, Yale University, University of California San Diego, Emory College (of Emory University), Massachusetts Institute of Technology (major is called mathematics with computer science), Middlebury College and the University of Illinois at Urbana-Champaign.

Other schools with joint majors in mathematics and economics include the University of Pittsburgh, Lafayette College, University of California San Diego, the State University of New York at Buffalo, New York University, Bowdoin College, Marquette University, and Mount Holyoke College.

Tracks Within the Major

Because only a small percentage of mathematics majors continue to graduate school in mathematics, a number of departments offer a range of tracks within the mathematics major to prepare students for careers in actuarial science, management consulting, financial planning, accounting, operations research, or medical professions. These normally include requirements in disciplines allied to mathematics.  For example, contracts developed by mathematics majors at St. Olaf College have included courses in economics, computer science, statistics, and biology.

The undergraduate program at the University of Chicago (click on “Concentrating in Math”) allows students to earn a bachelor’s degree in mathematics with a specialization in either computer science or economics. Both specializations are indicated on the student’s final transcript. Candidates are required to complete a yearlong sequence in calculus and in analysis, two quarters of abstract algebra, and a three-quarter sequence in either chemistry or physics. Computer science specialists take the third quarter of abstract algebra, an additional two mathematics courses, and seven courses in computer science. Economics specialists take probability, two additional mathematics courses, and eight courses outside the mathematics department, including one in statistics and seven in economics. Another option for students is the degree program in applied mathematics, in which candidates are required to take a three-quarter sequence in chemistry or physics plus an additional three-quarter sequence that is outside the mathematics department but within the Physical Sciences Collegiate Division.

The Actuarial Studies Program in the mathematics department at the University of Texas at Austin provides an opportunity for a student to earn a B.S or a B.A. degree in mathematics (actuarial option). In addition to the extensive general information available on the program’s website, news and information about the program is published each Spring semester in the Risky Business newsletter .

Smith College’s A Guide to Mathematics suggests sequences of mathematics courses tailored to students’ interests, whether in the sciences, computer science, economics, applied mathematics, theoretical mathematics, operations research, or teaching.

See also the report by Gilbert Strang, Massachusetts Institute of Technology, in Part 2, Section C.3.

 In order to recruit and retain majors and minors, mathematical sciences departments should:

• Put a high priority on effective and engaging teaching in introductory courses;
• Seek out prospective majors and encourage them to consider majoring in the mathematical sciences;
• Inform students about the careers open to mathematical sciences majors;
• Set up mentoring programs for current and potential majors, and offer training and support for any undergraduates working as tutors or graders;
• Assign every major a faculty advisor and ensure that advisors take an active role in meeting regularly with their advisees;
• Create a welcoming atmosphere and offer a co-curricular program of activities to encourage and support student interest in mathematics, including providing an informal space for majors to gather.

General References

Two useful references for information about what causes attrition among mathematics majors are the following: Talking About Leaving: Factors Contributing to High Attrition Rates Among Science, Mathematics, and Engineering Undergraduate Majors by E. Seymour and N. Hewitt, Bureau of Sociological Research, University of Colorado: Boulder, CO, 1994 and Talking about Leaving: Why Undergraduates Leave the Sciences by E. Seymour and N. Hewitt. Boulder, CO: Westview Press, 1997. (Reviews: 1  2)

A reference for what factors appear to retain mathematics majors is Math Education at Its Best: The Potsdam Model by D. K. Katta, Framingham, MA: Center for Teaching/Learning of Mathematics, 1993.

An article by Reuben Hersh in the Humanistic Mathematics Network Journal discusses the general issue of student retention in the context of general improvement of the educational experience for mathematics undergraduates. In the article Hersh quotes an MAA pamphlet from 1972 and the books by Seymour, Hewitt, and Katta that are mentioned above.

Designing Introductory Courses to be Effective and Engaging

The mathematics department at the University of Rochester increased the number of its mathematics graduates from 14 to 44 in three years.  A report written for the Rochester Review attributed part of the increase to changes that were made in the rigorous four-semester honors calculus sequence. While instituting the changes, faculty member Naomi Jochnowitz taught all four courses in the sequence. She told students that “math doesn’t close any doors; it only opens them,” and she encouraged “students to push themselves and to take on intellectual challenges, assuring them the faculty will be there to support them all the way.” In addition to three hours of class a week, students in the sequence are required to take a weekly two-hour workshop led by a teaching assistant. The purpose of the workshop is to get students interacting with each other while working on homework problems. Students report that the camaraderie which develops during the workshops continues into friendships outside of class.  The department provides an online Mathematics Survival Guide and has developed an online homework system, WebWork, which gives students instant feedback on their homework answers. 

For further information on effective and engaging teaching in introductory courses, see Part 1, Sections 3 and 5, and Part 2, Section A.

Encouraging Prospective Majors

The mathematics department of the State University of New York – Fredonia has created a course called Honors Problem-Solving. (Click the links for “Program” and “Courses.”) Each fall, just before course selection for the spring semester, a letter is sent to strong freshman calculus students telling them about the course and letting them know that they have been recommended by their calculus instructor for participation. The course is designed to help students learn how to attack a wide array of complex and open-ended problems, to work well as part of a team, to communicate proficiently with others about mathematical problems, and to appreciate mathematics as a discipline with exciting problems that can be fun to work on. Grades are based on class participation, write-up of problem solutions, and class presentations. One of the course goals is to keep good mathematics majors interested, and another is to encourage non-mathematics majors to take more mathematics. So during registration for the fall semester, students in the course are given information about mathematics majors and minors. The links from the department website for “Assessment,” “Alumni,” and “Events” contain additional information that are relevant to the success of the department in encouraging prospective majors.

The mathematics department webpage at Alfred University describes the interests of the faculty, some general information about the kinds of courses offered by the department, and the kind of post-graduate activities engaged in by graduates. It states: “Independent of the curriculum, we enjoy a level of personal interaction between faculty and students which is unusual even among small colleges. Faculty offices are located around the Math Common Room, a space with a large table and chairs where students feel free to drop in all day. Whether using this room for study or visiting a teacher, students quickly become accustomed to talking with faculty outside of class.”

Sandra O. Paur, director of the Mathematics Honors Program at North Carolina State University , has a set of slides from her workshop on Attracting and Retaining Majors. Another set of slides is on Fostering an atmosphere conducive to undergraduate research.

The article Experiences in Attracting Women to Mathematics at Spelman College by Sylvia T. Bozeman and Colm Mulcahy describes the way faculty attract and nurture majors at Spelman, where “upwards of one-third of the women at the college major in the natural sciences, mathematics, or a dual-degree engineering program.” The article discusses the innovative curricular approaches and activities taken by the department and includes concrete data about its success. In April 2005 the department “hosted the Infinite Possibilities Conference, the first-ever national gathering designed to celebrate, promote, support and encourage underrepresented minority women mathematicians.”

The Duke University website for the undergraduate program has links to a recruitment brochure as well as information about courses, course selection, activities, careers, and other resources.

Kaxem Mahdavi wrote “On Attracting Mathematics Majors” based on his experience at the State University of New York – Potsdam.

Some important resources for encouraging majors from under-represented groups are the websites of the MAA program Strengthening Underrepresented Minority Mathematics Achievement, the Association for Women in Mathematics, the National Association of Mathematicians, Advancing Hispanic Excellence in Technology, Engineering, Math and Science, Inc., the Society for American Chicanos and Native American Scientists, the American Indian Science and Engineering Society, and the Women and Mathematics Information Server. At an individual institution, Colm Mulcahy developed a website with information about Spelman College mathematics alumnae who went on to earn graduate degrees in mathematics-related fields.

Providing Career Information

The principal resource for nonacademic careers is Mathematical Sciences Career Information, jointly sponsored by the AMS, the MAA, and SIAM. 

There are many additional resources on the web that provide information about careers open to students majoring in the mathematical sciences.  Some useful ones are Careers from the MAA’s Information for Undergraduate Students webpage, the Career Center from the American Statistical Association, Careers in Mathematics from Purdue University mathematics department, and Careers in Mathematics from Eastern Washington University.

The Association for Women in Mathematics has a web brochure, called “Careers that Count: Opportunities in the Mathematical Sciences,” that contains information about the lives and careers of twelve women mathematicians.

Many mathematics departments invite their graduates back to give colloquiums on what they’re doing or include information about alumni careers on their websites.  One such department is St. Olaf (see, for example, the issue from March 19, 2007), and another is the State University of New York – Fredonia (click on “Alumni”). Having concrete information about people who came through the same program they are following can give students confidence that they will be able to follow a similar path.

The purpose of the MAA magazine Math Horizons is to introduce undergraduates interested in mathematics to mathematics outside the classroom. It regularly contains alumni profiles.

Mentoring and Advising Mathematics Majors

The American Mathematical Society website Undergraduate Mathematics Majors contains information about Graduate School, Summer Programs, REUs, Special Semesters, Math Links, Clubs, Conferences, On-line Journals, Competitions, Prizes, Honorary Societies, Careers, Jobs, Internships, and a Brochure entitled Resources for undergraduates in mathematics.

Advising and interacting with students at all levels is important for understanding the students and their goals, as well as for examining the effectiveness of one’s own program. Good advising starts with making sure that students are placed in appropriate courses. Descriptions of the St. Olaf College program, the University of Arizona program, and the University of Southern Mississippi program are available online through the MAA Supporting Assessment in Undergraduate Mathematics (SAUM) project. 

Contributors to the Project NExT session Preparing our Math Majors for the Future: Advice on Advising, organized by Lisa Marano, West Chester University, and Florence Newberger, California State University, Long Beach, offered suggestions for mentoring and advising mathematics majors. Versions of several of the suggestions are available on the Internet: General Information on Advising Math Majors by Lisa Marano; both a document and set of slides entitled “So You’re an Advisor – Now What?”  by Sandra O. Paur, North Carolina State University; information about the Advisor-Advisee Relationship from the Cornell University faculty handbook; and information about internships and co-op opportunities for undergraduates from the American Mathematical Society, ACM, Center for Talented Youth, American Statistical Association, and National Security Agency.

The Center of Excellence in Academic Advising at Penn State University offers a general description of the importance of advising, the role of the adviser, adviser’s tools and techniques, and references to advising resources.

Handbooks (either paper or web-based) can help inform students about a department’s programs, what course they should start with, various careers open to mathematics majors, and the paths of study required for those careers. The handbook Academic Advising for Mathematics Majors from the State University of New York – Potsdam includes descriptions of course requirements and expectations, the methods of teaching used in the department, and four levels of mathematical maturity and understanding.  Some other examples of web-based handbooks are those at Oberlin College, The College of William and Mary, Binghamton University (State University of New York), Queens University: Course Advice for Year 1 Students and Course Advice for Years 2 and 3 Students, Smith College, the University of Maine, Pennsylvania State University, Harvard University, the University of Washington, Duke University, the University of Iowa, Rochester University, the University of Oregon, and West Chester University (choose the “Prospective Students” pop-up menu and click on “Handbook for Mathematics Majors”).

Additional information on advising and mentoring mathematics majors for non-academic careers and for graduate study is in Part 2. Sections D.2 and D.3.

Co-Curricular Activities for Mathematics Majors

Many departments informally and formally involve mathematics majors in co-curricular activities. Providing mathematics students with an informal place to study, do group projects or socialize with other majors is a key form of support at many schools. Such a room might be equipped with books and magazines such as G. H. Hardy’s A Mathematician’s Apology, Albers and Alexanderson’s Mathematical People, copies of textbooks that could be used as course supplements, and the MAA journal Math Horizons.

The Northern Illinois University mathematical sciences department has a webpage devoted to alerting students to “non-curricular mathematics activities for undergraduates” It begins: “Hey, you! All work and no play makes for dull students. Why not participate in some of our non-course-related activities? They're open to all undergraduates, regardless of major.”

Xavier University’s NSF-sponsored STEM Exposure Program  “is designed to enhance this community experience for students majoring in the fields of Mathematics, Computer Science and Physics by first creating a casual and open learning environment through organized social events and peer mentoring, and secondly, by instituting learning experiences within this community through tutoring programs and summer bridge programs.”

The following are some examples of co-curricular activities supported in mathematics departments:
   * Informational Web Sites such as those at Hiram College and the State University of New York – Fredonia
   * Problem of the Week  such as that at Macalester College
   * Newsletters such as that at Carleton College or at North Carolina State University
   * MAA Student Chapters 
   * Pi Mu Epsilon Chapters
   * Sponsoring student participation at MAA Undergraduate Mathematics Conferences
   * Math club meetings, which often combine food (such as pizza or donuts) with problem or puzzle solving, a math talk, a math-related movie (such as  Flatland, N is a Number, and Julia Robinson and Hilbert’s Tenth Problem,  or To Dream Tomorrow about Ada Lovelace, or The Proof about Fermat’s last theorem, or popular films such as those listed on the Math in the Movies website)
   * Picnics, softball/soccer/volleyball games, bowling
   * Student-designed departmental T-shirts
   * Organized trips to research labs, conferences, businesses, etc.

Opportunities for students to serve as course assistants or mentors not only support students receiving the assistance but also motivate and encourage the students providing the help. Examples of such programs are Xavier University’s STEM Exposure Program, Rutgers University’s Peer Mentor Program, and Monmouth University’s Math Learning Center, where mathematics majors often begin tutoring students in college-algebra-level courses as second-semester freshmen while coming in for help themselves in calculus and which has become a central gathering place for mathematics majors throughout their studies. Helpful material for training undergraduates to be effective course assistants is available on the web page for the Peer-Led Team Learning project.