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 Monito*r,
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.

**C.1: Develop mathematical thinking and
communication skills**

*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.

**C.2: Develop skill with a variety of
technological tools**

*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.

**C.3: Provide a broad view of the
mathematical sciences**

*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.)

**C.4: Require study in depth**

*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.

**C.5: Create interdisciplinary majors**

*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.

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.

**C.6: Encourage and nurture mathematical
sciences majors**

* 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.