Mathematics Advising Guide
The Mathematics and Computer Science Department offers majors in
mathematics, mathematics education, and computer science. This
document is designed as a resource for students in navigating through
the requirements of the math or math education major. (There is a
companion guide for computer science.) It should also be valuable to
freshman advisors outside the department as well as advisors within
the department. The first section gives a brief overview of the
discipline; the remaining sections describe the major, the honors
program, and the minor.
As an additional resource, you can also use the Mathematics Major Form, which is
intended for students majoring (or considering majoring) in
Mathematics, and their advisors. This form will help you plan out
your mathematics courses.
Contents
Mathematics
Mathematics is a field that is rich in both theoretical analysis and practical
application. It is also quite broad in scope, encompassing subfields such
as statistics, applied mathematics, and the classical subjects of analysis,
algebra, and geometry. This diversity within mathematics makes most definitions
of mathematics either too narrow or too general. However, one can say that
mathematicians deal with objects (e.g. numbers, triangles, function), and
their patterns and relationships (e.g. prime numbers, isosceles triangles,
calculus of functions). The search for patterns and relationships involves
the process of abstraction, that is forming a generalization from a set
of examples that reflects shared properties of these examples. Mathematicians
use the skills of creative and analytical thinking to hypothesize the existence
of patterns and use logical argument to show the validity of these postulates.
Mathematics is also a science, a science of patterns and relationships.
Mathematicians experiment in various laboratories -- the mind, the computer,
and the natural world.
Mathematicians are always in demand in industry, business, government,
and academia. The breadth of subject matter and the logical, analytical
training required provide math majors with flexibility in their choice
of career.
Mathematicians are involved in activities such as:
-
mathematical modeling of semiconductors for a research laboratory
-
studying cryptology schemes for secure communication networks
-
teaching in the public schools
-
teaching at a college or university
-
researching the role that chaotic systems play in the regulation of the
heart
-
devising new fractal algorithms for the display of realistic natural objects
-
economic forecasting and model building for the government and industry
-
devising better ways to solve the differential equations arising in the
flow of turbulent fluids
-
working as an actuary for a large insurance firm
-
carrying out foundational mathematical research at a research university
or research laboratory
-
working as a statistician for a governmental agency
This is only the tip of the iceberg as far as career opportunities in the
mathematical sciences. For more information talk to someone in the Math/Computer
Science department or see the departmental secretary for brochures describing
careers in math.
Mathematics majors who are interested in research opportunities or in
teaching at the college level should choose a set of courses, in consultation
with a department member, that will prepare them for graduate study in
mathematics or a related field (such as economics, mathematical physics,
statistics, etc.)
The Major
This section lists the requirements of the math major and describes the
senior oral which is an optional component of the major. Qualified majors
may additionally participate in the honors program, which is described
in the next section.
A grade of C- or higher is necessary in all courses used to satisfy
the requirements of the major, which are as follows:
-
MCS-121, MCS-122 or MCS-132, MCS-220, MCS-221, and MCS-222, with a grade point
average of at least 2.333 in these five courses.
These five courses form the core of the major and should usually be taken
during the freshman and sophomore years.
-
MCS-142 and MCS-177. These two courses in cognate
fields to mathematics serve to give breadth to the math major.
-
At least four courses chosen from MCS-242, MCS-253, MCS-256,
MCS-265, MCS-303, MCS-313, MCS-314, MCS-321, MCS-331, MCS-332,
MCS-341, MCS-342, MCS-344, MCS-357, MCS-355, and MCS-358, subject
to the following constraints:
-
At least one sequence chosen from
(i) MCS-331 & MCS-321 (Real and Complex Analysis),
(ii) MCS-313 & MCS-314 (Abstract Algebra),
(iii) MCS-331 & MCS-332 (Analysis and Topology),
(iv) MCS-341 & MCS-342 (Probability and Statistics), or
(v) MCS-253 & MCS-357 (Differential Equations and Discrete
Dynamical Systems.)
-
Completion of at least one course from the classical core of mathematics:
MCS-321 (Complex),
MCS-313 (Algebra), and
MCS-331 (Real Analysis).
This course can count toward 3a.
-
Completion of at least one course from the applied areas of mathematics:
MCS-242, MCS-253, MCS-256, MCS-342, MCS-357, MCS-355, and
MCS-358.
This course can count toward 3a.
-
Either complete one of the capstone courses MCS-314, MCS-332, MCS-342,
MCS-344, MCS-357, MCS-358
(beyond any used for requirements 3) or alternatively pass a successful senior
oral examination covering the student's knowledge of mathematics. Successful
completion of an honors thesis also will
fulfill this requirement.
Senior oral
As described above, every math major must either take an additional upper
level math course from a specified list or alternatively submit to oral
examination during the Spring semester of their final year.
A student who chooses to take the oral examination selects, in consultation
with a faculty member, a topic to research. They then present a 20-minute
talk on that topic to an examining committee of three faculty members.
At the conclusion of the talk, the faculty question the student about the
talk, and also about fundamental topics from the student's full four years'
of courses. The goal is not to require recollection of details, but rather
to make sure that the student is leaving with the essentials intact.
The examination committee confers privately immediately after the examination
and delivers the results to the student at the conclusion of their deliberations.
The outcome is either that the student is deemed to have satisfied the
requirement or alternatively that the student is requested to retry the
examination at a later date. In the latter case, specific suggestions for
areas of improvement are provided by the faculty committee.
More information about the oral examination procedures and schedule
are provided routinely to those fourth-year majors who will likely choose
to take the examination.
Mathematics Education major
The requirements for the mathematics education major are met by the completion
of the mathematics major as listed above, with the additional requirement
that MCS-303(Geometry) and MCS-313 (Modern Algebra), must be completed. Math
education majors may substitute the sequence MCS-313-MCS-303
for the sequence requirement in part 3 of the math major.
Concentrations
Within the math major there are two concentrations or tracks available
to students with specific interests.
-
Applied Mathematics
-
This concentration is for those students interested in the scientific applications
of mathematics, and who are planning on entering fields that require training
in mathematical modeling and the analysis of physical problems. MCS-253(Diff
Eqns) and MCS-357(Discrete Dynamical Systems) form the core of this track. Other highly
recommended courses would be MCS-321(Complex), MCS-358(Math Model Building),
and MCS-355(Numerical Analysis).
-
Statistics
-
This concentration is intended for students who wish to pursue a career
in actuarial science, or who will do graduate studying statistics, biostatistics,
or a related field. MCS-242(Appd Stats), MCS-341(Prob Math Stat I), and MCS-342(Prob
Math Stat II) form the core of this track. For those interested in actuarial
science, MCS-355 (Numerical Analysis) is recommended. Also,
a minor in economics or management is recommended. For those interested
in graduate study in biostatistics, epidemiology, or public health, a minor
in biology is recommended. For those interested in pursuing a PhD in statistics,
MCS-331(Real Analysis) and MCS-332(Topology) are strongly recommended.
Sample student plans
All students should ideally lay out a schedules of their own showing what
courses they plan to take when. This schedule may not accurately forecast
the future, but it is helpful none the less. The sample plans below are
a useful starting point in developing such an individual plan. You can
select the sample plan that comes closest to fitting your own situation
and then tailor it as necessary. Note that these sample plans show only
courses within the Math and Computer Science Department, but in some cases
exceed the requirements of the major. Also note that certain courses are
offered on an every-other year basis; for example MCS-314 (Algebra II) is
offered in the spring of odd years and MCS-332 (Topology) is offered in the
spring of even years. Courses offered every other year include MCS 242,
313, 314, 331, 332, 341, 342, 344, 358, 385, and 394. Please keep these
course alterations in mind when planning out your major. Check the college
catalog for when the courses you are interested in will be scheduled.
| Traditional (MCS 313-314 (Algebra) sequence) |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221 |
| 3rd year |
222 |
321 |
| 4th year |
313
357 |
314 |
| Traditional (MCS 331-332 (Real Analysis) sequence) |
| |
Fall |
Spring |
| 1st year |
121
177 |
122 |
| 2nd year |
220 |
221
222 |
| 3rd year |
142
313 |
256 |
| 4th year |
331 |
332 |
| Traditional (MCS 341-342 (Prob Math Stat) sequence) |
| |
Fall |
Spring |
| 1st year |
121 |
122
142 |
| 2nd year |
220
177 |
221 |
| 3rd year |
222
331 |
256 |
| 4th year |
341 |
342 |
| Statistics concentration |
| |
Fall |
Spring |
| 1st year |
121 |
122
142 |
| 2nd year |
220
177 |
221 |
| 3rd year |
222
331 |
242 |
| 4th year |
341 |
342 |
| Applied concentration |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221 |
| 3rd year |
222
331
|
253 |
| 4th year |
357 |
355
321 |
Math Education
(Even Year Graduation; Student Teach in
Spring) |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221
242 |
| 3rd year |
222
313 |
253 |
| 4th year |
303
358 J-term |
student
teaching |
Math Education
(Odd Year Graduation; Student Teach in
Spring) |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221
256 |
| 3rd year |
222
358 J-term |
242
313 |
| 4th year |
303 |
student
teaching |
Math Education
(Even Year Graduation; Student Teach in
Fall) |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221
242 |
| 3rd year |
222
303 |
253 |
| 4th year |
student
teaching
358 J-term |
313 |
Math Education
(Odd Year Graduation; Student Teach in
Fall) |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221 |
| 3rd year |
222
303
358 J-term |
242
313 |
| 4th year |
student
teaching
|
253 |
| Traditional, grad. school bound |
| |
Fall |
Spring |
| 1st year |
121
177 |
122 |
| 2nd year |
220
142 |
221
222 |
| 3rd year |
331 |
332
321 |
| 4th year |
313 |
314 |
| Statistics grad. school bound |
| |
Fall |
Spring |
| 1st year |
121
142 |
122 |
| 2nd year |
220
177 |
221
222 |
| 3rd year |
331 |
242
332 |
| 4th year |
341 |
342 |
| Applied, grad. school bound |
| |
Fall |
Spring |
| 1st year |
121
177 |
122 |
| 2nd year |
220
142 |
221
222 |
| 3rd year |
331 |
321
253 |
| 4th year |
357
313 |
355 |
| Start with pre-calc |
| |
Fall |
Spring |
| 1st year |
120 |
121
177 |
| 2nd year |
122 |
220
221 |
| 3rd year |
222
142 |
321 |
| 4th year |
331 |
256
313 |
| Fall junior year abroad |
| |
Fall |
Spring |
| 1st year |
121 |
122
177 |
| 2nd year |
220
142 |
221 |
| 3rd year |
abroad |
222
256 |
| 4th year |
321
331 |
313 |
| Spring (Stats) junior year abroad |
| |
Fall |
Spring |
| 1st year |
121 |
122
142 |
| 2nd year |
220
177 |
221
256 |
| 3rd year |
331
222 |
abroad |
| 4th year |
341 |
342 |
| Junior year abroad |
| |
Fall |
Spring |
| 1st year |
121
177 |
122 |
| 2nd year |
220
142 |
221
222 |
| 3rd year |
abroad |
abroad |
| 4th year |
313
331 |
321
256 |
Honors Program
In order to graduate with honors in mathematics, a student must complete
an application for admission to the honors program, showing that the student
satisfies the admission requirements, and then must satisfy the requirements
of the program.
Admission to the honors program
The requirements for admission to the honors program are as follows:
-
Completion of MCS-121(Calc I), MCS-122(Calc II)
or MCS-132(Honors Calc II), MCS-220(Intro to Analysis), MCS-221(Linear Algebra),
MCS-222(Advanced Calculus), MCS-142(Intro Stats), and MCS-177(Intro CS I)
with a quality point average greater than pi.
-
Approval by the Mathematics Honors Committee of an honors thesis proposal.
(See the Mathematics Honors Thesis Guidelines, reprinted below.)
Requirements for
graduation with honors
The requirements of the honors program, after admission to the program,
are as follows:
-
Attainment of a quality point average greater than pi in courses used to
satisfy the requirements of the major. If a student has taken more courses
than the major requires, that student may designate for consideration any
collection of courses satisfying the requirements of the major.
-
Approval by the Mathematics Honors Committee of an honors thesis. The thesis
should conform in general outline to the approved proposal (or an approved
substitute proposal), should include approximately 160 hours of work, and
should result in an approved written document. Students completing this
requirement will receive credit for the course MC96 (Honors Thesis), whether
or not they graduate with honors. (See the Mathematics Honors Thesis Guidelines,
below.)
-
Oral presentation of the thesis in a public forum, such as the departmental
seminar. This presentation will not be evaluated as a criterion for thesis
approval, but is required.
Honors thesis guidelines
Mathematics honors thesis proposals should be written in consultation with
the faculty member who will be supervising the work. The proposal and thesis
must each be approved by the Mathematics Honors Committee. These guidelines
are intended to help students, faculty supervisors, and the committee judge
what merits approval.
The thesis should include creative work, and should not reproduce well-known
results; however, it need not be entirely novel. It is unreasonable for
an undergraduate with limited time and library resources to do a thorough
search of the literature, such as would be necessary to ensure complete
novelty. Moreover, it would be rare for any topic to be simultaneously
novel, easy enough to think of, and easy enough to do.
The thesis should include use of primary-source reference material.
As stated above, an exhaustive search of the research literature is impractical.
None the less, the resources of inter-library loan, the faculty supervisor's
private holdings, etc. must be tapped if the thesis work is to go beyond
standard classroom/textbook work.
The written thesis should sufficiently explain the project undertaken
and results achieved that someone generally knowledgeable about mathematics,
but not about the specific topic, can understand it. The quality of writing
and care in citing sources should be adequate for external distribution
without embarrassment.
The thesis must contain a substantial mathematical component, though
it can include other disciplines as well. If a single thesis simultaneously
satisfies the requirements of this program and some other discipline's
honors program, it can be used for both (subject to the other program's
restrictions). However, course credit will not be awarded for work which
is otherwise receiving course credit.
The Mathematics Honors Committee will maintain a file of past proposals
and theses, which may be valuable in further clarifying what constitutes
a suitable thesis. In order to provide some guidance of the sort before
the program gets under way, here are some possible topics that appear on
the surface to be suitable:
-
A student could study the history surrounding Fermat's last theorem, and
discuss and explain past failed attempts and the recent successful attempt
to prove this theorem.
-
A student could research the topic of knot theory and discuss the implications
of this theory to the study of DNA and other biological materials.
-
A student could study the use of wavelets in signal analysis, and the general
usefulness of orthonormal families of functions in signal analysis.
The Minor
The Mathematics minor
As with the major in mathematics, a minimum grade of C- must be attained
in all courses used to satisfy the minor. The necessary courses are
-
MCS-121(Calc I), MCS-122(Calc II) or MCS-132(Honors
Calc II), MCS-220(Intro to Analysis), MCS-221(Linear Algebra), and MCS-222(Advanced
Calculus), with an average grade of at least C+ in these five courses.
-
At least two course chosen from MCS-242(Appd Stats), MCS-303(Geometry),
MCS-265(Theory Computation), MCS-256(Discrete), MCS-321(Complex), MCS-358(Math
Model Building), MCS-313(Algebra), MCS-314(Algebra II), MCS-357(Discrete Dynamical Systems),
MCS-355(Numerical Analysis), MCS-331(Real Analysis), MCS-332(Topology), MCS-344(Topics
Adv Math), MCS-341(Prob Math Stat I), and MCS-342(Prob Math Stat II).