TO:
Beth Dobkin, Provost
FROM:
Keith Ogawa, Chair
Academic Senate
DATE:
April 3, 2014
RE:
Senate Action S-13/14-31CA
Proposal for a Modern Mathematics Major
At the March 27, 2014 meeting of the Academic Senate, the attached Proposal for a Modern Mathematics
Major was approved on the Consent Agenda.
The item was unanimously approved by the Undergraduate Educational Policies Committee and forwarded
to the Senate's Consent Agenda. This action was assigned Senate Action #S-13/14-31CA.
Attachment
Cc: President James Donahue
Vice Provost Richard Carp
Dean Roy Wensley
A Proposal for a Modern Mathematics Major
Introduction
Based upon graduating student surveys, the 2010 Program Review, conversations among
faculty and students, and a study of graduate school requirements the Department of
Mathematics and Computer Science (DOMACS) is proposing modernizing our major as outlined
below.
•
Remove the BA degree
•
Create three tracks within the BS degree: Pure, Applied and Computer Science
•
Remove the “two more science courses” requirement from the BS degree, replaced with
one or two additional mathematics courses, depending on the track
•
Remove two courses from the curriculum; Math 112 Abstract Algebra II and Math 130
Abstract Geometry
•
Add two courses to the curriculum: Math 103 An Introduction to Proofs and Math 131
Topology
•
Require Math 103 in the mathematics minor
•
Alter the prerequisites of various courses, in particular having Math 103 as the
prerequisite for several upper division courses
•
Rearranging in which semesters various upper division courses are offered.
Rationale for removing the BA degree
As far as we can ascertain there is no standard for whether a mathematics degree should
be considered a BA or BS degree. Currently the BA is (perhaps unfairly) considered a second-class
degree in mathematics by important stakeholders such as future employers of our students.
Given that we are a department in the School of Science, and the scientific nature of
mathematical study it seems natural that all our majors should receive a BS degree. We therefore
propose awarding all mathematics majors a BS degree, with three tracks described in what
follows.
Rationale for altering the BS degree
Currently the BS degree requires four lower division mathematics courses, eight upper
division mathematics courses, plus four science courses, for a total of sixteen courses. The science
courses must consist of Physics 1 and 3 or Chemistry 8 and 10, plus any two other science
courses. We have the notion that by taking science courses the math students are broadening
their mathematics education and gaining an appreciation for the applications of mathematics.
The reality is that many students are currently taking their year of physics or chemistry, and then
choosing two more “Area B” science classes, such as geology or astronomy. While these are
worthwhile classes, it is not apparent that they add anything to the mathematics education of our
majors. We therefore propose removing the “two additional science courses” requirement from
the BS and replacing these with one or two mathematics courses depending upon the track, thus
improving the mathematic acuity of our majors.
The three new tracks
We propose three tracks for the mathematics major, a pure, an applied and a computer
science track. The tracks are designed for the different goals that our majors have (attending
graduate school, teaching, or careers in industries such as banking, insurance, financial, or the
computer field). One of our current programs is a Computer Science “concentration” which can
be confusing to students and often leads them to believe that the computer science preparation
in this program is at the level of a minor. The new tracks will put this concentration on the same
footing as the two other mathematics major tracks. The three tracks are described on the
following three pages.
Pure Track
The pure track is similar to the current mathematics major and is a traditional
mathematics curriculum. This track is ideal for a student considering attending graduate school in
mathematics or a closely related science or planning a career in high school mathematics
teaching. By requiring both Algebra I and Advanced Calculus plus a choice of two of the courses
Number Theory, Topology and Complex Analysis successful students will be well prepared for
graduate level coursework and/or teaching.
This track consists of 4 lower division mathematics courses, 10 upper division mathematics
courses, plus two science courses.
Required Lower Division
Math 27-Calculus I
Math 38-Calculus II
Math 39-Calculus III
CS 21- Programming I
Science requirement
Physics 1+3 OR Chemistry 8+10 (with respective labs)
Required Upper Division
Math 103-An Introduction to Proofs
Math 120-Linear Algebra and its Applications
Math 111-Abstract Algebra
Math 150-Advanced Calculus
Math 193-Senior Seminar
Plus two courses from
Math 115-Number Theory
Math 131-Topology
Math 185-Complex Analysis
Plus three upper division mathematics electives
Applied Track
The applied track is designed for students planning on entering the workforce directly
from college. We encounter many mathematically strong students who have a utilitarian view of
the major and are less motivated and served by the more abstract courses. Students in this track
will study a wide variety of applications of mathematics in their courses.
This track consists of 4 lower division mathematics courses, 9 upper division mathematics
courses, plus two science courses.
Required Lower Division
Math 27-Calculus I
Math 38-Calculus II
Math 39-Calculus III
CS 21-Programming I
Science requirement
Physics 1+3 OR Chemistry 8+10 (with respective labs)
Required Upper Division
Math 103-An Introduction to Proofs
Math 120-Linear Algebra and its Applications
Math 193-Senior Seminar
Math 111-Abstract Algebra or 150-Advanced Calculus
Plus three from
Math 113-Probability and Statistics
Math 114-Mathematical Modeling
Math 134-Differential Equations
CS 102-Advanced Programming
Plus two upper division electives at most one of which is a CS (computer science) course
Computer Science (CS) Track
The computer science track is similar to the current computer science concentration. We
do not have a computer science major nor the wherewithal to offer one. We currently offer a
concentration in computer science, but students are often confused about exactly what this
means, particularly incoming or potential students. The small number of CS courses we offer
mean there are fewer elective choices for these majors.
This track consists of 4 lower division mathematics courses, 9 upper division
mathematics/computer science courses, plus two science courses.
Required Lower division
Math 27-Calculus I
Math 38-Calculus II
Math 39-Calculus III
CS 21-Programming I
Science requirement
Either Physics 1+3 OR Chemistry 8+10
Required Upper Division
Math 103-An Introduction to Proofs
Math 120-Linear Algebra and its Applications
Math 140-Combinatorics and Discrete Mathematics
Math 193-Senior Seminar
CS 102-Advanced Programming
Math 111-Abstract Algebra or Math 150-Advanced Calculus
Plus three upper division electives at least two of which are CS courses.
Mathematics Minor
The only change we propose to the current mathematics minor is the requirement that students
must take Math 103 Introduction to Proofs and this replaces one of the electives. The minor will
now be:
Required Lower Division
Math 27-Calculus I
Math 38-Calculus II
Required Upper Division
Math 103-An Introduction to Proofs
Plus two upper division mathematics electives
New courses
We propose two new courses: Math 103: An Introduction to Proofs and Math 131: Topology. A full
syllabus and schedule for each of these are included in an appendix. We give a brief description of
the courses here as well as the rationale for their proposals.
Math 103: An Introduction to Proofs:
Proofs are a central part of mathematics and we currently do not have a course in which
we explicitly teach the techniques of proof. Almost all mathematics courses use some level of
proof, however this is uneven depending upon who is teaching a given course and the audience.
We are currently unable to assume any particular prior knowledge of proofs in subsequent
courses. By creating a course in which proofs are a central pillar and having this course as the
prerequisite for several upper division courses we will be able to create a more developmental
major in which courses can build upon previous knowledge.
This will have some positive effects to other courses; currently Math 120, Linear Algebra and its
Applications is the place in which we primarily teach the technique of proofs. As a result the goals
of Math 120 are too many; the course is acting as both a major introduction to many
mathematical applications as well as being a bridge to the upper division courses in that
techniques of proof are being learned.
This new course, Math 103 An Introduction to Proofs in combination with the current Math 193
Senior Seminar is intended to be the department’s “Writing in the Disciplines” course. This course
has already been proposed and tentatively approved by the core curriculum committee.
Math 131: Topology
Topology is the study of continuity and the consequences of applying continuous functions
to sets. These consequences rest on properties of a space more basic than its geometry. A
topological space’s properties such as compactness, connectedness, and distance are based on
the abstract notion of open sets. Topological properties play an important role in many other
areas of mathematics such as analysis, differential equations, and dynamical systems and their
applications in computer science, biology, and physics in recent years has increased.
This course will be an excellent opportunity for students to further hone their proof writing skills
and to prepare themselves for graduate studies or teaching secondary mathematics. In addition
students who complete this course prior to Advanced Calculus will have a better understanding of
Euclidean space structure and properties.
Our department currently has two faculty members (Weiwei Pan and Kathy Porter) whose
research areas lie within the mathematical field of topology, and all current tenured/tenure-track
faculty are qualified to teach this course.
The prerequisite for this course will be Math 103, An Introduction of Proofs.
Courses to be removed
Math 130: Abstract Geometry
It is with some regret that we propose the removal of Abstract Geometry. This is an important
mathematical topic, and ideally would be part of a regular course rotation. However, prior to this
semester, the course has been cancelled the previous two times we have attempted to offer it
due to a low enrollment. Students enter college with a foundation in geometry from their high
school studies (all SMC students are required to have completed a year of high school geometry)
so those mathematics majors who would benefit from the study of geometry already have that
basic background. We plan to offer this course intermittently as a special topics course when the
demand arises.
Math 112: Abstract Algebra II
It is slightly misleading to state that we are removing Abstract Algebra II. A large portion of
the topics that are currently studied in Math 111 Abstract Algebra I will be studied in our new
proofs course Math 103. Thus, our Math 111 course will be able to begin farther along in the
Abstract Algebra curriculum allowing a solid education of this subject by the completion of Math
111, thereby not needed another Abstract Algebra course. The Proofs course plus the Math 111
course will be a solid algebraic preparation for those students planning to teach high school
mathematics or attend graduate school in mathematics.
Course Offerings
As currently, our courses will be offered on a two-year cycle. Below are the cycles we
anticipate starting in the fall of 2014. Based on the small increase in the number of courses taken
by majors as well as the current trend showing an increase in the number of students taking a
math major or minor we shall add one upper division course per year. We note that twice in
recent years we have been forced to offer two sections of an upper division course in order to
accommodate the number of students taking these courses. The course schedule will be as
follows
Fall (even years)
Spring (odd years)
140
114
Combinatorics/Discrete Math
Mathematical Modeling
190
185
CS102
103
Special Topics (Fall 2014 will be 115)
Complex Analysis
Advanced Programming
Introduction to Proofs
120
134
150
CS1xx
193
Fall (odd years)
113
103
131
190
CS102
Probability and Statistics
Introduction to Proofs
Topology
Special Topics
Advanced Programming
Linear Algebra and its Applications
Differential Equations
Advanced Calculus
Upper Division CS
Senior Seminar
Spring (even years)
111
120
134
115
CS1xx
193
Abstract Algebra
Linear Algebra and its Applications
Differential Equations
Number Theory
Upper Division CS
Senior Seminar
Changes from current offerings:
•
There is a special topics each fall (currently every other spring).
•
Math 111 is now in the spring.
•
Math 103 is offered every fall.
•
Senior Seminar is now in the spring.
•
To balance the schedule Number Theory has switched from even fall to even spring,
however it will still be offered as scheduled in the fall of 2014 so current juniors have the
opportunity to take the course.
Summary
These changes to the mathematics major will result in students having to take 14
mathematics courses in total. In referencing our most recent program review, the average
number of mathematics courses required at other WCC schools (as of 2010) was 13.57. The high
was 19 required at Loyola Marymount University and the low was the University of San Francisco
with 11. Thus requiring 14 math courses, plus the two lab sciences, one of which will satisfy the
science core requirement seems reasonable. In addition the creation of the Introduction to Proofs
course, and adopting it as the prerequisite for a number of the upper level courses creates a more
structured, hierarchical major benefiting those later courses.
A concern raised in discussion with the Dean and Chairs from the School of Science is the
whether we will be able to fill the classes being offered. We share this concern and have
conducted a survey of current junior and senior math majors to ascertain which classes they
would have taken if this new curriculum were to have been on offer during their time at Saint
Mary’s. The majority of the classes had enough interest that we are confident that we will be able
to run them. The addition of math minors and other students who take our upper division classes
(primarily physics and engineering majors) mean we should be able to have sufficient enrollment
for each class. We are, however, prepared to be flexible in our offerings and requirements so that
as time goes by we are not consistently offering small classes.
Courses, their prerequisites and their place in the new majors. (Prerequisite and
semester offered)
Math 103- Introduction to Proofs (Math 28 or 38. Every Fall)
This is the heart of the new major. It will be taken by all majors and minors preferably in the fall of
their sophomore year. All incoming transfer students interested in majoring in mathematics
should take it immediately upon entering. By offering this course early in the major and having it
at as the prerequisite for many courses (111, 115, 131, 150, 185, 193) we can make significant
assumptions in these later courses and have expectation regarding the levels of proof and writing
of the students in these courses.
Math 111-Abstract Algebra (Math 103 and 120. Even Spring)
Due to the topics being covered in Math 103, this course will cover roughly half of “old 111” and
half of “old 112” (the Abstract Algebra courses). This will remain a fairly traditional mathematics
course. Taken by all majors in the pure track and roughly half of those in the applied and CS
tracks.
Math 113-Probability and Statistics (Math 28 or 38. Odd Fall)
This course will likely be taken by most minor and many majors, probably in the fall of sophomore
or junior year.
Math 114-Mathematical Modeling (Math 28 or 38 AND CS21. Odd Spring)
Requiring CS21 (Introduction to Programming) as a prerequisite and moving this course (Math
114) to the spring will change the nature of this course. Previously it has been a course designed
with minors in mind. While some minors will have a programming background and take the
course it will now be more of a majors’ course. On the three occasions we have taught Math 114
the absence of a knowledge of programming on the part of the students has significantly
restricted the avenues of applications the class could explore. We anticipate on occasion allowing
students who do not have the formal programming background, but nevertheless have some
programming experience.
Math 115-Number Theory (Math 103. Even Spring)
The inclusion of Math 103 as a prerequisite will prepare the students for the level of proofs
necessary for the understanding of the natural theoretical aspects of this course. Taken by 2/3rds
of pure track plus several others.
Math 120-Linear Algebra and its applications (Math 28 or 38. Every Spring)
The creation of Math 103 frees this course to be closer to its description, with an emphasis on
application, and some numerical techniques. Taken by all majors and likely several minors.
Math 131-Topology (Math 103. Odd Fall)
Taken by 2/3rds of those in pure track plus a significant number of others. Useful for graduate
school and we have significant expertise to teach it.
Math 134-Differential Equations (Math 39. Every Spring)
Unchanged. Primarily a service course for physics/engineering students but also taken by many
majors, particularly in the applied track.
Math 140-Combinatorics and Discrete Mathematics (Math 28 or 38. Even Fall)
Course is offered in the fall to avoid conflicting with CS electives as it is a requirement for those in
the CS track.
Math 150-Advanced Calculus (Math 39 and 103. Odd Spring)
Course is unchanged.
Math 185-Complex Analysis (Math 39 and 103. Even Fall)
Course is unchanged.
Math 190-Special Topics (Dependent upon topic/instructor, likely at least 28/38. Every Fall)
The change in the programming prerequisite for Math 114 means we should on occasion consider
making this course appealing to minors, particularly in the even years when it does not coincide
with Math 113, Probability and Statistics. May also consider offering geometry semi-regularly
since it is no longer a regular course.
Math 193-Senior Seminar (Math 103. Every Spring)
Note the change in semester of this course. As a capstone it seems that the course’s natural place
is in the spring. There is also a practical consideration as our numbers grow. We now require that
students have a topic before the semester begins and Jan term seems a natural time to do this,
rather than choosing in the spring and having a large break to the fall. The course is now the
second part of the writing in the disciplines requirement.