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Mathematics
Prepared by Marshall
Cates, CSU Los Angeles, Lead Discipline Faculty for Mathematics
Summary of Identified Issues
Four areas of concern arose in nearly every meeting: 1)The
level of Differential Equations and Linear Algebra, which
is sometimes upper division and sometimes lower division;
2)The need and level for instruction in writing proofs; 3)The
need for early warning and informal contacts between community
colleges and four year institutions whenever curriculum changes
are planned; 4)The need for a description for Discrete Mathematics,
a course of great interest for Computer Science.
Identify Trends/Future
Directions
Physics is nearly universally required as part of the Mathematics
Major. It is seen as the most important way for students to
put the calculus to work, after all calculus was invented
to solve physics problems, and as a way to solidify the concept
of vectors. However, in the last 50 years other areas have
become more quantitative, using mathematics extensively. Economics,
biology, and computer science are but three of several areas
where mathematics majors could put their mathematics to work.
It was argued successfully at one of the regional meetings
that we should no longer cite physics as being required in
a model mathematics major. We should embrace the concept that
for some students, other math intensive applications would
better suit their needs. Thus we should allow for different
sciences or applications to play the role in the major, once
filled by physics.
At least in the CSU, a larger and larger percentage of mathematics
majors are planning on being high school mathematics teachers.
Consequently departments will have to examine their major,
not only allow for such preparation, which we all do, but
to really make it one of our core concerns.
Comments from Statewide
Meetings and the General Field
It was argued that physics should not be required of every
mathematics major. There are now several disciplines that
have intensive use of mathematics at the level accessible
to early mathematics majors. We should allow these in the
major in place of physics. This argument, when first proposed
was rejected by all community college instructors, in fact
by most all in attendance,
but as discussion continued giving examples of valid learning
objectives from different disciplines, more and more acceptance
followed, until in the end, a motion to this effect passed.
As a side note, there was a comment to the effect that the
host of biology prerequisites for upper division biology classes
has kept mathematics majors from contributing to the growth
of mathematical modeling in that field. Mathematicians would
have a lot to contribute in the modeling arena without having
to know the vocabulary of anatomy classes for example.
The computer science faculty presented their need for a course
in discrete mathematics. Their professional organization had
adopted a list of topics for such a course. At a joint meeting
between mathematics and computer science, we debated the design
of this course. Much of the debate centered on the prerequisite
for such a course. The prerequisite sets the level of the
class. This was a difficult decision since, a calculus prerequisite
would delay students taking the class in the major, while
the level or depth of the class would greatly benefit from
a calculus prerequisite. At the joint meeting with computer
science faculty the majority opinion settled on a calculus
prerequisite, that is the sophistication desired would require
a stronger background.
The discussion of the role that community colleges should
play in the development of competence in creating proofs started
with the desire by computer science for their majors to have
early training in this area. This is usually accomplished
in a class such as discrete structures. Since the construction
of proofs is the essence of upper division mathematics, clearly
mathematics majors are also in need of instruction. Most four-year
institutes have found it necessary to create specialized classes
to help their mathematics majors develop this essential skill.
All community colleges demonstrate proofs in their classes
and most ask students to prepare proofs, but few give actual
instruction on proof techniques. It was recommended that all
community colleges try to incorporate proof instruction into
key classes.
We spent a great deal of time looking at CAN descriptors.
In fact, most was spent on the description for the General
Education course for liberal arts majors. Finally, we admitted
that we had not come prepared to discuss in detail the description
of these service courses, but could make some headway with
major courses. Subsequently, we abandoned even that modest
goal, finally coming to believe that a separate conference
on CAN descriptors was needed to make consensus headway.
Recommendation
for the Discipline
If a four-year institution's requirement for differential
equations is at the upper division level and a transfer student
has taken a similar course (at the lower division level) the
four-year institution should try to give "content"
credit for the course even though they cannot transfer the
course.
The same for linear algebra, recognizing that there is a
strong difference between an early junior level and a senior/graduate
level linear algebra course.
Four-year institutions should keep in close contact with
their area community colleges. They should alert them early
in the process about proposed changes in the curriculum including
major course modifications. This is especially important if
the proposed changes could affect articulation agreements.
It is proposed that on a regional basis that there should
be a face-to-face meeting at least once per year. Institutions
could rotate hosting such meetings.
Proofs need to be part of the community colleges' curriculum.
Not just demonstrated in classes, but expected of mathematics
majors. It is unlikely that community colleges can support
a separate course for this purpose, but it is essential for
students to do proofs within existing courses.
Recommendations
for Support Courses
Discrete mathematics is an important issue for computer science.
Roughly speaking, finite mathematics is the non-calculus mathematics
useful to business majors and discrete mathematics is the
non-calculus mathematics useful or necessary for computer
science majors. Thus Boolean algebra, the fundamental mathematics
of and/or decision making and Lattice Theory, the structures
underlying data structures are topics to be included in discrete
mathematics courses. While the topics are not
fully prescribed in this emerging course, there is a great
deal of agreement. Computer science professional societies
have developed lists of suggested topics. What is not well
discussed is the desired level of such a course, and where
is the ideal position in the curriculum for such a course.
Unfortunately, the desired placement for the course (early
or later) is at odds with the desired level (freshman, sophomore)
for the course. Add to this mix the desired sophistication
of the problems considered in the course and there is no general
agreement.
Topics for Future
Discussion
- Engage in relevant discussions with Engineering. A review
of the topics needed from calculus as well as the need for
experience in modeling would be two starting points.
- Discuss with computer science the outcomes desired from
Discrete Mathematics as well as the level of prerequisites.
- Discuss the content of the service courses for prospective
elementary school teachers. There is a disparity between
the demonstrated performance in elementary topics such as
fractions, percentages, and number sense and the areas we
think should be learned such as geometry, number theory,
statistics.
Recommendations Forwarded/to be
forwarded to CAN
Convene a CAN discipline review committee to thoroughly review
CAN descriptors. Can descriptors for Discrete Mathematics
and for Mathematical Proofs need to be created. Our review
indicates that while the major courses are in pretty good
shape there is much disagreement in the descriptors for the
service courses such as the General Education courses for
non-quantitative majors, the math for elementary teachers,
and statistics. While we started this review at our statewide
meeting it was clear that the participants didn't feel empowered
to propose these descriptors. Participants for this review
committee should be empowered by their institutions to propose
changes. The process then should continue with a wide dissemination
of the proposed changes with feedback solicited.
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