Interdisciplinary Quantitative Curriculum Development:
Lessons from a Project in the Life Sciences
Louis J Gross
Professor of Mathematics and Ecology
University of Tennessee - Knoxville
gross@math.utk.edu
This is a talk presented at the American Mathematical Society/Mathematical
Association of America Workshop on "Changing Colleagiate Education:
Mathematical Sciences and their uses in other Disciplines", March 27-28,
1994 at the Washington, DC Marriott.
ABSTRACT: I review the progress on a curriculum development project
aimed towards increasing the quantitative skills of life science
students. This includes a summary of a workshop on quantitative
courses for life science students, a description of entry-level
quantitative course development, the software evaluation and gopher
site development component of the project, and a description of an
upcoming workshop for life scientists. Some general lessons are
abstracted from this with the emphasis on curriculum development
across disciplines. Included are some results of a survey of
mathematics faculty dealing with the issue of professional interest
in applications of mathematics to the real world. Some conclusions
are drawn regarding the difficulties associated with enhancing the
exposure of students to significant applications of mathematics.
Finally I discuss the general structure of interdisciplinary reform
of quantitative training, using the CPA approach, as well as
alternative paradigms for such reform.
Note that a brief summary of the goals of this project was
published in the February 1994 issue of BioScience 44:59.
BACKGROUND:
With support from the National Science Foundation (NSF Grant
USE-9150354), I have been working since 1991 to develop a
quantitative curriculum for life science students. The life
sciences include students in all pre-health professions (pre-
medical, pre-dental, pre-pharmacy, etc.), nursing, biology,
biochemistry, nutrition, agriculture, forestry, wildlife, and all
areas of natural resource management. The general project goal is:
To produce a flexible curriculum of quantitative courses for
undergraduate life science students, able to be integrated with the
biological courses these students take and utilizing examples from
recent biological research, thus creating a unified curriculum
which enhances a students appreciation of the utility of
quantitative approaches to address problems in the life sciences.
This would serve a serve a dual role of both introducing new
quantitative methods and reinforcing key concepts in modern
biology.
My procedure in carrying out this project has been:
a. Conduct a survey of quantitative course requirements of life
science students;
b. Conduct a workshop to bring together a group of researchers
and educators in mathematical and quantitative biology to discuss
the quantitative component of the undergraduate curriculum for life
science students;
c. Develop an entry-level quantitative course sequence based
upon recommendations from the workshop;
d. Implement the course in an hypothesis-formulation and testing
framework, coupled to appropriate software;
e. Evaluate the use of a wide variety of biological software in
quantitative courses through the use of individual student projects
and make these evaluations and software (if they are in the public
domain or Shareware) available via gopher/ftp; and
f. Conduct a workshop designed specifically for life science
faculty to discuss methods to enhance the quantitative component of
their own courses.
SURVEY OF QUANTITATIVE COURSE REQUIREMENTS FOR LIFE SCIENCE
PROGRAMS:
I conducted a survey to determine the current quantitative
requirements for undergraduates in the life sciences at many
institutions. The initial survey was done by looking at current
college catalog requirements in 1991, but this sample was expanded
by an open request on the ECOLOG-L listserve group for further
information. This information was compiled by Aaron Ellison of Mt.
Holyhoke College, and I below summarize the results.
The sample includes 86 life science programs from 47
institutions, of which 4 programs had no math requirements. The
majority of the institutions reported were research-level, doctoral
granting major universities, which may have biased the sample.
The average semesters required per life science program were:
Computer Science .128
Statistics .157
Calculus 1.26
Pre-calculus .337
Total semesters of quantitative courses 1.88
11 programs required computer science (13% of sample)
14 required statistics (16%)
68 required some calculus (79%)
26 required precalculus (30%)
There were 5 programs in which all requirements could be met just
by precalculus. Thus there were 9/86 = 10% of the programs in which
no quantitative skills above high school level were required.
The above assumes, if students had an option, the order in which
they took courses was precalculus, calculus, statistics, computer
science. Only 16 of the 86 programs allowed some options as to
which course to take.
In a subset of 34 programs at 21 universities included in the
above, obtained through my survey of college catalogs,
approximately 1/3 of the calculus courses required (or an option)
were of the social science/business/life science type.
SUMMARY OF CONCLUSIONS FROM THE WORKSHOP:
1. It is not sufficient to isolate quantitative components of the
curriculum in a few courses on quantitative topics, but rather the
importance of quantitative approaches should be emphasized
throughout the undergraduate curriculum of life science students.
2. As one means to foster the inclusion of more quantitative topics
within the curriculum, it is proposed that a Primer of Quantitative
Biology be developed to be used in conjunction with the General
Biology sequence typically included in most life science curricula.
3. Exploratory data analysis should be included in several ways as
part of a life science curriculum. Methods to do this would be as
(i) part of laboratory exercises within a biological course; (ii)
a short-course available for credit ; and/or (iii) a formal
biometry course.
4. An entry-level quantitative skills course should be developed as
a specialized year-long sequence for life science students.
Discrete methods should be the first topics covered in this course,
followed by the calculus, but the course should have a
problem-solving emphasis throughout.
5. Upper-division modeling and biological data analysis courses
should be encouraged, with extensive use of computers included as
an integral part.
THE ENTRY-LEVEL QUANTITATIVE COURSE - BIOCALCULUS REVISITED:
In response to recommendation #4 from the workshop, a pilot
version of an entry-level quantitative course for life science
students was constructed and has been taught over the past two
years to approximately 130 students, by three different instructors
(the author and two graduate students working in mathematical
ecology). The prerequisites assumed were Algebra, Geometry, and
Trigonometry.
Goals:
Develop a Student's ability to Quantitatively Analyze Problems
arising in their own Biological Field.
Illustrate the Great Utility of Mathematical Models to provide
answers to Key Biological Problems.
Develop a Student's Appreciation of the Diversity of Mathematical
Approaches potentially useful in the Life Sciences.
Methods:
Encourage Hypothesis Formulation and Testing for both the
Biological and Mathematical Topics covered.
Encourage Investigation of Real World Biological
Problems through the use of Data in class, for
homework, and examinations.
Reduce Rote Memorization of Mathematical Formulae
and Rules through the use of Software such as MATLAB and
MicroCalc.
Encourage Investigation of Quantitative Approaches in
Biological areas of Particular Interest to each Student
through Projects Utilizing Software from diverse of Bio-
logical areas.
THE PILOT FIRST-YEAR COURSE
Syllabus:
Semester 1:
Descriptive Statistics - Means, variances, using software,
histograms, linear and non-linear regression, allometry - 3 weeks
Matrix Algebra - Matrix algebra, using linear algebra software,
matrix models in population biology, eigenvalues, eigenvectors,
Markov Chains, compartment models - 4 weeks
Discrete Probability - Experiments and sample spaces, probability
laws, conditional probability and Bayes' theorem, population
genetics models - 3 weeks
Sequences and difference equations - limits of sequences, limit
laws, geometric sequence and Malthusian growth, linear first and
second order difference equations, equilibria, stability, logistic
map and chaos, population models - 3 weeks.
Limits of functions - numerical examples using limits of sequences,
basic limit principles, continuity - 2 weeks
Semester 2:
Derivatives - as rate of growth, use in graphing, basic calculation
rules, chain rule, using computer algebra software - 3 weeks
Curve sketching - second derivatives, concavity, critical points
and inflection points, basic optimization problem - 3 weeks
Exponentials and logarithms - derivatives, applications to
bio-physics, population growth and decay - 2 weeks
Antiderivatives and integrals - basic properties, numerical
computation and computer algebra systems, various applications - 3
weeks
Trigonometric functions - basic calculus, applications to medical
problems - 1 week
Differential equations and modeling - individual and population
growth models, linear compartment models, stability of equilibria,
phase-plane analysis - 3 weeks
From a student's perspective, mathematics and biology
typically appear to be disjoint subjects, with few interconnections
evident in the undergraduate curriculum. The above described course
was an attempt to provide these interconnections, within the
constraints imposed by a course specifically required in the
curriculum to be a quantitative one. In conjunction with this
however, it is important to keep in mind that perhaps the strongest
recommendation on which there was concensus at the Workshop was to
include quantitative concepts in both the General Biology sequence
typical of most curricula, as well as making use of quantitative
methods in many upper division courses in which there are natural
connections.
Many upper-division biology courses can include a quantitative
component:
Basic genetics - simple probability theory as well as difference
equations for gene frequency changes
Biochemistry - derivation of Michaelis-Menten kinetics and the
notion of a quasi-steady-state,
Molecular biology - discrete methods for sequence analysis,
Ecology - matrix methods to analyze population structure ,
difference and differential equations for species interactions and
population growth
Crop science - compartmental models utilizing linear and non-linear
systems theory
Ethology - matrix applications in developing evolutionary stable
strategy ideas.
A WORKSHOP TO FOSTER QUANTITATIVE CONCEPTS DIRECTLY IN LIFE SCIENCE
COURSES:
In an attempt to foster the inclusion of mathematical concepts
directly in life science courses, thus meeting recommendation #1 of
the workshop summarized above, a second workshop will be hed in May
of 1994. This Workshop is designed to bring together a group of
life science, mathematics, and statistical researchers and
educators to focus on the inclusion of more quantitative concepts
directly in life science undergraduate courses. The majority of
participants will be life science faculty, or faculty involved in
the quantitative training of life science students.
The basic tenet of the workshop is that it is not sufficient
to isolate quantitative components of the curriculum in a few
courses on such topics, rather, quantitative methods should be a
component of courses throughout the undergraduate life science
curriculum. The workshop will be held May 19-21 at the University
of Tennessee, Knoxville. Additionally, there will be a special
Symposium at the 1994 American Institute of Bological Sciences
Annual Meeting in August with two components. One will be session
of speakers who have developed quantitative curricula at their home
institutions. The second component will be an open computer lab,
displaying the wide range of software available to aid the
quantitative training of biologists.
APPLICATIONS WITHIN MATHEMATICS COURSES:
In his book "How to Teach Mathematics: a personal perspective"
(AMS, Providence, 1993), on the subject of applications of
mathematics Steven Krantz states ""don't get sucked into doing
trivial, artificial applications", and suggests that one should
"talk to experienced faculty in your department about what
resources are available to help you present meaningful applications
to your classes".
In regard to this, I was interested in knowing how faculty in
my home math department regarded the importance of knowledge of
some applications of math for themselves and their colleagues. This
was motivated by the NSF Initiative which I viewed as attempting to
foster major efforts by math departments to more directly couple
the mathematics curriculum to applications.
We continually make the argument that at a major university,
despite the fact that low-level math courses can readily be taught by
instructors with MS degrees, the undergraduates benefit by having faculty
teach these courses who themselves do research in math. A natural extension
of this argument, if one wants to do as the new NSF initiative in
Mathematical Sciences and their Applications Throughout the Curriculum
proposes, and include important applications of math at all levels of the
undergrad math curriculum, is that the math faculty doing this should be
researchers in areas of application. I thought myself that this is too much
for NSF to ask, but that it might be reasonable to expect
that the math faculty should at least be knowledgable in some area of
application, even if not actively pursuing research in it. Thus the below
question which I to the faculty in my department in an anonymous survey:
Do you consider the below to be an appropriate departmental
expectation of all professorial-level faculty?
All faculty members are expected to be knowledgable of the
application of an area of their mathematical expertise to
real-world situations. Knowledgable here does not mean that the
faculty member is necessarily expected to carry out research in
such an area of application. Rather, the faculty member shall be
sufficiently knowledgable so as to be able to read with
understanding the primary literature (e.g. journals) in the area of
application outside of mathematics, and be able to explain the
significance of the mathematical work in this application area to
other faculty and to students.
Please mark one response given below (or add your own if you
want!):
I consider this a reasonable request and support it 14 (45%)
I consider this a reasonable request, but would not
follow it myself 2 (6%)
I do not consider this a reasonable request, and
would ignore it 7 (23%)
I consider this an infringement of a faculty
members right to choose their own research program,
and would actively oppose it 6 (6%)
No classification checked 2 (6%)
The numbers and percentages above indicate the responses from
my colleagues. There were a number of comments expressed on this,
including: it should be an expectation that all faculty be able to
teach doctoral level courses in the areas of analysis, algebra and
topology; I thought all faculty engage in interpreting mathematical
applications when asked; this should be an expectation for all new faculty,
not the older ones; it is unreasonable to expect me to expend the
time and effort necessary to do this in my field of mathematical
interest; totally unreasonable, this is a mathematics department,
not a trade school; teaching applications to undergraduates is
remotely connected to knowing applications of our own research field;
I support this absolutely, emphatically, unconditionally; I consider
this a reasonable expectation, not necessarily a requirement for hiring,
tenure, promotion.
LESSONS FROM THE ABOVE:
It is unrealistic to expect most math faculty to have any
strong desire to really learn significant applications of math that
students will readily connect to their other course work in other
fields. The majority of math faculty do not feel particularly
comfortable with realistic applications, and cannot be expected to
expend the effort necessary to learn about them. Despite this,
there is a core group of faculty who support such efforts (who may
or may not be individuals involved in the subfields typically
associated with applied mathematics).
So what do we do to enhance quantitative understanding across
disciplines? Below is what I say to life science faculty:
Who can foster change in the quantitative skills of life science
students?
Only you, the biologists can do this!
You have two routes:
1. Convince the math faculty that they're letting you down
2. Teach the courses yourself
Note: Math faculty will not take you seriously unless you show them
how the quantitative topics you insist that they cover will be used
in your own courses!
This means biology courses must become less of a "litany of
conclusions", and more an exploration of how and why natural
systems came to be as they are. Unfortunately, this battle must be
fought over and over at each institution.
As a non-biology example in regards to the above, the Business
College faculty at my University recently decided to require all
their students to take the standard science and engineering
calculus sequence. This change was made despite the fact that the
math department had been asked by the Business college a few years
earlier to design a special sequence for their students, including
certain topics in multivariate calculus and optimization not
covered in the previous course. The change was also made despite
the fact that many incoming business students do not have
quantitative skills equivalent to the entering science and
engineering students. In fact, the Business College has had to
allow a sizable fraction of their students to continue to take this
prior course, due to their lack of preparation. My colleagues and
i viewed this change as an overt attempt to apply a filter to their
students, though the Business College swears that this is not the
objective. My colleagues and i would be much more willing to accept
this if there were any indication that the Business College had
made any effort to utilize the different skills which students get
in the standard science and engineering calculus course by revising
the business courses - no such effort has been made by any
department inthe College.
THE CPA APPROACH TO QUANTITATIVE
CURRICULUM DEVELOPMENT ACROSS DISCIPLINES:
As a summary of the approach I have taken in this life
sciences project, and in hope that this will be applicable to other
interdisciplinary efforts, I offer the CPA Approach:
CONSTRAINTS, PRIORITIZE, AID
1. Understand the Constraints under which your colleagues in other
disciplines operate - the limitations on time available in their
curriculum for quantitative training.
2. Work with these colleagues to Prioritize the quantitative
concepts their students really need, and ensure that your courses
include these.
3. Aid these colleagues in developing quantitative concepts in
their own courses that enhance a students realization of the
importance of mathematics in their own discipline. This could
include team teaching of appropriate courses.
Note: The above operates under the paradigm typical of most U.S.
institutions of higher learning - that of disciplinary
compartmentalization. An entirely different approach involves real
interdisciplinary courses. This would mean complete revision of
course requirements to allow students to automatically see
connections between various subfields, rather than inherently
different subjects with little connection. Such courses could
involve a team approach to subjects, which is common in many lower
division biological sciences courses, but almost unheard of in
mathematics courses.
Comments about the above should be sent to me at:
Louis J. Gross
Department of Mathematics
University of Tennessee
Knoxville, TN 37996-1300
gross@math.utk.edu
(865) 974-4295
(865) 974-3067 (FAX)
(865) 974-3065 (Secretary)