SUMMARY REPORT
WORKSHOP ON A QUANTITATIVE SCIENCES
CURRICULUM FOR LIFE SCIENCE STUDENTS*
KNOXVILLE, TENNESSEE FEBRUARY 6-8, 1992
OBJECTIVES: The Workshop was designed 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. The key goal of the overall project of
which this Workshop is a part involves the development of a
curriculum that would emphasize the great utility of
quantitative approaches in analyzing biological problems, use
examples from recent biological research, and serve the dual
role of introducing new quantitative methods and reinforcing
key concepts in modern biology. The main goal of the Workshop
was to develop a set of basic quantitative and biological
concepts that should be included in the curriculum. Specific
questions addressed at the Workshop included:
1. What are the possible goals for an entry-level quantitative
course for life science students?
2. What are the appropriate biological concepts we need to
include to meet these goals?
3. What are the appropriate quantitative concepts we need to
include to meet these goals?
4. How do we design a modular course flexible enough so that
it could reasonably be taught in three different modes:
through a mathematics department, a biology department, or co-
taught by both?
5. What are the most appropriate advanced topics around which
to build follow-up courses for upper-division students and
what should be the content of this set of courses?
6. What is the role of technology in all of the above, and are
there particular software packages/programs which will be most
useful in supporting various aspects of the curriculum?
SUMMARY OF CONCLUSIONS:
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. This implies that courses typically
considered part of the biology curriculum should contain
quantitative components appropriate for the topics addressed
in the course. Thus we should encourage the introduction of
quantitative skills at all levels in the life science
curricula.
2. As a means to foster the inclusion of more quantitative
topics in the curriculum, it is proposed that a Primer of
Quantitative Biology be developed to be used in conjunction
with the typical General Biology sequence included in most
life science curricula, with appropriate quantitative examples
developed for each section of the course. This Primer would be
at the level of high school mathematics, but would focus on
examples of non-intuitive results of biological importance
derived from quantitative approaches.
3. Exploratory data analysis should be included in several
ways as part of a life science curriculum. This can be done as
(i) part of laboratory exercises within a biology course; (ii)
a short-course available for credit ; and/or (iii) a formal
biometry course. The last option should be constructed around
key biological questions, rather than statistical methods.
* Supported by the National Science Foundation's Undergraduate
Course and Curriculum Program through grant #USE-9150354 to
the University of Tennessee, Knoxville
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
an integral part of such courses. Modules, based on diverse
biological topics, for use in illustrating key quantitative
concepts should be developed for these courses as well as for
the entry-level course.
General Project Goal:
To produce a flexible curriculum of quantitative courses
for undergraduate life science students, that can be
integrated with the biological sciences courses these students
take, thus creating a unified curriculum which enhances a
students' appreciation of the utility of quantitative
approaches in addressing problems in the life sciences.
A Brief History:
There were a number of courses developed during the late
1960's and early 1970's designed to provide an introduction
to the calculus specifically for life science students. These
courses were based on a number of texts which often included
topics neglected in a standard science and engineering
calculus course, such as basic probability theory, difference
equations, and qualitative theory of ordinary differential
equations. The inclusion of these topics, along with examples
of mathematical concepts using biological rather than physical
examples, came at some cost however. Namely, the level of
mathematical rigor as well as the number of techniques
discussed were lower than in the standard sequence.
Were these courses successful? Many would answer in the
affirmative, particularly those individuals involved in
constructing such courses. However, success was mixed given:
(i) the fact that almost none of the books on "Biocalculus" or
"Calculus for the Biological Sciences" are still in print, and
(ii) very few institutions either adopted or maintained these
courses in their curricula. There are several reasons why few
students benefitted from these courses. First, the courses
were often tied to particular mathematics faculty members with
research interests in biological areas. If the faculty member
moved on, the course disappeared because it was not intimately
coupled with the biology curriculum, but rather considered
essentially a mathematics course. Courses which were closely
coupled to requirements for biology students could survive a
change in faculty, as could courses taught by biology faculty
themselves. Second, mathematics programs with strong research
components in mathematical biology would also tend to maintain
these courses. These comments, though based on limited
information, point out the importance of close coupling to the
biology curriculum if a quantitative component is to be
successful.
Given that there are few programs offering quantitative
entry-level courses for life science students, what are these
students now taking? A survey of college catalogs of 21
universities, including 34 life science programs, indicates a
great diversity of requirements. Approximately thirty per cent
of the programs require two terms of science calculus, another
thirty per cent require one term of science calculus, and the
rest require calculus for business students or just algebra
and trigonometry. Only about ten per cent require a
statistics course. There is no evidence among the sampled
schools of courses designed specifically for life science
students. Are the courses offered at these schools serving the
needs of life science students? A major tenet of this
curriculum development project is that they do not.
Furthermore, we believe that we can do a much better job of
providing quantitative training for these students that makes
use of their interest in biology, and enhances their training
in biology by giving them additional exposure to the use of
theory.
In recent years, considerable attention has been paid
to reforms in the undergraduate mathematics curriculum,
particularly the calculus. The National Science Foundation has
extensively funded a variety of programs aimed at changing the
calculus from being a filter into a pump for upper division
science and engineering courses. It is perhaps too early to
say a great deal about the successes and failures of the
various reforms being attempted, many of which make use of
calculators and computer to aid the learning experience. If
there is one lesson that might be drawn thus far, it is the
success of a variety of models for mathematical training.
These appear to be at least as effective as the
lecture/recitation format that has been the standard model
over the past several decades. The success of a particular
model depend significantly upon the availability of resources
and willingness to carry out a change at a particular
institution. On the biology side, there also have been a
variety of attempts to carry out reform in the curricula.
These are not generally oriented towards improving the
quantitative reasoning skills or training of students, though
that is sometimes a component.
Topics addressed during workshop sessions:
A: Give examples of how quantitative methods allow
specific biological problems to be addressed, and how these
couple with experiment and observation to drive the field.
For faculty, provide reasons why their students, to
appreciate the scope of modern biology, need quantitative
skills. For students, point out the general role of
mathematics in aiding the development of biological theory.
Discussion summary:
The faculty, specifically the biology faculty, must
initiate the development or restructuring of introductory
quantitative courses for life science students. Yet there has
been very little effort by biology faculty at most
institutions to carry out this development. Why has there been
such a lack of interest in this? Some of the points made
included:
1) The lack of quantitative training of
agricultural/biological faculty.
2) The inherent difficulties when students become more
literate in quantitative thinking than the faculty.
3) Too few mathematicians realize the importance of
mathematical applications in biology, and many feel secure
only when teaching a theorem-proof type course.
One difficulty discussed was the lack of an adequate pool
of biologically-literate quantitatively-oriented teachers
capable of teaching quantitative courses which motivate
students by including important biological issues. To develop
a larger teacher pool for a more quantitative curricula
several ideas were mentioned.
1) Workshops and training sessions could be provided to
faculty to foster a higher degree of mathematical
sophistication among life science faculty.
2) Motivating examples should be provided for both life
science and mathematics faculty to illustrate situations in
which mathematics is directly applicable.
3) Encourage research across disciplines. It was stated that
some faculty had difficulty getting recognition for their
research simply because it was not directly related to their
field, and was not published in a journal of their discipline.
4) An obvious place to start incorporating more quantitative
ideas was in the standard statistics course.
Discussion on the topic "Should there be a
specialized entry level course for life science majors?" was
left unresolved, but did point out several difficulties that
a mathematics department would face in instituting a course
specifically for life science students.
1. There is little quality control. Some mathematics
departments have difficulty finding qualified instructors for
the introductory courses they offer. Many of the present
introductory courses are taught by adjuncts and graduate
students. Would this type of instructor be adequate for a
mathematics/quantitative biology course?. 2. There is a
lack of transportability. Assuming a freshman were to switch
majors, would a calculus for biology course be acceptable for
a management curriculum? 3. There are the bureaucratic
constraints. Can a diversity of introductory courses be
scheduled effectively?
4. Finally, there is the question: will mathematics
education still be as effective if business administration,
engineering, life sciences, social sciences, education, pure
mathematics, and honors programs all have separate
introductory courses?
From a student's perspective, mathematics and
biology typically appear to be disjoint subjects, with few
interconnections evident in the undergraduate curriculum. This
difficulty is not limited to life science students, but is
enhanced by the relative lack of quantitative emphasis in
biological courses in the curriculum. Perhaps the strongest
recommendation on which there was consensus at this Workshop
was to include quantitative concepts in both the General
Biology sequence and upper division courses in which there are
natural connections to quantitative methods. Suggested topics
for which quantitative methods could be utilized include:
basic genetic, incorporating the use of simple probability
theory as well as difference equations for gene frequency
changes; biochemistry, emphasizing the derivation of
Michaelis-Menten kinetics and the notion of a quasi-steady-
state; molecular biology, incorporating a variety of discrete
methods for sequence analysis; ecology, incorporating matrix
methods to analyze population structure in addition to
difference and differential equations for species interactions
and population growth; crop science, including compartmental
models utilizing linear and non-linear systems theory; and
ethology, including matrix applications in developing
evolutionary stable strategy ideas. Inclusion of regular
connections with quantitative topics at many different points
in the life science curriculum will rapidly reinforce for
students the importance of mathematical concepts, and do so
much more effectively than if such topics were isolated in a
specialized set of courses.
B: Are there key quantitative topics that we all agree
should be included in an entry-level course? Can we set
priorities for these and other concepts that might be
included? Similarly, are there biological concepts that
naturally are best introduced in a quantitative course, that
enhance a students exposure to biological topics, and
augment the topics covered in a general biology course?
How can we assure that the biology introduced to illustrate
quantitative topics doesn't add to confusion because the
students have little prior biological training?
Discussion summary:
Mathematical Concepts:
The following discussion assumes a prerequisite for an
entry-level course is a standard pre-calculus course,
including trigonometry. There were three general points of
agreement:
(1) The course should have a "problem-solving emphasis"
and should include simple modeling problems that can be
approached from many points of view. Thus in the beginning
of the course problems stemming from empirical biological
data could be analyzed from a descriptive statistics point
of view. Later, the same underlying biological problem might
be analyzed using difference or differential equations,
using the data to either estimate parameters or evaluate
model predictions.
(2) Discrete and continuous methods should be integrated
into the course, but discrete mathematics should take the
lead in the first course.
(3) The following broad course outline was developed:
Semester I:
1. Descriptive Statistics: including curve
fitting and non-linear, least-squares regression.
2. Matrix Algebra: up to and including
eigenvalues and eigenvectors.
3. Discrete Probability.
4. Difference Equations: limits would be
introduced in this context.
5. Differential Calculus: final 25% of course,
including limits, continuity and derivatives.
Semester II:
1. Differential Calculus: rates of change,
exponential growth and decay presented from both discrete
and continuous perspective.
2. Integral Calculus: closed-form solutions de-
emphasized and numerical integration given equal status.
3. Differential Equations: including numerical
techniques such as Euler, phase plane analysis, and
stability notions for equilibrium points.
Caveats:
One should guard against the course becoming a
hodgepodge of topics. Natural connections between topics
should be exploited. Modeling problems that can be
approached using different tools can serve to unify the
course. Also, the course should try to preserve precalculus
skills during the discrete math portion, for example using
logarithms in the curve fitting.
Biological concepts:
A general goal is to lower the life science
students' math anxiety level and help the student become
comfortable with looking at biological phenomena from a
quantitative perspective. Concepts which pervade all the
biological sciences were discussed. A reasonable list is:
1. growth
2. feedback
3. variation
4. interaction
5. time series
A key point regarding the utility of mathematics is that it
aids in reducing the volume of biological information
necessary to analyze a biological problem by providing
unifying conceptual formulations.
Quantitative methods should be included in a
General Biology course, and should focus on cases in which
the mathematics is necessary either to understand the
biological problem or to reduce the volume of information
presented, thus aiding comprehension. The following is a
list of motivating examples to consider for further
development. These could serve as a reserve that could be
drawn upon as necessary to illustrate either quantitative
points or explicate a biological problem. The list is not
intended to be complete.
1-locus, 2-allele with heterozygote inferior
2-species predator-prey (effect of a perturbation
such as pesticide on a system at equilibrium)
the counter-intuitive effect of pesticide
application when a natural enemy controls the pest
gene-flow migration through a population
enzyme kinetics (concept of reaction rates that
depend upon concentrations)
population structure (age and size)
competitive and non-competitive interactions (for
chemical as well as biological species)
faunal and floral equilibria
population growth and harvesting (from equilibrium
to dynamical situations)
alternative hypotheses for aging
morphology (effects of parameter changes on
developmental pathways)
population growth and maximum sustainable yields
Hardy-Weinberg and deviations from it
mutation-selection balance
sex-ratio and group selection arguments
A key point is that the introduction of
quantitative concepts in General Biology should not be done
in a way which increases the content, but rather helps to
decrease the amount of descriptive material covered.
Quantitative content might reasonably be increased, even
with the loss of biological content, because the earlier
that students see that quantitative methods are an integral
part of the life sciences, the more willing they will be to
pursue courses with a greater mathematical content. Also, a
typical curriculum contains many opportunities for
biological training to be enhanced in courses succeeding
General Biology, but very few opportunities to enhance
quantitative training.
Some general comments are that we need to find
methods to counter the widely held notion that biology is
"math free". Students are well aware that historically
biology has been taught in a non-quantitative framework. It
is viewed as the last math-free refuge for the science
major. We need to prepare students for the increasing impact
of quantitative methods in the life sciences. One means to
overcome mathematics illiteracy in the biology community is
to focus some efforts on increasing the mathematical
competency of current life science faculty. A series of
workshops or courses could do this, with appropriate
support.
C: How might the structure of an entry-level course vary
for general biology students versus those with more specific
life science interests? Are there particular topics/formats
which lend themselves well to an audience of general biology
students, but should be substituted in some way for pre-
health or agriculture students?
Disciplines including agriculture, pre-health, and
resource management are differentiated from general biology
by their inclusion of people, decision making, and social
and socioeconomic systems. Students in these applied
disciplines, therefore, require a quantitative curriculum
with an emphasis on decision making, data analysis,
probability, and risk analysis. Despite their interest in
applied biology, however, students in agriculture, pre-
health, and resource management (some felt life science
students in general) enter college poorly prepared in
mathematics, often have an aversion to mathematics, and
often have chosen to enroll in a life science or related
program specifically to avoid mathematics. The curricula in
these disciplines do not now incorporate very much
mathematics despite the fact that these disciplines are
highly involved with describing and interpreting data,
modeling, and dealing with uncertainty and complexity. These
disciplines have a tradition of using quantitative science,
especially statistics. Further, the increasing complexity of
these disciplines requires students to become more adept at
using quantitative analytical techniques.
The group felt that the specific curricular needs
in mathematics include: decision analysis, probability and
statistics, risk analysis, linear algebra, difference
equations, differential and integral calculus, and
differential equations. The focus in these areas should be
on data analysis and decision making, motivated through the
use of realistic examples, avoiding formalism and stressing
fundamentals, building conceptual understanding and basic
foundations for upper level courses, and avoiding
unnecessary computational complexity.
Such a mathematics course could be common for all
life science students and substantial integration with other
courses and years of the students' curriculum can easily be
obtained. Thus there is no need at the entry-level for
separate courses designed specifically for pre-health or
agricultural students. Rather, sections of a common
introductory course could provide the emphasis on
uncertainty and decision making required for these applied
students.
D: Provide some alternative methods to structure upper
division courses to build upon the biological training of
students as well as their quantitative skills developed in
the entry-level course. How can we best increase the
biological content of these courses to ensure that
biostatistics isn't merely a cook-book course, but rather is
closely tied to real biological problems? Can a modeling
course be run in a true problem-solving, exploratory mode
for the students?
An overall goal for upper division quantitative
courses is to prepare students to read, with comprehension,
the basic literature in their field.
Goals and Structures for a Modeling Course
1) WHAT IS MEANT BY MODELING?
The group examined a number of variants of the term
"model", including the statistical parameter fit-to-data
approach, conceptual models in which physical processes are
described from first principles, and models that capture the
essence of a some behavior in a more metaphorical way. It was
agreed that a modeling course should expose the student to a
variety of approaches.
The continuum from "strategic " to "tactical " models was
discussed. This refers mainly to the use to which a model is
put, but "stategic" models often emphasize possible mechanisms
and "tactical" models tend to focus on the detailed
understanding of a particular system, such as a pond. The
appropriate model depends on the specific aims or goals of the
modeler. A good mixture of general conceptual as well as
particular system models should be included. Any course on
modeling should include some problem-solving, some exposure to
"real data" and data analysis as well as modeling efforts by
the students on topics that they develop and work on
individually.
2) HOW SHOULD MODELING BE TAUGHT?
We considered three separate issues related to teaching
students how to model:
(A) HOW TO CONVERT VERBAL INFORMATION TO A MODEL
Students have difficulty with word-problems and the steps
through which a verbal description of a process is translated
into mathematics. The point was made that models need not
always consist of equations or mathematical symbols, but could
include graphical or diagrammatic (e.g. flow chart) models.
These structural modeling approaches can provide insight, even
for students who have weaker backgrounds in mathematics.
It was postulated that most problems are solved by
recognizing a connection with previously encountered problems,
and that the first part of such a modeling course should
introduce numerous examples. The ideas of proceeding from the
simple to the complex in gradual increments, and the process
of trial and error, were emphasized.
(B) HOW TO POSE THE RIGHT QUESTIONS
Even more difficult than the above is teaching the
approaches one might take when it is not obvious or
predetermined what the right questions are. We discussed the
fact that defining the right variables, selecting the
appropriate modeling strategy, and arriving at a sufficiently
tractable verbal caricature of a system are fundamental
problems - indeed, they are the same problems that we face as
researchers. There was conflicting opinion about whether the
four-step approach in Polya's "How to Solve It" truly depicts
the way we, as scientists, solve problems. In our limited
discussion time, these issues were not resolved.
We did, however, note the importance of good case studies
and "realistic" or captivating examples to motivate modeling.
Examples from the physical sciences include: (a) A Disney
movie of ping-pong balls and traps; (b) the drinking duck
oscillator; (c) a three-magnet pendulum; (d) the height of
foam on a glass of beer; and (e) a black-box containing an
electronic device (a nine-volt battery) to be analyzed by
external measurements of voltage. A similar list of examples
from the biological sciences could include cases in many of
the topic areas mentioned above under biological concepts.
(C) HOW TO ANALYZE THE MODEL
Our discussion did not emphasize this issue since,
presumably, one goal of the course would be to teach the
appropriate analytical techniques. Developing models that are
powerful, without unnecessary complexity is a talent that
evolves gradually, with increasing experience.
(D) HOW TO INTERPRET THE PREDICTIONS
A good model is one which contributes to understanding or
insight about the behavior of the system. Students need to be
able to convert their symbolism back to clear verbal
statements about the system being modelled. We strive for
examples in which the mathematics leads to conclusions that
might have escaped us had we been restricted to simple verbal
arguments, not those in which the predictions are trivially
related to the assumptions we made at the outset.
3) WHAT SHOULD BE INCLUDED IN A MODELING COURSE?
A list of possible mathematical topics, with the associated
biological case studies and examples appears below:
____________________________________________________________
____________
MATHEMATICAL TOPIC BIOLOGICAL EXAMPLES
____________________________________________________________
____________
Optimization chemical equilibrium (free energy)
Linear programming allocating resources
(root/shoot)
Decision analysis harvesting, fisheries,
bioeconomics
optimal foraging
medical decisions
-------------------------------------------------------------
-----------------------------------
Matrix Methods age distributions (Leslie Matrices)
life cycle models (see book by Hal
Caswell)
compartment models (see book on
linear models
by Michael Cullen)
ESS allocation of reproductive
effort (see book
by Eric
Charnov)
-------------------------------------------------------------
-----------------------------------
Dimensional analysis Michaelis Menten kinetics
chemostat
allometry (see book on Scaling
by Knut Schmidt-Nielsen)
Spruce Budworm models (see various
papers
by Donald Ludwig and collaborators
e.g. J. Anim. Ecol. 47:315-332)
-------------------------------------------------------------
----------------------------------
Phase Plane Analysis above examples in dimensional
analysis
grazing models (see I. Noy-Meir
J. Appl. Ecol. 15:809-835)
classical Lotka-Volterra models
spread of epidemics
temperature control in animals
-------------------------------------------------------------
---------------------------------
Nonlinear Behavior discrete logistic
Steady states population dynamics
Stability log logistic
discrete vs. continuous dynamical diseases
1 and 2D maps Leisch-Nyham defect
parameter sensitivity sickle cell anemia
chaos patch-dynamics models
-------------------------------------------------------------
---------------------------------
Finite Markov chains epidemic models
coding of point mutations
population genetics
migration
behavioral sequence analysis
density dependent succession
island biogeography
rainfall events
-------------------------------------------------------------
---------------------------------
Birth and Death Processes molecular evolution
coalescence theory
-------------------------------------------------------------
---------------------------------
Multivariate Models migration
Partial differential equations molecular diffusion
Diffusion pheromones
genetic drift
random walks (see book
by Howard Berg)
pattern formation (Turing models)
-------------------------------------------------------------
--------------------------------
Projective Geometry multivariate statistics
community descriptions
numerical systematics & taxonomy
-------------------------------------------------------------
--------------------------------
Data Analysis hare-lynx data
Meakin vs. Gompertz growth
dissociation curves (DNA)
drug clearance rates
-------------------------------------------------------------
--------------------------------
Model Evaluation successful vs. less successful
case studies
Goals and Alternative Structures for Biostatistics
Premise: Many biostatistics texts and courses are
carried out in a "cookbook" manner which teaches students a
number of tools but leaves them with little idea of overall
statistical principles
and with few data analysis skills. To remedy this we
propose the addition of a laboratory module or short course
to either a concept-based statistics course or a data
analysis rich
biology course. In some form such a course should be a
required component of a life-science curriculum.
Laboratory module contents: The module would be
constructed around an easy-to-use, flexible, statistical
computing package with superior graphics capability. The
focus would be on data analysis and re-analysis, both at
interim points and at the conclusion of a study. Ties to a
strongly conceptual statistics course or a biology course
would be necessary to provide a framework to avoid over-
evaluation of data. The links between biological principles,
experimental design and statistical analysis would be
emphasized. Topics which are not usually included in a
course at this level, but should be included in this course
are: multivariate statistics, presented from conceptual and
graphical viewpoints; nonlinear least squares estimation for
mathematical models. Some instructors may choose to
incorporate simulation or resampling based, computer
intensive methods for nonstandard problems (e.g.
bootstrapping). The corresponding statistics or biology
course would need to emphasize the following ideas:
sampling, experimental design, observational versus
experimental studies, bias and confounding.
A Sample Curriculum for Agricultural and Biological
Engineers
Premise: Essentially all curricula for these students
follow the same basic mathematics courses as generally
required for physical science and engineering students,
consisting of approximately two years of calculus, linear
algebra and differential equations. Thus the objective of
the curriculum briefly described below is to couple these
students' prior training in mathematics to realistic and
practical applications in biology. The curriculum would be
modified appropriately for students in particular programs,
such as biomedical engineering. The below list of topics
might represent distinct courses in the curriculum, or could
be combined in a shorter sequence of courses.
1. Properties of materials in biological systems
2. Transport processes in biological systems
3. Instrumentation and controls for biological systems
4. Biological science for engineers
5. Modeling and simulation of biological and
agricultural systems.
E: Is a computer-based course necessary? What are its
advantages, if any, over a course using graphing
calculators? How can instructors find appropriate software
and incorporate it in the courses?
Entry-level Courses:
The majority of students will use computers after they
graduate, therefore a computer element should be included in
any quantitative course. Furthermore, the use of calculators
and computers will enable an instructor to spend less time
on details such as techniques of integration. This allows
for relatively more time to be spent on conceptual ideas as
well as understanding and exploring more complicated non-
linear models of biological processes and their critical
evaluation. In particular, the use of graphing facilities
such as sketching and curve fitting, and the use of computer
algebra systems (Derive, Mathematica, Theorist, Maple, etc.)
was stressed. Although these packages are not perfect, and
may require a student to reformulate a problem to enable a
solution to be found by the algebra system, this provides
new incentives for a student to work on basic ideas such as
substitution methods for integration.
It was felt that both calculators and computers have a
role in an entry-level course. Relatively inexpensive
graphing calculators are readily available, and provide
advantages for in-class work when computer laboratory
facilities are in short supply. Often the abilities of
students in a class vary widely, from "Nintendo kids" to
those who suffer from technology anxiety. One solution is to
split the students into small work groups early in the
course, each group containing students with varying computer
experience.
Though there is an abundance of potentially useful
software for biological problem solving , the quality of
these is very mixed. A resource where faculty could obtain
reviews of software useful in entry-level courses would be
quite beneficial. Faculty as well as students would require
the time and training in such technology, and faculty would
perhaps need help in learning how to cope with more open-
ended, project oriented courses that the use of this
technology can foster.
Upper Division Courses:
Computers have become a valuable tool for
understanding and solving problems associated with modeling
biological phenomena. Therefore computing should be an
important part of a modeling course so that students have
an opportunity to observe the interplay between computation
and analysis that is typical of modern research in
quantitative biology. Computers will affect both the way
modeling courses are taught, and the order of included
topics. This is especially evident in the computer classroom
in which the traditional lecture style gives way to
individual exploration of problems, although most faculty
will find it appropriate to mingle the two teaching
approaches, either by holding computer laboratory sessions
or by assigning homework which requires computer use. The
use of calculators alone was thought to be inappropriate
in courses at this level.
There are several potential misuses of technology in
these courses:
(a) failure to think carefully enough about a problem
to ensure that the computer is a necessary tool to
investigate it;
(b) having too much confidence in the computer so that
one uncritically accepts misleading and incorrect output;
(c) having too little knowledge of algorithms used or
programming, so that they are used inappropriately; students
may well lack the ability or confidence to modify or develop
their own small programs.
(d) spending excessive time on programming details -
it is unrealistic to expect that life science students will
have extensive programming experience at the undergraduate
level, and the variety of languages and systems available
would make teaching a class that relied on the students'
abilities in this area very difficult. Despite this, some
knowledge of programming is extremely useful, if only to
reinforce the use of logical approaches to formulate the
solution to a problem.
There is a diversity of software available for use in
courses in the life sciences. Biologists and mathematicians
teaching upper division courses need access to a
clearinghouse or listing service for such software. Although
reviews appear regularly in a variety of journals, there
seems to be no compendium of these that biologists and
mathematicians might subscribe to.
F: Is a modular approach appropriate for the entry-level
course? If so, how can we aid an instructor in choosing
appropriate modules that cover key concepts and can be
unified, as opposed to a piecemeal assortment of biological
examples coupled to mathematical concepts? For upper
division courses, should these modules suggest avenues for
independent research by students, or quickly present
various biological topics that would often be glossed over
in non-quantitative biology courses?
I. In selecting examples to use in an entry level course,
it is important to keep the following points in mind:
a. remind the class that the mathematical structure
provides unity for the
course;
b. select examples that can be be approached by more
than one mathematical technique;
c. use some examples as problems for in-class workshop
sessions, or class projects that can be
feasibly completed by groups of students outside the formal
class sessions.
II. Examples should come from at least the following four
major areas of biology:
a. photosynthesis;
b. population dynamics and ecology;
c. genetics;
d. cell and organismal physiology.
III. The relationship between continuous and discrete
processes can be emphasized at many points; examples include
population dynamics, and photosynthesis (at the level of the
leaf vs. the chloroplast). In addition, for some
applications, discrete models are exact (seasonality,
population dynamics of organisms with synchronous patterns of
birth/death), whereas continuous models provide an
approximation.
IV. The following suggested examples are of biological
topics meant to accompany mathematical topics. Letters in
parentheses denote the biological areas described above.
Topics in square brackets are other examples that do not fit
into the area classification but could be included at the
instructor's discretion.
descriptive statistics, least squares regression
(a) rate of photosynthesis vs. light
(c) quantitative genetics
[allometry, scaling, log-transformations]
[classification of organisms, numerical taxonomy]
matrix algebra and eigenvalues
(b) life tables and stage structured populations
(b) plant succession
(b) ecosystem models
(c) population genetics
[pharmacokinetics]
discrete probability
(a) arrangement of leaves/chloroplasts in photosynthesis
i.e., the probability of a photon being utilized
in the photosystems
(c) mutation at the molecular level
(c) Mendelian genetics
(d) immunological response
[animal behavior models]
[Shannon-Simpson index]
difference equations and stability of difference equations
(b) population growth
(c) operons and genetics
limits
there are many applications on this list that
illustrate the process of going from a
discrete to a continuous formulation, e.g. genetic
variance, light hitting leaves vs.
chloroplasts, population dynamics
rates of change (discrete and continuous)
(b) population dynamics
(b) equilibrium biogeography
(c) divergence of DNA sequences and the molecular clock
(c) mutation-selection balance
(d) enzyme kinetics
(d) physiological homeostasis
(d) diffusion from the cell membrane
(d) uptake of substances from fish gills, chemical
exposure
exponential growth and decay
(b) population dynamics
(d) dye-dilution techniques, drug physiology, alcohol
clearance from the blood
integrals
(a) photosynthesis measured as a function of degree days
(other processes also)
(b) community productivity
(b) estimating depletion of natural resources
[volumes of objects such as bird eggs, tomatoes,
clams]
numerical integration
(d) areas under peaks of a curve in chromatography
[probability density functions]
differential equations and stability (including phase plane
analysis)
(b) population dynamics
(b) ecological interactions
(b) chemostat
[diffusion, as in diffusion vs. convection, or
movement through xylem or phloem]
G: Are there natural ways that the quantitative courses
might best be coupled to entry-level general biology, and
more advanced biology courses? How do we foster integration?
How do we increase the availability of appropriate material?
A theorem postulated at the Workshop was that if
mathematicians and biologists are brought together, great
wonders will occur. This general conclusion was based on the
experiences of many of the participants who found that
collaborative ventures lead to entirely new approaches to old
problems, in addition to pointing the way to entirely new
problems. A clear conclusion was that fostering an increase in
quantitative skills of life science students must come from
the life science faculty themselves - in general, this
increase will not occur if the impetus is left solely to
colleagues in mathematics. A variety of programs were
suggested to encourage life science faculty to increase their
own quantitative competence, and that of the graduate students
they are training. These include: setting up a modeling center
in which biologists might work jointly on problems with
mathematicians (in a similar manner to the statistical
consulting centers at many institutions), short-courses on
mathematical modeling in biology designed specifically for
life science faculty, post-doctoral and sabbatical programs in
quantitative biology tenable, and encouragement of team-taught
courses in which a biologist and a mathematically trained
colleague collaborate. Throughout, agreement was reached that
biology needs to develop its own cadre of mathematically-
competent researchers and teachers, and that this should be
viewed as a natural progression within biology.
Incorporating more quantitative concepts in biology courses
at all levels has the important effect of improving students'
capabilities to read with comprehension the literature in
their own field of interest. This in turn might entice
motivated students (i.e. those going on to graduate work in
life science) to invest the necessary effort in quantitative
topics, but it may not be sufficient for the general life
science student. For these students, regular examples are
needed in General Biology and upper division courses which
illustrate, using quantitative methods, that an individual's
naive biological intuition may be very wrong. Examples include
situations in which pesticide application actually increases
the density of pests, when projections of population growth
based on age structure are wrong because size or stage is the
key factor related to individual survival and fertility, and
when increases in harvesting effort expended on a particular
fish stock actually lowers the yield. Such counter-intuitive
results should be supplemented with examples of situations in
which mathematical approaches led to major new biologically
significant results, for example epidemiology theory providing
the basis for successful immunization programs and biocontrol
models leading to predictions as to when techniques such as
sterile male release will be successful. Examples related to
societal and human issues have a special power to convince
students of the efficacy of quantitative approaches, and
should be emphasized. Listed above in Section A are several
examples of quantitative methods which can be included in
upper division biology courses, with the general consensus
being that courses as diverse as Vertebrate Physiology, Plant
Physiology, Population Genetics, Population Biology, Ethology,
Evolutionary Biology, and Molecular Evolution could all be
taught in a manner which includes substantive quantitative
concepts. This is of course in addition to whatever
statistical methods might be used in these courses. An
additional suggestion was that the life science curricula
should include at least one upper division course with a
strong quantitative component, offering students the option to
choose from a list of courses such as: Biological Data
Analysis, Models in Biology, or Experimental Design. The
inclusion of such a requirement in the curricula would further
emphasize to students the importance of quantitative training.
In addition to the above, there are ways by which
professional societies can help to enhance the quantitative
training of life scientists. One suggestion which would serve
to enhance the prestige of quantitative research among life
science colleagues would be for a society, such as the Society
for Mathematical Biology, to offer occasional awards to
recognize significant recent accomplishments in a variety of
biological disciplines which relied upon quantitative methods.
There is no need for these awards to involve any amount of
funds, since the prestige alone may help to convince those
colleagues in biology who have some antipathy towards
mathematics that quantitative skills are essential for modern
biologists. Additional suggestions included efforts to revise
the quantitative standards required on such tests as the CAT,
since these were viewed as outdated by many participants.
PARTICIPANTS
Ha. Rest Akcakaya - Applied Biomathematics, Inc.
Dennis Baldocchi - Atmospheric Turbulence and Diffusion
Division, NOAA
Jaroslav Benedik - Department of Genetics, Masaryk
University, Czechoslovakia
Leslie Bishop - Department of Biology, Earlham College
Mahadev Bhat - Department of Agricultural Economics,
University of Tennessee
William Bossert - Division of Applied Science, Harvard
University
Russell Butler - Department of Biology, Vanderbilt
University
Charles Clark - Department of Mathematics, University of
Tennessee
Michael Cullen - Department of Mathematics, Loyola Marymount
University
Jim Cushing - Department of Mathematics, University of
Arizona
Lothar Dohse - Department of Mathematics, University of
North Carolina at Asheville
Jim Drake - Department of Zoology, University of Tennessee
Leah Edelstein-Keshet - Department of Mathematics,
University of British Columbia
Bard Ermentrout - Department of Mathematics, University of
Pittsburgh
Henry Foehl - Department of Mathematics, Philadelphia
College of Pharmacy and Science
Henry Frandsen - Department of Mathematics, University of
Tennessee
Lev Ginzburg - Department of Ecology and Evolution, SUNY -
Stony Brook
Anil Gore - Department of Statistics, University of Poona,
India
Louis Gross - Department of Mathematics, University of
Tennessee
Brian Hahn - Department of Applied Mathematics, University
of Capetown
Thomas Hallam - Department of Mathematics, University of
Tennessee
Art Heinricker - Department of Mathematics, University of
Kentucky
Carole Hom - Department of Mathematics, University of
California - Davis
Henry Horn - Department of Biology, Princeton University
Dan Hornbach - Department of Biology, Macalester College
Fern Hunt - National Institute for Standards and Technology
James Jones - Department of Agricultural Engineering,
University of Florida
John Jungck - Department of Biology, Beloit College
Denise Kirschner - Department of Mathematics, Vanderbilt
University
Mark Kot - Department of Applied Mathematics, University of
Washington
Suzanne Lenhart - Department of Mathematics, University of
Tennessee
Philip Maini - Centre for Mathematical Biology, Oxford
University
Chuck McCulloch - Biometrics Unit, Cornell University
Michael Mesterton-Gibbons - Department of Mathematics,
Florida State University
Dung Ba Nguyen - Department of Therapeutic Radiology, Yale
University School of Medicine
John Norman - Department of Soil Science, University of
Wisconsin-Madison
Stuart Pimm - Department of Zoology, University of Tennessee
Muriel Poston - Department of Botany, Howard University
Jon Seger - Department of Biology, University of Utah
Nicholas Stone - Department of Entomology, Virginia
Polytechnic Institute and State University
Marcy Uyenoyama - Department of Zoology, Duke University
For further information about the Workshop, or any aspect of
this Curriculum Development Project, contact:
Dr. Louis Gross
Mathematics Department
University of Tennessee
Knoxville, TN 37996-
1300
(615)974-4295
(615)974-2461
(Secretary)
(615)974-6576 (FAX)
gross@math.utk.edu
(INTERNET)
gross@utkvx (BITNET)