This continues the previous post.
2. Interdisciplinary Science
A good example of how many problems there are in trying to
build better freshman and sophomore courses that address
integration of material is to look at
my attempts to increase the integration of biology, mathematics and
computation in the training of new biological scientists.
This has made me focus on the much deeper questions of
how to train students to think in an interdisciplinary manner
not only in the biological sciences but also as part of a new
general educational curriculum.
I think it is clear that to do this, we will have to foster a new
mindset within both the students and the faculty related to interdisciplinary
interaction. In addition, we will have to introduce new metrics for faculty
evaluation and reward systems to ensure that the hard work
and creative effort to build such a curriculum is acknowledged.
This means also that new discussions of the tenure process are inevitable.
The following discussion will add some meat to the bones of this tale.
Let's focus on the interdisciplinary triad of ideas from
Biology, Mathematics and Computational tools
which for expositional convenience, I will denote by the symbol
BMC. The implementation of BMC will require the building
of bridges between mathematical, computer and biological sciences departments.
For convenience, let's call this integrative point of view Building Biological Bridges
or B3.
The typical biological sciences major thus must have a deeper
appreciation of the use of BMC as bridges are built
between the many disparate areas of biology and the other sciences.
The sharp disciplinary walls that have been built in academia hurt our
students chances at developing an active and questing mind that is able
to both be suspicious of the status - quo and also have the tools
to challenge it effectively. Indeed, I have longed believed that
all research requires a rebellious mind. If a student revers
the expert opinion of others too much, they will always be afraid
to forge a new path for themselves. So respect and disrespect are both part of
the toolkit of our budding scientists.
Blending many disciplines into one program, even when there are good reasons to
do so, is very hard.
2.1 Astrobiology As Metaphor For Biomath
Consider the
attempt to create a new Astrobiology program at the University of Washington.
Even a cursory examination of the new textbook
Planets and Life: The Emerging Science of Astrobiology
(W. Sullivan and J. Baross, editors. Planets and Life:
The Emerging Science of Astrobiology. Cambridge
University Press, 2007.)
illustrates the wealth of knowledge such a field
must integrate. This integration is held back by students who are
not trained in both BMC and B3.
Let's paraphrase some of the important points made by the graduate
students enrolled in this new program about interdisciplinary training
and research. Consider what the graduate students in this new
program have said ( page 548 in Sullivan's book )
`` ...some of the ignorance exposed by astrobiological
questions reveals not the boundaries of scientific knowledge,
but instead the boundaries of individual disciplines.
Furthermore, collaboration by itself does not address this ignorance,
but instead compounds it by encouraging scientists to rely on each
other's authority. Thus, anachronistic disciplinary borders
are reinforced rather than overrun. In contrast
astrobiology can motivate challenges to disciplinary
isolation and the appeals to authority that such isolation fosters.''
Indeed, studying problems that require points of view
from many places is of great importance to our society and
from (page 548 in Sullivan's book ), we hear that
``many different disciplines should now be applied to
a class of questions perceived as broadly unified and
that such an amalgamation justifies a new discipline
(or even meta discipline) such as astrobiology.''
Now simply replace the key word astrobiology
by biomathematics or B3 and reread the sentence again.
Aren't we trying to do just this when we design these new interactions?
Since we believe we can bring additional
illumination to problems we wish to solve in
biological sciences by adding new ideas from mathematics and
computer science to the mix, we are asking explicitly for
such amalgamation and interdisciplinary training.
We thus want to create a cadre of biomathematically literate majors
who believe as the nascent astrobiology graduate students do
( page 549 in Sullivan's book ) that
``Dissatisfaction with disciplinary approaches to fundamentally
interdisciplinary questions also led many of us to major in
more than one field. This is not to deny the importance of
reductionist approaches or the advances stimulated by them.
Rather, as undergraduates we wanted to integrate the results of
reductionist science. Such synthesis is often poorly
accommodated by disciplines that have evolved, especially in
academia, to become insular, autonomous departments. Despite the
importance of synthesis for many basic scientific questions, it is
rarely attempted in research...''
To paraphrase ( page 550 in Sullivan's book ), we believe that
``[B3 enriched by BMC] can change this by challenging the ignorance
fostered by disciplinary structure while pursuing the
creative ignorance underlying genuine inquiry. Because of its
integrative questions, interdisciplinary nature, ...,
[B3 enriched by BMC] emerges as an ideal vehicle for scientific
education at the graduate, undergraduate and even high school levels.
[It] permits treatment of traditionally disciplinary subjects
as well as areas where those subjects converge (and, sometimes, fall
apart!) At the same time, [it] is well suited to reveal
the creative ignorance at scientific frontiers that drives discovery.''
To address these needs and concerns,
a Quantitative Emphasis Area within the Department of
Biological Sciences at my university has been developed.
I have designed the mathematical components of this area
mostly alone because there is little interest within my
own Department of Mathematical Sciences for this particular
interaction. At my university, a reorganization some years
ago placed Mathematical Sciences into the Engineering College
and this profoundly altered my department's perceptions
about college level interaction. Biology is in another college
and, although it shouldn't make a difference, it does.
Interdisciplinary interaction is defined in adhoc
ways using engineering examples and since tenure and promotion
decisions are within the engineering college, there is a general
movement towards work with an engineering character.
With this said, my colleagues in biology and myself have labored to
design this area which will be a linchpin in our plan for the
interdisciplinary training of biomathematical majors.
My colleagues and I
believe,as do the astrobiology students, ( page 550 in Sullivan's book )
that
``The ignorance motivating scientific inquiry will never
wholly be separated from the ignorance hindering it.
The disciplinary organization of scientific education
encourages scientists to be experts on specialized
subjects and silent on everything else. By creating
scientists dependent of each other's specializations,
this approach is self-reinforcing. A discipline of
[ B3 enriched by BMC]
should attempt something more ambitious: it
should instead encourage scientists to master for themselves
what formerly they deferred to their peers.''
Indeed, there is more that can be said ( page 552 in Sullivan's book )
``What [B3 enriched by BMC] can mean as a science and discipline is yet
to be decided, for it must face the two-fold challenge of cross-disciplinary
ignorance that disciplinary education itself enforces. First, ignorance
cannot be skirted by deferral to experts, or by other implicit
invocations of the disciplinary mold that [B3 enriched by BMC]
should instead
critique. Second, ignorance must actually be recognized. This is not
trivial: how do you know what you do not know? Is it possible to understand
a general principle without also understanding the assumptions and
caveats underlying it? Knowledge superficially ``understood''
is self-affirming. For example, the meaning of the molecular
tree of life may appear unproblematic to an astronomer who has
learned that the branch lengths represent evolutionary distance, but will
the astronomer even know to consider the hidden assumptions about
rate constancy by which the tree is derived? Similarly, images
from the surface of Mars showing evidence of running water are prevalent
in the media, yet how often will a biologist be exposed to alternative
explanations for these geologic forms, or to the significant
evidence to the contrary? [There is a need] for a way to
discriminate between science and ... uncritically accepted results of
science.''
A first attempt at developing a first year
curriculum for the graduate program in astrobiology
led to an integrative course in which specialists
from various disciplines germane to the study of
astrobiology gave lectures in their own areas
of expertise and then left as another expert took over.
This was disheartening to the students in the program.
They said that (page 553 in Sullivan's book)
``As a group, we realized that we could not speak the
language of the many disciplines in astrobiology and that
we lacked the basic information to consider their claims
critically. Instead, this attempt at an integrative approach
provided only a superficial introduction to the major contributions
of each discipline to astrobiology. How can
critical science be built on a superficial foundation? Major gaps
in our backgrounds still needed to be addressed. In addition, we
realized it was necessary to direct ourselves toward a more specific
goal. What types of scientific information did we most need?
What levels of mastery should we aspire to? At the same time,
catalyzed by our regular interactions in the class, we students
realized that we learned the most (and enjoyed ourselves the most)
in each other's interdisciplinary company. While each of us
had major gaps in our basic knowledge, as a group we could
begin to fill many of them.''
Now we can place this comment into the biomath world by simply
paraphrasing as follows:
``Students can not speak the
language of the many disciplines B3 enriched by BMC
requires and they
lack the basic information to consider their claims
critically. An attempt at an integrative approach
that provided only a superficial introduction to the major contributions
of each discipline can not lead to the ability to do
critical science. Still, major gaps
in their backgrounds need to be addressed.
What types of scientific information are most needed?
What levels of mastery should they aspire to?
It is clear the students learned the most (and enjoyed themselves the most)
in each other's interdisciplinary company. While each
has major gaps in basic knowledge, as a group they can
begin to fill many of them.''
In many ways, our quantitative emphasis area is trying to address these
concerns. However, it requires a lot of infrastructure and culture changes
to make an impact. For example, simply writing lectures is a challenge.
From the comments above, it is clear we must introduce and tie disparate threads of material
together with care. I tend to favor theoretical approaches
that attempt to put an overarching theory of everything together into a discipline because it
offers insight into the basic building blocks hidden inside complexity.
So I pay a lot of attention to books that
approach various aspects of biologically theoretically
in order to enrich a biomathematically interdisciplinary approach.
Reading these books gives me insight into the design of biologically inspired algorithms
and models which, in later presentation, I use frequently
when I design undergraduate lecture and research experiences. A selection
of these books includes
Theoretical systems biology (U. Alon. An Introduction to Systems Biology: Design
Principles of Biological Circuits. Chapman & Hill:
CRC Mathematical and Computational Biology, 2006.);
Theoretical genomics (E. Davidson. Genomic Regulatory Systems: Development and Evolution. Academic Press, 2001 and
E. Davidson. The Regulatory Genome: Gene Regulatory Networks in Development and Evolution.
Academic Press Elsevier, 2006)
and
(M. Lynch. The Origins of Genome Architecture. Sinauer Associates, Inc., 2007);
Theoretical Neuroscience
(J. Kaas and T. Bullock, editors. Evolution of Nervous
Systems: A Comprehensive Reference Editor J. Kaas
(Volume 1: Theories, Development, Invertebrates).
Academic Press Elsevier, 2007),
(J. Kaas and T. Bullock, editors. Evolution of Nervous
Systems: A Comprehensive Reference Editor J. Kaas
(Volume 2: Non-Mammalian Vertebrates). Academic
Press Elsevier, 2007),
(J. Kaas and L. Krubitzer, editors. Evolution of Ner-
vous Systems: A Comprehensive Reference Editor J.
Kaas (Volume 3: Mammals). Academic Press Else-
vier, 2007) and
(J. Kaas and T. Preuss, editors. Evolution of Nervous
Systems: A Comprehensive Reference Editor J. Kaas
(Volume 4: Primates). Academic Press Elsevier, 2007);
Models of the thalamus (S. Murray Sherman and R. Guillery. Exploring The
Thalamus and Its Role in Cortical Function. The MIT Press, 2006);
and
Theories of organ development (A. Schmidt-Rhaesa. The Evolution of Organ Systems.
Oxford University Press, 2007.);
and
(J. Davies. Mechanisms of Morphogenesis. Academic Press Elsevier, 2005);
All have helped me to see the big picture and have helped motivate
our design of interdisciplinary training techniques.
However, it is not clear other mathematicians would be willing to invest
this much time into the development of the lecture material.
In fact, when I sent my book to a former mathematics mentor of mine, he was
quite dismissive of my efforts:
``no other mathematician would be willing to do this, Jim''.
But I am pretty stuborn so I forged ahead anyway. The problem,
of course, in order for a new approach to take off, more than one person
such as myself must become engaged fully. And my former mentor's comment
did not bode well! For now, let's end this section with
a nice quote from the astrobiology community (page 552 in Sullivan's book)
``Too often, the scientific community and academia facilely
redress ignorance by appealing to the testimony of experts.
This does not resolve ignorance. It fosters it.''
This closing comment sums it up nicely. We want to give the students
and ourselves an atmosphere that fosters solutions and the challenge is
to find a way to do this. The interdisciplinary training needed to foster thus
requires all of us to be aware of these underlying philosophical
changes that must be made.
2.2 The Quantitative Emphasis Area
Since 2006, my colleagues and I have developed a plan to enable an integration
between biology, mathematics and computational tools for placement
within a typical existing department of biological sciences.
Although we have built this program at my university,
we are aware that our experiences need to have a national perspective
and we have planned accordingly.
We posit that in a traditional college entity
80% to 90% of the tenured faculty have little interest
in the use of BMC in their own course development
and teaching. Hence, there are great challenges in implementing
this plan that revolve around both faculty retraining and
student training issues. In this traditional college
department, we see incoming majors who have an uneven training in mathematics
and computer science and who may even be biased against
mathematics and computer science as part of the discipline of
biology. There is also an existing mathematics requirement of two semesters of
engineering based calculus which does not serve the interests of biological
sciences well. Note, recent events in the Spring of 2013 have taken away the requirement for
the second semester calculus course for biology students which has essentially completely
undermined all these efforts!
Nevertheless, in the remarks that follow, we will be very detailed in our descriptions
of what we did in order to develop these courses. Hence, there is a fair
bit of mathematical detail, but if that is not one's cup of tea, just
try to get the overall flow. Note how complicated it is to merge
as seamlessly as possible, mathematical and computational ideas into
a traditional biology department's curriculum.
We start at the freshman level and try to encourage
existing faculty and students into a new found appreciation of
the BMC triad. This then permits integration of these ideas from
the freshman to sophomore year and prepares students for quantitative
biology in the junior and senior years. Our basic plan has
kept the existing First Semester Calculus Course
for engineers as a mathematical baseline (although, an argument
can be made for a retooling of this first course as well even though
we do not do that here). Since different universities use
different course numbers, we will call this
course Math-One-Engineering for convenience. However, we have
replaced the existing Second Semester Calculus Course, called here
Math-Two-Engineering, with a new
course specifically designed for Biology majors called
Calculus for Biologists.
We have also
added model based points of view to the first year
fundamental biology sequence which we call Biology-One
and Biology-Two. In these introductory biology courses,
the students write three papers per semester. Four of these six papers
have a required statistical analysis to determine the significance of
the difference between two treatments.
At the moment, this is confined to
the chi-square median test but other
statistical tests using Excel spreadsheets could be done.
In addition, there are two labs on bioinformatics. One is a general overview
that shows the students a lot of different tools, and the second one
specifically asks them to use bioinformatics tools to look at the
evolution of humans through mitochondrial DNA sequences.
Next, using Stella (ISEE Systems. STELLA, 2009), students build models of an equilibrium
chemical reaction and human weight regulation. In each of these
examples, more complex Stella models are then used to explore more
sophisticated questions.
There are also several embedded modules which go beyond the traditional
lectures to expose the students to beginning research concepts. These
include
A. the use of the Hardy-Weinberg
equation to predict equilibrium genotypic frequencies;
B. a population genetics simulation used to study the effects of
selection and genetic drift on allelic frequencies;
C. an elementary treatment of exponential and logistic population growth models;
D. simulation of complex populations with different age classes;
E. a large unit on cladistics in both lecture and lab.
So in general, despite the fact that in most topics the use of mathematics is minimized,
the course is quantitative in philosophy.
The Calculus for Biologists course
uses discovery based learning and introduces biomathematical research based topics
that are appropriate to the level of these beginning students.
It was run as a special
section of Math-Two-Engineering called Math-Two-Engineering B and enrollments
climbed steadily. Currently, we teach approximately 100
students per year in this course. This experimental course
was formally approved (Fall 2009)
as the new course Calculus for Biologists.
We developed this course using the following point of view.
We feel to serve the needs of the BMC point of view,
mathematics is subordinate to the biology: the
course therefore builds mathematical knowledge needed to study interesting
nonlinear biological models quite carefully.
We emphasize that more
interesting biology requires more difficult mathematics and concomitant
intellectual resources.
We choose a few nonlinear models and discuss them very
thoroughly; e.g.,
logistics, Predator - Prey and disease models.
Since graphical interfaces lose value with more variables, at the end of the
course, we explore
how to obtain insight from a six variable linear Cancer model.
We always stress how we must abstract
out of biological complexity the variables necessary to
build models and how we can be wrong.
We also focus on how to solve problems which have many steps
(so that in the modeling portions, typical assignments can have
3 - 4 pages per turned in per problem). We believe strongly that
this is a first step towards training them to see the larger picture
for long term projects. We therefore are preparing them for
extended research experiences.
We are successful at developing very sophisticated mathematical
ideas in these students. We achieve this by organizing our topics in
the following way. From the standard Math-Two-Engineering
we choose as topics simple substitution,
the Fundamental Theorem of Calculus (FTC),
the Logarithm and Exponential Function developed using the FTC,
integration by parts and partial fraction decomposition methods
and the ideas of
approximating functions by tangent lines and the error that is made.
Since we use these mathematical tools right away in logistic models,
the students always see a correlation between the mathematics we choose
to cover and its usefulness in our model building.
We need concepts from linear algebra (here, a sophomore or junior level
course) as well. So we cover
vectors and matrices, linear systems of equations,
determinants and what they mean and
eigenvalues and eigenvectors for 2 by 2 matrices.
This prepares them to study more differential equations.
From a traditional sophomore level differential course,
we can then cover
simple Rate Equations and their applications,
the idea of an integrating factor,
building a Newton's Cooling model from data,
the logistics model
complex numbers and simple complex functions and
second order linear homogeneous equations. Our discussion of the second order
linear differential equations cover all three cases: distinct roots, repeated roots
and complex roots.
In addition to these ideas, we have also been showing them the use of
computational tools a little at a time. We have chosen MatLab, but other
interpreted tools are of course possible to use instead. Hence, we
discuss approximations with tangent lines, Euler's method and MatLab
implementations concurrently with these topics. We also introduce
the ideas of Runga - Kutte methods as a more sophisticated numerical approximation
tool. We finish our discussions of linear models with linear systems
of differential equations, although we restrict our attention to
the distinct real root case. At this point, we also move toward qualitative
graphical means of understanding the solutions we are finding. This is all done
by hand to let them grow insight.
We then segue into nonlinear differential equations.
Little of this material is covered in a traditional differential equation course.
We thoroughly discuss the
Predator - Prey model without and with self interaction, a simple SIR disease model
and a cancer model at the end of the course. We stress heavily that
models can have explanatory power but need not unless there
is a careful interaction between the science, the mathematics and the
computational tools. We note graphical analysis and
numerical methods are not always useful and
determining the validity of the model is the most
important thing. We help them to realize that interesting problems
are inherently difficult; realistic biology requires nonlinearity
which implies more sophisticated mathematical and numerical tools to be
brought to bear.
In our implementation of this course,
we therefore interweave the calculus, differential equations, MatLab
and numerical methods, linear algebra and modeling
material in non-traditional ways. Roughly speaking
mathematics is introduced as needed for the modeling,
although not at all in a just in time paradigm. We believe
firmly that these difficult tools need to be carefully brought out
over time, not quickly.
The first part of the course introduces
specific mathematical material as they are not
quite ready for models. Hence, they find this more
challenging and great care must be taken to
keep them motivated. However, daily homework with lots of algebra and manipulation
helps as well as daily mention of where we will be going with
this material. We follow the general principle that doing 1 or 2 problems encourages
mimicry, while doing 5 - 6 of each type builds mastery.
Graders help with the daily busywork.
The ideas discussed above have been used to
develop a textbook which has evolved from handwritten notes
in Spring 2006 to the fourth edition ( Calculus For Biology: A Beginning – Getting Ready For Models
and Analyzing Models. Gneural Gnome Press: www.lulu.com/spotlight/GneuralGnome, 2012 ) .
2.3 The Current Quantitative Emphasis Approach
We have already begun the process of enhancing interdisciplinary training
by adding embedded undergraduate enhanced learning experiences
in a variety of ways to the freshman and sophomore years.
As we have discussed above, we achieve this by adding embedded undergraduate enhanced learning
experiences to the courses
Calculus for Biologists , More Calculus for Biologists, Biology I and Biology II.
We have taken advantage of these new programmatic tools
to implement a new four year
Quantitative Emphasis Area within the Biological Sciences B.S. degree
program. We think of it as a precursor to a new reasoned
approach to making these ideas a core part of the entire
biological sciences curriculum. In this section of the proposal, we specifically
address how we modify our current plan for the Quantitative Emphasis Area
to allow for undergraduate research experiences.
Since undergraduate research experiences will be a valuable tool for teaching the more
quantitative biology of the coming decades and for increasing retention of students in biology,
mechanisms must be found for fostering their development within existing curricula.
This requires increased mentoring of undergraduates by faculty.
Further, the traditional way to include such research experiences is to have
a capstone project in the senior year preceded by a quick research experience discussion.
Since the senior year is filled with many other requirements such as job search and
graduate school application preparation, this has also traditionally not been an optimal
path for many students. To get students involved from the Freshman year onward will therefore
require altering some aspects of the program of study. The Quantitative
Emphasis Area supports active student participation in the Freshman and
Sophomore year directly.
As a starting point, consider
the details of the Quantitative Emphasis Area which has been approved,
as seen in Table I and Table II below.
These tables shows the
basic format; as always, there are caveats and substitutions, but there are
not essential for our arguments. There are several important points to make here.
Note that essential ideas of Physics, say force and energy, which are so necessary to
many of the threads that run through quantitative biology ideas are not discussed formally
until the junior year.
Table I: A Typical Approved Quantitative Emphasis B.S. Freshman and Sophomore Year.
F and S denote the Fall and Spring semester, respectively. The number in
parenthesis behind the course is the number of credit hours.
Year Course Description
I (F) BIOL 110 (5) Biology I
BIOSC 101 (1) Frontiers in Bio I
CH 101 (4) Chemistry I
COMM 150 (3) Communication
MTHSC 106 (4) Calc I (Eng)
Total 17
I (S) BIOL 111 (5) Biology II
BIOSC 102 (1) rontiers in Bio II
CH 102 (4) Chemistry II
ENGL 103 (3) Composition
MTHSC 111 (4) Calc For Biology
Total 17
II (F) CH 223, 227 (4) Organic Chemistry
MTHSC 390 (3) More Calc For Biology
BIOSC 302, 303 (4) Animal Diversity
BIOCH 301, 302 (4) Biochemistry
Total 15
II (S) BIOSC 443, 444 (5) Ecology and Lab
EX ST 301 (3) Statistics I
BIOSC 304, 308 (4) Plant Diversity
GEN 300, 301 (4) Genetics
Total 16
Also, note the calculus in biology course gives students a full year of mathematical training during
their freshman year. Then in the sophomore year, there is a more advanced mathematical modeling
course based on a partial differential equation point of view and the first semester of statistics.
However, formal mathematical training, per se, stops at that point. Also, there is
no formal computer science course work which teaches some choice of programming language.
Hence, to develop the triad of mathematical, biological and computational science
absolutely requires large amounts of teacher based instructional materials and pedagogical
development to bridge the gap.
The junior year is quite crowded with necessary start up biology course work.
Currently,
the real freedom in the schedule is in the senior year with 12 credit hours of
advanced biology course to be chosen. We suspect this sort of schedule with
openings only at the end of the program is quite common.
For concreteness, we show the typical Calculus for Biologist syllabus
in Table III and a version of the More Calculus for Biologists syllabus in Table IV.
Note built into these courses are modules that introduce the students
to the use of mathematics in biology at a level much more advanced than usual.
In Calculus for Biologists, we have some modules already in place
There is a gene transcription rate model which shows
the students that a simple first order differential equation
model can give insight into why gene transcription has such
a low error rate. We also have
a standard SIR disease model with a discussion of
how to estimate the occurrence of an epidemic from hospital
admittance data. Finally, we go over carefully a
colon cancer model which expands the students mathematical
modeling viewpoint by using 6 variables, albeit in a linearized way.
They are shown that interpreting such a model requires them to
think carefully about mathematics, the science of genes and computational
modeling tools.
Table II: A Typical Approved Quantitative Emphasis B.S. Junior and Senior Year.
F and S denote the Fall and Spring semester, respectively. The number in
parenthesis behind the course is the number of credit hours.
Year Course Description
III (F) BIOSC 335 (3) Evol. Bio.
ENGL 315 (3) Science Writing
EX ST 311 (3) Statistics II
PHYS 122, 124 (4) Physics I and Lab.
BIOSC 401, 402 (4) Physiology
Total 17
III (S) BIOSC 428 (4) Quant. Bio.
BIOSC 461, 462 (5) Cell Bio. and Lab
PHYS 221, 223 (4) Physics II and Lab.
Social Science (3)
Total 16
IV (F) BIOSC 493 (2) Senior Seminar
GEN 440 (3) Bioinformatics
Arts and Humanities (3) Literature
BIOSC 3xx-4xx (7) More 300 + Bio.
Total 15
IV (S) BIOSC 491 (1) Under. Research
Arts and Humanities (3) Not Lit.
BIO 3xx-4xx (5) More 300+ Bio.
Social Science (3)
Total 12
Table III: The Calculus for Biologists Syllabus (4 credit course)
UNITS TOPIC
1 Riemann Sums, Fund. Theorem
1 Fundamental Theorem, continued
2 Defining ln and exp functions
2 Basic Rate Differential Equations
2 Gene Transcription Rates
1 Partial Fraction Decomposition
2 Integration By Parts
2 The Logistic Type Model
2 Euler's Method
1 Matlab for First Order ODE
1 Complex Numbers
4 Order 2 Linear ODE
6 Linear Systems
2 Solution Linear System
2 Matlab for systems
3 Phase plane analysis
9 Predator - Prey
3 Predator - Prey Self Interaction
3 Epidemiological Model
4 Cancer Model
Table IV: The More Calculus for Biologists Course (3 credits)
UNITS & TOPIC
2 Multivariate Functions
2 Partial Derivatives
3 Jacobians; Linearizations
1 Complex Eigenvalues
1 Repeated Eigenvalues
2 Nonlinear ODEs again
4 Basic BioPhysics (Ion movement etc)
2 Partial Differential Equation Models
4 Series and Series solutions
4 Boundary Value Problems
6 Matlab ( intertwined with other lectures)
4 Hodgkin-Huxley Models
4 Excitable neuron simulation
In the junior year, Quantitative Biology, continues to expose
students to critical BMC concepts. It applies quantitative methods to a wide range of biological problems.
The main focus is on building modeling skills using population, physiological, genetic, and evolutionary problems.
It also includes a review of statistical principles, and introduces basic bioinformatics techniques.
A typical syllabus is shown in Table V as a 3 credit lecture and 1 credit lab course.
Table V: Quantitative Biology Syllabus
UNITS TOPIC
1 Mathematical models
2 Excel, Stella, and MATLAB
5 exponential,logistic growth
2 Population growth - Leslie matrices
2 Optimal life history strategy
1 Optimal diet (linear programming)
2 Compartmental models (epidemics)
4 Predator-prey Models
2 Competitive interactions
3 Biochemical mathematics
3 Cell biology, development models
6 Physiological, biomechanical mathematics
2 Mathematics of genetics
3 Evolutionary mathematics
3 Statistical principles
2 Bioinformatics tools
To fully energize and engage current biology and mathematical majors in
beginning research experiences and to potentially recruit new students
it is important to develop a wide range of undergraduate research experience (UGRE) modules.
These can be deployed in a classroom lecture or a laboratory setting.
The development of such textual material is a highly nontrivial task
and requires a substantial investment of faculty time.
To make these new initiatives self-sustaining, it is important for these ideas
to become firmly entrenched
within both the Department of Mathematical Sciences and the Department of
Biological Sciences. This is both a retraining issue and an interest issue.
The courses Calculus for Biologists, More Calculus for Biologists, Biology I, Biology II and Quantitative Biology are not taught in a standard manner and require much from the instructor. To teach Calculus for Biologists effectively requires that the instructor both respects and loves both mathematics
and biology. Since this is not true in general of any department's teaching staff,
this is a significant faculty and instructor training issue.
If Calculus for Biologists enrollments were to grow, a seasoned teacher
would have to develop a mentoring strategy so that new teachers of
this material could be trained.
Similar mentoring can be done in the teaching of More Calculus for Biologists,
Biology I, Biology II and Quantitative Biology.
Indeed, training and mentoring would have to be a fundamental design component.
Of course, all these concerns are moot now as with the recent decisions by my biology department,
the entire quantitative emphasis we have outlined here will receive no university support and hence,
die away.
Great ideas dad!!
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