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\title{Teaching Calculus, Probability, and Statistics to Undergraduate
       Life Science Majors: A Unified Approach}
\author{Frederick R. Adler, University of Utah \thanks{adler@math.utah.edu}
\date{}
    }

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\hline
Institution& University of Utah \\
\hline
\hline
Size& about 2000 students\\
\hline
Institution Type & Large comprehensive state university offering
bachelors \\& through doctoral degrees \\
\hline
Student Demographic& Biology
majors (recommended in lieu of the standard first \\& two semesters of calculus, but taken
only by a minority \\& due mainly to scheduling constraints.\\
\hline
Department Structure&Mathematics and Biology are
separate departments\\
& in the College of Science.\\
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\subsection*{Abstract}
The Department of Mathematics at the University of Utah has developed an integrated modeling, calculus, and probability course for life science majors that emphasizes the central role
of dynamics in biological thinking, including statistical analysis.
The course and associated textbook use the themes of growth,
diffusion, and selection throughout, and we describe how diffusion
is presented in several contexts.  Although the University of Utah
has a large group of faculty and students in mathematical
biology, we discuss the challenges of serving all biology majors,
integrating the course with the biology and mathematics curriculum,
and motivating mathematically underprepared students.

\subsection*{COURSE STRUCTURE}

\begin{packed_item}
  \item Weeks per term: Two semesters, about 30 weeks
  \item Classes per week: Three 1-hour lectures
  \item Labs per week: One 1-hour lab
  \item Average class size: 25-50 students
  \item Enrollment requirements: For freshman, students should
have passed college algebra and trigonometry, or have Math SAT score
of at least 630, Math ACT score of at least 28, or AP Calculus AB score
of at least 3
  \item Faculty/dept per class, TAs: One math instructor
with one graduate student TA to handle the computer labs
  \item Next course: The purpose of this course is to prepare
students for the quantitative aspects of the biology curriculum
  \item Current website:
     \verb+http://www.math.utah.edu/~borisyuk/1170/+
  \item \underline{Archived website}:
     \verb+http://www.math.utah.edu/~adler/oldcourses/math1170_2008/+
\end{packed_item}

\subsection*{Scope and goals}

In 1990, the University of Utah received a grant from the Howard
Hughes Medical Institute; among the goals of the proposal was the
development of a new calculus course and associated textbook for
life science majors.  The University of Utah had, and still has, a
one-year mathematics requirement for biology majors.  To this end,
I was hired as a new faculty member into a joint position in the
Departments of Mathematics and Biology to develop the course.

I was given one semester for preparation, and had a committee
of experts in biology and mathematics to help with planning.  The goal was to create a course that could serve
the quantitative needs of the large majority of life science majors,
but without closing off the option of taking additional mathematics
courses.

After surveying the biology faculty, we found that modeling and
statistics were the two most important topics (along with the basics
of being able to read a graph effectively and other remedial skills).
The course was designed to emphasize these elements by covering
three areas under the quarter system then in effect:
\begin{enumerate}
  \item Discrete-time dynamical systems
  \item Differential equations
  \item Probability and statistics
\end{enumerate}
Within these broad headings, united by their focus
on modeling, the fundamental ideas of functions, differentiation,
and integration were to be seamlessly intertwined.

We chose to take a conventional approach toward teaching in
two ways.  First, the course would build from simple basic principles
toward more complex ideas (rather than confronting students with
realistic data to analyze right from the start).  Second, much of
the learning would be done in a traditional classroom setting, but
supplemented with an additional hour each week in the computer lab.

Because no textbook then existed that matched this model and
built from biological principles,
I began to draft what became \emph{Modeling the Dynamics
of Life: Calculus and Probability for Life
Scientists} (Adler, 1998; Adler, 2005).

The biological themes were simple and chosen so that they could be revisited in
growth, diffusion, and selection.  The goal was
to develop materials that would focus on understanding
the modeling process, with as much (or as little) attention on the
mechanics of algebraic calculation as was necessary to achieve this
goal.

The section on probability and statistics was to tie the work on
deterministic models together with data analysis using
the basic models of probability theory (the binomial, Poisson,
exponential and normal distributions).  Through computer experimentation, students
were to see how the probabilistic nature of population growth,
selection, and diffusion make patterns difficult to see without the
proper tools, namely a combination of basic understanding and statistical
insight.

\subsection*{Course Content}

The course begins with a four-week section that
introduces modeling by using discrete-time dynamical
systems, building or renewing students' familiarity and facility
with linear, exponential, and trigonometric functions and their
graphs.  Starting with the classic model of exponential growth,
we derive a linear diffusion model of gas exchange in the
lungs, and a non-linear model that shows how a selectively-favored allele spreads through a population.  Students
use graphical methods, such as cobwebbing, to
visualize and understand dynamics.

The central portion of the first semester introduces derivatives
and some of their applications.  After learning the meaning
of the derivative, its graphical interpretation, and computation rules
for biologically important functions, students
use the derivative to analyze stability of equilibria of
discrete-time dynamical systems.  Other applications include
optimization, using the derivative to understand
graphs of complex functions, and using the tangent line and
Taylor polynomials to approximate functions.  If time permits,
we tie together approximation and discrete dynamics tie together
with Newton's method for numerical solution
of nonlinear equations.

The first semester concludes with an introduction to integration,
starting with the antiderivative as the way to
solve simple pure-time differential equations, where the rate
of change depends only on time.  Riemann sums and the interpretation
of integrals as areas are delayed, and emphasize the importance of
the fundamental theorem of calculus in linking definite and indefinite
integrals.  The principal methods of integration are covered, although
in less detail than in a standard calculus course.

The second semester begins with autonomous differential
equations and shows how they parallel discrete-time dynamical
systems. The phase-line diagram, like cobwebbing for discrete-time
dynamical systems, provides graphical understanding.  The parallel
with discrete-time dynamics continues with the application of the
derivative to analyze stability, and the parallel with pure-time
differential equations continues with their solution using separation of variables.  The phase plane
forms the culmination of the deterministic portion of the course,
emphasizing the interplay between modeling and analysis in
the dynamics of a neuron as described by the Fitzhugh-Nagumo
equations.

The remainder of the second semester, 10-12 weeks,
is devoted to probability and statistics, with
an emphasis on modeling in data interpretation.  The first section develops the
concepts of probability, focusing on conditional probability
and independence, and includes the visual display of probabilistic
information.  The second section introduces probability
distributions through their derivation from discrete-time dynamical
systems (the binomial and geometric distributions) and from
differential equations (the Poisson and exponential distributions).
This section concludes with the
central limit theorem (presented without proof) and the normal distribution, along with the
difficult concept of the probability density function and its link
with the fundamental theorem of calculus.

The course concludes with an introduction to statistics designed to tie together the key ideas of modeling and calculus.
Maximum likelihood forms the backbone, showing how maximization
methods can be applied to data when interpreted in terms of probability distributions.  Although necessarily brief, the key
ideas of classical statistics, such as confidence limits and
hypothesis testing, are introduced with their application to
linear regression, analysis of variance, and contingency tables.

\subsection*{An Extended Example: Diffusion}

The course treats diffusion in several ways.
In the first chapter, before the introduction of
calculus, students develop a discrete-time model of gas exchange
in the lungs, \[
  c_{t+1} = (1-q) c_t +\gamma q ,
\] where $c_t$ is the concentration of some inert gas as a
function of the fraction of air exchanged, $q$, and the ambient
concentration.  The students derive this from first principles
(keeping track of air volumes), and then as a weighted average.
This allows students to develop more realistic equations for a gas,
like oxygen or carbon dioxide, that is used or created in the lung
with each breath.  After the development of the derivative, students
use stability analysis to show that the lung will indeed approach
the ambient concentration, and to study the effects of non-linearities.

The second portion of the course, on differential equations,
introduces both Newton's law of cooling and the formally identical
law for chemical diffusion between two containers in continuous
time, \[
  \frac{dC}{dt} = \beta (\Gamma - C) ,
\] where $\beta$ is rate of chemical exchange and $\Gamma$ is the
ambient concentration.  Students extend the derivation to
more interesting biological processes and use graphical
and algebraic methods to evaluate stability.

Finally, in the section on probability, students meet diffusion
from the molecular perspective as a stochastic process.  They see
that the probabilities that describe the location of a molecule
follow exactly the discrete-time dynamical system found with
macroscopic reasoning about volumes.  After deriving the binomial
distribution, they see that the ensemble of molecules, assuming
independence, obeys it, and in the limit the ensemble
behaves like its expectation.  They have then come full circle to
see that the deterministic discrete-time dynamical systems and
differential equations derived and studied in the first part of the
course are the equations for the expectation of a stochastic process.

The familiarity of diffusion allows students to use their intuition
to derive and understand mathematical models that can then challenge
and extend that intuition.  The multiplicity of modeling approaches
illustrates that the tool chosen to study a problem depends on the
problem and the question being asked, and shows
that apparently different methods can be closely related.

\subsection*{Successes and Failures}

Although no formal assessment has been done, students seem to enjoy
the course, and some show
evidence of having used the material in other courses.  The University
of Utah has a tradition of encouraging undergraduate research
in biology (the initial Hughes grant funding the development of the
course was focused on further strengthening these programs),
and many students have been attracted to this course because of the
benefits for research.

The course was to be taught to a single
section of students in the pilot phase and then scaled up to serve all
biology majors.  This has not occurred, primarily because of
scheduling difficulties, and the course remains at one or two
sections.  Strong advising from the biology department has maintained enrollment although the course is not required and is widely considered to be harder than the ordinary calculus
sequence.  The majority of the students
have not taken calculus before, and are comparable in mathematical
background to their peers who enroll in the ordinary calculus sequence.

We have found that separating biologists from physicists and
engineers has promoted a positive and collaborative atmosphere, which extends to the instructors.  Over the years, the
course has been successfully staffed by both faculty and advanced
graduate students, thanks to the large mathematical biology group
at the University of Utah.  The shared motivation of life science
students and the choice of biologically relevant topics provides
the best argument for such a course, with its
implementation depending on an institution's requirements and goals.

The course has faced many problems.  The pressure to develop a
second edition of the book took energy away from improving the
teaching of the course, leading to a period of relative stagnation.
Work on the second edition coincided with a switch to semesters
that broke up the elegant three part structure in an unnatural way,
with differential equations being divided over two semesters.

There is little doubt that the course is more difficult than the
standard calculus sequences because of the challenges
of modeling, the open-ended material, and the diversity
of topics.  Given the wide range of mathematical backgrounds and
abilities among students, the goals of the course must be adjusted
along a sliding scale.  The weakest students at least gain familiarity
with concepts such as models, differential equations, and
statistics.  The strongest students can apply what they have
learned to work in other courses and in research.  Those with the
most mathematical enthusiasm and ability can in principle move on
to advanced calculus, linear algebra, or mathematical biology, but the department advisors have discouraged this, assuming that students from this course will lack the background
needed for more advanced work.

In response to pressure from advisors and reviewers,
and in part for completeness, I added several topics to the third edition:
\begin{packed_item}
  \item Double-log graphs and an introduction to allometry,
  \item Implicit differentiation and related rates,
  \item Infinite series, Taylor series, and improper integrals,
  \item Integration by partial fractions,
  \item Trigonometric substitutions,
  \item Computing volumes of solids of revolution.
\end{packed_item}
Whether these additions will help
integrate this course with the rest of the mathematics curriculum
will depend on the idiosyncrasies of particular institutions.

Modernizing the course will probably take two directions.
First, the introductory part of the course needs to be closer to
real biological problems.  One of the difficulties is
balancing elucidation of general principles with the complexity of
real biology.  The themes of growth, diffusion, and selection
could be better introduced with real data, ideally student-generated.  A bit of preliminary statistical analysis would
help motivate the models and their analysis.  Second, the lack of
bioinformatics is frustrating.  At the University of Utah, we believe
that a course for beginning undergraduates must avoid methods presented as a black box, and methods for dealing with complex genetic data cannot be developed from scratch at this
level.  However, carefully chosen examples should reveal the methods,
challenges, and excitement of modern biology.

Even in the biology-friendly atmosphere of the University
of Utah Department of Mathematics, integrating this course with the rest of the curriculum
has been challenging.  We have failed to integrate
the material learned with the rest of biology curriculum, due
to the inability of the University of Utah to enforce prerequisites,
and to the challenges of incorporating any new material in
already over-stuffed biology courses.

\subsection*{Looking Ahead}

Today's college educator must contend with two conflicting problems.
Entering students generally have poor quantitative skills of all
sorts, from number sense through algebra and including
lack of real computer programming experience and knowledge of
statistics.  However, real biological
problems, whether in the realm of research or in medicine, are becoming
ever more complex.  Those of us trained in the step-by-step logic
of mathematics find it unacceptable to present students with a
series of black boxes that can be used to solve problems.  But the
jump from solving and understanding a linear discrete-time dynamical
system to appreciating the logic required to make inferences from
genetic data seems too large to make in a single year.  Much lip
service is paid to integrating quantitative material into biology
courses, but few institutions have succeeded in doing so in a serious
way due to the limitations of both faculty and students.

The  different sizes, emphases, and personalities of institutions
make a one-size fits allmodel for integrating calculus-level
mathematics with a life science curriculum inappropriate.  Each
institution needs to use its strengths (such as the large
mathematical biology group at the University of Utah) to advance
quantitative education and initiate the long-awaited generational
shift in biological thinking.

\subsection*{References}

Adler, F. A., 1998: \emph{Modeling the Dynamics of Life: Calculus
and Probability for Life Scientists}, Brooks/Cole.

\medskip
\noindent
Adler, F. A., 2011: \emph{Modeling the Dynamics of Life: Calculus
and Probability for Life Scientists, Third Edition}, Brooks/Cole.

\end{document}
