\documentclass[11pt]{article}
\pagestyle{empty}
\usepackage{longtable}
\usepackage{amsmath}
\usepackage{lscape}
\usepackage{multicol}
\usepackage{color}
\usepackage{url}
\setlength{\oddsidemargin}{-0.0truein}
\setlength{\evensidemargin}{-0.0truein}
\setlength{\textwidth}{6.8truein}
\setlength{\topmargin}{-0.0truein}
\setlength{\textheight}{9.0truein}
\setlength{\headsep}{0.0truein}
\setlength{\headheight}{0.0truein}
\setlength{\topskip}{-3.0truein}

\newcounter{MyEnum}
\newenvironment{MyEnum}%
{\begin{list}{\textsc{}}{\usecounter{MyEnum}%
\setlength{\labelsep}{0pt}\setlength{\leftmargin}{5pt}%
\setlength{\labelwidth}{0pt}%
\setlength{\listparindent}{0pt}}}%
{\end{list}}


\begin{document}
\begin{center}{\bfseries Implementation of First Year
    Biomath Courses at the Ohio State University}\\

\vspace{0.1in}
Laura Kubatko\footnote{kubatko.2@osu.edu}, Departments of Statistics and of
Evolution, Ecology, and Organismal Biology, The Ohio State University
\\
\vspace{0.1in}Janet Best\footnote{best.82@osu.edu}, Department of
Mathematics, The Ohio State University\\
\vspace{0.1in}
Tony Nance\footnote{nance.1@osu.edu}, Department of Mathematics, The Ohio
State University\\
\vspace{0.1in}Yuan Lou\footnote{lou.8@osu.edu}, Department of
Mathematics, The Ohio State University\\
\end{center}


\begin{center}


\begin{tabular}{|l| |l|}
\hline
\multicolumn{2}{|l|}{Name of Institution: The Ohio State University}\\
\hline
\hline
Size
    & about 38,000 undergraduate students \\
\hline
Institution Type &
    large state university with PhD program \\
\hline
Student
    & recent high school graduates with interests
    \\
Demographic & in mathematics, statistics, and/or biology
    \\
\hline
Departments & Six departments (Mathematics, Statistics,
    and four biology), plus \\
and Program & the biology major
    administered by the Center for Life Sciences \\

Structures & Education, all inside the College of Arts
    and Sciences \\

\hline
Web Pages & \url{http://www.stat.osu.edu/~lkubatko/CAUSEwebinar} \\
& \url{http://rumba.biosci.ohio-state.edu} \\
\hline
\end{tabular}
\end{center}

\section*{Abstract}
With approximately 38,000 undergraduates on the main campus in
Columbus, Ohio, The Ohio State University (OSU) is one of the largest schools in the
nation, providing a significant challenge in the creation of a
BioMathematics curriculum that will adequately serve the needs of this
undergraduate population. In this article, we describe our initial steps
toward this goal, namely,
the development and implementation of several first-year BioMath
courses for students majoring in the Biological Sciences at OSU.

\section*{STRUCTURE OF COURSES}\begin{itemize}
\item Weeks per term:
  \vspace{-3mm}10 weeks (will convert to semesters in 2012)
\item
  Classes per week/type/length: \vspace{-3mm} three 48-min lectures,
  two 48-min recitations
\item Average class size:  \vspace{-3mm}Calc I
  = up to 120;  Calc II = up to 60; Stats =  up to 40
\item Enrollment
  requirements: \vspace{-3mm} Calc = standard; Stats = Calc II
  prerequisite
\item Faculty per class, TAs: each course has 1 faculty and 2,
  1, 1 TA respectively
\end{itemize}


\section*{Introduction}
The Ohio State University (OSU) is one of the largest schools in the nation,
with over 50,000 students on the main campus in Columbus, Ohio,
approximately 38,000 of whom are undergraduates.  The administrative
structure of the University consists of eighteen distinct colleges, with
a total of 167 undergraduate majors.  The four biological science departments, housed within
the College of the Arts and Sciences serve approximately 2,300 undergraduates who obtain one of eight
majors.  Also, OSU houses the NSF-funded Mathematical
Biosciences Institute (MBI), which provides a link between the
mathematical and biological sciences.

In collaboration with the biological science departments at OSU, the Departments of Mathematics and of
Statistics have developed courses to provide a
quantitative foundation for the training received by undergraduates
majoring in a biological sciences (see accompanying article in this
volume for details about the development of these courses).  The courses include a two-quarter
sequence in calculus and a one-quarter course in statistics. The calculus courses are
taken by freshmen who have placed into calculus and who plan to obtain one of the
eight biological science  majors, and are structured so that
students those who complete them are prepared to continue with the more traditional Calculus III course
offered by the Mathematics Department.  Two sections of the calculus sequence
have been offered in  every academic year since 2006-07,
serving approximately 160 students each year.  The course in statistics is intended to serve as a follow-up elective course; students are recruited primarily
from the calculus sequence, though any student who has
completed two quarters of any calculus can enroll.  The statistics course has been
offered during the Spring quarter every year since 2007, with
approximately forty students per section.

In this article, we describe each of these courses in detail.  The next section describes
the structure of each course, the material covered, and projects and
class activities included.  The Discussion section includes
reflection on what aspects of the courses worked well and what needs
improvement. We then discuss some more advanced courses that have recently been added to
our BioMath curriculum. We conclude with a discussion of our long-term
vision for the BioMath curriculum at OSU.

\section*{Description of First Year BioMath Courses at OSU}

OSU followed the quarter system rather than the semester system until the Fall 2012 semester, with an academic year thus consisted of three 10-week quarters (Fall, Winter,
and Spring) with a week of final examinations following each.
Quarter courses generally comprised 3-5 credit hours of instruction
per week, and often consisted of a mixture of lecture' and
laboratory sections.
Each of the BioMath courses we developed was offered for 5 credit hours, with three 50-minute lectures taught by a full-time faculty member and two 50-minute laboratory sessions taught by
Graduate Teaching Assistants (GTAs) per week.  The lecture sessions were often
large (ranging from 40 to 100 students), while the laboratory
sessions generally had no more
than 30 students.  Laboratory sessions were devoted to hands-on
activities and problem-solving, while lectures covered the
introduction and discussion of new concepts.  We now describe these courses in detail.

\subsection*{Calculus I and II for the Life Sciences}
Both calculus courses were taught from the text \emph{Calculus
  for Biology and Medicine} by Claudia
  Neuhauser (2003). The Calculus I course covered the first five chapters of the text,
  including sequences and difference equations (2 weeks), limits and
  continuity (2 weeks), and differentiation and applications (6
  weeks). The Calculus II course covered material in Chapters 6, 7, and 8: basic integration (2 weeks), advanced integration
techniques (6 weeks), and differential equations (2 weeks).
Course grades were based on weekly
  homework and quizzes, a project, two midterm examinations, and a comprehensive
  final examination.

An innovative aspect of the courses was the inclusion of group
projects. During the first offering of Calculus I, ten projects were designed by the
instructors and GTAs.
In subsequent quarters, projects have been developed and
supervised by post-doctoral researchers associated with the
MBI.  Each quarter,
several post-docs developed projects and gave a lecture
advertising their projects to the students; students then selected one of the
projects  to study in a group of three to five students.
The post-docs
arranged two additional class meetings to work with the
students, who were also expected to spend time working
on their own outside of class.
Each project contained several questions to be addressed, and
each student was required to write the solution to at least one
of the questions.  The group was responsible for
combining the solutions to the questions to form a cohesive
written report for the project as a whole.  This has the advantage of
giving students experience in writing in a research-like setting.
The group reports contributed the equivalent
of two homework assignments to each student's grade in the
course.  Here are examples of the projects.

\paragraph{\bfseries Tumor growth and treatment.}  Differential
  equations can be used to model tumor growth. A tumor cell divides, generating
  two daughter cells with unlimited ability to reproduce. Initial
  rapid growth supported by abundant nutrients such as oxygen and
  glucose is followed by slower growth as the population size
  increases. Injecting anti-cancer drugs that kill a fraction of tumor
  cells per unit time may reduce the size of the tumor cell
  population. The project involves studying the differential
  equations to find the best strategy for reducing tumor size.

\paragraph{\bfseries Analyzing cartilage health.}  In an osmotic loading
  experiment, chondrons  (cartilage cells with encapsulating
  pericellular matrix, PCM) are extracted from cartilage and placed in
  different external concentrations of sodium chloride in
  water. Depending on the concentration of the sodium chloride, the
  chondron will either swell or shrink. We look at how the volume
  of the PCM changes with respect to the different external
  concentrations of sodium chloride in water.

\paragraph{\bfseries Fisheries management.} Humans catch fish for a variety of reasons, such as for food
  or for sport. While these purposes may have merit,
  it is important to realize that there is not an unlimited number of
  fish in the sea. The National Marine Fisheries Service was formed by
  the U.S. government to manage marine resources in the U.S. In order
  to do their jobs, their scientific research teams use many tools,
  including mathematical models. In this project we explore a
  hypothetical fishery model for a sea bass population within an
  enclosure.

The material given to the students consisted of an expanded verbal description followed by
the formulation of a
mathematical model for the situation.  The students were
then asked to solve a series of problems using the model
and to interpret their results.  Also, the students were
provided with information concerning expectations from the
groups, both for functioning of the group as a whole and
for individual interactions of each student with the group.
The principal mathematical tool for the Calculus I projects is the derivative;
differential equations, when they occur, are accompanied by solutions
that the students must verify.

In the Calculus II projects,  the
main tools are derivatives and integrals, and many involve solving some
ordinary differential equations.  Example projects are:

\paragraph{\bfseries Brain waves with noise.} An EEG
  (electroencephalogram) machine measures local field potentials in
  the brain by recording from electrodes placed on a patient's
  scalp. These potentials represent an electrical signal from a large
  number of brain cells called neurons. Monitoring the EEG recording is
  useful in assessing brain activity and in diagnosing conditions such
  as epilepsy, sleep disorders, and coma. This project
  examines brain wave power when the signal is noisy.

\paragraph{\bfseries Grass management.}  A modern method to raise
  cattle consists of preventing cows from grazing in a field until the grass has reached an optimal height. When the grass is too short, it cannot recover well from being cropped. Older grass grows more slowly, may become senescent, and is less tasty to cows.  In this project we examine
  grass growth and its optimal management.

\paragraph{\bfseries Drug absorption.} Understanding therapeutic and adverse drug reactions is important in the treatment of many diseases,
  particularly cancer. Two models for drug absorption and toxicity are
  examined to compare injected drug therapies with a
  polymer delivery system.

\subsection*{Statistics for the Life Sciences}
This course, intended as an optional course to follow
the calculus sequence, was taught from \emph{The Analysis of Biological Data}  by M. C. Whitlock and D. Schluter (2009).  Material
selected from Chapters 1 through 17 of the
text was included, with additional materials provided by the instructor that built on the students' knowledge
of calculus (the text does not assume Calculus).  Topics included
descriptive statistics and graphical methods (1 week),
probability, including Bayes Theorem (1 week),
discrete distributions and the analysis of categorical data (2.5 weeks),
one- and two-sample inference for means and variances (2.5 weeks),
experimental design (1 week), and
correlation and regression (1.5 weeks).

As in the calculus sequence, assessment was based on weekly homework and in-class examinations.
The lectures were in the
traditional style, though activities were
incorporated into a few of the sessions.  The recitation
sessions were divided between problem-solving
sessions and activity sessions.  In the activity sessions, students
used the StatCrunch software (StatCrunch, 2012) to analyze
biological data sets after being given an introduction to the
biological setting.  Results of their analyses were brought into the
lecture sessions and used as examples for subsequent topics in many
cases, creating continuity between the recitation sessions and the
lectures.  A short lab manual for the course can be found at \url{http://www.stat.osu.edu/~lkubatko/CAUSEwebinar}.

Examples of the data sets included in the course and with
the topics they illustrate are:

\paragraph{\bfseries Fisher's iris data.} A well-known data set
  studied by R. A. Fisher includes
  measurements on characteristics of three species of irises,
  including sepal length, sepal width, petal length, and petal width (Anderson, 1935; Fisher, 1936).
  These data were used in several ways.  When the
  normal distribution was introduced, students used it to explore the
  distribution of characteristics in the recitation
  session using the StatCrunch software.  They examined histograms and
  normal probability plots for the data when all three species are
  grouped together and for each species separately.  The data is later
  used in lecture to motivate hypothesis testing by
  asking whether the mean sepal width differs between
  pairs of species.  After an example is worked in lecture,
  the ideas are reinforced in recitation by carrying out
  hypothesis tests using the StatCrunch software.

\paragraph{\bfseries Population frequencies of human chemokine receptor
    gene variants.} A genetic variant of the human
    chemokine receptor gene appears to provide strong resistance to
    HIV infection.  This variant is found in all European
    populations with varying frequency. Lucotte and
    Mercier (1998) \nocite{lucottemercier1998}
    studied a sample of 2,522 people throughout Europe to determine
    their genotypes, and found an overall allele frequency for the
    gene variant of
    approximately 9\%. In the recitation session, students are given this information as
    background to the problem (with some added discussion about what
    is a genotype).  They then compute the expected frequency
    of this gene variant in various European subpopulations, and compare them
    to the observed data of Lucotte and Mercier (1998).
    Although hypothesis testing has not yet been formally introduced,
    the students compute quantities similar to $p$-values by examining the probability
    of observing the data given in the paper if the variant gene frequency
    actually is 9\%.  This
    example is used to motivate hypothesis testing later in the
    lecture sessions.

\paragraph{\bfseries Old-growth forest data.} This data set consists of
  an extensive survey of an old-growth beech-maple forest carried out
at the James H. Barrow Field
Station (owned and maintained by Hiram College in northeastern Ohio)
during the summer of 1993 by Laura Kubatko. The goal of the data collection was
to assess the diversity and distribution of tree
species in the forest.  The data were collected by
  subdividing the forest into 50m $\times$ 50m plots.  Within each of sixty-seven
plots, four 10m $\times$ 10m sub-plots were randomly selected to be
surveyed.  All tree species occurring
within the sub-plots were counted and recorded.  The diameter at breast
height (dbh) was measured for any tree larger than 3.0 cm dbh.
  An interesting feature of this data set is that the forest can be
  divided into four distinct areas that vary in composition of species
  and size-class distributions due to various external stresses (e.g.,
  chestnut blight and an area affected by a tornado in the early
  1970s) and physical characteristics (west-facing slopes).

These data are used in the course to give students experience with
  examining and comparing distributions.  When discussing summary
  statistics and graphical displays, students are asked to use the
  StatCrunch software to find graphical displays for
  the counts of the number of trees of each species in various areas
  of the forest and for the distribution of trees among size classes
  (based on dbh).  Later,
  the data are used to compare areas of the forest using
  hypothesis tests and confidence intervals.

\section*{Discussion}

\subsection*{Successes}

Feedback from the courses has been positive.  Students seemed motivated and interested in
what was taught, and many commented on its usefulness to them in their future careers. The current cohort of students in the courses is generally well-prepared
for this level of mathematics.  As we look to expand
the offerings to include a broader group of students,
our approaches may need to be adjusted to accommodate
differing levels of preparation among our students.

The class structure (three lectures and two recitations per week) worked
well for several reasons.  One is that it divides our
potentially large (40-100 student) lecture sections into smaller
groups, so that students can receive more individual attention.  This
allows for activities (such as the calculus group projects) and
computer exercises (such as analysis of real data using StatCrunch in
the statistics course) to be included. Another
advantage of this format is that the GTAs were able to
get experience teaching students from the biosciences.  The GTAs
were involved in developing course materials for
the recitation sessions, and were therefore able to learn how
to communicate with this group of students.  Since the GTAs are the next
generation of instructors in biomath, the early exposure to such
courses with the guidance of a faculty mentor is very valuable.  Although the GTAs
were not given special training in preparation for teaching the courses, we worked to ensure that
they were interested in teaching in this setting.  All GTAs were carefully mentored throughout the quarter.

Another positive feature was the inclusion of the MBI
post-docs in the calculus sequence through the development and
mentoring of the student projects.  The exposure of the undergraduate
students to a variety of research areas in BioMathematics and the
interaction with the BioMathematics
community at OSU is  beneficial in creating an interest in and
appreciation for interdisciplinary work.  Several of us have had discussions with students interested in
pursuing undergraduate research projects in these areas, and we expect
such interactions to continue to increase over time.

\subsection*{Challenges}
One challenge in designing and implementing the courses was that of finding textbooks
with an appropriate mix of
biological motivation and mathematical rigor.  We were happy with the
Neuhauser text used in the calculus sequence in that it was easy
to teach from, as it is similar to a traditional calculus
text but infused with biological examples.  However, other books
(e.g., Adler 2004) that are more oriented toward
biology-driven mathematics than calculus per se are also appealing.  An important point
to consider is that some students in these
courses may decide to continue in the traditional mathematics curriculum, and thus
the courses need to prepare them to do so.

It was also difficult to select the topics, both biological and
mathematical,  that should be covered in
the courses.  The topic coverage in both the Calculus and the
Statistics courses was similar to what would be covered in
traditional courses, though the time spent on
them varied somewhat.  On the biological side, the students consisted
largely of freshmen who, while
mathematically prepared for the courses, were just beginning their
study of biology.  Thus we often had to teach a
fair amount of the biology underlying our examples, as well as the
mathematics, which could be a bit overwhelming for the students (and
for us!) at times. In the future,  particularly as we convert our courses to the semester system, we hope to work more closely with faculty in
the biological sciences to understand the freshmen biology
curriculum better, which will allow us to use examples
that utilize biological concepts to which the students have
been exposed, and to gain a further appreciation for which
mathematical and statistical topics will be most relevant for
students in their future study and careers.

A final challenge is recruiting students to take the elective courses (Statistics and
the courses described in the next section) following the calculus
sequence. Because the students are
freshmen, they are also taking laboratory courses in biology and
chemistry, which is a substantial academic load in
the first year. It may be more convenient for students to return to
mathematics courses in their second or third year, when they have more
flexibility in their scheduling, and we therefore expect these courses
to grow.

\subsection*{BioMath Courses at OSU Beyond the First Year}
In addition to the first-year courses, we have developed two additional courses to enhance student interest in mathematical biology at OSU. The
first is an undergraduate seminar that has been developed as a result of our recently-funded NSF UBM grant for undergraduate research in mathematical
biology at OSU.  As part of our program, called RUMBA (Research for Undergraduates: adventures in Mathematical Biology and its Applications---see \url{http://rumba.biosci.ohio-state.edu} for more information), the seminar course is offered in each academic term.  It meets for one hour each week and consists of talks or discussions led by PIs for the program and speakers from within the OSU BioMath community.  The outside speakers have included faculty in Mathematics, Statistics, and Biological Sciences, Visiting Scholars at the MBI, and MBI post-docs.  The seminar enrolls approximately fifteen undergraduates at all acamedic ranks (freshmen to senior) and a variety of majors (mathematics, biology, environmental science, pre-medicine, etc.).  Student evaluations have been very positive, and we feel that the course is succeeding in its goal of increasing interest in and awareness of the field of mathematical biology.


The second new course is titled Introduction to Mathematical Biology and was taught for the first time during the 2009-2010 academic year.  The topics included population dynamics (logistic growth model and Lotka-Volterra predator-prey model), epidemiological modeling, modeling of competition, neuronal dynamics, and enzyme kinetics.  We expect that this course will be offered every year. We have also added a mathematical biology track within the math major.  All of these recent efforts build on our earlier curriculum development to strengthen the undergraduate BioMath program at OSU.

\subsection*{Future BioMath Curriculum Development at OSU}

Our most recent efforts in BioMath curriculum development at OSU are
focused on converting our courses to the semester system which will take effect at the start of the 2012-13 academic year.  Our goal is to refine them to meet
the needs of broader groups in the biological sciences and the university as a whole.
At present, our calculus sequence accommodates a maximum of 160
students per year, which is just a fraction of entering students
planning to major in biological sciences.  Students
majoring in health science fields may also benefit from
taking these courses rather than the traditional calculus
sequence.

One reason for our success has been the
presence of a group of faculty working in mathematical biology who interact with both the mathematical/statistical and
biological communities at OSU.  This has been facilitated by
the MBI, several recent joint hires between
departments (including such department as Mathematics; Statistics; Evolution, Ecology, and Organismal Biology; and
Molecular Genetics), and several hires of faculty within the Mathematics Department who work in mathematical biology.
 As we continue to assemble a community of interdisciplinary researchers, we hope to build our curriculum further.  Two areas of growth that have been discussed are the development of courses that
enable students in one field to obtain a
major or minor in the other, and the development of interdisciplinary
degree programs, such as the BioMath concentration. The recent addition of our RUMBA program has enabled us to establish
 an interdisciplinary research program for
undergraduates at OSU, and has thus facilitated the continued
enhancement and expansion of our curriculum
development.

Overall, we view our mission as helping undergraduate life
science majors learn to think in a quantitative manner,
whether for modeling biological phenomena or analyzing
experimental data.  With this goal in mind, we have enjoyed meeting
the challenges of cross-department collaboration on
curriculum development in a large university, and we look
forward to continuing our progress.

\section*{Acknowledgments}
We would like to thank the MBI post-docs who contributed projects and
mentored students in the calculus courses: Michael Rempe, German
Enciso, Brandilyn Stigler, Barbara Szomolay, Andrew Nevai, Huseyin
Coskun, Yangjin Kim, Judy Day, Paula Grajdeanu, Richard
Schugart, Andrew Oster, Julia Chifman, Shu Dai, Marisa Eisenberg,
Harsh Jain, Suzanne Robertson, Deena Schmidt, Dan Siegal-Gaskins, Rebecca
Tien, Yunjiao Wang, Chuan Xue, and Kun Zhao. In addition, Statistics GTAs Yonggang Yao and Lili Zhuang
developed much of the material
for the recitation sessions of the statistics course. Tony Nance's
participation in this effort is partially supported by the National Science
Foundation under Agreement No. 0112050 and Agreement No. 0635561.

\section*{References}

Adler, F. A., 2004: \emph{Modeling the Dynamics of Life: Calculus
and Probability for Life Scientists}, 2nd ed. Brooks/Cole.

\medskip
\noindent
Anderson, E., 1935: The irises of the Gaspe peninsula. \emph{Bulletin of the American Iris Society}, \textbf{59}, 2--5.

\medskip
\noindent
Fisher, R.A., 1936: The use of multiple measurements in taxonomic problems. \emph{Annals of Eugenics}, \textbf{7}, 179--188.

\medskip
\noindent
Lucotte, G. and G. Mercier, 1998: Distributions of the CCR5 gene 32 bp deletion in Europe. \emph{J. Acquir. Imm. Def. Hum. Retrov.}, \textbf{19}, 174--177.

\medskip
\noindent
Neuhauser, C., 2003: \emph{Calculus for Biology and Medicine}. 2nd ed. Pearson/Prentice Hall.

\medskip
\noindent
StatCrunch, cited 2012: StatCrunch: Data Analysis on the Web. [Available online at http://www.statcrunch.com/.]

\medskip
\noindent
Whitlock, M.C., and D. Shluter, 2009: \emph{The Analysis of Biological Data}. Roberts.

\end{document}
