Quantitative Education for Life
Sciences: BIO2010 and Beyond
Louis J. Gross
Departments of Ecology and Evolutionary
Biology and Mathematics, The Institute for
Environmental Modeling, University of
Tennessee – Knoxville
Financial Support: National Science
Foundation (DUE 9150354, DUE 9752339)
National Institutes of Health (GM59924-01)
www.tiem.utk.edu/bioed
Overview:
Summary of BIO2010 quantitative education
recommendations
Different viewpoints from the Math/CS Panel
summary
Overview of the University of Tennessee projects
Future directions: Suggestions to implement
BIO2010 ideas and impediments
Main components of quantitative life
science education:
(i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative  bio,
bio  quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.
Main components of quantitative life
science education:
(i) K-12 and teacher training.
(ii) Undergraduate intro biology courses.
(iii) Undergraduate intro quantitative courses.
(iv) Upper division life science courses.
(v) Undergraduate research experiences.
(vi) Graduate training: quantitative  bio,
bio  quantitative.
(vii) Faculty, post-doc, MD advanced training.
(viii) International cooperative training and
research.
Major BIO2010
Recommendations
1. Schools should reexamine current approaches
to see if they meet the needs of today’s
undergraduate biology students. Those
selecting the new approaches should consider
the importance of building a strong
foundation in mathematics, and the physical
and information sciences to prepare students
for research that is increasingly
interdisciplinary in character. This
implementation should be accompanied by a
process of assessment.
2. Concepts, examples, and techniques from
mathematics, and the physical and information
sciences should be included in biology courses,
and biological concepts and examples should be
included in other science courses. Faculty must
work collaboratively to integrate mathematics
and physical sciences into life science courses as
well as providing avenues for incorporating life
science examples that reflect the emerging
nature of the discipline into courses taught in
mathematics and physical sciences.
3. School administrators, as well as funding
agencies, should support mathematics and science
faculty in the development or adaptation of
techniques that improve interdisciplinary
education for biologists. This would include
courses, modules (on biological problems suitable
for study in mathematics and physical science
courses and vice versa), and other teaching
materials. Administrative and financial barriers
to cross-departmental collaboration between
faculty must be eliminated.
4. Laboratory courses should be as
interdisciplinary as possible, since
laboratory experiments confront students
with real-world observations do not
separate well into conventional
disciplines
5. All students should be encouraged to
pursue independent research as early as
is practical in their education. They
should be able to receive academic credit
for independent research done in
collaboration with faculty or with offcampus researchers
6. Seminar-type courses that highlight cuttingedge developments in biology should be
provided on a continual and regular basis
throughout the four-year undergraduate
education of students. Communicating the
excitement of biological research is crucial to
attracting, retaining, and sustaining a greater
diversity of students to the field. These courses
would combine presentations by faculty with
student projects on research topics.
7. Medical school admissions
requirements and the Medical College
Admissions Test (MCAT) are hindering
change in the undergraduate biology
curriculum and should be reexamined in
light of the recommendations in this
report.
8. Faculty development is a crucial
component to improving undergraduate
biology education. Efforts must be made on
individual campuses and nationally to
provide faculty the time necessary to refine
their own understanding of how the
integrative relationships of biology,
mathematics, and the physical sciences can be
best melded into either existing courses or
new courses in the particular areas of science
in which they teach.
Summary of Quantitative Recommendations
1.5 Biology majors headed for research careers need to be
educated in a more quantitative manner, which may
require the development of new types of courses. We
recommend that all biology majors master the concepts
listed below and that life science majors become
sufficiently familiar with the elements of programming to
carry out simulations of physiological, ecological, and
evolutionary processes. They should be adept at using
computers to acquire and process data, carry out
statistical characterization of the data and perform
statistical tests, and graphically display data in a variety
of representations. Students should also become skilled at
using the Internet to carry out literature searches, locate
published articles, and access major databases.
Concepts of Mathematics and
Computer Science
Calculus
• Complex numbers
• Functions
• Limits
• Continuity
• The integral
• The derivative and linearization
• Elementary functions
• Fourier series
• Multi-dimensional calculus: linear approximations,
integration over multiple variables
Linear Algebra
• Scalars, vectors, matrices
• Linear transformations
• Eigenvalues and eigenvectors
• Invariant subspaces
Dynamical Systems
• Continuous time dynamics — equations of
motion and their trajectories
• Test points, limit cycles, and stability around
them
• Phase plane analysis
• Cooperativity, positive feedback, and negative
feedback
• Multistability
• Discrete time dynamics — mappings, stable
points, and stable cycles
• Sensitivity to initial conditions and chaos
Probability and Statistics
• Probability distributions
• Random numbers and stochastic
processes
• Covariation, correlation, and
independence
• Error likelihood
Information and Computation
• Algorithms (with examples)
• Computability
• Optimization in mathematics and
computation
• '”Bits”: information and mutual
information
Data Structures
• Metrics: generalized 'distance' and
sequence comparisons
• Clustering
• Tree-relationships
• Graphics: visualizing and displaying data
and models for conceptual understanding
Additional Quantitative Principles
Useful to Biology Students
1. Rate of change
2. Modeling
3. Equilibria and stability
4. Structure
5. Interactions
6. Data and measurement
7. Stochasticity
8. Visualizing
9. Algorithms
Additional Quantitative Principles
Useful to Biology Students
Rate of change
This can be a specific (e.g., per capita) rate of change or
a total rate of change of some system component.
• Discrete rates of change arise in difference equations,
which have associated with them an inherent timescale.
• Continuous rates of change arise as derivatives or
partial derivatives, representing instantaneous (relative
to the units in which time is scaled) rates.
Modeling
• The process of abstracting certain aspects of reality to
include in the simplifications we call models.
• Scale (spatial and temporal) – different questions arise
on different scales.
• What is included (system variables) depends on the
questions addressed, as does the hierarchical level in
which the problem is framed (e.g., molecular, cellular,
organismal).
• There are trade-offs in modeling—no one model can
address all questions. These trade-offs are between
generality, precision, and realism.
• Evaluating models depends in part upon the purpose
for which the model was constructed.
Equilibria and stability
• Equilibria arise when a process (or several processes)
rate of change is zero.
• There can be more than one equilibrium. Multiple
stable states are typical of biological systems.
• Equilibria can be dynamic, so that a periodic pattern of
system response may arise.
• There are numerous notions of stability, including not
just whether a system that is perturbed from an
equilibrium returns to it, but also how the system
returns (e.g., how rapidly it does so).
• Modifying some system components can lead to
destabilization of a previously stable equilibrium,
possibly generating entirely new equilibria with
differing stability characteristics.
Structure
• Grouping components of a system affects the kinds of
questions addressed and the data required to
parameterize the system.
• Choosing different aggregated formulations (by sex,
age, size, physiological state, activity state) can expand
or limit the questions that can be addressed, and data
availability can limit the ability to investigate effects of
structure.
• Geometry of the aggregation can affect the resulting
formulation.
• Symmetry can be useful in many biological contexts to
reduce the complexity of the problem, and situations in
which symmetry is lost (symmetry-breaking) can aid in
understanding system response.
Interactions
• There are relatively few ways for system components to
interact. Negative feedbacks arise through competitive
and predator-prey type interactions, positive feedback
through mutualistic or commensal ones.
• Some general properties can be derived based upon
these (e.g., 2-species competitive interactions), but even
relatively few interacting system components can lead
to complex dynamics.
• Though ultimately everything is hitched to everything
else, significant effects are not automatically
transferred through a connected system of interacting
components—locality can matter.
• Sequences of interactions can determine outcomes—
program order matters.
Data and measurement
• Only a few basic data types arise (numeric,
ordinal, categorical), but these will often be
interconnected and expanded (e.g., as vectors
or arrays).
• Consistency of the units with which one
measures a system is important.
• A variety of statistical methods exist to
characterize single data sets and to make
comparisons between data sets. Using such
methods with discernment takes practice
Stochasticity
• In a stochastic process, individual outcomes cannot be
predicted with certainty. Rather, these outcomes are
determined randomly according to a probability
distribution that arises from the underlying
mechanisms of the process. Probabilities for
measurements that are continuous (height, weight,
etc.), and those that are discrete (sex, cell type) arise in
many biological contexts.

Risk can be identified and estimated.
• There are ways to determine if an experimental result
is significant.
• There are instances when stochasticity is significant
and averages are not sufficient.
Visualizing
• There are diverse methods to display data.
• Simple line and bar graphs are often not
sufficient.
• Non-linear transformations can yield new
insights.
Algorithms
• These are rules that determine the types of
interactions in a system, how decisions are
made, and the time course of system response.
• These can be thought of as a sequence of
actions similar to a computer program, with all
the associated options such as assignments, ifthen loops, and while-loops.
Potential curricula
Four examples were given:
1. First-year math, perhaps a mixture of
discrete and calculus
2. Quantitative emphasis with first year of
math, plus probability/biostatistics, DE plus
advanced course
3. Math I plus CS first year, then Math II plus
advanced math/CS course later
4. Calc and DE in year 1, biostatistics and 1 CS
course
Math/CS Panel Comments
•
•
There is a distinction between the
“quantitative biologist,” who works at the
interface of math/computer science and
biology, and the “research biologist,” who
needs familiarity with a range of
mathematical and computational ideas
without necessarily being expert. Thus, the
panel felt that flexibility in offerings is more
advisable than a fixed curriculum.
Consider offering “quantitatively intensive”
versions of standard biology courses, with
extra credit
• Consider new math courses which
condense much of undergrad math into
3-4 semesters
• Encourage interdisciplinary modeling
courses at both introductory and
advanced level, possibly as a first-year
and senior seminar
Key Points:
Success in quantitative life
science education requires an
integrated approach: formal
quantitative courses should be
supplemented with explicit
quantitative components within
life science courses.
Life science students should be
exposed to diverse quantitative
concepts: calculus and statistics
do not suffice to provide the
conceptual quantitative
foundations for modern biology.
We can’t determine a priori who will
be the researchers of the future –
educational initiatives need to be
inclusive and not focused just on the
elite. Assume all biology students can
enhance their quantitative training
and proceed to motivate them to
realize its importance in real biology.
The CPA Approach to Quantitative
Curriculum Development across Disciplines
As a summary of the approach I have taken in
this life sciences project, and in hope that this
will be applicable to other interdisciplinary
efforts, I offer the CPA Approach:
Constraints, Prioritize, Aid
Understand the Constraints under which your
colleagues in other disciplines operate - the
limitations on time available in their
curriculum for quantitative training.
Work with these colleagues to Prioritize the
quantitative concepts their students really need,
and ensure that your courses include these.
Aid these colleagues in developing quantitative
concepts in their own courses that enhance a
students realization of the importance of
mathematics in their own discipline. This could
include team teaching of appropriate courses.
Note: The above operates under the paradigm
typical of most U.S. institutions of higher
learning - that of disciplinary
compartmentalization. An entirely different
approach involves real interdisciplinary
courses. This would mean complete revision of
course requirements to allow students to
automatically see connections between various
subfields, rather than inherently different
subjects with little connection. Such courses
could involve a team approach to subjects,
which is common in many lower division
biological sciences courses, but almost unheard
of in mathematics courses.
Collaborators
Drs. Beth Mullin and Otto Schwarz (Botany),
Susan Riechert (EEB)
Monica Beals, Susan Harrell - Primer of
Quantitative Biology
Drs. Sergey Gavrilets (EEB) and Suzanne
Lenhart (Math) – NIH Short Courses
Drs. Thomas Hallam (EEB) and Simon Levin
(Princeton) – International Courses
Society for Mathematical Biology – Education
Committee – www.smb.org
Project activities:
• Conduct a survey of quantitative course
requirements of life science students;
• Conduct a workshop with researchers and
educators in mathematical and quantitative
biology to discuss the quantitative component
of the undergraduate life science curriculum;
• Develop an entry-level quantitative course
sequence based upon recommendations from
the workshop;
• Implement the course in an hypothesisformulation and testing framework, coupled to
appropriate software;
•
Conduct a workshop for life science faculty to
discuss methods to enhance the quantitative
component of their own courses;
• Develop a set of modules to incorporate within
a General Biology course sequence, illustrating
the utility of simple mathematical methods in
numerous areas of biology;
• Develop and evaluate quantitative competency
exams in General Biology as a method to
encourage quantitative skill development;
• Survey quantitative topics within short
research communications at life science
professional society meetings.
The Entry-level Quantitative Course:
Biocalculus Revisited
In response to workshop recommendations,
a new entry-level quantitative course for life
science students was constructed and has
now become the standard math sequence
taken by biology students. The prerequisites
assumed are Algebra, Geometry, and
Trigonometry.
Goals:
Develop a Student's ability to
Quantitatively Analyze Problems arising
in their own Biological Field.
Illustrate the Great Utility of
Mathematical Models to provide answers
to Key Biological Problems.
Develop a Student's Appreciation of the
Diversity of Mathematical Approaches
potentially useful in the Life Sciences
Methods:
Encourage hypothesis formulation and testing
for both the biological and mathematical topics
covered.
Encourage investigation of real-world
biological problems through the use of data in
class, for homework, and examinations.
Reduce rote memorization of mathematical
formulae and rules through the use of software
such as Matlab and Maple.
Course 1 Content – Discrete Math Topics:
Descriptive Statistics - Means, variances,
using software, histograms, linear and
non-linear regression, allometry
Matrix Algebra - using linear algebra
software, matrix models in population
biology, eigenvalues, eigenvectors, Markov
Chains, compartment models
Discrete Probability - Experiments and
sample spaces, probability laws,
conditional probability and Bayes'
theorem, population genetics models
Sequences and difference equations limits of sequences, limit laws, geometric
sequence and Malthusian growth
Course 2 Content – Calculus and Modeling:
Linear first and second order difference equations equilibria, stability, logistic map and chaos,
population models
Limits of functions - numerical examples using limits
of sequences, basic limit principles, continuity
Derivatives - as rate of growth, use in graphing, basic
calculation rules, chain rule, using computer algebra
software
Curve sketching - second derivatives, concavity,
critical points and inflection points, basic
optimization problem
Exponentials and logarithms - derivatives, applications
to population growth and decay
Antiderivatives and integrals - basic properties,
numerical computation and computer algebra systems
Trigonometric functions - basic calculus, applications
to medical problems
Differential equations and modeling - individual and
population growth models, linear compartment models,
stability of equilibria
Results:
This sequence is now taken by approximately 150 students
per semester, and is taught mostly by math instructors
and graduate students in math biology.
In many ways the course is more challenging than the
standard science calculus sequence, but students are
able to assimilate the diversity of concepts.
It is still necessary to review background concepts
(exponentials and logs), but this is eased through the
use of numerous biological examples.
Despite much experience with word-processing and game
software, students have difficulty utilizing
mathematical software and developing simple
programs.
Alternative Routes to Quantitative Literacy
for the Life Sciences: General Biology
Determine the utility of alternative methods to enhance
the quantitative components of a large-lecture format
GB sequence using:
Quantitative competency exams developed specifically to
evaluate the quantitative skills of students taking the
GB sequence for science majors;
Modules comprising a Primer of Quantitative Biology
designed to accompany a GB sequence, providing for
each standard section of the course a set of short, selfcontained examples of how quantitative approaches
have taught us something new in that area of biology.
Quantitative Competency Exams:
Multiple choice exams based upon the
skills and concepts appropriate for the
Organization and Function of the Cell
and the Biodiversity (whole organism,
ecology and evolutionary) components of
GB. Given at beginning and end of the
course to track changes in skills. Require
only high-school math skills, with
questions placed in a GB context.
Goals of Competency Exams:
(i) inform students at the beginning of a course
exactly what types of math they are expected to
already be able to do;
(ii) help students be informed about exactly what
concepts they don't have a grasp of, so they can
go back and refresh their memory; and
(iii) ensure that the class is not held back through
having to review material that the students
should know upon entering.
Pre- and post-testing were done
in GB sections taught by
collaborators on this project,
emphasizing quantitative skills,
and other sections taught by
faculty in a standard manner,
as a control.
Conclusion:
Inclusion of a quantitative emphasis
within biology courses can aid
students in improving their
quantitative skills, if these are made
an inherent part of the course and
not simply an add-on.
Do students retain the
quantitative skills developed?
We surveyed a sophomore level Genetics
class a year after the students had been in
the General Biology course, and determined
student performance on another quantitative
competency exam. We compared exam
scores of students who had been in a GB
course which emphasized quantitative ideas
to those who had been in a standard GB
course.
Thus the available evidence
suggests that students retain
quantitative skills obtained
within biology courses through
later courses.
Modules in GB
The objective is to provide, for each
standard section of GB, a set of short, selfcontained examples of how quantitative
approaches have taught us something new
in that area of biology. Most examples are at
the level of high-school math, though there
are some calculus-level and above examples.
A standard format for each module was
established and a collection of 57 modules
have been developed.
Use of Modules within GB
These modules have been implemented in a
variety of ways in GB.
(i) in lectures as a supplement to lecture
material.
(ii) assigned to students as outside reading
assignments.
(iii) students have been asked to turn in
formal reports as homework assignments
based around the additional questions to
be answered at the end of each module.
What quantitative topics are
used?
Surveys were done at annual meetings of
the Ecological Society of America and the
Society for the Study of Evolution. The
most important quantitative topic for
each poster was assessed as well as a
listing of all quantitative concepts used
for each poster.
ESA 2000 – Poster Quantitative Topics
SSE 2001- Poster Quantitative Topics
Some lessons:
1. It is entirely feasible to include diverse
mathematical and computational
approaches in an entry-level quantitative
course for life science students. This can
be successful, even though it is in many
respects more difficult than a standard
science and engineering calculus course,
if students see the biological context
throughout the course.
2. Inclusion of a quantitative emphasis
within biology courses can aid students to
improve their quantitative skills, if these
are made an inherent part of the course
and not simply an add-on. Evidence
suggests that students retain these
quantitative skills through later courses.
3. Instructors can utilize quantitative
competency exams to encourage students
early in a course to focus on skills they
should have mastered and see the
connection between these skills and the
biological topics in the course.
4. The key quantitative concepts that are used in
short scientific communications are basic
graphical and statistical ones that are typically
covered very little in a formal manner in most
undergraduate biology curricula.
Visualization/interpretation of data and results
are critical to the conceptual foundations of
biology training and we should give them
higher priority in the curriculum. This might
include a formal course on Biological Data
Analysis, but needs to be emphasized
throughout the science courses students take.
Future Directions:
The BIO2010 Report gives numerous recommendations
on quantitative skill development. Accomplishing these
above can be aided through:
a. Agreed upon quantitative competency testing across
courses.
b. Setting up teaching circles involving the key faculty
involved in appropriate groups of courses.
c. Encouraging projects either formally within courses or
as part of labs that require quantitative analysis
involving the concepts deemed critical for
comprehension.
d. Including key quantitative ideas from the beginning in
basic entry-level courses - expecting students to utilize
skills developed in high school and providing
mechanisms to aid those who need remediation.
Impediments to progress
Few math faculty at research universities have any
appreciation (or interest) in real applications of
math
Few biology faculty (not including many recently
hired) have strong quantitative skills except in
statistics
Multiculturalism of math departments creates
problems for faculty/student interactions
Cultures are different – few undergrads in math are
expected to work on research with faculty, while it
is expected that the better biology undergrads will
have some exposure to research in field/lab
situations with faculty
Math faculty prefer rigor (proof) over breadth