Instructor's Guide to Math 151-2 August 1994
Lou Gross
This course sequence provides an introduction to a variety of mathematical
topics of use in analyzing problems arising in the biological sciences. It
is designed for students in biology, agriculture, forestry, wildlife,
pre-medicine and other pre-health professions. Students who desire a strong
mathematical grounding, enabling them to take most advanced math courses,
should consider taking Math 141-2 instead. The general aim of the sequence
is to show how mathematical and analytical tools may be used to explore and
explain a wide variety of biological phenomena that are not easily
understood with verbal reasoning alone. Prerequisites are two years of
high school algebra, a year of geometry, and half a year of trigonometry.
Course 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.
Provide Experience using Computer Software to Analyze Data and
Investigate Mathematical Models
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 including MATLAB and
MicroCalc.
Encourage Investigation of Quantitative Approaches in
Biological areas of Particular Interest to each Student
through Projects Utilizing Software from diversity of
Biological areas.
To Tell the Students at the Beginning:
1. This is a hard course. In many respects it is more difficult that the
science/engineering calculus sequence (Math 141-2) since it covers a
wider variety of mathematical topics, is coupled to real data, and involves
the use of the computer.
2. Despite the fact that the course is hard, it has been specifically
designed for life science students, will include many more biological
examples than Math 141-2 or Math 121-2, and will introduce the students
to quantitative concepts not covered in these other math courses that
they should find useful in their biology courses.
3. Regarding the computer, there is no presumption that you have any
experience - we will provide handouts to guide you in the use of the two
main computer packages (MATLAB and MicroCalc) we will use. It will take
time to get used to these, but we will provide samples of how to do all
computer assignments.
4. For portions of this course we will not be following the textbook so it is
important that you attend class regularly.
Text: Mathematics for the Biosciences by Michael Cullen supplemented by
material done in class
Classroom: Should be assigned to a room with computer monitor and link to
the Math Lab, so you can demonstrate problems in class.
For Each Class:
1. Try to start each class by getting the students to come up with an
hypothesis (or several) regarding a biological or mathematical topic germaine
to that day's class. For example, after day 1 if the students went out to
collect leaf size data, ask them "Are leaf width and length related? Is the
relationship the same for all tree species? What affects leaf sizes? Why
do some trees have larger leaves than others?" Each of these questions could
generate many hypotheses, and you can then go on to use MATLAB to analyze
the data sets they collect to evaluate the hypotheses.
A hypothesis doesn't have to relate to a biological area, it could be
on mathematics alone. For example, after linear regression, ask "Can we
reasonably use this regression to determine the y-value for an x-value for
which we have no data?" This leads naturally to a discussion of interpolation
and extrapolation.
2. Include at the start of each day a brief description of how what is being
covered relates to biology. I suggest doing this by having a background
biological example used for each main mathematical topic being covered, and
refer to this biological example regularly as you develop the math. For example,
in covering matrices, you can introduce the idea of a matrix using:
"Suppose you are a land manager in the U.S. West, and you have satellite
images of the land you manage taken every year for several years. The images
clearly show whether a point on the image (actually a 500m x 500 m plot of
land) is bare soil, grassland or shrubland. How can you use these to help you
manage the system?" From this the students can themselves develop the key
notion of a transition matrix, you can then go on to matrix multiplication,
and eigenvalues and eigenvectors for describing dynamics of the landscape
and the long-term fraction in bare soil, grass, and shrubs.
3. Try to regularly include data (real, not made up) in class demonstrations,
project assignments, and exams. For example, use the northern hemisphere
monthly CO2 data for semi-log regression, and use allometry data for log-log
regression. Try to encourage the students to collect their own data for
appropriate portions of the course, particularly the descriptive statistics
section. Bring in scientific journal articles that use the math you are
talking about to illustrate its utility.
Syllabus Math 151:
Descriptive statistics - analysis of tabular data, means, variances,
histograms, linear regression - 3 weeks
Exponentials and logarithms, non-linear scalings, allometry - Text sections
5 & 15 - 2 weeks
Matrix algebra - Text sections 55 & 56 with supplementary material - addition,
subtraction, multiplication, inverses, matrix models in population biology,
eigenvalues, eigenvectors, Markov chains, ecological succession - 4 weeks
Discrete probability - Text sections 57 to 61 with supplementary material -
population genetics, behavioral sequence analysis - 4 weeks
Sequences and Difference equations - Introduction to sequences and limit
concept, Text section 47 with supplementary material - 1 week
Syllabus Math 152:
Difference equations - Text sections 48 to 49 with supplementary material -
linear and nonlinear examples, equilibrium, stability and homeostasis,
logistic model, introduction to limits - 2 weeks
Limits of functions and continuity - Text sections 6 & 7 - 1 week
Derivatives - Book sections 8-11 - 2 weeks
Curve sketching - Book sections 12-14 - 2 weeks
Exponential and logarithms - Book sections 16-17 - 1 week
Antiderivatives and integrals - Book sections 18-25 - 3 weeks
Trigonometric functions - Book sections 29-31 - 1 week
Differential equations and modeling - Book sections 33-39 - 2 weeks
Handouts:
There are handouts for the beginning of the course that describe the basics
of MATLAB and the Math Laboratory. Students can sign a form and take home a
copy of MATLAB (Student Version) to be used wherever they want. The Lab has
a directory with many MATLAB programs installed to do the various projects
for this course. You may want to hand out copies of these MATLAB codes to
the class when you use them in class.
See previous exams and quizes for guidance in what topics to emphasize. I
recommend handing out sample exams two class periods before each exam, then
spending the class period before the exam going over any questions on either
the assignments or the sample exam.
Grading:
My standard grading scheme for this course is:
The grade will be based on several components: (a) There will be a set of 10
minute quizes, generally given once a week at the end of a class period (in
weeks for which there is no exam scheduled); (b) There will be a set of
assignments based on the use of the computer to analyze particular sets of
data, or problems. These may be worked on within a study group, as long as
the instructor is notified, and each individual writes their own results;
(c) There will be a set of three exams during the term, in addition to a
comprehensive final. The exams will in general not be computer based, focusing
rather on the key concepts and techniques discussed in the course. Of the
three regular exams given, the one with the lowest score will be dropped.
The weighting of these components of the grade are: (a) 20%, (b) 20%,
(c) 60% (the final exam counts 30% of the course grade, and the two regular
exams not dropped will together count 30% of the grade). There will also be
opportunities for extra credit for those desiring this. One opportunity will
require the participant to evaluate one of a wide variety of software
programs we have available involving some area of biology - this requires
becoming very familiar with the program, and writing a formal review of the
software, in the same format as might appear in a scientific journal.
Participants will be expected to regularly work problems from the text, or
problems assigned by the instructor. These should be worked on individually,
but will not be graded.