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Entrylevel Math for Biologists: Some lessons from 30 years of teaching


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Title: Entrylevel Math for Biologists: Some lessons from 30 years of teaching

Entry-level Math for Biologists Some lessons
from 30 years of teaching
  • 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

  • Levels of quantitative life science education and
    MD program requirements
  • Competency exams, general biology and assessment
  • Possible objectives for an entry-level math
  • Biological data and content of texts
  • Case studies - 2 examples
  • Summary of how Ive changed my teaching of this

  • Drs. Beth Mullin and Otto Schwarz (Botany),
    Susan Riechert (EEB)
  • Monica Beals, Susan Harrell - Modules in
    Quantitative Biology
  • Drs. Sergey Gavrilets and Jason Wolf (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

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

  • We cant 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. Similarly, assume all
    math students can be enticed into research by
    including realistic applications in biology in
    their math courses.

Main components of quantitative life science
  • (i) K-12 and teacher preparation.
  • (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
  • (i) K-12 and teacher preparation.
  • (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.

The Art of College Teaching 28 Takes Edited by
Marilyn Kallet and April Morgan University of
Tennessee Press. 2005
A regularly expressed concern as to why calculus
should be required is that most of our biology
students are pre-meds and medical schools require
a standard calculus course
  • In Education for a Biocomplex Future (Science
    288807 May 5, 2000) I summarized the
    quantitative entrance requirements for US Medical
    based upon the 2000-2001 AAMC Medical School
    Admission Requirements Guide

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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
  • 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, self-contained 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

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.

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  • 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

Do students retain the quantitative skills
  • 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

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Thus the available evidence suggests that
students retain quantitative skills obtained
within biology courses through later courses.
Modules in General Biology
  • The objective is to provide, for each standard
    section of GB, a set of short, self-contained
    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 (see www.tiem.utk.edu/bioed/).

Possible objectives for an entry-level math for
biologists course
  • to get across math ideas that are assumed to be
    "most important" in biological applications (this
    changes with time, area of bio, etc.)
  • to get across these main math ideas motivated
    by explicit biological examples
  • to encourage biological understanding (e.g.
    introduce and enhance comprehension of key
    biological concepts) while doing either of the
  • to encourage understanding of science as
    hypothesis formulation, testing (by data
    analysis) and theory building enterprise

The role of data in this entry-level math course
  • John Jungck remarked that the GB texts have
    few equations, rarely have quantitative data, and
    mostly linear examples. Also, the challenges for
    2020 students involve Multivariate, Multicausal,
    Multidimensional, Nonlinear, Multiscale and
    Complex Data. If GB books dont do a good job of
    this, are the math texts any better?

The role of data in this entry-level math course
  • Regarding data in the texts available for
    this level of course (defined as data from an
    actual study, not made up data - so it must be
    referenced as to source, with associated scatter
    plot, list of values, etc.) - what has been the
    status of this in texts - at what point in the
    text do examples with data in this form appear?

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  • Levin (1975) - P. 39 (out of 434 in Vol I) has
    first data set (gamma irradiation and litter
    layer in pine oak forest) but after this and the
    "function description section (which has several
    data examples) there are essentially no other
    data in either Vol 1 or 2 (though there are many
    "made-up graphs" based upon reasonable biological
    assumptions) - this has 586 pages in very large
    double-spaced format in the 2 volumes

  • Batschelet (Intro to Math for Life Scientists)
    (1976) - P. 24 out of 571 has example of length
    and height of quadrapeds and very occasional data
    sets given throughout rest of the text.

  • Cullen (1983) - P. 185 out of 712 has a data
    table on radioactive half life followed by many
    other referenced examples for population sizes
    (in derivative section). Numerous other data
    tables occur in the text, but some are made up,
    and there are occasional referenced data examples
    (e.g. Beverton-Holt data, Gause data, bee
    population data) but not alot more.

Brown and Rothery (1993) (Models in Biology,
Mathematics Statistics and Computing) - P. 2 out
of 634 has data on leaf area vs LAI on orchids as
a linear function example and numerous referenced
data sets throughout text. This is not a calculus
text (assumes students are age 18, but this is
for UK students where they already have seen
  • Greenwell, Ritchey and Lial (Calculus with
    Applications for the Life Sciences) (2003) - P.
    17 out of 767 has first of numerous referenced
    examples of linear data- first referenced data
    table is on P. 19 (US accidental human death rate
    by year) - lots of other data examples in
    function section. Many other examples with
    referenced data sets throughout text, mainly in
    sections called applications after a math

  • Neuhauser (2004) - First data set occurs on P
    860 out of 917 pages on brachiopod shell length
    frequency distribution in probability
    distribution section (there are only a couple of
    other data examples in the problem section after
    this). There are no tables of data with
    references or data in graphs beyond these.

  • Adler (2005) - first "data" appear on P. 6 out
    of 802 pages for bacterial population through
    time, but no mention as to whether or not these
    are real or made up data and there are numerous
    other "data" examples in first section for which
    it is not at all clear whether they are real or
    made up - indeed there are many tables of data in
    Chapter 1 but no way to tell which are from
    models and which are from observations (if any).
    Adlers book has many, many tables of data, but
    nowhere in the book could I find a referenced
    data set

  • Would students learn biological concepts from
    the above texts? Is the connection to data not
    necessary to provide motivation for the
    importance of the math in these texts to actual
    biological application?

What do you think?
The Entry-level Quantitative Course Biocalculus
  •   In response to 1992 workshop recommendations, a
    new entry-level quantitative course for life
    science students was constructed and has now
    become the standard math sequence taken by about
    400 biology students per semester. The
    prerequisites assumed are Algebra, Geometry, and

  • 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

  •   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.  

Content of entry-level math sequence first
semester 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,
  • 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

Using Case Studies to Encourage Appreciation of
Mathematics/Biology Linkages
  • Within a math course, a brief set of case studies
    that relate to students experiences can serve as
    a means to introduce new concepts, building on
    students intuition. Using examples for which
    their intuition can provide an initial guide
    enhances student confidence, and as organisms
    ourselves biology provides a natural set of
    examples - ideally these can be returned to
    several times in a course to motivate different
  • Drug dosage Exponential decay for elimination
    from blood stream for dosage sequence, discrete
    time model of concentration immediately following
    each dose single compartment differential
  • Music ear response to varying sound levels as
    non-linear decibels and log-scaling waveforms
    and trig graphic equalizers power spectra

OSHA Regulations (Standards - 29 CFR)
Occupational noise exposure. - 1910.95
If the variations in noise level involve maxima
at intervals of 1 second or less, it is to be
considered continuous. TABLE G-16 -
Duration per day, hours Sound level dBA slow
response _____________________________________
90 6...........................
92 4...........................
95 3........................
... 97 2...................
........ 100 1 1/2
...................... 102
105 1/2 ........................
110 1/4 or less................
115 ________________________________
____________________________ Footnote(1)
When the daily noise exposure is composed of two
or more periods of noise exposure of different
levels, their combined effect should be
considered, rather than the individual effect of
each. If the sum of the following fractions
C(1)/T(1) C(2)/T(2) C(n)/T(n) exceeds unity,
then, the mixed exposure should be considered
to exceed the limit value. Cn indicates the total
time of exposure at a specified noise level,
and Tn indicates the total time of exposure
permitted at that level. Exposure to impulsive or
impact noise should not exceed 140 dB peak
sound pressure level.
Using a Case Study - Landscape change
  • Illustrate math concepts of vectors, matrices,
    Markov chain, equilibium, stability
  • Introduce biological concepts of succession,
    sere, climax, diversity indices
  • Step 1 show a LandSat image or aereal photo. Ask
    class as a whole what this shows (e.g. what can
    they learn from the image).

See www.myworldgis.org for free GIS access and a
variety of educational uses of GIS Newcombe, N.
A. A Plea for Spatial Literacy. Chronicle of
Higher Education B20. March 3, 2006. NRC. 2006.
Learning to Think Spatially.
Using a Case Study - Landscape change
  • Step 2 Using pair and share, ask students to
    develop methods to describe the image in
    mathematical terms (this is just after the
    descriptive statistics section of the course).
    Develop as a class a list of these descriptors.
    One suggestion is usually a list of how much of
    each image is in each type or color. Point out
    that this is a vector, illustrated by a bar
    chart, and when normalized gives a probability
    distribution (this is the first example of this -
    probability comes later in course).

Using a Case Study - Landscape change
  • Step 3 Ask what is missing from list of
    descriptors - they typically will not have any
    measure of spatial pattern. Motivate this by how
    different types or colors are clustered
    differently in the image - use board illustration
    of point clusters to illustrate overdispersed and
    underdispersed examples.

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Using a Case Study - Landscape change
  • Step 4 Pair and share to describe how theyd
    characterize the time-series of images.
    Eventually they come up with a time graph of some
    of the descriptors they already mentioned for a
    single image. They generally get the idea
    themselves that a changing vector of landuse is a
    way to characterize landuse change over time.

Using a Case Study - Landscape change
  • Step 5 Use EcoBeaker Intermediate
    Disturbance example to illustrate idea of a sere,
    succession, climax with different levels of
    description and point out loss of information as
    description changes. In later classes use a
    simpler 3-landscape type example to illustrate
    how equilibria vary with parameters (pair and
    share to guess response) and correspond to

Using a Case Study - student height measurements
  • Illustrate basic descriptive statistics, bar
    charts, scatter plots, regression utilizing
    personal data
  • Develop a data set that is easily understood, is
    multivariate, has potential multiple causal
    factors, illustrates problems with sampling and
    outlier effects

Step 1 Pair and share to develop hypotheses as
to what happens to student height overnight (e.g.
measure height before sleep and after sleep).
Summarize hypotheses on board and ask what data
need to be collected to investigate these.
Step 2 Assign this as a project for each student
to perform on 4 nights, collect the data and
email to instructor.
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Using a Case Study - student height measurements
Concepts illustrated
  • Look carefully at data sets for bad data and
  • Develop hypotheses before collecting data and use
    exploratory analysis to evaluate these and other
    possible hypotheses
  • Graph multivariate data in several ways and in
    particular partition the data set according to
    different factors
  • Simple data descriptors (measures of central
    tendency and dispersion) may not be adequate data
  • Importance of IRB (Institutional Review Board)
    approval for experiments dealing with human

  • This sequence is now taken by approximately 300
    students per semester, and is taught by me in
    large lecture format, 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
  • It is still necessary to review background
    concepts (exponentials and logs), but this is
    eased through the use of numerous biological
  • Despite much experience with word-processing and
    game software, students have difficulty utilizing
    mathematical software and developing simple

Summary of what Ive changed in this course over
  • Added brief discussion of recent journal papers
    at start of class illustrating some aspect of
    math topic (emphasizing warm and fuzzies, drugs,
    disease, sex and rock and roll)
  • Incorporation of projects involving the students
    in data collection and use of these data in
  • Greatly enhanced use of computer projects
    including some that are research-level such as
    structured population growth in random
  • Regular use of Pair and Share to develop
  • Emphasis of a few case study examples throughout
    course that are returned to several times to
    illustrate new ideas.

Impediments to progress
  • Few math faculty at research universities have
    any appreciation (or interest) in real
    applications of math
  • Few regular tenure/tenure-track faculty teach
    entry-level courses
  • Few biology faculty (not including many recently
    hired) have strong quantitative skills except in
  • 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

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