1 / 71

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

Outline

- Levels of quantitative life science education and

MD program requirements - Competency exams, general biology and assessment
- Possible objectives for an entry-level math

course - Biological data and content of texts
- Case studies - 2 examples
- Summary of how Ive changed my teaching of this

course

Collaborators

- 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

biology.

- 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

education

- (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

education

- (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

55

21

49

56

55

17

21

49

49

56

55

17

21

49

49

(No Transcript)

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, 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

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.

(No Transcript)

(No Transcript)

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.

(No Transcript)

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

above. - 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?

(No Transcript)

(No Transcript)

- 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

calculus).

- 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

section.

- 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

Revisited

- 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

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.

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,

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

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

ideas - Drug dosage Exponential decay for elimination

from blood stream for dosage sequence, discrete

time model of concentration immediately following

each dose single compartment differential

equation - 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

http//www.osha-slc.gov/OshStd_data/1910_0095.html

If the variations in noise level involve maxima

at intervals of 1 second or less, it is to be

considered continuous. TABLE G-16 -

PERMISSIBLE NOISE EXPOSURES (1)

__________________________________________________

____________

Duration per day, hours Sound level dBA slow

response _____________________________________

________________________

8...........................

90 6...........................

92 4...........................

95 3........................

... 97 2...................

........ 100 1 1/2

...................... 102

1...........................

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.

(No Transcript)

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

eigenvectors.

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.

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

(No Transcript)

Using a Case Study - student height measurements

Concepts illustrated

- Look carefully at data sets for bad data and

outliers - 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

descriptors - Importance of IRB (Institutional Review Board)

approval for experiments dealing with human

subjects.

Results

- 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

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.

Summary of what Ive changed in this course over

time

- 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

projects - Greatly enhanced use of computer projects

including some that are research-level such as

structured population growth in random

environments - Regular use of Pair and Share to develop

hypotheses - 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

statistics - 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

(No Transcript)