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Teaching Mathematics to Biologists and Biology to Mathematicians. Gretchen A. Koch Goucher College MathFest 2007. Introduction. Who: Undergraduate students and faculty What: Improving quantitative skills of students through combination of biology and mathematics

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teaching mathematics to biologists and biology to mathematicians

Teaching Mathematics to Biologists and Biology to Mathematicians

Gretchen A. Koch

Goucher College

MathFest 2007

introduction
Introduction
  • Who: Undergraduate students and faculty
  • What: Improving quantitative skills of students through combination of biology and mathematics
  • When: Any biology or mathematics course
    • Simple examples interspersed throughout semester
    • Common example as theme for entire semester
slide3
How??
  • Communication is key
    • Talk with colleagues in natural sciences
    • Use the same language
    • Make the connections obvious
      • Example: Why is Calculus I required for many biology and chemistry majors??
  • Case studies, ESTEEM, and the BioQUEST way
  • Have an open mind and be creative
what is a case study
What is a case study?
  • Imaginative story to introduce idea
  • Self-discovery with focus
    • Ask meaningful questions
    • Build on students’ previous knowledge
    • Students expand knowledge through research and discussion.
  • Assessment
case studies beware of
Case Studies – Beware of…
  • Clear objectives = easier assessment
  • Clear rubric = easier assessment
  • Focused questions = easier assessment
  • Too much focus = students look for the “right” answer
  • Provide some starting resources
    • Continue building your database
  • Have clear expectations (Communication!)
  • Be flexible
where do i start
Where do I start???

http://bioquest.org/icbl/

c 3 cal crabs and the chesapeake
C3: Cal, Crabs, and the Chesapeake
  • Cal, a Chesapeake crabber, was sitting at the end of the dock, looking forlorn. I approached him and asked, “What’s the matter, Cal?” He replied, “Hon – it’s just not the same anymore. There are fewer and fewer blue crabs in the traps each day. I just don’t know how much longer I can keep the business going. You’re a mathematician – and you always say math is everywhere…where’s the math in this???”
case analysis use for discussion
Case Analysis – Use for Discussion
  • What is this case about?
  • What could be causing the blue crab population to decrease?
  • Can we predict what the blue crab population will do?
  • Can we find data to show historic trends in the blue crab population?
  • How will you answer these questions?
learning objectives mathematics
Learning Objectives - Mathematics
  • Use different mathematical models to explore the population dynamics
    • Linear, exponential, and logistic growth models
      • Precalculus level
      • Continuous Growth ESTEEM Module
    • Predator-prey model
      • Calculus, Differential Equations, Numerical Methods
      • Two Species ESTEEM Module
    • SIR Model
      • Calculus, Differential Equations, Numerical Methods
      • SIR ESTEEM Module
learning objectives biology
Learning Objectives - Biology
  • Explore the reasons for the decrease in the crab population
    • Habitat
    • Predators
    • Food Sources
    • Parasites
    • Invasive species
  • In field experiments
  • Journal reviews of ongoing experiments
assessment and evaluation plan
Assessment and Evaluation Plan
  • Homework questions to demonstrate understanding of use of ESTEEM modules
  • Homework questions to demonstrate comprehension of topics presented in ESTEEM modules
  • Group presentations of background information
  • Exam questions to demonstrate synthesis of mathematical concepts using different examples
sources
Sources
  • Blue Crab
  • Chesapeake Blue Crab Assessment 2005
  • Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs
but what s the answer
But – what’s the answer??
  • Assessments and objectives vary
  • Knowledge of tools and structure
  • Adopt and adapt
first growth model
First Growth Model
  • Suppose you ask Cal to keep track of the number of crabs he catches for 10 days. He gives you the following:
  • Do you see a pattern?
linear growth model
Linear Growth Model
  • Simplest model:

where C is the number of crabs on day t, and D is some constant number.

  • Questions to ask:
    • What is D? Can you describe it in your own words?
    • What’s another form for this model?
    • Describe what this model means in terms of the crabs.
    • Does this model fit the data? Why or why not?
    • Is this model realistic?
esteem time
ESTEEM Time!
  • Continuous Growth Module
summary of manipulations
Summary of Manipulations
  • Entered data in yellow areas
  • Clicked on “Plots-Size” tab
    • Manipulated parameters using sliders until fit looked “right”
    • Asked questions about what makes it right
exponential growth model
Exponential Growth Model
  • Simplest model:

where C is the number of crabs on day t, and r is some constant number.

  • Questions to ask:
    • What is r? Can you describe it in your own words?
    • What’s another form for this model?
    • Describe what this model means in terms of the crabs.
    • Does this model fit the data? Why or why not?
    • Is this model realistic?
esteem time33
ESTEEM Time!
  • Documentation
  • Continuous Growth Module
compare the two models
Compare the two models…
  • Why can the initial population be zero in the linear growth model, but not in the exponential growth model?
  • Why do such small changes in r make such a big difference, but it takes large changes in D to show a difference?
  • What do these models predict will happen to the number of crabs that Cal catches in the future?
logistic growth model
Logistic Growth Model
  • Canonical model:

where C is the number of crabs on day t, and r and K are constants.

  • Questions to ask:
    • What are r and K? Can you describe them in your own words?
    • Describe what this model means in terms of the crabs.
    • Does this model fit the data? Why or why not?
    • Is this model realistic?
further analysis
Further Analysis
  • What does the initial population need to be for each of the three models to fit the data well?
  • Why is the logistic model more realistic?
  • How did the parameters (D, r, K) affect the models?
  • What does each model say about the total capacity of Cal’s traps?
  • Do these models give an accurate prediction of the future of the crab population?
let s kick it up a notch
Let’s kick it up a notch!
  • How do we model the entire crab population?
    • According to http://www.chesapeakebay.net/blue_crab.htm, blue crabs are predators of bivalves.
    • Cannibalism is correlated to the bivalve population.
predator prey equations
Predator-Prey Equations
  • Canonical example (Edelstein-Keshet):
    • Assumptions (pg 218):
      • Unlimited prey growth without predation
      • Predators only food source is prey.
      • Predator and prey will encounter each other.
why does multiplication give likelihood of an encounter
Why does multiplication give likelihood of an encounter ??
  • Law of Mass Action (Neuhauser)
    • Given the following chemical reaction

the rate at which the product AB is produced by colliding molecules of A and B is proportional to the concentrations of the reactants.

  • Translation to mathematics
    • Rates = derivatives, k is a number
    • What about [A] and [B]?
another version
Another version
  • Cushing:
  • What are the variables? Put them into context.
  • What’s the extra term?
  • Did the assumptions change?
esteem two species model
ESTEEM Two-Species Model
  • Isolation (discrete time):
  • What kind of growth?
  • What are the terms and variables?
esteem two species model43
ESTEEM Two-Species Model
  • Discussion Questions
    • What do the terms mean?
    • Which species is the predator, which is the prey?
    • What other situations could these equations describe?
    • Why discrete time?
    • For what values of the rate constants does one species inhibit the other? Have no effect? Have a positive effect?
    • Can we derive the continuous analogs?
esteem time44
ESTEEM Time!
  • Documentation
  • Two-Species Module
summary of manipulations45
Summary of Manipulations
  • Use sliders to change values of parameters.
  • Examine all graphs.
  • Columns B and C have formulas for numerical method.
discussion questions
Discussion Questions
  • How did one species affect the other?
  • What did the different graphs represent?
  • Did one species become extinct?
  • How can you have 1.25 crabs?
  • What would happen if there was a third species? Write a general set of equations (cases as relevant).
  • Can you determine the numerical method used?
simple sir model
Simple SIR Model
  • Yeargers:
  • Susceptible, Infected, Recovered
  • Given the above equations, explain the assumptions, variables, and terms.
connections to case study and beyond
Connections to Case Study and Beyond
  • Possible ideas for research projects
    • Parasites and crabs
    • Is there a disease affecting the crab population?
    • Pick an epidemic, research it, and model it.
      • Analytical or numerical solutions
      • Make teams of biology majors and math majors.
    • ESTEEM module…
sir esteem module equations
SIR ESTEEM Module - Equations

Hosts (S, I, R) are infected by vectors (U, V) that can carry one

of three strains of the virus (i=1, 2, 3).

esteem time50
ESTEEM Time!
  • Documentation
  • SIR ESTEEM Module
    • Red boxes are for user entry.
sir module discussion
SIR Module Discussion
  • Can you draw a diagram representing the SIR model?
  • What are all of the variables and parameters in the SIR model?
  • Can you find the continuous analog for the system?
  • Can you rewrite the system in matrix form?
  • What numerical method was used?
  • Why did some values of the parameters work, while others did not?
conclusion
Conclusion
  • Many, many ways to bring biology into the classroom
  • Build on students’ intuition and knowledge
  • Make obvious connections between ideas
  • Don’t be afraid to try something new.
  • Experiment and experiment some more!
  • Have fun!
works consulted cited
Works Consulted/Cited

Texts:

Cushing, J.M. (2004) Differential Equations: An Applied Approach. Pearson Prentice Hall.

Edelstein-Keshet, L. (1988) Mathematical Models in Biology. Birkhäuser.

Neuhauser, C. (2004) Calculus for Biology and Medicine. 2 ed. Pearson Education.

Yeargers, E.K., Shonkwiler, R.W., and J.V. Herod. (1996) An Introduction to the Mathematics of Biology with Computer Algebra Models. Birkhäuser.

Online Sources:

BioQUEST Sources

BioQUEST: http://bioquest.org

ICBL: Investigative Case Based Learning: http://bioquest.org/icbl/

ESTEEM Module Documentation

Continuous Growth Models Documentation (John R. Jungck, Tia Johnson, Anton E. Weisstein, and Joshua Tusin): http://www.bioquest.org/products/files/197_Growth_models.pdf

SIR Model Documentation (Tony Weisstein): http://www.bioquest.org/products/files/196_sirmodel.doc

Two Species Documentation (Tony Weisstein, Rene Salinas, John Jungck): http://www.bioquest.org/products/files/203_TwoSpecies_Model.doc

Blue Crab Resources

Blue Crab: http://www.chesapeakebay.net/blue_crab.htm

Chesapeake Blue Crab Assessment 2005: http://hjort.cbl.umces.edu/crabs/Assessment05.html

Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs: http://www.mdsg.umd.edu/issues/chesapeake/blue_crabs/index.html