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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 Mathematicians

  • 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


Teaching mathematics to biologists and biology to mathematicians
How?? Mathematicians

  • 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? Mathematicians

  • 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… Mathematicians

  • 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??? Mathematicians

http://bioquest.org/icbl/


C 3 cal crabs and the chesapeake
C Mathematicians3: 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 Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • Blue Crab

  • Chesapeake Blue Crab Assessment 2005

  • Maryland Sea Grant The Living Chesapeake Coast, Bay & Watershed Issues Blue Crabs


Blue crab http www chesapeakebay net blue crab htm
Blue Crab: Mathematicianshttp://www.chesapeakebay.net/blue_crab.htm


But what s the answer
But – what’s the answer?? Mathematicians

  • Assessments and objectives vary

  • Knowledge of tools and structure

  • Adopt and adapt



First growth model
First Growth Model Mathematicians

  • 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 Mathematicians

  • 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! Mathematicians

  • Continuous Growth Module


Summary of manipulations
Summary of Manipulations Mathematicians

  • 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 Mathematicians

  • 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 time1
ESTEEM Time! Mathematicians

  • Documentation

  • Continuous Growth Module


Compare the two models
Compare the two models… Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • 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! Mathematicians

  • 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 Mathematicians

  • 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.


Put it into context
Put it into context! Mathematicians


Why does multiplication give likelihood of an encounter
Why does multiplication give likelihood of an encounter ?? Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • Isolation (discrete time):

  • What kind of growth?

  • What are the terms and variables?


Esteem two species model1
ESTEEM Two-Species Model Mathematicians

  • 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 time2
ESTEEM Time! Mathematicians

  • Documentation

  • Two-Species Module


Summary of manipulations1
Summary of Manipulations Mathematicians

  • Use sliders to change values of parameters.

  • Examine all graphs.

  • Columns B and C have formulas for numerical method.


Discussion questions
Discussion Questions Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

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 time3
ESTEEM Time! Mathematicians

  • Documentation

  • SIR ESTEEM Module

    • Red boxes are for user entry.


Sir module discussion
SIR Module Discussion Mathematicians

  • 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 Mathematicians

  • 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 Mathematicians

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