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Incorporating Biology into Math Courses at the University of Richmond. Lester Caudill Kathy Hoke University of Richmond Department of Mathematics and Computer Science lcaudill@richmond.edu khoke@richmond.edu. New Math Courses at UR.
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Incorporating Biology into Math Courses at the University of Richmond Lester Caudill Kathy Hoke University of Richmond Department of Mathematics and Computer Science lcaudill@richmond.edu khoke@richmond.edu
New Math Courses at UR • Two-semester science-based calculus: Scientific Calculus I – II • One-semester Upper-division biomedical modeling course: Mathematical Models in Biology and Medicine
Main Goals • Sci Calc: • Rethink and revise course content to underscore relevance to the sciences • Help science students understand and utilize the role of math modeling in scientific investigation • Modeling: • Teach students to construct and analyze math models of scientific processes
Development of Sci Calc • Step 1:Identify the most relevant math topics for the sciences • Step 2: Make some room in our mainstream two-semester calculus sequence • Step 3: Fill resulting room with new topics (from Step 1), and organize in a logical and coherent order
Step 1: Absent/Underrepresented Topics • Consultation with science faculty AND • Researching modern uses of math modeling in the sciences LED TO • Some changes
Step 1: Absent/Underrepresented Topics • Multivariate calculus
Example: Chemical reaction data S2O82- + 2I-→ 2SO42- + I2
Step 1: Absent/Underrepresented Topics • Multivariate calculus • More emphasis on worst-case error estimates and practical estimation • Responsible data set management, including regression techniques • Discrete probability • Linear algebra and dynamical systems • Modern and relevant illustrations, examples, and applications
Step 2: Make room in mainstream calculus • Omit some less-relevant (to the sciences and to modern applied mathematicians) math topics: • Endpoint convergence tests for Taylor series • Old traditional physics/geometry applications (e.g. volumes of solids of revolution)
Step 2: Make room in mainstream calculus • Open course only to students who already have a good calculus background. • Relegate some simpler (review) topics to outside readings and assignments • Functions and other pre-calculus review, derivative shortcut formulas, vector basics, single-variable optimization SHOW FIVEFOLDPATHI.DOC
Step 3: Combine and Organize • Sci Calc I: • Fitting models to data • Fitting models to data and theory via ROC • AROC, IROC, and derivative • Multivariate differential calculus • Definite integrals
Step 3: Combine and Organize • Sci Calc II: • Applications of definite integrals • MIC-hour calculations • Distribution and density functions • Survival-renewal models • Probability and uncertainty • Polynomial approximation • Dynamical systems models • Difference equation models • Differential equation models • Discrete dynamical systems and linear algebra
Text for Sci Calc • Start with Calculus: Concepts and Contexts by Stewart • Heavily supplemented with • Handouts • Computer labs • Data sets • Assignments • Examples SHOW SURV-RENEW READING
Text for Sci Calc • ~40% of the material covered in the two semesters is not supported by the current text
Biomedical Modeling Course • Upper-level math course • Open to Sci Calc grads (and others) • Goal:Teach math-inclined science students to construct and analyze math models, using difference and differential equations, of scientific processes.
Biomedical Modeling Course • Strategy: Teach some modeling principles, then study DE and dfE models in various areas of biology and medicine. • Topics and their sequencing carefully planned to introduce successively higher-level model-building and analysis skills.
Biomedical Modeling Topics • Biological control of pest populations • Spotlight #1: Tumor growth dynamics • Pharmacokinetics • Spotlight #2: Models of chemotherapy • Epidemiology • Interacting populations • Spotlight #3: Leukemia dynamics • Immunology of the HIV virus • Enzyme kinetics
Biomedical Modeling Topics • Each unit begins with biomedical background info SHOW CANCERSPOTLIGHT1.DYN
Illustrating Concepts • Prey processing time and saturation effects
Modeling Projects • Antibiotic treatment of bacterial infections (including antibiotic resistance) • Drug dosing regimens • Spread of nosocomial infections (including infection-control methods) • Dynamic instability of microtubules (cytoskeleton components)