Designing Math Courses: Pedagogical Issues

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Designing Math Courses: Pedagogical Issues. Glenn Ledder Department of Mathematics University of Nebraska-Lincoln gledder@math.unl.edu http://www.math.unl.edu/~gledder1/Talks/. Key Issues to Consider. Course Goal Main purpose and place in curriculum Constraints

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Designing Math Courses: Pedagogical Issues

Glenn Ledder

Department of Mathematics

gledder@math.unl.edu

http://www.math.unl.edu/~gledder1/Talks/

Key Issues to Consider
• Course Goal
• Main purpose and place in curriculum
• Constraints
• Hours, class size, student background/ability
• Objectives
• What you want the students to learn
• Outcomes
• What you want the students to do to demonstrate their learning
• Goal:
• Empower engineering students with useful mathematics beyond linear algebra and differential equations
• Constraints:
• So many topics, so little time

50% vector calculus, 50% complex variables

Complex Variables (half-course)
• Objective:
• Be able to use the residue theorem to invert Laplace transforms
• Outcomes:
• Students will do homework problems and write solutions with explanations.
• Students will demonstrate techniques on exams.
Complex Variables (half-course)
• Course Content:
• Complex numbers
• Integration in the complex plane
• Laurent series and residues
• The residue theorem
A Challenge

I wrote an NSF grant for an interdisciplinary undergraduate research program in mathematical biology.

The proposal included “a 3-credit course to introduce young students to interdisciplinary research.”

In effect, I jumped off the Sears Tower with a bag of cloth and hardware, expecting to build a parachute on the way down.

Research Skills in Theoretical Ecology
• Goal:
• Introduce interdisciplinary research in mathematics/biology to talented students at an early stage in their careers.

“Early” means “between high school and college.”

Constraints
• The course must be self-contained.
• We cannot assume knowledge of calculus, statistics, or any specific biology topic.
• We cannot assume laboratory experience.
• The course must be integrated at different levels.
• Math and biology
• Theory and experiment
• Research design, conduct, and dissemination
Objectives
• Hard objectives: objectives that can be demonstrated with behavioral outcomes
• Soft objectives: objectives that are emergent properties of a broad whole
• The soft objectives are often more important for service courses. Don’t neglect them just because they can’t be measured.
Soft Objectives
• Experience the challenge and excitement of research.
• Appreciate the synergy between theory and experiment and between biology and mathematics.
• Developskills that will be useful in theoretical ecology research.
• Understand the theory developed through the experiments and analysis.
Hard Objectives
• Collectlaboratory data on real research questions using sophisticated techniques.
• Analyzedata using statistical methods.
• Construct mathematical models and use them to makepredictions.
• Prepare a poster to communicate research results.
• Design a research study.
Outcomes
• Students will work together to conduct experiments and record data.
• Students will do homework and quizzes on mathematical content.
• Students will build a mathematical model and use it to make predictions.
• Students will prepare a poster summarizing their research.
• Students will prepare a research proposal abstract to indicate possible future work.
Course Content
• Discrete linear stage-structured model:

xt+1 = Mxt,wherexis a vector giving the populations of the different stages and Mis a matrix of parameters