Designing Math Courses: Pedagogical Issues

1. 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/

2. 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

3. Advanced Engineering Mathematics • 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

4. 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.

5. Complex Variables (half-course) • Course Content: • Complex numbers • Integration in the complex plane • Laurent series and residues • The residue theorem

6. 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.

7. 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.”

8. 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

9. 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.

10. 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.

11. 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.

12. 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.

13. 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 • Research tasks: • construct the model • estimate the parameters • predict population growth • test the predictions • analyze the model