Glenn Ledder Department of Mathematics University of Nebraska-Lincoln Designing Math Courses: Pedagogical Issues

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

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

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 1.Experience the challenge and excitement of research. 2.Appreciate the synergy between theory and experiment and between biology and mathematics. 3.Develop skills that will be useful in theoretical ecology research. 4.Understand the theory developed through the experiments and analysis.

Hard Objectives 1.Collect laboratory data on real research questions using sophisticated techniques. 2.Analyze data using statistical methods. 3.Construct mathematical models and use them to make predictions. 4.Prepare a poster to communicate research results. 5.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: x t+1 = M x t, where x is a vector giving the populations of the different stages and M is a matrix of parameters Research tasks: –construct the model –estimate the parameters –predict population growth –test the predictions –analyze the model