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
– 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:
xt+1 = Mxt, 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