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Designing Math Courses: Pedagogical Issues. Glenn Ledder Department of Mathematics University of Nebraska-Lincoln Key Issues to Consider. Course Goal Main purpose and place in curriculum Constraints

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designing math courses pedagogical issues
Designing Math Courses: Pedagogical Issues

Glenn Ledder

Department of Mathematics

University of Nebraska-Lincoln

key issues to consider
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
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
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 course1
Complex Variables (half-course)
  • Course Content:
    • Complex numbers
    • Integration in the complex plane
    • Laurent series and residues
    • The residue theorem
a challenge
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
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.”

  • 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
  • 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
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
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.
  • 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
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