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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

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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

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