Matthias K. Gobbert
Math 621 - Numerical Analysis II
Numerical Methods for
Partial Differential Equations
Teaching Philosophy
Learner-Centered Teaching
This course will be taught according to the philosophy of
learner-centered teaching.
The key to this philosophy is to shift the focus from
the contents of the course to the students' learning.
For one thing, this means that we will not only discuss the material
of the course, but also the process of learning itself;
the true goal is to enable you to become self-directed learners
with good critical thinking and self-assessment skills.
To move into this direction, it is apparent that the students
will simply need to try to do more of their own learning
and that the instructor will need to provide something quite
different than a lecture presentation in class.
Concretely, the appearance of the contact hours in class should
be those of a seminar, in which every student is responsible
for preparing himself or herself for the class discussion
by doing the 'assigned reading.'
We will then use the valuable time in class to focus on the
reflection about and discussion of the material.
In addition, we will want to spend some time discussing
learning techniques and strategies; making these explicit
is an integral part of learner-centered teaching,
because it helps you develop those skills needed for
self-directed studying of any mathematical contents
beyond the material of this class.
Learner-Centered Teaching and Coverage of Material
Which brings us back to the material of this class.
The topic of this class is the numerical solution of
partial differential equations, applied to
linear prototype problems of
three classical types of second-order
partial differential equations (PDEs):
the Poisson problem for elliptic PDEs,
the heat equation for parabolic PDEs, and
the scalar transport equation for hyperbolic PDEs.
Since this class does not have any prerequisites
of PDE theory, we will start by defining PDEs as such
and by discussing the most relevant properties of
the solutions of the prototype problems.
Then we will study the two most important types of
numerical solution methods:
the finite difference method (FD) and the finite element method (FEM).
My goal is both to establish the basic convergence results
for each method rigorously and
to teach you how to use relevant software for these methods most effectively.
There are clearly many more details associated with programming
than one could fit into one semester.
So, the focus will be on those skills that will enable the student
to approach research in this area.
Clearly, these learning goals are 'visionary' for a one-semester course.
But I believe that by applying the philosophy
of learner-centered teaching, we will not just be able
to meet these goals, but we should also find time and opportunity
to talk about other important topics in class. Examples of those
might be professional development issues of peer-assessment or
techniques of teaching. To put this more concretely, consider
the placement of this class in the curriculum: It is a second-year
graduate course, and you are in the transition from traditional
first-year courses to independent research with qualifying exams
or thesis defenses; this course intends to help you in this
transition. Another example of an appropriate topic for class
discussion are the homework assignments. This is the level of class,
at which there is more than one correct answer. So, it should be
very profitable to discuss solution strategies in class.
This should also ensure that your overall time-commitment for
this class does not exceed the expected level of three hours of your
own time for each contact hour. That is, compared to a class
taught in traditional fashion, you should spend more
time in preparing for class and less time on homework.
More Information
I hope these remarks give you an idea of the approach that we
will take in this class. Feel free to ask me questions about it.
I am convinced that it will be an exciting experience for you and me.
For more information on learner-centered teaching,
I highly recommend the book by Maryellen Weimer on
Learner-Centered Teaching: Five Key Changes to Practice
(Jossey-Bass, 2002).
See my list of recommended literature on my homepage for additional
resources on learning, teaching, and other professional activities
(presentations, writing, etc.).
Copyright © 2003 by Matthias K. Gobbert. All Rights Reserved.
This page version 1.2, September 2003.