I’m going to use this blog as a way to get ready to teach partial differential equations (PDEs) for the first time at Harvey Mudd College. (I’ve taught a related course for physics majors for many years, but this is the first time I’m teaching PDEs for math majors.)
Here’s the official catalog description for Math 180: “Partial Differential Equations (PDEs) including the heat equation, wave equation, and Laplace’s equation; existence and uniqueness of solutions to PDEs via the maximum principle and energy methods; method of characteristics; Fourier series; Fourier transforms and Green’s functions; Separation of variables; Sturm-Liouville theory and orthogonal expansions; Bessel functions.” It’s really just a list of topics. It’s not even a complete sentence. Ick.
This course has historically been one of the more challenging courses. It’s typically taken by juniors and seniors, and beginning graduate students at Claremont Graduate University. All math majors have to take this course, so those who aren’t fond of applied mathematics often don’t have a rosy view of the course.
To begin, I’m going to do some backward design (Wiggins and McTighe, 2005).
Here is a first draft of my learning objectives:
Math 180 has been designed so that by the end of the semester, all students will
- be able to describe the typical behaviors of solutions to the three major classes of linear PDEs (elliptic, parabolic, hyperbolic) and explain why they behave the way they do,
- be able to select and carry out an appropriate solution strategy, when faced with a problem involving a linear PDE,
- appreciate the wide range of applications of PDEs and be able to describe a specific application of PDEs to colleague.
Along the way, students will encounter various solution techniques (separation of variables, method of images, method of characteristics, integral transforms), and mathematical ideas that enable the study of PDEs (function spaces, theory of distributions).
The three listed items are my “essential understandings.” The next step in backward design is to think about how students can demonstrate these understandings.
For #1, I’m thinking about having students write something mathematical. Perhaps analyze the behavior of a PDE that they had not yet encountered? Or maybe students will need to (re)produce a mathematical argument about the qualitative behavior of some PDE?
Goal #2 is the classic bread-and-butter of this subject. PDEs can be a very computational, procedural course, and while I would like the course to be more than that, there is no denying that there are lots of computations and procedures that students will need to master.
For #3, I’m thinking of having students do some sort of open-ended investigation on an application area of their choosing. Perhaps this might involve interviewing another scientist/mathematician who uses PDEs in her research or some digging through research articles? I’m wondering about the final product of this part of the course: a presentation or web page or paper?
As I’m crafting these learning objectives, I’m also thinking about the strands of mathematical proficiency.
Procedural Fluency: be able to separate variables and find eigenvalues and eigenfunctions for various differential operators, know how to construct the right orthogonality condition for a Sturm-Liouville problem, write a PDE in conservation form, use the method of characteristics to solve a PDE, etc.
Productive Disposition: see PDEs as an active area of research with a wide range of applications, see oneself as being able to learn PDEs
Conceptual Understanding: understand why there are three major classes of second-order linear PDEs and why their behaviors are different, understand why superposition is why these solution techniques for linear PDEs work, understand what it means to find a solution (including a weak solution) of a PDE
Strategic Competence: be able to select and carry out an appropriate solution technique given a particular linear PDE problem
Adaptive Reasoning: be able to explain the behavior of a linear PDE, be able to justify the solution technique for a given problem, be able to reflect on whether a solution seems reasonable or correct
For now, I think I have the different strands of mathematical proficiency well-represented in my course objectives.
(To be continued… Constructive comments from others on my course design or process are welcome!)