PDEs Course Design (Part 1)

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

  1. 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,
  2. be able to select and carry out an appropriate solution strategy, when faced with a problem involving a linear PDE,
  3. 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.

Five Strands of Mathematical Proficiency, taken from
Five Strands of Mathematical Proficiency, taken from “Adding It Up: Helping Children Learn Mathematics” edited by J. Kilpatrick, J. Swaford, and B. Findell, published by the National Research Council and National Academy Press in 2001.

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!)

Radical Inclusivity

Christopher Danielson encouraged us to seek out what we love and incorporate more of that in our teaching at TMC15. My answer came immediately: radical inclusivity. To me, radical inclusion involves making inclusion into a mathematical learning community the top priority in my classroom. It is based on the idea that a sense of belonging and connectedness is a prerequisite to students’ learning. I think the “radical” part also involves doing something to actively combat the injustices that exist in our world, and not just assuming that the little bubble of warm fuzzies that I create in my classroom are enough.

My paternal grandfather is Hakka, so that makes me Hakka too. The Chinese word Hakka literally means “guest family.” This subgroup of the Han people supposedly migrated to many different places in the world, making a home for themselves in each new place. Maybe I’m reaching too far back in history, but in our family it has always been very important to make others feel welcome in any situation. When hosting a party at our home, we always make 19 times more food than necessary. That’s just what we do. So, it feels natural to do the same in my classroom.

Having served as associate dean for diversity for the last four years at Harvey Mudd College, I have learned so much about diversity and inclusion in higher education, mainly because of my awesome friend and colleague Sumi Pendakur. You can’t “unsee” injustice once you realize it’s there. Those injustices propel me to want to broaden participation in STEM fields and to make my school and classroom as welcoming as possible to every individual.

My message to all educators: not attending to diversity and inclusion concerns in the classroom is the same as allowing your classroom to continue propagating the discrimination and bias that exists in our society. We have to actively combat discrimination and bias in our work as educators. Here are three reasons why.

1. Racism, sexism, classism, ableism (etc) are alive and well in our society. Our students are exposed to it all the time. Our school institutions mirror these practices in their policies and systems. If we don’t do anything, our students will continue to become indoctrinated in those things.

Example: Though we might wish for our world to be meritocratic, it isn’t. People don’t have equal access to opportunities to learn. In most schools, the demographics of “honors” or “advanced” classes don’t match the demographics of the rest of the school or community. Students internalize these patterns of belonging and that shapes their perceptions of themselves and others.

2. We all have implicit biases. They affect our thinking whether we like it or not. (Read this.) If we don’t keep these implicit biases in check, we risk letting them become manifest in our classrooms and cause students to feel alienated or marginalized. And, when students have low self-efficacy of themselves as mathematics learners, it doesn’t take much to make them feel alienated or marginalized.

Example: A few years ago a colleague pointed out that I tended to call on male and female students differently in class. When a male student raised his hand I was more likely to call on him by saying “Yes?” and when a female student raised her hand I was more likely to call on her by saying “Question?” Ack. The fix was simple. Now I just say “What questions, comments, or reactions do you have?” and I acknowledge students by name.

3. I also believe that our job as math teachers is much more than teaching mathematics. We are responsible for educating students about the ways in which our society is not fair and how we individually benefit from unearned privileges. The mission of Harvey Mudd College is to “educate engineers, scientists, and mathematicians well versed in all of these areas and in the humanities and the social sciences so that they may assume leadership in their fields with a clear understanding of the impact of their work on society.” Surely, understanding the impact of their work on society includes understanding who has access to and power in the American education system. This understanding will empower our students to do good in the world so we can multiply the effect of our work beyond our own classrooms.

I have so much more work to do in my own teaching to make my classroom radically inclusive. I think that in the past I had inclusion as a priority, but it wasn’t the top priority. The question I’m asking myself now is, what would it look like if that became the top priority in my teaching and what effect would that have on students?

I’ll be writing more this semester about my attempts to do this in a course on partial differential equations.

The power of a community

Attending TMC2015 jolted me out of complacency. It encouraged me to stop lurking in the MTBoS and participate more actively, and it reminded me about the power of being in a highly collaborative community of people who also love thinking deeply about teaching.

If you’re not already part of this community, here’s how you can explore what it has to offer.