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 "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.
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 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.

Using Google Docs and other online tools to create new modes of classroom interactions

(Note: This is post is part of something I wrote a few years back that never got published due to a flaky publisher. It might still be useful to some folks out there though.)

In recent years, I’ve been thinking about how to use Google Docs and other online tools that allow for synchronous editing to create totally novel ways of interacting in the classroom. This technology requires students to have laptop computers with them in class with reliable wifi, but this is pretty common these days.

Here are some examples of what I mean:

1) Birthday problem activity

There is a lovely, surprising result in probability called the “birthday problem” (http://en.wikipedia.org/wiki/Birthday_problem): the chance that 23 people share the same birthday (not year, just the month and day) is about 50%. That seems like an incredibly low number when there are 365 days in the year. But, an instructor can easily demonstrate this idea by having her students go to a Google spreadsheet like this during class:


Not only is it a fun, interactive way of showing that this result is actually true, it also gives the instructor a visual way of motivating the proof of the result. It’s also much faster than having everyone shout out birthdays and marking them one by one on a calendar.

2) Video commentary live mini-blogs

I teach a course for prospective high school math teachers in which we watch videos of other math teachers’ classrooms.  I have my students watch these videos using various “lenses” (for example, looking just at the kinds of questions the teacher asks and how that affects student learning, or looking for evidence of things that students understand or don’t understand).  I use a Google doc for everyone to comment on the video while we are watching it together.  I set up the document in such a way that each person has his/her own cell in a table and with a small enough font you can see lots of other’s comments in real time too.  Instead of having one large chain of responses, each person has his/her own space to keep track of observations, and we can look through this later.  I’ve also done this same thing but using Google hangouts instead so that everyone is watching videos remotely.

3) Simultaneous wordsmithing in class

Some of us at Harvey Mudd College are finding interesting uses for Google docs in our undergraduate writing class (Writ 1).  I’ve used Google docs as a space for students to post pieces of their writing and to “crowdsource” improvements.  For example, I have students pick out the sentence that they are most unhappy with in their essay and paste it into a Google doc.  Once all the sentences are there, we turn ourselves loose to create better versions of each sentence and we discuss our ideas.  The author of the original sentence gets to vote on the version that she/he likes the best, or to come up with a new version based on pieces of others’ work.

In all of these situations, the technology adds an interesting “layer” of interaction to classroom.  For example, the video commentary example above has three “layers” of human interaction: the video we are watching is one layer, the comments that we make to each other verbally in the room is a second layer, and the live-commenting that we do on Google docs is a third layer.  (You could also turn in the chat feature in Google docs for yet another layer.)  I’m fascinated by how my students can participate in multiple layers simultaneously and seemlessly shift focus from one layer to another.  I love that these technologies give teachers new possible ways for people to interact.

And, I believe it’s not just doing something for the sake of technology.  In each of these example, the interaction is far richer and quicker compared to having talking as the main form of interaction in class because only one (or a few) voices can be heard at a time in a room whereas the technology allows every person the chance to express her/himself at the same time. Most people can also type faster than they can talk, and there is the chance to edit what you’re typing.  I’ve noticed that students who are shy and not as likely to speak up in class find a new voice through this kind of technology.

I’ve used these technologies in other ways, but for the sake of brevity I’ll just stop here.  There are clearly lots of other ways for synchronous editing tools to enable these kinds of multilayered interactions in classrooms, and I’m interested in hearing other’s ideas on how to improve the technologies and how to make better use of them.

It’s all in the details

Going back to my home university has been a bit of a shock over the last few days. I finally figured a contributing reason: Last year I got used to an environment in which decisions didn’t seem to be made with a lot of careful thought. Administration and teachers (me included) just did the best we could and we didn’t sweat the small stuff. Lots of things and students fell through the cracks. So, the level of detail that goes into decisions made at our university seemed a bit overwhelming when I first stepped back on campus.

Example: Our school is small so one of our associate deans manually matches first-year students with faculty based on their personalities, interests and other things that they might have in common. This associate dean briefed me on some facts about my advisees and what kind of advising might need. A lovely thing to do, and completely worthwhile. But, this feels totally foreign to me right now given my experience over the past year. I immediately thought about a girl in my Geometry class who I discovered was enrolled in a second Geometry class by mistake. I discovered this about two months into the semester.

Student: “Oh Mister, I did this already in my other Geometry class.”


Another example: Today I talked with two other colleagues who are co-teaching a course. We talked about common themes between the different topics that we are planning on teaching and how to make more connections to bring these themes out. Awesome awesome awesome. I missed this kind of conversation. Made me sad that I never got to talk about such things at that level with my colleagues last year. And it’s not like that took a huge amount of time to have that conversation…

Overpayment !@#!$%! ????

I received this letter today in the mail from my school district:

“Dear xxxx,

Our records show that you have received an overpayment as a result of a change that was processed in June 2010. The total adjusted gross amount of your overpayment is $12,197.66. This letter is intended to advise you of your options in repaying the identified overpayment.”

This is ridiculous. I definitely have not been overpaid this amount.

Since September 2009, I received a total of $17,454.97 from the school district. There, now the entire world knows how much I made from my year of teaching high school (at a part timer’s rate, for someone with alternative credential).

Now the’re telling me I was overpaid by $12,197.66 and they want their money back?

That would mean that I would only have made $5,257.31 for the entire school year. Is that how much my effort was worth?

When I first read the letter, I was furious. After talking to a dear friend who went through something much worse with the district, now I find the whole thing so completely stupid and ridiculous.

My friend is right–I need to frame this letter.

Nothing like being around my peeps

I’m here nearing the end of a three-week professional development workshop for secondary school math teachers. I knew that it was going to be an excellent experience for me coming off of my year of teaching, and I was not disappointed.

Over the past three weeks, I have been surrounded by teachers that I trust and respect deeply who have asked questions and pushed me to think and articulate things that up to this point have been just too raw and recent to touch. And today, I got the chance to talk about my experiences over the past year with them. I won’t forget this one hour of sharing and conversation for a long time.

The encouragement I felt was incredible. One part of me felt like the whole thing was too indulgent–why should these people find it admirable for me to teach high school for a year when these same people do it themselves year after year? And yet, another part of me was soaking it all in. It was an emotional experience for me and many others in the room, I think because there was a shared experience and understanding that deeply connected us.

So what was gained from this hour? I realized that the emotional struggles of a being a high school teacher far outweighed the instructional, intellectual, physical, logistical challenges and that this was natural and not unusual. It was helpful to be reminded by other teachers that your reputation has a huge impact at school and that the first year at any school will always be difficult because of students’ uncertainty about you. It was helpful to be told that instructional changes take a long time and are supposed to happen slowly, and that I can’t expect to be good at teaching in a new setting in one year. It was helpful to be told that the fruits of a high school teacher are rarely observed by that teacher. It was helpful to be told that not being able to get off to a good start with your students on day 1 (let along week 4)  is very detrimental to classroom climate.

A colleague asked me, what lessons did I learn that will affect the way I lead professional development for teachers? I’m still trying to think beyond the obvious answer (that I now have much more empathy for and understanding of the work that teachers do). Stay tuned.