Proficiency assessments (PAs) are proceeding nicely in my partial differential equations (PDEs) course. I’m posting some of the logistical details here in case it helps other instructors. PAs are my attempt to allow students more flexibility in demonstrating their mathematical proficiency in my class. I wrote more about my intentions behind these PAs in this post.

First of all, I’ve reduced the number of PAs from 5 to 4. Here are the four PAs for this course:

• Be able to solve a first-order PDE using the method of characteristics, in both linear and nonlinear cases. (Nonlinear PDEs may involve shocks.)
• Be able to use the separation of variables technique to solve a homogeneous PDE problem.
• Be able to solve an inhomogeneous PDE problem using an appropriate change of variables, or the eigenfunction expansion method.
• Be able to use integral transforms (Fourier and Laplace) to solve PDE problems involving infinite or semi-infinite domains, and be able to identify how general solutions are convolutions involving Green’s functions.

The initial list of PAs was created while I was still constructing the class and once I got deeper into things I realized that one of the things that I listed on my previous list was better assessed on the final examination.

Scheduling 40 students to take these assessments has been a bit of a challenge, but I think we’ve solved that problem. Each PA is supposed to be completed during a 90-minute session without notes, calculators or outside assistance. But, students can use more than 90 minutes. That time is just a suggestion for the length of time to block out in their schedules.

I’m not terribly worried about cheating because of the strong honor code here at Harvey Mudd, but I want to be careful since the assessments can be taken at any time and I don’t want copies of them floating around. So, I needed to find a way for students to make appointments to take a PA. With the help of the students, I came up with this mechanism for taking PAs. There are four different ways to schedule an appointment:

1. Students can email me to set up an appointment. I have a calendar online so students can see when I have open blocks of time when they can sign up for an appointment.
2. Students can also contact our department administrative aide to make an appointment during business hours (Mon-Fri).
3. I have two graders/tutors for this course and they have additional tutoring time on Wednesday evenings. Students can arrange to take a PA under their supervision.
4. Finally, a group of students can get together and arrange to take an assessment together, at any time, even nights and weekends. These “group-proctored” sessions have turned out to be the most popular so far. I make arrangements to leave the assessments in some secure place, then they put the completed assessments in an envelope and slide it under my door if I am not at work at that time.

Some of you might be skeptical about students taking an exam/quiz on their own without supervision. it really does work here at Mudd, though. When students want to set up a group-proctored session, I also email the entire class to let them know about the opportunity. That gets groups of students together who aren’t in the same friend groups, so there is a bit more accountability.

So far, I’ve been meeting with each student after every PA to talk briefly about her/his performance. It is taking a lot of my time, but I really like having that personal connection with every student. I have far more information on how well each student is mastering the material than I have in previous classes.

Finally, I’m also really enjoying starting every class with 2-3 minutes on some cool application of PDEs: traffic simulation, tsunamis, bread baking, digital image restoration are some of the more interesting ones so far, in addition to the usual diffusion, advection/convection, Laplace problems. Students seem to really enjoy it too. If you have have more cool applications of PDEs to share, please let me know!

I had an interesting experience this week.

A student missed class because he was sick. He emailed me to let me know and I asked him to come by my office to catch up on things. When he came to my office for his appointment he was visibly nervous: uncomfortable body language, never made eye contact with me, spoke erratically.

After about 30 minutes, he told me that he’s never gone to a professor’s office hours before. I thought to myself, WHAT??? YOU’RE A JUNIOR MATH MAJOR! HOW IS THIS POSSIBLE????!!! I didn’t say that, of course. We just kept chatting and I encouraged him to email me with questions and to stop by again. Thankfully, he listened to my encouragement: we’ve been exchanging emails like rapid-fire IMs the last few days.

Perhaps this kind of thing is not unusual at larger institutions, but Harvey Mudd College is a small school with just 820 or so students. Faculty and students get to know each other well and it is a very close-knit community. In the 14 years I’ve taught here, I’ve not heard of a third-year student who has never gone to a professor’s office to ask for something. So on one hand it seemed to me a really unusual thing to happen.

But on further reflection, I started wondering if maybe it was more common than I realized. In every class I teach, I estimate that 1/3 of the students come to my office regularly, 1/3 come occasionally, and 1/3 never come at all.  I suspect that latter 1/3 group is mostly made up of students who never visit a professor’s office.

This interaction made me think about the “hidden curriculum” of higher education–those unspoken things about college that some of our students (mostly those with a lot of privileges) know, and that others don’t know. For example, one colleague shared about how she went to college and thought it was strange that there were so many people named “Dean” until she realized that “Dean Smith” meant “Dean of Something-Or-Other whose last name is Smith”, not that the person’s first name was Dean. The fact that students are supposed to go to office hours to interact with their professors is another thing that many students don’t realize. Some students think that going to ask for help is a sign of trouble or weakness, whereas other students have figured out that going to talk to a professor is a normal and routine part of going to college.

So to teach inclusively means that this welcome spirit must also extend to my interactions with students outside of the classroom. Am I making it clear to all students that they are welcome–no, expected–to come to my office to chat with me? If I don’t do that, then I am, deliberately or not, favoring some students who know about the “hidden curriculum” over those that don’t.

 

Together with Pam Mason, I help to design professional development for Math for America Los Angeles. Right now we have about 85 Teaching Fellows and Master Teaching Fellows in our program.

This year, we decided to implement something new to help “make thinking visible”–that’s my best version of what to call this thing. It’s not new, just new for us. But, I think it’s important enough to write about, hence this post.

Here’s what I mean by “making thinking visible.”

We teachers use phrases like “Say your ‘because’s” as a signal to students and each other that we care about student thinking. We want our students to understand mathematics, not just calculate, though that is important too. So when a student talks about mathematics in our class, we always press them to justify their answers. They need to say their “because”s. For example, we want students to say things like “I think the next step should be to square both sides of the equation because we are trying to find an equivalent equation without square roots.”

Well, the same thing should be true for us teachers too. When we talk to each other as professionals, we should also be making our reasoning and thinking more visible to each other so that we can learn from one another.

At least in Math for America Los Angeles, we haven’t been so good at that. We have been really good at sharing things with each other. We share our lessons, we share our resources, we share our struggles.

But when we say “I like this lesson”, do we explain why to each other?

When we say “I think students struggle here”, do we explain why?

When we say “This unit is going to take 2 more days than we expected”, do we explain why?

Some of the conversations behind these statements are actually the conversations we need to have with each other. They are meaningful and juicy because they reveal our beliefs, logic systems, and understanding about teaching and learning. If we want to really share our knowledge with each other, we should be sharing about these things rather than just sharing lessons and ideas.

Put another way, if we simply shared lessons with each other, that would be equivalent to students just telling us answers without justification. They might be arriving at the right answers for the wrong reasons! Of course we don’t want that to happen, with our students and shouldn’t let that happen for each other either.

I think making our thinking visible is not a complicated thing to do. We don’t need special training to do this, since we teachers are already in the habit of asking good questions and pressing for understanding with our students. We just have to be vigilant to do that we each other because we value each other as professionals in a craft that is deep and worthy of study.

 

Last week I used our exit ticket as a way to get feedback about the course so far. I’m very encouraged that all of the feedback was positive.

“Neat! I solved my first PDE and I’m proud.”

“This class is great so far…  I am enjoying this class more than I thought I would.”

“I like the set up of the class such that we can take the quizzes whenever and retake them.”

“I really like the class format. Still to experience this, but in theory I like the competence testing format you’ve chosen. Seems like a good system.”

“There’s a lot more focus on applications and examples than I expected. I really like this–it helps me understand why we’re doing this.”

“Good. I’m still nervous because DEs are my weak spot. But class is good.”

So I’m going to keep on truckin’.

I’d like to smack the person who came up with the idea of these proficiency assessments. It is turning out to be a huge amount of work to find several PDE problems that are similar enough in content area and difficulty. So many hours invested and I haven’t even finished one set of proficiency assessments yet. ARGH!!

OK, I’m done venting. Back to work…

I posted here about a question that I posed on the first day in PDEs.

I thought it went pretty well. Method of characteristics on Monday (tomorrow)!

My partial differential equations course has started!!  I have more students than I expected, but I have two amazing teaching assistants to help.

Students’ responses to the idea of the proficiency assessments has been all positive so far.

I’ve decided to structure each 75-minute class in the following way:

First 3 minutes of class: I highlight the work of some mathematicians, scientists, engineers who are using partial differential equations in some interesting way. I am trying to make sure to get a broad representation of people and some nice applications. (If you know of cool applications of PDEs, please let me know!) This is partly to help students see that this field of study is very large and active and that there are lots of people who make up this community of practice. Maybe that might even help to spark an interest or the feeling that they could contribute to this community too. This is also to give students some ideas for their application investigation paper/presentations (in which they have to investigate some application of PDEs).

Next 5 minutes of class: Short lesson on some aspect of Mathematica. We’re using Mathematica heavily in this class, and the learning curve is quite steep. In addition to this Mathematica training video that I created, I’ll highlight one command a day and have students try it out each class meeting. I’ve found this to be a good way to help students learn how to use mathematical software without boring them with long lectures about syntax.

Then we’ll launch into the main lecture for the day. My goal is to not talk for more than 15 minutes in one stretch and pause to include independent and group work time and small-group discussion.

We usually stop for a break halfway in the middle of class. (One student wrote to me that he needs to take a walk in the middle of class because he has ADHD.) Break usually involves this manatee video.

Another wonderful moment this week is that I got a lovely note from a former student with whom I had a very uncomfortable encounter that ended in tears. (Too difficult to explain here.) And, in addition to the note she came by my office to tell me that she is a very different person now from when I last knew her and that she was looking forward to class. It really made my day. ~~~

This post is part of a series (previous parts 1, 2, 3, and 4) in which I am blogging my way through a new course on partial differential equations (PDEs) that I am about to teach… OMG… tomorrow since it’s past midnight now. What am I doing still blogging instead of getting ready for class?

Anyway, I’m guessing that most people reading my blog are positively disposed to the idea that we should work toward an educational system that provides all students with access to high-quality instruction, resources, expectations, and support so that we can achieve more equitable and excellent outcomes for all.

That last part is the part that we sometimes quibble over, even those who are in the “diversity camp”. Sometimes we pit inclusion and excellence against each other, when we should be spending more time figuring out ways to structure our system so that inclusion and excellence go hand in hand.

Here’s my crude illustration of the difference between these two perspectives.

Diagram A on the left shows what happens when you try to create equitable outcomes without raising the bar for everyone: you end up improving outcomes for some and degrading them for others so it can seem like inclusion and excellence are pitted against each other. Diagram B on the right shows what happens when equity and excellence goals come into alignment. Everyone gets better outcomes. And though some might benefit more than others, the outcome is more equitable.

That’s why I’m really intrigued by studies describing interventions that improve outcomes for all and produce more equitable outcomes at the same time, like this, this, or this. (If you know of more studies like this, please let me know!)

---

 

During dinner last week, my dear friend Bill Thill (@roughlynormal) got me thinking about all of these issues. That led me to wonder, am I designing a class that pursues both equitable and excellent outcomes for all?

I’ve spent a few hours trying to sketch out a “theory of action” that links the different design elements of my course (proficiency assessments, active learning, video lectures, application project, etc.) to plausible student outcomes. I didn’t come up with much, but I assuaged myself by remembering that I was struggling to do something that the three studies linked above didn’t do either: they don’t really explain why students did better, they only observed that it happened.

So, I am about to start teaching this course with a strong reminder that it’s an experiment. I don’t know what the outcome will be, but I will try to be attentive and look for signs of trouble and success.

 

My conversation with Bill also reminded me that I also need to convey to my class this notion that inclusion and excellence don’t have to be at odds with each others. For example, inclusive teaching practices won’t result in a less rigorous class. I don’t intend to “cover” fewer topics than my predecessors did in previous versions of the class. In fact, my hope is for every student to achieve high levels of mathematical proficiency.

Here we go! Wheeeeeee!