PDEs Course: Wrap Up and Reflection

I haven’t been posting much because of the busy-ness of last semester. Now that grades have been submitted, I’ve been reflecting on the partial differential equations (PDEs) course that I taught. (All previous posts about this class can be found here.)

I firmly believe that we must evaluate our own teaching if we want to improve, and that one of the best ways to gather data on our teaching is to ask our students. Students aren’t always the best judge of how much they have learned, but I trust my Mudd students’ ability to tell me about their experiences and opinions about the course. Here is what students said in their comments on an end-of-semester evaluation survey.

Comments about students’ perceptions of the course and their overall experiences:

I really appreciate your comments at the beginning of class that you realized that most of the applications you were planning on presenting were thought up by dead white guys and that that might cause some people dismay. I think that recognizing that that’s a problem in math/science that permeates into classrooms is important, and you saying that out loud helped me feel more like I belonged in the classroom even if I am not a white male.

This class somehow made me enjoy solving PDEs even though the past three DE classes I’ve taken convinced me I just really hate DEs. Nice job.

I started solving random PDEs in my free time, from which I deduce the class was pretty interesting.

I also want to thank you for being such a dedicated teacher. Your lectures and notes in numerical analysis and PDEs were effective in delivering material and it never felt like there was anything “hidden” about the subjects that I couldn’t figure out without some closer reading. I also think that the structure of this class was awesome, because it encourages you to learn all aspects of the subject, even if you missed that specific part of the semester.

I feel that [Prof. Yong] is extremely understanding and is very approachable to students, regardless of how comfortable they are with the material.

Overall, I noticed that student engagement was high. I was really pleased that I was able to change some students’ opinions of differential equations.

On the end-of-course survey, I asked students to indicate their affinity for the following statements on a scale from 1 (strongly disagree) to 5 (strongly agree). (A total of 32 students responded to the survey, though not every student responded to every question.)

  • “The students and instructor for this course created a welcoming community of learners.” Average response: 4.59 / 5
  • “In this class, I was able to express myself (whether it was to answer a question, or to say that I didn’t know how to do something) openly without judgment or ridicule from my instructor.” Average response: 4.81 / 5
  • “I generally felt secure and confident to speak in this class (to answer a question or ask a question or something else) when I wanted to.” Average response: 4.44
  • “I feel secure and confident to speak in my classes at Mudd in general.” Average response: 4.13 / 5
  • “I feel like an outsider in this class.” Average response: 1.68 / 5
  • “The instructor was respectful to all students in the class.” Average response: 6.91 / 7

Student comments about the proficiency assessment system:

Since the proficiency assessment (PA) system was a big change for me and students, naturally there were lots of comments about this part of the course. I tallied up all of the types of comments that I received about the system:  16 favorable comments, 4 negative comments, and 1 mixed comment.

Here’s a selection of the positive feedback:

The proficiency assessments are spot on in encouraging learning–I feel like I’ve learned from them while satisfactorily representing my understanding.

I’ve never had a class where I could schedule tests like this one, but wish I had! I have gotten a lot out of this freedom to prepare when I have time, and to take more advanced versions once ready for them.

Also, there seems to be the philosophy that students deserve credits when he/she knows how to solve the problems and not only when he/she can solve the problems within the exam time. I feel like this grading philosophy is more applicable to the work in real life.

Since I could take it many times, I didn’t care much about getting the right answer. Instead I was able to focus more on improving myself each time I take the PA.

I like how the proficiency assessments encourage me to understand the material at my own pace in a less stressful way.

There were several classes that I’ve taken at Mudd where I didn’t learn the material by the time the exam came, and so there was no reason for me to learn it afterwards. I also like the ways in which [the PA system] discouraged cheating: since you have the opportunity to retake exams, there is less of an incentive to be dishonest about it. For me it was also great because it meant that it was never too late for me to try to catch up. One of the most depressing situations that I encountered at Mudd was having several exams during one week and feeling like I could have performed better if the schedule had just worked out differently. I personally think that your curriculum is the right direction for a lot of the classes at Mudd, so thanks for being willing to try something new out.

On the whole, I think I was able to meet my objective of coming up with a system to assess students’ understanding but in a way that gave students more agency and flexibility, and that promoted students’ growth mindset about learning PDEs.

On the end-of-course survey, I asked students to indicate their affinity for the following statements on a scale from 1 (strongly disagree) to 5 (strongly agree).

  • “The proficiency assessments fairly assessed my understanding of PDEs.” Average response: 4.56.
  • “I was able to demonstrate my understanding of the course material through the proficiency assessments and final exam.” Average response: 4.50

I will definitely use this system again in the future, but I need to figure out how to reduce the impact on my time and on students’ time. Also, I need to think more carefully about how students’ grades are calculated based on their assessment scores.

I ended up writing four sets of proficiency assessments, on four different subtopics. There were standard and advanced assessments for each subtopic. The maximum score attainable on a standard assessment was 23/25, and the maximum for an advanced assessment was 25/25. The standard assessments contained tasks that I would have used as final exam questions, and advanced assessments contained more challenging tasks that involved novel situations or challenges that they had not encountered, but that they could if they dug deeper into the course content. My rationale for this arrangement is that a standard level of attainment would correspond to a 92%, which in my mind is an A-. To get an A in a course, I feel that some effort above and beyond the normal set of expectations is required. I required students to reach the standard level of attainment before attempting a more advanced PA. This had the effect of requiring students to take at least eight separate assessments, if they wanted to get an advanced level of attainment for all four subtopics.

I’m really mixed about the PA format. On the one hand, I think I’ll probably have learned the material better than usual by the end of this course, simply because I’ll have had more tests on the subject scattered throughout the semester. However, I also feel like the PAs took up so much time (both for the student and the instructor grading them) that they caused more stress than a midterm would have.

I feel that the course had too many disparate requirements. The presence of PAs, homeworks, a final test, and a final project made it an overwhelming experience. I would advocate for a more efficient PA system that doesn’t require as much time outside of class, since the current PA system feels more like it’s testing students for how much of their free time they can sacrifice as opposed to their actual knowledge.

I liked the concept behind PAs, but found it somewhat annoying that in order to get full credit on all of the PAs you would have to schedule 8 different time slots. One idea I had to ease up on this sheer time commitment would be to have on each PA the option to choose between the 23 point problem or the 25 point problem.

I don’t feel that the PA system ended up fairly assessing my understanding. I feel that they got close to providing a good assessment, but with the time restrictions I’ve faced this semester, I was unable to take the advanced PAs. I had a very busy schedule this semester, and I was sick for multiple weeks in the middle of the semester, so I didn’t get to take my last regular PA until the end of finals week.

I got perfect scores on the 3 PAs I’ve gotten back so far, and I suspect I also did perfectly on the last PA I took this morning. But I just didn’t have the time to attempt the advanced PAs, especially given that my score isn’t even guaranteed to increase, so it was very hard to justify adding something else into my already exhausting schedule. I feel that I have a very strong understanding of the material, but my time restrictions only allowed me to demonstrate a “basic” understanding.

Personally, I feel that scheduling time for four 90-minute assessments isn’t a huge burden on students, but some felt that way. One thing I will try in the future is to set aside some class time for students to take the proficiency assessments, so they aren’t required to use out-of-class time.

Reflections

Overall, I’m really pleased with how the course went, even though it was my first time teaching the course and I was experimenting with lots of new ideas. I was trying to attend to students’ sense of belonging to the class all semester, and I think I was successful at that.

I consider the proficiency assessment experiment to be a strong success. I want to continue to refine and improve it. One important side effect of the proficiency assessment system is that I got to know all of my students much better than I normally would have. Another side effect is that the system enabled some students who would probably failed the course otherwise to pass and do well. For example, one student who had some family issues and was absent for almost 2/3 of the class was able to finish the course on time.

PDEs Course: Progress Update #4

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!

PDEs Course: Progress Update #3 (Early Student Feedback)

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’.

PDEs Course: Progress Update #2

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…

PDEs Course: Progress Update #1

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. ~~~

PDEs Course Design (Part 5): Inclusion and Excellence

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.

inclusion and excellence

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!

PDEs Course Design (Part 4): Letting Students Learn at Their Own Pace

This post is part of a series in which I am blogging my way through a new course on partial differential equations (PDEs) that I am about to teach in two weeks. (Links to parts 1, 2, and 3.)

My previous post in this series was about relieving time pressure during tests. This post is about relieving time pressure on a much longer time scale.

Last semester, one of my advisees told me something that still reverberates in my head. At the time, this student was not doing well in his classes at Harvey Mudd. We were talking openly about his performance at school when he told me, “I can learn everything here at Mudd, I just feel like I learn it two weeks after the quiz or exam.”

His retelling of his challenges got me wondering about lots of things. How intentional are we about helping students become more self-aware learners? Why is so much of our education system one-size-fits-all? Why do we require students to learn things by arbitrary deadlines, anyway? Some deadlines are out of our control: the date at the end of the semester when grades have to be entered into the grade system, for example. But most of the other deadlines that I place on my students’ learning are completely arbitrary.

After some thinking, I’ve drafted the following scheme to give students more flexibility to learn at their own pace (within the confines of the semester).

First, I took my Math 180 course objectives (previously identified in part 1) and broke them up into slightly smaller learning objectives:

  1. Given a PDE problem, be able to categorize and characterize it with enough detail so as to be able to understand how the solution might behave and to select an appropriate solution technique
  2. Be able to derive PDEs from an integral conservation law, and be able to solve a first-order PDE using the method of characteristics (nonlinear PDEs may involve shocks and fans)
  3. Given a PDE problem over a finite domain, be able to identify whether the associated eigenvalue problem is self-adjoint and determine appropriate orthogonality conditions if it is so
  4. Be able to use the separation of variables technique to solve PDE problem (including using the eigenfunction expansion technique for inhomogeneous problems)
  5. 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

I think I will ask students to demonstrate their proficiency on each the five objectives above primarily through proficiency assessments (PAs)–essentially quizzes with one or two tasks that are focused on only one objective above. Students can take a PA at any time during the semester. I will try to write several identical PAs for the same objective so that a student can retake a PA until s/he is satisfied with her/his mastery of that subject. Only the highest score for each objective will remain. I will try to come up with an advanced level PA for each objective so that students who want to push themselves can do so.

Students will also demonstrate their proficiency through a comprehensive final exam worth a relatively small portion of students’ final grades (15%). It will be administered as I described in part 3.

Weekly homework assignments will also make up a small portion (10%) of students’ grades, though I will encourage them to take the assignments seriously since the majority of their learning will come about through wrestling with problems.

Each student will also create a paper or presentation (thanks to Ed Dickey’s suggestion) on an application of partial differential equations that will account for 15% of her final grade. So to summarize: proficiency assessments (60%), application project (15%), final examination (15%), homework (10%).

No doubt many of my teacher friends will see some resemblance of this scheme to Standards-Based Grading (SBG), which seems to be catching on in the K-12 world. I don’t claim that my scheme is at all original. I’ve stolen ideas from conversations with many of colleagues. (Special shout out to Dann Mallet and Charisse Farr who shared their experiences using SBG at the university level in Australia.)

My primary purpose in designing the course this way is to let students learn at their own pace. I want them to be able to take proficiency assessments whenever they feel they are ready. And if they fail one of these proficiency assessments, I want them to realize that’s not the end of the story. My advisee’s experience made me realize that when I give my students only one (or two) chances to demonstrate their proficiency, (1) the pressure to get it right leads to cramming, and (2) once the test is over they don’t care about that knowledge anymore whether they learned it or not.

A true SBG implementation would also involve getting rid of summative letter grades or numerical scores in favor of a more comprehensive description of students’ abilities and understandings. I’m still stuck with letter grades here so that’s not going to happen. I’ll have to figure out a scheme for combining scores together to create a final grade. No definite plans yet on how each PA will be scored. Many SBG implementations use a rubric scoring scheme of 0/1 to 4. Right now I’m thinking that a baseline level of proficiency should correspond to a B- somehow, since this is an intro graduate level course and I need to come up with a grading scheme that works for both undergraduate and graduate students.

In my opinion, some SBG implementations are a bit too granular in the way course objectives are defined. I remember walking into one Algebra 1 classroom recently in which the teacher had maybe 15-20 mini objectives written on the wall for the unit. The objectives were small things (as in grain size, not in importance) like “express a line in point-slope formula” and “find the y-intercept of a line”. I feel this over-granulization is why some SBG implementations overemphasize the procedural nature of mathematics.

I was very conscious of that and tried to break up my course objectives into relatively large subordinate objectives. Some of the five items above are more procedural than others, but a lot of thinking and reasoning is still involved in each. The five objectives above should be relatively decoupled from each other so that students could take them in any order. (The exception is that you probably need to master #3 before #4.) I expect most students to take them in order, however, and to pass a PA once every two or three weeks.

There is still a final exam that will assess whether students can synthesize all of the information. I’m hoping that the relatively low weight of the final exam will help alleviate anxiety about the exam. The final exam will also help my students not feel completely at sea with a weird new system.

I think the biggest issue for me right now is writing all of these PAs. For each objective, I’m going to need several different but roughly equivalent mathematical tasks. That’s not an easy thing to do in PDEs. Small changes in a problem can change its complexity in large ways. For example, these problems on the surface look similar, but one is much more challenging.

Problem #1:pde1Problem #2:

pde2

Both problems involve separation of variables but problem #2 requires a change of variables first (u=v+w) where v is the solution to a Laplace equation problem also requiring separation of variables…so basically a separation of variables problem within a larger separation of variables problem.

Anyway, I’ve got lots to figure out. Your comments are appreciated!

PDEs Course Design (Part 3): Relieving Time Pressure During Tests

This post is part of a series in which I am blogging my way through a new course on partial differential equations (PDEs) that I am about to teach in a few weeks. (Links to part 1 and part 2.)

This post by Lani Horn reminded me that timed tests can be harmful for students’ self-efficacy as mathematics learners and their perceptions of mathematics as a whole. People have and will always argue for and against timed tests. In my view, most of the underlying disagreement is because we conflate automaticity and speed.

A friend told me that teaching basically involves moving students from

  • unconscious ignorance     to
  • conscious ignorance     to
  • conscious competence     to
  • unconscious competence.

Embedded in this pithy statement is some notion of automaticity: the state of having so deeply internalized a mathematical skill or concept that you know when to use it and can use/apply it correctly with relatively little cognitive demand.

A colleague of mine at Mudd often says that in calculus students finally learn all their pre-calculus skills well; in differential equation they learn all their calculus skills well. I think what he is saying here relates to automaticity too. As much as educators sometimes poo-poo it, practice is necessary for developing automaticity. The reason why automaticity is important in mathematics is that there are lots of things that build on top of each other.

Here’s an example: To correctly perform the integral
integralone has to (1) factor the denominator, (2) decompose the rational expression into simpler fractions (which also requires one to solve a system of linear equations), (3) use a substitution to integrate one of the pieces, and of course, (4) perform all of those algebraic manipulations without making any mistakes. If the cognitive demand of any of these subordinate steps is too high, a student can easily lose sight of the forest for the trees. In this problem there are many opportunities for tiny errors. To perform this integral correctly, it helps if those subordinate skills are automatic.

With the ubiquity of computer algebra systems and online services like Wolfram Alpha, some might wonder why automaticity is still important. But, think about how frustrating it would be to read a novel in an unfamiliar language. Yes, you could slog through it by looking up the definition for every other word, but you wouldn’t suggest that as a way for someone to learn a language–it would be too off-putting for most. If the goal is just to help someone learn how to get by in a foreign country then that is fine. My goal is to help students be deeply fluent in mathematics and so I believe some automaticity is desirable and necessary.

Whether you agree with me about the importance of automaticity in mathematics, the central issue of this post is that automaticity is not the same as speed, though they are closely related. The problem is that the speed at which one student completes a task with automaticity may be different from the speed at which another student completes the same task with the same level of automaticity. To disallow that variation in speed is to assume that all students think and do mathematics in exactly the same way.

Perhaps some of the arguments about timed tests can be resolved if instructors were more forthcoming and conscious about whether their underlying goal is automaticity. And if automaticity is a goal, instructors should find other ways of measuring it without using speed as a crude proxy for it. I believe it is the instructor’s job to give students enough practice (perhaps through homework or in-class tasks) so that students have the opportunity to develop automaticity, and to help students become self-aware enough to recognize when they have internalized a skill or concept to the desired level.

The best way to observe whether students have developed automaticity is to watch them doing those tasks. My suggestion is that we formatively assess for automaticity (perhaps through in-class tasks) rather than using timed tests to do so, and to reserve summative assessments for things that don’t rely as much on the automaticity of certain skills.

Another reason why I think that timed tests are harmful is that they introduce a non-trivial amount of anxiety (which leads to lower performance) for some students, particularly those who previously performed poorly in mathematics, or those who tend to doubt their skills. In college/university mathematics courses, these groups of students tend to overlap more with underrepresented minority and female students. If you agree with these two assertions, then is it not the case that timed tests can sometimes be a form of institutional racism or sexism? Let us not forget the theory of disparate impact, which holds that any practice or policy may be considered discriminatory if it has a disproportionate “adverse impact” on persons in a protected class.

(inserting a pause here so people can think about that…)


So back to PDEs. I was considering avoiding exams completely, but given my other plans for the class (more in another post), I think it would be best to give one comprehensive final exam that will contribute a relatively small percent of students’ final grades (maybe 10 to 15%). This goal of this PDEs final exam is to see whether students can synthesize the many skills and concepts that they will need to master in this course. It will focus on the first two learning objectives I listed in part 1. I can’t avoid computations on this final exam, but I can limit the complexity of the computations and focus on problems that ask students to synthesize or evaluate ideas instead of requiring them to have automaticity of certain computations.

At Harvey Mudd, we have the luxury of being able to assign take-home exams with relatively little concern about academic dishonesty. This is all the more reason that traditional timed exams can be replaced with something better on our campus. My current plan is to write a comprehensive take-home final exam.

Here’s my usual test-writing practice: After writing a test, I take the test and time myself. I multiply that time by 5 or 6 to arrive at a suggested duration for the test. That suggested duration is clearly indicated on the cover sheet of the exam, along with instructions to students that they can take more than the suggested time if they need it, without penalty. I ask students to take the exam in one contiguous block (with only short potty breaks), and to write the start and end time of the exam on the cover sheet. There are two reasons why I give a suggested duration for the test instead of just allowing for an unlimited time exam: (1) it helps students know about how much time to set aside in their schedule to take the exam, and (2) it helps students not use an excessive amount of time. There are some students (especially at Mudd) who, if given an unlimited amount of time, would use so much time that they would neglect other obligations (like eating, sleeping, or bathing–eewww). If I write my exam so that there are no “tricks” that require creative inspiration, then there should be some hard limit to the amount of time that students can productively spend on my exams. I don’t want them to use more time than that.

Love to hear your comments. In my next post, more on reducing time pressure, but on a much longer time scale.

PDEs Course Design (Part 2): Learning as Participation in Communities of Practice

Today I devoured Situated Learning: Legitimate Peripheral Participation by Lave and Wenger (1991, ISBN 9780521423748). I read the whole thing in several hours, but I will definitely need to revisit this book again to absorb more of it.

2015-08-02 12.30.06

Right now, here is my best attempt to summarize the main idea of this book:

Many theories of learning primarily define learning as a cognitive process on the part of the learner. Knowledge is usually gained by replicating the performances of others or internalizing knowledge transmitted through some form of instruction. These theories focus on the transmission and assimilation of knowledge and position the teacher as the source of knowledge. Instead, Lave and Wenger theorize that learning is by-product of a person seeking to participate more fully in a community of practice. The community of practice is the seat of the knowledge. “As an aspect of social practice, learning involves the whole person; it implies not only a relation to specific activities, but a relation to social communities–it implies becoming a full participant, a member, a kind of person. In this view, learning only partly–and often incidentally–implies becoming able to be involved in new activities, to perform new tasks and functions, to master new understandings.” (p 53)

Page 112 rocked my world:

When the process of increasing participation is not the primary motivation for learning, it is often because “didactic caretakers” assume responsibility for motivating newcomers. In such circumstances, the focus of attention shifts from co-participating in practice to acting upon the person-to-be-changed. Such a shift is typical of situations, such as schooling, in which pedagogically structured content organizes learning activities. Overlooking the importance of legitimate participation by newcomers in the target practice has two related consequences. First, the identity of learners becomes an explicit object of change. When central participation is the subjective intention motivating learning, changes in cultural identity and social relations are inevitably part of the process, but learning does not have to be mediated–and distorted–through a learner’s view of “self” as object. Second, where there is no cultural identity encompassing the activity in which newcomers participate and no field of mature practice for what is being learned, exchange value replaces the use value of increasing participation. The commoditization of learning engenders a fundamental contradiction between the use and exchange values of the outcome of learning, which manifests itself in conflicts between learning to know and learning to display knowledge for evaluation. Testing in schools and trade schools (unnecessary in situations of apprenticeship learning) is perhaps the most pervasive and salient example of a way of establishing the exchange value of knowledge. Test taking then becomes a new parasitic practice, the goal of which is to increase the exchange value of learning independently of its use value.

Will need to save my thinking on this paragraph for another time…

Lave and Wenger are very clear that their theory of learning “…is not itself an educational form, much less a pedagogical strategy or a teaching technique…It is an analytical viewpoint on learning, a way of understanding learning” (40). And yet, the teacher in me can’t help but grapple with the implications of these ideas on my own teaching practice and on my work with secondary school mathematics teachers.

For my partial differential equations (PDEs) course, I think I first need to situate the learning by defining the relevant community of practice for the course. I think the relevant community of practice should be the set of all individuals (mostly mathematicians and scientists) who use partial differential equations to solve problems in their professional lives. Since this is an introductory PDEs course, the ultimate goal of the course should be to introduce students to this community of practice and help them to see that they can be members of this community should they want to be.

The question is how to do that. Certainly I can explain this goal to my students, but to then go about teaching as usual seems to be missing the boat.