You Current Grade: It’s Complicated

By now, most US universities are 4-5 weeks away from the end of the fall semester. Whether it’s now, or just prior to the withdrawal deadline, student tend to demonstrate increased interest in their grade for their courses. They say that they want to know how they are doing. But they often prefer to know what grade they will earn at the conclusion of the course. The answer to the latter question could include all kinds of assumptions. But “What is my grade right now?” is a deceptively subtle question.

It seems direct. We could easily be curt and claim that it shouldn’t be complicated to tell a student what their grade is, and that it’s a failure of the teacher or of the education system writ large if it is complicated. While I entirely agree that a teacher should have an answer, it’s important to emphasize that “What is my grade right now?” is an ill-defined question. The problem is that a student can mean two different things when they ask about their grade.

Q1) What proportion of possible points have I earned so far?

Q2) What proportion of points will I have earned if my performance doesn’t change?

It’s important for teachers to ensure that their students understand which question is being answered.

First, I’ll illustrate when there is no distinction between the answers. Let’s say that there are two types of assignments: Exams, which are worth 75% of the course grade, and quizzes, which are worth 25%, of the course grade. So long as the two assignment types are identically distributed throughout the semester, Q1 & Q2 have the same answer. Below is a bar chart that illustrates a distribution of points over 4 weeks. The proportion of points for each assignment type is identically distributed over time (not necessarily uniformly distributed).

What is the student’s grade at the end of week 2 if they have scored 90% on the exams and 70% on the quizzes? By the end of week 2, there have been 30 possible exam points and 10 possible quiz points. The student has earned 34 of the 40 possible points so far. The math for Q1 is:

(0.9)(30)+(0.7)(10) = 27+7=34

34/40 = 85%

And, if they continue to perform identically in each assignment category, then they can expect to earn an 85% in the class. The math for Q2 is:

(0.9)(75)+(0.7)(25) = 67.5+17.5 = 85%

Both Q1 and Q2 have the same answer. And, honestly, principles or introductory courses have formats that often lend themselves well to having assignments distributed similarly over time. My own Principles of Macroeconomics class matches up pretty well with the above math. Each week, there is a reading, a homework, and a quiz. By the time students complete the first exam, they’ve completed about one third of all points in each assignment category.

Higher level classes or classes with projects tend *not* to have identical point distributions across time among assignments. Maybe there are presentations, projects, or reports due throughout the semester or at the culmination of the course. For example, my Game Theory class has two midterm exams, but no final exam. It has homework in the first half of the semester, and term paper assignments in the latter half.

The bar chart below displays a point-split among the same quizzes and exams, but they now are differently distributed throughout the semester. Quiz points have been frontloaded.

What is the student’s grade at the end of week 2 if they have scored 90% on the exams and 70% on the quizzes? By the end of week 2, there have been 30 possible exam points and 15 possible quiz points. The student has earned 37.5 of the 45 possible points so far. The math for Q2 is:

(0.9)(30)+(0.7)(15) = 27+10.5=37.5

37.5/45 = 83.33%

And, if they continue to perform identically in each assignment category, then they can expect to earn an 85% in the class. The math for Q2 is:

(0.9)(75)+(0.7)(25) = 67.5+17.5 = 85%

All I did was frontload 5 percentage points for quizzes and now the answers to Q1 and Q2 differ by 1.66 percentage points. That may seem like small potatoes. But consider that a) many students and universities use and care about the +/- system of grades, and b) a grade difference of 1.66 points was caused by a mere change of 5-point change in the distribution. Bigger changes result in bigger differences. Frontloading the remaining 5 quiz points from the end of the semester would result in a Q1 score of 82% – yielding a 3 point difference between the two calculation methods.

The differences between Q1 & Q2 illustrated above are even more pronounced once you begin to include extra credit. One point of extra credit has a smaller effect on the answer to Q1 as more and more possible course points have been earned.

If students only care about their ultimate grade in the course, then they will always prefer to receive the answer to Q2. But, students may also want to know how effective their recent study habits have been so that they can re-evaluate them conditional on the knowledge of the assignment point distributions. Q2 requires more assumptions if an assignment type hasn’t even occurred yet. Students can ask “Have I given this course the appropriate amount of attention given the types of assignments that we’ve had?”.

For example, my Principles of Macroeconomics course has the first exam at week 5. Students should have an average score that is greater than 90% by the end of week 4 because the reading assignments are simple, the homeworks are lenient, and the quizzes permit practice attempts. Students who have an 80% by the end of week 4 are going to have a rougher time once they encounter an exam.

Reasonable people can disagree about which calculation is more useful. And more mathematically inclined students can calculate their own grades anyway. Therefore, after every exam, I send a mail-merge email to each of my students in order to update them about their grade. I give them the answer to both Q1 & Q2, and I illustrate the impact of several alternative scenarios for their future performance. If there is information that a student wants about their grade, then it’s in that email.

In conclusion, teachers should take great care in making student grades and progress reports clear. Students should take great care to understand what they are asking and and what the answer means. Grades can be very important for students who are close to the margin for scholarships, academic probation, or failure. While students may care too much about their grades, teachers should be sensitive to the fact that the care is real none the less. Teachers owe their students a firm and clear indicator of performance.

*There is another case in which Q1 & Q2 have the same answer. It’s when the student earns exactly the same grade in each assignment category, regardless of whether the category points are distributed identically across time.

Reflections on Teaching in Fall 2020

As the Fall semester comes to close on college campuses, it’s a good time to reflect on and assess how the past semester went. Many universities went to almost exclusively virtual learning, but other schools tried to make Fall 2020 as normal as possible given the circumstances of the COVID-19 pandemic.

My school, the University of Central Arkansas, chose the route of trying to have things as normal as possible — by which I mean students live on campus, classes are mostly in-person — while still accommodating students and faculty that preferred a more physically distant atmosphere. For example, UCA increased the number of fully online courses available, roughly trying to meet faculty and student demand. I normally teaching one online course per semester anyway, and I continued that this semester. Other faculty had more online classes than usual, or moved their class to be partially online.

So what was my experience?

First, the students, the most important part of the teaching process. Overall, I would say my students did very well. At least in the classroom, they complied with all the rules the University set forth: wearing masks, physical distancing in classrooms (seen in the image below), even the one-way entrances and exits to the building. There were only 3-4 times I can recall this semester when a student entered my classroom without a mask, and they immediately asked me for one upon realizing their mistake (I kept a pack of surgical masks with me).

My classroom at the University of Central Arkansas, with chairs blocked off for physical distancing.

As far as academic performance of students, I was very pleased with the students. For those students that were able to stick with the class and keep up, which was most students, they perform as well or better than previous semesters. Some students, due to personal circumstances, had trouble keeping up. I tried as much as possible to accommodate students in these situations, by being flexible with deadlines, offering additional resources, and generally just trying to listen to them and empathize. It was hard for everyone.

On my end, I tried to make the teaching atmosphere of the classroom as normal as possible. I usually do have some interactive aspects of the classroom, where students work in small groups, talk to their neighbors, etc. Most of those activities didn’t happen, unfortunately. But otherwise, the classroom atmosphere operated as usual.

As my students did, I also wore a mask in the classroom while I lectured. For students that had to miss class due to quarantine, isolation, or other reasons, we were asked to record every lecture and have an option for students to watch the lecture virtually if needed. Making sure that the video was properly recording and the I had set up the Zoom link for students that needed to be remote added an extra element to think about at the beginning of each class, but it was the kind of thing that once you get used to it, it just became normal.

I will say that I often felt very exhausted after teaching each day. The mental load of making sure everything was working right in the classroom, combined with the constant sense of doom in the world around us, made this a challenging semester mentally. I’m sure this was even more true for some of my students. But, we made it.

Finally, how about the administration of my University. I’ll bite my tongue a little here: I am up for tenure this year! But really, I don’t have anything major to complain about. Guidance was communicated well, although sometimes big changes were rolled out a bit more quickly than the faculty liked. UCA provided isolation and quarantine dorms for students, though these never came close to capacity. Weekly updates on testing, cases, and related data were provided to everyone (and made publicly available, so I’m not revealing any secrets here).

Testing data for UCA students. This data excludes athletes, since they were required to get tested regularly, which could have skewed the data.

As you can see above, the general student body at UCA did report positive COVID cases every week. And some weeks the positive test rate was a little higher than I was comfortable with! But we never had a large spike in cases, and the University held firm to its commitment to offer in-person classes for everyone that wanted them, as long as the campus was generally safe.

All in all, I think it was the best semester we could have had under the circumstances. The only thing really weighing on my mind: we are going to do it all over in the Spring semester. And we’ll do it as well as we can.

Spontaneous Emergence of Property Rights in the Classroom

Last week I posted about Bart Wilson’s talk on his new book “The Property Species” and promised to share a class demonstration about the emergence of property rights in the classroom. But first let me tell you why I did this demonstration.

When I was a student I hated assignments that go through the motions of learning, but provide no learing. Building a paper maché volcano, while fun for some, teaches little about volcanic eruptions. Shaking and opening a soda bottle (pop?) is more instructive: it’s the fall in pressure as the bottle is opened that leads to the rapid release of the gas disolved in the liquid, the same thing happens to magma. And while being able to algebraically solve for the equilibrium price given supply and demand functions is a very necessary evil (to a point), it teaches little about the process of competition and price formation.

This is why I was reluctant to having my first Intro to Economics class write their own version of “I pencil”, quite a few years ago. Driving the point of how largely anonymous exchange and specialization, coordinated peacefully through property, prices, and profits and loss makes the modern world possible is very important. But how much can you really learn about this by watching and transcribing an episode of “How It’s Made”? For most students, not much at all. Partly in dread of reading and grading 80 versions of “I whiteboard marker”, or “I toothbrush”, and partly following my conscience I decided to throw in a twist.

The twist may seem evil and arbitrary at first. Students still had to choose a good and write their own version of “I _____” , but if two students wrote about the same good I would divide their grade by 2. If three students wrote about the same good I would divide their grade by 3 and so on. I did not give any additional prompts about how they should sort out potential conflicts or coordinate amongst themselves. These were just the rules of the assignment.

Without this seemingly arbitrary grading rule, goods to write about were not scarce. By changing the grading rules, goods to write about became scarce. While there are many more goods to write about than students, certain goods stand out in the mind, and extra effort must be devoted in thinking up a new good, and finding out if someone had already looked around their room and chosen the same good. Now students also had to coordinate amongst themselves or run the risk of a fairly severe penalty to their grade.

As expected, I have never had to enforce the the harsh grading penalties (anecdotal, I know). Students always find a way to coordinate and establish property rights over suddenly scarce goods. The point of the assignment was no longer about I pencil, but about the emergence of property rights and social coordination (and hopefully a little bit about I pencil as well). I didn’t act as a central authority that imposed and enforced property rights. I merely changed the incentives and constraints, hoping that the costs of coordinating and setting up agreements was smaller than the costs of not doing this.

When they turned in their assignment, we discussed how they had actually coordinated. Over the years I have seen multiple ingenious mechanisms. From class forums using the university platform, to a simple spreadsheet circulated amongst the students via email or WhatsApp. In the good old times before the pandemic they would sometimes meet after class and sort it out in person. Sometimes they created a common pool of goods and one of their classmates is chosen to distribute them among their peers. Leaders emerge to fill various roles from dispute resolution to registering claims. How this person is chosen also varies from class to class. Some students volunteer, others have it thrust upon themselves. The use of a homesteading rule is fairly common, first to choose gets the good in cases where there are multiple claims. In class we discuss why they use this rule, rather than last to choose gets the good, and the problems this alternative would entail.

I have only had one instance of a strong and contested dispute among “property owners”. That semester students had to not only write but present their work. Two groups (that semester “I _____” was a group assignment) wanted to do a good they thought would be amusing to present in class. I’ll leave it up to your imagination what good students in their late teens and early twenties might find to be amusing to present in class. The two groups of students underwent a rather complicated dispute resolution system with the rest of the class playing the role of arbiters of the multiple claims to the same good. Neither group wanted to budge, but one group ended up ceding the rights in the end.

What I like about this little classroom demonstration is that it makes it easier to teach the emergence of institutions as the products of human action but not human design. Order without design is a difficult concept to grasp, but maybe even more importantly it is a concept that is difficult to accept. But after this demonstration, not anymore, students experience the emergence of property rights. An added bonus is that in this case scarcity is clearly a product of the relation between their minds and how they relate to the world, not about objective quantities of goods.

Property rights emerge through their coordination but are not centrally imposed. They coordinate because a change in the environment turned a previously free good, the subject of their short “I ____” essay, into a scarce (economic) good. As you can probably tell Harold Demsetz is one of my favorite economists of all time. After the barrier of disbelief is breached, we can easily talk about the spontaneous emergence of money, cover a little about how property rights emerged in whaling on the high seas, and the spontaneous origin of law (very useful for future law students usually educated in the positivist tradition, as is the norm in Ecuador).

I later learned of the fish game (I am not an experimentalist). But, no disrespect intended, it seems a little contrived. I still like my assignment better. While the goldfish game teaches the tragedy of the commons, the “I _____” assignment teaches how the tragedy can be solved without a centralized authority by having students solve if for themselves and come to grips with the real limitations and problems they faced, albeit on a much smaller scale. I am still hoping for an experimentalist that thinks something serious can be made out of my little classroom demonstration.