# The exam feedback conundrum

By this point in the semester, I have given two exams in my Calculus class and one in my Real Analysis class. Grading is always a pain, but I have struggled much more recently with what to say as I hand out the exams. Giving statistics on the exam might be helpful for students so they can see how they stand with respect to the rest of the class. But is that really important? Isn’t it more important that the student only know how they did, and perhaps how they can improve? I really don’t know the answer to this question. I used to never give statistics on the exam. If my students asked, I would say that averages and medians were not a great way to summarize their performance, and that they should really only worry about their own performance. But I have recently caved, mostly because I realized I was one of the few people who did not give any stats on the overall class performance, and the students were sort of expecting it. After a couple of awkward incidents this semester, I’m considering going back to my “no stats” policy.

The first thing to think about is why would you want to know how everyone else did? Especially if the professor doesn’t curve, and so the performance of others will not affect your own individual performance. So what does it matter if you were the only A, or one of the many A’s? But of course, it matters.  Everyone wants to feel like they’re doing better than average, so they want to know the average. The paradox here is that there have to be people below the average! I remember telling my Calculus students, on their first exam, that the median was an 89. I was congratulating them, because as a class they performed well. But as soon as I explained (to the few who didn’t know) that the median was the cut-off between the top 50% and the bottom 50%, a lot of them looked very concerned. They realized that 50% of them had gotten an 89 or less. So somehow, this makes the people who didn’t get an A feel worse, and at the same time it makes the people who did feel less special. So my attempt to congratulate and praise kind of backfired.

And then we have the problem of what that praise does to the students’ performance later. For example, the average and median on the second exam were a lot worse. And I know in part it was because the material got harder (the material always gets harder), but I wonder if maybe they got a bit complacent and overconfident, and decided to blow off this exam. Of course, I have no way to prove this, and because I had done it for the first exam, I gave them the stats for the second. One student asked for more information, like what was the spread of the grades. This is a good statistical question, but I don’t like to give the highest and lowest scores, since then I make the student with the lowest score feel really bad. I gave the highest score, because I decided no one would mind, and then the student with the highest score came up to me and said that that made them a little uncomfortable. I didn’t say any names or indicate that they had been the student with the highest score, but then I realized that their friends could probably figure it out, and I guess that’s something that people are not always comfortable with sharing.

This reminds me a bit of my Real Analysis exam, where the median was an A-. I wanted to share that with the class because the exam wasn’t easy, Real Analysis is not easy, and I wanted to congratulate them on their accomplishment. But again, my mistake was in forgetting that there were still people who had not done very well, and thus I was essentially making them feel very bad about themselves. I am also not sure if this improved morale in the class, since an A in Analysis, which would normally be exciting, now didn’t seem like such a hard thing to attain and thus was made less special.

Of course, it could be much worse. When I was in college, my Linear Algebra prof liked to hand out exams in descending order. So if your name was called first, you knew you had the best grade in the class. As names were called, you got increasingly worried. Whoever was called last had to do the long walk of shame to the front of the class to pick up their exam, which we all knew was the worst. I was called first only once, and I was very pleased with myself, so I guess the method did improve my self-esteem. I’m not sure this method helped anyone in the bottom half, though. In any case, I believe that this would be illegal due to FERPA rules, so it’s not even something to consider.

All of this is a long way to say that I’m considering not giving any statistical feedback on the exam. Maybe just focus on common mistakes and things to work on for the final exam, and perhaps I could take a bit more time on each exam and write personalized feedback on the exam instead. I do tell everyone who got less than a 70 to come talk to me during office hours, so that’s another place to give feedback and comments or suggestions on a student’s individual performance. It is tempting to praise a class that did well, but it’s possible that I’m doing that more to stroke my own ego (look how great a teacher I am!) than to improve their morale. Each one of them knows how they’re doing, and I think now that that should be enough.

How about you, dear readers? How much feedback do you give to the class as a whole on their performance in an exam? Have you had uncomfortable situations stemming from this? Do you have suggestions for how to praise the students without making others feel bad? Please share your thoughts in the comments section below.

# Five (math) things to do before you die

1. My immediate response to the question was the Riemann hypothesis. Not that given 100 more years to live I would have any hope of solving this problem, but I would like to see it proved in my lifetime. Especially because we are all pretty certain that it’s true.

Of course, then one can go through the list of Millenium Problems and I would add two more things:

2. the Birch and Swinnerton-Dyer Conjecture, and

3. the P vs NP problem.

Again, I am not saying I have any chance of solving them, just that I would like to see the solutions to these problems. But this is where it gets tricky. I basically have a list of three things that probably anyone could have made (these are some of the most famous problems in math!). So how do I add two more things to it? Nothing will seem as important (nothing else I can think of would make you a millionaire!). OK, there are three other millenium problems, but I’m just not as interested in them. So then I started thinking about the math topics I would be sad not to have learned if I were to die tomorrow.

4. I have gotten interested in mirror symmetry and its relation to physics and number theory, so I guess I would be sad if I died tomorrow without learning more about it.

5. Arithmetic dynamics, since I am very interested but kind of new to it.

But doesn’t the list become weak after I add these two things? Anyway, please share your Top 5 in the comments below.

The original question got me thinking about other fun questions on might ask:

– Which 5 math books would you take to a desert island? The funny thing is that I can’t think of a top 5 but I can always think of at least one or two things. For example, I would bring Serre’s A Course in Arithmetic. But of course, if you asked me to bring just one I would be stuck.

– Who are your Top 5 mathematicians of all time? Gauss? Ramanujan?

-Slight variation: which 5 mathematicians would you take to a desert island? See, here I would probably pick some fun/handy mathematicians. I don’t know if Gauss would be very good at building a hut.

– What are the best 5 math formulas? Euler’s formula is widely regarded as one of the most beautiful formulas in mathematics. Do you agree? Can you think of others?

– What are your 5 favorite functions? I know one: hypergeometric functions!

As a final comment, I wanted to say the first question was suggested by my friend Casey Douglas, who is an Assistant Professor at St. Mary’s College of Maryland. He thought of this question as he was preparing a talk for the SMCM math department’s annual “MATH WEEK OF AWESOME”, which sounds, indeed, awesome.

So now I open it to you. Do you have answers to these questions? Do you also find it slightly frustrating when these questions are posed (if so, I apologize)? Can you think of other questions like this?

http://blogs.ams.org/phdplus/2012/03/23/five-math-things-to-do-before-you-die/