Final Countdown

Ah, end of April. The cruelest month is almost over. One more lecture, one more homework assignment to grade, and then the final exam. About this time of year, I start reading poetry and reevaluating my life choices and philosophies. Final exams are always on my angst list.

Summative assessment should assess a student’s learning at the end of a unit. This seems important to give a mastery-linked basis for assigning grades, and to see if the course accomplished the learning goals. My final exam probably did a reasonable job of both of those things. The students did fairly well overall, and I was really happy with some of their insights. However, an exam like this is far from a perfect tool. Some students have real test anxiety that never allows them to really demonstrate what they have learned in an exam setting. Other students just have a bad day or a bad week, and they give answers that even they don’t understand looking back at their papers. Some students work fast, others work slow, but go more deeply into the problem. Many students will excel regardless of what I do in the classroom—they’ve already taken calculus, they are very capable of learning on their own, without any help from me or my assignments. Does the final exam (or any exam) accurately assess these students’ mastery, or the effectiveness of my teaching?

To put this in context, let’s talk about some actual classes. In a couple weeks I will be writing more about my current classes, Modern Algebra and Foundations, so I will go back to my fall experience. I taught two sections of Biocalculus fall semester, with a total of about 60 students. Biocalculus is the main math class that all biological science majors take. The students are almost entirely freshmen. We cover discrete dynamical systems along with most of the material from a standard Calc I course, with more differential equations and modeling. Exams are worth 50% of their final grade.

In this course, I think of mid-semester exams as snapshots, opportunities for students to check their understanding and revisit weak areas. Many freshman students have not yet figured out how to assess their own learning. They haven’t figured out how to test themselves independently, review, and test again to find their own weak spots. They need an actual test to discover what they don’t really understand.

This is totally irrelevant in a final exam. Almost nobody looks back after their final exam—learning has to take place before or actually in the final. I do believe that the existence of a final exam can help facilitate learning. It can give the students motivation to study everything twice, which means they will retain much more of it. To me, growing mathematically seems to be a continuing process of learning, forgetting, and relearning (faster each time), so I figure the more cycles I can get into the class the better. Without a cumulative final, students have little motivation to revisit anything from the previous tests.

Of course there is the issue of how much labor the final exam involves for me. I wrote a cumulative final for those Biocalc classes, and then spent 20 hours grading those exams, a miserable experience. Four months later, I can look back and say it was all for the best and that the students probably learned a lot in that last round of studying, but I can also imagine that many of them have forgotten most of it by now anyway—not because they are bad students or anything like that, but just because they are now very busy with organic chemistry and physics lab, and haven’t thought about that material at all. Was it really worth those hours and hours of grading to give them one more round with those ideas?

Here are some possibilities that I consider:

  • Giving a long final that touches on as much material as possible is the fairest thing to do. If that means I need to suck it up and grade for 20 hours at the end of the semester, well, that’s why they call it work.
  • I could still give a cumulative final but make it short: somewhat randomly select a few topics from the course and ask questions on these. Students won’t know what questions will show up, so they will have to study everything.
  • I have been wasting my time–cumulative final exams are not worth the trouble. Give a normal last exam or final project during the semester and let the institution of long, hard final exams wither away into the past where it belongs.
  • Exams are the wrong tool for assessment. We need a different way of evaluating learning.

I went looking for some other perspectives on the matter. A 2010 Boston Globe report forecast the decline of final exams based on the fact that only 267 out of 1137 undergraduate Harvard courses had scheduled final exams. However, the same year, Psychology Today reported that cumulative finals did improve learning. But then this 2012 piece from the Chronicle of Higher Education calls into question the value of exams at all. On the practical side, another piece from Psychology Today gives some good perspective on how we should be writing our final exams (if, like me, you’re going to give one anyway), and some discussion of why it is actually very difficult for us as professors to write the exams that live up to these ideas. The Chronicle also just published this summary piece with some nice ideas, thoughts, and advice about finals.

These ideas are helpful, and I think it would be great to have a “finale” as the article describes (basically an end of semester experience which is not an exam) in some classes. At least all this reminds me that other people are struggling to write worthwhile exams, too. However, I wonder is there is still another magic end of semester option that I haven’t considered, that will assess students better, reduce my grading time, and let students learn even more?

What do you think? I would love to hear your philosophies and tactics, and how they vary for introductory vs advanced classes, majors vs non-majors.

PS While I was looking at the internet, thinking about exams, I found the following useful page: Ugh, this is a definitely a point against exams of any kind.

PPS If you are a major consumer of math blogs, you might notice that I used this title for a post on my personal blog a few years ago, but I think it’s time for some more Europe in everybody’s life.

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I hear your good advice

This spring has been the season of talks for me. This is great, because I love giving talks! Wait, I should say that I love giving good talks; the mediocre ones not so much. There are plenty of really good reasons to give talks (see this excellent post of Adriana’s, with its many excellent links), and a lot of good advice out there about giving math talks (here are three of my favorites, from Terry Tao, my first Algebra professor Eric Moorhouse, and Gizem Karaali).

I think that math talks are important and good talks are really powerful—talks that are enlightening instead of intimidating, that aim to communicate instead of impress, that bring out the fun and wonder of math. With this in mind, I usually spend a lot of time worrying about and planning my talks before I give them. Usually this works, and I have given many talks that I was really happy with!  However, even the best intentions can miss the mark, and mine have definitely missed on occasion. Then comes the agony of post-talk deconstruction. Of course this can become a black hole, a talk-related disaster of its own. Having already over-indulged in post-talk soul searching, though, I thought maybe I could save others some suffering by sharing a few of my talking disappointments with you. Alert: these disappointments were not, for me, fully preventable. But maybe some warning, and putting a name to these possibilities, can be at least a little helpful.

Sometimes it is just really hard follow the good advice. For example, the good advice is that you should give big-picture and context. Don’t be too technical. Some of my most unsatisfying post-PhD research talks were about brand new work that I didn’t have much context for, and honestly didn’t understand as well as I had understood my earlier work. I remember one talk especially where I know I gave more technical details than big-picture ideas. I remember it probably because there were multiple people sleeping in my seminar. I was embarrassed about this (though, in my defense, I’ve seen people sleeping in really good seminars), but at that point in my understanding it was just the best I could do. I understood the computations I had done but didn’t deeply get why I had done them. Preparing and giving the talk was also part of the learning process, and helped me understand the work better later on. So it was not my best talk, but I can console myself with the fact that at least I learned something from it even if some others did not. I think that it is still worth giving a less-than-perfect talk—you just have to respect your audience by really trying to give them the best talk possible. If you really care, your not-perfect talk will still be loads better than a talk from an expert who doesn’t care very much.

Another instance of not being able to follow good advice came up in the transition from graduate school to working life. People rightly say that you should practice your talk and get feedback on it. But in a new job, whom can you ask to watch your brand new and maybe very shaky talk? I practiced my first talks after grad school by myself in an empty room, but it took me a while to get used to this. I really missed my friends and their honest feedback (who knew I would miss hearing “Too much text! Boring! Give me an example!”?). Especially at first, my new talks were just not as well-organized as I might have wished. Looking back, I am really grateful for that community of people, who must have watched me give some awful practice talks.

Other disappointing talks happened when I thought hard about audience, as the advice suggests, but turned out to be wrong about who would be in the audience. For example, when I started giving undergraduate-focused talks, I didn’t realize that the audience for these talks usually includes a lot of professors. Though the organizer might claim you can choose any topic that the students might like, choosing certain mainstream topics will bore the professors in the audience nearly to death. This is not your fault as speaker, but it is very disappointing to *be* that speaker. I suffered this fate when I explained public key cryptography to one group. I could feel their silent screams as my talk went on. Now, I try to choose topics that include at least something that I think most professors in any audience won’t know very well. I still give the crypto talk, but if there are professors in the audience I start talking about lattice-based cryptography way earlier, so they have something a little different to think about.

One other issue that the good advice did not save me from: sometimes we just expect too much response. For example, after graduate school, I started to feel like my talks were not making much of an impression. Then, I realized I was used to people being extra nice to me because I was a graduate student. Of course people have higher standards and are a little more sparing with the praise after you get a PhD. I didn’t really realize how wonderfully kind and encouraging people had been until I was out of school, and noticed I really missed it! On the other hand, since I figured out what was going on, I adjusted my expectations and can even appreciate that people don’t feel the need to go easy on me anymore.

In another example of expecting too much, I realized that depending on the culture of the school, undergraduates may not be super responsive even in talks they really like. They don’t often start conversations with a speaker afterward. I had a lot of experience getting undergraduates to respond in the classroom, but didn’t realize it might be so different in talks until I started giving a lot of undergraduate focused talks. I love giving these talks and have had some great experiences connecting with undergrads at talks—I just now know not to feel bad if nobody introduces themselves!

Okay, so there are a few difficulties I’ve run into—I hope that they can serve to make someone’s talks better, or make some disappointment shorter-lived. More warnings or good ideas? Please share in the comments!

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Math Tea

My school has a longstanding tradition of a weekly math tea. And unlike other places I’ve been, where this time is a social hour before a seminar, at Hood it’s a time to play games, solve puzzles, or do some interesting math with students and faculty. When we divvied up departmental duties at the beginning of the year, I ended up as Math Tea co-czar (that’s the official title), and it’s ended up being one of the most fun parts of my week.

Each week, we choose some kind of activity. We’ve built up a pretty impressive of activities and games over the years, many of which are detailed on by our own Betty Mayfield on the MAA’s “Math Club in a Box” site.

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Our annual math pumpkin carving day. Carving done by student Justin, pumpkin gutting done by the author.

We have a shelf of games and puzzles, some of a mathematical bent like Set or Rush Hour, but some are just fun card games. We’ve also cribbed a few activities from the book Solve This: Math Activities for Students and Clubs, like one on cutting modified Möbius strips, and another on doing math on the surface of a donut (with the real thing for a snack, and a papercraft model for experimenting).

Bewildered by cutting weird Möbius-like constructions

Bewildered by cutting weird Möbius-like constructions

My favorite activities just explore whatever interesting topic I’ve found on the internet recently. Last week I’d seen Buffon’s Needle flying around, so we gave it a shot.

A small-scale test

A small-scale test

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Scaling up










The week before, we experimented with drawing 3.5-gons.

Our department chair's 3.5-gon.

Our department chair’s 3.5-gon.

And a while before that, we tried to figure out why coin flip probabilities are so frustratingly counterintuitive. I’m not sure we really resolved that one to everyone’s satisfaction, but it was fun to collect data.

The best part about our math tea is that the whole department comes, and we take care not to dominate the discussion. I think it’s incredibly helpful for students to see how professors approach difficult problems, particularly that we rarely know how to solve things immediately.

This takes very little time to put together, and it’s a valued part of our week. Since we do it in a public area of our building, I think it also gives our department visibility and makes us look fun, which probably helps attract new math majors. If you’d like to help develop a stronger sense of community with your department and your majors, I recommend starting a math tea.


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