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: http://www.wikihow.com/Cheat-On-a-Math-Test. 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.

I believe the research shows that the best way to remember something to be reminded of it exactly as you’re forgetting it (not before or after). I’ve always taken this as an excuse to flit from topic to topic revisiting them as I see fit. Of course the hard part is making sure that your class forgets things on schedule. Anyway, tests are a natural method of reminding people of things.

I’ve become a big fan of two-stage exams, including for finals. Students take an individual exam (taking a little over half the time available), and then get into groups (pre-planned groups either by them or by me). They retake the exact same exam in the group, discussing any problems on which they disagree / answered differently when they took it individually. Their final score is calculated as 80% individual / 20% group (or something similar).

If nothing else, it does mean that students will look back over the final exam and learn from at least some of their mistakes…