Thinking About How to End a Year

So, this morning both of my AP classes took their final exams. I have some questions for the world about this process, but first I want to share something fun from the Calculus BC final. You should know that my students understand that the word fun, when I use it in class, means that a problem is challenging, thought-provoking, unusual, or some other words that they might use but I won’t type. I believe that I mentioned that we ended the year after the AP test with a quick tour of some interesting topics that many high school students don’t get to see. I included a small unit on different number bases. I always start this by writing a series of addition and multiplication facts on the board. However, they don’t know that these facts are base 8 number facts. I usually reveal the secret by showing a picture of Lisa Simpson and tell them that these facts are how Lisa would compute. It’s a fun conversation to have. So, for our final exam I told my students that there would be ten problems. Nine of them would be Calculus problems taken from old tests. they have all of their old tests (all ten of them) so they could be very well prepared for that. I also told them that one problem would come from our last two weeks. After discussing this with my friend Richard – a former math teacher – he sent me the following passage from Alice in Wonderland

Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!    

I played around with this for a while and fell in love with this as a final problem for the year. This is how I presented it to my students:

For your final problem on your final Calculus test, we will play with number bases. Consider the following passage from Lewis Carroll’s Alice in Wonderland:

“Let me see: four times five is twelve, and four times six is

thirteen, and four times seven is — oh dear! I shall never get

to twenty at that rate!”

            Explain, in terms of your knowledge of number bases what is happening in this pattern. Explain how four times five is twelve, and how four times six is thirteen.    Guess what she will say four times seven is and make it clear to me why she won’t be able to get to twenty. 


I will sink my teeth into grading finals tomorrow since I am on dorm duty tonight. I did browse through four or five of the test as they were turned in and two students really nailed the problem and provided beautiful, detailed answers explaining the pattern. I won’t spoil it here, I’ll let you work through it if you wish to do so.


I don’t know how your school works, I do know a bit about the four schools where I have worked. All of them have been independent schools that emphasize the idea that we are college preparatory schools. Each school I have worked at has had a statement in their handbook about the importance and significance of final exams as a college preparatory experience. However, I know that tonight many of the seniors in the dorm will tell me that they do not have any finals left. Today was the first of three and a half days of final exams and many seniors won’t have any more after today. There is a pretty common feeling that final exams will not be pretty and are of questionable usefulness with our seniors who are days away from graduation. This is not just a feeling at my current school. But I really wrestle with this. We say we believe that taking a final exam, preparing and organizing a large body of information for a one-day thorough examination, is a useful skill AND one that is important for college. However, it is those students who are closest to college who are the most likely to have been excused from a final exam. In some classes the final experience is a paper or a presentation that happened last week. But our school, and others where I have worked, carve out quite a bit of time for final exam administration. I wonder whether we could use our time in a more meaningful way. I wonder whether the idea of a final exam makes sense only in certain disciplines or for certain age levels. Is it reasonable for us to ask our freshmen to take exams under the same circumstances that we ask of our juniors and (sometimes) seniors? I don’t see AP scores for my students until July. I like the idea of some capstone where I check in with them in one last, broad examination of ideas. I feel pretty old-school in that regard.

I want to make my assessments meaningful for me and for my students. I am really beginning to doubt whether exam week is such a positive way to do this. I would love to hear from others about how they deal with this question. Are many of you bound to a policy that your school or your department has mandated? I want to be smarter about this and I’d love your help.