Wednesday, January 7, 2009

Cumulative Final

In less than a week my Calculus 2 class will be starting up. As you may have noticed, I've been thinking about ways to run this class differently than I ran it last time. One of the things that has come up in my thinking is how to structure exams. Last semester we had 3 midterm exams and a cumulative final. What I hope to try this semester is two midterms and a non-cumulative final.

First, some background from last semester. Our first midterm covered 2 chapters, the second midterm covered 2 more chapters, and the third covered all but two sections of a single chapter (the chapter on sequences and series). The remaining two sections of that chapter were untested going in to the final. That insured a couple of problems from those sections on the final, as well as all the other material.

What I think I'd rather do is have 2 midterms, covering the same material as the first two midterms last semester, and then have our final exam cover that whole last chapter. Each of these exams will be given the same weight for the overall grade, instead of having a somewhat more heavily weighted final exam.

I think to explain why I like this idea, I need to mention the content that we cover in somewhat more detail. The first midterm covers techniques of integration and a few applications (arc length and surface area for surfaces of revolution). The next exam covers parametric and polar curves (derivatives, areas, and arc lengths) and iterated integrals. The third midterm, last semester, covered sequences and series (definitions and convergence tests) leaving Taylor series for the final.

So, why do I not want a cumulative final? Looking at the 'old material' that students would need to go back and learn, I see a lot of formula memorization. "Make this non-obvious trig substitution when an integral involves...", "to break up a function using partial fractions, do this strange procedure", "the formula for the (arc length, surface area) of a ((polar) function, parametric curve) is ...". These are things I want my kids to know about at the end of the semester. But if they forget formulas, or the appropriate substitutions, I'm perfectly ok with that. They should be allowed, after this semester, to look up all of these things in a book. I expect that the techniques of integration won't be used by hand by students again (unless they end up teaching calc), because they will be permitted to use computers or tables of integrals in the future. And if they are needed, that's what the textbook is for.

Ask any former calc student for a technique of integration. If you're very lucky they'll remember that sometimes something like a trig substitution is useful. Ask them how to find the arc length of a parametric curve. I'd guess you'll see a lot of blank stares. Same goes for polar areas. Why should I test my students twice on material I'm happy to let them forget details of, after demonstrating competence at least once? My vote is for non-cumulative finals.

Thoughts? Why am I wrong?

4 comments:

sam shah said...

My two cents...

I would suggest that you say you're going to have a final, but you don't specify what it's going to be like (cumulative or not). That way you'll get a choice, once you see what the tone of the class is and what would be most beneficial for the students once you see their strengths/weaknesses.

If you're concerned about a cumulative final really being a vehicle for rote formula memorization, maybe have 3 regular tests and 1 take home/open note/1-week investigative problem (e.g. http://www.amazon.com/Student-Research-Projects-Calculus-Spectrum/dp/0883855038) that counts as a test? (Or a particularly involved problem set, if not a project.)

Mitch said...

I would tend to agree with sam shah about saying there will be a final and more details provided at the end of the term. See how the first test goes. While some of those integration techniques seem to be not all that commonly used by hand, is integration by parts one of them? Students will be expected to know that if they move on to Calc III and DEs. Partial fractions will rear its head again for those who take DEs. I also like to use my cumulative final to assess which students struggled in the beginning but managed to put it together at the end. I also use it as a tie-breaker for borderline students. I look at it and say "Did this student do B work overall?" That's not something you can do with a non-cumulative final. The calculus sequence I teach in is structured differently than yours, but I think I'd still prefer my way in your system.

sumidiot said...

Thanks for the feedback, guys.

@sam shah It never really occured to me that I could delay deciding about cumulative versus non-cumulative finals until after the semester starts. And I also like this project idea.

I also like the idea of a two-part exam. One part where they are allowed to make and use their own cheet sheet for formulas, and one where they aren't allowed such a sheet.

@Mitch Integration by parts is definitely a weekness in my argument (and why I avoided mentioning it :)). I'm not surprised that partial fractions comes up again (can you believe I've avoided taking a DE class?), but I wouldn't guess that they're expected to know, without any review, partial fractions at that point. I do see that having the cumulative final makes students relearn things, and so even if they then forget it, relearning it for the third time when they get to a later class will be easier than relearning it only for the second time. And I hadn't thought about this difference you point out about cumulative versus not, where cumulative finals are a way to go back and account for 'put[ting] it together at the end'.

Lots to think about. Thanks again.

bevthebod said...

The "taking all semester to finally get it" was very applicable to me, as I typically would take a while to catch on to new ideas, but once I had them, I owned them. I loved cumulative finals as a Grad student for precisely that reason. B average going in, A or A+ after the final, and ultimately able to teach the material.