Week 1 of the 6-week Calculus 2 course is over, and of course it felt like 2.5 weeks of class because that’s the exchange rate between this course and a normal 14-week course. It was challenging for the students, but I feel like they are on board with what we’re doing. I was especially pleased with the outcome of one of the distinctives of this class: the in-class assessments which are called, er, Assessments.

I said at the outset that the key thing with this class was to force the issue on assimilation of material, and part of that was to engage in **early, small, and frequent assessment**. For formative assessment, we do daily online homework and clicker questions. There’s no requirement to get clicker questions right at all, and WeBWorK sets have no limits on number of attempts or the amount of collaboration or technology used. For summative assessment, we have a midterm exam and a final exam. But when designing the course I felt like we needed something in between — something that goes a little deeper than WeBWorK and clicker questions but not as high-stakes as exams — and that’s where Assessments came in. Rather than give three tests, which is what I usually do, I made Assessments basically half the length of a traditional test and twice as frequent. We have six of these — one every Wednesday morning. Here’s the first one, with solutions (PDF).

What makes Assessments different than the usual tests, other than their length, is that there is a collaborative portion to each one. Each is split up into multiple choice, mechanical calculations, and problems to solve. Assessments are 50 points with about 10, 20, and 20 points respectively assigned to those three areas. During the first 10 minutes of the Assessment, students are randomly assigned to groups of 3 or 4 students and given the “problems to solve” portion on a separate page. Then they brainstorm and try out solutions together for the duration of that 10-minute period. They may use whatever technology and notes they want. Then, when the 10 minutes are up, everything gets put away — including the notes they took during the group portion — and they get the entire assessment to work out individually for 25 minutes. After the individual portion is done, the work is handed in and we take 10-15 minutes for discussion and debrief (so they get some feedback immediately).

The idea here is that students get a chance to collaborate and share their ideas, but with limitations, and in order to really get anything useful out of that collaborative period, they have to participate actively and understand what’s going on. Otherwise, unless a student has a photographic memory, the student is not going to remember enough to give a fully fleshed-out solution in his or her own words on the individual portion. Students are graded on their understanding of the work they’ve contributed to, as evidenced by the quality of their individually-written solutions.

So we did the first Assessment this week, and I thought it went very well. The group portion was extremely loud and lively, with a lot of good mathematical work going on. I can vouch for the class that nobody was freeloading or just sitting back and consuming without producing. You really can’t do that, and expect to do well.

There were a few things that need to be changed. First, 25 minutes wasn’t enough for the individual portion of the exam. I gave them an extra 5 minutes and that was about right. Also, it took a lot of time to arrange 27 students into groups of 3-4 and seat them, so I’m going to arrange all of that in advance for the next assessment. With the extra time not taken up by getting seated, students said they could use a 5-minute Q&A session before the test, and that seems like a reasonable request to me.

The average on this Assessment was 84%, which I thought was decent. The median score on the multiple choice portion was 8 out of 10, which is pretty impressive, since I didn’t think the multiple choice questions were easy by any means. Perhaps not surprisingly, the median score on the problems to solve was around 17 out of 20 — more on that in a minute. What brought the scores down were the mechanical exercises. Students seemed to struggle particularly with problem 6 because they wanted to do the integral with antiderivatives, and the antiderivative of \( \sqrt{9 – x^2} \) is not something they know how to do — but a lot of them *thought* they knew. This problem was also what caused a lot of student to need extra time. It’s no surprise to me that an over-reliance on algebraic techniques is the root (pardon the pun here) of most of the issues.

With the scores on the problems-to-solve portion being so high, it’s fair to ask whether the collaboration was really leading to an improved individual understanding of the calculus, or whether students were just copying down what their peers did and inflating their grades. I think it’s the former. There were one or two instances where, on problem 9 (the one with the bicycle), students just wrote down the area values under the curve and did not show work — and those students lost almost all credit. That’s as it should be; if students just recall what their group partners got for those area values without reconstructing that work themselves, they shouldn’t get credit for it — and they didn’t. Students really have to understand what they do in groups well enough to write up a solution to the problem on their own, and there’s enough of a solution there that memorizing the solution would be pretty hard.

Still, though, I think a way to improve the Assessments would be to introduce a new part to each problem to solve on the individual portion, to see if students can individually perform some novel task with the information from the group portion. For example, on the last problem, I could have asked students on the individual portion to determine at what time the bicyclist made it back to her house. The solution to that part is similar to the solution to the final part, so it would test their abilities to map information onto new situations. I will probably try that this week.

This approach to assessment works well with the way I want the class to go. They provide a nice bridge between the daily work and the exams, and they extend the basic value of collaboration into the assessment process. An agreeable side effect is that students are getting to know each other better; I noticed that nobody wanted to go back to their original seats once the group portion of the Assessment was over. With some tweaks, I think this could be a permanent feature of the way I do classes, even in a 14-week semester.