Calculus III Mathematica Labs!
(compiled by Thomas Hull)

Table of Contents


In the Spring 2003 Quarter, while visiting the University of Cincinnati, I was assigned to teach 5 Calculus III lab sections. These were 1 credit courses that met for 50 minutes each week. They were co-requisites of their 4 credit Calculus III course, which covers Sequences and Series, Vector Geometry, Vector Functions, and a little bit of Partial Derivatives. (It's only a 10-week class.) The math department at UC had been running these labs (which use Mathematica in a very nice, 40-computer lab with projector, white board, printer, and plenty of space to walk around and see what people are doing) for the past three years. However, the department had no consistent policy as to how the labs should be run. After polling some of the faculty with lab experience, I learned that most had some kind of lecture in the lab, then a quiz (which the students did at their computers), then assigned homework, on which the next week's quiz would be based. I didn't really like this approach, and a number of students had previously complained to me about these labs (the main complaints were that it was a lot of work, they didn't feel they were learning a lot, and the labs weren't tying into the regular Calculus class very much).

Not knowing what I should do, I asked the ProjectNExT "older dot" list for any suggestions for lab ideas or where to find such ideas on the web. This web page is my attempt to present the resources I collected, as well as make the labs I ended up creating available to others for use.

Calc Lab Resources on the Web:

Note: I was only looking for labs for material that straddled Calc II/III. But most of the below links contain labs for all levels of calculus.

Non-Web Calc Lab Resources

My search turned up a few non-web resources for calculus labs. Now, I'm sure there are lots of books and such that I could have found. My intention here is NOT to compile a bibliography of calc lab books. No no no. I just wanted to share and give credit to the two texts that helped me quite a bit.

How I Set Up My Labs:

This quarter (Spring 2003) my teaching load was one 4-credit Honors Calculus section and five 1-credit Calculus Labs (one of which was for my honors class). During the previous two quarters I had used Blackboard to post documents and grades for my classes, so I wanted to try using Blackboard in the labs to make them "paperless labs." This started off kind of rocky. I had a grader (a graduate TA) for the labs, and together we figured out how to get the students to upload their completed labs onto Blackboard, how we could then download them, grade them, record the grades, and then upload them back to Blackboard to send back to the students. I think the students really appreciated this, since the printer in the lab can get pretty slow, and this avoided the bottleneck at the printer at the end of each lab. Plus, I had over 140 students in my labs, combined, so I think I must have saved a dozen reams of paper.

Each week the students would come into the lab classroom, download the lab from Blackboard, and get to work. I had them work in pairs, and this was an invaluable idea. (That is, they sat next to someone they wanted to work with (didn't have to be the same person each time), they both did the lab, but each pair submitted ONE lab for their grade.) I told them they should do this because (1) it makes the labs easier and faster to do, (2) I'm convinced that when people work together, they learn more, and (3) Mathematica sometimes crashes, so it's a big lifesaver if you have two people doing the lab at the same time. That way if one person's worksheet freezes the group can continue with the other person's work. In some of the labs that had lots of intense graphics rendering (like the Surfing Surfaces lab) this happened a bunch. It also meant less work for the grader, which was important because I didn't have time to grade too many labs, and the TA was only supposed to work 20 hours per week. (Which is NOT enough time to grade 140 labs...but 70, yes.) As the quarter rolled along, I noticed that some students in certain sections were opting to skip the lab and do it, by themselves, on their own time. Most of the students who chose this route did much worse than those who came to the lab (where they could ask me questions) and worked with a partner. At a big place like UC, I felt it was OK to let those who insisted on working alone continue to do so. (But when their grades slipped I tried to suggest that they find a buddy to work with.) I just wanted the majority to be working in pairs, which they did. If I were doing this again at a smaller school (like my home institution Western New England College), I'd penalize them for not working with a partner.

It was a 10-week class, so there were 10 labs. I tried to write them so that a pair of students who were reasonably proficient with Mathematica (which they all were supposed to be, since they had the "Intro to Mathematica" lab the previous quarter) could finish it during the 50 minute class. Sometimes I misjudged and made the lab too long. And other times I made it shorter on purpose, just to give them (and me and the grader) a break. But the general balance of difficulty versus the length of the class period seemed to be good.

As a result (and because they were working in pairs), there was virtually no cheating. I was worried about this, because other professors at UC who had taught the labs before warned me about the tendency of the students to try to cheat on the labs. I did warn them sternly at the beginning not to cheat, but I'm convinced that most students won't cheat if they feel the workload is reasonable and especially if I let them work in groups to begin with. (There was only one instance of cheating that I noticed: a student who missed a bunch of the labs submitted them late during the 2nd-to-last week of the course. (I allowed late labs, but I deducted points for being late.) One of his labs was copied from someone else's graded lab. I could tell because he forgot to delete some of the grader's comments. I killed him on that lab, but he was failing the lab anyway without any push from me.)

The Labs I Developed

Below are links to the Mathematica notebooks (labs) that I developed for the UC Calculus III course, Spring Quarter, 2003. There are two sets of labs: One for the regular Calculus III sections, and another for an honors section of the same class that I also taught. (The honors course went faster, and some of the labs are more extensive or rely on the students figuring more things out on their own.) I've also provided commentary for each of the labs.

Calculus III Labs:

Honors Calculus III Labs:

Tom's comments on this lab experience:

Grade-wise, in the regular lab sections most of the students received As and Bs. There were a smattering of Cs, Ds, and Fs, but there was almost a direct correlation between these students and those that chose to either work on the labs by themselves (without a partner) or outside of the lab class time. In the honors class virtually all the students got As. I was surprised by this, since their labs were harder and several times they were frustrated when I'd take off points for poor explanations. But the numbers ended up being stellar. There's a good chance that the real reason why they all did so well had to do with their willingness to meet my expectations, coupled with the fact that my expectations weren't impossible to meet. Their labs were harder and usually took longer to complete than the regular class'. But the honors kids were all willing to put in time outside of the lab class if the work demanded it. The other sections seemed much less likely to do this.

On the whole, I was very happy with how the lab classes turned out. As I watched the students working on them, it seemed like there was a lot of learning going on. Everyone was getting lots of hands-on practice with Mathematica. The conversations I'd overhear led me to believe that the labs were succeeding in reinforcing and building upon the material they were seeing in class. As far as I'm concerned, that's the whole point of having computer labs.

Since the labs were completely "project driven" (that is, each lab was a "project" they had to do, and their whole grade was based on this), I spent each class roaming the lab room, being flagged down by students, and being asked questions. I'd try to always answer their question with another question (in the whole Socratic method, make-them-teach-themselves manner), but half the time their questions had to do with Mathematica acting (to them) weird. Then I'd either have to remind them to clear their variables, or debug their code ("No, x times y has to be x*y or x <space> y, NOT xy!"), or sometimes restart their kernel. Being relatively well-versed in Mathematica was a must here, so that my time (and their time) was never sucked up too much by these little emergencies. (Someone ought to write a book: A Guide to Everything That Can Go Wrong and How Students Will Make Mistakes in Calculus Mathematica Labs.)


Feel free to email me if you have any comments or questions about these labs!

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