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Inside the inverted proofs class: What we did in class

March 18, 2013, 8:00 am

3095855157_7bf2df04fa_mI’ve written about the instructional design behind the inverted transition-to-proofs course and the importance of Guided Practice in helping students get the most out of their preparation. Now it comes time to discuss what we actually did in class, having freed up all that time by having reading and viewing done outside of class. I wrote a blog post in the middle of the course describing this to some degree, but looking back on the semester gives a slightly different picture.

As I wrote before, each 50-minute class meeting was split up into a 5-minute clicker quiz over the reading and the viewing followed by a Q&A session over whatever we needed to talk about. The material for the Q&A was a combination of student questions from the Guided Practice, trends of misconceptions that I noticed in the Guided Practice responses (whether or not students brought them up), quiz questions with a low success rate, and on-the-spot student questions if they had any. Usually we’d be done with this by 15 minutes into the class, leaving us 35+ minutes to work on Classwork.

Classwork from the inverted proofs class was largely the same thing as Homework in the non-inverted version of the class. It’s just that one was done in class and the other wasn’t. Instead of having weekly homework sets with 5–7 problems in each, we did daily classwork sets with 1–3 problems each, three times a week. And many of the problems I gave for Classwork were raided from last year’s Homework archives for the non-inverted version of the class. So there was really not much of a substantive difference in the kinds of work I asked students to do. Only the context was changed.

For the most part, this work consisted of proof writing. In a different class, this would not be the case, but here, I needed students working on their writing and reasoning skills constantly. Here is a typical Classwork assignment from about halfway through the course:

  • Let \(a\) be an integer and let \(n \in \mathbb{N}\). Prove that \(a \equiv 0 \, (\text{mod}\, n)\) if and only if \(n | a\).
  • Prove that for every integer \(a\), if \(a \equiv 3 \, (\text{mod} \, 8)\), then \(a^2 \equiv 1 (\text{mod} \, 8)\). Is the converse of this statement also true?

These are pretty basic exercises that involve taking the basic terminology and mechanics and doing something not-exactly-mechanical with them. Remember the students also were working outside of class on a Proof Portfolio worth 30% of their grade, so the Classwork was crucial in building up the skills they needed to work on the portfolio problems, which were legitimately hard. I saw the students’ work as a progression:

Introductory material —> Procedural understanding —> Done individually through Guided Practice

Intermediate skills —> Writing simple proofs —> Done collaboratively through Classwork

Advanced skills —> Coming up with complex arguments and proofs —> Done individually with instructor guidance through drafts and revisions in the Proof Portfolio

Getting this to work smoothly in the class was a messy and imperfect undertaking. One of the basic problems is that you have students with wildly different levels of facility with this material. There were some groups that could finish the assignment I put above in 15 minutes, leaving them with nothing to do for the entire second half of the meeting. Other groups struggled to know how to proceed at all with any of these proofs, sometimes because they didn’t prepare and sometimes because they did the readings and viewing but couldn’t transfer the knowledge to a new situation — and consequently there was no way they were getting done with this in class.

At first my policy was that each group was expected to hand in their solutions by the end of class — and if there were widespread issues with getting the work done, I could grant an extension on the spot if I saw things getting out of hand. This didn’t work because appeals to authority — “You MUST get your work done in 35 minutes or else!” — didn’t help students work better or faster. It just made the lower-performing groups give up faster. Also, for students struggling to learn proof, 35 minutes is not a lot of time, even assuming we had that much time (what if the Q&A session went longer than 15 minutes?). If one of the design challenges to the course was the stress level among students, this was not solving the problem.

So then I moved to this policy: You are encouraged to turn in a clean copy of your work by the end of class. If your group finishes all the problems, then you may hand in a single group writeup. If your group does not finish all the problems, each member of the group is responsible for completing the work individually prior to the beginning of next class. In other words, if your group doesn’t finish, the Classwork reverts to traditional Homework and it’s due next time. I liked this approach because it incentivized groups having their acts together and getting the work done, but it didn’t penalize groups for not completing — it only passed the responsibility on to the individual.

In practice, this didn’t go so well, and it was my fault. I’d have groups get most of the way through their work in class but not finish, then get confused as to whether they should hand in only the work that didn’t get completed in class, or the entire set — and some students who contributed to the group didn’t write up the stuff that got completed in groups. It was confusing and frustrating for all involved. Even when it worked sort of correctly, it added hugely to my grading load which was already sagging under the weight of twice-weekly Proof Portfolio submissions from 60 students.

The third approach to Classwork was the one that finally stuck. I was smart enough at the beginning of the semester to build in some “TBA” days in the course schedule, in case we fell behind or needed some extra time on a topic. By using those and by editing the schedule a little bit, I was able to free up an entire day about once out of every 5–6 class meetings during the last half of the semester. The policy became: Your group should try to finish up the Classwork during class. If you don’t, then we will use these “free” days as makeup days, where your group will hand in any outstanding work by the end of that class meeting. People using standards-based grading do stuff like this, and I think that’s where I got the idea. Groups were free to work outside of class in between meetings if they wanted, and some groups did this and came into the free days with no outstanding work to do — but this was sort of uncommon. This policy also, I’ll admit, gave me permission to give harder problems and more of them for Classwork (within reason).

In theory I like the second approach better than the third, but in practice the third approach worked best.

What about the groups that had nothing to do? Well, there was always something to do. I would sometimes give extra problems for bonus credit. For some groups who were just naturally curious, I could have them think about an extension to a problem and work on it for fun — and they would. For others, I’d let them use the time to work on their Proof Portfolios. Most of the time, though, if a group got done conspicuously early, I’d first ask each group member to explain in their own words the solutions that their group gave. And then if I was satisfied with each person’s answer, I’d make up something on the spot for them to work on. I got surprisingly little grief from students about this — but then again it didn’t happen very often either.

You’re probably wondering about grading at this point. I’ll get to that in the next post.

Image: http://www.flickr.com/photos/marcwathieu/

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