It’s been a while since I last wrote about the recently-completed inverted transition-to-proof course. In the last post, I wrote about some of the instructional design challenges inherent in that course. Here I want to write about the design itself and how I tried to address those challenges.

To review, the challenges in designing this course include:

- An incredibly diverse set of instructional objectives, including mastery of a wide variety new mathematical content, improvement in student writing skills, and metacognitive objectives for success in subsequent proof-based courses.
- The cultural shock encountered by many students when moving from a procedure-oriented approach to mathematics (Calculus) to a conceptual approach (proofs).
- The need for strong mathematical rigor, so as to prepare students well for 300-level proof based courses, balanced with a concern for student morale and emotional well-being in the process.
- The need to satisfy the university’s writing requirements, particularly in the form of the Proof Portfolio.

These are a lot of plates to keep spinning. Not all of these challenges are solvable by instructional design, either. But a sound design for the course, a good structure underneath it all, always makes other things easier to deal with.

First of all, it’s not a given that this course *has* to be done in an inverted format. But most of my colleagues who have taught the course before (including myself in Fall 2011) used a “nearly-inverted” structure in the course. Prior to a class meeting, students would be assigned readings from the textbook (written by my colleague Ted Sundstrom, who has been sort of the architect of this course for a long time) and asked to work 1–2 “Preview Activities” from the book in advance of the course. Then, course meetings would usually be organized mostly around active student work. What prevents this setup from being the canonical “flipped” classroom is that usually, students would still have take-home homework after class is over, and there was usually some amount of lecturing going on during the class meetings. This is, at least, how I taught the class the first time back in Fall 2011.

The first design decision I made when looking at this class for the second time was to make it fully inverted. I’ve already written about why I decided on a fully-inverted structure. In addition to the more philosophical reasons given in that article, inverting this class accomplished several practical objectives. First, by having students make first contact with the material before class, it made the class meetings more focused on the material that made the least amount of sense — which in turn helped me to not be as rushed through the whole body of content to cover. Second, by focusing class time on sense-making activities done in groups, we were able to work on building those peer-to-peer social networks that help students learn from each other and reduce the amount of stress in the class. These hit at least two of the design challenges I mentioned earlier — handling the diversity of material and dealing with cultural shock and personal stress in transitioning to doing proofs.

Third, and most practical of all, the work we did in class took the place of take-home homework — we called it “Classwork” instead of “Homework” to emphasize the point — and by doing most of it in class, it freed up time outside of class for students to focus on preparing for class really well and, importantly, to focus on their Proof Portfolios. In fact I made the point many times to students that what I wanted them to focus on outside of class was reading the book and watching the videos, and working on the Portfolio — and that’s it. In the previous version of the class, students had reading *and* homework *and* the portfolio to manage outside of class. By inverting, we were replacing homework with a greater emphasis on rigorous preparation — and it paid off handsomely in the form of better-prepared students and better portfolio writing.

This is an aspect of the inverted classroom I don’t think we stress enough: It helps students manage their time and tasks better. Instead of homework whose difficulty is often hard to gauge, let them do something simple — watching videos and doing basic exercises. Students are better able to manage this kind of cognitive load. How many of us have complained that students don’t have good time management skills? And yet we set up course structures that force students to deal with homework loads and time management issues that would be tough even for a seasoned academician to handle. This isn’t a skill anybody is born with — they have to be taught. *By us*.

Anyway, back to the course. So I designed the course around the idea that before class, students would have first contact with the material, then come to class and work in groups on challenging problems that required them to make sense of that material. And they’d be working on the Portfolio outside of class. To keep students honest, I reserved 5 minutes at the beginning of each class for a three-question multiple choice quiz over the reading. The first thing we’d do in class was take the quiz using clickers, so students would know their results right away, and I’d have data to work with in case we needed to discuss something before class started. We always went over the quiz right after taking it and folded in any recurring questions from the Guided Practice. This Q&A time was the closest thing to direct instruction we saw in the class, and it was effective — because it was targeted at specifically the one or two items that needed to be discussed. Having the Q&A time also reassured students that they weren’t being expected to learn all the material on their own — that questions were OK to ask.

I also felt it was important to make sure students weren’t skimming through the class on the strength of their group-mates, so I put in two hour-long exams and a final in the class over lower-level ideas (definitions, basic mechanics like writing down contrapositives, etc.). There would also be some proofs on the exams that were adapted, sometimes verbatim, from the Classwork.

The grade breakdown for the course I finally settled on was:

- Preparation, 5% as measured by Guided Practice exercises. I will have a lot to say about Guided Practice in the next post — it’s what made the whole course work, IMO.
- Quizzes, 5%
- Classwork, 20%
- Proof Portfolio, 30%. This is actually a bit lower than what the university mandates — the writing requirement says that one-third of the semester grade should come from writing assignments that involve drafts and revisions — but we also had writing assignments elsewhere.
- Midterm Exams, 2 at 10% each
- Final Exam, 20%.

There are two items for which I owe you an explanation. One is the details about Guided Practice, which is how I got students to do the work prior to class. That’s in the next post. The other is the details about Classwork. This is considerably messier, since the original way I had Classwork set up to work didn’t work out well, and we had to experiment with different configurations before we finally found something we were all happy with. That’s coming later too.

In the meanwhile — any questions or comments on this?