Inside the inverted proofs class: Why I did it

January 22, 2013, 8:00 am

It’s been a month or so now that the inverted transition-to-proofs class drew to a close. A lot of people, both here at my institution and online, have been asking questions about the design and day-to-day operations of the course, especially if they have ideas of their own and want to compare notes. So starting with this post, I’m going to publish a series of posts that describe exactly how this course was designed and managed throughout the semester. I’m not sure how many of these posts there will be. But the idea is to pull everything together so that people who want to try this sort of thing themselves will have a detailed accounting of what I did, what worked, what didn’t, and how it all went.

Some background on the course (MTH 210: Communicating in Mathematics) is in this post. The short version is that MTH 210 is a course on reading and writing proofs. It’s a cornerstone course in our curriculum as both a prerequisite for many upper-level courses and as a course that satisfies the university’s Supplemental Writing Skills (SWS) requirement. I taught it for the first time during my first semester here in fall 2011, but it’s been around for a long time and has been taught dozens of times by my colleagues in the past.

So why did I decide to mess with it?

Well, as I reported in the post linked above, MTH 210 has a history as a problematic class for a lot of students. Fully 1/3 of the students taking MTH 210 over the last academic year earned a B+ or higher – but the same proportion also earned a D or an F or withdrew. From having taught the class last fall, I had the sense that the reasons for this bimodality came from not so much a lack of mathematical preparation but a lack of cultural preparation. That is, students coming into MTH 210 often did OK in their prerequisite course, which is Calculus 2, but in their calculus courses they got the strong sense that what mathematics is all about is getting the right answer to a computational problem that has only one right answer, a single unambiguous statement, and no more than a couple of ways to legitimately work it. This is not what mathematics is about, it’s not what MTH 210 is about, and the students who ended up in the lower 1/3 are the students who, for whatever reason, don’t acculturate to this way of doing math. Conversely, it quite often happened that students whose grades were so-so in Calculus did well in MTH 210 because they liked it better than rote mechanical derivative or integral exercises.

So after teaching the class once and seeing the historical grade distribution instantiate itself, it seemed to me that if I wanted to do something about this acculturation problem, I needed to do three things that weren’t happening already:

  • Force the issue about students taking initiative over their learning. Where the acculturation issue really comes to a head in MTH 210 is in the changing role of the student versus host of the professor. Students are used to being passive recipients of information, lectured to by their teachers and asked to do things that people who listen to lectures are cognitively ready to do – which is to say, very simple things. In MTH 210, as in mathematics generally, students are required to be active learners: building examples of terms, seeking out the meanings of theorem statements, making conjectures of their own, asking questions and seeking knowledge. The students who end up in the bottom end of the course or withdraw tend to be the ones who hold on to their passive ways even if it kills them. And it usually does in the end – although in a traditional course design, they can hold on for a long time. I wanted to make passivity practically impossible.
  • Provide a robust social network of peers. For many students, MTH 210 is already stressful, even more so if I want to really make students active from day 1. The key in life to not cracking under stress is to have a strong network of people who want you to do well. This happens for some students but not for others. I wanted to change that so when students are asked to go places their mathematical training has never taken them before, they will have live people who have their backs – in class, online, at all times of the day and especially on the formative stages of learning the content and mastering the processes.
  • Be maximally available at the point of greatest need. One of those people in the social network is me, and if I am going to ask a lot of students – and I do – then I have to extend myself proportionally and be there when students need help the most. When is that time? It’s when students are applying basic concepts to actually solve complex problems (= writing proofs). Learning terminology and encountering material for the first time can be difficult, but not nearly as difficult in my experience as applying content to constructing a proof. So does it make sense to focus class time on the tasks that are the least difficult and then outsource the most difficult stuff to the students to work out when I’m not around? Or would it work better to switch this, and focus class time on the stuff where they are most likely to get stuck and need their social network and a professor to help?

From those three realizations it was obvious to me that I needed to apply the inverted classroom model, which I had deployed pretty successfully (but not without student resistance) at my previous school in an introductory MATLAB course. The way MTH 210 was taught already also made this an easy choice. My colleagues already had been teaching the course in a mostly-inverted way, requiring outside readings and preview activities before the class met and then focusing class time on active work. Going to a fully inverted model would just be an evolutionary, rather than revolutionary change.

In the next article in this series, I’ll go into some of the design principles that I kept in mind while getting the course ready. In the meantime, I welcome your comments and questions.


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