Inverting the transition-to-proofs class

July 10, 2012, 11:38 am

When I see the first back-to-school sales, I know it’s time, like it or not, to start prepping classes for the fall. This fall I am teaching two courses: a second-semester discrete math course for computer science majors and then two sections of “Communicating in Mathematics” (MTH 210). I’ve written about MTH 210 before when I taught it last fall. This fall, it’s going to be rather different, because I’m designing my sections as inverted or “flipped” classes.

If you’ve read this blog for any length of time, you know I’ve worked with the inverted classroom before (here, here, here, etc.). But except for a few test cases, I haven’t done anything with this design since coming to GVSU. I decided to take a year off from doing anything inverted last year so I could get to know the students and the courses at GVSU and how everything fits together. But now that I have the lay of the land, I think it’s time to start implementing.

For context: MTH 210 is a transition-to-proof class that, roughly speaking, is supposed to be taken between Calculus 2 and Linear Algebra (although that’s not a rule). It’s required for such upper-level mathematics courses as modern algebra and discrete mathematics, and it also serves as a “Supplemental Writing Skills” course for GVSU students. Here is a list of learning objectives [PDF] that I drafted for the course, and this should give you a sense of what we cover in it. There is a lot of content and a lot of process to learn.

MTH 210 is a big course in the math major here. We usually enroll about 80-100 students per semester in it and we even run a section in the summertime. But it’s also a course with some issues. I know you can’t put too much validity in course grades, but when we faculty received some grading data his past year that for the MTH 210 sections combined throughout the 2010-2011 academic year, some points really jumped out at me:

  • 25% of students enrolled in the course made an A or A- in the course. That’s pretty good, BUT:
  • 13% of students made a grade of F in the course, and 20% made either D or F. And,
  • We don’t know how many of the “A” students were repeating the course. It’s not uncommon to see students take MTH 210 twice or even three or four times before passing. So perhaps that 25% “A” rate isn’t what it looks like.

The concern here isn’t that grades aren’t sufficiently inflated. It’s that a significant portion of students aren’t learning what they need to the first time, and this leads to all kinds of problems — from delays in graduation times to problems downstream in modern algebra and discrete where students sometimes fail to show mastery of the mathematical skills they need for those courses.

So why am I flipping this class? I want to make clear that it’s not because I need to maintain my bona fides as a flipped classroom person or because I feel the need to show how clever I am at course design. I pass on flipped designs more often than not, because many times they are not the best choice for students. But here, with this course, I think the flipped design can work extremely well. Here’s why:

  • You really can’t learn proofs without doing lots of proofs. No more than I can learn how to be a great quarterback by watching Andrew Luck clips on YouTube. You have to get your hands dirty, and the flipped design is all about using time wisely to develop hands-on skills with the guidance of a coach (me).
  • The class is already mostly flipped as it is. We use a book written by my colleague Ted Sundstrom which is one of the few mathematics books I’ve seen that is really designed to be worked on prior to class. Each section is set up with two “Preview Activities” and fortified with “Progress Checks”; it’s historically common for MTH 210 profs to give the Preview Activities as pre-class work and then work on the Progress Checks in class and not lecture so much. My plans just consist of structuring the work students do outside of class a little more than we do already, and creating a bunch of videos to provide additional insight and examples for students as they read.
  • Having curated video resources will help students later on. In bouncing this idea around to my colleagues, one of the biggest benefits they see of creating the screencasts is so students can refer back to them easily in later courses. If you’re taking discrete math and you can’t remember anything about induction, for example, just go dial up the video(s) on that subject and have a quick refresher. For that matter, any person with an internet connection can get such a review.
  • This course is all about becoming independent as a learner. I think the most important thing students pick up in a class like this is the ability to sit down with new technical material and make sense out of it. You exercise that independence in a number of ways — evaluating your writing, instantiating new definitions, seeking out resources to help you understand a concept, and so on. If students can show ongoing evidence that they are independent learners, I consider the class to have been successful.

More on that last point. As much as college students want to be independent, becoming an independent learner is very tough for a lot of them. In fact, I tend to think that the cause of the 13% failure rate has a strong connection to whether students choose to move toward independence or not. Students who are coming out of the calculus sequence (or come in with AP credit) who want to remain in passive-observer mode when learning tend not to do well in MTH 210; students who come into the class with an open mind and a willingness to learn on their own tend to do well. Or so we think; we do not have data on this, which gives another angle on what I’m doing with MTH 210 this fall and I’ll tell you more about that later.

At this point, I have made up a lengthy list of highly-granulated learning objectives for the course that will be served out in small, frequent doses to students as they prepare for class — to let them know what, eventually, they must master. And I have made out a list of screencasts to prepare to augment the book. That list is currently at over 100 distinct videos, each of which will be short (they have to be!). Note that I will not be putting “lectures” in the video because I consider our book to do a good job at conveying basic information. The screencasts are going to be examples upon examples — and unlike some major purveyors of tutorial videos, I intend for those screencasts to be not only mechanical but also to illustrate the concepts and expert-learned decision-making processes that students can then begin to model. I hope to start work on those screencasts next week and finish half of them by the time school starts.

Any thoughts or suggestions here are welcome, and stay tuned for more.


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