Small proof of concept for inverted math classrooms

Geometric interpretation of the solution of a ...

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The very last topic in the linear algebra class this semester (just concluded) unexpectedly gave me a chance to test-drive the inverted classroom model in a mathematics course, with pretty interesting results. The topic was least squares solutions and applications to linear models. I like to introduce this topic without lecture, since it’s really just an application of what they’ve learned about inner products and orthogonality. Two days are set aside for this topic. In the first day, I gave this group activity:

The intent was to get this activity done in about 35 minutes and then talk about the normal equations — a much faster way of finding the least-squares solution than what this activity entails — afterwards. Then I meant to spend a second day on practice.

But, as they say, life got in the way — and we ended up spending the entire time just getting through the first 3/4 of the activity. We spent most of the second day debriefing the math in the activity, reducing my lecture on the normal equations to a 10-minute time slot… which I then screwed up hugely by making an intractable math error which I didn’t catch until time was up. To make things worse, this was the last day I had budgeted for covering material. So this was going to be on the final, but I had no time in which to teach it right.

Solution: Make a video containing the correct form of the lecture I was going to give, and a couple minutes shorter to boot:

…make another video with a more intricate example:

…and then give students some exercises to do and open the email/Skype/IM/office hours line for questions.

By the night before the final exam, the videos had about 15 hits each, some of which were from IP addresses my students would have. So they were being watched. On the final, I gave a mid-level exercise from the section on linear models in their textbook — an even-numbered one, so it was fairly unlikely they worked it.

The results? Out of 14 students taking the exam:

  • Two students had basically no progress on the problem — left it blank or just wrote down the problem.
  • Five students did NOT use the normal equations approach from the videos but the projection-oriented approach from the group work. Of those five, only two used the method correctly.
  • Seven students DID use the normal equations approach from the videos. Of those seven, six used the method correctly.

I’d like to conclude that the videos were more effective in conveying the subject matter than the group work was, and took less time. But of course there are confounding variables. The videos covered material that is less complicated than the group work covered. Perhaps only the most motivated students actually watched the videos, and their motivation rather than the video is what helped them to learn. But still, I think the inverted approach here had some kind of effect. Certainly the students who watched the videos were mostly (6 out of 7) able to pull off the method on the final, whereas less than half the students who relied upon what we did in class could do so with that other method.

Ideally, I’d put these videos out there, assign them along with some basic computation exercises, and then come in and do a group work assignment similar to what I used. With these kinds of results, I’m a little more emboldened to do so with some classes this fall. But more on that later.

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