This lecture was the second lecture on network flow. As anticipated, I got through about half the material in the first lecture on network flow. I had made the decision in the previous lecture to not rush the material, and if necessary, have a second student submission lecture. This lecture continued along the same line as the previous ones - hoping to have the student's understand the general concepts by working with specific examples. Network flow seems very well suited for this type of exploration.
Three activities were covered in the lecture - with one additional one in the slide deck, which was not presented.
A short introduction was given in the previous lecture - but this lecture was designed as a self contained introduction. The goal in the lecture was to give students a chance to work with flow graphs to make the concepts more complete. The lecture was designed with five activities. My expectation, before the lecture, was that about 90 minutes of material was prepared. The plan going into the lecture was to give the material at a standard pace, and then continue the lecture during the next lecture session, again with student submissions. (The once a week Tablet model has led me to rush through submissions - since if they were not done in class, they would be discarded.)
An initial analysis of student submissions was started before class took place!
Activity Goals: This activity was to identify the two properties we would need to establish for the augmenting path lemma to be true. The goals of the activity were to get people to think about the proof, and also to reinforce the basic flow properties.
Planned Use: Discuss the two general properties, based on student examples.
Actual use: The went pretty much as planned - solutions had the main point, so they allowed me to say these are the two main things we would show. I did not attempt to give the formal proofs - I just mentioned what we would need to prove.
Evaluation of activity: This was okay for an activity around a lemma - although a little bit forced. It's hard to involve students in proors (which may be an argument for this type of activity - even if it doesn't seem as beneficial as working through an example.
Activity Goals: Determine if students understood the cut capacity and cut flow. The example was chosen so that the cut was was slightly complicated and had edges going in both directions.
Planned Use: Show a correct solution and discuss misconceptions if they arose. Boxes for numerical entries were included to make it easier to assess the answers
Actual use: There was a moderate variety in the answers. For the cut capacity, I do not know how much of the variance was due to miscounting - some of the edges were a little tricky to figure out. The example did set the stage for making the point that the flow across a cut is the same for all cuts.
Evaluation of activity: THe activity was complicated enough that it was somewhat challenging, although I was concerned that the different answers arose from clerical mistakes. This example was rich enough to raise a number of issues. It was difficult to pick out good examples to talk about. I liked one exaple which showed a computation to get the flow, which was nice.
Activity Goals: Reinforce the concept of the minimum cut by having students look at a non-trivial example. This might also lead into the maxflow-min cut theorem because showing that there is a flow of the right size shows the cut is minimum
Planned Use: Possibly show several examples, if there are different examples, starting with incorrect examples. A space for the size of the cut was added to the slide to facilitate this.
Actual use: I walked around class as answers were coming in, and encouraged students to start working together. From reviewing the numbers, I was seeing about half the students submitting the right answer. This was also observed as I walked around the class. I told students the correct numerical answer before I went back to the slide display. My discussion of this activity was primarily on using the the next slide in the deck, as opposed to writing on one of the student examples.
Evaluation of activity: This activity was very successful - it is one of the first activities which seemed to achieve "peer instruction". After students worked on the example for a while, they were encouraged to discuss answers with their neighbors, and try to converge on an answer. This example seemed to hit the sweet spot on difficulty, where there was a roughly even split between 17 (the right answer) and 18. The discussion led to a number of related points such as how would you show that a cut is minimum, and how would you find a minimum cut from a maximum flow. Quite a bit of time was spent on the example, but it was time well spent. The activity design (including an extra slide, and having an answer box) were very helpful.
Activity Goals: This activity is to involve students in the proof by having them show that the cut is saturated when the residual graph is cut. The main point is to engage students in the proof, and assess whether or not they have understoon the basic idea of the proof
Planned Use: Show a correct answer to continue the proof
Actual use: This was not reached for lack of time. I will just use this slide in a standard lecture without the submission.
Evaluation of activity: Involving students in proofs is always hard - so this activity could have gone either way. I think it would have been early enough in the proof that quite a few students would have understood the point of it.
