I found this a challenging lecture to include activities in, although I was happy with the approach that was developed. For recurrences, I wanted students to come away with the understanding that the behavior of the recurrence is a geometric sum, and the geometric sums are dominated by dominated by the base. Qualitatively, all that matters is whether the base is less than one, equal to one, or greater than one. The design process of the lecture around activities, led me to emphasize this particular aspect of the topic, as opposed to mastering computational skills in working with recurrences.
The second part of the lecture was to talk about divide and conquer algorithms by showing particularly compelling examples. I ordered the examples to put the closest pair example before counting inversions (which will mean that the inversion example is skipped entirely). The closest pair example is a particularly elegant algorithm - but there was a challenge in designing activities around the exposition.
Four student activities were included in the lecture (one more was planned, but not reached for lack of time). Roughly twenty students were present for the lecture, along with two observers. All participants used HP TC1100 Tablet PCs on an Ad Hoc network. There were no observed technology problems.
Activity Goals: I wanted to put a strong emphasis on the geometric series - so I wanted students to recall what it was. I also wanted to assess whether or not it was a fact that they had ready access to (and make the point that it is one of a small number of formulae that they should know). The second part of the activity was to have them state the result that they should remember - the qualitative behavior of the series.
Planned Use: I planned to select correct examples to support the discussion. I was not planning to show incorrect answers - I just wanted to the result in someone's handwriting
Actual use: I showed a partially correct result, and then switched to the whiteboard to give, and derive the result.
Evaluation of activity: Students had more difficulty recalling this result than I had expected - the geometric series is not a result that many remembered. It was still valuable to have students work on it, since they did think about it. THe second half of the exercise was a failure - students did not understand what I was looking for. Maybe a more explicit: express the behavior using big Oh notation as a function of n would have been better. Several of the solutions involved absolute value - which suggested students were thinking of this using their calculus context. Even though the student results were not good - this was an informative activity that led to a good discussion.
Activity Goals: The goal of the assessment was to both evaluate if students understood how to unroll a recurrence, and how to solve the recurrence
Planned Use:Show a correct solution - unrolling along with the result
Actual use: The results showed that students understood the unrolling process, but not solving the recurrence - very few successful solutions were provided. I showed quite a few solutions quickly, to demonstrate that everybody was comfortable with the unrolling process - and then I presented the correct answer. This was done on a whiteboard slide.
Evaluation of activity: The activity worked well - there was a high partipation rate, and it gave a fairly clear result on what students understood. The format of the recurrence: T(n) = n + 5T(n/2) helped make it easy to unroll and collect terms. A post lecure review of the submissions showed that students had a better understanding of the recurrence than I thought they had in class.
Activity Goals: THis activity was to assess whether or not students had a qualitative understanding of recurrences.
Planned Use: The plan was to show a correct solution. If serious misconceptions were detected, I would go back over some material.
Actual use: I was running short on time, so I did not cover what happens when the work term is non-linear in much depth. I dropped the last two questions - indicating on the slide that students didn't need to do them. The results from the students showed that everyone understood the general point I was making. I displayed one or two solutions, but little discussion was necessary.
Evaluation of activity: This was very successful - it gave immediate feedback that the broad goal of the lecture was achieved. The answers were easy to assess, and the activity did not take much time. I was surprised at the number of students that did the unassigned portion of the activity.
Activity Goals: The purpose of the activity was to have students think about the "Packing principle" - you can't back too many things into a fixed area. This was setting the stage for a specific packing lemma that applied to the algorithm.
Planned Use: Show a range of solutions to support discussion.
Actual use: The answers were varied, which allowed me to show a range of different answers. The specific answer wasn't that important - it was more the general concept of packing that I wanted students to think about.
Evaluation of activity: The activity worked very well - it got students to think about one of the underlying concepts for the algorithm, without dealing with some of the technicalities.
Activity Goals: This activity was to be used to determine if students understood the algorithm by having them show where the comparisons took place.
Planned Use: The plan was to select a correct solution and talk about it. The students would need to guess at the value of delta coming from the recursive call - so there would not be a specific correct solution.
Actual use: I ran out of time during the lecture, so this activity was dropped.
Evaluation of activity: I am curious how well this would work - I have been trying to come up with assessments to gauge student understanding of algorithms - the idea is to have them show some specific action of the algorithm on data.
