Discrete Math - Reflection

Another semester of discrete math has ended. The final exam results were good and tracked how I expected things to go. The mean score out of 100 was 78 with a standard deviation of 10.1.

In problems attempted, the C-level questions had consistent numbers of attempts and showed no statistically significant variation (p=0.154). The same was the case for B-level problems (p=0.265). There were not enough attempts made on the A-level problems to conduct a statistical test.

In examining the attempts versus the number of successful completions and partial completions, I identified three questions that should be reworded or replaced.

In talking with students as they left or encountering them later in the hall, the students said they found the class challenging but interesting. They liked the change of pace from regular algebra and that they were allowed to explore and work through problems. This is consistent feedback with the first time I taught this course.

As for changes to next year:

1. I plan on eliminating Bayes Theorem from the probability section and to just spend a little time connecting counting outcomes and sample spaces to probability.
2. I plan to move the graph theory unit to follow the counting unit, enabling more connections to counting and developing proof.
3. Moving the graph theory unit will allow me to include an additional two weeks (approximately) to other graph theory ideas, such as Hamiltonian circuits, graph coloring, and spanning trees. I want to look through these topics to see which provide interesting math that is also satisfactory. If students investigate a problem to find there is no solution or resolution, I want to be sure that a at least it is an interesting investigation or a major topic of mathematical research.
4. I plan on removing the algorithm for solving linear congruences. As long as students understand that the slope for the solution set is the LCM of the divisors, that this value is an upper-bound for the initial value, and they have a strategy or process for finding the initial value it should suffice.
5. I will look at the number theory pieces to see if I can tighten things up just a bit more. 

My plan is to teach the course as is one more time and then look at building in new sections. I have a one semester class, but my goal is to develop a full-year of material. The counting unit and material I have for number theory and cryptography could easily encompass one semester. I've been revising these two units so that I can get a balance with the third unit in graph theory.

Pieces that I will be looking to add-in are set theory and Boolean algebra, logic and proofs, and excursions. The excursion section touches on game theory/decision theory, division and apportionment, and matrix representation and use. I envision that these could fill a second semester of work.

Ideally, I will have core material for each topic and then an expanded set of material. This way, if someone wants to delve deeper into, say, number theory, they can do this and then elect not to cover another topic, such as division and apportionment.

One of the issues I have with current high school textbooks that cover discrete math is that the coverage of division and apportionment tends to be extensive but the math itself is unsatisfying in that there are no optimal ways to apportion votes or divide fairly. As such, the math does not lead to insights or provide a base upon which to build. I'd like to try to avoid those pitfalls if I can.

I would love to hear what others are doing at the high school level with any of these topics. Leave a comment and we can get a discussion going.