NCTM Conference Days 2 and 3

Wow, what a couple of days. Day 2 was a lomg day. It started off with my presentation on discrete math at 8:00 a.m. I wasn't sure how many people would atend and was pleased to see so many people interested in discrete math. We worked through a few problems in number theory with direct connections to cryptography. I really had a fun time and greatly appreciated the enthusiastic participation of those in attendance.

After, I was able to relax and enjoy numerous sessions as a spectator. I focused on  seessions dealing with proof or statistics. The proof sessions tended to focus on justification more then proof; valuable but not a more structured, formal argument that I was looking for.

The statistics sessions were fantastic! I picked up some great investigation activities, some reference books to check out, and ideas on using software and presentations to capture student interest. There was an engaging activity that involved three strands of string, tying knots,looking at the probabilities of outcomes, and then conducting an inference on the results. What I like about this particular activity is that I could use it in my Discrete Math class while we are studying Combinatorics and Discrete Probability. For my Inferential Probability and Statistics class, I can use it have students run simulations to examine expected outcomes and make inferences, and in my AP Stat class, we can work through the probabilities as tree diagrams, look at conditional probabilities, and conduct a hypothesis test. Brilliant investigation.

In the evening there was an AP Statistics panel discussion. Three table leaders discussed scoring and responses from the 2012 exam. The panel reinforced the messages that I have been conveying to my class: determine the correct procedure, check your assumptions and conditions for that procedure, produce understandable work and calculations correctly, and draw a conclusion linked directly to the work that you produced. It was good to hear that I have been emphasizing things that I should; I still wonder how effectively I am sending the message and how well my class is absorbing the message. I'll know soon enough.

So this day was a very long day. I left my house at 6:45 a.m. and did not get home until 9:15 p.m. It was well worth the time but I was tired!

My tiredness showed the next morning. I just didn't want to get out of bed but there was a session I wanted to attend on cryptography that started first thing in the morning. I managed to get going but did arrive to the talk about 10 minutes late. Thankfully I had only missed the introductory piece that gave some background on the origins of what we were going to see. The talk focused on using Excel to create formulas to produce encryption and decryption. I am not sure I would use this in my discrete math class, but it is good to have it in the back of my mind as a possibility. The other piece I thought about was the number correspondence to letters of the alphabet. In the spreadsheet, a value of zero corresponded to a "space". This necessitated the use of mod 27 rather mod 26, which is what the Caesar Cipher activities from Shodor's Interactivate site. By the way, if you haven't been to this site, I recommend you look through the vast amount of activities and applets they provide for all math levels.

I saw the advantage of doing this since students are naturally inclined to use a=>1, b=>2, c=>3,...,z=>26. The issue I always ran into was that once modulo arithmetic was introduced, specifically mod 26, we end up with a zero value. I then had to re-orient students to code a=>0, b=>1, c=>2,...,z=>25. By introducing a space=>0 value, students do not have to re-orient their prior thinking, it provides a smooth transition to modulo arithmetic, and it provides a way to embed spaces into the message naturally rather than artificially.

I also looked at proofs in graph theory. There were several similar problems to those I use in class and a few new ones that I want to use. The proofs or construction of counter-examples were straight-forward. It got me thinking, maybe the sequencing for the discrete math course may be better served using the order of Combinatorics and Discrete Probability, Graph Theory, and then Number Theory and Cryptography. This allows all of the counting techniques to come into play, in fact in the few problems we worked through the possibility of using Gaussian summation, permutations, and the pigeon-hole principle could all come into play.  The proofs are simpler and more straight-forward than those in number theory, and this could provide a more solid foundation in proof that could be pushed forward in the number theory section. I have to think about this but it seems like the right way to go next year.

All-in-all it was a productive experience that was worth the time and effort. I'm glad the conference was in Denver; I'm just not sure I could swing the extra days for travel, especially with the AP Statistics exam looming so close.

Well, that's a wrap for the NCTM National Conference in Denver for 2013. I'm looking forward to the CCTM fall conference. I missed that it wasn't held last fall.