2009/09/23
Write-up #2
In order to continue with clear development of ideas, I created a person-specific lesson plan; I picked up on the particular point where Penelope and I finished a week ago. Her homework was to prove whether there is or there is not a Complete Graph of 5 vertices (denoted as K5). She seemed to have put some effort into it but she couldn’t reason exactly why there is not a solution (again J). I told her the process of thinking she has to go through (think about the K4 at first) and expanded our vocabulary by new terms: faces, cycles. She understood the explanation only after I drew it specifically on the blackboard and therefore I think that, as far as the Graph theory is concerned, Penelope is a visual learner. She told me after the lesson that she would like to learn mathematic writing of proofs as well and therefore her extra homework is to write everything we have discussed about down and bring it for the beginning of the next lesson. We will go over it and stress some basic ideas of structured thinking.
Right after introducing K5, I had planned to do some counting of the total number of edges in a complete graph and its arithmetic generalization developed by logic and recognizing patterns. We took, however, a different order of developing our ideas since Penelope wanted to make a parallel and prove impossibility of the Three Utility Problem. We introduced bipartite graphs and used our knowledge of vertices, edges and faces. As far as the development of that formula for edges in Kn is concerned, we will do it the next lesson. It is not a important formula to know but I want to see how Penelope thinks and if she can see patterns in numbers. I do not mind constantly adjusting our lesson curricula because the point of having one-on-one lesson is to facilitate the learner’s needs and her progressive way of thinking.
I realised that while my mind can take many discourses and keep thinking about different math issues independently, Penelope seems to see the clear picture better when we concentrate at one thing at the time. Moreover, I tend to draw connections a little bit easier and that is probably why I have to slow down a bit with my lesson expectations (By the way, again, we did not manage to cover everything I planned…). One of my current math professors said lately a humorous note: “Math is about making the least amount of mistakes. If you keep making a fewer mistakes than others, and if you can understand topics easily, they make you a teacher.” I would like to add a point that in order to be a math teacher, one has to manage thinking the way his students do. But that can be a tough thing to do in a setting with multiple people in a classroom. Thus I am happy I am starting with this project, not a classroom setting.
Anyways, back to our second lesson, we proved that K3,3 (the bipartite ‘utility problem’ graph) was impossible as well and so we mentioned Kuratowski Theorem about forbidden planar graphs. We introduced isomorphism, i.e. “equal shapes” from Greek, and we made a nice additional thought. Penelope asked me something I did not think of before and I had to improvise and prove planarity of K2,n. While doing it, I was not sure about the notation of faces and their degrees (I have not managed to find such notation yet!), thus I created our own notation “3xF2,2” for three isomorphic faces in K2,3 and proved possibility of K2,n. I stressed the point that the math is about personal approach and we can alter (or create) our notations as long as they keep some conventions (e.g. do not resemble or violate other general notations). I also realised that I am a visual-type learner, similarly as Penelope is. Great for this project!
Moreover, I gave Penelope a sheet with two pairs of graphs and her homework is to reason and write down if they are isomorphic. I helped her by mentioning that one pair has different faces and therefore is not isomorphic (although it looks like one). I hope she can notice it and understand it.
The end of the lesson was spent thinking of planarity again but this time in a bit different way. We defined planar graphs as finite, connected graphs holding the property of Euler’s formula. We run over its proof quickly because we were short of time. Therefore, I will ask Penelope to illustrate the same proof during the next lesson to see if she deep-processed the information presented in a traditional quick-lecturing way. Plus, I gave her 19 other proofs of the formula (including angles, electrical charges) to not to limit her to one way of thinking. I believe that there should be a degree of freedom in education.