2009/09/30

Write-up #3

Wow. This lesson was a little bit unexpected wake-up call for me. I planned to go over homework problems in first 15-20 minutes, ask Penelope if she could prove the Euler’s formula (to see if she really got the idea), introduce Eulerian and Hamiltonian Graphs, hand in new tasks and readings, and then continue with the Chinese Postman Problem during the lesson 4 and Travelling Salesman Problem during the lesson 5. I wanted the last sixth lesson to be a revision class to see how much we have really gained from this project. We needed, however, to move that revision class instead of the original lesson plan number 3 (well, at least I have a detailed lesson plan ready for the next, fourth lesson J).

The reason why we needed to spend this time clarifying confusions, and explaining mostly discourse language of the mathematics, was that I wrongly assumed certain facts about Penelope as a learner. Yes, I gave her a complex handout of proofs I did not expect her to understand but I hoped she could read through a textbook explanation at least. I wanted to test her if she, at all, could identify herself with proofs using mathematical induction, geometry, chemistry or physics applications and thus find out more about her mind. I wanted to base my further teaching strategies on that but now I see that we are at a different level. Penelope has not had a math lesson for more than 10 years and it is natural that she did not remember the concept of mathematical induction any longer or that she assumed that certain letters used as variables have only one meaning (a common misconception).

I was happy to explain other areas and examples of mathematics because I could expand our learning beyond what was planned. Moreover, I can see now that Penelope really understands the concepts of faces and planar graphs, that I though she had processed before. First, I asked her to define them (or describe) to me in a way as simple as she understood. She kept trying to reason them through algebraic applications that were developed after their clear definition. I let her know that she is correct as far as the description is concerned, but in order to understand the concepts she has to have a clear mind about definition. I used a cell phone and a table with its edges for an easy analogy of faces, and then I used several sheets of paper and pens for “squashing” 3D graphs onto planar 2D graphs. At this point, I am positive of Penelope’s understanding that ‘planar’ comes from the concept of a plane, which is 2-dimensional.

Afterwards, we moved onto reading our proof in the book. I asked Penelope to stop after each sentence or a section representing a point. She then needed to translate the mathematical discourse language of the textbook into her own words so that I could undoubtedly say she has processed the information. I realised that presenting a topic through ‘spoon-feeding’ method is effective only for introduction of simple concepts and basic glossary. Then a teacher has to alter the teaching style to be more than a dialogue and let the student speak out loud. When lecturing, students often nod and confirm their understanding with words “yeah…ehm…” but in fact, they are just understanding the single words being said, not the contextual meaning of sentences.

Therefore, I believe that students should be forced to re-explain theories in order to see whether they did not understand them at all, or they memorized proofs step by step categorically, or they really understood them and can reinterpret them using own vocabulary, or (hopefully not) they understood them incorrectly through some misconceptions. Then a teacher gets a real feedback and can alter teaching strategies. That, however, can be applied only in a small classroom setting where a teacher can concentrate on individual approach for students.

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 K_n 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 K_3,3 (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 K_2,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 “3xF^2,2” for three isomorphic faces in K_2,3 and proved possibility of K_2,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.