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.