Episode 4

2009/10/07

Write-up #4

Having thought about our last lesson and how it helped Penelope to visualize examples, I decided that we would be using mostly chalkboard for our next sessions. I wanted Penelope to do writing and thinking on her own instead of copying down information I usually present in a lecturing way. Using a set of smartly imposed questions, I planned to develop on our models and proofs of Euler’s formula, isomorphism, sum of total edges in complete graphs, Euler graphs and Hamiltonian graphs. If you think that there is already too much of information mentioned, you are right – we have managed to go through just the first two but that is all right. Once again, the aim of our project is deep understanding of some concepts (even if it was going to be only one) instead of shallow mentioning of as many concepts as possible.

Since my meeting scheduled prior to our session went a little bit over the time, I called Penelope and asked her to write down in those first 5 minutes a mathematical proof of Euler’s formula, which I have proven two sessions ago, and Penelope talked through a week ago. I wanted to see how exactly she approaches the task and I was surprised in rather disappointing way. But please, do not tell her, as I did not let her know. I believe that teachers should not show negative feedback too often; they should rather accept the way student presents a work. Then teachers should try to guide the student to the intended concept using topic related questions so that the student makes own discoveries. Instead of writing the proof in her words, Penelope used the book and technically took the notes of the text proof. That is all right. She even wrote down the name of some theorem used in the proof, which we have described in our own words a lesson ago and so we concluded that remembering the specific name is handy but not necessary. The understanding of it was though. By rewriting the proof step by step from the book, I guessed she was not completely comfortable with the proof. That is not her fault but the fault of me, a teacher.

I said well done because it looked neat. But I also asked her to talk through it in her own words and I saw that a little bit of additional help by posing questions was needed. Moreover, I made her draw some examples of the bucket effect on the board and I was happy she used her own mind to draw conclusions. Finally, I hope, she realised that mathematics is not a science of strict dogmatic theories with only one explanation (that is why I gave her once that handout with multiple explanations) but rather a field that requires personal approach and critical thinking. I have a feeling that Penelope is afraid to make errors (not mistakes!) in her own development of ideas and therefore she tries to memorize or follow what has been shown as appropriate. This is what I have thought of only now while writing the write-up. Hence, I will have to encourage Penelope to try thinking out loud and not to be afraid of making constructive errors.

During the rest of the lesson, we similarly approached the isomorphism topic. I think that the method of me asking topic-based questions has been working out the best so far and so I will keep doing that. I might have to mix some lecturing here and there again, because I gave Penelope quite enough of challenging material as homework. It was a short handout including my graphical notes and extra clarifications about Eulerian and Hamiltonian graphs. I decided to use this online printout because its discourse language was very friendly. Hence, hopefully, Penelope will have no troubles reading it. Also, she should try to solve my favourite of all basic math puzzles: Instant Insanity!

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