2009/10/16

Write-up #5 + Final Summary

Our last session was postponed due to the effects of the flu season; Penelope got sick and we could meet only on Friday, October 16^th, meaning that the gap in between the sessions was the longest one we had (10 days). Moreover, I could not expect from Penelope to be fully engaged in homework while being sick. Therefore, I reconsidered bringing up some new content and I decided to spend our last session with, at first, solving the instant insanity problem in a detailed way so that Penelope could solve similar puzzles on her own.

Actually, we did pretty great because Penelope spontaneously mentioned that she was happy she “got it” and so she could show the puzzle to her husband (and act clever by solving it quicker than him :P). Frankly, I was really satisfied with the teaching job I was doing overall as I engaged my student and solidified her interest. Moreover, Penelope was so involved in the graph theory that she showed her husband the idea of vertices and edges with charcoal and pieces of wood during one cosy evening. She told me this experience with such passion that I could not do anything but smile. And I actually forgot to tell her that a few days beforehand I had heard about it. I met her husband in front of the library and he was full of joy as well.

After we played with the colourful Instant Insanity cubes and Hamiltonian cycles for about an hour, we spent the second half thinking retrospectively what we have learnt. I made Penelope solve a few problems concerning Hamiltonian cycles (mentioning isomorphism) and Complete graphs which were and were not planar. We finished the lesson with deriving a general formula for the total number edges in a K_n (complete graph with N vertices). I wanted to see if Penelope’s mind manages to work with basic algorithms and see number patterns (teacher’s point of the view). Unfortunately, at this stage, she was not able to succeed but that does not mean that she will not ever be able to develop the ability. I tried to guide her through the process on her own but I saw that she needed a little bit of help towards the end. Thus, I finalised the process, clearly showing her both mathematical approaches – general and observational (empirical). The point of this final task was not really to find the answer, but to understand that there are more roads to the same destination/result. And that is the idea striking from the Penelope’s point of the view as a learner.

Doing this project taught me lot. I realised many important points outlined in my previous write-ups. Here follows a summary applicable for areas of knowledge other than math as well:

– Each student comes to a classroom with some prior knowledge, preferable learning style and expectations, all to be discovered by a teacher who also has favourite ways to approach content. However, in order to achieve better comprehension of a student, a teacher should prioritize student’s needs before his own.

– Tutoring a student one-on-one requires an individual approach, which is adjusted towards student’s needs since no individual comprehends information in the exactly same way.

– In order to do so, the teacher needs to first understand the student’s learning style, and only then the teacher can develop result-oriented lesson plans instead of content/activity-oriented lesson plans (i.e. Once I recognized that Penelope is a visual learner, I could focus on information presented graphically rather than abstractly. Although I wanted to do certain activities such as proper mathematical notation of 3D graphs, I decided to use visualisation and objects around instead. I altered the presentation but kept the idea behind the same; I was not too focused on how we were going to get to a final destination but I wanted to get there as easily as possible.)

– Although, there cannot be a unique way to approach different teaching styles, it is logical to start with lecture-type introduction developing discourse group vocabulary; and assessing the student’s prior knowledge with her response to content; continuing with person-specific/altered pace and lesson expectations presented in a preferred way (i.e. visually through a open dialogue for Penelope), until the student feels comfortable with new content presented.

-New topics, however, may require different approaches as the student’s understandings of different subjects vary. Then the teacher should start again with assessing the student’s initial response to a content (i.e. even though Penelope seems to be a visual learner for the first few topics of the Graph theory, she might actually learn better algorithmically through a strict table approach illustrating more complex ideas such as weighted graphs and Travelling Salesman Problem which we did not have time to include).

-Therefore, constant planning, revising, teaching, assessing, reflecting and adjusting is necessary.

-And lastly to mention, learning for pure understanding (qualitative) instead of subject coverage (qualitative) tends to strive the student’s interest more and builds a closer personal relationship. Then the student’s engagement may transfer outside of the classroom as well (I mean, I cannot imagine Penelope reflecting on our lessons and tutoring her husband in their free time unless I gradually slowed down after the first and the second session. Doing so gradually is important because student should never feel that the teacher’s initial – probably too ambitious – expectations are not met.).

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!