Genius Hour Update #2

The first book of Euclid's Elements is what I have decided to work on with the high school student that I tutor. This is actually a project that we began about three weeks previous to my choosing this as my Genius Hour project. It works out well because it gives me a longer period to see if any benefit came of it. Following the ordering one very useful resource, I decided to do a crash course in logic before tackling the geometry. With each successive tutoring session, I am seeing benefits from beginning in that context. My student is becoming more comfortable with proof, such that some of the propositions she can see the path that the proof will take before I show her. Sometimes, I let her attempt the construction first and then try to have her convince me that it works. This is results in an argument back and forth as she bolsters her approach why I try to tear it down. This is also a good test for my own understanding since I have to be ready to find any holes in her argument, no matter how trivial.

This debate would not be possible if we did not have a shared understanding of what makes a good argument. For example, we were arguing over whether a given triangle was a right triangle. I was arguing that it wasn't, she kept insisting that it was. My words didn't convince her, but a reductio ad absurdum did. What I am starting to see is that through our study of Euclid, logic and geometry are being developed simultaneously. The same logical tools are used to do proofs each week, and with each additional proof, the understanding of new material comes quicker. Interestingly, my student tells me that even though what we are doing is not related at all to what she is doing in school, her school work on math is becoming much easier. Word problems, for example, she is starting to see in terms of logic and arguments, which removes any illusion of arbitrariness (this is the same realization that was the turning point in my own understanding of physics).

I summary, I am very pleased with the benefits I am already seeing from this approach. Of course, one data point does not tell us anything very meaningful but it is certainly encouraging. In the future if I get the opportunity to repeat this "course" that I am developing with more students, I will have a more solid idea of its benefits. My finished product for the Genius Hour will be a package of materials can be used to reproduce the course. This is a very low-tech topic, and I am not sure at this point how technology can be meaningfully incorporated into it (as opposed to being merely slapped on top of it), although I imagine that some of the organizational materials that we have seen will be a good place to start. For example, the final proof of Book 1 is the Pythagorean Theorem. A web can be made showing the path of proofs from the first proposition to the Pythagorean Theorem (or perhaps even of the whole book). This could be an effective way of visualizing the broader argument Euclid is making, that encompasses the entire work.