Genius Hour Blogspot #4

I have decided after a week of banging my head against a wall that my original plan was too ambitious. I had hoped to create a set of stand-alone documents that would guide a student though the basics of logic and Euclidean Geometry. However, there is so much I would want to say on each subtopic and to put it all into words would make each document enormous. This surprised me because resources that I used to teach myself these concepts were not so verbose. I eventually realized that since I am a specialist of sorts in this field (my background is in mathematical physics), with more than seven years of university under my belt, I don't need as a complete an explanation to learn about these topics. I already have the foundation to teach myself. However, a grade 9 or 10 student would most likely not have this foundation (not these days anyway), and so a complete, tightly guided explanation would be necessary. This is particularly true with propositional logic, since a misconception in that field can do a lot of damage down the road. Over the course of my GH project, I've been incorporating logic and Euclidean geometry into my private tutoring which has left me with a big pile of lecture notes. I had hoped to turn these notes into a small book of sorts. I have decided instead to make the lecture notes into a set of SmartBoard lessons which I would use if I were to teach this subject to a class. Although I'm making these notes in a way that I would use them, I imagine any teacher may find them a useful starting point should they want to incorporate geometry and logic in their math class.

Genius Hour Update #3

I am currently in the process of consolidating my research for the Genius Hour project. I believe my final product will take the form of a small textbook. One of the challenges I am facing has to do with making figures. The original figures in Euclid are very good but they can be difficult to interpret. Mathematician Oliver Byrne produced a very colourful edition of Euclid which I find intriguing but I don't think would be good for my purposes. I want to produce an exposition of Euclid, not just a new edition. One solution I have arrived at, following Byrne's approach somewhat, is to colour-code my figures. Black represents "givens". Blue represents constructed lines. Red represents constructed circles. Green represents constructed points. My figure from proposition 2 (duplication of a line) is shown below as an example:

Another possibility that I found in class today is to use a live screen capture. This would allow me to show a construction step-by-step without producing a cumbersome amount of images. Using Screencast-o-matic I produced a video showing the construction of an equilateral triangle. Although this couldn't be done for a book in the normal sense of the word, it would be very useful to produce a set of these (perhaps one for each construction) to be a companion to the book.

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.

Genius Hour Update #1

To restate the topic of my Genius Hour project, I am researching Ancient Greek Mathematics (the geometry of Euclid in particular) and seeing how they may be appropriately incorporated into the Ontario Curriculum. Since this is a topic with which I am only superficially familiar, I decided that there are three concurrent steps I need to take:

1) To work through at least the first book of Euclid's Elements myself so as to have first-hand experience of his approach.

2) To research the history of mathematics to get a better understanding of where Euclid fits in, as well as to improve my understanding of the philosophy of mathematics.

3) To trace the development of the different branches of mathematics throughout grades 9-12 as dictated by the curriculum.

Hzenlic (2010, May 9). Euclid Elements 1573 [Online Image]. Retrieved from https://commons.wikimedia.org/wiki/File:Euclid's_Elements_1573_Edition.JPG

I have found several very good resources that have been guiding my progress through Book 1 of Elements (and the rest, should I have time to go further). For those unfamiliar, Book 1 of the Elements is the foundational work in which Euclid sets out his axioms and definitions. He postulates three geometric constructions: to draw a line between two points, to draw a circle about a line segment using the segment as a radius and one end as the centre, and to extend a line segment as far as one should wish. Using nothing but logic, he follows these propositions through 48 successive proofs and constructions, culminating in a proof of the Pythagorean Theorem. Along the way you derive important results on triangles such as the S.A.S and S.S.S uniqueness theorems and the result that the sum of all internal angles in any triangle is always two right angles. I need to verify it, but I believe that all of the (2d) geometric results that a student encounters are found in this book.

By a good coincidence, I tutor a grade 10 student in mathematics. Her parents don't consider the high school curriculum sufficient and specifically wanted me to teach her things above and beyond the bare curriculum. I originally started her on some set theory in order to properly build the theory of functions, but I have now decided to work through Euclid with her instead. There is a mutual benefit  here. Her foundational understanding of mathematics will be improved (if I do my job right), and I will be able to see first hand what pedagogical benefit may be gained from this approach. Before starting with geometry, we spent two sessions on propositional logic. This is because you can't do proof until you understand how an argument works. Although the use of it wasn't at first apparent to her, there was an immediate benefit because when we reached Euclid's first use of proof by contradiction, she picked up the idea very readily because I was able to translate Euclid's argument into propositional terms that she already understood. Another inadvertent benefit already seen is that the language of Euclid allows us to look at the "ambiguous case" of triangle construction without resorting to the Sine Law. The explanation is entirely geometry and very intuitive. This topic is often the bane of grade 11 students, yet she understood it immediately. In short, I am already seeing how effective this approach can be and am looking forward over the next weeks of fine tuning it.

As far as historical research, I only have a few weeks and so could not get too deep. I found a few interest books which I think will suffice at least for this project. God Created the Integers by Stephen Hawking is an anthology of historically important mathematical works. This starts with Euclid and follows the story through Descartes through Euler and Gauss, all the way to Cantor. I've been told to take the history given in this book with a grain of a salt, but it is a good resource of primary source material. Before Galileo by John Freely is a survey of the history of science focussing on the contributions of the Ancient and Medieval worlds. I like this book because it focuses on the philosophical development of science, and not just on the major discoveries. The King of Infinite Space by David Berlinksi focuses specifically on the philosophical foundation of Euclidean geometry and subsequent developments. Of particular interest is his Berlinski's contrast between shape and quantity, and how Davild Hilbert married the two in his own geometry (Grundlagen der Geometrie) is a side of mathematics that I had never considered before and which I hope to look further into.

I have not looked too deeply yet into the official curriculum since I want to get deeper into the Euclidean Geometry and mathematics philosophy first. As my supervisor used to say, "you can't find what you don't know you're looking for". 

Google Form Quiz: The Coal Thief

Internet Copyright

This week looked at copyright as it applies to classroom applications. Teachers are increasingly using technology and online applications in their classroom which makes familiarity with copyright and online professionalism essential. For example, a blog can be set up for classroom use (perhaps in order to share resources with the students) and if the teacher isn't careful, he could violate a copyright by sharing images or videos without properly citing them. Many educational materials are available under the Creative Commons (CC) license which is a public copyright used for works that are intended to be shared and or modified. Different levels of CC licenses exists specifying the ways in which a given material may be legally used. It is important to check the license before altering a material in any way. There are several media sites that CC material can be found, including Flickr, Wikimedia Commons, and the Petrucci Music Library.

GevorgyanAnna (2015, May 29). Computer [Online Image]. Retrieved from https://commons.wikimedia.org/wiki/File:Comp.png

Vincent Brown (2011, December 28). Child’s Play [Photograph], Retrieved from https://www.flickr.com/photos/vintuitive/ 

Welcome to my Blog

Welcome to my (hopefully to be renamed) blog - Mr. Vanderwoude's Blog. In in the coming weeks I'm looking forward to exploring how technology can be incorporated into the math and physics classroom to enhance the learning experience while retaining the benefits of directness and simplicity that is seen in a more traditional approach. In particular, I hope to learn how technology can be used to incorporate in my lessons the aesthetic and philosophical beauty of mathematics and science that I miss in the more utilitarian philosophy that has become so prevalent. My belief in an aesthetic/historical/philosophical approach to science education stems partially from C.S Lewis's educational philosophy outlined in "The Abolition of Man". I'm also interested in "Classical Education" and other forms of alternative education.