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.
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| 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".
