Purported as Opposed to Actual Goals

I have spent at least the last 9 hours reading math teacher blogs. NO SLEEP! YEAH! I have added most of the blogs to my list, Blogs I Read. I am happy that their experiences confirm my idea that writing about mathematics, mathematicians and how they are meaningful seems to benefit their students. It makes sense because I often figured out the most after I tried to distil something I thought I had figured out into an article for my fellow grad students to read. But reading their blogs has lead me to consider two questions: (1) Is it correct to say that "everything" in Geometry is from the time of Euclid/Archimedes and (2) How can I take advantage of the strategy of getting students to learn B by proposing that they solve A which requires B which seems to come from Out of the Labyrinth: Setting Mathematics Free? I will return to (1) later but will gather my current thoughts on (2).

I will be teaching Geometry, a class I've never taught before, and Pre-Calculus, a class which I have taught and TAed at a University many times before. I've been thinking about their relation to each other. Until recently, I had been of the mind that there is an obvious paradigm shift from Geometry to Pre-Calc but I am reconsidering my position. First, my copy of the Elements came bundled with several works by Apollonius of Perga on conics. So while I was reading, I was sort of primed to notice that Proposition 7 of Book I in the Elements more or less lays the groundwork for the idea that ellipses are well defined. This has me thinking that Pre-Calc probably naturally evolved from a classical class on conic sections to a modern class on basic function theory (transformations, symmetry, etc.). Motivate the modern material, functions, by classical questions about more advanced geometry concepts, conic sections. But thinking back on my high school math classes, I definitely did not get this impression. I never even thought about it in this way until now. Thoughts?

I have also had the interesting idea that it sort of explains why matrices are covered at the end of Pre-Calc. Geometry can be rigorously developed with linear algebra. Solving simultaneous equations is basically Gaussian elimination which gives us information about linear independence which tells us about co-linearity of points, orthogonality, etc. I don't think it's developed in that light though.