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"At this point the Greeks gave up algebra and turned to geometry."
Honestly? So did I! I am an artist, I think graphically. Geometry, Geography, Cartography, Orthography, etc. have always come to me easily. Irrational Quadratic Equations (IQE), as taught in most public schools in the United States of America, make absolutely no sense, and serve no discernible purpose in the real world.
My own instructors dedicated 50% or more of their courses to IQE, frustrating me to no end, because they wouldn't move on to anything else once they reached them. They constantly asked on written assignments to merely, "Solve.", equations. Then they always complained about the result I wrote, even when it was correct, because they wanted me to, "Show my work."
The process of going through the formula was more important to them than the result. None of them understood that I used a different means to get to the result, that was faster, and just as accurate. I didn't understand why they insisted upon writing mathematical expressions that were needlessly complex to denote an equation that was effectively upside down, backwards, and turned inside out. For them, algebraic notation was a mathematical puzzle to be taken apart and put back together, providing 'proof' that the expression was true at all points in the progression.
I skipped the algebraic notation and went directly to the result. I didn't need 'proof', I just wanted to get the work done. I knew in my heart that no one would actually write equations of the sort they expressed when attempting to solve real world issues in an expedited manner.
This article is very well written. I wish I had come across something of this sort thirty years ago, when it could have done me some good. Instead, it wasn't until I took classes in Trigonometry that it all fell into place. Trigonometry did for me, as an artist, what Algebra did for my high school instructors. Trigonometry acted as a mathematical bridge between Arithmetic, Geometry and Algebra, that I could traverse at will.
Unta Glebin Gloutin Globin
Red Ronin, The Cybernetic Samurai