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Prof. Clements: Your questions intrigue me because I love geometry at all levels and the teaching of it in high school in particular. I will try to answer some of your questions. •1: The NCTM Standards www.nctm.org/standards has a strain of geometry from K-12. At different levels different aspects from vocabulary, recognition, to properties of geometric figures, proof are discussed. At the NCTM site you should also be able to find references to research articles that have been published in the Journal for Research. Since your topic is so wide, I don't know where to begin in research. In the United States there has been much discussion on the how and why of proof in high school geometry. Van Hiele (from the Netherlands) has had extensive studies on five levels of geometric understanding. You might search the internet with "Van Hiele" as your source word. •2: I am not sure what you mean by circle theorems. Do you mean measures of angles formed by intersecting chords, by two chords meeting on the circle, by two secants to the circle meeting outside the circle, by a tangent and secant meeting outside the circle? If you do, yes, these are proved. If you mean lengths of intersecting chords within the circle, etc. these are also proved. •3: About this I have no idea. •4: Many teachers / texts do not think that these are important for high school students. For all high school students I think that they are not important. For some high school students I think that they are important. Who are the some? These are students who will spend time in mathematics / science / architecture. Surprisingly I solved a problem for a contractor whose son was in my class. He gave me the height of a window and the width of its cross-section. He wanted to figure the angle / location of the center of the circle so that he could cut the opening for a "Palladian-style" window in a wall. Yes I used my knowledge of angles, trig and chords of a circle! As these the most important theorems to be taught? No. Should they be mentioned? Yes. Should they be proved? Only to some students. -Marielouise, for the T2T service.
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