Plane Geometry - Bruce E. Shapiro

SECTION 1. PREFACE 3down in a formal manner the knowledge of his day. He started by layingout lists of assumptions and basic axioms, and then asking what couldbe derived, using the logical methods of his day, from those assumptions.The result has come to be known as Euclidean geometry. Euclid boiledthe basics down to five axioms (after writing a whole mess of somewhatconfusing definitions). And followed it with a zillion pages of theorems.In the past century or so other mathematicians have tried to re-formulatedEuclid’s axioms so that they are easier to learn. These other presentations,starting with Hilbert and continuing up to the common core standardsused today, are all completely equivalent, in the sense that the same massof mathematical knowledge is derivable from them. This process of redefiningEuclid to make things easier to teach – and learn – has not beenwithout controversy.On Standards. The Common Core State Standards Initiative has nowbeen adopted by 45 states, the District of Columbia, Guam, AmericanSamoa, and the U.S. Virgin islands. 1 These standards give teachers (orschools, or boards of education, depending upon where you end up teaching)a framework within which to begin your presentation. These recommendationsserve as a resource for educators, and help provide a guide for the developmentof curriculum, instruction, and assessment. Since the the choiceof textbook will, in all likelihood, be determined by factors outside yourcontrol, the formal method of presentation may also be pre-determined.The common core standards adopted in California for a high school geometryclass give a list of 48 specific topics that should be covered divided intocategories that include congruence, similarity, right triangles, circles, arclengths, measurement and dimension (among others). As you go throughthis course I want you to think about how what we are studying fits intothis material. Ask yourself questions like this: how can I put this togetherinto a coherent, logical sequence? How can I make it interesting and fun,challenging and not boring? How can I promote comprehension? How canI make my students more autonomous? How can I communicate high expectationsto all my students without bias? How can I treat all my studentsequally? How can I assess my students’ retention and understanding? Howcan technology help? What sort of examples would make this easier to understand?What sort of activities would promote retention? What shouldgo into my lesson plan?These are not easy question and there are no easy answers. Thinking aboutthem now, while you are learning geometry for the first time (or perhaps1 Holdouts are Alaska, Texas, Nebraska, Minnesota, Virginia, and Puerto Rico.Revised: 18 Nov 2012 « CC BY-NC-ND 3.0.