╨╧рб▒с>■ 13■ 0 ье┴` Ё┐bjbjцЗцЗ .ДэДэ дддддддд╕ААААМ╕╦╢ддддддддJLLLLLL$Бhщ
┌pддддддpддддЕ...дRддддJ.дJ..дд.дШp┐aNЙ┼АЎ".JЫ0╦.├├.├д.дд.дддддppддд╦дддд╕╕╕фЬф╕╕╕Ь╕╕╕дддддд Attached is a one-page synopsis of the planned course, that I prepared last winter. I am due to teach a pilot version of this course to Mathematical Education students this fall; unfortunately it has not been possible to run this course any earlier, so I won't be able to relate any experiences strictly regarding Math 360 until a couple of months from now.
Although I have obtained many ideas from reading other peoples' materials (in particular David Henderson of Cornell) I haven't distilled these into written form, as until I get into the classroom I won't know what will work and what might turn out to be impractical. The details will develop dynamically from the students' reactions. I am very enthusiastic about the idea of using Geometer's Sketchpad and maybe Povray to give the students hands-on experience of 2-dimensional and 3-dimensional geometry. In particular, the part of the brain that deals with 3-dimensional visualization is rather dormant on average and it would be good to give it some stimulation.
This summer I had a kind of warm-up in that I taught our 460 geometry course to students, most of whom are probably comparable in ability to the 360 clientele (quite a few were Education students.) The flavor of 460 is usually a little bit more formal than what I have in mind for 360, but on this occasion I adopted a very "hands-on" approach, with an emphasis on experiencing the worlds opened up by various geometries rather than presenting a stuctured edifice of axioms and theorems. We took the Klein approach to geometry whereby a particular geometry is considered to be characterized by its isometries. We spent some time looking at Euclidean isometries and how to write them down and compute them using matrices, and then looked in some detail at inversions in circles. Of course this was a sneaky preparation for hyperbolic geometry in two dimensions which we moved onto fairly painlessly. I was pleasantly surprised how the students readily accepted that it was valid to consider a metric different from the usual one, although I tried to help in that regard by drawing attention to the spherical metric we use on the surface of the Earth. In figuring out how to write things down and compute things for inversions and for the hyperbolic plane we got a little bit into complex numbers, which I think was useful for broadening the students' mathematical experience. Attached are a couple of pdf's from the 460 course. I think some of the students thought they would never be the same again after doing questions 2 and 3 of h2sol.pdf! An integral part of my teaching style is that I make myself available, even late in the evening, for questions by email about the homework assignments - very often students seem to get emotionally involved with their assignments in the evening, and they are then very receptive to the odd well-placed hint. Doing geometry in an email can be challenging, but I learned a while back just what can be done by seeing some of John H. Conway's geometry forum contributions.
With hindsight I think that partly subconsciously I steered Math 460 in the direction it went this summer because I was mindful of figuring out how to organize Math 360 this fall, so maybe my experiences are fairly relevant. There is no set syllabus for Math 460; the content is entirely at the discretion of the instructor. Therefore, for the purposes of the NSF review I think it would be appropriate to regard that particular instance of the course as a "pre-pilot" for 360. The component that was missing of course was the computer workshop aspect.
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