This is an example of the sort of discussion that takes place on the newsgroup geometry.pre-college. The first article, posted by Mark Saul (via his America Online account) asks a question. The subsequent articles are replies to that message, either via mail or news. They all appear in a "thread," which is a string of related articles.
From: Marksaul@aol.com Date: Tue, 28 Feb 1995 21:29:01 -0500 To: email@example.com Subject: What next for geometry? I would like some feedback. I'm teaching an excellent ninth grade honors class (following, more or less, a tenth year curriculum). The kids are bright and work hard. They form cooperative groups spontaneously and smoothly. When the class is teacher-centered, they listen and are free to interrupt with questions or comments. The whole thing is a joy. Without going full speed ahead, but just through the kids' own momentum, I will come to the end of the planned curriculum some time in March. We've had axiomatics in the context of simple group and field theorems, and in a more or less traditional, Euclid-based geometry. We've looked at Euclid bare, in the Heath translation, and traced through some of his proofs. We did an intensive unit on geometric inequalities (well beyond the usual 10th year texts). We did two weeks of informal geometry using Mira. We did the standard analytic geometry of the line and parabola. We don't have access to computers. :( ! My question is, what should I do with the time left? I'd like to do more geometry, because I know that other teachers in my department won't do as much. It's not an emergency: I've got lots of stuff. I would love to do more classic geometry (Ceva, Menelaus, Simson, Brahmagupta, Ptolemy, nine-point circle, and so on). I've done this often and could do it in my sleep. One of my kids is already about to discover the circles of Appolonius. Or, I could do the rest of the conics a la Appolonius. Or, I could do transformations in space--that'll keep them busy. Or, I could do graph theory. Or combinatorics. Or begin projective geometry. Or... My question is, what's new in geometry? In algebra, the reforms seem to be centered around data analysis. Graphing of lines is treated as related in some way to lines of regression. Parabolas and sine curves are also fit to data. It's one enormous way in which technology can be applied: getting kids to think about functions as objects, not as "something that you do with a number." It also fits in with the current idea of education as preparing kids to be productive in the "real world" so we can beat Japan. Well, now you can guess that I'm not in complete agreement with the above sort of thing. I find it as rigid as the old style. But I'm glad I've seen it, and will certainly use some of what has been developed through this movement in my teaching. But what corresponds to this in geometry? I'm not talking about just about technology--I can't get all the kids on the Sketchpad at once (as I'd like to). But what would technology, or other re-thinking of curriculum, lead us to in geometry? I'd like to hear what ideas others have on this question. From: firstname.lastname@example.org (bo3b Overkamp) Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: 1 Mar 1995 09:27:43 -0500 Organization: Odyssey Ultraware Inc. Marksaul@aol.com is trolling for geometry class ideas, asking specifically " what's new in geometry?". My answers are motivated not by newness but by recollections of students' excitement in the classroom. How about a comparison and contrast of Euclidean and nonEuclidean geometries? In particular, what do Saccheri Quadrilaterals (and Lambert Quadrilaterals and Biperpendicular Quadrilaterals) offer to sort out the differences as well as the similarities? In any plane which can measure angles and lengths, the summit angles of a Saccheri Quadrilateral (SQ) have equal measures, the diagonals of a SQ have equal lengths, and the line joining the midpoints of the summit & base is perpendicular to both summit and base. Any side of any triangle in any such plane has a naturally associated SQ for which the side you chose is the summit, and the base is collinear with the midpoints of the other two sides of the triangle; the area of the SQ and the sum of the measures of the summit angles of the SQ have intimate relationships with the area of the original triangle, and the angle measure sum of the original triangle. In a Euclidean plane, there is no difference between a SQ and a rectangle, but in a Lobachevskian plane or a Riemannian plane, the differences between a rectangle and a SQ are evident. In order for the differences to be evident to students, you probably need some models. The 19th century geometers Beltrami, Klein, and Poincare provided descriptions of models of the nonEuclidean planes. Even more concretely, bring a basketball to your class and investigate the geometry of its surface. Construct a SQ there, and observe that the summit angles are necessarily obtuse. Construct a triangle there, and consider the SQ associated with one of its sides. Or what about considering the properties of functions defined by the power series representations of sin, cos, exp? Or what about spending some time folding polygons and braiding polyhedra (_Build Your Own Polyhedra_ by Jean Pederson and Peter Hilton, Addison-Wesley, 1988)? What about tessellations of the plane(s both Euclidean and nonEuclidean)? The fundamental principle of mathematics which is least communicated to nonmathematicians is "you could have figured this out yourself." The best chance we have to communicate this to our students is in our geometry classes. From: email@example.com (Dan Hirschhorn) Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: 1 Mar 1995 17:41:11 GMT Organization: Illinois State University In article <firstname.lastname@example.org>, Marksaul@aol.com wrote: > > I would like some feedback. > > > My question is, what should I do with the time left? > > But what corresponds to this in geometry? I'm not talking about just about > technology--I can't get all the kids on the Sketchpad at once (as I'd like > to). But what would technology, or other re-thinking of curriculum, lead us > to in geometry? I'd like to hear what ideas others have on this question. As you know Mark, but others may not; finishing Geometry in March is a rare occurrence and it is clear that you have plenty to do. I'm speaking and attending the NCTM Regional in Chicago this weekend & the question you pose is one I am interested in. In my talk tomorrow morning "Geometry and Curricular change" I mostly focus on transformations as the dynamic software does them so nicely and the notion that geometry is dynamic leads to a necessity for transformations. An example that I do at the end is how translations and size changes of data relate to the geometryt of translations and size changes. My pretty conclusion at the end is to take some 2-d data and then apply a scale factor of 1.25 to both the x and y variables. The resulting figure is a blow-up of the original data & the regression lines are parallel. This is visually very nice. Doing the same thing with functions nicely connects geometry with future ideas in mathematics as well as the visual and technological changes that are going on. Relating plane transformations with 2 x 2 matrices is another nice connection. On a totally different note, I always thought the vector approach to analytic geometry is very pretty and most college-bound students never get a chance to see it. Dan From: email@example.com (Doris Schattschneider) Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: 1 Mar 1995 18:38:39 -0500 I don't see any 3-dimensional geometry in your list of topics covered. Why not spend some time on geometric relationships in three dimensions? Study polyhedra and their nice properties-- Euler's and Descarte's theorems, special polyhedra (Platonic and Archimedean, deltahedra), symmetry properties, etc. You can do it very informally with lots of hands-on discovery activities; you can also do some real proofs. Students should know that geometry is important in the world in which we live-- from the basics, they can branch out to other topics of interest. There are activity books, videos, and manipulatives available from Key Curriculum Press, Dale Seymour, and other distributors. Kids love to build models-- either with ready-made components like Polydrons or die-cut nets, or make their own. Doris Schattschneider From: Anthony@ynot1.demon.co.uk (Anthony Hugh Back) Subject: Re: What next for geometry? Organization: Myorganisation Date: Thu, 2 Mar 1995 21:47:56 +0000 In article: <firstname.lastname@example.org> Marksaul@aol.com writes: > > My question is, what should I do with the time left? I'd like to do more > geometry, because I know that other teachers in my department won't do as > much. > How about transformation geometry or an introduction to topology? -- Anthony Hugh Back From: email@example.com (Bill Marthinsen) Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: Thu, 02 Mar 1995 14:35:14 -0800 Organization: Key Curriculum Press In article <firstname.lastname@example.org>, Marksaul@aol.com wrote: > My question is, what's new in geometry? Some possible directions include: 3-D geometry--build and study polyhedra, make them different ways, unit origami is a nice approach, branch out and look at the history of the shapes, the etymology of their names, occurrances in nature, applications, not to mention their properties an related theorems. Or go the cross-disciplinary route History--great chance to do some math history, focus on the people who have done the math. Have your class research and present information about mathematicians who made major contributions in geometry; then let them debate who they would judge to be the most important/significant geometer in history. From the description of your class they might really get into this. Literature--search out, have them search out examples of geometry in literature or geometry-based literature. This is a stretch that might be intriguing; then again it could go no where. Debate--Hot topics in geometry, historical hot topics in geometry (you could ask on the Forum for suggestions for the topics). More of a straight geometry focus could take you through vectors, projective geometry, or transformations. Good luck and enjoy your class. Bill Marthinsen Key Curriculum Press From: email@example.com (Peter Appelbaum) Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: Sat, 04 Mar 95 14:41:10 GMT Organization: William Paterson College of New Jersey Here are two open-ended ideas: 1. Have the students decide, based on what they have done this past year so far, whatthe students in teh course NEXT YEAR should learb/do/etc. They have to talk about what is IMPORTANT, what made the BEST IMPACT, what areas of geometry need more or less focus, ... 2. Have them plan THEMSELVES what questions theyhave in geometry, and make a plan for what they want to learn about. Peter M. Appelbaum Curriculum and Instruction The William Paterson College of New Jersey (201)595-3123 firstname.lastname@example.org From: Marksaul@aol.com Newsgroups: geometry.pre-college Subject: Re: What next for geometry? Date: 5 Mar 1995 22:23:43 -0500 > Have the students decide, based on what they have done this >past year so far, whatthe students in teh course NEXT YEAR >should learb/do/etc. Great idea! I often give the kids a choice of curriculum, but never in just these terms. The problem with a choice of curriculum is that some of the Great Things To Do are not things that kids would fantasize about, without having seen them before. Let's see what happens.
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