As someone who teaches geometry and practices as an applied geometry, I find it very disturbing to consider introducing Congruence as some form of abstract 'equivalence' not related to transformations.
There is so much to be learned from using transformations, so many connections to be made to 'measurement' and to the skills which are embedded in the k-8 curriculum. The transformational insights are also key to later use of geometry. I recommend looking at Henderson, Experiencing Geometry and Plane and Sphere. The unifying theme is the use of appropriate transformations in both contexts, to prove shared properties (and to explore the differences). My university students (often future teachers) find it quite possible to prove the theorem about base angles in both contexts, several ways. It is not hard after they have explored why SAS and ASA do guarantee there is a transformation of one to the other (ie. are sufficient for congruence).
I recall a conversation with someone who developed a Tangram software to help elementary students explore plane transformations. The only permitted moves were reflections and translations. The students in elementary school did not find it too hard to work with that. There were different levels of interfaces, some of which offered more scaffolding (e.g. indicating the ghost where the object would be, as GSP does), In fact, what seems to work well was gradually decreasing the scaffolding so the student developed internal, visual reasoning skills.
One nice story was a grade 3 student (age 9) who played for an hour and then observed: I can always get an object from one place to the other in at most 3 moves (reflections). Wonderful geometric thinking. That student would be ready to explore congruence of the type we are talking about, without the concrete experience of congruence as transformation (which it is).
I am reminded of the article 'Why do we teach so little math in elementary school'!. In fact, this type of experience is perhaps harder for a 15 year old who has been cut off from what they learned as younger children, and is struggling with geometry as this strange game played by weird people.
On 26-Nov-05, at 8:38 PM, NealAgMan@aol.com wrote:
I agree with you that geometry students can grasp the concept of corresponding parts being congruent. And if you define congruence of triangles in that way (corresponding angles and sides congruent), I think they understand it fine. It's the transformational definition that bothers me as the **initial** definition (rather than something they take up after alot of experience with congruence).
I agree with you also that proving that an angle bisector of a triangle intersects the 3rd side is very difficult. If memory serves, Moise offers a proof of this proposition, which he calls the "Crossbar Theorem" (Actually the crossbar theorem is more general, but the angle bisector situation would be a special case in which the theorem applies). This is a theorem far beyond the scope of a high school course in geometry. This is one reason, among several, why the "traditional" proof (introducing the bisector of the vertex angle) is deficient.(This is the reason given by Moise, in his famous textbook, for rejecting this proof.) In one or both of her geometry books (Modern Geometry: Structure & Method, and Modern School Mathematics: Geometry), Prof. Dolciani actually confronts this deficiency, and tells students that she is postulating that the angle bisector will intersect the 3rd side. To me, this is a superior approach than rejecting this proof entirely in favor of the proof where triangle ABC is congruent to triangle CBA, which I find just too ethereal. Sorry, but I just do.
OK!!! Have dug out the reference...Modern Geometry, Structure & Method (Houghton Mifflin 1965), Jurgensen, Donnelly, & Dolciani, page 208. "In the proof of Theorem 31 (base angles of isosceles triangle congruent), and again in the proof of Theorem 32 (the converse), the statement that the bisector of angle C intersects segment AB could be supported by this postulate: The bisector of an angle of a triangle intersects the opposite side. The postulate is used in this section without further mention."
Now, right on point!! In her Teacher's Manual, Prof. Dolciani states: "The question as to whether or not the bisector of an angle of a triangle intersects the opposite side (segment) is not interesting or meaningful to most high-school students. The problem can be avoided by using the method shown on p. 226 [ie, triangle ABC congruent to triangle CBA] . . . That method seems elusive to many students, however. The authors prefer to present proofs that are meaningful to typical students, mentioning -- but not emphasizing -- the assumption that is being made." I think the authors are exactly right here, but, of course, you are free to disagree.