"There is no one way"

Friday, April 19, 2013

Transformational Geometry

One of the features of the Common Core content standards in secondary school is a change in the foundations of geometry. Instead of basing everything on congruence and similarity postulates, as is traditional, the idea is to build  on a basis of geometric transformations: translation, rotation, reflection, and dilation. This is an interesting change, but it is so fundamental that it may meet with stiff resistance.

Here is a pedagogical argument for the change: congruence postulates are pretty technical and far from self-evident to a beginner. In fact, many of us explain the basic idea of congruence by saying something like "if you can superimpose two figures, they are congruent." Well, that is not very far from saying "if you can move one figure to land exactly on top of the other, they are congruent." In other words getting at congruence on the basis of transformations is probably more intuitive than going in the other direction.

There are also mathematical arguments for the change: geometric transformations tie in nicely with such concepts as functions, composition of functions, inverse functions, symmetry, complex numbers, matrices, and basic group theory.

One of the consequences of this change is the need for a lot of professional development to acquaint teachers with the mathematics and pedagogy of the new approach, and to make the connections with old and new parts of the curriculum. This summer, I will include a small amount of transformational geometry in my Hands-On Geometry and No Limits workshops. I will also help present a one-week workshop on Transformational Geometry which is being offered by the Bay Area Math Project.

As it turns out, I have taught transformational geometry and its connections to a number of related topics as part of my Space course for more than 20 years. Partially in preparation for this summer's work, I just recently updated a series of worksheets on the computation of images under various transformations. (The worksheets start from complex numbers, and end with matrices, making extensive use of CAS technology along the way.)

--Henri

(Check out the Transformational Geometry page on my Web site.)

3 comments:

  1. You're exactly right. When I wish to informally justify the SAS and ASA triangle congruence principles, I first ask students whether the triangles could be made to coincide. We then talk about how this would be done and what then follows about the triangles. The Common Core Standards really just ask us to formalize this.

    Here's how I think I'll do my congruence chapter next year:

    1. Begin with definitions of the rigid motions. Give students an opportunity to familiarize themselves with them. (They're supposed to have done this in Middle School, but often they haven't.)
    2. Define congruence as the possibility of coincidence after a sequence of rigid motions. Present the Rigid Motion Postulate: the rigid motions preserve segment length and angle measure.
    3. Have students prove, among other results, that if the sides and angles of two triangles can be paired up in such a way that paired angles and paired sides have the same measure, then those triangles are congruent. This is done by a description of those rigid motions that will make them coincide.
    4. Have students prove the triangle congruence principles of SAS, SSS, ASA, AAS and HL. SAS and ASA lend themselves to transformation-type proofs. The others don't. The Isosceles Triangle Theorem turns out to be a powerful tool here. It's key to the proofs of SSS and HL. AAS is shown to be a corollary of ASA by use of the Triangle Angle Sum Theorem.
    5. I'll end with applications of the usual sort - proofs and problems. By this point, it doesn't matter how we got to the triangle congruence principles. All that matters is that we have them.

    ReplyDelete
  2. I'd love to hear how this plays out. I have not thought this through at this level of detail, so for me, your steps 3 and 4 seem to require the most working out. In particular, I wonder if the postulate you state in Step 2 is sufficient. Please keep me posted as you implement your plan!

    ReplyDelete
  3. You can check out the materials I've created at my site: https://sites.google.com/site/beautyrigorsurprise/home/courses/elementary-euclidean-geometry. (Google Drive is a nice way to distribute notes and assignments.) I'd love to hear any suggestions you might have.

    I plan to ditch the textbook. Hopefully I can get the other teachers in my corporation on board. The wheels are in motion . . .

    ReplyDelete