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.)