Saturday, May 19, 2012
Teaching Geometry According to the Common Core Standards
Teaching Geometry According to the Common Core Standards, by H. Wu.
from the preface:
This document is a collection of grade-by-grade mathematical commentaries on the teaching of the geometry standards in the CCSS (Common Core State Standards) from grade 4 to high school. The emphasis is on the progression of the mathematical ideas through the grades. It complements the usual writings and discussions on the CCSS which emphasize the latter's Practice Standards. It is hoped that this document will promote a better understanding of the Practice Standards by giving them mathematical substance rather than adding to the verbal descriptions of what mathematics is about. Seeing mathematics in action is a far better way of coming to grips with these Practice Standards but, unfortunately, in an era of Textbook School Mathematics, one does not get to see mathematics in action too often. Mathematicians should have done much more to reveal the true nature of mathematics, but they didn't, and school mathematics education is the worse for it. Let us hope that, with the advent of the CCSS, more of such e fforts will be forthcoming.
there is a seamless transition from the geometry of grade 8 to high school geometry in the CCSS. The concepts of rotation, reflection, translation, and dilation taught in grade 8--basically on an intuitive level--become the foundation for the mathematical development of the high school geometry course. In the process, students get to see, perhaps for the fi rst time, the mathematical signi ficance of rotation,
reflection, translation, and dilation as well as the precise meaning of congruence and similarity. Thus the latter are no longer seen to be some abstract and shadowy concepts but are, rather, concepts open to tactile investigations.
Because rotation, reflection, translation, and dilation are now used for a serious mathematical purpose, there is a perception that so-called "transformational geometry" (whatever that means) rules the CCSS geometry curriculum. Because "transformational geometry" is perceived to be something quaint and faddish--not to say incomprehensible to school students--many have expressed reservations about the CCSS geometry standards.
The truth is di fferent. For reasons outlined above, the school geometry curriculum has been dysfunctional for so long that it cries out for a reasonable restructuring. The new course charted by the CCSS will be seen to ful ll the minimal requirements of what a reasonable restructuring ought to be, namely, it is minimally intrusive in introducing only one new concept (that of a dilation ), and it helps students to
make more sense of school geometry by making the traditionally opaque concepts of congruence and similarity learnable. One cannot overstate the fact that the CCSS do not pursue "transformational geometry" per se. Transformations are merely a means to an end: they are used in a strictly utilitarian way to streamline the existing school geometry curriculum. One can see from the high school geometry
standards of the CCSS that, once reflections, rotations, reflections, and dilations have contributed to the proofs of the standard triangle congruence and similarity criteria (SAS, SSS, etc.), the development of plane geometry can proceed along traditional lines if one so desires.
Such knowledge about the role of reflections, rotations, etc., in plane geometry is fairly routine to working geometers, but is mostly unknown to teachers and mathematics educators alike because mathematicians have been negligent in sharing their knowledge. A successful implementation of the CCSS therefore requires a massive national e ffort to teach mathematics to inservice and preservice teachers. To the extent that such an eff ort does not seem to be forthcoming as of April 2012, I am posting this document on the web in order to make a reasonably detailed account of this knowledge freely available.