"There is no one way"

Thursday, August 31, 2017

Transformational Proof

Prior to the publication of the Common Core State Standards for Math (CCSSM), transformational geometry was rarely seen in geometry courses. It certainly was missing from the one I taught. Still, I have always been interested in this topic, and it provided the backbone of my "Geometry 2" class, a post-Algebra 2 elective which I called Space.

The CCSSM has changed the landscape, because of its emphasis on transformations in 8th grade, and the idea of basing the definitions of congruence and similarity on transformations. While I have concerns about the CCSSM in high school, I support that particular change, and wrote about it here. The CCSSM, however, is silent on the role of transformations beyond that, and in fact on many related questions. It raises more questions than it answers.

I have tried to help in two ways. I have been offering Transformational Geometry summer workshops for teachers every summer. And I have shared some introductory curricular materials for eighth grade, as well as some materials from my Space course, on the Transformational Geometry page of my Web site.

The largest section of that page is addressed to teachers and curriculum developers. It includes:
  •  an introduction to the glide reflection
  •  the epic proof that any figure congruent to a given figure in the plane is its image in a single translation, rotation, reflection or glide reflection. The proof is accompanied by dynamic applets in lieu of illustrations.
  • worksheets on the fact that all parabolas are similar, as are all exponential graphs

Filling a Curricular Hole

This is all well and good, but in relation to transformational geometry, the biggest hole in the US curriculum is at the grades 9-10 level. How should a transformational approach affect the geometry course? (or the integrated curriculum at that level?) Many teachers wrongly believe that a transformational approach is by necessity not rigorous. Many are only familiar with the least geometric approach to the topic, involving special cases on the coordinate plane. Many curriculum developers have reinforced these misconceptions. And finally, the creators of some standardized tests have revealed their cluelessness on this topic.

To help address that, I teamed up with Lew Douglas, another teacher who like me has retired from the classroom after several decades, and who shares my passion for geometry. We set out to fill the grades 9-10 gap by first clarifying the underlying mathematics for teachers and curriculum developers. We've worked on this project off and on for a few years, often spurred to action by the fact one or both of us would be presenting a talk or workshop on the topic. (Don't miss Lew's talk at NCTM in DC this April!)

We have finally put our work together in a 45-page document, titled Transformational Proof in High School Geometry: a guide for teachers and curriculum developers. In it, we present a logical sequence from definitions and a short list of axioms through many standard (and some less standard) theorems of the traditional geometry course. We are certain this is more accessible and/or rigorous than previous attempts we have come across, and we hope it will be useful to its intended readership.

Our approach was to prove theorems strictly by transformational means. This is not because we think there is anything wrong with the congruent-triangle-based approach. Our intention is to present an alternative to the traditional routine . Teachers and curriculum developers can choose which to emphasize in different cases, and/or present both approaches, and/or lead discussions of which is preferable in different situations.

We would love some feedback on this! Let us know what you think in the comments or via e-mail.


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