"There is no one way"

Friday, July 11, 2014

Rethinking Triangle Congruence and Similarity

As you may know, I have discussed transformational geometry and the Common Core State Standards for Math (CCSSM) several times on this blog. This summer, I have already been a presenter in two professional development workshops on transformational geometry, and will be involved in two more. This has pushed me to move forward on writing and editing some curriculum materials. Some of it can be found on my Math Education site already, and some is in progress, with the current version available on my temporary summer workshop site.

My latest writing in this area is about what may be the biggest curricular bombshell at the secondary level in the CCSSM: a shift from proving properties of geometric transformations on a foundation of congruent and similar triangles, to proving the triangle congruence and similarity criteria on a foundation of transformational geometry.

As I said before, I support this change. I have now developed a specific way to do it, which I spell out in this document. What I like about the approach I came up with is:

- The write-up is reasonably short, and thus is more likely to be read and understood by many teachers and curriculum developers than a 100-page opus. (Note: in its current version, it is not intended for students.)

- Definitions and assumptions are spelled out clearly. I also wrote an introduction about pedagogical and curricular implications.

- The transformational proofs for the congruence criteria are not unlike each other, which means that each one helps the reader understand the next one. Likewise for similarity.

- The proofs rely heavily on geometric construction, which provides an accessible hands-on foundation to the needed logical arguments. Done right, with the help of both old and new technology, construction is a very motivational arena for student exploration and sense-making. (See a unit I developed.)

- Because of its fairly strict adherence to CCSSM guidelines, the approach can be adopted widely. I hope that it will reach curriculum developers who are working on next-generation curricula. (I intend to use it as I develop curriculum myself, but as in other curricula I have developed, I hope the reach of these ideas is greater than what I can achieve on my own.)

If you have the time and inclination to read the document, I'd love to hear your comments.

Also: if you have ideas about a venue where I might publish this, please let me know!


Update! There is a new version of this paper, co-authored with Lew Douglas. Read about it here.

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