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.