Kate Nowak blogged about it first.  Evan Abbey asked for this post.  Here goes nothing.

I teach Geometry therefore I also teach proofs, specifically two-column proofs about congruent triangles.  It's a Geometry student's nightmare.  Ask a Geometry alumnus what they didn't like about Geometry, chances are it's proofs.

Aside from showing the "girls are evil" proof, it used to be one of my least favorite topics to teach.  Honestly, who cares if two triangles are congruent?  Furthermore, who cares to prove two triangles are congruent "given" some sort of figure marked up with congruent sides and overlapping angles?  Yeah, yeah, we need to expose our students to this type of recipe-like logical reasoning.  Yeah, yeah, we should be exposing our students to mathematical proof in the unlikely event one or two go on to become math majors.  For the super-majority of my students, Geometry is the first and last time they'll see congruent triangles as well as two-column proofs.  It's up to me to make it a little bit interesting.

Here's how it all shook down a few months ago:
Me: Let's say I tell Tommy to go home and cut out a triangle with three sides: 3 cm, 4 cm and 5 cm.  He's really good with his protractor, ruler and scissors and won't mess up the triangle.  I give the same instructions to Sandy - go home and cut out a triangle with three sides: 3 cm, 4 cm and 5 cm.  Sandy is also really good with her protractor, ruler and scissors.  Would they come back tomorrow with the exact same triangle?

Student A: No way!  They will definitely be different.  Just because they have the same sides doesn't mean the two triangles will be exactly the same. 

Student B: Yes!  The same three sides means it will be the same exact triangle.

Me: Interesting thoughts.  Who thinks Student A is right?  Who thinks Student B is right?  How about this scenario:  If I tell Randy to cut out a triangle with three angles...40 degrees, 45 degrees and 95 degrees and Leslie to do so as well, will both of these students come back tomorrow with the same triangle?

Student A: No way!  They will be different, just like Tommy and Sandy's triangles.

Student B: Yes!  There's only one way to draw a triangle with those angle measures.

Student C: Yes!  I've done it before, Mr. Townsley.

Me: Wow.  Who thinks Randy's triangle will be the same as Leslie's triangle?  Of those of you that voted yes, how many of you also think Tommy and Sandy's triangles will be the same?  How many of you think that there's a difference between these two pairs of triangles?

...and the conversation continues until a student finally says, "let's try it out!!"
Over the next class period, pairs of students cut out 12-15 triangles and test some conjectures I propose.  Are two triangles with congruent and corresponding angles always going to be congruent?  What about two triangles with congruent and corresponding sides?  What about a combination of angles and sides?

Will many of my students (and yours) ever think about triangle congruence theorems and/or two-column proofs again outside the math classroom?  Probably not.  In my earlier years of teaching, I used to motivate this kind of lesson with some sort of artificial "real-world" problem.  I'm starting to learn that its often more engaging and meaningful to address the math for what it is rather than pretending it is immediately applicable to adolescents' lives.   Pure mathematicians:  can/should a high school math teacher get away with this type of rationale or am I providing a disservice to my students?