How do we know students are learning proofs?

The beginning of geometry is rough - right from the start there is a ton of vocab and new structures that students need to learn.  And then we go to proofs.  I honestly believe these first proofs are the most difficult for me to do as a mathematician.  Proving triangles are congruent has a nice flow to it, but proving that random angles and segments are congruent when it seems obvious is painful for students.

As a teacher, it is difficult for me to diagnose if students are really learning the material in class.  Furthermore, while doing proofs, it is really easy for a kid to smile and nod along when really they cannot do it independently.  While I believe in collaborative problem solving, with proofs, there is no "intriguing problem" to hook students with.  Dan Meyer talks about if proof is the aspirin, doubt is the headache we want to create.  But once again with these beginning proofs, there is not really any doubt for students.

In our unit, we start with some algebraic proofs just so students start to get the hang of some basic properties and the general structure of a two column proof.  This year students did very well with these.  I printed out notes for them so they had the structure ready to go which, while very basic, seemed to make a big difference for students.  I also took an extra half day to do a jigsaw activity in class - it is amazing how much a little extra time helps students learn.

From there we introduce geometric proofs.  We started with this packet of introducing the idea of what you can do with "given" information.  Hopefully this would allow students to get started on the proof which seemed to be the biggest issue.

From there, we actually did proofs - I walked through some, and then we did another jigsaw activity. I tried using color and arrows but I am not really sure if that really helped.

I have tried getting students to convince themselves they are sure that what we are trying to prove is actually true first, and THEN start to prove it.  The hope is that by convincing themselves it is true, they have a mini outline in their head ready to put in the structure of the proof.  

With all that, when I am walking through proofs or helping students with proofs, I still am not really sure if students are truly learning how to do a proof.  Sometimes I feel like it something they need to wrestle with themselves, however, I am trying to identify signs that students are learning in class.  So far, these are some signs I have developed but I would love to know others thoughts:

• Students who think there is a better/shorter way to do a proof.  Even if they are wrong that we cannot take a short cut, at least they are looking for one!
• Students who wonder why we did a substitution a specific way.  Usually I can do a substitution two different ways in these proofs but only one way is needed - students who notice the two different ways are showing that they get it. 
• Students who are able to convince themselves it is true first and can explain why it is true.  That's the beginning of the proof - now if I could just get them to write it down!
• Any student who asks clarifying questions.  I feel like hearing student questions is one of the best ways to understand what students know.

Unfortunately, with all of these ways to know if students really are learning the content, I need students to talk, and I cannot have 36 students ask all of these things over the course of the hour.  

I wonder if "translating proofs" from a paragraph format, to two column, to flow chart proof might be another way to help understand their abilities... this might be something I want to add next year!