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!

My First Foray with Abolishing Grading

Inspired by Joe Bower's call to abolish grades, and after being tired of having students "play the game" of school and being tied to PLC grading procedures, I decided to throw out grades in the only class I could.  It was/is just a pilot, but I have learned so much already.  I would love to hear other people's thoughts on this.

Background

I work in a context, and my work is influenced by this context:

• I could not throw out grades entirely.  I need to submit 6-week grades with the rest of my school, but that did not mean they needed to be determined in the same way.
• We have 80/20 grading practices in my school - it is a policy.  So, my 80% is really just 100% in my class.  After each unit, a summative grade is determined. 
• Grades need to be tied to content. Again, school policy. 
• My goals for students were 1) to increase student interest in computer science and 2) to provide students with opportunities to engage in collaborative, challenging, and creative tasks.

What Abolishing Grades Looked Like in Class

• Day to day, the goal was to have students reflect on their learning.  Students created a portfolio with artifacts and daily reflections from the unit.  These were not graded.  I looked at them once through the semester and provided written feedback on their portfolio and in class contributions.  No grades.
• From day 1 students were given a "single point" unit rubric with learning objectives on them.  This was modified during the semester and as the "abolishing grading" philosophy became a reality. I added "Essential Knowledge" statements below each learning objective to make the learning objectives more accessible to students. 
• Students were given one day in class to work on filling out their rubric - they were given the example below and encouraged to put links into their rubric that went to their portfolios.
• The next day, students met with me individually to discuss their grade.  I had created some questions ahead of time to help guide our conversations - students did not see these, but rather they were for me to use since this was my first time teaching this content, I did not feel like I could come up with these questions on the spot. 
• At the end of this discussion, students told me their grade they think they earned and why.  Students were told that they could change that grade by learning what they did not know (their own assessment of what they did not know - this was shaped also by the questions I asked during our conference).  I documented all of our conversations in a google doc so I could go back and refer to it. 

What I have learned (thus far)

I just had my first round of conferences with students yesterday - I only got through 4 students which tells me I need to up my efficiency.  The conversations were good, perhaps a bit uncomfortable at times.  Sometimes it felt a bit like a oral exam where I asked them a question and they tried to answer it to the best of their abilities.  If student's did not know information I was asking about, I felt a little conflicted between wanting to tell them the answer and knowing that they should have documentation that they know this before they see me.  I realized that when I give a test, I stop being a teacher and I start being more of an evaluator.  With the "throwing out grades" model, I still put on pause the "being a teacher" part but the "evaluator" role becomes the students responsibility.

I also have very mature and motivated students in this class so I wonder how their ability to self- reflect and set goals for themselves would be different from students in my other classes.   Is this a model that could work for all students?

During this time I also learned that I am entirely uncomfortable with the "new system" that I created.  In theory, I am entirely in favor of focusing on learning rather than grades, however, I was born and raised in a grading system.  I thrived in that schooling system with grades in high school - or at least, I thought I thrived.  In college, grades were not my focus and I still believe I was better off for that. As a teacher, I feel like throwing out grades would be liberating, and it was to some extent teaching without "the test" looming over students was fantastic.  Students were engaged throughout the unit.  However, this new system presents an entirely new set of challenges.  I have a lot of concerns that I had not considered before.  For example:

• Are grades being assigned by personality?  For example, my most confident student may assign themselves an A but a less confident student may not be comfortable with that grade.
• What values are being imposed upon students with this system?
• Is this system truly helping my students meet my goals for them?
• How does this grading system impact the level of rigor in my class?  Students seem to be very willing to take risks which I am appreciative of, but would some students learn more if they knew there was a test on the material on a given day?
• Am I expecting too much from students when I put them in this system?  They have not thought about assessment and grading systems in such a broad way before.  
• If students aren't taking tests, how do I decide what is valued in the content?
• Did I just "change the game" of school?  Am I truly focusing on learning?  How do I know that I am focusing on learning vs. grades?

I also learned that from an accountability perspective, I do want to have documentation on why this grade was assigned.  The single point rubric ensures that we are talking about the learning that the college board has said is important and when I ask students what they would need to do to improve their grades, I have a "re-learning" plan already set for them.  This way, if families come to me asking about grades, I have a response.  Also, allowing students to change these grades by addressing the concerns they had, I think students felt able to more accurately grade themselves.  This wasn't then end of the conversation. 

Finally, I learned that I need to increase the amount of feedback I give students.  I gave them a 3 week feedback report, but I don't know if students really knew what to do with that information.  We never really discussed what they should do with that information.  Even when I think about it, I suppose I assumed that students would make changes to their portfolios based on the feedback they received, but that only happened for a few students.  I think having this policy requires me to be more clear with students on what my expectations are.  One way to do that is to give more frequent feedback, even if the feedback is brief.

Now What?

I have 15 more students to talk to over the next few days and get their self-assessment.  I learned that I need to do a bit more prep for these conversations and look over their self-assessment rubric before hand so I could make the most of our conversations.  From there, I need to get student feedback on how this first unit went from a content perspective but also how the "no grades" system worked for them.  Ultimately, I want to live and work in a democracy as much as possible so if students decide this system was not for them, I want to know why and see how I could tweak it.