I really hate the idea of handing out a syllabus and going over class business on the first day. My #1 reason is because that's what every other teacher in the building does which means all of us start to run together in the minds of the students. #2 is probably more of an adaptation to the way my building is run. Student schedules are often not firmed up until the second week of school, let alone the first day, so why waste time going over something I'm going to have to repeat to a handful of new kids tomorrow?
So this year I tried something new. A coworker had forwarded an image of nested squares to the math dept staff as his plans for a generic first day activity. I wanted to connect it to my teaching philosophy and what goals with modeling for the class, so I adapted it to become a class long activity.
I started by passing out a copy of the image to every student and asking that they take ~5 mins to count the squares *by themselves.* That last part needed emphasis because on the first day with what was obviously a blow-off assignment, the kids wanted to see what their neighbor got and talk about it. When they were done, I informally polled the class to get a range of responses - generally from 16 to 36 if no one has seen the image before. I also asked students to rate how confident they were in their response.
Stage #2 involved giving each group (my desks are set for groups of 4) a laminated copy of the same image and a wet-erase marker. Repeat. This time I asked questions like "how much *more* confident are you?" and "what else could I have done to make this easier?" (access to different colored markers was the common answer to the second one).
Then we go through the image as a group and make sure we account for every square. I honestly didn't know the right answer when we started and I make sure the kids know that too. The concluding discussion revolved around the importance of the process vs. the correctness of the answer. I wanted the kids to buy in to the idea of working collaboratively and taking full advantage of the resources at their disposal. And if there is a "right" answer to be found, there shouldn't be an arbitrary end point by when it has to be given.