First, micro: do actual math on the first day. It's easy to get caught up in going over rules, procedures, grading standards, etc. But why do we implement those systems in the first place? For the most part, the point is to ensure that our students do, actually, learn some math. How ironic to then spend the first day not doing math in order to communicate that math is the main thing! Much better is to communicate that math is the main thing by actually spending a chunk of the period doing math!
(For the record, I doubt that going over lists of rules or procedures is even helpful. I myself have a personal capacity of learning about five rules, procedures, or standards at a time--tops--and maybe ten per day. I doubt most students are much better. Even at just two rules per period, the students are effectively done by lunchtime. So there's no point in going through every detail; instead, provide a handout with the fine print, mention the big picture, and make sure to go over rules as (just before) they come up.)
This math doesn't have to be hard. In Geometry, we go through a sequence of folds on a circle and talk about angles, symmetry, terminology, etc. It's a great pre-assessment of what kids know coming in, and it allows us to "get through" a lot of vocabulary in a context where that vocabulary is meaningful. In Calculus, we watched a video of a speedometer:
and a video of my dog's ability to do calculus while fetching a ball--which wouldn't upload, so here's the original "Dogs know Calculus" video.
Second, macro: do something new. Obviously (?) do NEW math with the class on the first day: spending several weeks "just reviewing", as far as I can tell, only communicates to kids that it doesn't really matter whether they learn it the first time around. But mostly I'm talking about you, the teacher. Try something new this year. Change your grading system. (My department did: regular nightly homework, as such, is no longer counted towards student grades; instead, we'll quiz more often, and assign more interesting out-of-class work that we can collect and grade thoughtfully.) Or change your pedagogy:
- use videos (á la Dan Meyer), or
- incorporate more formative data into your lesson planning, or
- plan quizzes and tests collaboratively, or
- create tiered work on which students can select for themselves the appropriate level of challenge (aka Challenge by Choice), or
- design (or borrow) writing prompts that get students to think mathematically, or
- anything else you haven't done very well before, like in that cool presentation you heard at last spring's conference.
Good luck, and do some great math! In the comments: what math did you do on the first day?