But if you think you really get them, try a little experiment: ask a small group of math teachers what they think "Attend to Precision" means. What does it look like if a classroom task requires it? What does it look like when a teacher is facilitating it? What does it look like when students are doing it? Here are some responses you might hear:
- Rounding correctly according to the directions
- Rounding sensibly based on the problem's context
- Being careful when plotting points
- Labeling axes and diagrams correctly
- Drawing sketches and diagrams to scale
- Using an appropriate number of sig figs based on the precision of the measuring device
- Using precise mathematical terms in written and verbal communication
- Defining variables and symbols
I've spoken to teachers who express their understanding with numbers 2, 6, and 7, but I've talked to teachers whose understanding hews closest to numbers 1 and 3. Which is not to pass judgment, but is to say: it might be wise to be aware that you and your colleagues could have different, and potentially incorrect, assumptions about the SMPs.
And "Attend to Precision" seems like one of the more concrete ones. See what your colleagues have to say about "Look for and express regularity in repeated reasoning," and I bet the answers will be even more all over the place.
Another observation: it can be really hard to evaluate which SMPs are highlighted or emphasized in a classroom task. When I try, I tend to go "uummmm...all of them...?"
So what kind of task lends itself to "Modeling with Mathematics"? What does it look and sound like when teachers and kids "Look for and Make Use of Structure"?
I'd like to point you to a recently published resource: A Rubric for Implementing Standards for Mathematical Practice. It was written in July of 2011 by Danielle Maletta, Mimi Yang, and Mariam Youssef as part of the Visualizing Functions working group at PCMI. It gives an observer specific items to look for in a task, as well as specific teacher behaviors, to help evaluate how faithfully a standard is being met in a particular lesson. The accompanying Resources document will also give you a deeper understanding of each standard.
Also, heads up that Illustrative Mathematics, in addition to the Herculean undertaking of trying to illustrate every K-12 content standard, has put a significant amount of effort into illustrating the Standards for Mathematical Practice using both sample videos and classroom tasks.
Check them out. Share widely.