We want our students to have problem solving methods that are general enough to work in any situation. If we assign a series of exercises that are too similar to each other, it becomes easy for students to lock onto the wrong pattern, or to use a 'trick' that works just frequently enough to seem worth the effort to learn it.
One thing I tried this year was to prompt students to make themselves aware of the spectrum from generality to specificity. What works for solving specifically this question? What general ideas apply to answering all of the problems on the page?
I used my randomized question generator to help create problems that worked this way. Here's an example:
I only started a deliberate effort to prompt these conversations at the middle of the second semester. I wish I was doing it all year.
Social work is important but social work will require, by its nature, more wait time than automated work.
--p. 131, Functionary: Learning To Communicate Mathematically In Online Environments by Dan Meyer
This quote from Dan's dissertation gets to a theme of my lesson design this year. The time requirements of social interactions in the classroom are critical to honestly working them in to classroom routines. Dan is referring to the time required waiting for another students to refactor and resubmit a verbal description online. My takeaway from this point gets at a reality of making student socialization a tool for learning in the classroom.
Conversations about learning take time.
Exit tickets at the end of the class are quick ways to assess specific skills presented during a class period, but they are essentially one way channels since they can't be acted upon until next class. Time in class for lightly structured conversation around a lesson reveals understanding (or a lack thereof) is not just interactive for students, but allows me to hear a range of responses and parse them for what my students have learned. This conversation can be limited to small chunks of one or two minutes, so the payoff to investment ratio is big if those conversations are carefully designed and motivated.
Identifying what is and is not useful in those conversations is essential to working in an environment with peers. This is a valuable skill for students to develop. It's
difficult impossible to plan for every possible response students will have to everything that is said, and there will always be unexpected or off topic elements. This 'noise' can be managed but shouldn't be eliminated. Doing so denies the ebb and flow of real conversations that students have outside our classrooms all the time. If we are to leverage socialization in our classrooms for learning, we have to acknowledge that the efficiency will never be perfect. This is especially the case as Dan's research suggests that students best learn to communicate mathematically through revision and feedback.
I could go much faster through material if all I used was direct instruction. My students would be forced to be compliant to such a structure, and probably wouldn't enjoy my class as much, which I've decided is important to me. It is satisfying as a teacher to see students working through their understandings without my help, and this can only happen if I provide time for it during class. Scheduling time for it is a way to show students that I value what comes out of these conversations.