"Anticipating likely student responses to challenging mathematical tasks" is the first practice of 5 Practices for Orchestrating Productive Mathematics Discussions by Mary Kay Stein and Peg Smith.
You're using standards to develop learning goals for your students. You have found a mathematics task, such as the Ticket Booth from Illustrative Mathematics for example, that promotes problem solving and reasoning. Now you give the task to your students, right? Before doing that, ask yourself, "What's my next move?"
To answer that question, I am finding the 5 Practices to be helpful, particularly when supporting teachers' planning. I've begun collecting tasks that promote reasoning and problem solving (see anticipatingtheirideas.blog) where I've started anticipating how students might approach the tasks. Feel free to visit that site and provide your predictions of student thinking.
When using the 5 Practices framework for orchestrating mathematical discussions, there are two timeframes these practices occur.
The last four practices (monitoring, selecting, sequencing and connecting) occur during a live classroom lesson
The first practice, anticipating, occurs prior to the lesson during a planning session
Anticipating how students will attack a task is the action of this practice. Some general questions I consider about student thinking on tasks that encourage multiple strategies, are:
What are some typical strategies students will use?
What is a possible, out of the box, approach a student might take?
How have students in the past approached this task?
What are some typical conceptions students bring to this task that could lead students astray?
What are students' prior knowledge that this task requires?
Because this thinking occurs during the planning phase rather than during the live action of a classroom, in theory, teachers have time to consider these types of questions. That's easier stated than what is often the reality for teachers.
That said, spending time on your own, or better yet with a colleague during a PLC, to anticipate student thinking on a task is the foundation for orchestrating meaningful mathematical discourse (one of the mathematics teaching practices from Principles to Actions, NCTM 2014).
To see how I began anticipating student thinking on the "Ticket Booth" problem, go here. I welcome your thoughts.