The Case of Nick Bannister 5 Practices for Orchestrating

Productive

Mathematics Discussions The critical starting point for planning is to specify clearly and explicitly the mathematical goals for the lesson or unit; what are the understandings that the student should come away with? This is "practice 0" or the "foundation on which the five practices are built." The next step is to select a task that involves high-level thinking and reasoning and aligns well with the goal of the lesson. Intro: The 5 Practices Anticipating: This next step requires that the teacher monitors the students' actual responses to the task while they are working in pairs or small groups. It will be made easier if a quick chart is made to jot down responses and keep them organized. Monitoring: Anticipating In their thought-provoking approach to teaching (and learning) mathematics, Margaret Smith and Mary Kay Stein lay out their 5 practices for creating discussions that lead to a deeper understanding of key ideas behind the content and skills of math. Their approach is student-centered and inquiry-based and begins with a clearly stated goal for understanding and 5 steps to orchestrating meaningful "give and take" among students and teacher. The 5 steps are:

1. Anticipating

2. Monitoring

3. Selecting

4. Sequencing

5. Connecting Laying the Groundwork: Practice "0" Calling Plans: the Case of Nick Bannister

Ninth grade Algebra teacher Nick Bannister wants his students to understand 3 algebraic ideas: 1) that there is a point of intersection between two non-parallel linear equations that is shown where the two functions have the same x and y values; 2) that the two functions "switch positions" at this point and that the function that was on top before the point flips to the bottom after the point and vice-versa; 3) to know how to make connections between tables, graphs, equations by identifying the slope and y-intercept in each form of representation. Nick anticipates these possible solutions:

1. Students will make a table with columns:

a) number of minutes, b) Cost A, and c) Cost B. The minutes will start at 0 and go up in increments of either 10 or 20; 2. Students will write equations for Companies A and B, where y=0.04x + 5 (Co. A) and y=0.10x + 2 (Co. B);

3. Students will graph the two equations using values from a table or plugging the equations into the graphing calculator. Nick also considered the challenges that students might run into including: a) have trouble finding the point of intersection in the table if the number of minutes increased by a number not a factor of 50; b) start the table at some # of minutes other than zero; c) have notational difficulties; and d) confuse what was fixed and what was changing. Anticipating Students' Responses This is the first practice, in which the teacher anticipates likely student responses to the challenging mathematical task that is given. The more thorough and thoughtful the teacher is at this step, the easier the next 4 steps will be. Calling Plans Task: Long distance Company A charges a base rate of $5.00 per month plus 4 cents a minute that you're on the phone. Long distance Company B charges a base rate of $2.00 per month plus 10 cents per minute used. How much time per month would you have to talk on the phone before subscribing to Company A would save you money? Monitoring Student Groups Nick places students into groups of four and has them create a poster which shows their work on the task and the answer at which they arrive. He asks questions as needed to keep students on track and to initiate deeper thinking. Nick makes a table for himself to keep track of different answers of groups and their various approaches. This will help him to decide how to select and sequence the answers to be presented to the class at the end of the task. This will be the discussion phase of the lesson.

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# 5 Practices for Orchestrating Productive Mathematics Discussions

Summary of the 5 practices using the example of Nick Bannister's "Calling Rate Plan"

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