Chapter 2 5 Practices for Orchestrating Productive Mathematical Discussions

1. Laying the Groundwork: Setting Goals and Selecting Tasks Chapter 2 5 Practices for Orchestrating Productive Mathematical Discussions

2. Setting Goals for Instruction The key is to specify a goal that clearly identifies what students are to know and understand about mathematics as a result of their engagement in a particular lesson.

3. Identifying a Math Goal Practice 0- foundation on which the 5 practices are built

4. Practice 0- Identifying the math goal “Without explicit learning goals, it is difficult to know what counts as evidence of students’ learning, how students can be linked to particular instructional activities, and how to revise instruction to facilitate students’ learning more effectively. Formulating clear, explicit learning goals sets the stage for everything else.” Hiebert and colleagues (2007, p.51)

5. Goal statements for 8th grade students • How are the goal statements the same, and how are they different? • How might the differences in goal statements matter? • Students will learn the Pythagorean Theorem (c2=a2+b2) • Students will be able to use the Pythagorean Theorem (c2=a2+b2) to solve a series of missing value problems. • Students will recognize that the area of the square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on the legs and will conjecture that c2=a2+b2.

6. Tasks that align with the goal • What kind of task might align with each of the three goal statements?

7. See Tasks on Page 17

8. Leaves and Caterpillars: The Case of David Crane • What was the learning goal for David Crane? • What goal could have lead to quality mathematical discussion? • What did students learn in David Crane’s lesson on leaves and caterpillars?

9. The Case of David Crane What if his goals were: • Students will recognize the relationship between caterpillars and leaves as multiplicative, not additive. • Students will recognize that the leaves and caterpillars need to grow at a constant rate (for every 2 caterpillars, there are 5 leaves; for each caterpillar, there are 2.5 leaves).

10. Selecting an Appropriate Task • It is crucial that the task that a teacher selects align with the goals of the lesson. • Consider the corresponding goals and tasks in figure 2.4 • Students will be able to use the Pythagorean Theorem (c2=a2+b2) to solve a series of missing value problems. • Students will recognize that the area of the square built on the hypotenuse of a right triangle is equal to the sum of the areas of the squares built on the legs and will conjecture that c2=a2+b2.

11. Levels of Demand • Take 5 minutes to read and discuss the levels of demand chart. • What makes a task lower level? • What makes a task higher level?

12. Tiling a Patio page 19 • What level demand does the tiling problem fall under? • What is your rational?

13. Time to Sort • Review the Task Analysis Guide • As a table, review each task. Determine if each task falls under a Lower-level demand or Higher-level demand. • Where do the tasks that you have used in your classes last year fall under?

14. Conclusion • Select a clear and specific goal with respect to the mathematics to be learned. • Select a high level mathematical task. • All tasks selected do not have to be high level, but productive discussions that highlight key mathematical ideas are unlikely to occur if the task on which students are working requires limited thinking or reasoning.