1. 1 Problem of the Day (work on this in your notebook) Today is 9/18. How many different 3-digit numbers can you make with the digits 9, 1, and 8? Can you show that you have all the possibilities?

2. 2 Overview of Class #3 Continue work on place value with bundling sticks: representing quantities, notation, careful mapping between manipulatives and notation Studying records of practice: context and rationale Classroom culture –Teachers’ questions –Study video clip (September 12) –Mathematical tasks Wrapping up -- 7 people to leave notebooks today

3. 3 Core Ideas of Place Value LAST WEEK Distinction between quantity and numeration Grouping (tens, but could be other) Fundamental “ten-ness”: “decimal notation” Directionality Representational materials (bundling sticks) WHERE WE’RE HEADED Close mapping between structure of grouping and notation we use Modeling the meaning of algorithms Efficiency for representation and computation

4. 4 What stood out to you today about place value and the work of modeling it?

5. 5 Purposes for Teacher Questions Get students “into” a task Probe students’ understanding Scaffold students’ work or thinking Get students back on task Management of student error (correction, probing, converting) Focus students to listen and respond to others’ ideas Guide students to reason mathematically Strengthen students’ thinking by posing a “what if” or incorrect idea for them to respond to Support students to take intellectual risks Demonstrate interest in students’ thinking Encourage students to develop mathematical expression and fluency Extend thinking or make connections to other situations

6. 6 Studying Records of Practice: Context and Rationale WHAT DO THESE RECORDS OF PRACTICE OFFER? Continuous record of “real practice” Regular public school classroom with diverse student population Serious effort to teach with respect for mathematics, for student thinking, and for collective work, but not “model” teaching Responsible for covering the curriculum, making sure students did well on testing WHY STUDY RECORDS IN LEARNING TO TEACH? Common “text” Record offers more access than some Learn to look and listen more closely Study mathematics, students, teaching Expand pedagogical “taste” and critical sophistication and skill Develop ideas for one’s own practice: e.g., moves, problems, questions, pacing HANDOUT: FREQUENTLY-ASKED QUESTIONS

7. 7 Video Clip – September 12, 1989 Problem of the day: Today’s date is 9/12. How many three-digit numbers can you make with 9, 1, and 2? Second day of school, day after two-coin problem; this problem designed to help them develop systematic way to prove that they had all the solutions Setting up and clarifying the problem: Start with 2- digit numbers you can make with 5 and 9; Jillian: 14 Clip: We have six solutions. How can we be sure we have them all?

8. 8 Focus for Viewing What other kinds of teacher questions do you see? What specific questions? The mathematics task and its extension: What do they offer?

9. 9 Criteria for Tasks at the Beginning of the Year Offer the teacher a way to learn about students’ mathematics concepts, skills, practices, dispositions Convey to students what doing mathematics in this class is going to be like Accessible to students who are at different levels Will not immediately distinguish conventional conception of “good at math” from “not good at math” and perhaps even change those ideas Help to build norms for independent and collective mathematical work Avoid math problems that are win-lose, competition

10. 10 “Problem Package” 1.A problem and questions that can be used to launch, develop, and discuss it profitably 2.Re-scaling of the same problem, accompanied by analyses of their relative difficulty compared with the base problem 3.Cautious points: difficulties students might have, and ways to handle those 4.Examples of students’ solutions to the problem

11. 11 Classroom Culture: What Have You Learned? Subtlety and importance of teachers’ questions to shape norms, talk, values, participation, what it means to be “good at math” Some particular questions to use The role of mathematical tasks in shaping norms, talk, values, participation, what it means to be “good at math” Some particular tasks and criteria for others Pacing and allowing time for students to talk, think, express themselves Developing sense of collective work of verifying mathematical ideas and solutions (basis in reasoning) -- confidence, retention of concepts and skills

12. 12 Wrapping Up Assignments on website later today Next week: begin work on students’ thinking (start thinking of a student for your project) 7 people to leave notebooks Week 1 assignment and notebooks available for pick up outside of my office Friday afternoon Office hours: Mondays from 3:30-4:30 and by appointment

13. 13 End of Class Comments How are things going so far? What are you finding useful and not as useful? What questions do you have about what we’ve done or where we’re headed?