1. Analyzing Program Design and Implementation through Mathematics Learning Theory - Exploring Representations of Addition and Subtraction – Concepts, Algorithms, and Mental Math - Exploring Algebraic Reasoning through Arithmetic, Geometry, and Data Management using manipulatives and graphing calculators - Relating theories of learning and adolescence, lesson design principles, and components of an effective learning environment for effective mathematics teaching of adolescent students Creating learning environments to support adolescent students in learning mathematics Creating learning environments conducive to the intellectual, social, emotional, physical, linguistic, cultural, spiritual and moral development of the adolescent Understanding and implementing Ministry of Education curriculum expectations and Ministry of Education and district school board policies and guidelines related to the adolescent ABQ Intermediate Mathematics Spring 2010 SESSION 13 – June 3, 2010

2. Sample Bansho Plan 11” 8-1/2” AFTER Highlights/ Summary -3 or so key ideas from the Discussion For TI grade AFTER Practice -Problem -2 solutions - focused on TI grade Knowledge Package Gr 7 to 10 -codes and description -lesson learning goals in rect highlighted Math Vocabulary list BEFORE Activation -Task or Problem -2 solutions Relevant to TI grade DURING -Lesson (bus) Problem -What information will WE use to solve the problem? List info AFTER Consolidation Gr7 Gr8 Gr9 Gr10 4 different solutions exemplifying mathematics from specific grades labels for each solution that capture the mathematical approach -Math annotations on and around the solutions (words, mathematical details to make explicit the mathematics in the solutions -Mathematical relationship between the solutions

3. Is it possible to create two patterns that have the same pattern rule, but look different? a.Create two growing patterns, using square tiles. b.Record your growing pattern on square grid strip of paper. c.Explain and record each other’s pattern rule on the paper. d.Discuss how your patterns are mathematically similar and different. We usually have students write and compare pattern rules to concrete models; When do we have students re-present a pattern rule in different ways? Such develops a conceptual model of equivalent algebraic expressions BEFORE - Same Pattern Rule, Looks Different

4. DURING - Pool Border Problem A How many one-by-one square tiles are required to surround a 5 x 5 square unit pool? Show 2 different solutions. 5 x 5 Pool 5 square tiles

5. Video : Jo Boaler

6. Original Pool Border Problem How many square tiles are needed for the border of any square-shaped pool? a. How many one-by-one tiles required to surround other square-shaped pools? b. Determine a rule to predict the number of tiles required to surround a square- shaped pool of any size. c.How does your rule relate to the number of tiles for the pool and for the pool border? Pool 5 square tiles Same problem -,but different focus - to choose the arithmetic solution that has the potential for generalizeability for any size square-shaped pool)

7. Engage students in the same mathematics (demonstrating achievement of the same expectations) but give them choice about − Strategy − Numbers that they use to do the math Allow teachers to ask “Common Questions” of ALL students that make explicit some of the mathematical relationships in the question prompting students to reason and defend their thinking Parallel Tasks

8. 8 Lisa : $1.65 in quarters and nickels Amy : $1.05 with half the quarters and twice the nickels How many of each coin does Lisa have? Parallel Tasks Choice 1 Use equations to model the problem and then solve it. Choice 2 Solve the problem using only number thinking.

9. 9 Did you need to know how much each coin was worth or just the relationship between them? How did you know that Amy had an even number of nickels? Common Questions

10. 10 Could you be sure that Amy had an even number of quarters? How do you know that each girl had fewer than 7 quarters? Common Questions

11. 11 How did you solve the problem? How do you know your solution is correct? Common Questions

12. Parallel Tasks Choice 1: The track is 24 cm long. How long could the pieces be? Choice 2: The track is 264 cm long. How long could the pieces be? 12 Ian put together equal length pieces of train track.

13. 13 How do you know that the pieces are not 10 cm long? How do you know that the pieces are not 25 cm long? Common Questions for Parallel Tasks

14. 14 Could the pieces be 2 cm long? How do you know? Could there be 12 pieces of track? How do you know? Common Questions for Parallel Tasks

15. An open question provides valuable information about the range of knowledge in your classroom. The student responses will help you know how to proceed with your lesson. 15 Open Questions

16. Open: The area of a rectangle is about 400 square units. What could its dimensions be? 16 Not open: The area of a rectangle is 432 square units. The length is 12 units, what is the width? Contrast

17. Open: A pattern begins like this 3, 6, … How might it continue? 17 Open Questions

18. 18 How might such open questions be effective in diagnosing student differences? Do you see them more as exposing or evoking thinking? Purpose Open Questions

19. 19 Assessment for learning is your focus. An open question should be accessible to all students. Open Minds On