1. Understanding Diversity of Knowing and Learning Mathematics – Mathematics for All Students - Exploring Representations of Addition and Subtraction – Concepts, Algorithms, and Mental Math (Integers, Fractions/Rational Numbers) - Exploring Algebraic Reasoning through Arithmetic, Geometry, and Data Management using manipulatives and graphing calculators - Making Sense of Student’s Differentiated Responses to Solving Problems within Inclusive Settings - Understanding and implementing Ministry of Education curriculum expectations and Ministry of Education and district school board policies and guidelines related to the adolescent Understanding how to use, accommodate and modify expectations, strategies and assessment practices based on the developmental or special needs of the adolescent ABQ Intermediate Mathematics Spring 2010 SESSION 11 – May 26, 2010

2. Preparation for Wednesday May 26, 2010 Treats – Dawa and Cathy Due Sat May 29 – Annotated bibliography for teacher inquiry Due Wednesday June 2 – Learning Theories paper Read all learning theories papers and bring along Behaviourism and Constructivism: - Funderstandings. Behaviourism, - Constructivism (Piaget, Vygotsky) - Clements, D. & Battista, M. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), 34-35 Complexity Theory: - Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An International Journal of Complexity and Education, 2, pp. 85-88. - Davis, B. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education (34)2, pp. 137-167

3. Remember to log your hours … At least 60 h

4. Analytic Writing – this is what you have to write about in your Learning Theories paper… 1. Quote: What does the author say? (author, date) 2. What do you think s/he means? 3. How will you act on the information? 4. What does it mean for the way you will work with the mathematics? 1. Each of you will be “expert” on one topic. 2. Review and answer – what ? (most important ideas) - so what ? (what does that mean for a math teacher) 3. Then pick 2 other conversations you want to have and gather around those people. Discuss “Now what?”

5. Learning Theories 1. Behaviorism 2. Constructivism 3. Vygotsky 4. Piaget 5. Maslow 6. Communities of Practice 7. Complexity theory 8. Adolescent learning 9. What every Middle School teacher should know What? So what? Review and answer – what ? (most important ideas) - so what ? (what does that mean for a math teacher) Then pick 2 other conversations you want to have and gather around those people. Discuss “Now what?” Now what?

6. What Can We Learn From TIMSS? Problem-Solving Lesson Design BEFORE Activating prior knowledge; discussing previous days’ methods to solve a current day problem DURING Presenting and understanding the lesson problem Students working individually or in groups to solve a problem Students discussing solution methods AFTER Teacher coordinating discussion of the methods (accuracy, efficiency, generalizability) teacher highlighting and summarizing key points Individual student practice (Stigler & Hiebert, 1999)

7. Criteria for a Problem Solving Lesson Content Elaboration- developed concepts through teacher and student discussion Nature of Math Content - rationale and reasoning used to derive understanding Who does the work Kind of mathematical work by students - equal time practising procedures and inventing new methods Content Coherence - mathematical relationships within lesson Making Connections - weaving together ideas and activities in the relationships between the learning goal and the lesson task made explicit by teachers Nature of Mathematics Learning - seeing new relationships between math ideas Nature of Learning first struggling to solve math problems − then participating in discussions about how to solve them hearing pros and cons, constructing connections between methods and problems − so they use their time to explore, invent, make mistakes, reflect, and receive needed information just in time-

8. 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

9. Math Task 2 - Bus Problem 1 Design an Before (activation) task for your TI grade level (Before problem) - activate students’ knowledge and experience related to the task and show 2 different responses 2.Develop curriculum expectations knowledge package – overall, and specific for grades 6 to 10 3.4 solutions (grade 7, 8, 9, and 10) to the problem (precise and clear in your mathematical communication) 4.Bansho plan (labels at the bottom, categories of solutions, mathematical annotations, and mathematical relationships between solutions) with your anticipated solutions to the problem 5.Design an After (Practice) problem for students (grade level of TI) to practise their learning and provide 2 different responses

10. There are 36 children on school bus. There are 8 more boys than girls. How many boys? How many girls? a)Solve this problem in 2 different ways. b)Show your work. Use a number line, square grid, picture, graphic representation, table of values, algebraic expression c)Explain your solutions. 1 st numeric; 2 nd algebraic Bus Problem

11. Did you use this mathematical approach? (Takahashi, 2003)

12. Did you use this mathematical approach?

13. (Takahashi, 2003)

14. Did you use this mathematical approach?

15. (Takahashi, 2003) Did you use this mathematical approach?

16. How are these algebraic solutions related to the other solutions? … Knowing Math on the Horizon (Takahashi, 2003)

17. Which order would these solutions be shared for learning? Why?... Knowing MfT (Takahashi, 2003)

18. Math Task 2 - Bus Problem 1 Design an Before (activation) task for your TI grade level (Before problem) - activate students’ knowledge and experience related to the task and show 2 different responses 2.Develop curriculum expectations knowledge package –overall, and specific for grades 6 to 10 3.4 solutions (grade 7, 8, 9, and 10) to the problem (precise and clear in your mathematical communication) 4.Bansho plan (labels at the bottom, categories of solutions, mathematical annotations, and mathematical relationships between solutions) with your anticipated solutions to the problem 5.Design an After (Practice) problem for students (grade level of TI) to practise their learning and provide 2 different responses

19. Preparation for Saturday May 29, 2010 Treats – Amjad, David, and Kappa Due Sat May 29 – Annotated bibliography for teacher inquiry Due Wednesday June 2 – Learning Theories paper Read all learning theories papers and bring along Behaviourism and Constructivism: - Funderstandings. Behaviourism, - Constructivism (Piaget, Vygotsky) - Clements, D. & Battista, M. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), 34-35 Complexity Theory: - Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An International Journal of Complexity and Education, 2, pp. 85-88. - Davis, B. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education (34)2, pp. 137-167