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 - Collection of Data for Teacher-Based Teacher Inquiry (solving the lesson problem and analysing the design of the lesson) 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 Fall 2009 SESSION 8 – Nov 4, 2009

Preparation for Wednesday Nov 4, 2009 Treats – Donovan and Brenda Reminder - Gathering math topic articles for teacher inquiry Read and Record for Nov 4 (Point form / chart or table) See sample next page a. Describe 2 characteristics of each theory: behaviourism, constructivism, and complexity theory. b. Infer how these theories can be used to analyze a mathematics teaching/learning experience. Behaviourism and Constructivism : - Funderstandings. Behaviourism, Constructivism (Piaget, Vygotsky) - Clements, D. & Battista, M. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), Complexity Theory - Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An International Journal of Complexity and Education, 2, pp Davis, B. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education (34)2, pp Assignments Oct 28 - Technology Webquest Nov 4 - Math Task 2 Nov 7 - Annotated Bibliography Nov 18 - Learning Theories paper

Teacher Inquiry - Annotated Bibliography 6 articles (one person), 8 articles (two people or more) 1st paragraph - describe key math idea 2 nd paragraph explains how you are using the key idea to to design your lesson Due Sat Nov 7, 2009

Analytic journals TI plan Math Task 1 Math Task 2 Learning Theories paper Annotated Bibliography

TI Oral/Visual One bansho plan for one of the 2 lessons (only 2 lessons, not 3) Two lesson – 2 different lesson problems Display bansho of original student work on the whiteboard (if in groups, put in grade level order to see development) – take digital pictures of blackboard and student work for ppt, but bring originals (put before, during, after on chart paper) Organization of the solutions is mathematical (to see how the idea you are teaching develops) NOTE difference (NOT across grades – that was only for math task 1 and 2 to get you see development) Focus for 35 minute OV Presentation - Rationale (what did you want to find out); description the 3 part lesson; have us do the problem and discuss solutions; analyze student solutions through your whiteboard bansho (math task type – lesson problem), description of math that student learned (practise solutions – evidence of learning from lesson); conclusions Paper is due either Dec 2 or (last class) or Dec 5 (delivered to MLK’s 45B Benlamond Ave #3 Toronto ON M4E 1Y8) – SUMMATIVE ASSESSMENT

Teacher Inquiry Topics 1. Sarjeet – Gr7 – Addition and Subtraction of fractions 2. Christina – Gr7 – Area of trapezoid 3. Joe – Gr7 – Patterning and using a table to represent a sequence 4. Donovan – Gr7 – Dividing Fractions 5. Brian – Gr8 – Geometry ?? 6. Maria – Gr9Applied – Collecting and organizing data using charts, tables, and graphs 7. Elina and Marijana – Gr9 Basic – Area and perimeter 8. Spencer – Gr9 – Area composite shapes 9. Yudhbir – Gr7 – Area of composite shapes 10. Jim – Gr9 Applied – Adding and Subtracting Integers 11. Brenda – Gr 8 – Using Algebraic Expressions to describe pattern 12. Michelle – Gr 8 – Multiplying and Dividing fractions RED Oct 21 GREEN Oct 28 BLUE Nov 4

Topic Discussions - Schedule & Readings Oct 14 - Adolescent Learning, (BLUE) Oct 28 - Maslow’s Hierarchy of Needs and Communities of Practice, Behaviourism and Constructivism, Vygotsky and Piaget (Green) Nov 4 - Complexity theory (Yellow) Nov 7 – Comparisons of learning theories Adolescent Learning Jensen, E. (1998). How Julie’s Brain Learns. Educational Leadership, 56(3), pp Knowles, T., and Brown, D. (2000). What every middle school teacher should know. Portsmouth, NH: Heinemann. Reinhart. S. (2000). Never say anything a kid can say. Mathematics Teaching in the Middle School. Pp Stahl, R. (1994). Using think-time and wait-time skilfully in the Classroom. ERIC Clearinghouse of Social Studies/Social science Education, Bloomington, IN. Behaviourism, Communities of Practice - Funderstandings, Wenger Maslow’s Hierarchy of Needs – Funderstandings Vygotsky and Piaget Constructivism Clements, D. & Battista, M. (1990). Constructivist learning and teaching. Arithmetic Teacher, 38(1), Complexity Theory Davis, B. (2005). Emergent Insights Into Mathematical Intelligence from Cognitive Science. Delta-K, 42(2), pp Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An International Journal of Complexity and Education, 2, pp Davis, B. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education (34)2, pp Topic Discussion Process 30 minutes (usually start here) 1 facilitator per 3 or 4 colleagues (one from each of the other groups) Preparation (by all) - do the readings, viewing of webcast, website search Facilitator - develops some thought provoking questions or a task to stimulate discussion of the topic, making reference to preparation Colleagues - participate in task to be prepared to share learning with group members in record learning in journal

Complexity Theory – discussion 1

Compare your solutions. How are they similar? How are they different? 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

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 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

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

Lesson Analysis Using Learning Theories 1. Intro statement identifying the focus of the paper 2. Description of the lesson flow in your MAIN lesson 3. Lesson Analysis (At least 4 examples) a) Aspect of lesson that does align with a learning theory principle (summary statement) b) Detail of the lesson aspects explained in relation to learning theory principles (APA referenced) c) Aspect of lesson that does not align with a learning theory principle (summary statement) d) Detail lesson aspects explained not aligned to learning theory in relation learning theory principles (APA referenced) 4. Suggestions for Improving Lesson - using learning theory principle (APA referenced) 5. Conclusion Adolescent Learning theory Behaviourism Communities of Practice Complexity Theory Constructivism Maslow’s Hierarchy of Needs

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)

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-

Teacher Inquiry Lesson Analysis Using Problem Solving Bring your ONE TI Lesson Plan - 9 copies for June 3 class to get analysis feedback

Pool Border Problem - What Should the revised lesson look like? Lesson Description - What the students do to learn -Include math details -Framed within a 3-part problem solving lesson

Preparation for Saturday Nov 7, 2009 Treats – Joe and Maria Due – Annotated bibliography for teacher inquiry 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), Complexity Theory - Davis, B. (2005). Teacher as “consciousness of the collective’. Complicity: An International Journal of Complexity and Education, 2, pp Davis, B. (2003). Understanding learning systems: Mathematics education and complexity science. Journal for Research in Mathematics Education (34)2, pp Assignments Oct 28 - Technology Webquest Nov 4 - Math Task 2 Nov 7 - Annotated Bibliography Nov 18 - Learning Theories paper