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 Winter 2010 SESSION 12 – March 3, 2010

Preparation for Wednesday Mar 3, 2010 Treats – Cory-Ann and Rashi Due – can we make LT paper due Mar 6?? We will practice Mar 3 Read all learning theories papers and bring along Last of the assigned readings Smith, M. (2004). Beyond presenting good problems: How a Japanese teacher implements a mathematics task. In R. Rubenstein and G. W. Bright (Eds.) Perspectives on the Teaching of Mathematics, pp Reston, VA: NCTM. Ertle, B. and Fernandez, C. (2001). What are the characteristics of a Japanese blackboard that promote deep mathematical understanding? Lesson Study Research Group. Retrieved from Yoshida, M. (2003). Developing effective use of the blackboard through lesson study. Online publication. Retrieved from Assignments Feb 24 - Annotated Bibliography Feb 24 – Math Task 2 in class Mar 3 - Learning Theories paper Assignments Feb 24 - Annotated Bibliography Feb 24 – Math Task 2 in class Mar 3 - Learning Theories paper

Lesson Plan Template

Assessment for Learning (Seating Plan) Tool

A Sample of a Completed Lesson Plan with advisory notes (from MLK) in red

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

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

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

Consider our banshos… Gallery Walk…

Highlights / Summary What did you learn by doing Math Task II? What did you learn by reading the bansho of others who did Math Task II? How to think about it Extremely valuable to work together Very practical Answered all questions re TI Math Task 1 good practice too

Lesson Analysis Using Learning Theories 1. Identifying the focus of the lesson 2. Describe the lesson flow Lesson Analysis: 3a) State one aspect of the lesson that does align with a learning theory principle (summary statement) 3b) Describe part of the lesson in relation to a learning theory principle (APA referenced) 3c) Describe part of the lesson that does not align with a learning theory principle (summary statement) 3d) Describe the part of the lesson related to your example where the instruction is not aligned to learning theory principles (APA referenced) 4.Suggestions for Improving lesson - using learning theory principle (APA referenced) Conclusion: Lesson Description - What the students do to learn -Include math details -Framed within a 3-part problem solving lesson

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) Original Pool Problem

Key Characteristics of our Community and Links Theories and our treatment of Math Communities of Practice Social community in a math classroom - interacting with the peers; think-pair-share - structure discussion; What to do when others are not interested in listening; ask another student to paraphrase what another student said - develops sense of belonging; How to teach students to listen and interact - model it in your teaching - its about having the kids experiencing appropriate interactions, but not how to do it: Knowledge is inseparable from practice (communiites of practice) Our communities of practice - snacks, talk to each other, not just each other, help each other; show case each other’s work - bansho; establish a support network; asking questions valued; sharing dilemmas; sitting in groups, journal writing, sense of belonging by the group name; Maslow’s Hierarchy of Needs Basic needs and moves its way up - physiological and safety; Bullying - “you don’t get it” - that was really easy, why don’t you get that, I don’t want him …” -> Solution - different ways to get to a solution and a step to get towards it - collective knowledge Comfortable class environment - encouraging multiple solutions - creates a safe learning environment - not about right/ wrong; we don’t show them how to do it Positive response - identified aspects of their solution; ask them how they got their solution Belonging - initial prayer (catholic) ; share their solutions, Esteem - positive reinforcement - goal to be intrinsically motivated; more engaged Self Actualization - driven by student - sense of how they feel about themselves; self- efficacy; needs others to affirm ourselves

About Learning Theories (4 groups again) Respond with respect to the video of the border problem… Group A What ideas of which philosophers (Piaget, Vygotsky, Maslow, or Communities of practice) did you see reflected in the mathematics lesson on the video? Groups B, C, D a. What is the basic premise of each learning theory – behaviourism (B) / constructivism (C) / complexity theory (D)? b. What is the role of the teacher and the student in each? c. What role does the mathematics play in each theory (what would instruction look like)?

Lesson Analysis Using Learning Theories 1. Intro statement identifying the focus of the paper 2. Description of the lesson flow 3. Lesson Analysis 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

About Behaviourism and Constructivism

Whole Number Addition and Subtraction – – 183 a. Show using base-ten materials. b. Show using movement on a number line. What will be our “narrative” for addition and for subtraction?

Integer Addition and Subtraction (+4) 5.+3 – (+4) (-4)6.+3 – (-4) (+4)7.- 3 – (+4) (-4)8.- 3 – (-4) a. Show using 2 colour counters. b. Show using a number line. How do our narratives for addition and subtraction hold for this set of expressions?

Addition and Subtraction Integers – Our Conjectures, Tests, and Generalizations

Re-presenting Rational Numbers What different ways could these rational numbers look like using: Square grid Rectangular grid using paper folding Number line two colour counters for +1/2, -1/2, +2/3, -2/3 How does the idea of compose/join, decompose, separate, compare, and part-whole relationships related to rational number operations?

Rational Number Addition and Subtraction 1. +1/2 + (+2/3) 5. +1/2 – (+2/3) 2. +1/2 + (-2/3)6. +1/2 – (-2/3) /2 + (+2/3)7. - 1/2 – (+2/3) /2 + (-2/3)8. - 1/2 – (-2/3) a. Show using 2 colour counters. b. Show using a number line. How does the idea of compose/join, decompose, separate, compare, and part-whole relationships related to rational number operations?

Algebra Tiles Subtraction is separating/decomposing OR comparison Represent (3x 2 – 2x) – (x 2 – 3x + 3) Simplify (2x – 3) (x + 4) = use area model Factor 4x 2 – 10x + 6 = Graphy1 = 3x 2 – 2x y2 = x 2 – 3x + 3 y3 = y1 – y2 = 2x 2 + 1x – 3

Differentiating Fraction Subtraction 1a.How could you use base ten blocks to model 1/2 - 1/3? 1b.How could you use base ten blocks to model fraction subtraction? 2a.Don and his friends at 1/2 of a vegetarian pizza and 2/3 of a pepperoni pizza. Which has more? How much more? 2b. Ian and his friends ate part of a vegetarian pizza and part of a pepperoni pizza. Choose the size of your parts.

Differentiating Fraction Subtraction 3. I subtract 2 fractions and my answer is one half. 4. If you subtract thirds from fourths, what would the difference be? 5. Describe a situation where you might need to figure out 3/5 – 1/ /3 of the students like apples better than oranges. How many students might be in the class? 7. Why is it sometimes easier to 4/9 – 1/9 rather than 4/9 – 1/5? 8. How is subtracting fractions like subtracting whole numbers? How is it different?

Before - Analyzing a Student Solution 1. What dots in this model can be represented by (1+2t) 2 – (2(t)) 2 ? Represent this algebraic model, graphically and numerically using a table of values, quasi variable, and coordinate pairs. 2. What dots in this model can be represented by 4t + 1? How do you know? Colour code constant and variable

Prom Dress Problem A few days ago, Veronica and Caroline were both asked to the prom. That night, they went out to shop for dresses. As they were flipping through the racks, they each found the perfect dress, which cost $80. when they showed each other their dresses, they realize they both wanted the same dress! Neither of them had enough that night, but each went home and devised a savings plan to buy the dress. Veronica put $20 aside that night and has been putting aside an additional $5 a day, since then. Caroline put aside $8 the day after they saw the dress and has put in the same amount every day since. Today, their friend Heather asks each girl how much she has saved for the dress. She says, “Wow! Caroline has more money saved.” How many days has it been since Veronica and Caroline began saving?

Palette of Problems Ernesto drives from home to the library at 60 km/h. Then he drives home. His average speed for the round trip is 50 km/h. At what speed did he drive home from the library? A about 40 km/h B about 43 km/h C about 50 km/h D about 55 km/h

What are the differences between the American and Japanese lessons?

Preparation for Wed Nov 25, 2009 Treats – Joe and Spencer For OralVisual Presentations: Bring the following (if applicable): powerpoint on USB stick (not on your own computer) student work samples from research lesson to display on board for white board bansho Read Smith, M. (2004). Beyond presenting good problems: How a Japanese teacher implements a mathematics task. In R. Rubenstein and G. W. Bright (Eds.) Perspectives on the Teaching of Mathematics, pp Reston, VA: NCTM. Ertle, B. and Fernandez, C. (2001). What are the characteristics of a Japanese blackboard that promote deep mathematical understanding? Lesson Study Research Group. Retrieved from Yoshida, M. (2003). Developing effective use of the blackboard through lesson study. Online publication. Retrieved from

Preparation for Wednesday Nov 25, 2009 Treats – Joe and Spencer