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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

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Understanding Diversity of Knowing and Learning Mathematics – Mathematics for All Students

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  1. - 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 Understanding Diversity of Knowing and Learning Mathematics – Mathematics for All Students ABQ Intermediate Mathematics Fall 2009 SESSION 11 – Nov 18, 2009

  2. Treats – Maria and Brian Readings Due 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. 96-106. 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 http://www.tc.columbia.edu/lessonstudy.html. Yoshida, M. (2003). Developing effective use of the blackboard through lesson study. Online publication. Retrieved from http://www.rbs.org/lesson_study/conference/2002/index.php Preparation for Wednesday Nov 18, 2009

  3. Tonight • What I have • The problems of others • Computer lab

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

  5. Integer Addition and Subtraction 1. +3 + (+4) 5. +3 – (+4) 2. +3 + (-4) 6. +3 – (-4) 3. - 3 + (+4) 7. - 3 – (+4) 4. - 3 + (-4) 8. - 3 – (-4) a. Show using 2 colour counters. b. Show using a number line. How do our conjectures hold for these set of equations?

  6. 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?

  7. 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) 3. - 1/2 + (+2/3) 7. - 1/2 – (+2/3) 4. - 1/2 + (-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?

  8. Algebra Tiles Subtraction is separating/decomposing OR comparison • Represent (3x2 – 2x) – (x2 – 3x + 3) • Simplify (2x – 3) (x + 4) = use area model • Factor 4x2 – 10x + 6 = • Graph y1 = 3x2 – 2x y2 = x2 – 3x + 3 y3 = y1 – y2 = 2x2 + 1x – 3

  9. 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.

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

  11. What are the components of effective blackboard use for teaching mathematics?

  12. Background - Components of Japanese Teaching Blackboard is used to show the flow of the lesson process and to • keep a record of the lesson to help students remember what they need to do and think • help student see connections of different parts of the lesson and the lesson progression • contrast and discuss ideas students presented from their solutions • organize student thinking and develop new ideas Bansho – means board writing – technical term created by Japanese teachers; rarely erase what they write on the blackboard … all recordings have a mathematical meaning and purpose Hatsumon – asking a question to stimulate students' thinking Neriage – teacher coordinating student sharing, analysis, and discussion Matome – teacher summarizing key mathematical ideas from the lesson (Yoshida, 2003; Shimizu, 2005)

  13. Background - A Sample Japanese Blackboard Plan Prior knowledge Problem (drawing) Prior knowledge Problem (drawing) Today’s story problem (in words) What today’s problem is asking (in words) Student drawing of solution to problem Student mathematical expression Student mathematical expression Area for student ideas and presentations of solutions (Yoshida, 2002)

  14. 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? Before - Analyzing a Student Solution Colour code constant and variable

  15. DURINGWhat Does It Look Like? • Choose a set of 4 equations that you predict have a mathematical relationship. y=4x + 1 y=2x + 1 y=4/3x + 1 y=4x – 1 y=2x – 1 y=2/3x + 1 y=4x + 6 y=2x + 6 y= - 4x + 1 y=4x – 6 y=2x – 6 y= - 4x - 1 • What do the expressions you chose look like, pictorially, in a table of values, and graphically? • What are key characteristics of a linear relation? • What’s the mathematical relationship between the graphs?

  16. 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?

  17. 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

  18. What are the differences between the American and Japanese lessons?

  19. 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. 96-106. 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 http://www.tc.columbia.edu/lessonstudy.html. Yoshida, M. (2003). Developing effective use of the blackboard through lesson study. Online publication. Retrieved from http://www.rbs.org/lesson_study/conference/2002/index.php Preparation for Wed Nov 25, 2009

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

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