1. Micro Teaching and Operations with a Variety of Number Sets Intermediate / Senior Mathematics Winter 2011 SESSION 9 – Jan 27, 2011

2. Upcoming Dates Actions Monday Jan 24 – Gizmoes Wed and Thurs Jan 26, 27 – Micro Teaching Monday Jan 31 – Assessment and Evaluation − Bring Growing Success electronic or paper version Wednesday Feb 2 – Assessment and Evaluation Due Dates Monday Jan 31 – Micro Teaching due and Lesson Plan due Wednesday Feb 2 – Teaching around the World; Assessment and Evaluation

3. Peer Feedback

4. Assessment of MicroTeaching

5. Integer Addition and Subtraction 1.+4 + (+2) 5.+4 – (+2) 2.+4 + (-2)6.+4 – (-2) 3.- 4 + (+2)7.- 4 – (+2) 4.- 4 + (-2)8.- 4 – (-2) a. Show using 2 colour counters. b. Show using a number line. How does the idea of part-whole relationships, compose/join and separate/decompose related to integer operations?

6. Fraction and Rational Number Addition and Subtraction 1. +1/3 + (+2/5) 5. +1/3 – (+2/5) 2. +1/3 + (-2/5)6. +1/3 – (-2/5) 3. - 1/3 + (+2/5)7. - 1/3 – (+2/5) 4. - 1/3 + (-2/5)8. - 1/3 – (-2/5) 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?

7. Algebra Tiles Represent (3x 2 – 2x) – (x 2 – 3x + 3) Subtraction is separating/decomposing OR comparison Simplify (2x – 3) (x + 4) = 2x 2 + 8x – 3x – 12 = 2x 2 + 5x – 12 – area model Factor 4x 2 – 10x + 6 = (2x – 2 )(2x – 3)= 2(x – 1)(2x – 3)

8. 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 t=0t=1t=2 t=3

9. 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?

10. 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?

11. 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?

12. 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 =

13. 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.

14. 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. 6. 2/3 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?

15. Try This With Colleagues … Arranging Desks Problem Sally works out how many people can sit around a row of desks. 1 desk 5 people 2 desks 8 people 3 desks 11 people 4 desks 14 people Sally says that 39 people can fit around 12 desks. Without drawing the desks, how do you know that Sally is wrong? What mathematics are you using to solve the problem?

16. Let’s Do Math as a Teacher! Pick a number. Double it. Add 6. Double again. Subtract 4. Divide by 4. Subtract 2. What’s your final number? Analyze and Explain Why did you end up with that number? How does it work? 32 26 13 ?

17. What Does It Look Like Algebraically? X 2 + 6 2 ( ) - 4 _____________ 4 ( ) - 2 Pick a number, double it, add 6, double again, subtract 4, divide by 4, and subtract 2. = X

18. Understanding the Problem … The Teaching Gap “As important as it is to know how well students are learning, examinations of achievement scores alone can never reveal how the scores might be improved. We also need information on the classroom processes – on teaching – that are contributing to the scores.”

19. “Teaching is a cultural activity. We learn how to teach directly, through years of participation in classroom life, and we are largely unaware of some of the most widespread attributes of teaching in our own culture.” “If we wish to make wise decisions, we need to know what is going on in they typical classrooms.” “Video information can shake up the way we think and let us take a fresh look at classrooms.” The Teaching Gap - TIMSS Video Study (Stigler & Hiebert, 1999)

20. Background – the TIMSS Study Why U.S., Japan, Germany? Goal – Through video study, describe and compare 8 th grade mathematics instruction in United States, Germany, and Japan TIMSS study largely funded by US gov’t Japan – scored near the top in all international comparisons of mathematics achievement for decades Germany – an important comparison country, because like Japan, it is a major economic competitor of United States (Stigler & Hiebert, 1999)

21. Background – the TIMSS Study Why Videotaping … How? problem definition – − “If we wish to make wise decisions, we need to know what is going on in typical classrooms.” different teachers use the same words to mean different things data collected – random sample of 238 grade 8 mathematics classrooms − videotape of a grade 8 lesson − teacher questionnaire about their day before and day after teaching plans − collection of teaching materials used (textbook pages, worksheets) (Stigler & Hiebert, 1999)

22. Background – the TIMSS Study Observation Codes and Analysis taped a typical lesson scheduled for that teaching day to avoid bias – they developed a set of standard observation codes to identify and quantify the frequency of specific teaching events a team of 6 coders independently watched the 238 videos and identified images of teaching and assigned observation codes and recorded frequency (Stigler & Hiebert, 1999)

23. A mathematician’s perspective “In Japan, there is the mathematics on one hand and the students on the other. The students engage in the mathematics and the teacher mediates the relationship between the two.” - structured problem solving “In Germany, there is the mathematics as well, but the teacher owns the mathematics and parcels it out to students as she sees fit, giving facts and explanations at just the right time.” - developing advanced procedures “In the U.S. lessons, there are the students and there is the teacher. I have trouble finding the mathematics. I just see interactions between the teacher and the students.” − learning terms and practising procedures (Stigler & Hiebert, 1999)

24. Nature of Content coded quality of content based on level of challenge and how the content was developed definitions vs. rationale and reasoning used to derive understanding What is The Teaching Gap? Mathematics Instruction (Stigler & Hiebert, 1999)

25. Content Elaboration stated concepts by teacher and student developed concepts through teacher and student discussion Threats to Coherence switches in topics interruptions from external events (e.g., PA announcements never occurred in Japanese lessons, 13% of the time in German lessons and 31% of the time in U.S. lessons) What is The Teaching Gap? Mathematics Instruction (Stigler & Hiebert, 1999)

26. What is The Teaching Gap? Mathematics Instruction Content Coherence connectedness or relatedness of the mathematics across the lesson (e.g., a well-formed story consists of a sequence of events that fit together to reach a final conclusion) Making Connections weaving together ideas and activities in the relationships between the learning goal and the lesson task made explicit by teachers − 92% of Japanese teachers − 76% of Germany teachers − 45% of U.S. teachers (Stigler & Hiebert, 1999)

27. What is The Teaching Gap? Engaging Students in Mathematics Who Does the Work? who controls the solution method to a problem predominately student controlled − 9% of the time in the U.S. − 9% in Germany − 40% of the time in Japan classes What Kind of Work is Expected? German and U.S. students spent most of their time practising routine procedures Japanese students spend equal time practising procedures and inventing new methods (Stigler & Hiebert, 1999)

28. Why Does The Gap Exist? Beliefs about Mathematics Teaching Nature of Mathematics U.S. (math is a set of procedures and skills) Japan (math is about seeing new relationships between mathematical ideas) Nature of Learning U.S. students become proficient executors of procedures Japanese students learn by − 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 (Stigler & Hiebert, 1999)

29. 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 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 - seeing new relationships between mathematical 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-

30. 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)

31. BEFORE – Angle Measures Problem 1. Show these angles in any way that you know: 0 o, 90 o, 180 o, 270 o, 360 o 2a.Fold paper along 180 o in halves, to create different angles. 2b.Label the measure of each angle you make. 2c.What’s the relationship between the angles you made and 180 o.

32. DURING - Angle Relationship Problem What are 2 possible solutions for x?

33. AFTER – Angle Relationship Problem What should be the next problem for students to practise? Different angle measures Real context/situation

35. What is Japanese Lesson Study? How is Japanese lesson study similar to our teacher Inquiry? How is it different? <GER: Lesson Study: An Introduction>

36. What are the Range of Solutions for This Problem? 2. What mathematics did we use in our solutions? 3. Record your ideas as a network (knowledge package), showing the interconnections among the curriculum expectations from grades 6 to 9. 4. What is the sorting criteria for these solutions? 1.1.

37. Analysing Classroom Situation using Learning Theories Describe Communities of Practice and Maslow’s Hierarchy of Needs in relation to this classroom situation. Let’s take a classroom field trip …

38. Growing Dot and Bus Problem - Using a Graphing Calculator Use a graphing calculator to: Input data into a table of values create a graph Apply “regression analysis” to get the algebraic equation Growing Dot Problem Bus Problem

39. What Can We Learn From TIMSS? Problem-Solving Lesson Design BEFORE (ACTIVATING PROBLEM 10 min) Activating prior knowledge; discussing previous days’ methods to solve a current day problem DURING (LESSON PROBLEM 20 min) Presenting and understanding the lesson problem Students working individually or in groups to solve a problem Students discussing solution methods AFTER (CONSOLIDATION the REAL teaching 30 min) Teacher coordinating discussion of the methods (accuracy, efficiency, generalizability) Teacher highlighting and summarizing key points Individual student practise (Stigler & Hiebert, 1999)

40. 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-

41. 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

42. 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. c)Explain your solutions using one or more operations. What’s the Mathematical Relationship to the Previous Problem? …. Knowing MfT

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

44. Did you use this mathematical approach?

45. (Takahashi, 2003)

46. Did you use this mathematical approach?

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

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

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