1. Learning through Problem Solving – Studying Mathematics Instruction in Japan Intermediate / Senior Mathematics Winter 2011 SESSION 10 – Jan 31, 2011

2. Upcoming Dates Actions 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

3. 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?

4. 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 ?

5. 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

6. 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.”

7. “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)

8. 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)

9. 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)

10. 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)

11. 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)

12. What is the Teaching Gap? Placemat Activity Individual ideas Common thoughts

13. What is the Teaching Gap? Describe and compare teaching and learning mathematics in Japanese, German, and US schools How is content − developed? − elaborated? − coherence maintained? What is the nature of − the work done in the classroom by the students? by the teacher? − the mathematics? − the learning?

14. What is teaching and learning through Problem Solving?

15. 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)

16. 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)

17. 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)

18. 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)

19. 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)

20. Criteria for a Problem Solving Lesson Content is challenging rationale and reasoning used to derive understanding through teacher and student discussion developed as the teacher weaves together ideas and activities making certain the relationships between the learning goal and the lesson task are explicit − 92% of Japanese teachers − 76% of Germany teachers − 45% of U.S. teachers

21. Criteria for a Problem Solving Lesson Nature of mathematical work by students equal time practising procedures and inventing new methods seeing new relationships between mathematical ideas learning by first struggling to solve math problems then participating and discussing how to solve the problem – listening to ideas of others pros and cons, adapting thinking, constructing connections between methods and problems they use their time to explore, invent, make mistakes, reflect, and receive needed information just in time

22. 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)

23. 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.

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

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

26. 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)

27. 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

28. 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

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

30. Did you use this mathematical approach?

31. (Takahashi, 2003)

32. Did you use this mathematical approach?

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

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

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

36. Growing Success: Assessment, Evaluation, and Reporting in Ontario Schools. 2010 The Basics

37. Seven Fundamental Principles – What do they mean for Mathematics Education? Work with a partner and complete this form. Be specific about mathematics tools and strategies.

38. Assessment and Evaluation

39. Success Criteria based on Expectations

40. Practicum

44. See WIKI Return to ML Kestell