2. What is a super even number?  Is 2,097,152 supereven? How can you decide in an efficient way? (Roodhardt et al, 1997)  Is 2,097,152 supereven? How can you decide in an efficient way? (Roodhardt et al, 1997)

3. Reflecting on practice: Week 1  What does it mean to teach? How do you know?  What does it mean to productive observe and discuss teaching?  What are the benefits of observing and discussing what goes on inside our classrooms?  What does it mean to teach? How do you know?  What does it mean to productive observe and discuss teaching?  What are the benefits of observing and discussing what goes on inside our classrooms?

4. What does it mean to teach?  To facilitate, plan and guide student learning  To have visible evidence of student learning  To create an environment where students can learn  To stimulate student thinking using rigorous and meaningful tasks  To express passion and excitement and encourage students to do the same  To facilitate, plan and guide student learning  To have visible evidence of student learning  To create an environment where students can learn  To stimulate student thinking using rigorous and meaningful tasks  To express passion and excitement and encourage students to do the same

5. You know when students can  explain their thinking or their work to their peers  communicate their thoughts and reasoning  apply concepts correctly to novel situations  make connections  explain their thinking or their work to their peers  communicate their thoughts and reasoning  apply concepts correctly to novel situations  make connections

6. How do you know students understand? When they  Recognize how the concept/idea is related or connected to other things we know  Consciously reflect on experiences with the concept/idea (know what does and does not make sense  Communicate with others about the concept/idea  Recognize how the concept/idea is related or connected to other things we know  Consciously reflect on experiences with the concept/idea (know what does and does not make sense  Communicate with others about the concept/idea Hiebert et al, 1999

7. To facilitate learning teachers can  Ask what they themselves expect to see/hear from the students  Observe the students during a lesson, measuring understanding, guiding and adapting as needed  Structure opportunities for interactions among teacher-student, student-teacher, and student-student  Ask what they themselves expect to see/hear from the students  Observe the students during a lesson, measuring understanding, guiding and adapting as needed  Structure opportunities for interactions among teacher-student, student-teacher, and student-student

8. Teachers Report (ES/MS) Hudson et al, 2003 Rarely or never Once/twice month Frequently or always Make connections 10%/4%34%/37%57%/59% Students explain to each other 9%/8%26%/20%66&/72% Open ended tasks 51%/29%33%/40%16%/31%

9. The vehicle for teaching: the lesson

10. Reflecting on practice:Weeks 2/3  In your group: design a lesson related to a given topic.  First draft of the lesson will be due on Friday with an electronic copy sent to a lesson study consultant who will provide feedback over the weekend on the lesson.  Revise the lesson Monday and Tuesday.  Teach a selected portion of the lesson to the larger group in one of the three rooms on Wednesday and Thursday.  In your group: design a lesson related to a given topic.  First draft of the lesson will be due on Friday with an electronic copy sent to a lesson study consultant who will provide feedback over the weekend on the lesson.  Revise the lesson Monday and Tuesday.  Teach a selected portion of the lesson to the larger group in one of the three rooms on Wednesday and Thursday.

11. Typical flow of a class United States  Demonstrate a procedure  Assign similar problems to students as exercises  Homework assignment United States  Demonstrate a procedure  Assign similar problems to students as exercises  Homework assignment Japan  Present a problem to the students without first demonstrating how to solve the problem  Individual or group problem solving  Compare and discuss multiple solution methods  Summary, exercises and homework assignment Takahashi, 2005

12. Introduction: Hatsumon Thought provoking question Key question – shu hatsumon Individual or small group work Walking among the desks – kikan- shido Anticipated student solutions Student solutions - Noriage Massaging students’ ideas Summing up- Matome Bass et al, 2002 The Lesson

13. The Teaching Principle Effective teaching requires understanding what students know and need to learn and challenging and supporting them to learn it well. (NCTM, 2000)

14. The Teaching Principle Effective teaching requires understanding what ALL students know and need to learn and challenging and supporting them to learn it well. (NCTM, 2000)

15. Research suggests that students learn when they are doing something they consider worthwhile are actively involved in choosing strategies get feedback on what they are thinking before it is too late struggle with a concept before they are given a lecture NRC, 1999

16. The lesson  Context  Explicit mathematical goals  The investigation  Launch  Tasks  How students will work  Managing discussion  Questions  Student work  Evidence of learning- begin with the end in mind  Context  Explicit mathematical goals  The investigation  Launch  Tasks  How students will work  Managing discussion  Questions  Student work  Evidence of learning- begin with the end in mind

17. Our work Be explicit about the grade level and context of the lesson Include all of the components of the lesson Be explicit about the grade level and context of the lesson Include all of the components of the lesson

18. Mathematical objectives “Explore the Pythagorean Theorem” “Deepen their understanding of proof” “Learn about lines” “Section 2.4” “Learn to work in groups and share” “Explore the Pythagorean Theorem” “Deepen their understanding of proof” “Learn about lines” “Section 2.4” “Learn to work in groups and share”

19. How will you gain and maintain student attention: A launch  Invites students into the mathematics  Engages students  Connects to the mathematics in the lesson  Short How might you launch a first discussion of solving systems of equations?  Invites students into the mathematics  Engages students  Connects to the mathematics in the lesson  Short How might you launch a first discussion of solving systems of equations?

20. Choosing Tasks - a purpose  Master routine procedures  Develop conceptual understanding  Explore new mathematical terrain  Secure understanding  Do mathematics  Master routine procedures  Develop conceptual understanding  Explore new mathematical terrain  Secure understanding  Do mathematics

21. Opportunities for discussion Tasks have to be justified in terms of the learning aims they serve and can work well only if opportunities for pupils to communicate their evolving understanding are built into the planning. (Black & Wiliam, 1998)

22. In the figure below, what fraction of the rectangle ABCD is shaded? A B D C a)1/6 b)1/5 c)1/4 d)1/3 e) 1/2 NCES, 1996 D

23. Dekker & Querelle, 2002

25. Characterizing Tasks & Knowledge Cognitive DemandKnowledge Needed Doing Mathematics Strategic Choosing, formulating strategies Procedures with Connections Schematic explaining, justifying, predicting Procedures without Connections Procedural performing procedures Memorization Declarative defining, giving examples (Stein, et.al., 2001) (Shavelson et al, 1999) Increasing Demand

26. Which shape will hold the same amount of spaghetti and be the most economical? Area of base Surface area VolumeRatio of surface area to volume Cylinder Rectangular prism Shape 3 MDoE, 2003

27. The questions  “… the only point of asking questions is to raise issues about which a teacher needs information or about which students need to think.”  The responses a task might generate and the ways of following up have to be anticipated  Questions are a significant part of teaching where attention is paid to how they are constructed, used to explore, then develop learning  “… the only point of asking questions is to raise issues about which a teacher needs information or about which students need to think.”  The responses a task might generate and the ways of following up have to be anticipated  Questions are a significant part of teaching where attention is paid to how they are constructed, used to explore, then develop learning (Black et al, 2004)

28. Using Inquiry  Engaging in mathematical thinking  Justifying, thinking procedurally and reflecting  Making and explaining connections  Doing procedures  Knowing facts  Engaging in mathematical thinking  Justifying, thinking procedurally and reflecting  Making and explaining connections  Doing procedures  Knowing facts

29. Inquiry Questions  Compare and contrast: How are they alike? How different?  Predict forward: “What would happen if.. ?”  Predict backward: “How can I make.. happen?” “Is it possible to... ?”  Analyze a connection/relationship: “When will... be (larger,equal to, exactly twice, …) compared to...?” “When will.. be as big as possible?”  Generalize/make conjectures: “When does... work?” “Under what conditions does … behave this way?” “Describe how to find...?” “Is this always true?”  Justify/prove mathematically: “Why does... work?”  Consider assumptions inherent in the problem and what would happen if they were changed  Interpret information, make/ justify conclusions: “The data support… ; “This… will make ….happen because…”  Compare and contrast: How are they alike? How different?  Predict forward: “What would happen if.. ?”  Predict backward: “How can I make.. happen?” “Is it possible to... ?”  Analyze a connection/relationship: “When will... be (larger,equal to, exactly twice, …) compared to...?” “When will.. be as big as possible?”  Generalize/make conjectures: “When does... work?” “Under what conditions does … behave this way?” “Describe how to find...?” “Is this always true?”  Justify/prove mathematically: “Why does... work?”  Consider assumptions inherent in the problem and what would happen if they were changed  Interpret information, make/ justify conclusions: “The data support… ; “This… will make ….happen because…”

30. Scaffolding  Framing of tasks  Implementing the tasks - the questions posed - the way questions are answered  Framing of tasks  Implementing the tasks - the questions posed - the way questions are answered

31. Types of math problems Stigler & Hiebert, 2004

32. How teachers implemented math problems Stigler & Hiebert, 2004

33. Managing learning- about the details  How will students work? How will they be arranged and for what reason?  What tools will be useful and how should they be made available?  How will the work be recorded?  When will you break for a discussion and why?  How will they share their work?  How will students work? How will they be arranged and for what reason?  What tools will be useful and how should they be made available?  How will the work be recorded?  When will you break for a discussion and why?  How will they share their work?

34. Managing Student Responses “…when teachers learn to see and hear students’ work during a lesson and to use that information to shape their instruction, instruction becomes clearer, more focused, and more effective.” (NRC, 2001, p.350 )

35. What and how the work is recorded matters 2x 15 + 25x3 = 150 15 25 x2x3 30 +75 105 2x.15 + 3x.25 = 1.50

36. What and how the work is recorded matters 2x15 + 3x25 = 30+ 75 = 105 3x15 + 2x25 = 45 + 50 = 95 4x15 + 2x25 = 60 + 50 = 110 Goal: Ax+By = C

37. Strategies for learning  Brainstorming discussion  Drawing/artwork  Field trips  Games  Humor  Graphic Organizers, maps,  Manipulatives/labs/ experiments  Music  Movement  Metaphor/analogy/simile  Mnemonic devices  Brainstorming discussion  Drawing/artwork  Field trips  Games  Humor  Graphic Organizers, maps,  Manipulatives/labs/ experiments  Music  Movement  Metaphor/analogy/simile  Mnemonic devices Projects/problem based instruction Reciprocal teaching (peer teaching) Role play Storytelling Technology Visualization Visuals Apprenticeships Writing/journals (M. Tate, 2005)

38. Our work Area of a quadrilateral Slope Similarity Area of a quadrilateral Slope Similarity

39. Our work Tuesday: Specific mathematical goal and context selected; tasks chosen Wednesday: Initial framing of lesson, anticipated student solutions Thursday: Description of how the class will be organized and managed, key questions identified Friday: First draft produced Tuesday: Specific mathematical goal and context selected; tasks chosen Wednesday: Initial framing of lesson, anticipated student solutions Thursday: Description of how the class will be organized and managed, key questions identified Friday: First draft produced

40. Suggestions Tuesday: Brainstorm with whole group but come to consensus Everyone does the task(s) and shares solutions - thinking about their own approach and how their students might work Divide the work on Thursday so a subset is responsible for each segment; share the last 20 minutes of the hour Tuesday: Brainstorm with whole group but come to consensus Everyone does the task(s) and shares solutions - thinking about their own approach and how their students might work Divide the work on Thursday so a subset is responsible for each segment; share the last 20 minutes of the hour

41. Working together  Gain insights  Better able to understand students  Accountability  Troubleshoot  Stimulate our own teaching  Gives you perspectives you may not have thought of  Gain insights  Better able to understand students  Accountability  Troubleshoot  Stimulate our own teaching  Gives you perspectives you may not have thought of

42. References  Bass, H., Usiskin, Z, & Burrill, G. (Eds.) (2002). Classroom Practice as a Medium for Professional Development. Washington, DC: National Academy Press.  Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp. 139-148.  Black, P., Harris, C., Lee, C., Marshall, B. & Wiliam, D. (2004). Working inside the black box: Assessment for learning in the classroom.. Phi Delta Kappan. (86,1) p8  Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., & Murray, H. (1992). Introducing the critical features of classrooms. in Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann  Dekker,T. & Querelle, N. (2002). Great assessment problems (and how to solve them). CATCH project www.fi.uu.nl/catch www.fi.uu.nl/catch  Bass, H., Usiskin, Z, & Burrill, G. (Eds.) (2002). Classroom Practice as a Medium for Professional Development. Washington, DC: National Academy Press.  Black, P. & Wiliam, D. (1998). “Inside the Black Box: Raising Standards Through Classroom Assessment”. Phi Delta Kappan. Oct. pp. 139-148.  Black, P., Harris, C., Lee, C., Marshall, B. & Wiliam, D. (2004). Working inside the black box: Assessment for learning in the classroom.. Phi Delta Kappan. (86,1) p8  Hiebert, J., Carpenter, T., Fennema, E., Fuson, K., Wearne, D., & Murray, H. (1992). Introducing the critical features of classrooms. in Making sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann  Dekker,T. & Querelle, N. (2002). Great assessment problems (and how to solve them). CATCH project www.fi.uu.nl/catch www.fi.uu.nl/catch

43.  Hudson, S., McMahon, K., & Overstreet, C. (2002). The 2000 National Survey of Science and Mathematics Educators: Compendium of Tables. Chapel Hill NC: Horizon Research, Inc. Michigan Department of Education. (2003). MMLA Lesson Study Project. Burrill, G., Ferry, D., & Verhey R. (Eds). Lansing, MI..  National Center for Educational Statistics. (1996). National Assessment for Educational Progress Released Item.  National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston VA: Author  National Research Council. (1999). How People Learn: Bain, mind, experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press.  Roodhardt, A., Kindt, M., Burrill, G., & Spence, M. (1997). Patterns and Symbols. From Mathematics in Context. Directed by Romberg, T. & de Lange, J. Chicago IL: Encyclopedia Britannica  Hudson, S., McMahon, K., & Overstreet, C. (2002). The 2000 National Survey of Science and Mathematics Educators: Compendium of Tables. Chapel Hill NC: Horizon Research, Inc. Michigan Department of Education. (2003). MMLA Lesson Study Project. Burrill, G., Ferry, D., & Verhey R. (Eds). Lansing, MI..  National Center for Educational Statistics. (1996). National Assessment for Educational Progress Released Item.  National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston VA: Author  National Research Council. (1999). How People Learn: Bain, mind, experience,and school. Bransford, J. D., Brown, A. L., & Cocking, R. R. (Eds.). Washington, DC: National Academy Press.  Roodhardt, A., Kindt, M., Burrill, G., & Spence, M. (1997). Patterns and Symbols. From Mathematics in Context. Directed by Romberg, T. & de Lange, J. Chicago IL: Encyclopedia Britannica

44.  Shavelson, R., Ruiz-Primo, M.A., Li, M., and Ayala, C.C. (August 2003). Evaluating New Approaches to Assessing Learning, CSE Report 604, National Center for Research on Evaluation, Standards, and Student Testing (CRESST), Center for the Study of Evaluation (CSE), and the UCLA Graduate School of Education & Information Studies  Stigler, J. and Hiebert, J., Improving Math Teaching. In Improving Achievement in Math and Science. 2004 (pp. 12-17).  Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York NY: Teachers College Press.  Takashi, A. (2005). Presentation at Annual Meeting of Association of Mathematics Teacher Educators.  Tate,M. (2006). Designing Lessons for Learning. Presentation at TI International Technology Conference. Denver CO.  Shavelson, R., Ruiz-Primo, M.A., Li, M., and Ayala, C.C. (August 2003). Evaluating New Approaches to Assessing Learning, CSE Report 604, National Center for Research on Evaluation, Standards, and Student Testing (CRESST), Center for the Study of Evaluation (CSE), and the UCLA Graduate School of Education & Information Studies  Stigler, J. and Hiebert, J., Improving Math Teaching. In Improving Achievement in Math and Science. 2004 (pp. 12-17).  Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing standards-based mathematics instruction: A casebook for professional development. New York NY: Teachers College Press.  Takashi, A. (2005). Presentation at Annual Meeting of Association of Mathematics Teacher Educators.  Tate,M. (2006). Designing Lessons for Learning. Presentation at TI International Technology Conference. Denver CO.