Learning Mathematics for Teaching Kathy Kubota-Zarivnij & Laurie Moher August 22, 2006

1. Learning Mathematics for Teaching Kathy Kubota-Zarivnij & Laurie Moher August 22, 2006

2. Learning Mathematics for Teaching presented by Kathy Kubota-Zarivnij Student Achievement Officer, Literacy and Numeracy Secretariat Laurie Moher Numeracy Coach, Kawartha-Pine Ridge District School Board

3. What Do You Think? • What have you improved in your mathematics teaching and learning practices? 2. What are you still working on to improve? 3. Why would teachers want to improve their mathematics teaching? Teachers would be willing to engage in improvement if they: • recognized a link between poor student learning and their instructional practices • studied and problem solved ways to improve their practice with other educators on a regular basis • believed that it would help improve student learning Purposes for Improving

4. What Do You Know About Coaching? What Do You Want to Learn? KWL Coaching Chart MetacognitiveJournaling

5. How is learning mathematics for teaching different than learning mathematics content or learning mathematics pedagogy? In this session, numeracy coaches will: develop a understanding of learning mathematics for teaching through a variety of strategies. develop some planning and implementation strategies for use during the school year. Job-embedded professional learning approaches: Co-teaching Coaching Teacher inquiry/ study Learning Goals Setting Learning Goals

6. Introductions … Who’s Here? PART B Sort your group’s patterning and algebra examples. Describe your sorting rule. Connecting informal, lived, embodied mathematics to formal mathematics Examples of Patterning & Algebra in Our Daily Lives label 1 label 2 label 3 label 4 Co-Teaching

7. Co-Teaching • What are some key aspects of co-teaching? • What does the numeracy coach do? • What do the teacher(s) do? • How does co-teaching help teachers learn mathematics for teaching? • How does co-teaching help students learn?

8. Co-Teaching What’s the Difference? … from Coaching binder • “Co-teaching is an informal professional learning arrangement in which teachers with different knowledge, skills and talents have agreed to share responsibility for designing, implementing, monitoring and/or assessing a curriculum program for a class of students on a regular basis (e.g., biweekly, monthly, or per term).” • “The purpose of co-teaching is to enable groups of teachers to improve their instruction and their understanding of students’ thinking and learning through shared observation, and analysis of student work.” • “Co-teaching makes it possible for teachers to engage in teaching as collaborative problem-solving.” Same/Different

9. Coaching What’s the Difference? … from Coaching binder • “Coaching is a formal relationship that is established by a third party organizer (e.g., principal, curriculum leader or supervisory officer) or between two parties (e.g., an inviting teacher and a coach) to meet a particular learning goal.” • “Because the coach is assigned a formal role, it is that role which defines the relationship.” • “Coaching involves teachers in processes in which they collaborate, refine, reflect, conduct research, expand on ideas, build skills and knowledge and problem solve in order to improve student learning and achievement.” • “Yet coaching needs to be non-evaluative and build upon a foundation of mutual respect.” Same/Different

10. Teacher Inquiry/StudyWhat’s the Difference? … from Coaching binder • “Inquiry engages teachers in working on dilemmas and difficult situations that will appear in every teacher’s practice. Participants learn that solutions depend on contextual information and that there is never only one answer to any one problem.” • “They re-examine their strategies and are supported and stimulated to try fresh ways of dealing with dilemmas in practice.’ • “Instead of an expert or theoretician telling teachers what works and should be carried out in practice, case participants are able to revisit scenarios in their own classrooms, draw on their experiences of success or failure and share expertise from a variety of perspectives.” Same/Different

11. What Do You Think? Why might teachers be interested in: • co-teaching • a coaching relationship • teacher inquiry/study? Teachers’ immediate, short-term, and long-term classroom interests, dilemmas, and needs must be to professional learning approaches within a systematic, strategic, and collaborative within a learning job-embedded framework that is doable.

12. What Are Teacher’s Classroom Dilemmas, Interests, And Needs? • Understanding the sequence and relationship between math strands within textbook programs and materials within and across grade levels • Knowing the relationship between mathematical ideas, conceptual models, terms, and symbols • Generating and using strategic examples and different mathematical representations using manipulatives • Developing students’ mathematical communication - description, explanation, and justification • Understanding and evaluating the mathematical significance of students’ comments and coordinating discussion for mathematics learning

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

14. What do you think a teacher needs to know and be able to do, mathematically, to teach mathematics? Let’s take a classroom field trip … What Do You Think? Analyzing Student Learning

15. Organize into groups of three. Follow the four steps. Write down two numbers. Write down another two numbers which total 36. Write down a third set of two numbers with a difference of 8. Write down a fourth set of two numbers with a total of 36 and a difference of 8. How could you adjust your pairs of numbers to meet the criteria #4. Coaching focuses on students’: Drawn, modelled, and written mathematical work learning actions and interactions Oral, modelled, and written evidence of learning LET’S DO MATH as a Teacher! Analyze Math Thinking

16. Pedagogical Content Knowledge(Schulman,1985)compared toMathematics for Teaching (Ball, 2006) Learning Mathematics for Teaching

17. HOW and WHY Should We Be Learning Mathematics for Teaching Learning Mathematics for Teaching

18. Deborah Loewenberg BallMathematics for Teaching • Expert personal knowledge of subject matter is often, ironically inadequate for teaching. • It requires the capacity to deconstruct one’s own knowledge into a less polished and final form where critical components are accessible and visible • Teachers must be able to do something perverse: work backward from mature and compressed understanding of the content to unpack its constituent elements • Teachers need to make mathematical ideas accessible to others and understand others’ mathematical thinking • Teachers must be able to work with content for students in its growing, unfinished state Learning Mathematics for Teaching

19. Brent Davis and Elaine SimmtMathematics for Teaching Teachers have embodied, tacit knowledge of mathematics for teaching. Mathematics knowledge is recursively elaborated across the curriculum. Learning Mathematics for Teaching

20. Mathematics Curriculum as Recursively Elaborate Visualize Patterning and Algebra curriculum from Kindergarten to Grade 12 as a growing fractal tree. Iteration 2 Seed Generator Iteration 1 Iteration 2 Iteration 3 Iteration 4 Iteration 5 Iteration 6 Learning Mathematics for Teaching

21. Curriculum – Recursively ElaborateP & A - K to Grade 2 Seed Generator Iteration 1 Learning Mathematics for Teaching

22. Curriculum – Recursively ElaborateP & A - Grades 3 to 5 Expectations Iteration 2 Iteration 3 Iteration 1 Iteration 4 Learning Mathematics for Teaching

23. Curriculum – Recursively ElaborateP & A Grades 6 to 8 Expectations Iteration 5 Iteration 6 Learning Mathematics for Teaching

24. Which Grade Level is Most Appropriate?Choosing Appropriate Tasks

25. Teachers often relate mathematics to every day life events so that students’ mathematical thinking and doing is situated within a meaningful context. Chinese teacher’s mathematical knowledge was rooted and intertwined with real world contexts. American teachers identified contexts for problems that were often superficially connected to the mathematics. Liping Ma Mathematics for Teaching Learning Mathematics for Teaching

26. Fosnot and DolkMathematics for Teaching • Context problems need to be connected closely to the students’ lives … see mathematics in context • Problems should be designed to anticipate and develop students’ mathematical modeling of the real world • Problems for mathematics learning have built-in constraints to support and stretch the students’ mathematizing Learning Mathematics for Teaching

27. Brent DavisUniversity of British ColumbiaCoach/Teacher as “Consciousness of the Collective” Consciousness of the Collective

28. Numeracy Coaching GoalsLearning Mathematics for Teaching • Be precise and consistent with the use of oral and written mathematical language – drawings, symbols, terms • Figure out why procedures work, not just how to do them • Try to solve problems in more than one way, using different representations • Listen to and probe others’ thinking, especially when struggling • Study students’ thinking and work • Analyze which manipulatives are/not appropriate for students’ modelling their mathematical thinking and problem solving • Learn to listen and make sense of student thinking • Coordinate students sharing their mathematical solutions for learning Setting Learning Goals

29. THINK BACK - What have your learned so far during this coaching institute? RECORD three coaching implementation strategies that you will use in September. GIVE one strategy. GET one strategy. What Do You Know?Coaching Implementation Strategies Coaching Implementation Strategies GIVE AND GET Give and Get

30. Let’s Do Math !! • What questions would you ask about it? • How would you have the students do math with it? • How would you facilitate student thinking and discussion? Here’s a partial number chain pattern. Developing Math Tasks/Prompts

31. Questions and Prompts … • Extend the number patterns. • Record the number patterns. • What patterning rule(s) did you use? Explain. • Look at a set of numbers in the pattern. How are these numbers related to each other? Developing Math Tasks/Prompts

32. Coaching • What are some key aspects of coaching? • What does the numeracy coach do? • What do the teacher(s) do? • How does coaching help teachers learn mathematics for teaching? • How does coaching help students learn?

33. LET’S DO MATH as a Teacher! There are 36 children on school bus. There are 8 less boys than girls. How many boys? How many girls? • Solve this problem in 2 different ways. • Show your work. • Explain your solutions using one or more operations. Compare your solutions. How are they similar? How are they different? Solving in Different Ways

34. Constructing a Collective ThinkpadJapanese Bansho • What will the teacher do to understand the range of student responses? • What will the teacher do to organize class discussion so it builds mathematical knowledge from student responses? Japanese Bansho

35. What does Differentiated Algebraic Knowledge Look Like?

36. What does Differentiated Algebraic Knowledge Look Like? boys

37. What does Differentiated Algebraic Knowledge Look Like?

38. What does Differentiated Algebraic Knowledge Look Like?

39. What does Differentiated Algebraic Knowledge Look Like? Why were these solutions chosen to share with students? Which order should these solutions be shared? Why? Japanese Bansho

40. Why is It Important to Understand Differentiated Mathematical Knowledge? From “The Teaching Gap” (Stiger & Hiebert, 1999) … Japanese teachers • view individual differences as a natural characteristic of a group; provides a range of ideas and solution methods for students’ discussion and reflection; • tailoring instruction to specific students is seen as unfairly limiting and prejudging what students are capable of learning; • all students should have the opportunity to learn the same material

41. THINK BACK - What have your learned so far during this coaching institute? RECORD three coaching planning strategies that you will use in September. GIVE one strategy. GET one strategy. What Do You Know?Coaching Planning Strategies Coaching Planning Strategies GIVE AND GET Give and Get

42. Teacher Inquiry and StudyCollaborative Inquiry for Learning Mathematics (CIL-M) • What are some key aspects of teacher inquiry/study? • What does the numeracy coach do? • What do the teacher(s) do? • How does teacher inquiry/study help teachers learn mathematics for teaching? • How does teacher inquiry/study help students learn? It is a job-embedded professional learning framework that is comprised of: Co- teaching, Coaching, and Teacher inquiry/ study

43. What Did You Learn About Coaching? KWL Coaching Chart Metacognitive Journaling

44. References • Ball, D. et al (2006). NCSM and NCTM presentations • Ball, D. & Bass, H. (2000). Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics. In J. Boaler (Ed.), Multiple perspectives on the teaching and learning of mathematics (pp. 83-104). Westport, CT: Ablex. • Ball, D. Hill, H., and Bass, H. (2005). Knowing Mathematics for Teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, pp. 14,16,17,20-22,43-46. • Davis, B. and Simmt, E. (2005). Mathematics-For-Teaching: An ongoing investigation of the mathematics that teachers (need to) know. Educational Studies in Mathematics, pp. 1-27. • Fosnot, C.T. & Dolk, M. (2002).Young mathematicians at work: Constructing fractions, decimals, and percents. Portsmouth, NH: Heinemann. • Ma, L. (1999). Knowing and teaching elementary mathematics. NJ: Lawrence Erlbaum. • Schulman, L.S. (1987). Knowledge and teaching: Foundations of the new form. Harvard Educational Review, 57, 1-22. Read Professional Resources

45. A Few Algebra Resources • Lawrence, A. & Hennessy, C. (2002). Lessons for algebraic thinking. CA: Math Solutions. Pearson Education • Mason, J., Graham, A., Johnston-Wilder, S. (2005). Developing Thinking in Algebra. UK: Open University Chapters Indigo • Small, M. (2005). PRIME: Patterns and algebra. Toronto: Nelson-Thomson Nelson-Thomson • Thompson, V. & Mayfield-Ingram, K. (1998). Family math: The middle school years – Algebraic reasoning and number sense. CA: EQUALS Spectrum Educational Products • Wiebe, A. et al. (2001). Multiplication the algebraic way. CA: AIMS Spectrum Educational Products Find & Analyze Resources

46. Planning Identifying purposes for improving Setting learning goals Read professional resources Find and analyze resources Implementation Same/Different Analyze and explain Analyze student learning Being the consciousness of the collective Developing math tasks and prompts Solving in different ways Japanese Bansho Give and Get Metacognitive journaling Learning Mathematics for Teaching - Strategies Used • Job-Embedded Professional Learning • coaching • co-teaching • teacher inquiry/study

47. It was GREAT to MEET you and LEARN with you! Have a GREAT year! Keep in touch with us and with one another. KKZ … kathryn.kubota-zarivnij@edu.gov.on.ca Laurie …