1. FACILITATORS’ SESSIONS K-1, 2-3, and 4-6 Number and Computation JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

2. Before We Get Started….  What are your questions/concerns/joys about acting as facilitators for these courses?  Write your questions/concerns/joys on an index cards. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

3. ….that’s one of the purposes of the courses. So, what is number sense? Building Number Sense JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

4.  “…good intuition about numbers and their relationships.” It develops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms” (Howden, 1989).  Flexibility in thinking about numbers and their relationships.  “Two hallmarks of number sense are flexible strategy use and the ability to look at a computation problem and play with the numbers to solve with an efficient strategy” (Cameron, Hersch, Fosnot, 2004, p. 5). Number Sense JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

5. The History of the NCTM and Standards  Curriculum and Evaluation Standards for School Mathematics (NCTM, 1989)  Professional Standards for Teaching Mathematics (NCTM, 1991)  Assessment Standards for School Mathematics (NCTM, 1995)  Principles and Standards for School Mathematics (NCTM, 2000)

6. Principles for School Mathematics  Equity  Curriculum  Teaching  Learning  Assessment  Technology

7. Content Standards for School Mathematics  Number and Operations  Algebra  Geometry  Measurement  Data Analysis and Probability

8. Process Standards for School Mathematics  Problem Solving  Reasoning and Proof  Communication  Connections  Representation

9. Standards for the Professional Development of Teachers of Mathematics  Experiencing Good Mathematics Teaching  Knowing Mathematics and School Mathematics  Knowing Students as Learners of Mathematics  Knowing Mathematical Pedagogy  Developing as a Teacher of Mathematics  Teachers’ Role in Professional Development

10. Mathematician at Work  Think of a mathematician at work. What is this person doing? Where is this person? What tools is this person using?  Draw what you “see”. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

11. If mathematics were an animal….  What would it be, and why? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

12. Core Beliefs  About mathematics, mathematics teaching and learning.  List 3-5 core beliefs about these ideas. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

13. Kathy Statz, Third-Grade Teacher I got good grades in high school algebra. I learned the procedures that the teacher demonstrated. I thought that was what mathematics was about; if you could memorize a procedure then you could do math. I didn’t even know that I didn’t understand math because I didn’t know that understanding was part of math….I have become a confident problem solver by working to understand my kids’ strategies (Thinking Mathematically, pp. 74-75). JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

14. Jim Brickweddle, First/Second-Grade Teacher There are things [in math] that I don’t understand. I am a learner along with my students. Most of us teachers don’t use the invented algorithms that the kids do. Teachers who don’t have a broad understanding of math might end up restricting kids who are thinking outside the box. I saw a teacher ask a child to solve 20 x 64. The child said, “It will be easier if 20 is 4 times 5, then I can find what 5 64s are then add that 4 times.” The teacher wasn’t sure if this would work. I tell teachers, you need to be comfortable with feeling uncomfortable (Thinking Mathematically, pp. 111-112). JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

15. How would you respond to this student who answered the following task as shown? Which of the following helps you with 12 – 7 = ? a. 12 + 7 = 19 b. 2 + 5 = 7 c. 5 + 7 = 12 d. 4 + 5 = 9

16. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Possible Responses… depends on how you are listening  No. You should have chosen (c) because it is part of the fact family.  No. Choose another answer.  What is 12 – 7? How did you get your answer?  Explain how 2 + 5 = 7 helps you find the answer.  Can you show me your reasoning using materials?

17. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 How you respond gives a glimpse into  how you are listening,  what you are listening for,  what you ignore, and  your beliefs about mathematics, mathematics teaching and learning. It also sends a message to students about what is important in your classroom.

18. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Child’s explanation: 12 - 7 = ?  L: What is 12-7?  C: 5  L: How did you get that?  C: Well, I know 12 is 2 away from 10, so I broke 7 into a 2 and a 5. Then I took away 2 from both 12 and 7 so that I had a 10 and a 5. I know what 10 - 5 is. It’s 5.  L: Why did you choose (b)?  C: Because it’s 2 + 5 = 7 and I used those numbers to find the answer.

19. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Different kinds of listening  Evaluative  Response seeking  Listening for particular responses  Set learning trajectory  Interpretive  Information seeking  Making sense of students’ sense-making  Listening for particular responses  Set learning trajectory  Hermeneutic  Moving with the students  Mathematical ideas are locations for exploration  Student contributions essentially direct the learning trajectory of the class Davis, B. (1997). Listening for differences: An evolving conception of mathematics teaching. Journal for Research in Mathematics Education 28(3), 355-376. Reston, VA: NCTM.

20. Beliefs and Listening JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011  Evaluative  Math is about getting answers  Strategies to get answers are decided by the teacher  Interpretative  Math is about making sense  Reasoning is big part of mathematics learning and assessment  Strategies to get answers could be decided by the student  Hermeneutic  Math is about exploring ideas and making sense  Teaching math is about capitalizing on students’ ideas  Strategies to get answers are decided by the student

21. Challenging Beliefs  Making beliefs explicit  Supportive environment  Where in the sessions do you remember either your beliefs or a colleague’s beliefs were challenged?  Engaging in class activities, reading Young Mathematics at Work, responding to blogs, observing videos, interviewing children, etc. – all in attempts to develop different perspectives about mathematics, mathematics teaching and learning. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

22. Mathematical Proficiency  Conceptual understanding  Procedural fluency  Strategic competence  Adaptive reasoning  Productive disposition National Research Council. (2001). Adding it up: Helping children learn mathematics. J. Kilpatrick, J. Swafford, and B. Findell (Eds.). Mathematics Learning Committee, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

23. Think about the following problem: 40,005 – 39,996 = ___. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 A student with weak procedural skills may launch into the standard algorithm, regrouping across zeros (this usually doesn’t go well), rather than notice that the number 39,996 is just 4 away from 40,000 and 5 more mean the difference is 9.

24. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Constructivism…  What does this mean to you?

25. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Continuum of Understanding Relational Instrumental Understanding

26. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Mathematical Example of Instrumental Understanding 7 x 8 = ? Knows the number 5,6 and 7,8 go in that order. So, remember that those numbers “go together.” 7 x 8 = 56 Instrumental Understanding

27. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Relational Understanding: 7 x 8 = ? 7777777777777777 14 x 2 28 x 2 56 Seven 7’s are 49, so all I need is one more 7. I know five 8’s is 40 and two 8’s is 16. 40 and 16 are 56. 4 times 7 is 28. Double that would be 56. 10 x 8 is 80. Take away 3 8’s or 24 is…60, then 56. Relational Understanding 3 times 8 is 24, double that to get 48. I need one more 8 to get 56.

28. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Continuum of Understanding Relational Instrumental Understanding Perturbation = Disequilibrium = Learning

29. How do you deal with resistant teachers? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 What is the origin of the resistance and fear?  Overwhelming to make (any) changes.  Uncomfortable with not knowing. Possible approaches  Make it less overwhelming.  Choose a part of your practice to focus on one semester.  Change is a process, not an event.  The process is a marathon not a sprint.  Share what we know from research and international studies  Research shows….  International comparisons….  Keep the focus on the students and what is best for them in terms of learning for understanding.

30. What if teachers have questions I am unable to answer?  Let’s brainstorm ideas! JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

31. Standards for Teaching Mathematics JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011  Worthwhile Mathematical Tasks  Teacher’s and Students’ role in Discourse  Learning Environment  Analysis of Teaching and Learning

32. Hiking Club Problem JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011 Two-thirds of the students in the school’s hiking club have climbed Massanutten Mountain, one-half have climbed Afton Mountain, and one-fourth have climbed both of these mountains. Only two students in the club have not climbed either mountain. How many students are in the club?

33. Questions to consider: JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011  What is the purpose of the problem?  What prior knowledge and experiences can students draw on to solve the problem?  What mathematics do the students need to know to solve the problem?  How will I present this problem?  What questions will I ask struggling students?  How might students solve the problem?

34. Procedural Steps JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

35.  2 students represent of the hiking club.  2 x 12 = 24 students in the hiking club.

36. Your Joys and Concerns/Questions What concerns/questions do you need more support with? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011