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

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

  4. Number Sense • “…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). 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 ofTeachers 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 Share your ideas with a neighbor. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  16. 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? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  17. 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. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  18. 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. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  19. 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. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  20. Beliefs and Listening • 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 JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  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 = ___. 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. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

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

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

  26. Mathematical Example ofInstrumental 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 JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

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

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

  29. How do you deal with resistant teachers? 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. JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

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

  32. Hiking Club Problem 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? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

  33. Questions to consider: • 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? JMU MSP K-3 and 4-6 Grants, Facilitator Module, 2010-2011

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

  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