1. Brief Description: The NCTM “Focal Points” emphasize place value, addition, and subtraction in second grade. Learn how to use chip trading and a story about the hidden kingdom of Fiveton to get students truly invested in building understanding of these vital concepts. Mark Roddy – Seattle University Master in Teaching Program NCTM 2008 Annual Meeting & Exposition April 10, 2008 Salt Lake City, Utah

2. Preview Defining the problem Introduction to Chip Trading A quick look at the research Imagining a solution A few days and several activities in Fiveton Questions, comments, conversation

3. Abdalla’s Pre-test

4. Chip Trading A white is worth one. It takes five whites to get a red. It takes five reds to get a blue. It takes five blues to get a green etc.

5. Fuys, D. & R. Tischler (1979). Teaching Mathematics in the Elementary School.

6. NCTM Focal Points 1.Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts 2.Developing an understanding of whole number relationships, including grouping in tens and ones 3.Composing and decomposing geometric shapes NCTM (2006). “Focal Points for Pre-Kindergarten Through Grade 8 Mathematics: A Quest for Coherence,” 1.Developing an understanding of the base-ten numeration system and place value concepts 2.Developing quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction 3.Developing understanding of linear measurement & facility in measuring lengths

7. NCTM Focal Points 1.Developing understandings of addition and subtraction and strategies for basic addition facts and related subtraction facts 2.Developing an understanding of whole number relationships, including grouping in tens and ones 3.Composing and decomposing geometric shapes NCTM (2006). “Focal Points for Pre-Kindergarten Through Grade 8 Mathematics: A Quest for Coherence,” 1.Developing an understanding of the base-ten numeration system and place value concepts 2.Developing quick recall of addition facts and related subtraction facts and fluency with multi-digit addition and subtraction 3.Developing understanding of linear measurement & facility in measuring lengths

8. “Children develop, discuss, and use efficient, accurate and generalizable methods to add and subtract multi-digit whole numbers.... They develop fluency with efficient procedures including standard algorithms for adding and subtracting whole numbers, understand why the procedures work (on the basis of place value and properties of operations) and use them to solve problems.” NCTM Focal Points NCTM (2006). “Focal Points for Pre-Kindergarten Through Grade 8 Mathematics: A Quest for Coherence,”

9. In Grade 2, instructional time should focus on four critical areas: (1) extending understanding of base-ten notation; (2) building fluency with addition and subtraction; (3) using standard units of measure; and (4) describing and analyzing shapes. In Grade 1, instructional time should focus on four critical areas: (1) developing understanding of +, -, and strategies for + and -within 20; (2) developing understanding of whole number relationships and place value, including grouping in tens and ones; (3) developing understanding of linear measurement and measuring lengths as Iterating length units; and (4) reasoning about attributes of, and composing / decomposing geometric shapes.

10. Marilyn Burns on Developing Understanding “Problem situations should be the starting place for developing understanding of each of the four basic operations of arithmetic – addition, subtraction, multiplication, and division – thereby establishing the need and context for computation skills. Children need to see that learning to compute serves a purpose – for solving problems.” (p. 14) Marilyn Burns (2007). About Teaching Mathematics: A K-8 Resource. (3 rd Edition)

11. National Research Council on Developing Understanding “Research indicates that students’ experiences using physical models to represent hundreds, tens and ones can be effective if the materials help them think about how to combine quantities and, eventually how these processes connect with written procedures. The models, however, are not automatically meaningful for students; the meaning must be constructed as they work with the materials.” (p. 198) National Research Council (2001). “Adding It Up: Helping Children Learn Mathematics.” J. Kilpatrick, J. Swafford, and B. Findell, (Eds.)

12. “For learning to occur, three conditions are necessary: maturity, physical experience, and social interaction.” Maturity “The older children are the more likely their mental structure will act in coordinated ways.” Physical Experience “The more experiences children have with physical objects in the environment, the more likely related understanding will develop…. Testing an idea with physical materials provides valuable feedback for understanding relationships.” Social Interaction “The more opportunities children have to interact with peers, parents and teachers, the more they hear other viewpoints that help them gain perspective on their own ideas. Social interactions stimulate children to think through their own ideas and to approach objectivity.” (p. 31) Marilyn Burns on Developing Understanding Marilyn Burns (2007). About Teaching Mathematics: A K-8 Resource. (3 rd Edition)

13. Fiveton Learning Targets 1.Students develop their understanding of place value to three places. 2.Students are able to use renaming and the standard algorithms for addition and subtraction for 2 and 3-digit numbers. 3.Students enjoy math in this context and feel confident in their grasp of Targets 1 and 2

14. Fiveton http://www.artist.omsk.ru/

16. The Dragon

17. Fiveton in schools

19. Mark Roddy – Seattle University Master in Teaching Program NCTM 2008 Annual Meeting & Exposition April 10, 2008 Salt Lake City, Utah