1. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Supporting Rigorous Mathematics Teaching and Learning Tennessee Department of Education Elementary School Mathematics, Kindergarten December 7, 2012 Developing Students’ Sense of Quantity: A Means to Mathematical Understanding

2. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Rationale 2 Number sense refers to a person’s general understanding of numbers and operations and the ability to handle daily life situations that include numbers. This includes the ability to develop useful, flexible, and efficient strategies (i.e., mental computation or estimation) for handling numerical problems. Howden, 1989; McIntosh, Reys & Reys, 1992; Reys, 1991; Reys & Yang, 1998; Sowder, 1992a, 1992b; Treffers, 1991; Yang, 2002a, 2002b Students who participate in well-designed activities are more likely to develop number sense than students who receive instruction focusing on the development of standard written algorithms and computation proficiency. Sowder, 1941; Reys, 2001 In this session, we will learn to recognize indicators of students who are developing number sense and instruction that is fostering number sense. We will also consider the role of the principal in supporting teachers so they can develop students’ number sense.

3. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Session Goals Participants will learn: The Common Core Standards for Mathematical Content. Methods of developing students’ sense of quantity. Indicators of number sense. Activities for developing number sense. 3

4. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Overview of Activities Read and discuss the CCSS Standards, specifically the Counting and Cardinality Standards and the Operation and Algebraic Thinking Standards. Read classroom cases and discuss teaching and learning that supports and illustrates the development of Number Sense. Read and discuss ways in which hands-on activities support students in developing Number Sense. 4

5. Counting and Cardinality K.CC Know number names and the count sequence. K.CC.1 Count to 100 by ones and by tens. K.CC.2 Count forward beginning from a given number within the known sequence (instead of having to begin at 1). K.CC.3 Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects). Count to tell the number of objects. K.CC.4 Understand the relationship between numbers and quantities; connect counting to cardinality. K.CC.4a When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. K.CC.4b Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. K.CC.4c Understand that each successive number name refers to a quantity that is one larger. The CCSS for Mathematical Content: Kindergarten Common Core State Standards, 2010, p. 11, NGA Center/CCSSO 5

6. Counting and Cardinality K.CC Count to tell the number of objects. K.CC.5 Count to answer “how many?” questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1–20, count out that many objects. Compare numbers. K.CC.6 Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies. K.CC.7 Compare two numbers between 1 and 10 presented as written numerals. The CCSS for Mathematical Content: Kindergarten Common Core State Standards, 2010, p. 11, NGA Center/CCSSO 6

7. The CCSS for Mathematical Practice 1.Make sense of problems and persevere in solving them. 2.Reason abstractly and quantitatively. 3.Construct viable arguments and critique the reasoning of others. 4.Model with mathematics. 5.Use appropriate tools strategically. 6.Attend to precision. 7.Look for and make use of structure. 8.Look for and express regularity in repeated reasoning. Common Core State Standards, 2010, p. 6-8, NGA Center/CCSSO 7

8. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH What is Number Sense? Read the excerpts on Number Sense. What are the authors describing? In what ways do you give students opportunities to develop Number Sense? 8

9. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Recognizing Opportunities that Develop a Sense of Quantity (Private Work Time) Read each mini-case. In what ways is the teacher giving students opportunities to develop Number Sense? Do the students have a sense of quantity? If so, why do you say they have a sense of quantity? What in the description indicates a sense of quantity? 9

10. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Sharing Our Observations (Small Group Discussion) Share your observations with each other. Make a poster with a list of strategies that you and other teachers can use to develop students’ Number Sense. Identify the indicators of Number Sense. 10

11. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Sharing Strategies for Developing Number Sense (Whole Group Discussion) Listen to one team’s observations. Be prepared to add teacher moves or student indicators that group one did not identify. 11

12. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Strategies for Developing Number Sense Read the list of strategies for developing Number Sense. Check the strategies that you have identified already. Circle those that we did not discuss. Are there other strategies that you would add to the list now that you have read the excerpts from research? 12

13. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Early Numeracy Strategies Developing spatial relationships involving hands-on experiences (i.e., provide the sensory input that helps students develop mental imagery). Focusing on the meaning of sets in the context of problems. Developing visual cues such as dot cards and patterns on the die help students see relationships. Building mental imagery expands children’s ability to think in flexible ways. Recording students’ ideas as they share them can reinforce concepts and help students make the connection between the concrete items and the abstract numbers. Asking students to compare quantities (i.e., Which is more? How do you know? Which is less? How do you know?). 13

14. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Early Numeracy Strategies 14 Solving problems involving joining, separating, grouping, and sharing helps students see how sets come together and are taken apart. Counting and showing objects to 120 helps students to hear the number pattern and to see quantities. Counting forward and backward. Ordering/sequencing sets, pictures, and numbers from least to greatest. Matching numerals to objects. Being exposed to part/whole relationships. Showing students sets and asking them to make estimations of the quantities. Talking about the estimates that are closest and furthest from the amount.

15. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Hands-On Activities In what ways might working with hands-on activities help students to develop a sense of quantity? Use each activity and discuss what students might learn from the activity. 15

16. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Numeracy Activities Oral Counting and Writing Numbers and Noticing Patterns Dice Recognition Ten-Frame Recognition and Addition Beads Dominos Spill the Pennies 16

17. LEARNING RESEARCH AND DEVELOPMENT CENTER © 2012 UNIVERSITY OF PITTSBURGH Setting Goals What are some ways you will work to develop students’ sense of quantity? 17