CHAPTER 7 Counting and Number Sense in Early Childhood and Primary Grades Tina Rye Sloan To accompany Helping Children Learn Math9e, Reys et al. ©2009 John Wiley & Sons  

Focus Questions Why is subitizing an important early step in number sense? What characteristics are associated with the different counting stages? How can calculators help children develop number sense? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

NCTM(2000). Principles and Standards for School Mathematics. Understand numbers, ways of representing numbers, relationships among numbers, and number systems Count with understanding and recognize “how many” in sets of objects Develop understanding of the relative position and magnitude of whole numbers and of ordinal and cardinal numbers and their connections. Connect number words and numerals to the quantities they represent, using various physical models and representations Excerpts of Expectations from the Number and Operations Standard Grades Pre-K-2 NCTM(2000). Principles and Standards for School Mathematics. Master 7-2: Number and Operations Excerpts (number concepts) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Number Sense Number Sense includes: An understanding of number concepts and operations on these numbers. The development of useful strategies for handling numbers and operations. The facility to compute accurately and efficiently, to detect errors, and to recognize results as reasonable. Master 7-1: Number Sense Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Number Sense The ability and inclination to use this understanding in flexible ways to make mathematical judgments. An expectation that numbers are useful and that work with numbers is meaningful and makes sense. Master 7-1: Number Sense Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Number Sense Stages Number sense is not a finite entity that a student either has or does not have. Its development is a lifelong process, and in early childhood and elementary school number sense development involves several stages: Prenumber Concepts Early Number Development Number Development Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Number Sense Stages Classification Patterns Prenumber Concepts are those that do not rely on numbers but provide a foundation for later number concepts and skills. Classification Patterns Master 7-3: Prenumber and Early Number Concepts Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Classification Attribute blocks can be used for classification and reasoning. Size: large, small (L, S) Color: blue, red, green (B, R, G) Shape: square, triangle, pentagon, circle (S, T, P, C) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Patterns Each car in a train is like the car it follows in one or two ways, or it is different from the car it follows in one or two ways. Find the alike-and-difference pattern in each train, and describe the missing car: Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Extending a Pattern Ask students to complete number 5 Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Early Number Development Conservation Group Recognition Comparisons and One-to-one Correspondence Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Conservation of Number Groups of 5 counters are arranged in the following 3 patterns Master 7-4: Conservation of Number Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Conservation of Number Student A T: Are there more red, blue, or yellow counters? S: More blue. T: How do you know? S: I can tell by looking. T: How many of each? S: One, two, three, four, five... five red. One, two, three, four, five...five blue. One, two, three, four, five...five yellow. T: Five of each? S: Yes. T: Do you still think there are more blue? S: Yes, I can just see there's more blue. Master 7-4: Conservation of Number Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Conservation of Number Student B T: Are there more red, blue, or yellow counters? S: They're the same. T: How do you know? S: I counted them. T: How many of each? S: One, two, three, four, five...Five red. Five blue. Five yellow. T: Five of each? S: Yes. Master 7-4: Conservation of Number Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Group Recognition How many birds do you see in each group? Is one group easier to subitize (recognize) than another? Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Comparisons and One-to-One Correspondence Figure 7-5 Models for making comparisons by (a) counting; (b) physically comparing without counting; and (c) one-to-one correspondence Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Number Development - Connecting groups with number names, including oral and written cardinal and ordinal numbers - Group recognition - Counting forward and backward - Skip counting - Establishing benchmarks of quantities, such as 5 or 10 - Place value Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Principles One-to-One Rule Stable Order Rule Order Irrelevance Rule Cardinality Rule Master 7-5: Counting Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Counting Stages Rote Counters may know the proper counting sequence, but they may not always be able to maintain a correct correspondence between the objects being counted and the number names. Rational Counters not only uses one-to-one correspondence but also are able to answer the question about the number of objects being counted. Master 7-5: Counting Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Counting Strategies Counting On Counting Back Skip Counting 3,6,9,12,15…. Master 7-5: Counting Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Counting Practice Counting practice should include counting on and counting back. A calculator is a valuable instructional tool that helps improve children’s ability to count, but it is also a powerful counting tool that they love to explore. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Developing Number Benchmarks The five-frame (5 x1 array) and the ten-frame (5 x 2 array) use these early benchmarks. Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting: Making Connections Figure 7-15 Connections form representations on the ten-frame Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Cardinal, Ordinal, and Nominal Numbers Cardinal- “How many?” Ordinal- “Which one?” (i.e. first, second, third) Nominal- “Name” (i.e. phone number or address) Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Writing Numerals Children should begin by tracing the digits and here is a recommended stroke sequence: Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting Principles Student A T: Here are some blocks in a row. Start with this one on the end and count them. S: One, two, three, four, five, SIX. There are six blocks. T: What if you start at the other end of the row and count them? S: One, two, three, four, five, SIX. There are six. Master 7-6: Counting Principle Examples Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting Principles (cont’d) Student B T: Here are some red blocks in a row. Start with this one on the end and count them. S: (Touches each of the 5 blocks) One, two, three, five, six. Six red blocks T: Now count these blue blocks. S: (Touches each of the 4 blocks) One, two, three, five. Five blue blocks. Master 7-6: Counting Principle Examples Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting Principles (cont’d) Student C T: Here are some blocks in a row. Start with the one on this end and count them. S: One, two, three, four, five, six. There are six. T: What if you start at the other end of the row and count them? S: I already counted them! There are six! Master 7-6: Counting Principle Examples Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Counting Principles (cont’d) Student D T: Here are some red blocks (4) in a row. Start with this one on the end and count them. S: (Points to each but says two numbers with each point) One, two, three, four, five, six, seven, eight. Eight red blocks. Master 7-6: Counting Principle Examples Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Construct a Pattern The color spinner will be used to select the colors to be used in constructing the pattern. Using the colors chosen, make a pattern that repeats at least twice in the time allotted. When the time is up, compare patterns with others. Did you come up with a unique pattern idea? Master 7-7: Construct a Pattern Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Construct a Pattern Try another pattern with new colors or different types of manipulatives. Why should students have an opportunity to participate in patterning activities? Do you have other suggestions for patterning activities? Master 7-7: Construct a Pattern Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Chad: Third Month of Kindergarten Student Work Samples Chad: Third Month of Kindergarten Master 7-8: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Cody: Third Month of Kindergarten Student Work Samples (cont’d) Cody: Third Month of Kindergarten Master 7-9: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Steven: Third Month of Kindergarten Student Work Samples (cont’d) Steven: Third Month of Kindergarten Master 7-10: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Elizabeth: Third Month of Kindergarten Student Work Samples (cont’d) Elizabeth: Third Month of Kindergarten Master 7-11: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Andrea: Third Month of Kindergarten Student Work Samples (cont’d) Andrea: Third Month of Kindergarten Master 7-12: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Jason: Third Month of Kindergarten Student Work Samples (cont’d) Jason: Third Month of Kindergarten Master 7-13: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009

Joshua: Fifth Month of Second Grade Student Work Samples (cont’d) Joshua: Fifth Month of Second Grade Master 7-14: Writing Sample Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math, 9th Edition, © 2009