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Counting and Number Sense in Early Childhood and Primary Grades
CHAPTER 7
Tina Rye SloanTo 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
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.
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.
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.
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
Prenumber Concepts are those that do not rely on numbers but provide a foundation for later number concepts and skills.
•Classification•Patterns
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Number Sense Stages
Classification
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
Attribute blocks can be used for classification and reasoning.
Patterns
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
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:
Extending a Pattern
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Ask students to complete number 5
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Groups of 5 counters are arranged in the following 3 patterns
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.
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Conservation of Number
Group Recognition
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
How many birds do you see in each group? Is one group easier to subitize (recognize) than another?
Comparisons and One-to-One Correspondence
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Figure 7-5 Models for making comparisons by (a) counting; (b) physically comparing without counting; and (c) one-to-one correspondence
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
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.
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….
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Figure 7-15 Connections form representations on the ten-frame
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
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Children should begin by tracing the digits and here is a recommended stroke sequence:
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.
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.
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!
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.
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?
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?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Chad: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples
Cody: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)
Steven: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)
Elizabeth: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)
Andrea: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)
Jason: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)
Joshua: Fifth Month of Second Grade
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,9th Edition, © 2009
Student Work Samples (cont’d)