Counting and Number Sense in Early Childhood and Primary Grades CHAPTER 7 Tina Rye Sloan To...

Counting and Number Sense in Early Childhood and Primary Grades

CHAPTER 7

Tina Rye SloanTo accompany Helping Children Learn Math9e, Reys et al.

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?

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.

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.

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.

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

Prenumber Concepts are those that do not rely on numbers but provide a foundation for later number concepts and skills.

•Classification•Patterns

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)

Attribute blocks can be used for classification and reasoning.

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:

Extending a Pattern

Ask students to complete number 5

Early Number Development

•Conservation•Group Recognition•Comparisons and One-to-one

Correspondence

Conservation of Number

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.

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.

Group Recognition

How many birds do you see in each group? Is one group easier to subitize (recognize) than another?

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

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

Counting: Principles

•One-to-One Rule

•Stable Order Rule

•Order Irrelevance Rule

•Cardinality Rule

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.

Counting: Counting Strategies

•Counting On

•Counting Back

•Skip Counting 3,6,9,12,15….

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.

Counting: Developing Number Benchmarks

The five-frame

(5 x1 array) and the ten-frame (5 x 2 array) use these early benchmarks.

Counting: Making Connections

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)

Writing Numerals

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.

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.

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!

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.

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?

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?

Chad: Third Month of Kindergarten

Student Work Samples

Cody: Third Month of Kindergarten

Student Work Samples (cont’d)

Steven: Third Month of Kindergarten

Elizabeth: Third Month of Kindergarten

Andrea: Third Month of Kindergarten

Jason: Third Month of Kindergarten

Joshua: Fifth Month of Second Grade

Counting and Number Sense in Early Childhood and Primary Grades CHAPTER 7 Tina Rye Sloan To...

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