1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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

9. 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

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

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

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. 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

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

21. 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

22. 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

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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

37. 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

38. 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