teaching number and number sense.ppt

number sense

1. Counting and Number
Sense in Early
Childhood and
Primary Grades
CHAPTER 7
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,
9
th
Edition, © 2009

3. 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,
9
th
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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,
9
th
Edition, © 2009

7. 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,
9
th
Edition, © 2009
Number Sense Stages

8. 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,
9
th
Edition, © 2009
Attribute blocks can be used for classification
and reasoning.

9. Patterns
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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:

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

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

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

13. 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,
9
th
Edition, © 2009
Conservation of Number

14. 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,
9
th
Edition, © 2009
Conservation of Number

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

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

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,
9
th
Edition, © 2009

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

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

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

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,
9
th
Edition, © 2009

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

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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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!
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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.
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
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?
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009

32. Chad: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples

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

34. Steven: Third
Month of
Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples (cont’d)

35. Elizabeth: Third
Month of
Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples (cont’d)

36. Andrea: Third Month of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples (cont’d)

37. Jason: Third Month
of Kindergarten
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples (cont’d)

38. Joshua: Fifth Month of Second Grade
Reys/ Lindquist/ Lamdin/ Smith, Helping Children Learn Math,
9
th
Edition, © 2009
Student Work Samples (cont’d)