Helping teachers to learn about number sense. For use in the Teacher's Colelge elementary education

1. Dr. Harriet Thompson

2. Number Sense
A “good intuition about numbers and their
relationships. It develops gradually as a result of
exploring numbers, visualizing them in a variety of
contexts, and relating them in ways that are not
limited by traditional algorithms” (Howden, 1989).

3. Number Sense
The NCTM Standards call for students in Pre-K through
grade 2 to understand numbers, be able to represent them
in different ways and explore relationships among
numbers. Flexible thinking with regard to numbers
should continue to be developed as students in the upper
grades work with larger numbers, fractions, decimals, and
percents. But number sense development must begin in
kindergarten, as it forms the foundation for many ideas
that follow.
Number sense refers to a person's general understanding
of number and operations along with the ability to use this
understanding in flexible ways to make mathematical
judgments and to develop useful strategies for solving
complex problems (Burton, 1993; Reys, 1991).

4. Number and Operations Standard
Grades Pre-K-2
Understand numbers, ways of representing numbers,
relationships among numbers, and number systems
Understand meanings of operations and how they relate to
one another
Compute fluently and make reasonable estimates

5. 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

6. The meanings for the number five suggested
by young children…

7. Prenumber Concepts
Patterning
Sorting
Classifying

8. Prenumber Concepts
Many of the experiences that young children need in
order to be successful with number, do not rely on
numbers per se. Such experiences are called
prenumber experiences. Certainly, mathematics is
sometimes defined as the study of patterns, and
before one can count, they have to sort the items that
are to be counted. If I want to count the females in
the room, first I must sort the males from the females.
These concepts are really important to the
development of number.

9. Early Number Concepts
Early number concepts begin with counting. To adults the
act of counting seems natural and simple. In reality it is
the culmination of a lengthy developmental process.
Fortunately, development of this process is fostered by
every day social interaction. Children come to
kindergarten with a basic understanding of what counting
is all about, that is, that there is a set of fixed number
names, said in a specific order, which are matched one-to-
one with things, and that the last word in the sequence
tells "how many." Although children who have been
fortunate enough to live in a complex, supportive
environment possess these basic understandings, their
ability to carry out the process free of errors generally is
not fully developed and formal teaching is needed to
complete their development.

10. Counting Principles
One-to-one correspondence: Each object to be counted
must be assigned one and only one number name.
Stable order rule: The number-name list must be used in a
fixed order every time a group of objects is counted.
Order irrelevance rule: The order in which the objects are
counted doesn’t matter. The child can start with any object
and count them in any order.
Cardinality rule: The last number name used gives the
number of objects. The cardinality rule connects counting
with how many. Regardless of which block is counted first
or the order in which they are counted, the last block
named always tells the number

11. Counting Stages
Rote Counting

12. Rote Counting
 Rote counting involves only the ability to recite the number names in sequence…
 Some typical rote counting errors are displayed in the Reys text (p. 150).
 A child using rote counting may know some number names, but not necessarily the proper
sequence. Consequently, the child provides number names, but they may not be in the
correct order. (A)
 Rote counters may know the proper counting sequence, but may not always be able to
maintain a correct correspondence between the objects being counted and the number
names. (B)
In the second example, the rote counter is saying the number names faster than she is
pointing, so that number names are not coordinated with the shells being counted.
 It is also possible that the rote counter points faster than saying the words. This rote counter
is pointing to the objects but is not providing a name of each of them.
 Rational counting uses the ability to rote count, but goes one step farther. Rational counting
by ones requires the child to make a one-to one-correspondence between each number name
and one-and-only-one object. In addition, the child must realize that the last number said is
the total number of objects in the set. Children must also be taught to use partitioning
strategies, that is, to systematically separate those objects counted from those that still need
to be counted.
 Rational counting is an important skill for every primary-grade child.

13. Counting Stages
Rational Counting

14. Rational Counting Stages
Rational counting uses the ability to rote count, but
goes one step farther. Rational counting by ones
requires the child to make a one-to one-
correspondence between each number name and one-
and-only-one object. In addition, the child must
realize that the last number said is the total number
of objects in the set. Children must also be taught to
use partitioning strategies, that is, to systematically
separate those objects counted from those that still
need to be counted.
Rational counting is an important skill for every
primary-grade child.

15. Counting Strategies
 Counting On & Counting Back: In Counting On, the child gives correct number names as counting proceeds and can
start at any number and begin counting. For example, the child can begin with 7 pennies and count “eight, nine, ten”
or begin with 78 pennies and count “79, 80, 81). Counting on is an essential strategy for developing addition.
***Many children find it difficult to count backward, just as many adults find it difficult to recite the alphabet
backward. The calculator provides a very valuable instructional tool to help children improve their ability to count
backward.
 Skip Counting: In skip counting, the child gives correct names, but instead of counting by ones, counts by twos, fives,
tens, or other values. In addition to providing work with patterns, skip counting provides readiness for multiplication
and division. (Counting change…start with the largest value coin and then continue skip counting by the appropriate
value).

16. Relationships Among Numbers
 Spatial Relationships: Spatial relationships – Children can learn to recognize sets of objects in
patterned arrangements and tell how many without counting. Prior to counting, children are aware
of small numbers of things: one nose, two hands, three wheels on a tricycle. Research shows that
most children entering school can identify quantities of three things or less by inspection alone
without the use of counting techniques.
 One and Two More, One and Two Less: One and Two More…The two-more-than and two-less than
relationships involve more than just the ability to count on two or count back two. Children should
know that 7, for example, is 1 more than 6 and also 2 less than 9.
 Number Benchmarks: Benchmarks or anchors give students a reference point. Since 10 plays such a
large role in our numeration system and because two fives make up 10, it is very useful to develop
relationships for the numbers 1 to 10 to the important anchors of 5 and 10. (e.g. 8 is 5 and 3 more or
two away from 10)
 Part-part-whole Relationships: To conceptualize a number as being made up of two or more parts.

17. Writing Numerals
Start with very clear, very strong models.
Focus on one number at a time.
Provide maximum guidance at first.
Be accepting of initial efforts.
Gently reduce the amount of guidance.
Reward correct performance.
Review previously-learned material at regular
intervals.