Dr. Harriet Thompson

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).

Number SenseThe 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).

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

Understand numbers, ways of representing numbers, relationships among numbers, and number systems Count with understanding and recognize “how many” in

sets of objectsDevelop 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

The meanings for the number five suggested by young children…

Prenumber ConceptsPatterning

Sorting

Classifying

Prenumber ConceptsMany 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.

Early Number ConceptsEarly 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.

Counting PrinciplesOne-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

Counting StagesRote Counting

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.

Counting StagesRational Counting

Rational Counting StagesRational 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.

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).

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.

Writing NumeralsStart 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.