Developing Number Sense and Early Number Concepts

1. Developing Number Sense and Early Number Concepts  EDN 322

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). Number sense has been described as � a good�(It�s an intuitive feel for numbers.) While this definition was recorded more than a decade ago, the NCTM Standards still 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 sense has been described as � a good�(It�s an intuitive feel for numbers.) While this definition was recorded more than a decade ago, the NCTM Standards still 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).

3. Number and Operations Standard�Grades Pre-K-12 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 Master 6-1: Number and Operations Standard Master 6-1: Number and Operations Standard

4. Excerpts of Expectations from the Number and Operations Standard Grades Pre-K-2 Principles and Standards for School Mathematics National Council of Teachers of Mathematics 2000 Master 6-2: Number and Operations Excerpts (number concepts) Master 6-2: Number and Operations Excerpts (number concepts)

5. Excerpts of Expectations from the Number and Operations Standard Grades Pre-K-2

6. 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 Master 6-2: Number and Operations Excerpts (number concepts) Master 6-2: Number and Operations Excerpts (number concepts)

7. The meanings for the number five suggested by young children�

8. Prenumber Concepts Patterning Sorting Classifying 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 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.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 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. Counting Principles One-to-one correspondence Stable order rule Order irrelevance rule Cardinality rule 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. Each object to be counted must be assigned one and only one number name. The number-name list must be used in a fixed order every time a group of objects is counted. The order in which the objects are counted doesn�t matter. The child can start with any object and count them in any order. 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. These principles help teachers recognize the levels of children�s counting skills. Careful observation of children, coupled, with a good understanding of these principles, will pinpoint counting errors. Once the trouble is diagnosed, instruction can focus on the specific problem. 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. Each object to be counted must be assigned one and only one number name. The number-name list must be used in a fixed order every time a group of objects is counted. The order in which the objects are counted doesn�t matter. The child can start with any object and count them in any order. 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. These principles help teachers recognize the levels of children�s counting skills. Careful observation of children, coupled, with a good understanding of these principles, will pinpoint counting errors. Once the trouble is diagnosed, instruction can focus on the specific problem.

10. Counting Stages 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. 119). 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. 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. 119). 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.

11. Counting Stages Rational Counting 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. 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.

12. Counting Strategies Counting On Counting Back Skip Counting 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. When children count back, they give correct number names as they count backward from a particular point. At an early stage, counting down, five, four, three two, one) can be related to rockets blasting off; later it becomes helpful in developing subtraction. 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. 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).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. When children count back, they give correct number names as they count backward from a particular point. At an early stage, counting down, five, four, three two, one) can be related to rockets blasting off; later it becomes helpful in developing subtraction. 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. 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).

13. Relationships Among Numbers Spatial Relationships One and Two More, One and Two Less Number Benchmarks Part-part-whole 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�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. 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 � To conceptualize a number as being made up of two or more parts. 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�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. 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 � To conceptualize a number as being made up of two or more parts.

14. 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.