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# Number sense

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

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### Number sense

1. 1. Dr. Harriet Thompson
2. 2. Number SenseA “good intuition about numbers and theirrelationships. It develops gradually as a result ofexploring numbers, visualizing them in a variety ofcontexts, and relating them in ways that are notlimited by traditional algorithms” (Howden, 1989).
3. 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 persons 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. 4. Number and Operations StandardGrades 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. 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. 6. The meanings for the number five suggested by young children…
7. 7. Prenumber ConceptsPatterningSortingClassifying
8. 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. 9. Early Number ConceptsEarly number concepts begin with counting. To adults theact of counting seems natural and simple. In reality it isthe culmination of a lengthy developmental process.Fortunately, development of this process is fostered byevery day social interaction. Children come tokindergarten with a basic understanding of what countingis all about, that is, that there is a set of fixed numbernames, said in a specific order, which are matched one-to-one with things, and that the last word in the sequencetells "how many." Although children who have beenfortunate enough to live in a complex, supportiveenvironment possess these basic understandings, theirability to carry out the process free of errors generally isnot fully developed and formal teaching is needed tocomplete their development.
10. 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. 11. Counting StagesRote Counting
12. 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. 13. Counting StagesRational Counting
14. 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. 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. 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. 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.