How to Define Place Value, Where Does It Sit within Number Sense and Numeracy - Coursework Example

Title: Name: Department: Subject: Mathematics Tutor: Institution: Date of Submission: The purpose of the paper The aim of this study is to assist students in the classroom to understand the number system. Number system is important because it leads to the development of higher levels of numeracy. Relational understanding is the process of connecting mathematical concepts and relationships (conceptual knowledge) with the symbols, rules and procedures that are used to represent and work with mathematics (procedural knowledge). Relational understanding is evident when the rules or procedures of mathematics have a conceptual and meaningful basis and the concepts can be represented by appropriate symbolism. The study also addressed recognizing that a base structure underlies a counting or numeral system, realizing that the numeral system that we commonly use has a base number of ten, and to show that imagery and patterns recognition are important in learning place value in number sense and numeracy. Definitions and Introduction Place value is one of the most important topics in primary mathematics. Students require a thorough understanding of place value when learning the formal processes of addition, subtraction, multiplication and division. The place value knowledge is used by students when thinking about number and computing mentally. Notions of place value underpin decimal currency and metric measurement. Place value refers to how the position of a digit within a number determines the value of that number. Place value numeral systems have a regular structure or organization underlying the values. Our numerical system has an underlying structure of ten. Other numerical systems exist that use a different number as their base. By experiencing systems with differing base numbers place value understanding is enhanced (page 43). The place value manipulative is designed to be made or assembled by students. Ideally, they should be made from scratch, as this enhances student ownership and the learning experience. (Ron, 2007). Place-value chart is a chart that identifies the place values of the digits in a number. Numeracy also called number sense entails knowledge of numbers, how they are applied and their involvement in everyday life. In other words, it involves understanding of mathematics ideas, concepts and acquaintance with numbers (Morgan, 2002). Making sense of Numbers Number sense refers to ‘flexibility’ and ‘inventiveness’ in strategies to calculating and is a reaction to overemphasis on computational procedures devoid of thinking. It refers not only to the development of understanding but also to the nurturing of a positive attitude and confidence that have been lacking in more dated curricula. By watching, listening and copying, children will initially learn isolated facts, but from the earliest stages they can be encouraged to see different ways that numbers are used and the connections that underlie their use. The number 6, for example, will not only be known as the name associated with a set of six objects, but also as the number after 5 and before 7 (as in children’s ages), and 6 may also be recognized as a pattern of ‘2 and 2 and 2’ (as seen on a dice) or ‘4 and 2’ (counting wheels on a car and a bike) or ‘double 3’ (as another way to see the pattern on a dice). As their experiences of individual numbers progress, children can quickly become aware of how each number can be related to many others. Number sense is highly personalized and relates not only to those ideas about number that have been established but also to how these have been established. It involves a way of thinking that enables children to identify quickly important relationships (Stubbs, 1995: 90). When older children struggle to solve? – 4 = 9 or 100 ÷ 25 =? Instead of recognizing the number relations involved, it is probable that they are seeking to identify an appropriate procedure to solve the problem whilst other children are able to use facts they already know. It is not only effort that gives some children a facility with numbers, but an awareness of relationships that enable them to interpret new problems in terms of results they remember. Children who have this awareness and the ability to work flexibly to solve number problems are said to have a ‘feel’ for numbers or ‘number sense’. What characterizes children with ‘number sense’ is their ability to make generalizations about the patterns and processes they have met and to link new information to their existing knowledge. How is number sense developed? From the earliest stages of learning about numbers, children can be making connections that will establish flexibility in their thinking that is characteristic in developing number sense. Some links between numbers are established through patterns in counting. Counting does not mean only the ‘unitary’ counting in ones that is a child’s first experience, but includes counting in 2s, 5s, 10s and 100s, starting with any number and moving forwards and backwards. The patterns of numbers established in such counting can encourage the development of strategies that are powerful when they are connected to the arithmetic operations. There are also connections to be established among the operations and links to be made with the particular numbers that appear in problems. If the reader takes time to do the three calculations: 25+26, 39+17 and 12+35, it will probably become clear that a different strategy is used for each one, using results and relationships that are relevant. The first may be derived quickly from the ‘known fact’ that 25+25 + 50. The second may be ‘transformed’ to 40+16 =56 while the third is most likely to involve ‘partitioning’ the numbers to find 10+30+2+5, or some similar procedure. Applying a standard procedure would not be as efficient as this selection of appropriate strategies based on number sense. There are three key areas where number sense plays a key role: Knowledge of and facility with numbers-a sense of orderliness of number; multiple representations for numbers; a sense of relative and absolute magnitude of numbers; and a system of benchmark referents for thinking about numbers; Knowledge of and facility with operations-an understanding of the effects of operations; an awareness of the rules that apply and an awareness of the relationships between the operations; and Applying this knowledge and facility with numbers and operations to problems requiring reasoning with numbers-understanding the relationship between a problem context and appropriate solution strategies; awareness that multiple strategies exist; inclination to utilize an efficient representation and/or method; and an inclination to review data and results. Knowledge of numbers involves an understanding of the structure and regularity of the number system; starting with whole numbers and extending to rational numbers (that is, those that can be close represented as fractions and decimals) and the way these systems are linked. Facility with operations involves the links among them, for example doubling being the same as multiplying by 2. It includes an appreciation of certain rules and knowledge of when these can be used, for example, in addition and multiplication the numbers can be reversed so that 8+3 is equal to 3+8 and 8x3 is equal to 3x8. For subtraction and division, on the other hand, reversing the numbers will give different calculations with different solutions. Utilizing numerical knowledge involves an understanding of which operations to use in problem-solving, knowing when it is appropriate to work with approximations and making sense of calculation procedures and results in terms of original problem. Some relationships are the crucial building blocks for working with numbers and these may be identified as ‘benchmarks’, for example the number pairs that make 10, or the equivalence of 0.5, ½ and 50%. So when presented with a new problem, whether it is in words or symbols, an effective solution strategy will involve careful consideration of the numbers involved to identify links with any facts or relationships that are already known (Morgan, 2002). Classroom resources that help develop learning place value in number sense and numeracy Mathematical understanding involves progression from practical experiences to talking about these experiences, first using informal language, and then more formal language. Later, children will learn to use the symbols that characterize the conciseness and precision of a mathematically reasoned argument. Beginning with the manipulative of real objects, sorting and rearranging different collections, children are introduced to patterns that will be identified with number words (Flegg, 2002). The number ‘three’, for example, can be represented as a collection of 3 buttons, or as 3 toys or 3 steps, but ‘three’ itself is an abstraction from each of these situations and ideas such as ‘3 years old’ will take some time to understand. These experiences will help children establish the significance of number words that relate to the visual patterns that they consistently meet and provide the foundations for mental imagery (Ron, 2007). Mental imagery can also be encouraged through patterns and symbols. Sometimes patterns are more evident in written form than in the spoken words, for example, starting with 13 and adding tens will result in the pattern ’13, 23, 33, 43, 53, 63…’. Here the counting pattern ‘1, 2, 3, 4, 5, 6….’ Appears in the first ‘place’ for each number and will continue ’73, 83, 93, 103, 113..’ although the words to be matched to these higher numbers give less of a clue that this pattern continues. In the classroom, beads or cubes are introduced because they can be linked together and so have the potential to represent numbers in a structured way. Cubes are sometimes favored because they can be used not only t model different numbers, but also to show the place value structure of the number in ‘tens’ and ‘units’ Understanding place value Ideas about place value are build on the broader ideas about the number system, and these ideas are complex and difficult for children to learn. Many children are not aware, even at the basic level, of the usefulness of the place value system, and need more active help in developing a satisfactory understanding of the structure of the number system. The ability to put out the correct number of units for a number such as 18, or writing the correct numeral 18 for eighteen objects does not constitute place value understanding. They might see the 18 as 18 separate entities and not as being made up of one ten and six ones. Even though many students in Reception or Year 1 can tell that 81 is greater than 18, they might do that because they know that 81 comes after 18 in the counting sequence – they don’t see that 81 is greater because it is more than 8 tens, and 18 is less than two tens. Most children master place value in numbers with up to four digits by Year 3. Many students in Years 2 to 6 are unable to use the structure of the number system and are poor in the visualization of the array structure of the number system 1-100 to help them count in recursive groups of tens. Many students counted on from one number to another only in ones and don’t count on in tens and ones, even when presented with a 100 chart. The teacher has a responsibility of ensuring that students are constantly using multiplicative relationships (see Figure 1) such as one hundred being ten times ten. Figure 1. Multiplicative relationships in the base 10 number system Developing knowledge about the number system If the mental images and relationships to which the student is trying to connect new knowledge are poorly developed, future constructs are weak and confused, and misconceptions may arise. Secondly, if students can execute a procedure but do not have understanding, it is impossible that they will use this knowledge when required in a new situation (Heizen and Ewind, 1995: 78). The growth in constructing knowledge about the number system is “unpredictable, involving regressions as well as progressions and being, more often than not, non-linear”. Whilst it remains important for teachers to know about broad stages of development, it should be noted that children differ in development time. An assumption that the same early indicators and sequencing are equally appropriate for all children should not be made. When this is not recognized, it is easy to label these children as “behind” or as being “at risk (of not meeting the national numeracy goal).” relevant knowledge that plays a crucial contributory role in the development of an understanding of place value are; Student’s levels of counting are significantly related to their knowledge and ability to explain place value. An understanding of the multiplicative structure of the base 10 number system is a determining factor in differentiating high performance. For students to understand decimal numbers, they are supposed to reconstruct their ideas of whole numbers, and fractions, to include decimal fractions (Morgan, 2002: 56). Use of actual representations Use of concrete materials; for example, pop-sticks and Multi-Base Arithmetic Blocks (MAB), in developing understandings about place value is very important. Children will learn to interpret certain concrete materials (bags, blocks, bundles, etc.) as representing the number system and will be able to reach the general level of understanding needed to interpret unfamiliar groupings (e.g., circled marks) in the same way”. The use of place value blocks (e.g., MAB) may result in developing powerful models of the number system. The configuration of the number system is constructed in the minds of the learner. Better understanding can be achieved by representations of numbers by everyday materials, structured or semi-structured materials as concrete embodiments of numbers, which “reflect the structure of the concept where learners can use the number system structure of demonstrations to mentally create a model of the concept”. Practice in exchanging unit materials for sets of ten is an important activity for children in understanding the base 10 system. To ensure that students to benefit from using concrete materials; they must be sufficiently familiar with the materials that the mapping of representations into the base 10 system of numbers is automatic. Regular use and discussion of the models will assist students in this process of mapping the concrete materials and the number names into numeric symbols and conventional names. The broad picture of the teaching/learning environment to recognize reasons for the success or failure of instructional use of actual materials must be given attention (Flegg, 2002: 23). Student assessment strategies Performance • Create cards with various number words on each as shown below: Students will be asked to rearrange the cards to create different number combinations. Students will record the created numbers create, in diverse ways for example in standard form, expanded form, on a place value chart, and in words. Students will then order the numbers using a number line and give an explanation on how they selected the endpoints and benchmarks, and how they placed their numbers on the line. Students will have to create the maximum and the minimum numbers using all the available cards. Ask students to use the number 519 742 766 to answer the following questions. a) What is the value of the 9 in the number? b) What is the value of the 4? c) Select a digit and demonstrate how that digit is ten times greater than the digit to its immediate right. Student-Teacher Communication Write a number, such as, 42 785 325 on the board and ask students the number of millions, ten thousands, thousands, and hundreds in the number. Then allow the students to justify their thinking. Journal The following problem may be posted to students: Noah said 5 450 000 is greater than 27 450 000 because five is greater than two. Students will decide on Noah’s answer by explaining in words, numbers and/or pictures. Conclusion To enable children understand the relationship between the place value and the number sense or numeracy, it is important that images are used and the relationships between the numbers in made clear. The configuration of the number system should be constructed in the minds of the learner. Better understanding can be achieved by representations of numbers by everyday materials, structured or semi-structured materials as concrete embodiments of numbers, which “reflect the structure of the concept where learners can use the number system structure of demonstrations to mentally create a model of the concept”. References Flegg, G. Numbers: their history and meaning. New York: General Publishing company, 2002. Heizen, D,E and J.H Ewind. Numbers. New York: Springer- Verlag Berlin Heidenberg, 1995. Morgan, S. Place Value. Chicago: Zephyr Press, 2002. Ron, Smith. Cornerstone in Number place Value. NSW, Australia: Hyde Park Press Ltd, 2007. Stubbs, D.L. Numbers. New york: Baker Publishing Group, 1999. Read More

Making sense of Numbers Number sense refers to ‘flexibility’ and ‘inventiveness’ in strategies to calculating and is a reaction to overemphasis on computational procedures devoid of thinking. It refers not only to the development of understanding but also to the nurturing of a positive attitude and confidence that have been lacking in more dated curricula. By watching, listening and copying, children will initially learn isolated facts, but from the earliest stages they can be encouraged to see different ways that numbers are used and the connections that underlie their use.

The number 6, for example, will not only be known as the name associated with a set of six objects, but also as the number after 5 and before 7 (as in children’s ages), and 6 may also be recognized as a pattern of ‘2 and 2 and 2’ (as seen on a dice) or ‘4 and 2’ (counting wheels on a car and a bike) or ‘double 3’ (as another way to see the pattern on a dice). As their experiences of individual numbers progress, children can quickly become aware of how each number can be related to many others.

Number sense is highly personalized and relates not only to those ideas about number that have been established but also to how these have been established. It involves a way of thinking that enables children to identify quickly important relationships (Stubbs, 1995: 90). When older children struggle to solve? – 4 = 9 or 100 ÷ 25 =? Instead of recognizing the number relations involved, it is probable that they are seeking to identify an appropriate procedure to solve the problem whilst other children are able to use facts they already know.

It is not only effort that gives some children a facility with numbers, but an awareness of relationships that enable them to interpret new problems in terms of results they remember. Children who have this awareness and the ability to work flexibly to solve number problems are said to have a ‘feel’ for numbers or ‘number sense’. What characterizes children with ‘number sense’ is their ability to make generalizations about the patterns and processes they have met and to link new information to their existing knowledge.

How is number sense developed? From the earliest stages of learning about numbers, children can be making connections that will establish flexibility in their thinking that is characteristic in developing number sense. Some links between numbers are established through patterns in counting. Counting does not mean only the ‘unitary’ counting in ones that is a child’s first experience, but includes counting in 2s, 5s, 10s and 100s, starting with any number and moving forwards and backwards.

The patterns of numbers established in such counting can encourage the development of strategies that are powerful when they are connected to the arithmetic operations. There are also connections to be established among the operations and links to be made with the particular numbers that appear in problems. If the reader takes time to do the three calculations: 25+26, 39+17 and 12+35, it will probably become clear that a different strategy is used for each one, using results and relationships that are relevant.

The first may be derived quickly from the ‘known fact’ that 25+25 + 50. The second may be ‘transformed’ to 40+16 =56 while the third is most likely to involve ‘partitioning’ the numbers to find 10+30+2+5, or some similar procedure. Applying a standard procedure would not be as efficient as this selection of appropriate strategies based on number sense. There are three key areas where number sense plays a key role: Knowledge of and facility with numbers-a sense of orderliness of number; multiple representations for numbers; a sense of relative and absolute magnitude of numbers; and a system of benchmark referents for thinking about numbers; Knowledge of and facility with operations-an understanding of the effects of operations; an awareness of the rules that apply and an awareness of the relationships between the operations; and Applying this knowledge and facility with numbers and operations to problems requiring reasoning with numbers-understanding the relationship between a problem context and appropriate solution strategies; awareness that multiple strategies exist; inclination to utilize an efficient representation and/or method; and an inclination to review data and results.

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