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# Whole Number: Theoretical Background

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by

Tweet## Stevie Cassidy

on 28 September 2012#### Transcript of Whole Number: Theoretical Background

The numbers in the set {0, 1, 2, 3, 4, 5, 6, 7, . . . } are called whole numbers. In other words, whole numbers are the set of all counting numbers plus zero. Whole numbers are not fractions, and not decimals. Whole numbers are non-negative integers. (I Coach Math. n.d.) Whole Number: Theoretical Background The most fundamental concept in elementary school mathematics is that of number, specifically whole number. (Kilpatrick, et al. 2001)

Research suggests that children begin developing an early unstructured understanding of numbers from an early age (pre-school) within their own home and community through play and social experiences.

Children, even at this early age, can categorise a few items at a time before they can actually count with understanding. (Kilpatrick et al, 2001; Gelman & Gellistel, 1978; Van de Walle, 2004.) In order to understand whole number concepts, skills and strategies, children must first become competent in 6 key

pre-number skills.

These are:

Determine Attributes

Match Attributes

Sort Attributes

Compare Attributes

Order Attributes

Patterning Each of these will have their own concept, skill and strategy. E.g. Sort Attributes talks about the 3 D's:

The child will Decide how they will sort

The child will Do the sorting

They will Describe what they sorted. Once the children have been introduced to the concepts, skills and strategies in pre-number, we can then look at Early Number concepts, skills and strategies, more specifically:

Counting Principles

Subitising

Place Value In order for a child to be deemed competent in counting, they must be able to do all the following 5 principles:

1 - One to one correspondence

2 - Stable order (1…9)

3 - Cardinal principle (last number counted tells how many in the set)

4 - Abstraction (what it is possible to count e.g., number of people vs liquid in a glass)

5 - Order irrelevance (left to right or right to left?) This also incorporates the 3 types of numbers children will come to understand and utilise:

Cardinal numbers for counting

Ordinal numbers to determine position

Nominal numbers as an identifier Moving forward we develop children's number sense, and numeration.

Numeration is knowledge consisting of formal ideas related to numeration and

place value, and;

Number sense utilises informal ideas that make sense of numbers we use in our everyday living. Teachers need to understand the big ideas of mathematics, and be able to represent mathematics as a coherent and connected enterprise. (NCTM, 2000, p.17) Big Ideas in

Number

A Big Idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole. (Charles, 2005, p.10)

John Van de Walle (2004, p.178) notes the Whole Number Big Ideas:

The major principle of the Base 10 system is student's being able to classify a set of ten (and beyond) as 'single entities', and using this as a way to determine quantities.

Understanding that place value numeration is based upon students being able to identify the 'position of digits' to understand their value An integral part of developing student's numeracy and understanding of operations is using the correct Mathematical Language. This language is evolving as the student progresses through 4 stages:

1. Children's Language 2. Materials Language 3. Mathematical Language 4. Symbolic Language

The Australian Curriculum advises this requires that teachers:

identify the specific numeracy demands of their learning area

provide learning experiences and opportunities that support the application of students’ mathematical knowledge and skills

Use the language of numeracy in their teaching as appropriate.

Understanding mathematical terminology and the specific uses of language in mathematics is essential for numeracy. Therefore, teachers should be aware of the correct use of mathematical language in their own learning areas.

(ACARA, n.d) Language

& the numeracy connection The Australian Curriculum introduces Prenumber in Foundation Year, moving onto Numeration in Year 1 , 2 and beyond building upon the previous year's key learning's and achievement standards.

Student's are primarily introduced to Pre and Whole number in the Number and Algebra strands, however, there is also key learning's pertaining to Number in the Patterns and Algebra, and Fractions and Decimals strands.

(ACARA, n.d.) Whole Number

and the Curriculum It is important that teachers use the Language Model as shown below, when introducing concepts, skills and strategies when teaching Number. Understanding of the patterns numbers create when formed certain ways.

The grouping and partitioning of numbers can be varied when working with base systems Large numbers are more easily comprehended when contextualised in a meaningful way with students. Once children understand place value, they can move onto developing a working understanding of subsets of numbers; exponents; integers; and real numbers (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) It is imperative that teachers spend ample time introducing the pre-number. early number and whole number first using concrete materials; at children's stage using real world materials, that then move onto the material, then mathematical stage, allowing student's to completely immerse themselves in concepts, skills and strategies, before they move onto the symbolic stages of numbers. Jamieson-Proctor (2012) advises in addition to the use of concrete materials, teachers need to ensure as they move through the different concepts, skills and strategies in Number, they introduce and use the correct language - An example of this being imperative when teaching place value, as students can easily interpret and describe the values as represented in the houses as incorrect based on an auditory response e.g 419 could be recorded as 4 hundred and 19 = 40019; and when referring to values in houses using the terminology: 4 hundreds, 1 ten, and 9 ones.

Full transcriptResearch suggests that children begin developing an early unstructured understanding of numbers from an early age (pre-school) within their own home and community through play and social experiences.

Children, even at this early age, can categorise a few items at a time before they can actually count with understanding. (Kilpatrick et al, 2001; Gelman & Gellistel, 1978; Van de Walle, 2004.) In order to understand whole number concepts, skills and strategies, children must first become competent in 6 key

pre-number skills.

These are:

Determine Attributes

Match Attributes

Sort Attributes

Compare Attributes

Order Attributes

Patterning Each of these will have their own concept, skill and strategy. E.g. Sort Attributes talks about the 3 D's:

The child will Decide how they will sort

The child will Do the sorting

They will Describe what they sorted. Once the children have been introduced to the concepts, skills and strategies in pre-number, we can then look at Early Number concepts, skills and strategies, more specifically:

Counting Principles

Subitising

Place Value In order for a child to be deemed competent in counting, they must be able to do all the following 5 principles:

1 - One to one correspondence

2 - Stable order (1…9)

3 - Cardinal principle (last number counted tells how many in the set)

4 - Abstraction (what it is possible to count e.g., number of people vs liquid in a glass)

5 - Order irrelevance (left to right or right to left?) This also incorporates the 3 types of numbers children will come to understand and utilise:

Cardinal numbers for counting

Ordinal numbers to determine position

Nominal numbers as an identifier Moving forward we develop children's number sense, and numeration.

Numeration is knowledge consisting of formal ideas related to numeration and

place value, and;

Number sense utilises informal ideas that make sense of numbers we use in our everyday living. Teachers need to understand the big ideas of mathematics, and be able to represent mathematics as a coherent and connected enterprise. (NCTM, 2000, p.17) Big Ideas in

Number

A Big Idea is a statement of an idea that is central to the learning of mathematics, one that links numerous mathematical understandings into a coherent whole. (Charles, 2005, p.10)

John Van de Walle (2004, p.178) notes the Whole Number Big Ideas:

The major principle of the Base 10 system is student's being able to classify a set of ten (and beyond) as 'single entities', and using this as a way to determine quantities.

Understanding that place value numeration is based upon students being able to identify the 'position of digits' to understand their value An integral part of developing student's numeracy and understanding of operations is using the correct Mathematical Language. This language is evolving as the student progresses through 4 stages:

1. Children's Language 2. Materials Language 3. Mathematical Language 4. Symbolic Language

The Australian Curriculum advises this requires that teachers:

identify the specific numeracy demands of their learning area

provide learning experiences and opportunities that support the application of students’ mathematical knowledge and skills

Use the language of numeracy in their teaching as appropriate.

Understanding mathematical terminology and the specific uses of language in mathematics is essential for numeracy. Therefore, teachers should be aware of the correct use of mathematical language in their own learning areas.

(ACARA, n.d) Language

& the numeracy connection The Australian Curriculum introduces Prenumber in Foundation Year, moving onto Numeration in Year 1 , 2 and beyond building upon the previous year's key learning's and achievement standards.

Student's are primarily introduced to Pre and Whole number in the Number and Algebra strands, however, there is also key learning's pertaining to Number in the Patterns and Algebra, and Fractions and Decimals strands.

(ACARA, n.d.) Whole Number

and the Curriculum It is important that teachers use the Language Model as shown below, when introducing concepts, skills and strategies when teaching Number. Understanding of the patterns numbers create when formed certain ways.

The grouping and partitioning of numbers can be varied when working with base systems Large numbers are more easily comprehended when contextualised in a meaningful way with students. Once children understand place value, they can move onto developing a working understanding of subsets of numbers; exponents; integers; and real numbers (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) (Jamieson-Proctor, 2012) It is imperative that teachers spend ample time introducing the pre-number. early number and whole number first using concrete materials; at children's stage using real world materials, that then move onto the material, then mathematical stage, allowing student's to completely immerse themselves in concepts, skills and strategies, before they move onto the symbolic stages of numbers. Jamieson-Proctor (2012) advises in addition to the use of concrete materials, teachers need to ensure as they move through the different concepts, skills and strategies in Number, they introduce and use the correct language - An example of this being imperative when teaching place value, as students can easily interpret and describe the values as represented in the houses as incorrect based on an auditory response e.g 419 could be recorded as 4 hundred and 19 = 40019; and when referring to values in houses using the terminology: 4 hundreds, 1 ten, and 9 ones.