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Unit 5: Counting and developing an understanding of the number system This is Unit 5 of the Mathematics Guide for Teacher Educators developed by the Cape Peninsula University of Technology. Unit 5 and the accompanying DVD 1 deal with counting and its role in developing young children’s understanding of the number system. Unit 6 and Unit 7, accompanied by DVDs 2 and 3, deal with Representing and Manipulating Number, and Operating with Number respectively. In Grades R and 1, children spend most of their time working with positive whole numbers. So this unit focusses on how to develop children’s understanding of the system of natural numbers.

DVD 1 focuses on six different aspects of counting: 5.1.

DVD 1.1: Oral counting (see Section 5.3) DVD 1.2: Resultative counting (see Section 5.4) DVD 1.3: Subitising (see Section 5.5) DVD 1.4: Number sequences (see Section 5.6) DVD 1.5: Estimation (see Section 5.7) DVD 1.6: Profile of a number (see Section 5.8)

At the beginning of each section that follows, we suggest that students do a general viewing of the relevant clip of DVD 1 to orientate themselves to the concepts addressed.

5.2. Counting in the Learning Pathway for Number (LPN) framework In Unit 3 you can find an explanation of the LPN, within the framework of trajectory learning and realistic mathematics education. In the table below we indicate those topics that relate to counting in the Foundation Phase in the Learning Pathway for Number. The columns highlighted in red below summarise Counting in Stages 1 and 2 of in the LPN.

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Counting in the Foundation Phase in the Learning pathway for Number

Number knowledge

Stage 1 Emergent and growing number concept

Stage 2 Counting-and-calculating

Stage 3 Calculating

Stage 4 Advanced calculating

DVD clip

Number range up to 10 and beyond

up to 20 and beyond

up to 100 and 1000 and beyond

up to 10 000 and beyond

Models of representing numbers

Group models Line models Combination models

Ways of dealing with numbers

Contextualising Positioning Structuring

Distinguishing numerosity

Oral counting (Acoustic Counting)

Resultative counting

Subitising

Sequencing

Estimating

Learning different number meanings and functions (Profile of a number)

Working with number patterns

Pre–R R–Grade 1 Grades 1–2 Grades 3–4

DVD 1.1

DVD 1.2

DVD 1.3

DVD 1.4

DVD 1.5

DVD 1.6

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5.2.1 Counting in the NCS Mathematics Curriculum and Assessment Policy Statement (CAPS)

In South Africa teachers employed by the government are required to implement the Curriculum and Assessment Policy Statement (CAPS). It might be an interesting exercise for students to compare how counting is dealt with in the CAPS compared to how it is presented in this guide.

Foundation Phase Mathematics CAPS: Grade R (DBE, 2011a).

Section 3: x 3.2 Foundation Phase Overviewpp19 – 21 x 3.4.1 Grade R Overview per term pp 41 - 45 x 3.5.1 Content Clarification Notes with Teaching Guidelines pp 63 – 65, 70, 75,

80, 86, 104, 107 – 109, 112 – 117, 119 – 121, 122- 123, 129, 134 – 136, 139 – 141, 145 – 146, 148, 150, 152, 154, 159, 161 – 165, 167, 169, 172 – 173, 176- 178, 181 – 184, 186 – 187, 191 – 192, 197 – 198, 201 – 202, 206, 207 – 209, 212 – 213, 219 – 222, 226, 228, 232 – 233, 237 – 238, 241 – 245, 250 – 251, 255 – 256, 263

Foundation Phase Mathematics CAPS: Grades 1 – 3 (DBE, 2011b).

Section 3: x 3.2 Foundation Phase Overviewpp18 – 20 x 3.4.1 Sequencing and Pacing of Grade 1 per term pp 40 - 41 x 3.5.1 Clarification of Grade 1 Content:

o Term 1 pp 94 – 101 o Term 2 pp127 - 132 o Term 3 pp159 - 162 o Term 4 pp 183 – 187

5.3 Oral or Acoustic Counting Children’s understanding of number is rooted in counting. (MacLellan, 2008: 80). Oral counting is the first form of counting. Initially young children may say lists of numbers out of order, e.g. 1, 2, 3, 5, 7, 11, 6 ….. Counting rhymes and counting songs help children to learn to recite number words in order. Reciting the number name sequences is called oral or acoustic counting. Oral counting can be defined as the ability to produce in speech a correctly ordered string of numbers. (Threlfall, 2008) There are two different ways of thinking about counting numbers: on the

If you wish to read more about Counting in Stages 1 and 2 of the Learning Pathway for Number, you can refer to Van den Heuvel-Panhuizen, M, Kühne, K, Lombard, A.P. 2012. The Learning Pathway for Number in the Early Primary Grades. Northlands: Macmillan South Africa. :

x Glossary entries on counting: pages 195 - 204 x Stage 1: Emergent and Growing Number Concept pages 15 – 26 x Stage 2: Counting and Calculating pages 38 – 43

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one hand, they form an ordered list, and, on the other hand, they describe cardinality, that is, how many things are in a set (Cross,Woods & Schweingruber, 2009). Acoustic counting is an important part of developing an understanding of ordinal position, namely ordinality. An ordinal number tells where in the ordering a specific number lies (e.g., first, second, third, etc.) (Cross, Woods & Schweingruber, 2009). While reciting the verbal number sequence, pre-school children start to learn about which contexts are appropriate for counting such as number rhymes (Munn, 2008). It is uncommon for children to understand why they are counting. A four year old girl was asked why she counted at home? Her response was: “But counting’s just saying the words isn’t it?” (Munn, 2008). Children start by simply learning syllables in a rhythm. Young children do not always perceive the number words as separate from each other. So they may not understand, for example, that 6 represents something different to 7. Oral counting on its own does not necessarily help children to learn about cardinality of numbers or the principles of counting. There is more to counting than learning the number names in order. However Munn (2008) cautions by saying that young children’s first attempts at oral counting must be taken seriously. She states:

“Counting without quantification is not necessarily meaningless, and is certainly not meaningless to young children. The purely verbal counting seen in young children has received particularly bad publicity because it has been seen as a trap for the unwary teacher who might otherwise attribute far too much number knowledge to young children… Young children’s counting often has a social rather than a numerical goal. For instance, children may recite the number sequence to imitate an admired family member, to show that they can count, or to take part in a well-established ritual. It is important not to negate non-numerical counting, as it can provide a point of contact with a child’s social being. Adults also have count sequences with a purely social goal. Take, for example, the teacher who says, ‘I want quiet please! One, two, three, four, five!’ This is every bit as much a social and ritual use of the number sequence as the young child’s counting. Such counting provides a framework for the development of properly numerical counting and as such it should be gently encouraged” (Munn, 2008: 30-31).

Numbers can be represented in different ways. More sophisticated mental representations of number are associated with better mathematics performance and an ability to faster understand mathematical concepts (Williamson, Edward & Voutsina, 2011). Young children often use fingers, apparatus or manipulatives to help them learn to count. These can be loose counters or strings of counter beads or counting frames.

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Counters represent numbers in a discrete way: each number is separate from the next. Number cards and number tracks also represent numbers in a discrete way. Cross et al (2009) uses the term ‘number path’, as in childhood games in which numbers are put on squares and children move along a numbered path. Such number paths are count models.

A number line however, represents numbers in a continuous way. It is important to remember that Grades R & 1 learners who do not have rich informal number and counting experiences or those with limited skills in the language of learning and teaching (LOLT) will need a range of experiences and more time to develop. Sadly, learners who need more practical experiences are sometimes rushed into an abstract mode before they are ready (Kennedy, Tipps & Johnson, 2008:151).

5.3.1 Description of the DVD clips The focus in DVD 1.1 is on the progression of oral counting from the LPN stage 1 to stage 2. In DVD 1.1 teachers lead learners in oral counting, but they include various

visual supports which assist learners to begin to learn more than just the number names in order. The aim is for learners to learn more than simply chanting the sequences of whole numbers in order. The two clips (A & B) in DVD 1.1 illustrate the successor function of Natural Numbers: that one can generate the next whole number by adding 1 to the previous number.

Teaching Notes Towards the end of DVD 1.1 are still frames of the clips which can be used for the purposes of reflection

1 2 3 4 5 6 7 8 9 10

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Summary of DVD clips on oral counting: DVD 1.1 Elements Clip A Clip B Grade Grade R Grade 1

Timing (00:33 – 01:27) (01:27 – 02:11)

Approach Teacher modelling Teacher modelling Progression / transition

From counting rhyme with images illustrated to counting on the number line

Manipulatives Counting back using picture and rhyme

Counting on using structured number line

Everyday contexts Bottles Monkeys

Mathematics

Successor function for Natural numbers: generating the preceding whole number, in the number range 10 – 1

Successor function for Natural numbers “generating the next consecutive whole number, in the number range 1 – 20

The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with teaching oral counting.

Student Activity 5.3.1: Oral counting Do this alone and in pairs.

1. Read the introduction to oral counting in Section 5.3. 2. View DVD 1.1 (Clips A & B) all the way through. 3. While you are viewing think about what you have read, and think about the

following questions. a. There is more to counting than reciting numbers in order. What else is

involved in counting? b. In which contexts do young children count? c. What are the implications for teaching and learning to count? d. What could be similar or different in the counting activities selected for

Grade R and Grade 1? 4. When you have finished viewing the DVD clip, discuss the reading and the

questions with your fellow students. Compare your thinking and discussion with our description of the DVD clips, below. Clip A: Grade R Oral Counting The first DVD clip deals with oral counting. The Grade R class sings the counting song “Ten green bottles”. The arrangement and actions of the children holding bottles represent an image of a

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changing amount. The counting activity contains elements of both cardinality and ordinality. The song and the actions help the children to focus on the number decreasing by one each time. The actions accompanying the song attempt to foreground counting back from ten in ones. The number of bottles is not represented in numerals in this activity. However, the arrangement and actions of the children modelling the song contain elements of an image, a number track. A number track is often used as a precursor to a number line.

Clip B: Grade 1 Oral Counting In the section of the lesson preceding the recorded clip learners sing a counting song about monkeys. The Grade 1 class counts in ones from one to 20. There is a visual display of a row of 20 monkeys; above each monkey is the appropriate number, the numeral 0 precedes the numeral 1. This connects the pictorial representation with the symbol. The learners count as the teacher points to the monkey below the number. The teacher’s gesture focusses learners on each successive count. The oral counting activity contains elements of both cardinality and ordinality. You may need to alert students to the fact that there is a row of objects beneath the numbers, and that learners are counting the objects. When learners get to count from 13 – 19 they emphasize the teen part of the number. Learners are looking at the numeral whilst emphasising how one pronounces the word in English.

5.3.2 Helping students to reflect on DVD 1.1: teaching oral counting The two clips in DVD 1.1 (Clip A and Clip B) are designed to be viewed together.

Focus points for student activities on oral counting

Student Activity 5.3.2: Images and Mathematical representation View the two DVD clips, and then discuss the questions with a partner.

1. Why did the Grade R teacher do the following? a. arrange 10 children in a row in front of the class b. give each child a large image of a green bottle c. ask the child who was on the class’s right hand side, to sit down each

time the song reached “accidently fall”

x Visual images and ways of representing mathematics x Models and modelling x Level Principle: Progression in oral counting

Numbers can be represented in different ways. This is a focus of Unit 6 and DVD 2. You can read the notes on models in Unit 6.3, and watch DVD 2.1 - Representing and symbolising number.

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2. Discuss how the mathematical representation would have been different if the teacher had let the whole class sit in a circle and sing together while showing the number of green bottles on their fingers.

3. Which principles of counting are foregrounded by the arrangement and actions of learners during the song?

View the DVD again. Consider the following questions: 4. Are all the children actively singing all of the time? What things do you think

might be distracting them? 5. Relatively how much time do you think the class is watching the row of

children holding pictures of bottles in relation to the amount of time they are watching the teacher? What is the significance of the times?

6. If you were to repeat this activity, how would you position yourself as a teacher to ensure a greater focus by the rest of the class on the row of children with the bottles?

7. When watching the other DVD clips (especially Clip D of DVD 1.2 and DVD 1.4 Number sequences) look for how teachers position themselves differently in relation to a row of children representing numbers.

Many teachers ask learners to do oral counting without any focus on a visual model, for large periods of time. Both teachers in this DVD 1.1 used visual models (pictures, arrangement and actions of learners, pictures and the number line) to support and highlight how one constructs the system of consecutive whole numbers.

Student Activity 5.3.3: Models, modelling and visual aids Do this activity in pairs.

Imagine that the Grade 1 teacher had shown learners flashcards with each numeral as learners had counted. 1. What aspects of the structure of the Natural Number System would have

been less visible? 2. What aspects of the system of whole numbers are highlighted by the visual

models used in DVD clips? An aspect of the level principle is the idea that teaching is aimed at supporting or scaffolding a transition from simpler to more complex ways of dealing with number concepts.

Student Activity 5.3.4: Level Principle: Progression in oral counting You can consider these questions alone or in pairs, or both.

Both activities in the DVD consisted of whole class oral counting. The Grade R activity consisted of counting backwards in ones. The Grade 1 activity consisted of counting forwards in ones. 1. What was different about the number range of the counting activities?

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In both activities there were visual representations supporting the oral counting. 2. What was different about the visual representations in the Grade R example

when compared to the Grade 1 example? 3. In what ways does the row of 10 children in front of the Grade R class visually

scaffold the transition to using a number line for counting? 4. You could also think of the question this way:

How does the activity in the Grade R clip prepare learners for doing the kind of activity in the Grade 1 clip?

Teaching Notes The following Applied Task will help students to apply what they have learned from doing the focussed DVD activities.

Applied Task for Students 5.3.5: A file on counting rhymes On your own:

1. Make a file of counting rhymes and songs in different languages. These can include: a. Counting rhymes/songs that you source from elsewhere b. Counting rhymes/songs from elsewhere that you adapt c. Counting rhymes/songs that you make up

2. Make up or identify existing rhymes/songs that can be used to familiarise learners with additional languages.

3. Find or make up some counting rhymes that relate to other Grade R or 1 school subjects, to show integration.

5.4 Resultative Counting Counting objects to determine how many there are, is called resultative counting. Children learn more from resultative counting than oral counting. They learn the principles of counting (see below) and they learn to understand the structure of the natural or counting numbers (see below).

Principles of counting Despite often having accurate and extensive counting ability, young children do not have an adult definition of counting. To them, counting is a recreational activity which they usually do not associate with quantification, leading to a difference between their counting behaviour and their beliefs about counting (Munn, 2008). In the 1970s, while researching the possibility of children’s arithmetic competence prior to them attending pre-school, Gelman and Gallistel (1978) identified five principles of counting.

x One-to-one correspondence x Stable order principle x Cardinal principle x Order irrelevance principle x Abstraction principle

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They argue that once learners know and use these five principles they are able to count. Staves (2013a) categorised the five principles into two groups. The first group consists of three principles which involve how to count, and the second group consists of the remaining two principles which are applied when counting. The three principles that relate to how to count are:

x One-to-one correspondence Children need to learn that each number word is applied to a distinct object in the set. Each item gets only one number tag. Young children sometimes count the same object twice, skip an object when counting or are unclear which objects have been counted. It is useful for children to start by touching or moving an object as they count. Later they can form a mental image of this. x Stable order principle The number name tags are always used in the same order: the sequence of number names is stable. This, however, needs to be understood in relation to the order irrelevance principle (see below). For example the list of numbers in English is always one, two, three, four and never four, one, three, two

x Cardinal principle The last number number of the count is the total number of the set. This is the cardinal number of the set. It describes how many have been counted.

Staves (2013a) argues that when learners understand and can work with the three counting principles above they will be able to

x understand that the next number in the sequence is a larger quantity, x count on, x use counting to compare the number of objects in sets.

Staves (2013 a) states that the remaining two “counting principles” of Gelman and Gallistel (1978) are applied when counting. These are:

x Order irrelevance principle The cardinal value of a set will be the same no matter which object you start with and in what order you tag objects with a number name. So the number names are given in a stable order, but the order in which these names are applied to objects can vary. There is no one particular item that must be labelled “one” or “two” etc.

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x Abstraction principle It is not only physical objects that can be counted, but imagined objects, or events, or ideas. Very young children start by counting only physical objects that are in front of them. Sometimes they will even omit one or more objects because they look physically different to the rest of the set.

When children understand that all of the other four principles can be applied to any collection of objects they have understood the abstraction principle (Sarama, & Clements , 2009: 57).

Natural numbers, counting numbers, whole numbers Most children learn to count by starting with 1. Until the nineteenth century natural numbers were also defined as starting with 1. During the nineteenth century this changed and 0 was included. Now some people include 0 in the set of Natural Numbers, and some do not. Also, some people include 0 in the set of Counting Numbers, but others do not. Haylock and Cockburn (2008: 43 - 47) state:

“The set of natural numbers consist of those that we use for counting: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. … The basic properties of these numbers are that there is a starting number (one) and that for each number there is a next number, also called its successor…. The next number after any given number is always one more.”

Children in Grade R are mostly working with the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10. In Grade 1 this is extended to 20 and beyond. However, through the exposure to these numbers and the work they do with these numbers learners should learn about the structure of the natural number system and the rules for operating with addition, subtraction, multiplication and division with natural numbers.

one

two three

one

two

three

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Learners need to connect the idea of “the next number” with the idea of “one more”. When working with whole numbers we move from one number to the next consecutive number by adding one. This is called the successor function. For any set with a cardinal number of n, the next numeral in the list will have a cardinal value of n + 1. For example if you have a set of 3 objects and you add an object you will have four objects: you will count 1, 2, 3 for the first set, and 1, 2, 3, 4 for the second set and 3 + 1 = 4

Adapted from: Sarama, J & Clements D.H. 2009. Early Childhood Mathematics Education Research: Learning Trajectories for Young Children. New York. Routledge

Le Corre and Carey, (2007: 660) write about the system of natural numbers that:

“Children discover a structural similarity between two very different representations of linear order: next in a list of symbols, and next in a series of sets related by +1”.

While this may seem obvious and automatic, it does need to be learned and it is not true for all types of numbers. For example it is not true for fractions. Learners in the Intermediate Phase sometime count in fractions. An example could be

;

. If we look at fractions on a number line, we can see that next consecutive quarter is 4/4,

but halfway between ¾ and 4/4 is the fraction 7/8, and halfway between ¾ and 7/8 is the fraction13/16, so there is no number that is the next consecutive number to ¾.

0 1/4 2/4 3/4 4/4 5/4

6/8

12/16

7/8 7/8

16/16 13/16

14/16

1

2

3

4

4 3 2 1

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Stages in resultative counting In the Learning Pathway for Number in the Early Primary Years, resultative counting in ones is associated with Stage 1: Emergent and Growing Number Concept, while resultative counting in groups is associated with Stage 2: Counting and Calculating. When learners count in groups they are either imposing a structure on the objects or using a structure that has been already imposed on the objects.

Mathematization Hans Freudenthal in Gravemeijer and Terwel (2000:781) regarded Mathematics as a human activity that involves looking for problems, solving problems and organising subject matter. This subject matter can be everyday subject matter which is organised mathematically or mathematical subject matter that is re-organised according to new mathematical insights. He called this process mathematization. Realistic Mathematics education uses both real contexts and contexts or mathematics that learners are capable of imagining as the starting point for the process of mathematization. Van den Heuvel-Panhuizen (2001: 52) writes that models are a useful device to help learners move through various levels of mathematical understanding. Models provide learners “with a foothold during the process of vertical mathematization (2001: 52). Initially a model may be context dependent or be a “model of” a situation, but in order for it to help learners access mathematics more deeply it eventually needs to become a “model for” more general, less context dependent mathematics. A “model of” can also be substituted by a similar but more generalizable “model for”. An example of this is shown in the Grade 1 lesson of DVD 2.4, particularly Clip E. Walkerdine (1988) writes about this process in a slightly different way. She states that everyday practices become mathematical practices through a series of transformations. An idea is represented in some way, either a word, and image or an object. Walkerdine states that one representation or sign is replaced by another, which is replaced by another. In this way learners are drawn more deeply into mathematics through linked chains of representations. The initial idea is transformed through chains of signification: where one representation is replaced by another. In this way the learners are inducted more deeply into Mathematics.

5.4.1 Description of the DVD clips on resultative counting The focus of DVD 1.2 is on the progression of resultative counting from the LPN stage 1 to stage 2.

Resultative counting helps children to learn the principles of counting. It develops their awareness of both ordinality and cardinality of numbers, but develops the cardinality aspect more strongly. Clip B shows Grade 1 resultative counting, and illustrates how one model can be replaced by a second model which is itself replaced by a third model in a process that helps induct learners more and more deeply into mathematics. DVD 1.2 has a total running time of 36 minutes and 12 seconds: this is considerably longer than DVDs 1.1, 1.3, 1.4, 1.5, 1.6. Clip C shows two different teachers and two different classes of children, both using number cards and counters.

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Teaching Notes Towards the end of DVD 1.2 are still frames of the clips which can be used for the purposes of reflection

In several of the DVD clips learners are also asked to compare numbers. You will notice that the children find this difficult.

Summary of DVD clips on resultative counting: DVD 1.2 Elements Clip A Clip B Clip C Clip D Grade Grade R Grade 1

Timing (00:00 – 02:32)

(02:32 – 07:56) (07:56 – 12:57; 13:00 – 17:31)

(17:39 – 36:14)

Approach Teacher modelling

Teacher modelling

Teacher modelling

Teacher modelling

Progression / transition

Recognising oral number words and representing these with counters

Recognizing oral number words and representing these with fingers 1-5, then 6 - 9

Recognizing number symbols and representing these with counters Number range 1 – 9

Counting in multiples Number range for 5s: 5 - 50 Number range for 10s: 10 - 80

Manipulatives

Counters Fingers Number cards Fingers Picture cards Number line

Everyday contexts

Counting objects to find out how many there are.

Mathematics

Resultative counting to develop cardinal aspect of a number

Resultative counting in multiples. Introducing language related to multiplication

The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with resultative counting.

Comparing numbers is the main focus of Unit 6, Section 6.5, DVD 2.3. The notes in Section 6.5 explain why comparing numbers is a challenging task for young learners.

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Student Activity 5.4.1: Resultative Counting Do the first part alone, and then discuss with a partner or small group

1. Read the introduction to resultative counting in Section 5.4. 2. View DVD clip 1.2 all the way through. 3. While you are viewing think about what you have read about resultative counting, and

think about the following questions. a. Which of the principles of counting can you identify in the DVD clip? b. Do you think the children are beginning to connect the idea of “the next

number” with the idea of “one more”? Why do you say that? c. Are the stages in resultative counting evident in some way in the DVD? d. How are the resultative counting activities sequenced so that there is a process

of mathematization from informal context-bound tasks to less informal activities?

4. When you have finished viewing the DVD clip, discuss the reading and the questions

with your fellow students. Compare your thinking and discussion with our description of the DVD clips, below.

Clip A: Resultative Counting Grade R using counters The first DVD clip shows a group of Grade R learners doing resultative counting using counters. The number range is between 2 and 6. The teacher starts by explaining what she is going to do. She is going to place counters on a towel in the middle of the group. The children must say how many counters she has placed. She involves all six learners all of the time. In the first two examples she places the counters slowly and deliberately one-by-one. This allows the children to count as she places. Example 1 5 counters in a pentagon formation.

Example 2 6 counters in a hexagon formation.

Working with the initial set of 5 counters helps learners to count the second set of 6 counters more easily. The teacher and the learners check that there are six counters in the second group. They count as the teacher places them one-by-one in a row. In the second set of two examples the teacher places the whole collection of counters down simultaneously. Children count the already-positioned group. This is more difficult than

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counting each counter as she places it. This is why it is done second, and why the teacher restricts the number of counters. The teacher and the learners check that there are three counters in the first group. They count as the teacher places them one-by-one in a row. Example 3 3 counters in a random formation.

Example 4 4 counters random formation.

Working with the set of 3 counters helps learners with the last set of 4 counters. Counting a small group of already-placed counters draws on subitising skills. The cardinal value of the four counters in the second frame could be determined by recognising the arrangement of three and mentally counting “three, four”. You may like to link this DVD clip with the clips in DVD 1.3, which is about subitising.

Clip B: Resultative Counting Grade R using fingers The second DVD clip shows a group of Grade R learners doing resultative counting using their fingers. Young children find it easy to count using their fingers. Movement and using one’s body is an important way of learning for young children. It can be called kinaesthetic learning. Using fingers is both a visual and kinaesthetic representation of number (5 on one hand) which is readily available to children (Cross, et al., 2009). Howard Gardener (1983) identified 8 different kinds of intelligences:

Verbal/ Linguistic

Visual/ Spatial

Musical/ Rhythmic Existential

Bodily/ Kinaesthetic

Interpersonal

Intrapersonal

Naturalist

Multiple Intelligences

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Wright (2008:200) describes the mathematical significance of finger patterns in the early learning of number. The form of finger representations of numbers ranging from one to ten, correspond to the patterns inherent in the bead string and ten frame. There is also a homomorphism (correspondence in form) between finger patterns and standard configurations on a ten frame. For example four can be shown as two fingers on one hand and two on the other hand, so four can be shown as two red beads and two yellow beads or on the ten-frame as two dots in each row of five on the number frame. At first children work with numbers 1 to 5. Once the teacher is sure that they are secure within this number range she moves on to numbers between 6 and 9. When learners show fingers on both hands, they are building mental images of numbers being 5 + …. At the end of the clip children compare numbers. This DVD provides a useful classroom cameo of progression and sequencing of learning on a micro-level. In the Realistic Mathematics Education framework this is called the level principle. Van den Heuvel-Panhuizen (2001: 52) states that:

“Learning Mathematics means that students pass various levels of understanding: from the ability to invent informal, context-based solutions to the creation of various short cuts and schematization, to the acquisition of insights into the underlying principles and the discernment of even broader relationships. The condition for arriving at the next level is the ability to reflect on the activities conducted. This reflection can be elicited by interaction. Models serve as an important device to bridge this gap between informal, context-related mathematics and more formal mathematics.”

The teacher starts by using only numbers five and less: 5, 2, 1, 4. Once all learners have shown the number on their fingers, the whole class counts their fingers one by one: touching and saying the number. This physical tagging develops several of the counting principles. The teacher demonstrates counting starting at the thumb side of her right hand. Once learners are comfortable with this, she explains that there is going to be a change and that learners will need to use both hands. The teacher is aware that working with the numbers 1 – 5 (showing fingers on one hand) is easier than working with the numbers 6 – 10 (showing fingers on both hands). She alerts learners to the fact that there will be a shift in the content that they are learning. The teacher is drawing on Vygotsky’s theory to promote effective scaffolding. First, she determines the “actual level of development” of the children (working with 1-5) so that she can guide the learners step by step to reach their potential level (using numbers 6-10) through guidance form the teacher and interaction with peers. Through an awareness and assessment of the actual and potential levels of the learners’ development, the teacher assists the transition of the “zone of proximal development” (Cross et al., 2009:255). This time when they check they start from their smallest finger. Some communities in South Africa tend to count from their smallest fingers. Other communities tend to count from their thumbs. She models both these ways of counting. After counting she consolidates the image

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of 5 and …. By saying “so we have 5 on one hand, and how many on the other hand?” This is done with appropriate hand gestures which she uses as a scaffolding technique. The clip ends with a section in which learners compare numbers. This is done in the form of a game (“more-I-win/ less-I-win”) between pairs of learners. They place their hands behind their back and when they show their hands, they say who has the biggest or smallest number. Learners enjoy the game, but they still find comparing numbers difficult. To compare the number in two sets, children must count each set, keeping track of the two numbers, and then compare these numbers’ ordinal values (Sarama & Clements, 2009: 83). The teacher is supportive in the way that she corrects learners’ misunderstanding of more and less. In some cases it is visually evident that there are more items in one collection than in another, But in other cases it is not immediately clear which collection (if either) has more items in it. Sarama and Clements (2009:84) claim that research suggests that multiple factors influence children’s use of counting to compare collections and state:

“They [children]do need to develop sufficient working memory to make a plan to compare, count two collections, keep the results in mind, relate these results, and draw conclusions about the two collections. To do so, their counting scheme itself first must be developed to a particular level” (p.84).

One way of comparing two collections is to directly match them as in the middle of Figure 2-4 below. (Cross et al., 2009:31-32).

Here each black bead is placed with only one white bead. If there is at least one extra white bead, the white collection has more objects. If at least one extra black bead is present, there are more black beads. If there are none left over, then the two sets of beads have the same number. It is not necessary in the early stages for the child to be able to say what the particular number is. If it is not possible to use the technique of direct matching, then each set of beads can be counted to work out which collection has more objects. When counting

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is used to compare which set has more objects, the child knows that the number that is said later in the number sequence relates to a larger collection than a number which is said earlier in the sequence. Using counting to compare collections of objects is a more advanced and more flexible way of comparing sets than one-to-one correspondence. Counting as comparing requires knowing how numbers compare to each other, that is: linking the number list to cardinality. In addition, collections which cannot be physically matched can be compared by counting (Cross et al., 2009:31-32). These kinds of mathematics learning opportunities help children learn to mathematize or engage in processes that involve focusing on the mathematical aspects of an everyday situation, learn to represent and elaborate a model of the situation, and use that model to solve problems (Cross et al., 2009:255).

Clip C: Resultative Grade R Counting using number cards This clip shows two different teachers working in different Grade R classes. In both situations the Grade R learners are doing resultative counting using number cards and counters. Each teacher works with different looking number cards and different looking counters. In the first situation the teacher holds the number cards and all learners work with the same number. In the second situation the learners start to choose their own number cards and so the numbers are not the same. This allows learners to compare the relative size of their sets.

Situation 1: reading number symbols and packing out the corresponding number of counters The number range is between 2 and 9. Children need to read the number symbol, and then pack out the correct number of counters. This is a more demanding task than saying how many counters are shown. The teacher starts by modelling and explaining what she is going to do. She is going to show a number card, for example, 2. Each child needs to pack out the corresponding number of counters. The examples are carefully chosen to get progressively more difficult. ; ; When learners pack out 3 counters, one learner uses a triangle formation. All the other learners pack their counters in a line. The teacher asks learners to check their partner’s counters. This they do enthusiastically. When learners pack out 6 counters, three girls initially make two rows of 3: one of these learners changes her formation to a straight line and then back to a cluster. The other children place their counters in a straight line. Children check each other’s sets. One learner notes, “six counters and six in the packet”. This is echoed by other learners. When learners pack out 9 counters, some children make 3 rows of 3. One child makes a column of 4 and a column of 5, one child makes a ring of 6 and an arc of 3. When they are

3 6 9

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asked to check, one of the learners changes her formation into a row as she counts. This reflects what the teacher did at the beginning of the lesson. Learners are again asked to check each other’s counters. The boy in the group tells the girl next to him that she should put one counter back. By the time the teacher checks her work, she only has 8 counters displayed.

Situation 2: threading number tags and the corresponding number of counters Here too, the teacher works with a small group of learners on the mat. Learners are given sets of threading beads and a group of number tags. The teacher has prepared a piece of string that has a number tag 3 on it. Learners are asked to read the tag, say the number and then pack out 3 beads. They are asked to check each other’s set. They do look at the set of the person next to them, but it is unclear whether they do in fact check. They then thread 3 beads onto the string. Learners then choose their own number tag. They read their numbers aloud in turns. The teacher then asks them to compare numbers of beads: “who has the biggest amount”, “who has the least amount”. Children find this difficult. In all clips in this unit, children find it difficult to compare numbers. As in so many of the DVD clips, we see children’s engagement with the task and apparatus is informed by what has been modelled earlier in the lesson. The teacher’s aim was for learners to pack out the number of beads stated on their chosen tags, to thread first their number tag and then the corresponding number of beads onto their strings. In the initial activity, the number tag was already on the string. It appears that children understand that the purpose of the activity is to thread the correct number of beads on the string. The teacher has not anticipated that the children will proceed before she gives the instructions. The children appear to be assuming that they should do what they did with the initial three counters. The teacher in this situation is aware that the second task is different, and has good reasons for changing the task. This time, the learners are asked to pick a random card themselves that is not on the string already, and thread the corresponding number of beads on the string. In doing so, each child has a different number from the other learners. In the first activity, all learners had the number three. However, she does not explicitly alert learners, before they start doing the second task, that it will be different. In Clip B of DVD 1.2 (finger counting) the teacher alerts learners to the fact that there will be a shift in the content that they are learning. You may like to compare that situation, in which the teacher is explicit about the content and procedural change with this one, where the teacher does not mark out this difference before learners engage with the second activity.

Comparing numbers: which is more, which is less? When learners work with different numbers of counters, they can compare how many counters each has. If numbers are different one can compare them: which is more, which is less? This is what was done in the last activity in both the “threading counters” clip and the previous one, in the game “More-you-win / Less-you-win”.

3

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Checking your peers collection of counters When learners are asked to recognise and pack out the same number of counters, it is easy for them to check each other’s collection. It also allows opportunities to understand conservation of number: different arrangements can represent the same cardinal number. This is what was done in the first part of the activities where children pack out counters and where children thread counters.

Clip D: Resultative Counting Grade 1 This clip deals with resultative counting in groups with a Grade 1 class. The teacher works with the whole class. In the Grade R classes learners were counting in ones. In this Grade 1 class, learners are counting in tens and fives. The teacher has chosen to work with groups of 10 first and groups of 5 second. This is because learners find counting in tens easier. It is also easier to hear from the spoken number word and easier to see from the written symbol how many groups of 10 there are in any particular multiple of 10. The teacher uses different media for counting: fingers, picture cards, number lines. The groups of numbers are symbolised in three different ways. She has sequenced these different representations of groups of numbers from ones that she believes are easier for learners to access to ones that she believes are more difficult for learners to access. This DVD allows for discussion and exploration of how learners can be brought more deeply into mathematics, through a series of activities in which one representation is replaced by another. The teacher aims to help learners understand how to generate multiples of 10 and 5. First the numbers are represented by fingers and verbal numbers. Then the numbers are represented by pictures of grouped objects, written numbers and the verbal language “groups of” is introduced. Finally the groups of numbers are represented by “hops” on a number line, while the verbal language “groups of” is consolidated. Using fingers To introduce counting in tens, the teacher first focuses learners on their fingers. Then she asks learners how many fingers they have on both hands. She selects a group of seven children to stand at the front. She arranges them with spaces between them and asks them to stand one step away from the writing board. Instead of just doing counting, she poses a problem: if each child has 10 fingers, how many fingers will seven children have altogether? The children modelling the activity lift both hands as she touches them on the back. The rest of the class counts: saying the next multiple of 10 as the corresponding child lifts his or her hands and shows 10 fingers. When the class get to 70, the teacher confirms 7 children have 70 fingers. She invites another learner to join the row and asks “what if we have eight children? She moves the focus from counting to learners reflecting on the numbers. “Did the numbers get bigger or smaller?” “Did the numbers get more or less?”

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“How many more fingers each time as we counted?” Whilst asking this, she gets the front row of learners to raise their hands each time. “So in groups of what are we counting?” She has changed the focus from counting, to being conscious that they are counting in groups of 10. This lays the basis for multiplying by 10. This activity is repeated with the focus on groups of five: starting with focusing on the number of fingers on one hand and ending with the focus on counting in groups of five. Using picture cards The teacher shows a card with five apples on it. She asks them how many apples are on the card. She explains that she has many such cards and that as she places the cards on the board, they will count. She asks them “in groups of what will we count?” This consolidates the language “groups of” developed in the previous activity. After 10 cards are placed on the board, the learners have counted to 50. She asks learners to tell her, how many apples there are at each stage. As they count again, she constructs a sequence of multiples of 5. This lays the basis for the number line image that she will work on later. When learners make a mistake, she writes up the incorrect number and then asks learners to check it. In this way she ensures that learners do keep focussed on the task. To get learners to reflect, she asks “in groups of what are we counting again?” Then she returns to the questions asked after the finger exercise: “The number of apples: is it becoming bigger or smaller?” “Is it getting more or less?” “Who can tell me, by how many apples is it becoming more each time?” Learners find this question quite difficult to answer. It is possible that learners may have found the question easier to answer if she had left spaces between each picture of 5 apples.

However, even where there are spaces between the pictures of groups of 5 apples, the number sequence and verbal counting still foreground the cardinal number of the set, rather than the size of the group added each time.

5 10 15 20 25 30 35 40 45 50

5 10 15 20 25 30 35 40 45 50

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Using a number line The teacher has introduced the language “groups of”, she has focussed learners on the fact that they are counting in groups of 5, and that each time they get 5 more. She tells them that they are going to help her to count on the number line: “We are going to use this number line and we are going to make groups of 5”. She then models the process they are going to use. She asks “When we count in groups of 5, where do we start?” The children say 5. She explains that she needs “to add one group of 5” whilst she draws a “jump” from 0 to 5 above the number line. She is explicitly modelling that a group of five is shown with an arc from one multiple to the next. Despite this when she asks learners “How many jumps do you see?” they give her random answers. It takes a while for learners to come up with the answer 1. She explains that she will “add another group of 5.” She asks “if I add another group of 5, which number will I land on?” This is easy for the learners to imagine. They have an automated awareness of counting in 5s, but are not secure in thinking about the number of “groups of 5.” This is the focus of the lesson. The teacher continues to link number of jumps with number of groups with the multiple “….. groups of 5 makes …..” She is moving from counting in groups to the language of multiplication. In order to check that learners understand the process, she asks: “Who can tell me what I will do next?” She then shifts the focus to a series of questions: “…. hops of 5 makes …..” These questions are not posed in increasing or decreasing order. The teacher then focusses on making groups of 10 which are represented as jumps on the number line. The language focus is “…. groups of 10 makes….” Sometimes we assume that children will automatically connect skip counting with multiplication. It is however, better to scaffold these connections rather than to leave learners to make the connection by chance. When counting in groups the focus is most strongly on the sequence of multiples. In this DVD the teacher explicitly focusses on the multiple, the number of groups and the size of the groups. The aim is to connect, for example, that 8 x 10 = 80, means the same as “eight times a group of ten is eighty”.

5.4.2 Helping students to reflect on DVD 1.2: teaching resultative counting

The four clips in DVD 1.2 are designed to be viewed together.

Teaching Notes The total running time of DVD 1.2 is about 36 minutes, so you may like to work with the three Grade R clips first. When students have a good grasp of the issues

raised in those three DVDs, then you could work with Clip D. Finally discuss and compare the three Grade R DVD clips and the Grade 1 DVD clip.

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Focus points for student activities on resultative counting

Resultative counting helps learners to understand the principles of counting. It also builds awareness of both ordinality and cardinality. Acoustic counting may help children to learn about ordinality, but it does not help them to learn about the principles of counting or about cardinality.

Student Activity 5.4.2: Principles of Counting Do this with a fellow student

1. Remind yourself of Gelman and Gallistel’s (1978) Principles of counting and their implications for teaching, in Section 5.4 above.

2. Which of the principles of counting are being reinforced in DVD 1.2? 3. Go back to DVD 1.1 and

a. identify which principles of counting are being developed b. identify the visual and other support provided in the oral counting snippets. c. Describe how these supports help to develop the principles of counting.

These DVD clips are designed to focus attention to ways of supporting or scaffolding a transition from simpler to more complex ways of doing resultative counting. The aim is to see that apart from increasing the number ranges and the change from counting in units to counting in groups, there are several other ways to increase learners’ capacity to do resultative counting. When children start to do resultative counting, it is easier for them to touch or move the objects that they are counting. With practice children are able to imagine doing this in their head and so no longer need to physically touch or move objects. You can explore this idea a bit more in the next activity.

Teaching Notes The next activity is very long. You might want to break it up in some way. You can either do some parts of the activity in different class sessions, or you can divide the class into groups and get different groups to discuss different questions. Do what is most suitable to your situation.

x Principles of Counting x Level principle: Progression in resultative counting

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Student Activity: 5.4.3: Level Principle - progression in resultative counting Think about the questions alone while you are viewing and then discuss your thoughts with a fellow student or in small groups.

1. View DVD 1.2 Clip A: resultative counting with counters 2. You will see that the teacher asks the children to say how many counters she puts down

on the towel. 3. First she places 5 counters down and then she places 6 counters down

a. Why does she initially place the counters down one-by-one? b. Why does she place the counters in a pentagon and hexagon formation and not

in a straight line? (DVD 1.3 deals with this in more detail). c. When she puts the set of 3 and then 4 counters down she puts the whole

collection down at once. d. Why did she use more counters with the first two sets and fewer counters with

the last two sets? e. When she counts the second and third set she moves the counters into a line

formation. Why does she do that? 4. View DVD 1.2 Clip B: resultative counting with fingers 5. You will see that there are three separate activities here:

x Showing five or fewer fingers x Showing between five and ten fingers x Pairs of children decide a number to show and then they compare the size of the

numbers. a. Which of the three activities do the learners find most challenging? b. When you watch the other clips, and DVDs see whether this kind of task is always

challenging to learner. c. Discuss the difficulties which the learners are experiencing with comparing

numbers in terms of the theories of Cross et al. (2009) and Sarama and Clements (2009)

d. Wright (2008: 201) raises the question: “When a child builds finger patterns … (a) Does the child need to look at their fingers? (b) Can the child raise fingers simultaneously (e.g. for seven, simultaneously raising five fingers on one hand and two on the other)?” Discuss the above questions in terms of the clip in DVD 1.2.

e. How does the teacher get learners to check their answers? Why does she teach them to check in this way?

6. View DVD 1.2 Clip C: resultative counting using number cards

a. What is different about what the learners do in the two situations in DVD Clip C? b. The task in the first class allows the learners to do something that they can’t do

in the second class. What is that? c. The task in the second class allows the learners to do something that they can’t

do in the first class. What is that? d. In Situation 2, the teacher first provides the children with a tag already threaded

on the string. Then she deliberately changes the second instruction for threading. Why does she do this? What are the implications of this?

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7. Comparing DVD 1.2 Clips A to C. a. List the apparatus that is used in each of the clips A, B and C. b. Apart from the apparatus, what is different about the mathematical tasks in each

clip A, B and C? c. In what ways do the tasks get more complex from Clip A to Clip C?

8. DVD 1.2 Clip D: resultative counting using fingers, picture cards and number line

a. What are the mathematical differences between the counting task that learners in Grade R do and the counting task that learners in Grade 1 do?

b. The learners first count using fingers, then pictures where the teacher records the sequence of numbers, finally she records their counts on the number line.

i. Why did the teacher sequence the tasks in that order? ii. Why did she not start with the number line and end with fingers?

9. The teacher let the learners count all the fingers on the hands of 7 children. Once they

had an answer, she added another learner, and asked “how many fingers now?” Adding an additional learner was planned. The teacher was not correcting an earlier mistake.

a. Why did she add an extra learner? What was the mathematical purpose? What was the pedagogical purpose?

10. When we multiply we often talk about “….. groups of …….”

a. Did the children find it easier to count and give the total number, or easier to say how many groups they had counted, or easier to say how big the group they were counting was?

11. We often say that skip counting lays the basis for multiplication. Watch the DVD again

and check whether learners make this connection on their own.

12. Write down the questions and the language that the teacher uses to make learners aware:

a. That they are counting in groups. b. How many groups they have counted. c. How big the groups are that they counted.

Teaching Notes The following Applied Task will help students to apply what they have learned from doing the focussed DVD activities.

Applied Task for Students 5.4.4: Progression from oral counting through resultative counting to the beginning of multiplicative thinking This is an activity to do alone in order to apply what you have learned from

doing preceding the activities. 1. DVD 1.2 Clip D illustrates the level principle for resultative counting in Grade 1. It shows

the series of transformations from oral counting to awareness of multiples.

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2. Make a flow diagram in which you show how the teacher takes the class from oral counting through resultative counting to using the image of hops on the number line to focus learners on groups. In groups of what are they counting; how many groups of …. make … Your flow diagram should include parallel lines for

a. Mathematical focus b. Focus of questions c. Focus on development of mathematical language d. Focus on visual representation including written recordings

5.5 Subitising Le Corre and Carey (2007) suggest that very young children who are not yet fluent with the principles of counting are able to recognise between 1 and 4 objects. The ability to recognise and name the number of objects in a collection without counting is called perceptual subitising (Clements, 1999). Learners can practise subitising of amounts up to 4, then extend this to 5. Children can learn to use perceptual subitising to identify sub-sets of objects and combine these subsets into a whole. This is known as conceptual subitising (Clements, 1999). Here children use the structuring or layout of the objects in the set to chunk them into immediately recognisable amounts. It helps to develop learners understanding of part-part-whole relationships. “The different arrangements suggest different views of a number” (Clements, 1999). The arrangements below depict the following relationships: four and one make five, two and one and two are five, two and three are five, and five and zero make five.

The idea of building up and breaking down numbers is developed further in Unit 6, DVD 2.4.

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These groupings are useful, first in recognising the number, and then using the composition as a model for calculation. (Flexer, 1989). Understandings of addition and subtraction are developed. Addition combinations such as 4 + 1 = 5; 2 + 2 + 1 = 5: 2 + 3= 5 (or 3 + 2 = 5); 5 + 0 = 5 are implicit in these arrangements. In the same way, the inverse operation, subtraction, is implied. 5 – 4 = 1; 5 – 3 = 2; 5 – 2 = 3 and later, 5 – 0 = 5.

“The spatial arrangements of sets also influence how difficult they are to subitise. Children usually find rectangular arrangements easiest, followed by linear, circular and scrabbled” (Sarama & Clements, 2009: 45). Subitising small numbers appears to precede and support the development of counting ability (Le Corre et al., 2006). Thus it appears to form a foundation for all learning of number….but it is not the only way children think and learn about number. Counting is ultimately a more general and powerful method (Sarama & Clements, 2009: 50 - 51).

5.5.1 Description of the DVD clips on subitising The focus of the two DVD clips in DVD 1. 3 is on the progression from the perceptual subitising in LPN stage 1 to conceptual subitising in stage 2. Subitising activities focus

strongly on the cardinal number of the set: how many objects are in each group.

Teaching Notes Towards the end of DVD 1.3 are still frames of the clips which can be used for the purposes of reflection.

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Summary of DVD clips on subitising: DVD 1.3 Elements Clip A Clip B Grade Grade R Grade 1

Timing (00:00 – 01:48)

(01:48 – 05:26)

Approach Teacher modelling Teacher modelling

Progression / transition

From perceptual subitising to conceptual subitising

Manipulatives Dot cards Grouped pictures of kinds of items of clothing

Dot cards

Everyday contexts Clothing

Mathematics Cardinality of Number Cardinality of Number Part-part-whole relationships

Teaching Notes The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with subitising.

Student Activity 5.5.1: Subitising Read alone, watch the DVD alone or as a group, and then discuss the questions.

1. Read the introduction to resultative counting in Section 5.5. 2. View DVD clip 1.3 all the way through. 3. While you are viewing think about what you have read, and think about the following

questions. When you have finished viewing the DVD clip, discuss the reading and the questions with your fellow students

a. Does subitising focus on the cardinal or ordinal aspect of number? Explain your answer.

b. What are the differences between perceptual and conceptual subitising? c. How do the visual representations of the collections presented by the teachers

encourage or discourage subitising? d. A question about mathematization? What is the role of subitising in the process

of mathematization? Compare your thinking and discussion with our description of the DVD clips, below.

Clip A: Perceptual Subitising The first DVD clip deals with perceptual subitising in a Grade R class. The number range is between 2 and 4. The teacher works with dot cards with a small group on the mat. They recognise at a glance the number of dots on each card she shows.

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The teacher also shows the whole class pictures of clothing arranged in three groups. Each group has both a different item and a different quantity of clothing: 2 shoes, 3 caps, 4 shirts. She asks learners “how many ____ are there?” The pictures are large enough for the whole class to see. Once learners have answered, she draws a ring around the set and re-iterates the cardinal number and what objects they are. Her actions foreground the group as a unit and its cardinality. Learners work only with images and words representing numbers, not with numerals.

Clip B: Conceptual Subitising The second clip deals with conceptual subitising in a Grade 1 class. The teacher works with the whole class. She shows them cards with dots on them. The arrangement of dots into sub-groups is quite distinct on the first two cards (4 dots is 2 dots and 2 dots; 7 dots is 5 dots and 2 dots). On the final card the dots are presented as a single group, without a clear spatial separation into sub-groups. This makes it difficult for the children to recognise how many dots are on this card. First the teacher asks how many dots they see, and then asks how they know that they saw that number of dots. She allows learners to explain the patterns of sub-groups that they see. She allows different learners to explain that they see different sub-groups of dots. She consolidates that different sub-groups exist and uses gesture to ensure that all learners can see that there are different sub-groups and by implication that it does not matter which sub-groups they focus upon. Learners are being oriented to decompose the total number of dots into smaller recognisable groups of dots. The decomposition strategy has been privileged as the preferred strategy in the lesson, both by the arrangement of dots on the cards and by the amount of time the teacher spends asking learners how they identified the number of dots. In the first card learners answer by describing sub-groups of dots placed in a line: either the horizontal rows “2 on top and 2 at the bottom” or the vertical columns “2 on the left and 2 on the right”. When the second card is shown, some learners appear to have assumed that they should try to see different arrangements and appear to be looking for dots positioned in a line: “3 on the right and 3 on the left and one in the middle.” The teacher appears momentarily surprised by the answer.

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The decomposition of the dots into sub-groups develops a familiarity with part-part-whole relationships. It lays the basis for aspects of number-based resultative counting and for addition and subtraction. The last card, which has dots evenly spaced in a diagonal row, is considerably more difficult to recognise that there are 5 dots. There is no spatial separation of the dots into sub-groups. Fewer learners offer to provide an answer. Some learners try to apply the decomposition-into- sub-groups-(parts) strategy that has been privileged earlier in the lesson, although the dots themselves are not been spatially arranged into distinct sub-groups. Some learners try to read the answer off the back of the card, this is because the arrangement of the dots makes it too difficult for them to just see that there are five dots.

5.5.2 Helping students to reflect on DVD 1.3 teaching subitising The two clips in DVD 1.3 are designed to be viewed together.

Focus points for student activities on subitising

Student Activity: 5.5.2: Visual images and ways of representing mathematics Discuss these questions with a partner.

1. Some teachers think that subitising can / should only be done with dot cards.

a. Name the two different kinds of images that the Grade R teacher uses? b. What number range did the Grade R teacher work with? c. What dot-card did the Grade R teacher show first? d. What did she show second? e. What did she show third? f. Do you think that the sequence was random or deliberate? g. What number of objects / dots are easy to subitise? h. What arrangements are easier to subitise and what arrangements are more

difficult?

x Visual images and ways of representing mathematics x Aspects of number x Level principle: from perceptual to conceptual subitising x Modelling and classroom practice

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2. When the Grade R teacher worked with the whole class she showed a group of shoes, a group of caps and a group of shirts.

a. Imagine that she had shown three different groups of shirts. How could this have made the task less clear?

3. Each group of clothes is spatially separated from the others. a. Why did she circle each set of objects as she stated the cardinal number of

the set e.g. she circled the shoes as she said “2 shoes”?

4. Imagine that she had had the 2 shoes, 3 caps and 4 shirts all mixed up. She could still have asked: “How many ________ are there?” How would the task have been different?

An aspect of the level principle is the idea that teaching is aimed at supporting or scaffolding a transition from simpler to more complex ways of dealing with numbers. Let’s explore this in the next activity.

Student Activity 5.5.3: Level Principle: from perceptual to conceptual subitising

Think about these questions on your own first, and then discuss your ideas with a partner. 1. Subitising is done in both the Grade R and the Grade 1 class.

a. What number range is used for perceptual subitising in the Grade R class? b. What number range is used for conceptual subitising in the Grade 1 class?

2. Think about the first two dot cards shown in the Grade 1 class.

a. How do they build on the work done with perceptual subitising in Grade R? b. What is perceptual subitising? c. How is conceptual subitising different to perceptual subitising?

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3. How can practice with conceptual subitising lay a foundation for: a. estimating numbers of objects? b. resultative counting? c. Addition? d. Subtraction?

At any one moment in a classroom teachers are managing the interplay between many different content issues, learners with many different needs and lots of classroom management issues. When the student teacher who taught this lesson reflected on the DVD she said the following:

x I tend to have a time management problem. I am working on this and getting better at it. The pace of the lesson was too slow

x I asked too many learners to say what they saw. I should just have flashed the card at them briefly. I should not have shown it to the class again whilst they spoke about what they saw. I could rather have drawn what they were talking about on the board.

x Did you notice that when I was repeating what a learner saw on the first card, I hesitated when I tried to show “two on the left and two on the right”. I find it difficult to remember how to show the mirror image of left and right for learners.

Student Activity: 5.5.4: Modelling and classroom practice Discuss these questions with a partner.

Take into account the student teacher’s reflection above on her own teaching, when you discuss the following questions:

1. What is subitising? 2. How can looking at a dot card for an extended period of time change the focus of an

exercise from subitising to something else?

The way in which learners respond to questions, instructions, problems, calculation and texts is sometimes influenced by their reading of classroom practice in earlier parts of the lesson or earlier lessons.

The second card that Grade 1 learners were shown had 7 dots on it. The first learners, who explain how they knew it was 7, say that they saw 5 dots and 2 dots. The teacher repeats this as “5 dots on the top and 2 dots on the bottom.” Other learners then justify the answer by saying they saw “3 on the right and 3 on the left and one in the middle”. The teacher appears momentarily surprised by the answer.

3. Why do you think that the teacher may have been surprised by this answer? 4. Why do you think learners offered this answer?

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In the Grade 1 lesson, the last card that learners were shown had 5 dots equally spaced in a diagonal. When learners explained how they knew it was 5 dots, they decomposed 5 into 2 and 3 or 4 and 1. This decomposition was not reflected in the arrangement of the dots.

5. Why do you think learners offered these kinds of justifications for seeing 5 dots on

the final card?

5.6 Number sequences The terms ordering, sequence, and seriation are sometimes confused. Kennedy et al (2008: 143) define the terms as follows: “Order has a beginning, middle, and an end, but placement within the order can be arbitrary. When threading beads on string, children may put a red bead first, a yellow bead second, and a green bead last, but they could easily rearrange the beads. In sequence, order has meaning. Days of the weeks have fixed sequence, as does the counting sequence. Seriation is an arrangement based on gradual changes of an attribute and is often used in measurement. For example: children line up from shortest to tallest.” Counting is a sequence of words related to increasing number: 1, 2, 3, 4, 5. . . . (Kennedy et al 2008: 143). One of the basic properties of the system of whole numbers is that for each number there is a next number, also called its successor. The next consecutive whole number is always one more. One moves from one whole number to the next by adding 1. In the same way one can get from one whole number to the previous whole number by subtracting 1. Counting backwards is a process of repeatedly subtracting 1. “A sequence of events has two orders: beginning to end and end to beginning, or forward and backward. Developing a sense of reversibility, or opposite order, is an important thinking skill” (Kennedy et al 2008: 398). Young learners find it relatively easy to count forwards in ones. Counting backwards in ones is more difficult for young learners than counting forwards so teachers need to provide a range of both types of number sequences. In the Learning Pathway for Number, counting forwards and backwards between 1 and 10 is regarded as a Stage 1: Emergent-and-Growing-Number-Concept skill. In Stage 2: Counting-and-Calculating, learners are expected to first count forwards and backwards in ones between 1 – 20, and are then introduced to skip counting. Counting in tens, fives and twos are also regarded as being within the capabilities of young learners. Counting backwards in tens, fives and twos is more difficult for young learners. Learning to count in ones or to skip counting is not the same as understanding number sequences. Learners’ understanding of Mathematics and number systems can be deepened considerably if they are also asked to focus on how to generate particular number sequences. We have already seen that the set of natural numbers is generated by adding one to each successive number. Similarly when you counts backwards in ones, you are subtracting one to generate the preceding number.

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In Mathematics education there have been concerns that learners do not connect what they learn about arithmetic in primary schools with what they learn about algebra in high schools. One response has been to take a more algebraic approach to the way arithmetic is taught. This means focusing on the structure and patterns that underpin arithmetic (Carraher, Schliemann & Schwartz. 2008). An aspect of this is to for learners to examine and generalise patterns, including number patterns. Learners work towards understanding and being able to say how the patterns are generated. This goes beyond counting or reciting multiples. The aim is not for learners to learn set procedures for finding patterns, but at times it is useful to guide learners’ approaches. In the diagram on the next page, Roberts (2012a: 312) suggests a guideline to help learners move from counting and extending number patterns, to generalising towards a rule or function for generating the number pattern.

5.6.1 Description of the DVD clip on Number Sequences There is one clip in DVD 1. 4. It focusses on backward counting sequences in a Grade 1 lesson. The mathematical focus of the lesson and the song is on how to generate

the backward counting sequence. It is about finding the rule that allows one to count backwards in ones. This clip illustrates that one can generate the previous whole number by subtracting one. The clip also reinforces the following three counting principles:

x One-to-one correspondence x Stable order principle x Cardinal principle

Teaching Note Towards the end of DVD 1.4 are still frames of the clip which can be used for the purposes of reflection

4. HOW TO CARRY ON

3. JUMPS

2. DIRECTION

1. START

How big are the jumps between numbers? Are they the same or different?

What must you do to get to the next number?

Does it get bigger or smaller? Is it increasing or

Where does the pattern start? What is the first number? Roberts (2012a: 312)

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Summary of DVD clip on Number Sequences Elements 1 clip Grade Grade R

Timing (00:00 – 3:48)

Approach Teacher modelling

Progression / transition

DVD 1. 4 shows number sequences for Grade R. Learners count backwards in 1s. Students need to imagine a suitable task for Grade 1 learners. They need to think about progressions themselves. This is a possible student task.

Manipulatives Number line with removable pictures of cookies

Everyday contexts Buying cookies

Mathematics Successor function for Natural numbers: generating the preceding whole number, in the number range 12 – 1

The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with number sequences.

Student Activity 5.6.1: Number sequences Read alone, view the DVD alone or as a group, and then discuss the questions.

1. Read the introduction to resultative counting in Section 5.6. 2. View DVD clip 1.4 all the way through. 3. While you are viewing think about what you have read, and think about the following

questions. When you have finished viewing the DVD clip, discuss the reading and the questions with your fellow students.

a. What is the concept that this DVD clip illustrates? b. What is the context that the teacher uses? c. How do the learners respond to the context and resources used? d. What are the complexities which the teacher experiences regarding the use of

mathematical language? e. What progression do you notice (or not) in the sequence of questions posed by

the teacher? f. How do the questions promote mathematization?

Compare your thinking and discussion with our description of the DVD clip below.

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Clip A: Grade R Number Sequences Unlike the other Grade R clips, this teacher uses a number line to support the notion that one can get to the previous whole number by subtracting 1. She has made a number line from 1 to 21. Learners work with the numbers 0 to 12. She has placed a removable picture of a cookie at each number 1 – 12. The cookie at 10 looks different to the other cookies. The teacher frames most of the lesson within the context of the song: buying cookies from a bakery shop. However, she first establishes the mathematical focus of the lesson. At the end of the story song, she elicits from learners first a summary of the story context and then focusses them again on the embedded Mathematics. The teachers first checks that learners can read numbers from 12 down. From 12 to 9, she asks “what number comes before…?” She uses hand gestures to indicate that when counting back one is moving to the left on the number line. When asking what comes before 10, she changes the direction of her hand gesture. After singing the song she again uses hand gestures when talking about counting forwards or backwards. Throughout the lesson she asks different learners questions and provides constant positive feedback. She chooses a different learner as the character for each verse of the song. Learners enjoy being chosen: it increases learner participation in the lesson. After singing the song, the teacher asks questions which allow learners to reflect on the mathematical focus of the lesson. She is training learners to think about their learning. Developing metacognition can start from a young age. In her last two questions the teacher incorporates one learner’s answer “numbers”. Generating questions based on learners’ answers or partial answers is a useful skill for teachers. When trying to incorporate learners’ responses and their language into her subsequent questions, her language becomes unclear and she deviates from the focus of the lesson. By the simple slippage of “one number” instead of “the number one” or simply “one” has been taken away from eight to get seven, the focus shifts from the cardinal aspect of number and the successor function, to the ordinal aspect of number. After the song she asks learners whether they were counting forwards or backwards. She teaches interactively and reinforces with hand gestures, instead of using only words that counting forwards moving (on a number line) to the learners’ right and counting backwards is moving to the learners’ left. The use of gesture and facial expressions in teaching can also be used as a scaffold for second language learners (Kennedy et al, 2009:35).

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5.6.2 Helping students to reflect on DVD 1.4 teaching addition with object-based counting-and calculations

The clip in DVD 1.4 focusses on number sequences in Grade R.

Focus points for student activities on Number Sequences

Student Activity 5.6.2: Backwards number sequences Think about these questions and then discuss your ideas with a partner.

1. How can you tell that the focus of this lesson is on backwards number sequences? 2. The research of Griffiths (2008:49) on the mathematical home practices of parents with

young children showed that all counting activities involved forward number sequencing. It had not occurred to them that counting backwards might be useful. This is also sometimes true of Grades R and 1 teachers.

a. What are possible reasons for this? b. What is your experience of backwards counting activities during the teaching

practicum in the Foundation Phase? Young children learn many things when singing. We have already discussed the value of singing counting songs.

Teaching Notes Again, the next activity is very long. Plan if you want to break it up in some way. You could do it differently to how you did the first one, perhaps, to provide some variety. Think about creative and challenging ways to organise it, to get the students thinking.

Student Activity 5.6.3: Incidental, implicit or explicit learning Discuss the questions in this activity with a fellow student.

1. Compare the song used in DVD 1.4 with the counting rhyme: “1, 2, 3, 4, 5; Once I caught a fish alive, 6, 7, 8, 9, 10, then I threw it back again”

x Backwards number sequences x Incidental, implicit or explicit learning x Everyday contexts x Language x Gesture

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2. The rhyme above has sequences of numbers listed next to each other. The song “12 little cookies in a bakery shop….” has only one number in each verse.

a. Discuss how this difference may impact on what children might learn. b. Discuss the benefits and possible limitations of the rhyme and the song for

teaching aspects of counting.

3. In this lesson the teacher wants learners to learn more than just order of the numbers. a. What feature of the system of whole numbers is the main focus of this lesson?

4. The DVD clip is about 3½ minutes long. Learners spend about 2 minutes singing the song.

a. What visual aids has the teacher prepared before the lesson? b. 11 of the 12 cookies look the same.

i. The cookie at which number looks different to the other 11 cookies? ii. Is this deliberate or accidental?

iii. Why is it useful to have a different looking cookie at this number? c. What features of the visual aids, help learners to focus on:

i. Cardinality? ii. Ordinality?

iii. How to generate numbers in the whole number system? d. What does the teacher do before she introduces the song? e. What does the teacher do once the song has ended? f. Discuss different ways of introducing this counting rhyme. You will remember

our reference Gardner’s Multiple Intelligences in Section 5.4. You can use this as a starting point for your discussion here.

5. The teacher asks the following questions. Several of these questions build directly on

learner responses to her previous questions. x What happened to all our cookies? x Why are they gone? x What did we do first? x How much did we pay, can you remember? x Who can remember what number we started at? x What was the last number we said? x If we were at number eight, what number would come next according to our rhyme? x So were we counting forwards or backwards? x So what happens when you count backwards? x How much do you take away each time? x How many numbers? x If we go from eight to seven, how many numbers did we go back?

a. Which question marks the shift from the song to the focus of the mathematics?

Motivate your answer. b. What answer is the teacher expecting to the following question: “How much do you

take away each time?” a. What are the various answers the learners give to this question? b. Are any of them the answer the teacher is hoping to get?

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In her last two questions the teacher incorporates one learner’s answer “numbers”. Generating questions based on learners’ answers or partial answers is a useful skill for teachers. However, one needs to be careful that one does not lose the focus of the questioning. This is particularly important in mathematics because Mathematics is a precise science, and so language needs to be used carefully.

Student Activity 5.6.4: Use of language Think about the questions in the next set of activities. Discuss them with a fellow student or in a small group.

1. What is the difference, in the context of the song, between the following two

questions: a. “How many numbers” (do we take away)? b. “What number do we take away?”

2. What is the difference between asking, in the context of the song, the following questions:

a. “If we go from eight to seven, how many numbers do we go back?” b. “If we go from eight to seven, what number do we subtract?” c. “Seven is how much less than eight?”

3. Teachers are often quite surprised at the language that they use when posing questions. It may be an idea to DVD or audio-tape one of your mathematics lessons and consider how you use language in mathematics teaching. Is it possible that the questions you pose cause misconceptions in the learners? What is the response of the learners when you ask questions? How is the mathematics evident in your questions? You could share these experiences with your peers (adapted from Zevenbergen, 2001:48).

Before the song the teacher engages learners with questions about “what number comes before ….” View the DVD to see her deliberate gestures to indicate that when counting back one is moving to the left on the number line. After the song she asks learners whether they were counting forwards or backwards. She reinforces with hand gestures, that counting forwards is moving (on a number line) to the learners’ right and counting backwards is

moving to the learners’ left.

Student Activity 5.6.5: Use of gesture

1. At which point in the DVD clip does she lose focus and gesture in the wrong direction?

2. What are the possible reasons for this mistake? 3. What impact might this have on the leaners?

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Student Activity 5.6.6: Everyday contexts

The teacher embeds the lesson in an everyday context of children buying biscuits from a bakery shop.

1. Is this context suitable for the class she is teaching? What makes the context of the song suitable or unsuitable for these learners?

2. If you are a practicing Grade R or Grade 1 teacher: a. Say whether this context is suitable for your class. Explain why or why not. b. Talk about how you could adapt this song or suggest another song that uses

repeated verses for counting back 3. If you are a student teacher,

a. Think about contexts in which this song may not be suitable. Explain why it may not be suitable in that context.

b. Talk about how you could adapt this song or suggest another song that uses repeated verses for counting back.

Teaching Notes Once again, the following Applied Task will help students to apply what they have learned from doing the focussed DVD activities on number sequences. The lesson plans that they produce can be used in their practical work in schools, and/or in their assessment.

Applied tasks for students 5.6.7: Progression in Number Sequences: Planning Grade 1 lessons on Number Sequences Do this on your own.

In the Grade R DVD clip, learners count backwards in 1s from 12 to zero. The activity is supported by a counting song, and a number line with removable images 1. Write a list of number sequences suitable for Grade 1. 2. Order these from easiest to most difficult. Consider both the number sequences (e.g. 1s,

2s, 3s, 5s, 10s,) and the medium of production (oral counting, oral rhymes and songs, written sequences – here consider various possible ways of recording the sequences).

3. Think not only about how to help learners extend the pattern, but also ways in which you can help them to find and talk about the rule for the construction of the number sequences.

4. Write lesson plans or sections of lesson plans that contain number sequences. Indicate at what point in the year you think it will be suitable to use each lesson or section of lesson.

5. About how much time do you think should be spent on number sequences in Grade 1?

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5.7 Estimation “According to (Kennedy et al 2009:176), estimation “is an educated or reasonable guess based on information, prior knowledge, and judgment”. Sarama and Clements (2009: 87) refer to estimation as “a process of solving a problem that calls for a rough or tentative evaluation of quantity.” They add that different types of estimation have been identified, but amongst those it is important for young children to learn to include:

x Estimation of numerosity x Estimation of measurements x Computational estimation

DVD 1.5 deals with estimation of numerosity. Primary school teachers sometimes ask questions which indicate that they are not sure how to handle estimation in class. Teachers often ask one of the following questions:

x What do I do when learners’ estimates are either considerably more or considerably less than the actual number of objects or the actual measurement?

x Should I assess estimations? o If so, how do I rank/ evaluate how good estimation is? o If not, why am I teaching it?

Behind the questions is perhaps a belief, that exact answers are always better than approximated answers. Do teachers give learners mixed messages regarding the importance of estimation and approximation in relation to getting the answer exactly correct? Learners need to be assured that estimation is about “reasonableness” and being “close enough” Useful vocabulary to reinforce the concept are words such as: ‘about’, ‘more or less’ and ‘roundabout’. In Stage 1 estimation skills and understanding of what it means to estimate can be encouraged by tasks that ask for a ‘qualitative’ comparison of the numerosity of two collections. This means that children only have to say which of two collections contains the larger number of objects. This approximate resultative counting also occurs when children use qualitative number terms such as ‘a lot’, ‘very many’, ‘more than enough’ or ‘too many’ describing collections in terms of more/less or big/small based on the visual image. – noting the quantities taking up more or less space (LPN). Estimation is about relationships. Using language such as: “Does the brown bird have as many eggs as the yellow bird?” or “Nande, bring just enough spoons for everyone to have one.” helps learners to think about approximate quantification. In Stage 2, children build upon their experiences in Stage 1 and move beyond simply describing collections in terms of more/less or big/small and begin to make comparisons of collections based on reasonable estimates. However, mathematics is not always about exact answers. Calculating by approximation, as well as working towards answers from increasingly improved upper and lower bounds are recognised and useful mathematical practices. Estimation and approximation are useful to

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check whether one’s answers are reasonable: do answers fall within an appropriate number range. Estimation is also useful when budgeting; to assess whether one has sufficient money or time or petrol or other resources. Estimation is useful in developing

x number sense or numerosity: how does a collection of 14, 4, 40, 400 objects look different from each other?

x a sense of unit of measurement: how is 1 cm different to 1 m, etc.? x judging the reasonableness of one’s calculations

Research also indicates that “the ability to quickly tell which of two numerals, 0 to 20, represented the larger number” is “one of only three [abilities] that predicted later mathematics achievement (Chard et al. cited in Sarama and Clements, 2009: 97).

5.7.1 Description of the DVD clips on estimation The focus of the two clips in DVD 1.5 is on estimation in stage 1 and estimation in stage 2 of the LPN.

Teaching Note Towards the end of the DVD are still frames of the clips which can be used for the purposes of reflection

Summary of DVD clips on estimation: DVD 1.5 Elements Clip A Clip B Grade Grade R Grade 1

Timing (00:00 – 01:57)

(01:57 – 08:34)

Approach Teacher modelling Teacher modelling

Progression / transition

DVD 1.5. deals with the teaching of estimation based on stages 1 and 2 of the LPN. Learners progress from comparing collections through qualitative descriptions (e.g. more or less) to making reasonable estimates.

Manipulatives Children Counters

Number line

Everyday contexts

Classroom environment- estimation of the number of learners standing in a line

Classroom environment – use of counters, number line to indicate actual and estimated counts

Mathematics Estimation Estimation using a referent

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Teaching Notes The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with estimation.

Student Activity 5.7.1: Estimation Do the reading and viewing of the DVD alone, and then discuss in small groups or with a fellow student.

1. Read the introduction to estimation in Section 5.7 above. 2. View DVD clip 1.5 all the way through. 3. While you are viewing think about what you have read, and think about the following

questions. Then discuss the reading and the questions with your fellow students. a. The DVD clips deal with the estimation of numerosity. Design an estimation

activity that deals with the estimation of measurement b. Name the terms you would use to to foster young learners’ understanding of

estimation of measurement c. Why is the development of young learners’ understanding of relevant

terminology important when nurturing their estimation skills? d. In developing the estimation skills of very young learners, estimation is firstly

introduced using qualitative terms and thereafter quantitative estimates are introduced. How does this approach reflect the concept of mathematization?

Compare your thinking and discussion with our description of the DVD clips, below.

Clip A: Introducing Estimation in Grade R The first DVD clip deals with estimation in a Grade R class. The teacher has arranged a number of learners in a line facing the rest of the class. She is careful to introduce the term “estimation” and to explain what it means. She asks the learners to estimate how many children are in the row at the front of the class. Some learners immediately suggest 10. There are no other offers until she asks for other suggestions. One learner suggests 2. The teacher then separates 2 learners from the line and asks “Are there more or less than 2 learners?” The teacher is starting to introduce the idea of the relative size of numbers and implicitly the relationship between numbers. She is alerting learners to the possibility of using an amount that they can subitise as a referent for judging the size of an amount that they cannot subitise. She is helping learners to find strategies for using what they know to estimate or get a feel for the size of an unknown amount. Both the Grade R and the Grade 1 class appear to struggle to answer the “more or less” questions.

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Clip B: Estimation using referent sampling technique The second clip deals with estimating in a Grade 1 class. The teacher works with the whole class. She too, is careful to introduce the term “estimation” and to explain what it means. Before asking learners to estimate, she shows them a referent amount: she shows the learners 4 counters in her hand. She places on a tray the 4 counters together with extra counters from those in a packet. She gives learners an image of 4 counters to use as a frame of reference. At the least they will see that the second collection of counters is more than 4. Estimating is about getting a feel for an amount or a unit of measure. She asks learners to write down about how many counters they think there are on the tray. She then asks a learner to count the counters in 2s. They find out that there are 14 counters. Learners begin to change their estimates. She explains that there is no need to do this. Young children often try very hard to please the teacher. The Grade 1 teacher does not reject any estimates, but uses the estimates as an opportunity to compare numbers. She encourages learners to see whether they have over-estimated or under-estimated. She helps them to evaluate their estimates in a non-judgemental way. She places a coloured peg on 14. She elicits learners’ over-estimates. One learner suggests 10 as an estimate that is bigger than 15. He is unsure whether 10 is more or less than 15. She chooses 15 as an example of an over-estimate and places a white peg on 15. Learners see easily that 15 is 1 more than 14. She elicits learners’ under-estimates and chooses 10 as an example. She places another white peg on 10. Again she asks how much less is 10 than 14. The learner, who is chosen to answer this question, struggles to give the answer. The number line provides a visual support to help learners see the difference between the actual number of counters (14) and the samples of over-estimates and under-estimates. When learners were struggling to understand whether their estimates are more or less than the actual number of counters earlier in the lesson, the teacher could have used the number line to help them to get an image of which numbers are more than 14 and which numbers are less than 14.

5.7.2 Using DVD 1.5 to help students to reflect on teaching estimation The two clips in DVD 1.5 are designed to be viewed together.

Focus points

x Modelling and classroom practice x Connections between mathematical topics x Level principle x Language

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Teaching Notes The activities and questions below are designed to help students reflect on particular issues that arise in DVD 1.5 Estimation.

Student Activity 5.7.2: Modelling and classroom practice Discuss these questions with a partner or in small groups.

1. The Grade R teacher lines up 10 learners in front of the class. She asks learners to estimate the number of learners.

a. Why do you think she chose a number more than 5? 2. The Grade 1 teacher uses counters, and a referent amount. She chose 4 as her

referent amount, and then shows them 14 counters. a. Why is it useful to use a referent amount when estimating?

3. The teacher shows 4 counters on her hand. She shows 14 counters on the tray. a. Would the task have been easier or more difficult if she had shown both 4

and later 14 counters on the same tray? Explain your answer. b. What would be the advantages or disadvantages of showing a group of 4

on the tray, and then adding the remaining 10 counters to a different place on the tray?

c. Why does the Grade 1 teacher ask learners to write down their estimates?

d. Why do learners try to change their estimates, once the counters have been added?

e. Do you think that learners often try to change their estimates to the correct amount? What can you do to try to prevent learners from changing their answers?

4. Young children often try very hard to please the teacher. Notice how the Grade 1

teacher does not reject any estimates, but uses the estimates as an opportunity to compare numbers. She encourages learners to see whether they have over-estimated or under-estimated. She helps them to evaluate their estimates in a non-judgemental way.

a. Why does the Grade 1 teacher not ask which learners estimated 14 counters?

b. Why does she not say whose estimations are closer and whose estimations are further from the actual number of counters?

c. How does comparing the actual number of counters with the under and over estimates help learners?

d. How does showing on the number line the over-estimates and under-estimates help learners?

5. Some learners seem to find it difficult to know whether their estimate was more

or less than 14. a. How could you use the number line earlier in the lesson, to help learners

see which numbers are more than 14 and which numbers are less than 14?

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Student Activity 5.7.3: Connections between different topics in Mathematics Think about this alone for a minute, and then share your thoughts with someone else.

1. You have already discussed subitising. a. Could subitising be used when estimating a quantity? b. If so, how could it be used?

An aspect of the level principle is the idea that teaching is aimed at supporting or scaffolding a transition from simpler to more complex ways of dealing with numbers. The Learning Pathway for Number in the Early Primary Years (Van den Heuvel-Panhuizen et al., 2012) suggests that comparison is the starting place for estimating in Stage 1: Emergent and Growing Number Concept. Even before learners can count, they can be asked which of two collections has more. Subitising a sub-group in each collection and seeing about how many are outside of the subitised sub-group can help learners to compare collections of objects. In Stage 2, learners can go beyond comparing estimates with vague descriptions like “more” or “less”; “a lot” or a little”. They can compare collections by making reasonable estimates of each amount.

Student Activity 5.7.4: Level Principle Discuss the questions with a partner.

1. The clips show estimating in the Grade R and the Grade 1 class. a. What number range was used for estimating in the Grade R class? b. What number range was used for estimating in the Grade 1 class? c. Apart from the number range, what other differences were there

between the Grade R and the Grade 1 class? Think about: i. the models used: compare the level of symbolism.

ii. Oral versus written forms of communication iii. Planned versus incidental use of a referent group.

Student Activity 5.7.5: Language Share your thoughts about these questions with a partner.

Both the Grade R and the Grade 1 teacher repeatedly use the word “estimate”. Neither of them use the word “guess”. They both explain that they want learners to say “about how many there are” without counting. 1. Is there any difference between guessing and estimating? If so, what is the

difference?

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Both the Grade R and the Grade 1 teacher explain that they want learners to say “about how many” they see.

2. Do you think it is important to introduce and explain mathematical terminology

in Grades R and 1, or do you think teachers should try to keep language simple and exclude mathematical terminology where possible? Explain your answer.

5.8 Profile of a number We use numbers in different ways. In daily life, numbers are often used as a label, e.g. telephone numbers (073 512 3774); road numbers (M3; R44; N1); bus or train numbers (0219). Here the digits are often pronouced separately e.g. train number ‘oh-four-one- five’. When we use a number as a label i.e. to name something, we call it the nominal aspect of number. We seldom focus on or even acknowledge the nominal use of number in Mathematics classes. We can also use numbers to say how many objects there are in a set. This is the cardinal aspect of number. There could be 35 learners in a class, there are 7 days in the week, but 5 days in a school week. Children might ask for 3 apples from a fruit vendor. Think of other examples where we use cardinal aspects of numbers. Another way in which we use numbers is to indicate order or position. We say:

x April is the fourth month of the year, and September is the ninth month of the year. x Samkele came second in the sprint. x We are on the fifth floor.

This is the ordinal aspect of number. However, we should not think of ordinal as only the “th-type” words first, second, third, fourth, fifth etc. Haylock and Cockburn (2008:34) write that page numbers, numbers on an analogue clock, house or flat numbers are ordinal numbers; for example page 3 is the page between pages 2 and 4. We should remember that acoustic counting, and counting on number lines help to focus learners on the order of numbers. These activities focus on the ordinal aspect of number. Haylock and Cockburn (2008) argue that the image of numberlines and the ordinal aspect of numbers lay the basis for an understanding of negative numbers. So care should be taken to not only focus on the cardinal aspects of numbers. Van den Heuvel-Panhuizen, Kühne & Lombard (2012) refer to two additional meanings of numbers: numbers that indicate measures e.g. she is 5 years old and calculation numbers: 2 + 3 = 5.

From counting to knowledge of numbers, operations and relationships In South Africa, as in many parts of the world, it has become common practice to talk of changes in Mathematics Education. Roberts (2012b, 288) writes that there have been a number of shifts in approach to Mathematics Education. She draws on Yackel (2001: 21-30) when she describes that one of the important shifts has been to use counting rather than sets as the basis for arithmetic. The idea of counting being the basis for arithmetic is embedded in the Learning Pathway for Number (LPN) in the early grades. Stage 1 of the LPN

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is called “Emergent and growing number concepts”; Stage 2 is called “Counting and Calculating”; Stage 3 is called “Calculating” and Stage 4 is called “Advanced calculating”. In other words counting is used both as the basis for the meanings attached to numbers and as the basis for early calculations.

5.8.1 Description of the DVD 1.6 on profile of a number There is one clip in DVD 1.6. The focus of this DVD is different meanings of a number. The clip shows a teacher questioning learners about where the number 5 can be found in everyday contexts. The focus of the everyday contexts is the classroom environment and on the learners’ bodies.

Teaching Note Towards the end of the DVD are still frames of the clip which can be used for the purposes of reflection

Summary of DVD clip on profile of a number

Elements 1 DVD clip Grade Grade R

Timing (00:00 – 5:24)

Approach Question and answer Progression / transition

DVD 1.6 shows a section of a Grade R lesson on different meanings of numbers.

Manipulatives classroom clock, puzzle, toes, fingers, number symbols on clothing

Everyday contexts Classroom environment and learners’ bodies.

Mathematics Different meanings of number

Teaching Notes The purpose of the following activity is to help students to begin thinking about some of the important concepts associated with number sequences.

Student Activity 5.8.1: Number profile Do the reading and viewing alone, and then discuss the questions.

1. Read the introduction to number profile in Section 5.8. 2. View DVD clip 1.6 all the way through. 3. While you are viewing think about what you have read, and think about the following

questions. When you have finished watching and reading discuss the reading and the questions with your fellow students.

a. Which of the DVD clips that you have seen before have a connection to the profile of a number?

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b. What is the effect on the development of the learners’understanding of the questions posed by the teacher?

c. What do you notice about the responses given by the learners? d. What progression or mathematization, if any, do you see in this activity?

Compare your thinking and discussion with our description of the DVD clip below.

Clip A: Grade R profile of a number The teacher introduces the lesson by asking learners to identify the numeral (5) she has written on the board. She writes slowly and deliberately forms the symbol in the way that she intends to teach the children to write it. All the children appear to recognise the symbol. She explains that she is going to teach them how to write the number 5. Many of the children call out that they can already write five, but she blocks their responses. She shows and explains to them how to write the symbol. She asks the children to practice writing the symbol in the air while she gives instructions. They then practice writing it on the palms of their hands. The writing of the symbol 5 takes about 90 seconds and makes up the complete introduction to this lesson snippet. The teacher asks the children “Now, I want to know what you know about the number 5?” She does not, however, give them a chance to respond and quickly adds a second question “Who’s seen the number 5 somewhere?” This is a different question, which resonates with the focus on the symbol 5. The first learner (Stella) says that she is 5 years old. This is a response to the first question “what do you know about 5”? The teacher records this in a drawing, as she does with all the other contributions. The children then all want to share how old they are. The teacher disallows further contributions around age. She asks them to have a look at their bodies and find the number 5. She repeats this questions several times. One child very excitedly says that there is the number 5 on his T-shirt. The teacher again asks them to look around for the number 5. For the children this connotes finding the symbol 5 in their immediate surroundings. The further responses given by the children indicate this. “The number 5 on the clock”; “There’s also a 5 on my watch”; “There’s a 5 on the bottom there on the puzzle.” All of these responses are about visible examples of the symbol 5. Both the teacher’s questions and the initial focus on the symbol 5 have oriented most of the learners in the class to look for the symbol 5. She then says “Have a look at your hand. Who can tell me…?” The chosen learner responds “5 fingers”; other children respond “5 fingers.” Then she says “Whose sitting beautifully again?” and asks Stella (who previously said she was 5 years old) for her contribution. Stella says “A Grade 5.”

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The teacher continues: “Now, I said we had 5 fingers, where else can you find 5. Where else have we got 5? Have a look at your body; where else have you got 5?” A learner responds “5 toes.” Half of the responses offered by the learners related to where they can see the symbol 5. Two responses given by the same learner (Stella) relate to more abstract usage of 5 (5 years old and grade 5). By the end of the snippet the class have come up with different meanings of number:

x Numbers as measure: clock + watch; age. x Number as label (nominal): puzzle number 5 x Number as quantity (cardinal): fingers, toes x Number as pure symbol: symbol printed on t-shirt.

5.8.2 Helping students to reflect on DVD 1.6 teaching about the profile of a number DVD 1. 6 deals with profile of a number in a Grade R class.

Teaching Notes The topics and questions below are designed to help students reflect on particular issues that arise in DVD 1.6 Profile of a number. We suggest that students

complete a task which focusses on different meanings of numbers in general and ways of working with these different meanings of numbers with Grade 1 learners.

Focus points for student activities on profile of a number

Student Activity 5.8.2: Using prior knowledge Work alone, and then share what you have done with another student.

1. View the DVD again. This time look for and make notes of the incidents of the different

categories in the table below.

Teacher draws on learners’ prior knowledge

Teacher ignores learners’ offering to share their prior knowledge

x Using prior knowledge. x Different meanings of numbers x Use of questions x Introduction and sequencing ideas.

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2. Examine each incident in which she ignores the learner’s suggestion. State whether and why you think it would have distracted from the focus of the lesson if she had incorporated it, or whether and why it would have enriched the lesson if she had incorporated it.

Different meanings of numbers Van den Heuvel-Panhuizen, Kühne and Lombard (2012) refer to five different meanings of numbers: a magnitude, an order number, a measure number, a label number, calculation number. Staves (2013b) also explores what it is to “know” a number and refers to the fact that there is a social element of knowing “what three is” which entails being able to describe or name the concept of “three”- using language.

Student Activity 5.8.3: Different meanings of number Work with a partner for this activity.

1. Record all the learners’ responses in DVD 1.6 which was their way of using language to understand the concept of a number.

a. Group them according to the different “meanings of a number” raised in the readings.

Student Activity 5.8.4: Use of questions Continue working with a partner on this activity.

As a teacher it is sometimes difficult to remember that you have a clear plan of the lesson focus in your head when you start the lesson. When you plan the lesson you are spending time focusing on that topic. Learners generally do not start a lesson with this strong focus even after having already spent time on a topic. This has many ramifications including:

x that it is important to give learners time to think before answering questions and to be comfortable with silences as they think,

x that whatever you do earlier in the lesson, or have done in previous lessons is likely to impact on how learners respond.

1. Think about how the teacher asks the questions. a. Which questions does she repeat? b. How many times does she repeat these questions? c. When does she ask a question and then change it into a different

question, before learners have time to respond?

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2. The following questions used by the teacher are all related to 5. x “Now I want to know, what do you know about the number 5?” x “Who’s seen the number 5 somewhere?” x “Have a look at your hand. Who can tell me…?” x “Now, I said we had 5 fingers, where else can you find 5?” x “Where else have we got 5? x “Have a look at your body; where else have you got 5?”

a. Explain how the questions are different to each other. Explain why learners respond to them differently.

Student Activity 5.8.5: Introduction and sequencing of ideas Think about this alone for a bit, and then discuss your thoughts with a partner.

The teacher introduces this lesson snippet by focusing on how to write the symbol 5. This takes almost 1/3 of the time on the DVD clip.

1. How does the focus on writing the symbol 5, impact on the way learners answer the

questions? 2. Half of the children’s responses are where they have seen the number 5 in symbol

form. a. Which questions and what words in those questions orient learners to

looking for the symbol 5? b. Think of a different way to introduce a lesson that focusses on the

different meanings of a number; other than the symbol. c. Write a list of questions that could help learners to think of how numbers

are used differently.

Teaching Notes As always, after the student activities, this next Applied Task will help students to apply what they have learned from doing the focussed DVD activities on Number Profile.

Applied Task for Students 5.8.6: Progressively Building Awareness of Different Meanings of Numbers Work on this applied task alone.

The Grade R DVD 1.6 clip showed the teacher attempting to alert learners to the different meaning of numbers. Van den Heuvel-Panhuizen, Kühne & Lombard (2012) refer to five different meanings of numbers: a magnitude, an order number, a measure number, a label number and a calculation number. You can read the suggested readings at the end of this chapter to get a clearer idea of how these meanings of number are different and why it is important to be aware of them.

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1. Illustrating the different meanings of numbers a. Find printed (from newspaper, magazines, etc.) or digital images

(downloads or your own photographs) and illustrations of the different meanings of a number.

b. Sort these according to the five different meanings of numbers mentioned above.

c. Make either a poster, or zig-zag (concertina) book, or a PowerPoint presentation to illustrate examples of each meaning and use of number.

2. The implications of different meanings of numbers for teaching Foundation

Phase Mathematics. For the next questions choose either to focus on Grade R or Grade 1.

a. Write a plan for how to introduce learners to the fact that numbers are used in different ways and have different meanings.

b. Write a list of incidental ways that you could use to reinforce the different meanings of numbers during the year. Include examples that you would use both during and outside of Mathematics Lesson time.

c. Write a list of part-lessons to show how you can build learners’ awareness of each of the meanings of number over the year.

i. Estimate about how much time you would spend on the part lesson.

ii. Estimate about how much time you would spend on focused development of each meaning of number over the year. Would you spend more time on some meanings of number than others?

iii. Estimate for each part lesson, when in the year you imagine using it. Show that there is progression between the parts of lessons.

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5.9 References

References & Readings Counting and the Number System

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Page references

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