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Unit 6: Representing and manipulating number Unit 6 and the accompanying DVD 2, focus on ways to develop young children’s ability to manipulate numbers. Counting (together with subitising) is considered an important basis on which to build numerical and arithmetical understanding (Gray 2008: 83; Sarama & Clements 2009: 50 – 51). Comparing numbers, building up numbers and breaking down numbers assists learners’ understanding of the myriad of relationships between numbers. Composing and decomposing numbers lays the foundation for part-part-whole relationships and operating with numbers. When learners manipulate numbers they develop an informal understanding of different ways to combine and separate numbers. Learners develop a sense of which actions will result in a number with a greater value and which actions will result in a number with a smaller value. Learners’ understanding of numbers is deepened as they manipulate numbers: it prepares them for understanding operations on numbers.

6.1 The focus of DVD 2 DVD 2 focuses on four different issues: two relate to the social knowledge of how we talk about and represent numbers and two relate to manipulating numbers. Different representations of numbers make it easier or more difficult to manipulate numbers

DVD 2.1: Expressing and representing numbers (see Section 6.3) DVD 2.2: Recognise, identify and read number symbols and names (see Section 6.4) DVD 2.3: Comparison: more, less and equal (see Section 6.5) DVD 2.4: Building up and Breaking down numbers (see Section 6.6)

6.2 Representing and manipulating number in the LPN framework

The focus of this Mathematics Support Guide is on progression in the learning and teaching of number in Grade R and Grade 1, this is covered by Stage 1 and Stage 2 in the Learning Pathway for Number. The focus of DVD 2 is on representing and manipulating number in Stages 1 and 2 of the Learning Pathway for Number. In the table below we indicate those topics that relate to representing and manipulating number in the Foundation Phase. The columns highlighted in red below summarise representing and manipulating number in Stages 1 and 2 of in the LPN.

Counting Manipulating Numbers Operating with Numbers

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Number knowledge Stage 1 Emergent and growing number concept

Stage 2 Counting-and-calculating

Stage 3 Calculating

Stage 4 Advanced calculating

Number range up to 10 and beyond

up to 20 and beyond

up to 100/1000 and beyond

up to 10 000 and beyond

Models of representing numbers

x Group models x Line models x Combination models

Ways of dealing with numbers

x Contextualising x Positioning x Structuring

Representing and symbolising numbers

Building up and breaking down numbers

Pre–R R–Grade 1 Grades 1–2 Grades 3–4

If you wish to read more about representing and manipulating number in stages 1 and 2 of the Learning Pathway for Number, you can refer to Van den Heuvel-Panhuizen, Kühne, Lombard. 2012 The Learning Pathway for Number in the Early Primary Grades. Northlands: Macmillan South Africa. :

x Glossary entries page 203tage 1:Emergent and Growing Number Concept pages 26- 28;

x Stage 2: Counting and Calculating pages 43 - 46

6.2.1 Representing and Manipulating number in the NCS Mathematics Curriculum and Assessment Policy Statement (CAPS)

Teachers are required to implement the Department of Basic Education Curriculum and Assessment Policy Statement (CAPS). Below we have outlined references to counting in the Grade R and Grade 1 Mathematics CAPS documents (Department of Basic Education [DBE], 2011a; b). Students may like to compare how counting is described in CAPS compared with how it is described in this support guide. The relevant sections in the FP CAPS mathematics documents are mentioned below:

DVDs 2.1 & 2.2

DVDs 2.3 & 2.4

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

x Section 3: o 3.2 Foundation Phase Overview pages 20, 21, 22, 23 o 3.4.1 Grade R Overview per term pages 42 - 45 o 3.5.1 Content Clarification Notes with Teaching Guidelines pages:

65, 104, 109, 112 – 115, 117, 119 – 121, 123 – 124, 136, 140, 146, 149, 152, 154, 157, 163 – 164, 169, 178, 183 – 184, 187, 192, 198, 201- 202, 208, 209, 220, 222, 228, 232-23, 237, 238, 245,250 – 251, 256, 258 - 259

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

o 3.2 Foundation Phase Overview pages 19 - 21 o 3.4.1 Sequencing and Pacing of Grade 1 per term pages 40, 43 o 3.4.2 Sequencing and Pacing of Grade 2 per term pages 57, 59 o 3.4.2 Sequencing and Pacing of Grade 3 per term pages 74, 76 o 3.5.1 Clarification of Grade 1 Content

� Term 1 page 100 – 103, 109 � Term 2 pages 130 – 134, 138 - 139 � Term 3 pages 161 – 162, 165, 166 � Term 4 pages 185 – 187, 189, 192

6.3 Expressing and representing numbers

Young children use numbers to describe how many objects there are in a collection. Later they learn to think of numbers as objects in themselves. This change in perception is accompanied by changes in ways of representing and symbolising number. DVD 1 shows learners developing an awareness of numbers through a range of counting activities. Whilst doing this, they are also exposed to different ways in which numbers are represented and symbolised. In DVD 2.1, 2.2, 2.3 and 2.4 learners work with a range of ways of symbolising number. This is done in conjunction with building number sense, generally through manipulating number. In the Foundation Phase (FP) CAPS Mathematics Subject content area an awareness of number names and number symbols is specified separately to other work on number. This does not, however, mean that it can or should be taught and learnt in isolation from other number knowledge. Before young children can fluently compare numbers or fluently operate with numbers they need to change the way they think about numbers. This change happens over an extended period. What change happens in the way young children view numbers as they move from working with apparatus and counting to working with symbols and calculating? Before they can operate with abstract numbers, children learn to “model” situations with their fingers or some form of apparatus that can represent the real objects” (Anghileri, 2006: 9).

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“Early counting experiences associate numbers with things …” (Gray, 2008: 85): the numbers “are a property of the set” of objects. (Gray, 2008: 85): for example, six, red crayons.

Here both “six” and “red” describes the collection of crayons. In this way numbers are used as adjectives, which describe other nouns. This is one reason why it is useful for young learners to first work with apparatus as models for numbers.

However, the objects alone do not necessarily provide a model for numbers. It is the guidance that teachers provide when they set up situations and questions for learners, along with manipulating these models that help children to learn about numbers, the relationships between numbers and ways of working with numbers. Learners will develop images of these models which they can manipulate in their heads. Learners can attach verbal names, then written numerals and finally written number words to their meanings of number. Anghileri (2006: 9) writes “later, children will learn to use the symbols that characterise the conciseness and precision of a mathematically reasoned argument. Just as speech is transferred to writing over an extended period of time in school, with complexities arising in manipulative control and spelling, so transfer from concrete experiences to mental methods and symbolic representation takes time in mathematics” (Anghileri, 2006: 9). Gray (2008: 85) contrasts the way young children use numbers as adjectives with the way that adults use numbers as nouns: they do not necessarily relate numbers to real objects. When we say that 6 can be made of 5 and 1 or 2 and 4 or 3 and 3, we are using six as a noun. Gray adds that when we view numbers as objects, we compress ideas and that this compression of ideas “allow[s] us to see other relationships” Gray (2008: 85). “Children’s growing sophistication in handling counting procedures may be seen as a steady compression which can eventually permit the choice between these [counting procedures] and the use of number concepts” (Gray, 2008: 86).

6.3.1 Description of DVD 2.1: Expressing and representing numbers The four DVD clips in DVD 2.1 deal with ways of expressing and representing numbers

x Clip A, B and C focus on apparatus as models for number x Clip D is a vignette of how apparatus can serve as a model for number in a learning

and teaching situation. The focus of DVD 2.1 is on the ways of expressing and representing numbers in the LPN stage 1 and stage 2 using:

x Apparatus as models for number x Spoken language x Written representations of number

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These four clips illustrate apparatus that can usefully serve as models for number development in the Emergent and Counting-and-Calculating stages, as described in the LPN. Numbers can be represented in different ways. Models for representing numbers can be categorised into three types:

x Group models x Line models x Combinations of group and line models.

Teaching Notes Towards the end of DVD 2.1 are some still frames of the clips which can be used for the purposes of reflection

Summary of DVD clips on expressing and representing numbers Elements Clip A

Clip B

Clip C

Clip D

Timing 00:37-01:19 01:20-02:09 02:10-03:15 03:16-07:25

Manipulatives

Fingers, Beads

Sticks, tubs and number

cards

Line models for number

i.e. Beads, number

tracks, Structured number

lines and unstructured number lines

Children work independently

with manipulatives

Counters Display board

Mathematics apparatus as models for number Conservation of number

Student Activity 6.3.1: Expressing and representing numbers Do this alone and in pairs.

5. Read the introduction to expressing and representing numbersin Section 6.3. 6. View DVD clips A, B, C and D all the way through. 7. While you are viewing the clips, think about what you have read, and think about

the following questions. a. Explain the importance of using apparatus/ manipulatives in facilitating

learners’ development in expressing and representing number? b. Is the choice of an apparatus important when sequencing lessons to

advance learners’ understanding towards abstract representations of number such as representing number by writing number symbols?

c. Besides providing apparatus, how else does the teacher support learners’ development in expressing and representing number?

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d. What possible challenges could learners face when asked to firstly represent a certain number on a number track and then the same number on a structured number line?

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

Compare your thinking and discussion with our description of the DVD clips, below. The clips below provide examples of group models and line models.

Clip A: Group Models with a 5 structure – using fingers and beads Fingers are one example of a group model of number. Group models tend to represent numbers as a set. Group models tend to foreground cardinality more than ordinality. Counting beads is an example of line models. Line models tend to represent numbers as a sequence. For this reason they tend to make the ordinality aspect of number more obvious than group models. However, group models and line models can both be used to facilitate learners’ understanding of ordinality and cardinality of numbers. If learners are exposed to both group and line models, and particularly if they translate number images from group to line models, vice-versa, it is likely to help them to connect “next in a list of symbols” to the “next in a series of sets related by +1”. As mentioned in Unit 5.4 in the discussion on DVD 1.2: Resultative Counting, this lies at the heart of understanding whole number and the system of Natural Numbers. The DVD shows stills of learners representing the same numbers with their fingers and on counting beads. In the teaching incidents from which these photographs are drawn, the teacher asked two learners to each show a number first with fingers and then on counting beads. Although they are different kinds of models, both learners appeared to use the same strategy to identify each number using the different apparatus. Kairo uses his fingers to represent 4. Below is the discussion that took place when the teacher asked Kairo to show 4 using his fingers. (This discussion is not recorded on the DVD clip.)

Teacher: Kairo, can you show me four fingers? Kairo: [Kairo shows four fingers held up on his left hand, folds thumb back

behind his palm.] Teacher: How do you know it’s four? Kairo: Because it’s four fingers [Kairo places the four finger of his left hand

on the palm of his right hand] and you just count them. Kairo had shown the four fingers immediately. It appears that he is able to recognise four fingers instantly: a process of subitising. He did not appear to count them. He was able to immediately fold back his thumb and in the process take one (the thumb) away from five (the fingers on one hand). It appears to be a process of conceptual subitising. People modify what they say and how they say it according to who they are speaking to. You might tell the same story to your partner and your granny, but you might tell the story differently. Interviewees often modify what they say to the interviewer, based on what they think the

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interviewer expects them to say. This may partially account for Kairo’s focus on counting. However, he probably also does not have the vocabulary to talk about subitising. He knows that counting enables one to state the cardinal number of a group. He is able to rationalise a process, and proceeds to explain what he understands by counting:

You count them like this. [Kairo raises the small finger of his left hand] This is the f…. one [He keeps this raised and also raises the ring finger of his left hand] two [He keeps both these fingers up and raises the middle finger of his left hand] three [He keeps the 3 fingers up and raises the index fingers of his left hand] four

Teacher: Oh, and you’ve got 4. Thank you. Teacher: Now can you show me 4 on the beads. Can you show me four beads?

[The teacher provides a string of 10 beads, which has 5 yellow beads and 5 red beads. It is a visible 5 structure imposed on the 10 beads]

Kairo: [Takes the string of beads. He holds the string after the 1st bead with his right hand, and then places his left hand on the 1st bead] One. He holds the string after the 2nd bead with his right hand, and then places his left hand on the 2nd bead] Two. He holds the string after the 3rd bead with his right hand, and then places his left hand on the 3rd bead] Three. He holds the string after the 4th bead with his right hand, and then places his left hand on the 4th bead] Four.

Teacher: Ok, you got that pretty quickly, four yellow beads. Now if I ask, can you do something else? Let’s shuffle the beads. [Teacher takes the beads and shakes them] Can you show me 4 in one go?

Kairo: [Takes the string of beads. First he separates off one yellow bead at the end of the string. He looks at the remaining beads and then he changes his strategy. He moves the last yellow bead in the set closer to the set of red beads. He has now separated four yellow beads and shows these four yellow beads to the teacher. He lets the remaining beads on the string hang down].

Teacher: How did you do that so quickly? Kairo: Because you do this. It’s easy Teacher: It’s easy? Kairo: [Takes the string of beads, and lets the four yellow beads hang down.

Then he holds the four yellow beads horizontally in front of the teacher and lets the remaining six hang down].

His focus is on keeping 4 beads together in a group. He is not focussed on answering how to identify a group of four beads, but rather his focus is on how you keep the group of four together as one group.

Teacher: But how did you know to put your finger there? [Teacher indicates to where Kairo is holding the string after the 4th yellow bead]

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Kairo: Because I had to, so that that one does come into all one [indistinct] [He spins the string about the fulcrum he has created at the position where he is holding the beads. He demonstrates that he is keeping the two groups apart.] It just won’t come.

He remains focussed on keeping 4 beads together in a group. The teacher has seen that he gets four by taking one away from a group of five. She tries to elicit his thinking in another way.

Teacher: You just knew it was four. Kairo: Ja Teacher: And if I said to you find 5 quickly. Can you do that? Kairo: He moves his finger and then he holds up the string in the middle of

the string: letting the 5 yellow beads hang down on one side and the 5 red beads hang down on the other side.

Teacher: Gee, that was so quick. How did you know that? Kairo: Because 4 [he holds four beads with his right hand, but does not

relinquish his hold on the middle of the string showing 5] and you just need one more [releases his hand that covered the 4 yellow beads, to show the 5 yellow beads and 5 red beads hanging down one either side of the middle of the string]

Teacher: and you’ve got five altogether. Kairo: Ja Teacher: Thank you very much.

Kairo has demonstrated the reverse of his original strategy, which is a form of perceptual subitising and can lay the basis for calculating by modifying known number facts. Seth shows the number 7 using his fingers. The 5 structure is evident. Seven is seen as composed of five and two. Below is the discussion that took place when the teacher asked Seth to show 7 using his fingers. (This discussion is not recorded on the DVD clip.)

Teacher: Can you show me seven fingers? Seth: [holds up 5 fingers of his left hand and 2 fingers of his right hand] Teacher: Seven fingers. How do you know it’s seven? Seth: Because you count your fingers.

Seth appears to draw on the fact that he knows counting is used to determine the cardinal number, when he refers to counting, in explaining how he could show seven fingers. However, Seth had shown the seven fingers immediately. He did not appear to count them. He appeared to know immediately that five fingers (one full hand) and two more fingers make seven fingers. It appears to be a process of conceptual subitising. The teacher appears to be aware of this. She probes further to see whether she can get him to explain his process more clearly.

Teacher: OK, show me seven again?

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Seth: [holds up 5 fingers of his left hand and 2 fingers of his right hand] Teacher: What’s it made of? Seth: Fingers

Seth’s focus is on the apparatus, but the teacher wants him to focus on the numbers. She redirects his focus through further questioning.

Teacher: Fingers OK, how many fingers? How many here? [Teacher touches learner’s left hand]

Seth: 5 Teacher: and how many here? [Teachers touches learner’s right hand] Seth: 2 Teacher: and altogether? Seth: 7 Teacher: Wow, thank you

Seth shows that counting beads can also represent number. Seth represents 7 on a string of beads that is designed to foreground numbers in a 5 structure. Below is the discussion that took place when the teacher asked Seth to show 7 using the counting beads (This discussion is not recorded on the DVD clip).

Teacher: Can we use the beads now? Can you show me 7 beads? Seth: OK

[Seth places the fingers of his right hand after the seventh bead and holds the end of the string with his left hand. The string contains 20 beads arranged in a five structure: one group of five red beads, followed by one group of 5 yellow beads, followed by one group of five red beads, followed by one group of 5 yellow beads. The group of 7 beads that Seth shows consists of 5 red beds and 2 yellow beads]

Teacher: OK you’ve got seven beads. How do you know it’s seven? Seth: … mmm …… because …… mmm …… because you count them.

[but Seth had clearly not counted them. The teacher probes further] Teacher: You count them, OK. Seth: And ….. Teacher: You can see how many here?

[Teacher indicates that Seth should consider the 5 red beads] Seth: And you can plus them ... [indistinct] 5 and 2 Teacher: 5 and 2, OK.

Once again this reveals that a five structuring (whether it be fingers on a hand, or beads on a string) helps learners with the process of perceptual subitising.

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Teacher: Anything else you can tell me about 7? What can you tell me about 7? Seth: There’s days of the week. Teacher: How many are there? Seth: Seven days in the week. We have the 2 for the weekend and the 5 for

the school days. Teacher: Oh well done. Two for the weekend and five for the school days.

Clip B: Using symbols to represent number There is a progression towards abstraction when learners no longer need a concrete representation but are able to represent numerosity using symbols. Kairo is asked to put the correct number of sticks into tubs which are labelled with the number of objects they hold. These kinds of tasks help learners “to make the transition from representing numbers using countable objects to representing numbers using a symbol” (van den Heuvel-Panhuizen et al. 2012: 26). Below is the discussion that took place when the teacher asked Kairo to read the number symbols on the cards and place the correct number of sticks in each tub. (This discussion is not recorded on the DVD clip). The teacher has provided containers (margarine tubs). Each container has a number card on the top of it.

Teacher: Kairo, can you see what I’ve got here? Kairo: [Kairo points to each tub respectively as he answers.] Papers what’s 7 and 4 and 2 and 6. Teacher: Alright [provides another tub, filled with counting sticks] Now I

wonder, if you can put the correct number of sticks into the container? [points to container]

Kairo: [Places two sticks on the number card labelled 2.] Teacher: Shall we lift the lid? Shall we lift the lid up and place them inside the

container? Kairo: [Removes the number card to drop the sticks into the tub. Replaces

the number card] Teacher: OK great. How many in there? Kairo: Two Teacher: But I can’t see them. Kairo: [Removes the number card and peers inside the tub. Moves the tub to

a position where the teacher can see them. ] Teacher: You sure they’re in there. Kairo: ja Teacher: Close it up again. Kairo [Kairo moves the tub back to its original position and places the

number card back on top of it] Teacher: You sure that’s two. Kairo: ja Teacher: OK, let’s do another one.

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Kairo: [Scans the number cards on the remaining tubs. He then dips into the tub of sticks and counts out sticks silently. Before placing them in the tub he separates them by holding three in one hand and two in the other hand. He looks at the sticks and then he reaches back into the tub of sticks and takes another one]

Teacher: That was a clever way to check. How did you check that there were six?

Kairo: [lifts the number card 6 and places the sticks in the tub.] Because 5 is not 6. [He replaces the number card]

Teacher: You quite right there. Kairo: [He replaces the number card] Teacher: Excellent, and what about this one. [Teacher points to tub with

number card 4 on it] Kairo: [Counts out sticks, checks, removes number card, places sticks in

bottom of tub, and replaces number card] Teacher: And there’s one more. [Teacher moves tub with number card 7 on it,

to in front of Kairo] Kairo: [takes sticks out of the tub and looks at them, places them in his right

hand] Three plus [dips back into the tub of sticks with his left hand] a three and a one [puts all the sticks in his right hand]

Teacher: That was clever, what did you do there? Three? Kairo: plus a three and a one equals seven [places the sticks one by one into

the tub] Teacher: That’s a clever way of making seven. [As he places the last stick] Yay.

Shall we cover it up? Kairo: [He replaces the number card] Teacher: Can you see seven sticks here? [points to the number card 7] Kairo: Ja Teacher: Are there 7 sticks here? [points to the number card 7] Kairo: Ja Teacher: Look carefully at this green piece of paper here. [points to the number

card 7] Kairo: [Makes “binoculars” with his hands over his eyes] Yes, there are. Teacher: OK, how do you know that there are seven? Kairo: Because I put in Teacher: But on this green piece of paper here. [points to the number card 7]

Are the sticks on this green piece of paper? Kairo: I think that’s a “yes”…… But that’s a “no”. Teacher: OK, but on the green piece of paper. Kairo: I see 7. Teacher: seven. Kairo: That’s all I see. Teacher: That’s all you see. But you know that there’s 7, don’t you? Kairo: Ja. Teacher: you know that there’s 7. Kairo: [nods] Teacher: Why?

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Kairo: Because, because I put seven sticks. And, and I plussed them like this. Three [holds up three fingers] plus three [holds up three more fingers] plus one [holds up another finger] makes seven

Teacher: Wow, yes I remember, that was a very good way of making seven Kairo And four [raises 4 fingers] plus three [raises another 3 fingers] makes

7 Teacher: Oh my goodness, you know a lot about 7 don’t you? Kairo: Ja. Teacher: Alright, OK, Thank you.

From this DVD clip it is evident that Kairo is able to read and interpret numerals less than 10. Once learners are able to read number symbols, they learn to write the symbols: first individually, then in a sequence. In the teaching episode below, the teacher asks Kailib to write two forwards counting number sequences and one backwards counting number sequence. (These instructions are not recorded on the DVD clip.)

Teacher: I want you to take your khoki and write some numbers. Could you start at 5 … and carry on until 8.

Kailib: [Writes 5 6 7 8 in a string] Teacher: OK thank you. Now we’re going to do another one. Can you start at 8

and carry on until 12? Kailib: [Writes 8 and then 9 slowly and then 10 11 12 quickly] Teacher: Now I wonder if you can start at 9 and go until 5: nine to five. Kailib: [Writes 9 and then each successive number to the left of it. Each pair

of numbers (except for 6 and 7 is separated out by a comma: which he adds after completing both numbers in the consecutive pair. He produces a string: 5, 6 7, 8, 9]

Clip C: Line models The string of beads is a linear model. They represent number in a discrete way. A number track also represents numbers in a discrete way. There is no zero on a number track.

A number line represents number in a continuous way. A number line starts with zero whereas there is no zero on a string of beads.

Structured number line

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On a number track the numbers appear to be represented by the spaces, while on a number line the numbers appear to represent the intersection of the gradation line with the number line.

Number track Learners can first be introduced to a structured number line. Later, the number line becomes more abstract, such as a semi structured or empty number line.

Semi-structured number line

To build an understanding of number, learners must be exposed to several representations of number. They gradually learn to recognise that a number such as 14 or 12 can be shown using different models.

Clip D: Conservation of Number in Grade R

This clip starts with the teacher showing learners a subitising card with 5 dots arranged in the classic domino 5 pattern. The teacher asks “who can remember?” The learners all say “five”, some learners also hold up 5 fingers. The teacher confirms “five”.

The teacher hands out bags of counters and asks learners to place 5 counters anywhere in the circle. She repeats several times that learners can place the counters anywhere in one of the circles. The focus of her lesson is on conservation of number. She aims to clarify that “fiveness” is not dependent on any particular arrangement of the items in a collection, and that the arrangement does not affect the value or total count of the collection. Four of the six learners place 5 counters in the domino-5 pattern they saw initially. The visual memory of that pattern appears to be stronger than the teachers instructions “You can place them anywhere in the circle”.

1 2 3 4 5 6 7 8 9 10

10 0

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One learner who works quickly re-positions her counters twice. She starts with the arrangement alongside. She changes the arrangement to this one alongside. Finally she places her counters in the domino-5 pattern she sees that most other learners have used. One learner, who works more slowly than the others, arranges her counters in an arc around the inside of the circle. The teacher notices that this learner has too many counters. The teacher intervenes in the following way. T: “Just check that you have 5.” The learners who sit next to her, check the number of counters. T: “how many do you have?” L: “eight”. T: “So do you need to put some back?” The teacher then asks the children to “look at the special way that Yasira put them out”. “Why did you put them out like that?” The learner does not know what to answer. The teacher adds “that’s O.K. I like them like that”. The teacher asks “Who else put them out in a different way?” But no-one has a different arrangement at this stage. The teacher gets the learners to count their counters to confirm. As learners are counting, two more learners change their arrangement to the arc around the outside of the circle: presumably because the teacher brought their attention to this arrangement by saying “I like them like that”.

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x Are the learners focusing on counting in this interaction? x What questions could the teacher have asked to focus the learners’ attention on the

act of counting the objects?

6.3.2 Helping students to reflect on DVD 2.1: Teaching the expressing and representing of number

The four DVD clips in DVD 2.1 are designed to be viewed together. The topics and questions below are designed to help students reflect on particular issues that arise in DVD 2.1; Expressing and Representing Number.

Focus points for student activities on expressing and representing numbers

Student Activity 6.3.2: Level Principle - Progression in representing and symbolising number (DVD Clips A and B)

1. Why is it useful for learners to first use apparatus and then diagrams to represent numbers, before reading and writing numbers in symbols? Use what you know about how children learn about numbers (and the notes provided in 6.3 Expressing and Representing Number and 5.4 Resultative Counting) to explain your answer.

2. In Clip A, the teacher asks each learner to show (represent) a number using first

their fingers and then counting beads. a. Why does she ask them to work with two different models? b. Why does she ask them to first show with their fingers and then show

with the beads?

3. In Clip B, the teacher first asks learners to read and represent numbers and later asks them to write number sequences.

a. Describe how the learner is asked to read and represent numbers? b. Write a list of other activities that you can use to let learners read and

represent numbers? c. How does the activity that the teacher chooses to use, help learners to

make the transition from using countable objects (numbers as adjectives) to representing numbers as symbols (numbers as nouns)?

4. In Clip B, the teacher asks the learner to represent two forward counting sequences and one backward counting sequence “9 to 5”.

x Level Principle: Progression in representing and symbolising number x Linking counting with beads to counting on a number line x Factors that can influence how learners set out their work

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a. What surprised you about the way the learner wrote the latter sequence?

Student Activity 6.3.3: Line Models for representing number (DVD Clip C)

1. A string of beads can be used as a model for representing numbers. A number track and a number line can also be used as a model for representing numbers.

a. In what ways are strings of counting beads, number tracks and number lines similar?

b. What are some of the differences in the ways number is represented on strings of counting beads, number tracks and number lines?

c. Why is it useful for young learners to use hops or jumps to represent counting or the movement from one number to the next or to find the difference between numbers?

Student Activity 6.3.4: Factors that can influence how learners set out their work (DVD Clip D)

The teacher has provided learners with a mat that has two circles, and a bag of counters. She requests learners to put out counters in any way that they like.

1. What important mathematical

aspect of counting and number can be reinforced if learners see that the same number can be represented by different arrangements of counters, objects or images?

Most of the learners set their counters out in a typical domino-5 arrangement.

2. How does this arrangement relate to the lesson introduction?

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3. The aim was for learners to see that the same number can be represented by different arrangements of counters.

a. Suggest a different way in which the lesson could be introduced that would encourage learners to arrange (set out) their counters differently from each other.

4. The learner who packs out her counters in an arc on the inside of the circle

packs out more than the requested number of 5 counters. a. How does the teacher respond to this situation? b. Why does the teacher respond in this manner? c. How might it have affected the learner if the teacher had instead said “You

have more than 5 counters, put 3 back in the bag”?

5. The teacher praises the learner who has arranged her counters in an arc on the inside of the circle.

a. Why does she do this? b. Two other learners immediately change their arrangements to the arc-

formation. c. Why do they do this? d. What does it alert you to in terms of praising learner’s work. e. Think about your own teaching experiences.

i. Discuss instances where there have been unintended consequences when you praised learners.

ii. Brainstorm why learners reacted in ways that you did not expect? iii. Plan what you would do differently next time?

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

Applied tasks for students: 6.3.5 These tasks focus on applying what you have learned from doing the focussed DVD 2.1 activities.

1 How numbers are represented in learning material used at school

Describe how numbers are represented in learner material? Look through the:

a. DBE Grade 1 workbooks for Terms 1 & 2 b. Grade 1 Learner Textbooks and other Grade 1 Workbooks

2. File of number recognition games Make a file of number recognition games. These can include:

x Number recognition games that you source from elsewhere

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x Number recognition games from elsewhere that you adapt x Number recognition games that you make up

6.4 Recognise, identify and read number symbols and names Counting objects is an important part of learning to count and learning the value of numbers. According to Clements and Sarama’s (2007: 478) analysis of literature:

‘Early numerical knowledge includes four interrelated aspects (as well as others); x recognising and naming how many items are in a small configuration (small

number recognition, and when done quickly subitising); x learning number names and eventually the ordered list of number words to

ten and beyond, x enumerating objects (i.e. saying the number word in correspondence with

objects) and x understanding that the last count refers to how many items have been

counted.’ Two of these aspects (subitising and recognising and naming how many) involve the construction of mental objects for each of the numbers to 10, and includes the notion of more than/ less than and part-part-whole. The remaining aspects can be seen as social practices (e.g. reciting the number names, matching objects to number names, and counting to find out how many) (Siemon, Beswick, Brady, Clark, Faragher & Warren, 2011: 273). Piaget (1952) identified three different types of knowledge:

x Physical knowledge This related to knowledge learned directly through observing or acting on the physical environment.

x Social knowledge Children learn an enormous amount of social knowledge by interacting with various members of society. In Mathematics the following social knowledge is important

o Number names and other terminology o Symbols used to represent numbers and operations, and other notation o Conventions such as the order of operations

x Logico-Mathematical knowledge

Mathematics is not inherent in objects. When learners reflect logically and mathematically about the relationship between objects, they begin to mathematize. One starting place is to compare objects and to consider what about them is the same and what is different. Mathematics is based on the construction of abstract mathematical relationships such as the properties of numbers and the properties of operations.

Learners use both physical knowledge and social knowledge as they develop logico-mathematical knowledge.

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One aspect of social knowledge is learning to read and write number symbols and names. Clip B of DVD 2.1 and Clip C of DVD 2.2 show learners simply reading number symbols. In Clip C of DVD 2.2, there is nothing to indicate that learners are attaching any meaning to the symbols. Similarly at the start of DVD 1.6 learners are writing the number 5. This part of the lesson is simply a writing exercise. Learners need to attach meanings to number symbols. Most of the other DVD clips show ways of helping learners attach meaning to numbers and operations. A starting point is to develop learners’ understanding of number through working with meaningful contexts and working with manipulatives. This allows learners to develop both an understanding of the different aspects of numbers and to develop mental images that they can use when thinking of and working with numbers.

6.4.1 Description of the DVD 2.2: Recognise, identify and read number symbols and names

The four DVD clips in DVD 2.2 focus on recognising, identifying and reading number symbols and names.

The focus of DVD 2.2 is on recognising, identifying and reading number symbols and names. The first three clips were filmed in Grade R classes, but the activities done in those classes could also be done early in the year in Grade 1.

Teaching Notes Towards the end of DVD 2.2 are some still frames of the clips which can be used for the purposes of reflection

Summary of DVD 2.2 clips on recognising, identifying and reading number symbols

and names Elements Clip A Clip B Clip C Clip D

Grade Grade R Grade 1

Timing 0:14 – 3:35 3:35 – 9.30 9:30 – 11.12 11.12 – 14.47 Classroom management

Whole class teaching

Whole class teaching

Small group teaching

Whole class teaching

Progression / transition

From recognising numbers represented by dots and numerals, to reading numbers represented in numerals,

to matching number words to number symbols

Manipulatives

Dot cards and number cards

Dot cards and number cards arranged in a number track

Number cards

Number names on cards in words Number names on cards in symbols

Everyday contexts

Fingers

Mathematics

Recognise, identify and read number symbols and names.

Relationships between numbers Recognise, identify and read number symbols and names.

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Student Activity 6.4.1: Recognise, identify and read number symbols and names

5. Read the introduction to recognising, identifying and reading number symbols and

names in Section 6.4. 6. View DVD clipa A-D all the way through. 7. While you are viewing the clips think about what you have read about recognising,

identifying and reading number symbols and names. a. Identify instances in the clips in which children appear to be using any of the 3

types of knowledge as described by Piaget (1952). b. Evaluate the effectiviness of the activities in all four clips in scaffolding learners’

abilities to recognise, identify and read number names and symbols over the course of Grdes R and 1.

c. How would you have improved/changed the activities seen in the DVD clips to make them more effective?

8. 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: Recognising, identifying and reading numbers shown as a collection of dots and as a numeral – attaching meaning to numerals in Grade R. In the first DVD clip the teacher shows the Grade R learners a series of cards. The number range is 1 – 10. Some of cards have number symbols, some have collections of dots. The teacher requests individual learners to: “say what number you see” on the card. When she shows the card with the numeral 9 on it, the learner says “eight”. The teacher then asks the rest of the class whether they agree. She asks them all to show the number 9 with their fingers. She sees that some children are having some problems and a not able to count to and show nine. She asks them all to count on their fingers with her. Some of the children do not co-ordinate their fingers with the number words while counting: one boy does not raise his hands and randomly holds up fingers. When some children continue to count on to 10, she takes them back to count to 9 to ensure the link between the meaning of 9 and the representation of 9.

Nine

9 Eight

8 Five

5 Seven

7

Ten

10

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Learners struggle to see that there are 9 dots on the card. The teacher focuses learners’ attention on the sub-set of 5 dots and the sub-set of 4 dots. She then asks them to show 5 fingers on one hand and 4 fingers on the other. She models this. Then she asks, “how many fingers do we have?” She is modelling conceptual subitising with fingers and then asks the children to apply this to the dots on the card. Conceptual subitising builds on the notion of part-part-whole relationships, and although subitising does not rely on counting, learners need to recognise and name small collections of three, four, two, etc. Teachers can then consolidate learners’ intuitive and perceptual skills of subitising and scaffold activities in which learners combine two collections of up to 5. In the DVD the teacher uses real objects (fingers). Gradually learners will understand that the collections of objects, the last count in the collection (e.g. ‘9’), the numeral (symbol) and the number name (word) all represent the cardinal value of the number. Learners eventually need to be able to recognise numerals and number words in any font, but initially it is important to choose a font that has the least distracting features. Teacher’s Pet is typical of the type of font that is used in the Foundation Phase as it is simple and clear to read.

Clip B: Using the counting sequence to assist learners recognise, identify and read numbers shown as a collection of dots and as a numeral In the second DVD clip the teacher has displayed two rows of 10 cards. Some are blank. On the top row three cards have collections of dots to represent the numbers 1; 3; 5. On the bottom row three cards have numbers shown in numerals: 1; 7; 10.

The teacher gives cards with either a numeral or a collection of dots to a select number of learners. She instructs the learners sitting next to children who received cards to assist them with the activity. She asks learners to identify the number of dots on the card in the top left hand side of the display. She asks “what comes next? ..... Who has the card with two dots?”

After the card with two dots has been placed, the teacher asks how learners know that the card with two dots comes after the card with one dot. The teacher’s initial line of questioning indicates that she is attempting to draw learners’ attention to

the successor function: the next in the sequence is the set of numbers plus one more.

1 7 10

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She asks: T: What is different from the first to the second card? T: What is different between the two cards? L1: “The first card is one. The second card is two.” T: Does this one have more dots or less dots? Ls: More T: To get more dots what do we do? Ls: Make the number bigger. T: How do we make the numbers bigger? L2: You must have a lot of numbers to put on T: A lot of numbers? L3: Add 100. T: Here’s one dot and here’s two dots. So did we add a dot or take a dot away? Ls: Add a dot.

She adapts her question at least 5 times before she gets to the answer “we add one more dot” The arrangement of cards encourages learners to focus on the counting sequence. Learners use the counting sequence to position both the dot-cards and the numerals. The children’s responses indicate that they are only focussed on the counting sequence. This is confirmed when she asks “how do you know that number 3 comes after number 2.” A Learner responds by saying “You must count the numbers”. At this point the teacher goes with the learner’s counting focus. The cards arranged in sequence focus learners’ attention on the ordinal aspect of the numbers. However, it is possible that at a subconscious level learners may be seeing the relationship between the next in a list of symbols, and next in a series of sets related by +1. The learners show that they are able to recognise numbers represented both by a collection of dots and by a numeral when these are placed in order. An extension activity could be to ask learners to match the numeral with the appropriate dot card when they are not placed in order.

Clip C: Reading numerals without support in Grade R. In this clip learners are asked to read number symbols. They are shown number cards with only the numeral written on it. There is nothing in this clip which builds the meaning of numbers. This means that the activity is simply a reading activity; no mathematics is being built. Think back to all the activities that have been done before learners were asked to read the number symbols (See DVD 1). The teacher has very carefully planned the progression in this lesson. The initial part of the lesson focuses on 1 to 5; learners are asked to respond individually. When the teacher adds the numbers 6 to 10, she not only alerts learners to this, but also allows for group responses. It is a carefully scaffolded progression.

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The teacher shows the number cards 1 – 5; but not in order. Learners cannot therefore rely on the counting sequence to support their reading of the numbers. Learners respond individually 3, 4, 5, 2, 1, The teacher then focuses on the numbers 6 – 10. There is a clear break between the two number ranges. She first finds the new number cards, and then changes the way in which the learners respond: learners respond in a group. 6, 8, 9, 10, 7, 6, Finally she includes the full range 1 – 10. She alerts learners to this by saying “Now I going to mix them all up … “6, 8, 9, 10, 7, 1, 3, 4, 5, 2.”

Clip D: Recognise, identify and read number symbols and names In this clip learners were each given an envelope with numbers in words and numerals from 1 to 10. The teacher decides not to just let learners pack out all the numbers and all the words and then match them. Instead she builds in comparisons between numbers. She starts with “one more” and “comes after”. She proceeds to “two less than”. She explicitly asks learners to imagine and hold a number in their head, and then find another number relative to this:… “You’ve got the number 6 in your mind. … Now find two less than 6” Once learners have found the correct number symbol, they are asked to find the corresponding number name.

6.4.2 Helping students to reflect on DVD 2.2: teaching learners to recognise, identify and read number names and symbols

The four DVD clips in DVD2.2 are designed to be viewed together.

Teaching Notes You may, however, 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 teaching learners to recognise, identify and read number names and symbols The topics and questions below are designed to help students reflect on particular issues that arise in DVD 2.2 Recognise, identify and read number names and symbols.

Student Activity: 6.4.2 - Attaching meaning to numerals/number symbols (DVD Clip A) 1. In DVD Clip A, the teacher asks learners to read numbers represented on cards

with collections of dots and on other cards with number symbols. One learner looks at the numeral 9, and says that it is 8.

a. What is the first way the teacher assesses whether the rest of the class knows the cardinal value of 9?

b. What does she do when she sees that some other learners do not know the cardinal value of 9?

2. When she asks learners to all count on their fingers, some learners go beyond 9 to 10. Some people might say that if you can count to 10, you can count to 9.

a. Why does she stop, and make all learners count again?

Student Activity: 6.4.3 - Using the counting sequence to support reading numerals and dot cards (DVD Clip B)

1. What responses do learners give that indicate that they are using counting to support their reading of the picture cards and numerals?

Student Activity: 6.4.4- Number cards, number tracks, number lines (DVD Clip B)

In DVD Clip B, the teacher refers to two rows of sequenced dot cards and numerals as “our number line”.

1. Draw a number track going from 1 – 10. 2. Draw a number line going from 1 – 10 3. In what ways do they look similar? 4. In what ways do they look different? 5. Number lines represent a different aspect of numbers to number tracks. What is

this difference?

x Attaching meaning to numerals x Using the counting sequence to support reading numerals and dot cards x Number cards, number tracks, number lines x Reading number symbols – a mathematical activity? x Level principle x Transition

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Student Activity 6.4.5: Reading number symbols – a mathematical activity? (DVD Clip C)

1. In DVD Clip C learners are asked to read number symbols that are presented without a context, or visual supports. The numbers are also not presented in the counting order.

a. Is this a mathematical activity or a reading (labelling) activity? Explain your answer.

b. Towards the end of Clip C a learner is heard exclaiming “Yo! She put it the wrong way”. Why does the learner make this statement and what number property(ies) could the learner referring to?

Student Activity 6.4.6: Level principle (DVD Clip D)

1. What form of representing numbers is used in the Grade 1 class (Clip D) that is not used in the Grade R classes (clips A, B, C)?

2. What ways of representing numbers were used in some of the Grade R classes (clips A & B) that are not used in the Grade 1 class?

3. In this activity learners are matching number symbols to written number names. At one point the Grade 1 teacher says “you’ve got the number 6 in your mind, now find 2 less than 6……” Learners need to know number symbols and names to do this activity.

a. What else do they need to know in order to complete the activity?

Student Activity 6.4.7: Transition

1. DVD clips A, B and C of DVD 2.2 were filmed in Grade R classrooms. Which of these activities is suitable for Grade 1 learners? Explain.

Applied tasks for students 6.4.8 These tasks focus on applying what you have learned from doing the focussed DVD activities.

Matching representations to numerals

1. Design an activity which allows you to check that Grade R learners can match random dot cards or other pictures representing collections of objects to numerals. Make sure that you do not give the picture cards in order.

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Linking the next in the sequence to the set +1. 2. In DVD Clip B the teacher first tries to get learners to link the idea of ‘next in the

sequence’ to ‘the collection’ +1. Later she no longer does this. a. Design an activity in which learners focus on linking the next in the sequence

to ‘the collection’+1.

3. Create a list of activities you can do with learners to help them understand the value of each number 1 to 10. These can include:

a. Activities that you source from elsewhere b. Activities from elsewhere that you adapt c. Activities that you develop on your own

6.5 Comparison: more, less and equal Comparing collections Learners usually learn to compare collections or objects before they learn to compare numbers represented in symbols or words. According to Sarama and Clements (2009: 82) young children find it easier to identify equivalent rather than non-equivalents sets. They cite Cooper (1984) and Sophian (1988) in positing that infants and pre-schoolers are more successful at comparing sets of equal rather than unequal numbers “presumably because there are many ways for collections to be unequal (difficulty is also increased if the task demands unequal collections be ordered in size.)” (Sarama & Clements, 2009: 82). In Clip C of DVD 2.3, one can see how easy it is for learners to say that each child gets an equal number of khoki’s. This can be compared with how learners struggle in other DVD clips, to answer questions about whether numbers are more or less than each other. Van de Walle and Lovin (2006: 38) suggest that young children find it easier to say which collection has more; but find it more difficult to say which collection has less. Young learners can compare the number of objects or pictures of objects in two or more sets or collections. If the number of objects is small this can be done by subitising (for more about subitising see DVD 1.3 and the Mathematics Content Notes in section 5.5 of this guide). As the number of objects in the sets increases learners are more likely to use matching or counting. One-to-one correspondence is a powerful way for young children to check whether two groups contain more, or less or equal numbers of objects. The assumption is that children will match one object with a corresponding object of another set. If learners have worked with subitising, they may use the idea of recognising / counting / making a group of a particular size in each set, and seeing how many objects remain in each collection. Rather than matching objects one by one, they may match sub-groups of objects in each set. Some examples are provided below.

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Comparing the number of objects in 2 collections by subitising

Comparing two collections of objects using matching (one-to-one correspondence) Learners need to be fluent in counting before they can compare the size of collections by counting. Children “need to develop sufficient working memory to make a plan to compare, count two collections, keep the results in mind, relate these two results, and draw conclusions about the collections …….the development is less about creation of entirely new procedures and more the interplay among existing components of knowledge and processes” (Sarama & Clements, 2009: 84).

Comparing numbers Once learners have developed an understanding of the size of numbers, they can begin to compare numbers when presented with the numeral (symbol) or number word. Young learners tend to develop an understanding of smaller whole numbers first. This makes it easier for them to compare numbers that have smaller values. However, it is also easier for learners to compare collections of objects that have a larger difference in the number between them, than collections that have a smaller difference in the number of objects between them (Cowan & Daniel cited in Sarama and Clements, 2009, 86). Learners do not only need to understand what individual numbers mean, they also need to understand the successor function of natural numbers, i.e. that each successive number represents a collection that contains one more element and that one generates adjacent numerical values by adding or subtracting 1. “Children develop the ability to order numbers over several years by learning the cultural tools of subitising, matching and counting” (Sarama & Clements, 2009: 86). When whole numbers are put in order or sequenced one is focussing on the ordinality aspect of the number. Saying forward and backward number sequences, and working with number tracks and number lines helps to focus learners on the ordinality aspects of numbers. These activities help learners to develop mental images of number lines: a mental image of spatial linear arrangement of numbers in order, which may be used when comparing numbers. Learners compare amounts and/ or numbers in several of the DVD clips: DVD 1.2 (Clip B, C, & D); DVD 2.1 (Clip A); DVD 2.2 (Clip D); DVD 2.3 (Clips A, B, C, D). Learners are also asked to change the number of counters in a collection: making the collection either more or less than the original collection: DVD 2.3 Clip B.

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The focus of DVD 2.3 is on activities that are related to comparison. However, it is not necessary for these activities to be separated out from other learning tasks. They can be embedded in both counting and calculating activities, as is shown in DVD 1 and DVD 3. The aim is for learners to develop an understanding of the relative size of numbers. For example, 6 is more than several numbers, but less than several other numbers.

6.5.1 Description of DVD 2.3: Comparisons: more, less and equal The four DVD clips in DVD 2.3 focus on comparing numbers and amounts. One-to-one correspondence is an important process when young learners compare sets or

groups of small amounts. The DVD clips in DVD 2.3 deal with ways to develop learners’ understanding of the relative size of number. Learners’ understanding of number will be deepened by activities in which they compare the size of numbers (DVD 2.3) and work with part-part-whole relationships (DVD 2.4). These activities help learners to develop networks of connections between numbers. They help learners to begin to understand the relative size of numbers.

Teaching Notes Towards the end of DVD 2.3 are still frames of the clips which can be used for the purposes of reflection

Summary of DVD clips in DVD 2.3 Comparison: more, less and equal

Grade Grade R

Elements Clip A Clip B Clip C Clip D

Timing 00:00 – 3:46 3:46 – 6:23 6:23 – 7:55 7:55 – 11:25

Approach Group/working in pairs Group Whole class Group

Progression / transition

Comparing amounts

Making an amount more or less

Making equal amounts

Stating numbers more or less

Manipulatives Fingers Counters and display board

Khoki pens and children

Learners work mentally whilst looking at, but not manipulating a model of a number line and columns representing the numbers of counters.

Everyday contexts Fingers fingers Equal sharing

of pens

Mathematics Comparison: more, less and equal

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Student Activity 6.5.1: Comparisons: more, less and equal

1. Read the introduction to Comparisons: more less and equal in Section 6.5.1. 2. Watch DVD clip 2.3 (A-D) all the way through. 3. While you are viewing the clips think about what you have read about comparisions and

think about the following questions. a. Which of the principles of counting can you identify in the DVD clips? b. Children seem to firstly use qualitative descriptions then later develop the ability

to use quantitative descriptions when comparing number sizes. What does this mean?

c. In all the clips A-D, notice the teachers’ attempts at scaffolding learners’ understanding using language and gestures. Identify instances in the clips where you may have phrased probing questions differently or altered the activity to improve teachers’ scaffolding attempts.

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: Comparing amounts to develop an understanding of relative size of numbers: more or less The aim is for learners to develop an understanding of the relative size of numbers. For example, 5 is more than 4, but also more than 3, 2 and 1; 5 is also less than 6, 7, 8, 9, 10, etc. The game “more I win / less I win” is used. In this game learners work in pairs, each learner decides how many of their fingers to hold up, they show their partner and then compare who has more or less fingers up, according to the teacher’s instruction. More I win

The first time it is played, each pair is able to say which of them has shown more fingers. However, when the teacher asks “How do you know who has more?” the children are not able to answer. The teacher continues to probe through questioning. “Why do you say that you have more fingers or less fingers?” It is unclear what the answer should be or how learners were expected to arrive at the answer. Eventually she reduces the question to one that requires a ‘yes’ or ‘no’ response: “Is 9 bigger than 6?” This is in fact just a restatement of “who has more fingers”, which the children could answer. What they are unable to answer is why it is more, or why it is bigger. One option might have been for pairs of learners to place their raised fingers against each other and to do one-to-one-correspondence to see who has more fingers

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Less I win The second time the game is played learners are also able to answer who has less fingers up. It appears as if the teacher wants them to be able to translate what she said in the previous game “It is more because it is bigger” into “it is less because it is smaller”. Learners do not do this.

Clip B: Changing an amount: making it more or less The teacher’s aim here is for learners to experience how they can change their collections of objects to make them more or less. The teacher alternates between focussing on the actual numbers and focusing on growing and shrinking a collection in general terms. Her aim is for learners to understand that when you place additional counters into a collection you make it more, and when you take counters away from a collection you make it less. The teacher also focuses on vocabulary that she associates with more and less; namely ‘add’ and ‘take away’. Teacher: “Now I want you to make that group of 5 counters more, using your counters.

All learners place additional counters in their circles.” Teacher asks one learner: “How many counters do you have?” Learner1: “7” Teacher: “What did you do to make the 5 more?” Learner1 “I took 5 fingers” [gestures with all 5 fingers up on one hand]

“and I added two more fingers” [gestures with 2 fingers up on other hand] This is an example of how learners are translating between mental images of one kind of manipulative into number in order to solve a problem that involves numbers represented by another kind of manipulative.

Teacher: “How do we do make something more?” Learner1: “Add it” Teacher: “What do you do to the counters?

There was 5, and what did you do to the counters? What did you do? Learner2: “I put more in.”… Teacher: “Now I want you to make that number less.”………

What did you do to make it less?” Several learners: “Took away some” / “Took some away.” Teacher: “You took some away. Good.

How many did you take away? Can you remember?” One learner miscalculates, another learner explains correctly. One learner has simply re-arranged her counters. The teacher tries to get her to acknowledge that she has not reduced the size of her collection because she has not taken counters away. The learner appears to want to please the teacher, by saying that she did make her collection less and nodding when asked if she took counters away.

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Clip C: Making equal amounts Six pens have been stuck on the board. Two learners stand in front of the board.

Teacher: “I have 6 khoki’s on the board. I must share them equally between Hloni and Joshua. Who can tell me, how many khoki’s must Hloni get?” Learner1: “3” Teacher: “3? Why do you say 3? Learner1: “Because 3 and 3 make 6”

The speed with which the learner responds to the question indicates that she is not sharing one by one, but has progressed beyond one-to-one correspondence in this number range. The learner is also translating the image of pens into numbers. She does not say 3 pens and 3 pens make 6 pens. She is using known number facts: 3 and 3 make 6. The teacher asks the learner (Hloni) to take 3 khoki’s.

Teacher: “Who can tell me, how many khoki’s Joshua must take?” Learner2: “3” Teacher: “3? Why do you say 3, he must take 3 khoki’s?” Learner2: “Because 3 and 3 make 6” Teacher: “3 and 3 make 6” Teacher: “Show everyone your khoki’s” Teacher: “Who has more khoki’s?” Learners together: “No-one” Teacher: “Who has less khoki’s?” Learners together: “No-one” Teacher: “So what can you tell me about the amount of khoki’s each of

them has?” Learner3: “Equal”

Clip D: Stating numbers that are more or less than a given number The teacher sets out a display of columns of counters from 1 to 10 alongside each other. The number of counters in each column is indicated by a numeral below the column. In the previous activities numbers were represented by manipulatives: Clip A – fingers; Clip B – counters that learners set out; Clip C – children and khoki pens. Here learners are looking at number symbols which are associated with

a column of counters. You will notice that the form of representation is becoming increasingly symbolic.

1 2 3 4 5 6 7 8 9 10

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Learners are positioned at a distance from the counters. They are not able to move or touch them. The representation is fixed. The teacher first asks learners to identify one number that is less than 2. This has only one answer. She then asks learners to identify two numbers that are less than 6. Learners could give any combination of 1, 2, 3, 4, and 5. This allows for more than one answer (number pair) that is less than 6. The teacher aims to dispel the misconception that it is only the number 5 that is less than 6. This point would have been more strongly made if she had allowed other learners to answer this question. Her later questions do not allow for multiple answers. When she asks for one number more than 7 one learner answers “10” and one learner answers “2”. She points to the column of 2 on the display as she asks the learner to reconsider whether 2 is more or less than 7. The teacher then generalises the relationships between numbers on a number line, when she asks:

“When we look at numbers more than 1 are we going to look this way (pointing left) or that way (pointing right). Some learners do not respond, most learners indicate to the right. This is perhaps obvious to the children for two reasons:

x The display does not show number less than 1 x These learners have not worked with numbers less than 1.

The children’s concentration is disturbed by the siren for interval. The teacher extends this question, and asks

“When we look at numbers that are more than 6 are we going to look that way (pointing left) or that way (pointing right). The three learners on the teacher’s right all point to their left. The three learners on the teacher’s left point to their right. The teacher’s voice intonation indicates that there was something incorrect about the answer. “Which way? More! More!”

All the children indicate to their right. She then asks individual learners to answer

Give me a number more than 9? Give me a smaller than 3, less than..? Give me a number more than 8? The learners answer: “7”. The teacher does not notice that this is incorrect.

She then asks for two numbers less than 7. Some learners seem to be aware of (‘know’) answers when they are not asked to answer individually, but struggle to give an answer when they are singled out. In this example the learner who is asked to give the answer remains quiet. Other learners try to tell her the answer. Eventually she says “8” and then “9”.

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The teachers repeats the question, and indicates on the display, the numbers on this side (to the left of seven are all less than 7). Another learner says “1”. The learner who is singled out to answer then says “1”. The teacher asks for another example. The learner says “2”. The teacher opens the question to other learners. One learner volunteers “4”. In this way she returns to her original focus, that there can be many numbers that are more and many numbers that are less than a chosen number.

6.5.2 Helping students to reflect on DVD 2.3: teaching comparison: more, less and equal The four DVD clips in DVD 2.3 are designed to be viewed together.

Focus points for student activities on teaching comparison: more, less and equal The topics and questions below are designed to help students reflect on particular issues that arise in DVD 2.3 Comparison: more, less and equal.

Student Activity 6.5.2: Questioning and/or probing (DVD Clip A)

1. In DVD Clip A, learners play “More I win/ Less I win”. Learners are able to say who in the pair shows more fingers, and who in the pair shows fewer fingers. The teacher wants them to justify their answers. Learners struggle to answer the teacher’s questions.

a. What questions does the teacher ask in her attempt to get learners to justify how they know who is showing more or less fingers?

b. How do you understand the difference between “Which is more?” and “Which is bigger?”

2. Many Grade 1 workbooks contain exercises in which learners are shown diagrams of

sets of objects, and are asked to compare the numbers of objects in sets. From our observations in classrooms learners will complete the activity by doing one-to-one correspondence. Probing does not always need to take the form of verbal questioning alone. In what practical way could the pairs of learners have done a one-to-one correspondence activity to check their answers of more or less?

a. How could this have consolidated or extended learners’ understanding of the relative size of numbers

b. What possible additional language could have been developed from this?

x Questioning and/probing x What is the mathematical purpose of comparing numbers? x Mathematizing: from working with manipulatives to using mental images to

solving problems x Comparing numbers: many possibilities x Level principle

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Student Activity: 6.5.3: What is the mathematical purpose of comparing numbers?

1. What mathematical concepts can be developed, consolidated or extended when learners compare numbers?

Student Activity 6.5.4: Mathematizing - from working with manipulatives to using mental images to solving problems (DVD Clips B & D).

1. Even before children can write they can use mental images - either graphical images or images of numerals to solve calculations in their heads. It is assumed that posing guiding and reflective questions to learners, who are working practically with apparatus, will help them to develop mental images that they can use to solve problems.

a. DVD 2. 3, Clip B: Find an example of where a learner explains his/ her solution by making references to mental images that are different to the apparatus the child is currently using.

b. DVD 2. 3, Clip D: Find an example of where a child’s response indicates that he/ she solved the problem by doing a mental calculation with number symbols.

Student Activity 6.5.5: Comparing numbers: many possibilities

1. Teachers sometimes ask learners “what number is more than ….?” or “what number is less than ….?” The answer that they are looking for is often the number that immediately follows the number given (successor number) or the number immediately preceding the number given.

c. In which DVD clips is the focus broadened to include more than just the number immediately preceding or following the number given.

d. Why is it important to distinguish between the numbers one more and one less, and any number more or less than the specified number.

Student Activity6.5.6: Level principle

1. Draw a flow diagram in which you sequence the activities in DVD 2.3 from the easiest to the most complicated.

a. Explain in detail why you think activities are sequenced in this order.

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Applied tasks for students 6.5.7 These tasks focus on applying what you have learned from doing the focussed DVD activities.

Applied Task 1: Design activities to help Grade R learners to compare amounts.

1. Brainstorm ways for Grade R learners to compare numbers. Use sticky notes, post-its or strips of paper to record your ideas.

a. Include activities in which learners make amounts more, less and equal. b. Decide which of these activities will be whole class activities and which will

be small group activities c. Check that you have included a range of ways of representing numbers and

amounts. Consider the language you intend to use and teach d. Now sequence the activities so that they get more complex. Arrange your

sticky notes to order the activities. Remember that: i. it is easiest for young children to recognise equal groups using one-to-

one correspondence. ii. Making and recognising a group that has more seems to be easier

than recognising a group that has less. iii. It is easier for learners to recognise which amount is more or less if

there is a bigger difference between the amounts. e. Make sure that you can explain why you sequenced the activities in the way

you have. f. Choose one member of the group to report for your group. g. Listen carefully to the report backs from other groups. Compare their

suggested sequencing with that of your group. Remember that there may be several appropriate ways in which to sequence similar tasks. There is not only one correct way to sequence activities.

Applied Task 2: Write a series of lessons to develop Grade 1 learners’ understanding

of comparing numbers. 1. The lessons should

a. include a focus on more, less and equal. b. get increasingly more complex c. include a mixture of whole class activities and small group work d. include a range of ways of representing numbers e. include careful consideration of language you intend to use and teach f. include an explanation of why you sequenced the activities in the way you

have.

Applied Task 3: Micro-teaching Young learners find it difficult to compare amounts and numbers. One-to-one correspondence is at the heart of comparing small amounts and small numbers. Part of the difficulty in teaching this, is that the language and images you use is different when you are comparing amounts (counters or other manipulatives) than when you are comparing numbers. You will have noticed in DVD 2.3 that one needs to carefully

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choose the activities, apparatus and language that one uses when assisting learners to compare numbers.

1. Choose one of the lessons you developed from Applied Task 2 above. 2. Micro-teach it in front of the rest of the class. 3. Help your peers by providing constructive feedback on their micro-teaching

6.6 Building up and breaking down numbers Working with apparatus such as loose counters, strings of beads, dot patterns, money, number cards and number lines provides models and scaffold the development and understanding that numbers can progressively be built up or broken down into a series of smaller numbers. Conceptual subitising, (see DVD 1.3: Clip 2 and the description of this in Unit 5.5.1) provides early experiences of part-part-whole relationships between numbers, and helps to build images of some of the relationships between numbers. Composing and decomposing numbers adds to learners’ understanding of the relationships between numbers and the relative sizes of numbers: for example if 7 is split into 2 and 5, it means that 7 is 2 more than 5. Building up and breaking down numbers can assist learners to think flexibly about numbers. For example 7 can be broken up into 1 and 6, or 2 and 5 or 3 and 4. If learners work with a range of ways of building up and breaking down numbers, it can help them with combining other numbers. For example to add 7 and 5 it is useful to break 7 into 5 and 2, and then recombine 5 and 5 to make 10 and then to add the remaining 2 to make 12. Composing and decomposing numbers also lays the basis for operating or calculating with numbers.

Commutative Property There are three properties of operations that are used when calculating. Learners do not need to know the names of these properties until they study Algebra in high school. However, it is useful for learners to be able to work with these properties. Here we discuss the commutative property, in Unit 7 we will also look at the associative property and the distributive property. The commutative or order property of addition and multiplication means we can change the order in which we add or multiply two numbers without changing the result.

x Example: 4 + 1 = 1 + 4, x The commutative property is expressed algebraically as

o a + b = b + a – the commutative property for addition o a x b = b x a– the commutative property for multiplication

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Young children can easily see that the order in which you add two numbers does not affect the value. This is easily demonstrated with unifix cubes.

x Join unifix cubes of different colours e.g. 3 red and 2 yellow. Describe these verbally.

x Turn the set of 5 joined unifix cubes through 180˚. Describe the set again.

Alternatively learners place counters on a piece of paper or card, they can split up the group of counters, into two groups. For example, 5 is 3 and 2. The card or piece of paper can be turned around to show that this is equivalent to 5 is 2 and 3.

Canobi et al summarised in Sarama and Clements (2009) state that children as young as 4 years old can make sense of the commutative property if it is represented physically with objects. Sarama and Clements suggest that working practically with the commutative property with young children is beneficial as it helps them to create one mental image for the two linked addition number sentences.

Counting on Over time children begin to use increasingly sophisticated counting procedures when building up and breaking down numbers. Carpenter and Moser cited in Thompson (2008: 98) describe five different “levels of addition strategies

x Counting all x Counting on from the first number x Counting on from the larger number x Using known number facts x Using derived number facts”

5 is 3 and 2

5 is 2 and 3

5 is 3 and 2 5 is 2 and 3

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When young children combine two groups of objects, for example: a group of 4 counters and a group of 3 counters, they may initially count the first group: 1, 2, 3, 4; and then count the second group: 1, 2, 3 and then put the counters together and count them all: 1, 2, 3, 4, 5, 6, 7. This is counting all. At a later stage the children may count or subitise the first group, and then keep the cardinal number of this group in their heads as they count on with the second group: 5, 6, 7. Thompson (2008: 99) writes that in order to count-on learners need be able to understand and work with:

x Cardinality: the last count gives the total of the collection x the counting sequence as a breakable chain: counting can start from any point in this

chain x it is not only objects that can be counted; numbers themselves can also be counted

Thompson (2008: 99) adds that when counting on “a cardinal/ordinal switch is followed by an ordinal/cardinal switch”. For example, when using counting on to find the total of a group of 5 and a group of 6, a learner needs to take the cardinal number of the first set, i.e. 5, and change “this to an ordinal number so that the count can be continued to 11; then the ordinal number 11, is converted back to a cardinal quantity to give the answer.” Gray (2008: 86) concurs that switch of compressing the lengthy count-all procedure into a more compact count-on procedure is “not as simple as it seems. Count-on is a sophisticated double counting process.” When combining 4 and 3, the count-on procedure “requires not only counting on beyond 4 in the number sequence, but also keeping a check that precisely three numbers are being counted.”

Count: 1, 2, 3, 4 5, 6, 7

Cardinality 4 7 In the example above 3 counters are added to the 4 counters, the first counter added gets the number name 5, the second counter gets the number name 6 and the third counter gets the number name 7. Initially some learners get confused because the sets of numbers in the double count are different:

1 2

3 4

1 2 3

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In Clip J of the Grade 1 lesson on DVD 2.4, one can see Mikaeel counting on from 6 to 9. This clip shows clearly how challenging this “double count” is for learners.

Numbers simultaneously represent a count and a cardinal value; they are both adjectives and nouns Gray and Tall cited in Gray (2008: 89) refer to symbols that ambiguously represent both process and concept as “procepts.” Gray (2008: 89) states:

“Numerical symbols do not represent either a process or an object; they represent both at the same time. Consider, as an example the symbol ‘5’. It can be spoken and it can be heard. The symbol ‘5’ represents the fusion of the number name with a counting process. We can recreate the counting process whenever we see the symbol or hear its name. But we can also use the concept of ‘five’ without any reference to countable items. Many different processes give rise to the object five. Not only the process of counting ‘one… two… three… four… five…’ but also the process of adding four and one, of adding three and two, two and three, of taking three away from eight, or two away from seven, of halving ten and so on. All these processes give rise to the same objects. The symbol ‘5’ represents a considerable amount of information, not least the counting process by which it is named and the concept or idea by which it is used.”

Young children who are good at counting may be able to use counting to add small numbers and they may do this more quickly than children who use more sophisticated methods, but those who use numbers as objects are more likely to be able to adopt a more flexible approach to composing and decomposing numbers, and operating with numbers. “If a number is seen as a flexible procept, evoking a mental object or a counting process, whichever is more fruitful at the time, then children are likely to build up known facts in a meaningful way …what we need to do is to help all children achieve the flexible form of thinking developed through compressing number processes into concepts” (Gray, 2008: 90).

x Counting helps young children to learn about both the cardinal aspects of numbers (their value) and their ordinal aspects (the position or value in relation to each other).

x Comparing numbers, knowing which numbers have a greater value and which have a lesser value, builds on ordinality and helps learners to understand the relationship between numbers.

x Composing and decomposing numbers helps learners to build an image of a stronger network of connections between numbers. They begin to understand that there is a myriad of connections between numbers. This in turn allows them to choose flexibly which of these relationships to draw on when calculating.

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6.6.1 Description of DVD 2.4.1: building up and breaking down numbers in Grade R

DVD 2.4 deals with building up (composing) and breaking down (decomposing) numbers in stages 1 and 2 of the LPN:

x It extends the focus on relationships between numbers done in DVD 2.3. x It builds on much of the work shown in DVD 2.3. x It lays the basis for much of the work shown in DVD 3.

Building up and breaking down numbers, helps learners to develop a good sense of the value of numbers and their relationships with other numbers. It also lays the basis for additive thinking. There are 15 DVD clips in DVD 2.4.

x DVD 2.4.1: Composition and decomposition of number in Grade R (five clips – about 12 minutes of footage)

x DVD 2.4.2: Composition and decomposition of number in Grade 1 (ten clips – 20 minutes of footage)

The focus of DVD 2.4.1 is on the progression of building up and breaking down numbers in Grade R.

Teaching Notes Towards the end of DVD 2.4.1 are still frames of the clips which can be used for the purposes of reflection

Summary of DVD 2.4.1: DVD clips on Building up and Breaking down numbers

Elements Clip A Clip B Clip C Clip D Clip E

Timing 00:00 – 02.06 02:06 – 6:29 06:29 – 07:34

07.34 – 08:56 08:56-11:23

Approach

Teacher models work with manipulatives

Children work independently with manipulatives

Teacher models recording

Teacher guides transition to mental mathematics and mental images

Children do independent recording

Progression / transition

From working with apparatus, to understanding a way of recording this work, to working mentally with the amounts / images, to recording own work

Manipulatives

Counters Display Board

Everyday contexts

Hands hands hands Hands hands

Mathematics Part-part-whole relationships, building towards number bonds, additive thinking

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Student Activity 6.6.1: Building up and breaking down of numbers in Grade R Do the first part alone, and then discuss with a partner or small group

1. Read the introduction to building up and breaking up of numbers in Section 6.6. 2. View all 5 clips of DVD 2.4.1. (A-E) all the way through. 3. While you are viewing the clips think about what you have read about building up and

breaking down of numbers. a. Look at how the activities are sequenced. Comment on the effectiveness of

these activities in progressively developing Grade R learners’ understanding towards more abstract ways of building up and breaking down of numbers.

b. Listen to the questions and statements phrased by the teacher during the activities. Identify the important terms/phrases used by the teacher in her attempts at getting learners to understand the building up and breaking down of numbers.

c. Think of other activities that you would have done in a Grade R class to develop learners’ understanding of building up and breaking down numbers.

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. Grade R Clip A: Teacher models breaking up 5 with manipulatives. The teacher holds out her hands and shows learners that she has 5 counters. She asks learners to say how many counters she has.

Teacher “How do you know I have 5 counters?” Learner1 “Because I can see it looks like 5” Learner2 whilst pointing “1, 2, 3, 4, 5” The teacher nods All the learners respond by pointing and counting “1, 2, 3, 4, 5”

The teacher demonstrates the shake-and-break game. She asks learners to say how many counters she has in each hand and then how many she has altogether.

Grade R Clip B: Children work independently breaking up 5 with manipulatives. The children are given counters. The initial instruction is for learners to shake and break and then put the counters on the circles in the board. But the teacher asks the first child to show her how many counters he has in each hand. After the child has shown her, he is asked to put the counters on his board. This is repeated with each child. This increases the difficulty of classroom management as children are not able to look at the counters that their peers show in their hands, whilst simultaneously placing their own counters on the board.

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The teacher then asks who has the same counters. All the boys have split their counters into 2 and 3. All the girls have split their counters into 4 and 1. The children play shake and break again. The teacher asks the children to show her what they have in each of their hands. Learners keep their counters in their hands as she asks

“Let’s look at what Keene has? What does he have? How many has he got in this hand? And how many does he have in this hand” And how many altogether? Who else has 4 and 1? OK let’s have a look” Keene put yours down in your circles …. Everyone put yours down in your circles.

The teacher selects three of the boys display boards, and places them next to each other. The fourth boy is keen to add his to the group, but the teacher does not allow it. She hands back two of the boards and takes the fourth boy’s board and places it next to the remaining board.

She asks: “Are they the same?” The children shake their heads and chant “noooooooooooo” The boy who made the grouping (and thought they were that they were the same) holds his head in his hands. Teacher asks: “Why are they not the same?” Learner: “Because, because you can, you can put it in your own way” The teacher points to the respective circles and asks: “How many’s here?” Teacher: “So, what can you tell me about them?” Learner: “They’re different” Teacher: “They’re different” This is the answer the teacher wants.

The teacher does not engage learners with the commutative property of addition, even though one of the children seems to have an intuitive understanding of it.

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Grade R Clip C: Teacher models recording in diagram The teacher asks learner by learner how many counters they had in each hand. While she re-iterates the process, she models how to record this on the display boards. She links the action with a verbal description and a graphical recording.

Grade R Clip D: Teacher guides transition to mental mathematics and working mentally with images to calculate. The teacher asks learners how many counters she has altogether in both hands. Learners tell her 5. Then she shakes and breaks, and place one hand behind her back. She opens the other hand and learners tell her that there are 4 counters in the hand. She asks how many counters she has behind her back. Learners are being challenged to make mental images based on the apparatus, using the language and models. They are expected to use these mental images to solve the problem. The teacher records learners’ answers:

2 1 3 1

The teacher shows them again that she has 4 in the one hand, and then brings the hand with 1 counter from behind her back. She reminds them, “because remember, 4 and 1 Makes 5.”

Grade R Clip E: Children independently record breaking up of number 5.

Learners are given recording sheets. They play “shake and break” and record their results diagrammatically.

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6.6.2 Description of DVD 2. 4.2: building up and breaking down numbers in Grade 1 The focus of DVD 2.4.2 is on the progression of building up and breaking down numbers in Grade 1.

DVD 2.4.2 is split into 10 different clips. These clips form part of a continuous lesson. It has been split into separate clips to alert the viewer to the different stages in the lesson, and to show how the teacher works with learners’ Zones of Proximal Development in a continuous process of changing aspects of the lesson. The DVD also highlights the way the teacher attempts to scaffold the learning, by including the forms of questioning and recording. The flow diagram below highlights the stages in the lesson.

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Grade 1 Clip A: Teacher models breaking up 9 with manipulatives

Grade 1 Clip B: Children work independently breaking up 9 with manipulatives

Grade 1 Clip C: Teacher models a way of recording the language of breaking up

Grade 1 Clip D: Teacher discusses different combinations of 9

Grade 1 Clip E Teacher introduces social context and diagrammatic “model of”

Grade 1 Clip F Teacher models how practical work with counters can be written as number sentences

Grade 1 Clip G: Learners work with counters and record in table and number sentences

Grade 1 Clip H: Teacher consolidates process and supports learners to examine equivalent number sentences

Grade 1 Clip I Teacher explains number bonds task

Grade 1 Clip J Learners fill in missing numbers in number bonds table. Some learners support their reasoning with finger counting, others do mental calculations

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Summary of DVD 2.4.2: DVD clips A – E on building up and breaking down numbers Elements Clip A Clip B Clip C Clip D Clip E

Timing 11.25-13:55 13: 55 – 15:47 15:47 – 16:30

16.30-17:08 17:08 -17:41

Approach

Teacher models breaking up, demonstrates working with manipulatives

Learners work independently with manipulatives

Teacher models recording of language used

Discusses different combinations of 9

Teacher introduces social context and “model of”

Progression / transition

From working with apparatus, to understanding a way of recording this work, to recording own work

Manipulatives Counters counters counters Counters counters

Everyday contexts

Hands hands Hands hands

Reference to rooms in a building

Mathematics Part-part-whole relationships, building towards number bonds, additive thinking

Summary of DVD 2.4.2: DVD clips F – J on building up and breaking down numbers Elements Clip F Clip G Clip H Clip I Clip J

Timing 17:41 –21:58 21:58 – 23:42 23:42 – 26:38 26:39 – 26:55

26:55– 31:15

Approach

Teacher models how practical work with counters can be written as number sentences

Learners work with counters and record in table and in number sentences

Teacher consolidates process and supports learners. Examine equivalent number sentences equal to 9

Teacher explains number bonds task

Learners fill in missing numbers in table “model for” Some learners support reasoning with fingers, others do mental calculations

Progression / transition

From working with apparatus, to understanding a way of recording this work, to working mentally with the amounts / images, to recording own work using mathematical symbols for addition and equal

Manipulatives Counters & display board Recording board without counters “model for”

Everyday contexts

Reference to rooms in a building

hands Fingers

Mathematics Part-part-whole relationships, building towards number bonds, additive thinking

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Student Activity 6.6.2: building up and breaking down of numbers in Grade 1

1. Read the introduction again about building up and breaking down of numbers in

Section 6.6. 2. View all 10 clips of DVD 2.4.2 all the way through.

3. While you are viewing think about what you have read about building up and

breaking down of numbers. a. Which of the mathematics strategies do learners use when composing and

decomposing numbers? b. Whilst viewing the clips, take note of the language used by the teachers in

trying to facilitate learners’ understanding of composition and decompostion of numbers. Reflect on the ability of the teachers to scaffold learning through verbal language and discuss the challenges that the teachers experience in trying to get learners to understand what their (the teachers) are saying or asking or teaching.

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. Grade 1 Clip A: Teacher models breaking up 9 with manipulatives. The teacher models the game “shake, shake, break” using 9 counters. She stresses repeatedly that the total number of counters is 9. She shows learners the number of counters in her right hand first. Learners tell her that there are 6 counters in her right hand. She then shows learners the counters in her left hand. Learners tell her that there are 3 counters in her left hand. She repeats this. Then she says “and if I put them together, I have 9” She changes the language to “so 6 and 3 is 9.” She writes this on the board: 6 and 3 is 9. This is her first form of recording.

Grade 1 Clip B: Learners work independently breaking up 9 with manipulatives. Learners each play the game under close guidance from the teacher. She encourages learners to make a mental record: “keep it in your mind”. The teacher asks learners to explain the combinations of counters that were made from 9. She stresses repeatedly that when both hands are placed together, the learners have 9 counters. The teacher takes time to develop language to describe the mathematics embedded in the activity: Count in your mind, Keep it in your mind

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Grade 1 Clip C: Teacher models recording of language use The teacher gestures and asks various learners:

“What do you have in this hand?” and

“What do you have in that hand?” and

“When you put them together?” When learners have reported on the part collections that they have in each hand, the teacher changes the language to: “so….. and … is 9” The teacher has encouraged the learners to talk about the part-part relationships of 9. She writes the latter spoken language on the board. After recording this once, she asks the learners to tell her what to write on the board.

Grade 1 Clip D: Teacher discusses different combinations of 9 In this clip the teacher marks out what is considered as different.

Teacher asks a learner: “what do you have? Learner: “5 and 4” Teacher: “We already have 4 and 5 is 9, but 5 and 4 is DIFFERENT.

It is interesting to note that both the Grade R teacher and the Grade 1 teacher choose to ignore the commutative property of addition. They do not discuss that the order of addition is irrelevant. The teacher asks all the learners to say what she should write on the board. The learners chant together “5 and 4 is 9”. The teacher is careful to write pairs of sentences with the same parts, specified in a different order, next to each other on the board. Later (see DVD 3.1.3 Teaching the meaning of symbols: equality) this approach appears to impact on what learners consider to be the same and different about mathematical statements.

6 and 3 is 9 4 and 5 is 9 7 and 2 is 9 8 and 1 is 9

3 and 6 is 9

6 and 3 is 9 4 and 5 is 9 7 and 2 is 9 8 and 1 is 9

3 and 6 is 9 5 and 4 is 9

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Grade 1 Clip E: Teacher introduces social context and diagrammatic “model of”. The teacher introduces a social context: “Now let’s pretend that the counters on the carpet are 9 people. Now these nine people live in a building” The teacher then produces a diagram which links both to the context created and to bonds of 9 (the mathematical focus). This diagram is a model of the context which can become a model for number bonds. Notice the point at which the context is introduced.

1. The teacher has modelled the concept and learners have already worked with apparatus, which has allowed them to build clear images of starting with 9, breaking it down into two parts and putting it back together again.

2. The teacher has modelled the concept and learners have described the underlying mathematical principles of the part-part-whole relationship as “…… and …… is 9”

3. The teacher has written up this form of recording on the board. 4. At this point a context and a model of recording the numbers related to this context

are introduced.

Grade 1 Clip F: Teacher models how practical work with counters can be written in

a number sentence. The teacher models how the practical work with counters can be re-interpreted into the ‘house’ context and also how this can be recorded in the model of the house.

Bonds of 9

Bonds of 9 1 8

9

9

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While she does this she also checks whether learners are able to imagine / create a mental image of / mentally solve bonds of 9. Learners are able to work out in their heads that 9 can be split into 1 and 8. The teacher then models how to write this as a number sentence. The teacher consciously introduces the terminology:

“Number sentence”; “plus”; “equals” and “total” She explicitly draws learners’ attention to the new symbols + and =. She asks learners to explain what + means, and what = means. Although she is happy with their descriptions of “plus”, the answer they give for “is equal to” is not the answer she is looking for. She does not want them to understand that it indicates that the answer is to follow. She demonstrates, using counters, that = means: “has the same value as.” She explains this verbally as “is the same as.” However, earlier she told learners that 5 and 4 is 9 is DIFFERENT to 4 and 5 is 9. She is establishing potentially conflicting views of when things are the same and when they are different mathematically. At this stage of number development it is important to introduce mathematically precise terminology using appropriate language while extending the learners’ understanding of the mathematics concepts being learnt.

Grade 1 Clip G: Learners work with counters and record in table and in number sentences.

As learners continue to play “shake, shake, break”, the teacher checks that they understand how to record their results in the table.

Grade 1 Clip H: Teacher consolidates process and supports learners to examine equivalent number sentences equal to.

Once learners have recorded at least one example, the teacher elicits some of the learners’ results. She records a few of these in her table on the board. This confirms for learners both the relationship between the game and the recording, and how to record their results. She encourages learners to write down as many examples as possible. The clip shows that one learner has completed his/her activity of filling in the missing combinations of number and writing the associated number sentences.

Bonds of 9 1 8 1 + 8 = 9

9

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In this example the learner demonstrates and understanding of the required form of recording: a number in each “room” of the house and that the number sentence is made up of adding the two numbers in each room and getting a total of 9. However, the learner recorded the total of 9 in the right-hand side “room” on three “floors”/ rows in the table, so his/her corresponding number sentences are thus incorrect.

Grade 1 Clip I: Teacher explains number bonds task. At this point in the lesson the task changes. The model of the block of flats is now being used as a model for number bonds of 9.

Grade 1 Clip J: Learners fill in missing numbers in number bonds table. Some learners support reasoning with fingers, others do mental calculations. Learners are working with bonds of 9. You will notice that one of the addends is missing in the example. Learners need to calculate this missing addend. This process illustrates how the teacher attempts to clarify for the learners the part-part-whole relationship and how to record it both in the model/ table and in the number sentence. Observe the learners work in Clip H. The intention behind this kind of activity is to assist learners’ progression away from a dependency on apparatus, and towards being able to calculate with number independently. Learners are encouraged not to work with counters. Some learners are able to do the calculation mentally, whilst others resort to relying on the use of their fingers. The clip shows a boy calculating on his fingers, by counting on to 9. This clip also shows the teacher assisting a learner to use her fingers to subtract the given amount from 9. Whilst the latter process may assist the child to see that there is a connection between addition and subtraction, it may also confuse the child as she is doing one operation (subtraction) practically but recording it as a different operation (addition).

Bonds of 9 1 8 1 + 8 = 9 2 9 2 + 9 = 9 3 9 3 + 9 = 9 4 5 4 + 5 = 9 5 4 5 + 4 = 9 1 9 1 + 9 = 9 3 6 3 + 6 = 9 8 1 8 + 1 = 9 9 0 9 + 0 = 9

9

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6.6.3 Helping students to reflect on DVD 2.4: teaching building up and breaking down of numbers

The DVD clips in DVD 2.4 focus on building up and breaking down numbers. These clips are designed to be viewed together. However, because of the length of the DVD clips, you may want to first let students view, make notes on and discuss issues arising from the DVD 2.4.1 (the Grade R clips). Then view, make notes on and discuss issues arising from DVD 2.4.2 (the Grade 1 clips). You could then discuss the progression in number range, activities, ways of reflecting and ways for recording from the Grade R situation to the Grade 1 situation.

Focus points for student activities on teaching building up and breaking down of numbers. The topics and questions below are designed to help students reflect on particular issues that arise in DVD 2.4: Building up and breaking down numbers

Student Activity 6.6.3: Modes of representation (Grade R)

1. In the Grade R class the learners first show how many counters they have in each hand. Later they use the display boards to show how they have split 5. At times in the lesson the teacher wants learners to look at how other learners partitioned 5 counters.

a. Is this easier to see when learners show the counters in their hands, or when learners show the counters on the boards?

x Modes of representation x Teacher- learner interactions versus teacher-learner-learner interactions x Forms of recording in Grade 1 x Number sentences to describe part-part whole relationships x What is different? What is the same? x Progression in forms of recording (Grade R to Grade 1)

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Student Activity 6.6.4: Teacher-learner interactions versus teacher-learner-learner interactions (Grade R – Clip B)

1. List the instances where the teacher in the Grade R class directs interactions to occur:

a. between herself and the whole group of learners b. between herself and an individual learner c. between learners

2. Once learners have shown the teacher how they have split their 5 counters, she asks them first to show her how many counters they have in each hand and then to place the counters on the circle on the board.

a. How does the timing of when learners are asked to place counters on the board impact on learners’ ability to see how other learners have split up 5?

b. If you want to increase learner-learner interactions in this activity, when would you have asked the learners to place their counters on the circles?

Student Activity 6.6.5: Forms of recording (Grade 1)

1. Below is the record of one learner’s work shown in DVD Clip H of the Grade 1 class.

a. Examine this record. Identify and describe the errors. Discuss a suitable form of remediation.

b. Discuss how the final task in this Grade 1 lesson could help to correct the misconceptions that this learner may have.

Bonds of 9 1 8 1 + 8 = 9 2 9 2 + 9 = 9 3 9 3 + 9 = 9 4 5 4 + 5 = 9 5 4 5 + 4 = 9 1 9 1 + 9 = 9 3 6 3 + 6 = 9 8 1 8 + 1 = 9 9 0 9 + 0 = 9

9

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Student Activity 6.6.6: Number sentences to describe part-part whole relationships (Grade 1)

1. Part-part whole relationships between numbers can be represented in different

diagrams: some examples are given below.

a. Discuss the various representations above in terms of

i. Clarity and accessibility ii. Level of symbolism

2. We can write number sentences to represent part-part whole relationships. For

example, the relationship above can be written as 9 = 6 + 3. This is sometimes called a closed number sentence. If one of the parts is not specified e.g. 9 = 6 + F, it can be called an open number sentence.

a. Write as many different open number sentences to describe the part-part whole relationships shown in the diagrams above.

Student Activity 6.6.7: What is different? What is the same? (Grade R and 1)

1. In the Grade R lesson 3 boys have displayed on their boards their efforts to make 5

by breaking up the collection into 1 and 4. The fourth boy in the group says that he has also broken up the number this way. The teacher explains that 4 and 1 is not the same representation as 1 and 4, and that 5 has been broken up differently, although the value is the same.

a. Does the teacher expand on this discussion? b. How could she have extended this discussion with the learners to draw

their attention to the fact that 1 and 4 is the same (value) as 4 and 1?

6 3

9

3 6

9

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2. In the Grade 1 class the teacher says “We already have 4 and 5 is 9, but 5 and 4 is DIFFERENT.

a. What is the commutative property of addition?

3. Learners can be shown that 1 and 4 is equivalent to 4 and 1 by rotating an arrangement that breaks up 5.

a. Discuss: i. the benefits of learners knowing about and understanding the

commutative property of addition. ii. when, in your view, should learners to be exposed to this property

of addition.

Student Activity 6.6.8: Mathematization - progression in forms of recording (Grade R to Grade 1)

1. Write or draw: a. The form of recording used in the Grade R class. b. The first form of recording used in the Grade 1 class. c. The two forms of recording used in the hand-out in the Grade 1 class.

2. Explain how each form of recording is more abstract than the previous form, and how each form draws learners more deeply into Mathematics.

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Applied tasks for students 6.6.9 These tasks focus on applying what you have learned from doing the focussed DVD activities.

Level principle: Progression and sequencing in the Grade R lesson There are 5 DVD clips that focus on building up and breaking down numbers in the Grade R lesson. Each of these represents one stage in the sequence of learning in the lesson.

1. Draw a flow diagram of the sequencing and progression in the lesson. Explain the purpose of each stage in the lesson. Focus on:

a. Teacher guidance b. Learner activity c. Development of mathematical language d. Development of forms of recording mathematics e. Development of mathematical content

Level principle: Progression and sequencing in the Grade 1 lesson There are 10 DVD clips of the Grade 1 lesson. Each of these represents one stage in the sequence of learning in the lesson.

1. Draw a flow diagram of the sequencing and progression in the lesson. Provide an explanation of the purpose of each stage (DVD clip) in the lesson. You can focus on

a. Teacher guidance b. Learner activity c. Development of mathematical language d. Development of forms of recording mathematics e. Development of mathematical content

2. Discuss the transition from each stage to the next stage (DVD clip) in the lesson.

Level principle: progression in “breaking up numbers” from Grade R to Grade 1 1. Complete the table below by:

x Comparing the lessons on building up and breaking down numbers in Grade R and Grade 1.

x Think particularly about the progression from Grade R to Grade 1. Remember that there is more to progression that increasing the number range.

Grade R Grade 1 Number range Classroom arrangement Modelling, instructions and language

Mode(s) of reflection Development of mental calculations Form(s) of recording

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6.7 References

References Manipulating number

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Anghileri, J. 2006. Teaching number sense. London, New York: Continuum International Publishing.

8 – 12 31

36-41 43-46

Clements, D.H., & Sarama, J. 2007. Early childhood mathematics learning. In F.K Lester, Jr. Second handbook of research on mathematics teaching and learning: A project of the national council of teachers of mathematics, Volume 1, (pp. 461-556). Charlotte, N.C.: Information Age Publishing Inc.

461-556

Gray, E. 2008. Compressing the counting process: Strength from flexible interpretation of symbols. In I. Thompson (Ed.), Teaching and learning early number, (2nd ed.) (pp.82-94). Maidenhead: Open University Press.

82 – 92 82 - 92

Piaget, J. 1952 The child's conception of number. London: Routledge and Kegan Paul Ltd.

Sarama, J. & Clements D.H. 2009. Early childhood mathematics education research: Learning trajectories for young children. New York. Routledge.

82-86

Siemon, D, Beswick, K., Brady, K., Clark, J., Faragher, R., Warren, E. 2011. Teaching Mathematics: Foundations to Middle Years. Melbourne. Oxford University Press.

278 281 - 283

267

Thompson, I. 2008. From counting to deriving number facts. In I. Thompson (Ed.), Teaching and learning early number, (2nd ed.) (pp.97-109). Maidenhead, Berkshire: Open University Press.

97-108

Van de Walle, J.A. & Lovin, L.H.2006. Teaching student-centered mathematics grades K-3. Vol.1. USA. Pearson Education.

37-38 44-45 63-64

44-53

Van den Heuvel-Panhuizen, Kühne, K, & Lombard, A.P. 2012 The learning pathway for number in the early primary grades. Northlands: Macmillan South Africa.

32-37

26-28 43-44

28 44-46

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6.8 Suggested Readings

Students and teacher have found the following resources useful when planning to teach activities and lessons related to manipulating number: Readings Manipulating number

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umbe

r na

mes

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less

and

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al

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up a

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Kennedy L.M, Tipps, S., & Johnson, A. 2008. Guiding children’s learning of mathematics (11 Ed.). Belmont, USA: Thomson & Wadsworth

151 - 154 149 149, 155

Staves, L. 2013. What is three? [ONLINE] . Available at: http://www.veryspecialmaths.co.uk/downloads/ What-is-three.pdf [Accessed 25 July 2013].

1-2

Troutman, A.P, & Lichtenberg, B.K. 2003. Mathematics a good beginning. Ontario, Canada: Thomson. Wadsworth.

111, 112 111, 113 109 -112, 114

117, 118, 160 -

Wright, R. J., Stanger, G., Stafford, K.A. & Martland, J. 2006. Teaching number in the classroom with 4 – 8 year olds. London: Sage.

35, 36, 42

Wright, R. J., Ellemor-Collins, D. & Tabor, P.D. 2012. Developing number knowledge: Assessment, teaching and intervention with 7 – 11 year olds. London: Paul Chapman Publishing.

54-57

Zaslavsky, C. 2001. Zero. Is it something? Is it nothing? Number sense and nonsense. Chicago, USA. Chicago Review Press.

35 - 50