CHILDREN INVENTING SIGNS FOR MENTAL NUMBER STRATEGIES

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CHILDREN INVENTING SIGNSFOR MENTAL NUMBERSTRATEGIESDavid Womack aa University of Manchester ,Published online: 14 Apr 2008.

To cite this article: David Womack (2000) CHILDREN INVENTING SIGNS FOR MENTALNUMBER STRATEGIES, Research in Mathematics Education, 2:1, 91-104, DOI:10.1080/14794800008520070

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7 CHILDREN INVENTING SIGNS FOR MENTAL NUMBER STRATEGIES

David Womack

University of Manchester

The research study described here was conducted with a small group of 5- and 6- year-old children in a 35-pupil rural school in the Langdale Valley (Lake District, UK) over a period of 14 weeks. It considers the theoretical implications of children inventing their own signs for their counting actions instead of using the culturally inherited signs of conventional arithmetic. It also raises questions concerning 'transformational addition' in which ordinals ('position numbers ') are distinguished

from cardinals ('size numbers 7, and suggests that the invented signs may have a consistency and validity comparable with the signs of conventional arithmetic.

BACKGROUND

This paper is part of an ongoing investigation to build a theory which (in the tradition of grounded theory) is faithful to and illuminates the area of children's intuitions about numbers and operations (Strauss, 1990, p. 24). Although children's intuitions about number are necessarily rooted in their experience, little is known of the reasons for their beliefs, except that they are not based on formal instruction and are often incompatible with forrnal education (Sinclair and Sinclair, 1986). In a recent work citing many experimental findings in cognitive neuroscience, Dehaene (1997) has come to the conclusion that a quantitative representation, inherited from our evolutionary past, underlies our intuitive understanding of number. Dehaene believes a major upheaval in the child's mental arithmetic occurs during the primary school years, when children are often pressured to shift from an intuitive understanding of number supported by simple counting strategies, to what Dehaene considers to be a rote learning of arithmetic. However, leaving aside the question of the 'arithmetic competence' of neonates, there is certainly a great deal of evidence for the numerical competence of pre-school children - evidence which is largely bypassed by current curriculum initiatives. For example, Gallistel and Gelman (1992) show that pre- school chlldren can already give verbal answers to verbally posed addition and subtraction problems by using vocal or subvocal counting algorithms, and claim that children seem to possess the ability to enter the positional representation for a number upon hearing its name. Klein and Starkey (1987), also believe that children beginning school possess a latent mental number ability which is considerably undervalued in schools. Discussing the reasons for children's failure to maintain this momentum, Ginsburg (1977) saw the ability to translate between concrete and written representations as fundamental to children's understanding of arithmetic. Martin Hughes (1986) demonstrated this elegantly, showing that when children see a collection of objects added to or taken from, they rarely see the connection with the conventional operation symbols of addition and subtraction. Hughes stressed that children's understanding of the addition sign and equals sign points to serious

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Research in Mathematics Education Volume 2

mismatch between the system of symbols which they are required to learn, and their own spontaneous conceptualisations. Indeed it seemed to Hughes (p. 78) that the whole notion of representing these transformations on paper was something which children found very hard to grasp, though he cautiously concluded that the exact reason for this difficulty was not yet entirely clear. A further input to the present investigation was the belief that although chldren's understanding of conventional mathematical symbols is greatly overestimated, they can invent and use their own symbols with some facility (Gardner and Wolf, 1983; Hughes, 1986; Resnick et al, 1990; Atkinson, 1992; Neuman, 1993).

The subject of this paper is the intuitive number knowledge which children bring to learning situations prior to receiving instruction through the socially transmitted conventional symbols of mathematics. This is not to ignore the social nature of symbols; in fact, far from dismissing social influences on the teaching of mathematics, the present research draws attention to the considerable power which symbols (social or otherwise), have on the thinking of young children [I].

RATIONALE AND AIMS FOR THE INVESTIGATION

As part of a larger enquiry, (including the theoretical and hstorical basis of the counting sequence), the writer has come to believe that research shows that children's understanding of numbers is primarily one in which the number symbols represent positions in a hierarchical sequence, rather than sizes of whole collections. In children's 'theory of numbers', size is identified not with the unordered collection of objects in a set, but with the extent of the ordered sequence of counting actions between two positions - effectively the 'distance' which the subject covers in order to proceed from one position-number to another (Womack, 1995). It is W h e r proposed that this intuitive theory of numbers is not replaced when conventional definitions are introduced but remains a covert model on which mental calculations are made throughout adult life.

It was decided that a model in which numbers represented positions would be a more intuitive basis for mental calculation than a model in which numbers represent the cardinal property of an unordered collection of objects. The cardinal model is ideal for dealing in the subitizable range (say, 1 - 7), but beyond that, numerical size cannot be mentally visualised successfully, except as sequentially arranged numerals. Therefore rather than pursuing the cardinal model, the investigation familiarised children at the outset with a scenario which supported the mental model they used intuitively - irrespective of their prior learning experiences. Children also tend to consider the size of a number to be associated more with the 'length' of number sequence rather than the Cantorian definition of cardinality. The idiosyncratic nature of children's (and adults') number imagery hardly needs to be pointed out; however, as teachers we strive (though never quite succeed) to create a learning environment suitable for as many children as possible.

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Children Inventing Signs for Mental Number Strategies

Therefore, after informal experimentation with ordinal models such as staircases of ascending numbers and a study of world-wide counting techniques, eventually a model was chosen homologous to the body-counting systems often used by non- literate peoples. The model consisted of a series of stepping stones, unevenly distributed across a stream - a familiar situation to the rural Lake District children and the researcher - in which number symbols represented positions, so that stepping actions (similar to body-counting actions) were required to pass from one number to the next. Effectively, this was an external representation of the hypothesised mental model of numbers in which children's mental strategy of counting from one number to the next is anticipated by the physical action of moving from one stone to another (Womack and Williams, 1998).

For example, if each stone is numbered in sequence, then a child standing on stone number 10 can be asked to walk back 4 steps. Alternatively, children can be asked to walk from stone 10 to stone 4. These activities are analogous to the strategies of count-back and count-down between numbers respectively (see Figure l), and are well-documented in the literature (e.g. Fuson, 1988) [2].

A count-back stratem ... 9, 8 ,7 ,6

I t 4 - Start position 10, . . - Count-back 4. n 10 Result: position n (n=6)

A count-down stratew ..... 9, 8, 7 ,6 ,5 ,4

I e m - Start position 10,

I - . .- Count-down to position 4.

I 4 10 - Result: m count-actions (m=6)

Figure 1. Mental Counting and Walking strategies to calculate 10 - 4

Mum (1997) has argued that the reason for chldren's alleged shortcomings in counting lies in socio-emotional factors and our failure to take account of the purpose (or lack of purpose) of much of chldren's counting. The stepping stone activities carried out by the children do not consist in counting miscellaneous collections of objects, but in the verbal recitation of numbers in correspondence with their walking actions in the context of games [3].

Children's 'pre-tuition' ideas about numbers are not the result of a school socializing process and so a guiding principle of the sessions was to nurture and develop children's intuitive ideas rather than 'reshape' or socialize them too soon. The particular aims of the study included determining children's ability to count-on (or back), and count-up (or down) between positions (both physically and mentally), and their ability to use and understand written symbols as instructions to perform these

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skills and find a number. A further aim [4] was to determine the viability of using the stepping stone model in some form to provide a basis for mental counting strategies.

PROCEDURES

Sessions took place in an open area away from the immediate classroom environment, to minimise associations with formal 'school work'. Within each session the researcher initiated various scenarios from which questions about numbers arose and in which the need for children to communicate questions about numbers was created. Questioning was generally carried out informally and responses were tape-recorded with a small dictaphone left in a box containing various items of interest in the centre of the table. Relevant transcripts were made immediately after the session. Children seem to work more naturally in a less focussed environment than that provided by the normal classroom, and so under no circumstance was pressure brought on children to produce a 'correct' answer. For t h s reason, questions were sometimes directed to pairs of children.

The activities included games, reading stories, conversations, play with magnifjring glasses, magnets etc. in which the researcher participated. This minimised the risk of questions becoming 'disembedded' (Donaldson, 1978) from the communal sub- culture. Care was also taken that the language used related only to the context of 'here and now'. The pedagogical philosophy of the sessions could therefore perhaps best be likened to a form of 'continuous interaction' (Vygotsky, 1962) of adult with the children - which sometimes extended to week-end social encounters. Although a provisional framework for each session was planned beforehand, the questions asked, and the directions subsequently taken were contingent on children's earlier responses. Following each session, the structure of activities for the following session was loosely decided.

SIGNS USED BY THE CHILDREN

Communication was either oral, written, or by using signs. The signs consisted of written or drawn instructions such as a word or an arrow sign. Whether the instruction was given by a child or by the researcher; whether the sign was prepared earlier as a 'card' to be chosen, or whether created during the course of the activity, all were carefully sequenced to facilitate incremental learning steps.

Written instructions moved towards a linear format similar to that used in conventional number sentences, to facilitate the later posing of questions in a more conventional format. However, no reference was made at any time to conventional mathematical terms such as add, subtract, equals; nor were the formal signs used or referred to in any of the sessions [ 5 ] . The five children were in their first year at the primary school and regularly completed pages of very simple written addition and subtraction 'sums' in their school classes. However, on only one occasion was it noticed that a child used the symbol '+' in the context of a written stepping stone question.

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THE STEPPING STONE QUESTIONS

Two basic types of question were asked (and answered) within the 'stepping stone' framework: Where will you reach? and How many steps (did you take)? Where will you reach? questions require a child to count-on (or back), whereas to answer the question, How many steps ... ? requires a child to count-up (or down). Although count-back and count-upldown are all subtraction strategies, conventionally we have only one sign (-) for this (see Figure 1). The conhsion sometimes caused by using a single 'subtraction7 sign to represent distinct number-related situations can be clearly seen when negative integers are introduced. Some researchers have made this distinction very clear (e.g. Rowland, 1982; Haylock, 1995, p. 95). Therefore in order to communicate about these two different situations, children were encouraged to invent their own signs. The signs used were as follows:-

Situation A

Where will you reach? questions were asked using vertical arrows invented by the children in Session 3. Up-pointing arrows (?) instructed children to move up the number sequence the indicated number of steps; down-pointing arrows (A) instructed children to move down. Therefore both types of vertical-arrow sign referred to actions children were required to take in order to reach a new (initially unknown) position. These arrow instructions were written on cards. (Underling indicates signs and numbers on the same card). For example:

5 meant From 5, walk-on 3 (or later, From 5, count-on 3).

5 J.3 meant From 5, walk-back 3 (or later, From 5 count-back 3).

Situation B

How many steps? questions were asked much later in the sessions, using another sign invented by the children - a horizontal arrow which asked/ instructed children to find the number of steps linking two numbers. For example, 9 t 6 meant Walk to 9 from 6. Effectively, this was a finding-the-counted-on-steps question and so these cards were not greatly used since the same question can be asked implicitly in a rnissing- count-on-card activity (see Session 12). Signs therefore both described and instructed; significantly, they defined not only the nature of the question, but also instructed a particular counting strategy. They also encouraged children to think of number operations as transformations of one number into another, by means of a thrd number (c.f. Hughes, 1986). The relation between count-on signs and count-up signs is more fully discussed in Womack (1 998a).

THE STEPPING STONE MATERIALS

The initial activity was designed to familiarise the children with the number sequence through numbered stepping stones. Stepping stones outdoors or on the floor led to table-top games in which children could move coins or a doll from one number to another. Scenarios included houses in a lake connected by bridges, ladders connecting balconies in a block of flats, and a self-made vertical strip of paper with

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the numbers 1 to 20 (but with no other lines or marks present), sloping up from the table, to give practical meaning to the instructions to move 'up' and 'down'. Activities were also undertaken briefly with an alphabetic letter sequence. Letters avoided the common confusion caused by number representing both positions and the movements between those positions (see also Womack, 1998a).

A number scroll 1 to 100, having no dividing lines other than the numbers themselves, was also introduced on whch children could practice mental calculations (see session 9). Later in the sessions, for the more confident children (6-year-olds), a 'missing number board' was introduced on which the arrow-cards and position-cards could be displayed. An initial (green) card indicated the starting number and a final green card the resulting position. By turning over any one of these cards, children could be required to find either the final position-number, the starting position- number or transforming-number (arrow-card). The latter two 'problems' are considered as 'identical' subtraction problems in an aggregative addition model; however, the distinction between transformation (or a similar concept) and position is a necessary distinction when integers are meaningfully introduced. The introduction of this distinction at this stage may therefore be considered both intuitive and preparatory to later algebraic ideas. Stepping stones introduce no extraneous concepts such as rectilinearity or equidistance between points, and engender only the idea of succession from one number to the next. In this respect it was considered to be more intuitive than a conventional number line (Klein and Beishuizen, 1998). [6].

DESCRIPTIONS OF SAMPLE SESSIONS

Approximately 60 separate activities were carried out by the children, in addition to various drawing and story-reading activities, of which only a small illustrative sample can be given here. Various 'transformational' scenarios were introduced throughout the 14 weeks and were returned to periodically as 'themes' throughout the fourteen 45-minute sessions.

In all sessions, the intention was not to assess the children but to learn from them, to surmise how and why they responded in the way they did. For this reason, activities were only undertaken whch it was felt children could 'handle' in a natural way. When it was clear that an activity was becoming too cognitively complex or tiring, it was aborted until a new step or approach could be devised to develop an interesting line of enquiry. 'Results' therefore show conspicuously few 'incorrect' answers and only indicate in a general way the point of understanding the children had reached. The following notes should therefore be interpreted only in this sense. Transcripts from tape recordings are selected as indicated.

Activity: Unmarked stepping stones on a table top (Session 2)

Stones are arranged in a meandering line from one side to another of a make-believe river. A large centre stone is chosen as 'starting point'. Chldren use a miniature doll to 'walk' from the large stone and stop. An arrow is casually drawn by the researcher to show the directions of 'On' and 'Back'. ('On' means across the river towards the

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far bank.) Instructions to walk-on or walk-back were written on cards. Cards were shuffled and distributed randomly to each child, who read them and moved the doll the appropriate number of steps across the stepping stones. Initially, cards were written by the researcher in sight of the children; later the activity was repeated but the children wrote their own instructions. The aim of the activities was to determine the extent to which children could follow the instructions given by the cards as intended.

Outcome: Instructions were read and understood correctly by all children. All children walked the doll forward and back with ease - though with some prior hesitation.

Instructions were interpreted correctly for cards showing:- 'Walk-on 3' , 'Walk-on 6'' 'Walk-back 4', 'Walk-back 7' , 'Walk-on 8'.

Activity : Inventing arrow signs as abbreviations for 'upldown' cards (Session 3)

In these transcripts 'Res' is the researcher and 'P', 'Pk' etc are children. The researcher sat among the group of children (P, Pk, J, G and T):

Res: Who can think of some signs we can use instead of writing these words which take you so long?

P: Use arrows.

Res. to P: Arrows, alright. You try. Show us what you would do; how you would do 'up 3'. P is going to show us.

(P draws hls first card - three small vertical arrows in a vertical column.)

J : One arrow then put the 3.

Res. to Pk: OK, he's got 3 arrows. What do you think Pk? How would you show it? You show us on your card. Up 3 with an arrow.

(Pk draws on a card - three vertical arrows and the symbol 3.)

Various types of arrow signs were drawn on the cards (both 'up' and 'down'). Cards were made by both children and researcher and these were used in various game situations at different times. The arrow cards were distinguishable from other printed (green) cards which indicated only ordinal numbers along the sequence. After using two arrows for 'up 2 ', three arrows for 'up 3 ' etc, the format eventually decided upon by the children was a single vertical arrow followed by a number, ?3 meaning 'up 3' and 44 meaning 'down 4' [7].

Activity: Using the arrow cards in number sentence format (Session 4)

Res: I'm coming round to write a question on your paper.

(The researcher writes a question on each child's paper in the same form as that which the child has written in the previous activity.)

Res: Now you write down the answer.

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Outcome: Questions and answers were as follows (all correct) given to children in the order G, J, T, Pk, P:-

Child

- ppppp

Activity: Checking meaning of the arrow signs. (Session 4)

The chldren had before them a sequence of numbers from 1 to 20 which they had drawn themselves (i.e. each of the children had constructed their own individual number sequence).

(i) Interpreting the cards

Res: I'll show you a green card to say which number to start at.

(One green card is distributed to each child.)

Res: Remember your up/ down arrow cards which tell you how many to count- on.

(All the upldown arrow cards in the range 1 to 5 are checked with each of the children.)

Outcome: All were remembered from the previous week and all children could read them accurately.

(ii) Carrying out instructions

Res: If I was to give that card to you. So if you started at 10 and did that (?2), what number would you get to?

J: Twelve.

(The cards presented to the children in turn were: 14 $2; 7 ? 1; 5 ?2; 16 ?'5; 10 ?3.)

Outcome: It was found that all children could carry out the instructions successfully for the above numbers. (Child T showed some uncertainty - or preferred to count-on in an almost inaudible whisper).

Activity: 'Puzzle-questions' - finding the transform. (Session 5)

Children observed the researcher take an arrow card, read it and move a coin along a sequence of numbers written on a strip of paper. The number on the arrow card could not be seen by the children. The aim was twofold: to assess children's ability to infer the value and direction of the upldown arrow card from observing the movements,

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and to check whether children could write this transformation in a number sentence format.

Game 1 - Watch me jump - then draw it!

The researcher places his coin on the number 5 on the vertical number-sequence strip, selects an upldown arrow card from the pack and carries out the card's instruction in discrete jumps. The card is not seen by any of the children enabling all children to participate at the same time as the child chosen to answer. The researcher then places the card face down in fi-ont of one of the children and asks; 'What is on the other side of the card?' The child is then asked to 'draw the card' on paper - by means of an arrow and a number. [8]

Outcome: Each child was asked one question each and answered correctly.

Game 2 - Watch me move - then draw it!

Game 1 was repeated for another two 'rounds' but the researcher's movement of the coin was shown not in jumps but in one complete movement of the coin. This meant that children could no longer rely on counting any jumping movements of the researcher's hand.

Outcome: Out of 10 questions (two questions per child), two errors were made.

Activity: Mental counting strategies with numbers 1 - 100. (Session 9)

The chldren stand facing a 3-metre-long number strip numbered 1 to 100.

Res: You are at 34, step back 3, where do you come to? ; You are at 56, go on 4, where do you reach? 30, count-on 1; 15, count-on 1; etc.

The researcher initially demonstrates how to arrive at an answer. Subsequently, children walk up to the scroll and give their answers in the same way.

Outcome:. 90 percent of questions asked were answered correctly. By this stage, answers were given immediately, by walking to the chart and pointing to the answer. Some children preferred 'hop' to 'walk' or 'count'. For example, to the question '97 hop-on 3', Tara immediately shouted out '100' without even looking at the chart. Some chldren wanted to close their eyes while they (slowly) produced the answers. It was not determined whether this involved some form of visual imagery.

Activity: Finding the Missing Number - initial number only. (Session 12)

A folding display board was constructed on which the green position-number cards and white, differently-sized transformation-number arrow cards could be placed. On the reverse of each card is a question mark.

Since the cards were arranged linearly in the format of a number sentence, the activity allowed children to test each other's skills at finding the missing number on the reverse of chosen cards.

No 'equals' signs were used in the activities; the result (answer) card was spaced apart from the instruction cards to indicate its status.

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In this sample, only the initial term was found - and with 'positive' transforms only. The researcher arranged the three cards as shown below. In the first example, the three cards are: unknown number card, transformation card ('r2), and green position- number card 7. This could be more formally written as y f 2 = 7 (y is the unknown).

The most able child was asked first, to 'cue' the other children into the activity.

Outcome: Children's comments are included.

The 3 cards

? 'r2 7 G's answer 5 G: I knew it was 5 because I thinked it.

? 'r3 9 P's answer 6 T: I want to do one ... ? ?4 10 T's answer ... T: Bit hard that one ... (too difficult)

? 1'3 9 Pk's answer 6 J: I 'm playing it!

All children express delight.

? 'r1 7 J'S answer 6

The 6-year-olds were significantly more successfd in finding the missing number than the 5-year-olds in this activity. In the previous 'Puzzle questions' session, children had successfidly solved problems of the form 5 & y = 9 etc. (i.e. fmding the transform but in a more intuitive context). Problems involving a 'negative' transform (e.g.. y &3 = 5 ) were found to be more difficult than the 'positive' transforms [9], although results were not collected systematically in this session, since children wanted to 'play the game' themselves without adult participation or supervision.

DISCUSSION

How helpful are the conventional operation symbols in helping children achieve numeracy? The answer to t h s question depends on the priorities of learning. The conventional written symbols (+, -, =) developed historically in response to the need to express the result of adding or subtracting two numbers on paper. However, to calculate mentally requires a counting technique dependent on understanding and using the relationship between cardinal and ordinal concepts of number. For example; children must decide which counting skill to use in any particular case - a decision not made solely on the basis of the semantic structure of the question, but also on the relative size of the two numbers concerned. (Womack, 1998b) [lo]. The symbols used in the stepping stone scenario do not require children to make such a decision, since they are created by the children themselves; are introduced as an instruction for action (both physical and mental) and are indicative of both the semantic structure of the question being asked and the calculation strategy required.

The invented signs also allow valuable 'symbol-manipulating' skills to be acquired in a familiar context. This effectively bridges the gap perceived by Hughes and others who suggested that chldren seem to find representing numerical transformations on

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paper hard to grasp. The results of this study suggest that the difficulties which children find in arithmetic do not lie in the representing of actions or transformations on paper, but in the fact that the traditional written signs do not relate to the mental counting processes.

From the results of this small-scale investigation, it would seem that chldren can cope very well with written signs for their mental counting strategies. If these signs were used initially, then the transition to conventional signs could be made at a later time when the operation signs can be understood as standing for both count-back and count-up. The alternative is to introduce and use signs such as '+' and '-' whose meaning is not yet understood and which require an understanding of the functional interplay between cardinal and ordinal numbers [ l 11.

NOTES

1. However, de-emphasizing the social aspects of mathematics education need not imply the infallibility and uniqueness of 'orthodox' mathematics, and Ernest (1991) has ably defended the charges of 'arbitrary' and 'relative' often levelled at social constructivism.

2. Count-back is a similar skill to count-on but in a 'backwards' direction. The other type of counting skill is the converse of count-on and count-back; that is, the start and end points are given and children must count how many steps are required to pass from start point to end point. If the end number is greater than the start number, then the activity is one of counting-up; otherwise (as in the example in Figurel), the activity is counting-down.

3. The stepping stone games are currently being informally trialled in rural South Aflcan schools courtesy of University of South Africa in Pretoria.

4. According to Piaget (1962), the mental image is initially a form of interiorised (though deferred) imitation of children's previous actions and is therefore a form of symbol rather than a direct copy of a previous perception (Womack, 1979).

5. Whether to call the invented signs 'operations' is debatable and I have elsewhere called them 'operactions' ( Womack, in preparation).

6. One modification of the conventional number line is known as the empty number line. The advantage claimed over number rods or blocks is that it incorporates the 'counting' (ordinal) aspect of number rather than the 'numerosity' (cardinal) aspect (Gravemeijer, 1994) and is 'empty' to discourage 'primitive counting strategies' which the 'full' number line may invoke. It also relates better to informal solution procedures such as counting-back and counting-up, and is a useful vehicle through which to talk about such strategies. However, there are major practical and theoretical difficulties with this model (Gravemeijer, 1993).

In particular, the root of the problem seems to be that for problems such as 63 - 46, once the 63 is entered by the child onto the line, the decision as to whether to enter 46 as a position on the line must be made. Placing 46 at a position provokes a counting-up strategy, whereas commencing to 'count-off 46 'positions' from 63 is a counting-back strategy. This decision must be taken mentally before pencil is put to paper so making the completion of the empty number line essentially redundant (except in cases where thousands or hundreds need to be set down to aid the memory).

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However, in this case, the empty number line is serving merely as an algorithmic device similar to traditional column subtraction. Gravemeijer (1994, p. 464) recognises this fact when he states that "The student does not make a model of a situation, but for the calculation." In contrast to this, the purpose of the stepping stones is to strengthen the intuitive mental 'stepping' actions from one number to another, referring back to the physical stepping actions children initially took. The stepping-stone numbers are not infinitely small points on a line but real objects of attention.

7. Sinclair, Siegrist and Sinclair (1983) found children inventing similar notation.

8. The imaginative power of children seems to be much greater when a written numeral is present but hidden, rather than hypothetical.

9. Vergnaud (1982) has given a comprehensive account of different types of transfornative addition and subtraction problems.

10. A similar case for multiplication is noted by Mulligan and Mitchelmore (1997).

1 1. The intuitive status of the activities might be questioned in view of the prescriptive nature of the instruction cards. I would answer this criticism as follows. Children receive numbers initially as positions. Sets and formal signs do not emphasise this aspect whereas the stepping stones and arrow signs do. The arrow signs prescribe what to do, but soon become redundant as children gain confidence, until eventually particular kinds of number problems become associated with particular kinds of (mental) actions. For example, perceptive teaching can arrange that examples such as 57 - 54 will be associated with a count-up or 'comparison' strategy, whereas 57 - 4 will be associated with a count-back or 'take-away' strategy.

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Documents

CHILDREN INVENTING SIGNS FOR MENTAL NUMBER STRATEGIES