dynamic imagery in children’s representations of number

Mathematics Education Research Journal 1995, Vol. 7, No. 1, 5-25

Dynamic Imagery in

Children's Representations of Number 1

Noel Thomas Joanne MulliganCharles Sturt University Macquarie University

An exploratory study of 77 high ability Grade 5 and 6 children investigated linksbetween their understanding of the numeration system and their representations ofthe counting sequence 1-100. Analysis of children's explanations, and pictorial andnotational recordings of the numbers 1-100 revealed three dimensions of externalrepresentation—pictorial, ikonic, or notational characteristics—thus providingevidence of creative structural development of the number system, and evidence forthe static or dynamic nature of the internal representation. Our observationsindicated that children used a wide variety of internal images of which about 30%were dynamic internal representations. Children with a high level of understandingof the numeration system showed evidence of both structure and dynamic imageryin their representations.

The development of counting, grouping and notation skills within anorganised base-ten structure is essential for an understanding of the numerationsystem. Yet there is widespread evidence that children lack an organised structurein the formation and representation of numeration. This causes further difficultieswhen dealing with decimal notation and place value in later years (Hiebert &Wearne, 1986; Resnick, Nesher, Leonard, Magone, Omanson & Peled, 1989). Itappears that many children lack base-ten structure because they focus solely onseparate unrelated components of the number system such as counting by ones,grouping in tens, face value of digits and value of place.

Children's representations of number as some form of physical, pictorial, ornotational recording have been exemplified in many studies analysing children'sstructural development of number and understanding of the numeration system(Davis, Maher & Noddings, 1990; Goldin & Herscovics, 1991; Hiebert & Wearne,1992; Hughes, 1986; Pengelly, 1988; Rubin & Russell, 1992; Thomas, 1992). Thomas(1992) reported a wide variety of mental pictures of the number sequence 1 to 100which were used by 40 children in Grades K-4. Although some aspects of structureappeared in the imagery of Grade 2 children, most Grade 4 children had not yetdeveloped the structural flexibility with number to enable them to be successful intheir manipulation of 2-digit numbers.

In a further cross-sectional study of 166 children (K-6) and 79 high abilitychildren (Grades 3-6) it was found that the children's internal representations ofnumbers were highly imagistic, and that their imagistic configurations embodiedstructural development of the number system to widely varying extents, and oftenin unconventional ways (Thomas, Mulligan & Goldin, 1994). Children's active

1. This is a revised and extended version of a paper presented at the 17th AnnualConference of the Mathematics Education Research Group of Australasia, held at theSouthern Cross University, Lismore, 1994.

6 Thomas & Mulligan

processing of internal images was found to be either static or dynamic in nature.The term static is used to describe a fixed representation, and dynamic arepresentation that is changing and moving. In the cross-sectional sample, 3% ofthe children displayed dynamic images of the number sequence whereas 29.9% ofthe high ability children used dynamic images.

Research on the counting sequence has been well documented, with theprimary focus of attention usually on young children (Fuson, 1988; Kamii, 1989;Steffe, 1991; Wright, 1991). It might be thought that asking older children, andparticularly high ability children, to describe their image of counting from 1-100would be somewhat inappropriate. However, results of our preliminaryinvestigations yielded a very much richer data source than we had anticipated.Children's representations of the counting sequence, in conjunction with a range ofnumeration tasks revealed a wealth of information about their understanding ofthe counting sequence, their understanding of the structure of numeration, theorder and magnitude of numbers, and their use of pattern and notation. Moreover,it revealed something of their creative structural mathematical development.

In this paper, we provide an in-depth descriptive analysis of high abilitychildren's dynamic internal representations of the counting sequence 1-100. Weraise the question of how children's representations of particular numbers arelinked with their understanding of the numeration system. We question whetherdynamic and static images are conventional or uniform in nature, and whetherchildren have a range of available images. More broadly, can highly developedrepresentational capabilities influence the way children apply their knowledge inmathematical problem-solving situations, and if so, what implications can bedrawn for instruction?

Background

Construction of the Number SystemBoulton-Lewis (1994) showed that children's levels of counting were

significantly related to their knowledge of, and ability to explain, place value in thefirst three years of school. Research into children's counting strategies (Kamii, 1986;Steffe, 1991; Steffe, Cobb & Richards, 1988; Wright, 1991) and conceptualdevelopment of numeration (Bednarz & Janvier, 1988; Cobb & Wheatley, 1988;Denvir & Brown, 1986a, b; Fuson, 1990; Hiebert & Wearne, 1992; Kamii, 1989; Ross,1990) has highlighted children's construction of number systems. We cannotassume that all children develop a uniform understanding of the Hindu-Arabicbase 10 system. Lean's (1994) work suggests that there are thousands of numbersystems, and that children's development of number concepts must clearly belinked to culture and language.

Rubin and Russell (1992) asserted that children's counting, grouping,estimating and notating skills are essential elements in developing numberstructure. They described these elements in terms of "landmarks in the numbersystem." These landmarks appear to be related to additive structure, multiplicativestructure, the generation and analysis of mathematical patterns and mathematical

Dynamic Imagery in Children's Representations of Number 7

definitions. Rubin and Russell (1992) suggested that people who are adept withnumber operations—for example computing, comparing, and estimating—have anon-uniform view of the whole number system.

Hiebert and Wearne (1992) investigated children's structural development ofnumeration. They described children's understanding of numeration as "buildingconnections between key ideas of place value such as quantifying sets of objects bygrouping by ten, treating the groups as units ... and using the structure of thewritten notation to capture the information about grouping" (p. 99). Other studieshave focused on structural aspects of numeration, identifying the child's ability togroup and re-group composite units of ten and relate this to a general structure ofthe base-ten system (Cobb & Wheatley, 1988; Kamii, 1989).

The Role of Imagery in RepresentationThe role of imagery in the representation of mathematical ideas has been

described by a number of researchers (Bishop, 1989). For example, Hershkowitzand Markovitz (1992) emphasised the importance of visualisation of mathematicalconcepts and the development of advanced visual thinking. Hershkowitz (1993)and Bobis (1993) investigated the role of visualisation in estimation of number.Hershkowitz showed that visual imagery played a vital role in the process whichchildren used when performing numerical tasks, especially in problem solving.Bobis found that, with practice, Kindergarten children were able to use visualisingstrategies to combine and separate patterns mentally.

Recent work (Brown & Presmeg, 1993; Brown & Wheatley, 1990; Presmeg, 1986,1992) in which students' thinking was probed in clinical interviews indicated thatstudents use imagery in the construction of mathematical meaning. Brown andPresmeg (1993) asserted that learning frequently involves the use of imagery and itmust include very abstract and vague forms of imagery. Presmeg (1986) identifiedfive types of visual imagery used by students: (a) concrete, pictorial imagery(pictures in the mind); (b) pattern imagery (pure relationships depicted in a visual-spatial scheme); (c) memory images of formulae; (d) kinaesthetic imagery(involving muscular activity such as fingers "walking"); and (e) dynamic (moving)imagery involving the transformation of concrete visual images.

Recent findings have revealed wide differences in the types and facility ofimagery used by students in problem solving. Owens and Clements (1993)suggested that the process of visualising is important "in the sense that itinfluences the idiosyncratic construction of meaning" (p. 1). Students with agreater relational understanding of mathematics tended to use more abstract formsof imagery such as dynamic and pattern imagery while students with lessrelational understanding tended to rely on concrete and memory images (Brown &Presmeg, 1993). Further, imagistic processes are extremely important in theproblem-solving process and have a role "in establishing the meaning of theproblem, in channelling the problem-solving approaches of the students, and ininfluencing students' cognitive constructions" (Owens, 1994, p. 11). In this paperwe draw upon Brown and Presmeg's notion of a dynamic image in the context ofhigh-ability children's imagistic representations of the number sequence.

8 Thomas & Mulligan

Dynamic and Static RepresentationsMason (1992) suggested that images can be viewed as either eidetic (fully

formed from something that is presented), or constructed (built up from otherimages). The process of constructing meaning continues as the mental picture isdescribed, drawn, compared and discussed. He asserted that it is not just thecreation of a mental image but the use of that image as a mental "screen" in visualthinking that constitutes mathematics learning. He questioned the relationshipbetween the relative accessibility of static and dynamic images and questionedwhat makes some images enduring and others transitory. Mason further suggestedthat, in order for students to access images, they need to be active in processing theimages. In other words, learners should be "looking through" rather than "lookingat" the "mental screen," regardless of the mode of external representation. In thispaper we extend Mason's notion to analyse children's active processing of internalimages as static or dynamic in nature.

Representational SystemsWe distinguish carefully between external representations (a structured

environment with which the child is interacting that may include, for example,actual physical objects to manipulate and actions in response to that environment),and internal imagistic representations (a theoretical construct to describe thechild's inner, cognitive processing). Goldin's model of competence in mathematicalproblem solving is based on the idea of cognitive representational systems internalto problem solvers, as distinct from (external) task variables and task structures(Goldin & McClintock, 1979). We consider three of the five types of internalrepresentational systems discussed by Goldin: (a) verbal/syntactic systems (usingmathematical vocabulary, developing precision of language, self-reflectivedescriptions); (b) imagistic systems (non-verbal, non-notational representations,such as visual or kinaesthetic); and (c) formal notational systems (using notation,relating notation to conceptual understanding, creating new notations). Thesesystems develop over time through three stages of construction: (a) inventive/semiotic, in which characters in a new system are first given meaning in relation topreviously-constructed representations; (b) structural development, where the newsystem is "driven" in its development by a previously existing system; and (c)autonomous, where the new system of representation can function independentlyof its precursor.

The analysis in this paper is based on Goldin's (1987a, b; 1988) model ofproblem-solving competency structures because our data consistently reflectedaspects of internal representation previously described by Goldin. Goldin's modelalso contained features which are helpful in describing the development ofchildren's conceptual understanding of numeration (Goldin & Herscovics, 1991).Further, we analyse the role of imagery in representation and in the construction ofrelational understanding in mathematics (Brown & Presmeg, 1993; Brown &Wheatley, 1990; Mason, 1992; Presmeg, 1986, 1992).


MethodThis study is an exploratory investigation which aims to describe children's

representations in as much detail as possible. Data obtained from a sample of high-ability children provide a rich source of creative imagery. This data will becompared and contrasted with that obtained in earlier studies which used a crosssectional sample of children with a range of abilities (Thomas, 1992; Thomas,Mulligan & Goldin, 1994).

SampleThe sample consisted of 77 high ability children from Grades 5 and 6, assessed

by teachers for participation in a program for gifted and talented students from 58country and city State and Independent schools in New South Wales. The childrenwere participants in a mathematics enrichment program conducted by the one ofthe authors (Thomas) over five 2-day workshops over a three-year period. Theworkshops were conducted on weekends in May 1992, February, July and October,1993, and May 1994. There were 31 girls and 46 boys in total who attended theworkshops. All students were selected by their classroom teachers on the basis oftheir high achievement in school mathematics. Students were interested inparticipating in the workshops and displayed positive attitudes towardsmathematics learning.

Task-Based InterviewsIn order to describe individual representations of number in as much detail as

possible, the study employed structured, task-based interviews based on themethod used by Goldin (1993) and Davis and Maher (1993). We inferred thediverse concepts, structures and/or processes that occur inside children's mindsfrom the observation, classification and description of the various kinds ofbehaviour displayed. This method is valuable because it provides a means ofinvestigating the interplay between the children's (internal) representations andexternal representations that are constructed during the interviews.

Interview ProceduresThe children were interviewed individually on the visualisation task,

questioned verbally about their attitudes towards learning mathematics, and givena group-administered written test on eight numeration tasks. The countingsequence task was asked prior to the numeration tasks so that responses could notbe influenced by experiences with these questions. A selected set of six numerationtasks was administered to each group of children at each workshop. Each groupcomprised approximately 15-16 children who were seated individually in arandom pattern. The children were provided with paper and pencils, and told thatthey could record their thinking and their responses if they chose to do so. For thevisualisation task, each child was encouraged to draw or explain in writing his orher mental image.

A follow-up interview was conducted with each child by one of the authors

10 Thomas & Mulligan

(Thomas) during the first day of the workshop series to allow the child to explaintheir responses to the visualisation task. The interviewer recorded the children'sresponses as they were asked to explain how they solved the visualisation task. Theinterviewer recorded how each child explained his or her drawings and recordingsand noted any gestures and finger movements. When a response was unclear,additional probing was used by the interviewer. A standard set of initial andfollow-up questions were used without prompting or helping the child. If a childgave an explanation at interview which conflicted with his or her drawing orwritten explanation, then the child was asked to describe what he or she wasthinking when the recording was made. If the child did not show any array-typestructural features in what was recorded, the researcher asked the child to think ofthe numbers from 1-100 in rows and columns, and then to draw or describe thatimage. Interviews lasted approximately ten minutes. All children attempted toexplain their thinking and no interviews were terminated.

Interview TasksThe numeration tasks and the visualisation task were selected from the 92

numeration tasks asked of children from the cross-sectional sample. We designedtasks which encouraged explanation of visualisations and problem-solvingprocesses used by the children.

Counting Sequence TaskThe children were asked to close their eyes and to imagine the numbers from 1

to 100, and then asked to draw whatever they saw in their minds. They were askedto explain the image and their drawing in writing.

Numeration Tasks1. Re-naming hundreds as thousands

Write the number which is given by 2 thousand, 16 hundreds, 1 ten and 4ones;

2. Re-naming tens as hundredsWrite the number which is given by 6 hundreds, 14 tens and 3 ones;

3. Renaming hundreds as tensWhat is the missing number in the following statement?

2 643 is 2 thousand, hundreds, 14 tens and 3 ones;

4. Digit correspondenceThe picture shows twenty-six sticks in bundles of four.

How many sticks would you take to show the meaning of the part of thenumber which is circled?


5. Subtraction where minuend involves zerosWrite the number which is 16 less than half a million.

6. Renaming ten thousands as tensHow many tens are there altogether in 3042?

7. Naming of the ten thousand place5214 The 2 stands for 2 hundreds521 400 The 2 stands for 2 ...

8. Count which takes a number to the next hundredThis meter counts the people going into a football stand.

6399After one more person has gone in, it will read:

The digit correspondence task was adapted from that used by Ross (1989). Thenaming the ten thousand place and the count-on one questions were taken fromBrown (1981). The other questions were designed to extend the probing to higher-level applications of place-value knowledge.

Analysis of External RepresentationsThe interview transcripts and the pictorial and notational recordings of the

counting sequence 1-100 task for the 77 high-ability Grade 5-6 children wereanalysed. The external representations were categorised and coded according tothree dimensions: (a) the type of imagistic representation identified by the pictorial,ikonic and notational recordings; (b) the level of creative structural development ofthe number system, and (c) evidence of a static or dynamic nature of the image.Table 1 describes the classifications of representation for each of these dimensionsby mode, type of structure, and nature of image. If more than one mode ofrecording for the imagery was observed, it was classified according to the mostsophisticated mode.

The static or dynamic nature of the image was defined according to whetherthe recordings and the children's explanations of their representations describedfixed or moving (or changing) entities. We discuss features of the dynamicexamples of these representations making reference to the theoretical perspectivedescribed earlier.

Results

Table 2 indicates that 29.9% of the children displayed dynamic images of thenumber sequence, with 87% of these dynamic images being notational, althoughseveral children saw the numerals in association with pictures. There was verylittle difference between the distribution of the mode of representation or the typeof structure for this sample of high achievers compared with that of the cross-


Table 1Classification of Representations by Mode, Structure,and Nature of Image

Mode of RepresentationConcrete/pictorial—imagery where the child draws or describes objects which do nothave any quantitative relationship to the numbers.

Ikonic—pictorial imagery which relates to a quantity.

Notational—conventional numerals are used to represent numbers.

Type of StructureNo structure—objects, pictures or numerals which have no apparent relationship to equalgroupings or sequence.

Linear structure—a linear formation (straight or curved), numbers appearing in sequence.

Emerging structure—one hundred represented by equal groups of objects, or linearsequence broken into equal segments.

Emerging structure—related in some way to multiplication including multiple count andmultiplication grid.

Partial array structure—objects, pictures or numerals in rows and columns but not apattern of ten tens.

Array structure—objects, pictures or numerals in a pattern of ten tens

Nature of the ImageStatic—the image as a fixed representation.

Dynamic—the image as a representation that is changing and/or moving.

Table 2Percentage of Imagistic Representations by Mode and Structure (N = 77)

Representation % Responses

Mode of RepresentationConcrete/pictorial

Ikonic

Notational

3.9

6.5

89.6

Type of StructureNo structure 19.5

Linear structure 27.3

Emerging structure— groups of ten objects, linear broken into tens 3.9

Emerging structure—multiple count, multiplication grid 5.2

Partial array structure—not a pattern of ten tens 10.4

Array structure—pattern of ten tens 33.8

Nature of the ImageStatic 70.1

Dynamic 29.9


sectional sample. There were however, many more dynamic and creative, non-conventional images among the high achievers. Also, when children wereprompted to think of the numbers in rows and columns all but two gave somearray-type structure. Examples of dynamic images included groups of numeralsmoving around, numerals going past in order, numerals appearing one at a time ona moving screen, sequences of numerals in line appearing and then being replacedby other sequences, and whole arrays of numerals moving. Evidence of a widerange of highly structural imagistic representations for the children's numerationsystems were found (80.5% of children showed structure of some sort). In somecases these representations revealed idiosyncratic, highly individualistic images.Although initial images may not have produced the conventional array structure,when children were asked to think of the numbers in rows and columns, 97.4%drew pictures of arrays.

Table 3 indicates that children with dynamic imagery performed better onnumeration tasks than those with static imagery. The numeration skills of our high-achieving sample on the "naming the ten thousand place" and the "count to thenext hundred" tasks were very much higher than the results reported by Brown(1981) in the large CSMS study. The item facility of each task for 12-year-oldsreported in the CSMS sample which was representative of all English children wasgiven as 22% and 68% respectively. This observation is consistent with the specialnature of our sample which was based on a selection of high-achieving children.

Table 3Percentage of Imagistic Representations by Mode and Structure (N = 77)

Representation

0/0

ResponsesDynamicN= 23

ResponsesStaticN = 54

ResponsesTotalN=77

Renaming hundreds as thousands 52 41 44

Renaming tens as hundreds 65 56 58

Renaming hundreds as tens 74 52 58

Digit correspondence 91 74 79

Subtraction where minuend involves zeros 74 57 62

Renaming ten thousands as tens 91 72 78

Naming of the ten thousand place 91 93 92

Count to the next hundred 96 94 95

The discussion which follows will focus on the dynamic and structuralcharacteristics of the children's representations.

Linear structureFigure 1 shows Clint's initial response to the visualisation task where "the

numbers were moving around like people" while Figure 2 shows his response tothe prompt to think of the numbers in rows and columns. Evidence of linearstructure was demonstrated in Clint's external imagery. His initial response

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showed the linear sequence in a highly creative context and his prompted responsedemonstrated his conventional experiences with arrays but in a creative wayshowing a separation of odd and even numbers. Clint gave incorrect answers tothe first two renaming questions but was readily able to give the number which is16 less than half a million and the total number of tens in 3042. David (Grade 6)described a pictorial representation of groupings of objects, "... as I thought,everything suddenly became more numerous ... like first there was one ofeverything, then there was two of everything and then there were three trees, suns,cats, dogs, people and it kept on going." Figure 3 shows the 6 by 6 grid withnumerals following a Fibonacci pattern that David drew when prompted to thinkof numbers in rows and columns. David's initial imagery focused on the cardinalaspect of the numbers as they appeared in sequence and the structure appearing inthe array was creative and individualistic. There was no evidence of a numerationstructure based on groupings of tens. David gave incorrect answers for the first tworenaming questions, gave 499 884 as the number 16 less than half a million and3040.2 as the numbers of tens in 3042.

Figure 1. Clint (Grade 5)

Rosalie (Grade 6) described "a screen moving one number at a time" (Figure 4).When asked to think of the numbers in rows and columns Rosalie drew a picture ofa 10 by 10 grid with the numerals 1 to 100 in rows of ten. The initial visualisationwas dynamic but this did not suggest advanced structural development. Whenlater prompted to think in rows and columns Rosalie produced a conventionalstatic array image for the number sequence. From this we infer that she isdeveloping an autonomous structure of numeration. Rosalie answered allnumeration questions correctly.

Colin (Grade 5) described the numbers moving along a wave-like line (Figure5) and Christopher (Grade 5) explained "I saw all the numbers from 1 to 100beaming at me and lighting up like neon signs ... the numbers did this in order thendisappeared when I opened my eyes ... they were moving around ... then theywould flash. ... Once they had done that they disappeared." Figures 6 and 7 show

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Figure 2. Clint (Grade 5) Figure 3. David (Grade 6)

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the later prompted responses from Colin and Christopher. Colin started at thebottom left corner of the grid and then filled in diagonals of increasing size movingup to the top right corner and Christopher displayed the numbers backwards from100 to 1 in rows of ten. Both boys gave dynamic linear imagery for the initialrepresentations and filled in the ten-by-ten grid in non-conventional ways. Colinand Christopher answered all numeration questions correctly showing soundunderstanding of numeration.

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Figure 5. Colin (Grade 5)

Tessa (Grade 5) described the numbers "walking round in circles ... circlesgetting bigger and bigger" and when prompted to think of the numbers in rowsand columns, described them "like soldiers marching into rows with ten as thecommander" (Figure 8). When prompted, Tess readily changed her linear structureto dynamic imagery with a tens structure. All renaming questions were answeredcorrectly but Tessa gave 484000 as the number 16 less than half a million and 6700as the number one more than 6399.

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Figure 6. Colin (Grade 5)

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Figure 8. Tessa (Grade 5)


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The drawing produced by Tim (Figure 9) did not reveal anything about hisstructure of the number system. He explained his image as "3-D numbers flyingacross in front of my eyes in order ... of 1 first ... and 100 last" and this was classifiedas a linear notational image. When Tim was asked to close his eyes and think of thenumbers in rows and columns he drew a conventional grid with the numbers 1 to100 in rows of ten. He answered all numeration questions correctly except for thenumber 16 less than half a million (9999984).

Figure 9. Tim (Grade 6)


Both Tessa and Tim showed a developing structure of numeration which is notapplied consistently in the use of large numbers. Although they displayed aflexibility to think of the numbers in ten tens structure, operated successfully withnumbers up to one thousand and could rename numbers based on noncanonicalrepresentations, they did not make the connections which would have enabledthem to work consistently with large numbers.

Emerging structureMichelle (Figure 10) gave an array structure for her initial visualisation of the

numbers 1 to 100. She described the columns of six as "moving up and down" suchthat she "was getting dizzy." Michelle's emerging structure was the sequence ofnumbers separated into columns of six. She gave incorrect answers for the threerenaming questions, and subtracted 16 from five thousand instead of from half amillion.

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Figure 10. Michelle (Grade 6)

Leah saw the sequence of the multiples of ten presented in two columns. Thischanged to a sequence counting by tens from five. Other sequences were thenobtained by starting from all other one-digit numbers. Leah drew a picture of a treeand explained that "without all the features of a tree, there would be no leaves,trunks, branches, etc ... without numbers no hundreds." The developing auto-nomous nature of Leah's numeration structure was reflected by her understandingof the importance of tens and hundreds in the numeration system and by hersuccess on all numeration tasks.

Array StructureFigures 12 and 13 show drawings produced by Renee and Ben. Renee (Grade 5)

initially recorded a standard array which was as a "board ... moving to the right."Ben explained that he "saw ten rows of numbers each containing the set of tennumbers that come next in the sequence. ... The rows move along to be replaced bythe next few." Renee gave 340 as the number of tens in 3042 and 2 thousands as thevalue of the 2 in 521 400. Ben did not rename in order to calculate the correct

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Figure 11. Leah (Grade 6)

numeral for each position in the numbers in the first three questions but simplyread the numerals in order of presentation. Renee and Ben gave dynamic images ofthe number sequence which were otherwise conventional, and this reflected theirclassroom experiences. Their responses to the numeration tasks indicated that theywere not able to apply numeration structures consistently.

Figure 12. Renee (Grade 5)

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Figure 13. Ben (Grade 5)

In contrast Edward (Figure 14) who described the numbers as "floating downin rows" showed a tens pattern in this initial visualisation and also gave correctresponses to all 8 numeration tasks.

Dario (Figure 15) gave an array structure for his initial visualisation of thenumbers 1 to 100. Dario described how he "first saw all the whole numbers ... then... all the fractions all in a line doubling, tripling." Dario's imagery changed from astandard tens array to a further tens array based on tenths. The only error whichDario made on the numeration tasks was with the third renaming question.

No Structure

Michael (Figures 16) and Joel (Figure 17) both produced an apparently randomcluster of numerals moving all over the place, although Joel showed his in thecontext of a road. When both children were asked to close their eyes again and


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think of the numbers in rows and columns, Michael (Figure 18) showed nostructure whereas Joel (Figure 19) produced columns for each decade as troopsmarching behind the general, giving one hundred altogether. Michael did not givecorrect responses to the first two renaming questions and calculated the number oftens in 3042 by adding 300 and 40 together to give 340. Joel demonstrated a higherlevel of understanding of the numeration system than Michael through hisawareness of the ten tens structure in one hundred and his success with thenumeration tasks.

Figure 20 shows that Keryn (Grade 5) used no structure in either the initial orprompted visualisations of the numbers 1 to 100. For the initial visualisation, sheexplained that "they keep changing from plus to times." When Keryn was giventhe rows and column prompt she placed the numerals in a grid like formation andthen ruled some grid lines but did not really demonstrate a full understanding ofthe grid formation. Keryn could not rename hundreds to thousands or tens tohundreds and wrote down one million correctly but then attempted to subtract 16from 5000 (instead of half a million). Through their lack of structure whenprompted to think of rows and columns, and their problems with the numerationtasks, Keryn and Michael demonstrated that they had not developed a relationalunderstanding of the numeration system.


Figure 16. Michael (Grade *5) Figure 17. Joel (Grade 5)

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All examples of representations reported here were made by children identifiedby their teachers as high-achievers in mathematics and with attitudes reflectingself-confidence in their abilities. Although some of spontaneous dynamic imagerydid not reveal structure, when the children were prompted, a range of examples of .

structure appeared; these were often highly creative. These children showedevidence of having access to a variety of internal images and of developing arelational understanding of the numeration system.

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Figure 20. Keryn (Grade 5)

Discussion

Conclusions and LimitationsThis investigation raises several questions about the role of imagery in

children's representation of number. We conjecture that children's internalrepresentation of number gives clues about their structural development of thenumeration system. Our data show that a dynamic external representation issomething which stands for motion.

We would ultimately like to be able to describe in detail children's internalrepresentations of the numeration system, and how these representations develop.From the external representations produced by the high-ability children, we haveattempted to infer dynamic aspects of their internal representation, and from thisto infer a description of the structural development of the system that has takenplace. We found, in general, a wider diversity of representations of the countingsequence 1-100 than might have been expected and a higher percentage of dynamicimagery than for the average/lower ability children observed in the earlier studies.

We have described the children's responses to the visualisation tasks aspresented to them. Those children with dynamic visualisations were shown tohave higher achievement on the numeration tasks than those with staticvisualisations (Table 3). What might be important here is the unconventional,creative nature of the visualisations rather than whether the image is dynamic orstatic. It should be noted that other responses might have occurred if the childrenhad been prompted in other ways, for example, to imagine moving numbers, or togroup the numbers in tens. Also, other representations may have been available tothe children, with just one of several possible internal image configurations havingbeen selected for recording or elaboration. Thus we are probably gaining only apartial description of each child's internal representational capabilities.

Many questions have been raised by the data described in this study. Forexample, why do some children seem to be able to visualise the counting sequencespontaneously and in a dynamic way? Can static external representations represent("carry the meaning of") dynamic internal representations? Further research isneeded to shed light on how children construct their personal numeration systems,and how they structure them over time. On the basis of our study, we hypothesise


that the more developed the structure of a child's internal representational systemfor the counting numbers (for example, kinaesthetic, auditory, or visual/spatialrepresentation of the counting sequence that embodies grouping-by-tens), themore coherent and well-organised will be the child's externally-producedrepresentations, and the wider will be the range of numerical understandings. Weaim to undertake further longitudinal investigations into whether children whohave access to several forms of imagery with which to represent their internalstructures, will be more capable of developing a relational understanding of thenumeration system.

Implications for the ClassroomFurther research should develop our understanding of how children represent

meaning in the structure of the numeration system. If a lack of structure existsbecause children focus solely on separate—and to them unrelated—components ofthe number system then the encouragement, use and discussion of imagery mayassist in helping them to develop an understanding of the system of numeration. Ateacher needs to know the framework of thinking of each child when he or she isworking with number. As children develop a framework of thinking andknowledge in the form of mental representations, the teacher needs to probe theserepresentations. Maher, Davis and Alsatian (1992) pointed out the complexity facedby teachers in trying to identify children's representations. The process of buildingup a mental image is an "extremely arduous task" (p. 260) for any learner. At timesit involves shutting out most incoming signals and engaging in deep thought.Because of the value and fragility of a mental representation it is essential that theteacher recognises the imagery so that further development of understanding canoccur. If a child's representation of the number sequence reflects flexibility andstructure, the teacher needs to build on and strengthen the relationalunderstanding, but if the representation is fixed and structure is absent, then theteacher needs to help the child to develop the basic components and relationshipsof the numeration system.

The extension of our knowledge about children's representations will haveimplications for:

• designing alternative teaching strategies focusing on representation andimagery;

• determining an appropriate curriculum scope and sequence; and• suggesting assessment strategies that might identify landmarks in the

development of understanding the structure of the numeration system.

Learning approaches need to be organised which focus on children's internalrepresentations and understandings rather than on teacher- or textbook-constructed representations.

AcknowledgementThe authors wish to acknowledge the value of discussions with Gerald Goldin

of Rutgers University on his model of problem-solving competency structures andits implications for children's visualisations of the number system.


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AuthorsNoel Thomas, School ofWales 2795, Australia.

Joanne Mulligan, SchoolWales 2109, Australia.

Education, Charles Sturt University, Bathurst, New South

of Education, Macquarie University, Sydney, New South

dynamic imagery in children’s representations of number

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