Mathematics Education Research Journal, Vo12, No.1, 1990.

LANGUAGE AND COUNTING: SOME RECENT RESULTS

Garry Bell, University of New England, Northern Rivers

It has long been recognised that the language ofmathematics isan important variable in the learning ofmathematics, and therehas been useful work in isolating and describing the linkage.Steffe and his co-workers at Georgia, for example, (Steffe, vonGlasersfeld, Richardson and Cobb, 1983) have suggested thatyoung children may construct verbal countable items to countobjects which are hidden from their view. Although there hasbeen a surge ofresearch interest in counting and early childhoodmathematics, and in cultural differences in mathematicsattainment, there has been little work reported on the linkagebetween culture as exemplified by language, and initial conceptsof numeration. This paper reports on some recent clinicalresearch with kindergarten children of European and Asianbackground in Australia and America. The research examinesthe influence that number naming grammar appears to have onyoung children's understandings oftwo-digit numbers and placevalue. It appears that Transparent Standard Number WordSequences such as Japanese, Chinese and Vietnamese whichfollow the numerical representation pattern by naming tens andunits in order ("two tens three"), may be associated withdistinctive place value concepts which may support sophisticatedmental algorithms.

The Conservation DilemmaThe stark, predictable simplicity of the Piagetian conservation of number

task is a persuasive paradigm. If two equally numerous rows of counters aredisplayed in a matched regular array, and a 4-6 year-old child is asked to countand compare the rows, the chances are that the child will be able to state equality.If one row of the display is then stretched out without changing its numerosityand the child is again asked to compare, the chances are that the child will nowsay that the stretched row has more. Moreover, the child is likely to maintain herconviction of difference even if the differently spaced rows are subsequently

re-counted. The anomaly led Piaget (1952, p.45) to dismiss counting as a poorindicator of numerical and operational development.

'Mis apparent deficit in numerical knowledge implied by the conservationtask has evoked a voluminous literature, both replication and commentary.Donaldson (1978) criticised the level of language used in the task. Markman(1979) noted that children's performance was better if collective nouns (forest,army) were used as against singular nouns (trees, soldiers). Hudson (1983) usedbirds and worms as counters, asked the child "How many birds won't get aworm`?", and found that children presented with this new format clearlyoutperformed those on the standard format. Estes (1979) pointed out that theword "more" may have different collocations in the field of children's discoursefrom those of mathematics researchera, Asked to choose between "more" and"less", the young child prefers "more" with greater frequency than chance wouldlead us to expect. ("Do you want more or less ice creamT")

Through the 1970's, a major focus was on conservation training, for itseemed that Piaget's conservation theory, with its implied developmental deficit,would be called into question if children could be trained to conserve. But aplethora of successful conservation training experiments (e.g., Smither, Smiley& Rees, 1974; Botvin & Murray, 1975; Field, 1981; Golomb & Bonen, 1981;Acredolo, 1980) seemed merely to balance studies (e.g., Benziger, 1970;Schenck & Canaday, 1974) which supported the potion of some kind ofdevelopmental cognitive barnier.

Research Emphasis on CountingThese inconsistent results, and other results (Pennington & Wallach, 1980)

and comment (e.g., Clements & Callahan, 1983) which question the necessity ofconservation for subsequens numerical development, prompted major researchinterest in counting as it relates to number conceptualisation throughout the1980's. Most notable among these research initiatives were those of Saxe, 1980;Ginsburg, 1982; Fuson, Richards and Briars, 1982; Resnick, 1983; Steffe, vonGlasersfeld, Richards and Cobb, 1983, and Steffe and Cobb 1988; and Fuson,1988.

Steffe and his colleagues (1983) worked from a constructivist position andtried to identify the conceptaal structures that children leem to develop for use insolving simple numerical problems. Initially, children's preferred strategy iscounting, but eventually, most can operate on abstract numeric entities to makebasic combination facts meaningful, and it was this progression which theGeorgia workers attempted to clarify. `l'hey used a clinical problem solvingparadigm which required the children to solve simple addition and subtraction

Language and Counting 3

tasks on dot arrays, part of which were hidden from view. So, the child mightbe shown an array of 5 dots, told that there were seven more dots concealedunder a cover, and asked to find the total. The children's responses seemed tosuggest three major stages of development which Steffe called Perceptual,Figural and Abstract Counting Types. Perceptual counters could not find thetotal when one addend was hidden, apparently needing perceptual material onwhich to superimpose their counting scheme. Figural counters were able to forma total by devising visual, motor or subvocal images of concealed arrays andcounting the images. Abstract counters appeared to have reached the significantconclusion that numbers do not necessarily have to be constituted by counting,and were able to use this meaning structure to, for example, count on from thehidden array. The work has subsequently been extended (Steffe & Cobb, 1988)to explain more sophisticated part/whole concepts.

While these initiatives have made a significant contribution ·to ourknowledge of young children's numerical understandings and capabilities, itmust be said that they all seem to take the Standard Number Word Sequence(SNWS) as an invariant. This assumption seems contrary to the advice of asuccession of writers including Laney (1983) and Bishop (1988), who advocatecultural explanations of mathematical cognition. From the latter viewpoint, itappears that there has been a failure to consider the SNWS as a cultural variablewhich may influence the numerical understandings of children within a particularlinguistic context. Linguistically, the SNWS is a grammatical system in whichwell formed elements are generated by ordered concatenation of the base digitsequence ("one, two, .... nine") on units of different value (tens and ones).These compounded elements are mathematically isomorphic to the numericrepresentations of numbers, which concatenate positionally after subsequences of10 and powers of 10. It seems reasonable to suggest that in cases where thegrammatical structure of the SNWS maintains linguistic as well as mathematicalisomorphism, the child might come to conceptualise the number system as asystem of decimal concatenations.

SNWS Structure and Number ConceptsConversely, if a culturally specified SNWS contains linguistic

discontinuities which impede fluent recitation, or obscure decimal repetition, orconstrain symbolic representation, there may be significant effects on the

subsequent development of numerical cognition of children from that culture.There is ample evidence of cross-cultural differences in attainment on

arithmetic tasks (Mcknight, et al., 1987, p.lx), but it is never easy to isolate

variables in cross-cultural research. As McKnight and his colleagues point out,observed differences may, indeed, emanate from structural differences betweeneducational systems rather than from linguistic influences.

The SNWS's of different languages are clearly different (refer Figure 1),although languages of similar origin (Germanic, or Romance) exhibit similarstructures. The "teen" rule, for example, emerges after 12 in German andEnglish, whereas it does not emerge until 17 in French and Italian. Onedimension of difference derives from the presence of what Thompson (1982)calls "homonymic transformations" which specify modifications to certainelements - "three" to "thir" in English, or "harom" to "harminc" in Hungarian.Another dimension of difference derives from the combinatorial roles, as, forexample, the pre-war French for 99 ("quatre vingt dix neuf '), which combinesfour twenties, ten and nine. Many Asian SNWS's such as Vietnamese, Mandarinand Japanese, are so structured as to preserve linguistic and mathematicalisomorphism, and allow us to speak of "transparent" number naming grammars.

ENGLISH VIETNAMESE MANDARM JAPANESE HUNGARIAN

one mot yi ichi egytwo hai er ni kettothree ba san san haromfour bon si shi negyfive nam wu go otsix sau liu roku hatseven bay qi shichi heteight tam ba hachi nyolcnine chin jul ku kilencten muoi shi ju tizeleven muoi mot shi yi juichi tizenegytwelve muoi hai shi er juni tizenketthirteen muoi ba shi san jusan tizenharomfourteen muoi bon shi si jushi tizennegyfifteen muoi nam shi wu jugo tizenotsixteen muoi sau shi Iiu juroku tizenhat

twenty three hai muoi ba er shi san niju san huszonharom

thirty seven ba muoi bay san shi qi sanju shichi harminchet

forty bon muoi si shi shiju negyvenf i ft y nam muoi wu shi goju otvensixty sau muoi Iiu shi rokoju hatvanseventy bay muoi qi shi shichiju hetveneighty tam muoi ba shi hachiju nyolcvanninety chin muoi jul shi kuju kilencven

Figure 1. Number naming grammars.

n guage and Counting 5

In addition to the evidence of cultural differences on numerical tanks thereis evidence that young children can attend to the structure of utterances and draw

meaning from that structure (McKoon & Ratcliffe, 1979), and that nursery schoolchildren are able to produce more correct recall when stimuli are linked with asyntactically lawful language (Moeser & Olsen, 1974). Results such as thesehave led the author (Bell, 1988) and another worker in the United States (Miura,1987) separately to investigate the role of Asian number naming grammars in thechild's numerical development.

The American StudyMiura and her colleagues (Miura, Kim, Chang & Okamoto, 1988) studied

the number conceptualisations of five groups of first-graders from the UnitedStates, Peoples' Republic of China, Japan and Korea. The average age rangelfrom 71 months for the Korean kindergarteners to 85 months for the Koreanfirst-graders, and the children were interviewed individually, with the procedurebeing translated into their own language. The children were shown theequivalence between a base 10 long and 10 shorts, and were then asked to read atwo-digit numeral on a card, and to show the number using some of the 100shorts and 20 longs available. Coaching was permitted on two practice itemsbefore the test items (11, 13, 28, 30 and 42) were presented in random order.Immediately after the first Trial, the children were reminded of the equivalencebetween a long and 10 shorts, shown their first constructions and asked if theycould show the number in a different way. Responses were classified as one-to-one (for example, using 24 shorts for 24), canonical (using 2 longs and 4shorts for 24), or noncanonical (using 1 long and 14 shorts for 24). All childrenin the study could read the numerals correctly.

Figure 2 shows the percentages of correct representations in each category.Significantly (p<.001), the U.S. group preferred to use unit constructions,whereas the other groups (except Korean kindergarteners) preferred canonicalconstructions. More generally, all of the Korean kindergarteners, 98%® of Koreanfirst-graders, 76% of those in the Chinese sample, and 75% of the Japanesesample were able to show all five numbers correctly in two ways. Only 13% ofthe U.S. children could do so; half of the U.S. sample could not make twoconstructions for any of the numbers. Across the two trials, 84%® of the childrenin the P.R.C., 67%® of those in Japan, 75%® of Korean first-graders and 60%® ofKorean kindergarteners used a canonical representation to construct all fivenumbers; while only 8%® of the U.S. children did so. One half of the U.S.children used no canonical constructions at all.

U.S. P.R.C. JAPAN KOREA 1 KOREA K(N=24) (N=25) (N=24) (N=40) (N =20)

TRIAL 1One-to-one collection .91 .10 .18 .06 .59Canonical base 10 .08 .81 .72 .83 .34Noncanonical base 10 .01 .09 .10 .11 .07

TRIAL 2One-to-one collection .10 .43 .59 .56 .33Canonical base 10 .71 .16 .12 .09 .48Noncanonical base 10 .19 .41 .29 .35 .19

Figure 2. Percent of correct representations in each category for the five groups.(after Miura, et al., 1988).

The evidence was not conclusive, but Miura concluded that althoughsocialisation could account for some of the cultural differences in mathematicalattainment, there also appeared to be differences in the basic mental representationof numbers as they are linguistically represented. These resuits were generallysupported by the present study.

This report details results found when working with Vietnamese children inAustralia. 'Me author compared the numerical development of 4 monolingualEnglish-speaking (ME) children (mean age 5.8) and 4 bilingual Vietnamese-speaking (BVE) children (mean age 6.2) in a six month teaching experiment. Thechildren were in the same class in their first year of schooling, and were given 15individual weekly instructional sessions using base-10 materials on simpleadditive and subtractive tasks. A difference in treatment was that the BVEchildren used the transparent Vietnamese SNWS in all teaching sessions, whereasthe ME children used the English SNWS exclusively. The sessions emphasisedthe development of SNWS skills and the application of these to simple additionand subtraction tasks. As an example, in an investigation of the Decade NumberWord Sequence (DNWS) the investigator might hide 3 bundles of 10 sticks undera cloth, and display 2 bundles and 5. The instruction to a Vietnamese childwould then be "There°s hai muoi nam there (points to 25), and ba 'mooi there(points to cloth). flow many altogether?"

Following the Theory of Counting Types developed by Steffe and his co-workers at Georgia (Steffe, et al,, 1983), the teaching sessions also emphasisedthe development of figurative and abstract counting schemes by systematicallyrequiring each child to construct and operate on images of hidden countable

Language and Counting 7

items. These terms refer to the observed capacity of children to mentally re-present patterns or images (or, in the case of abstract units, implied counts) anduse these as countable unit items in numerical problem solving.

Steffe and von Glasersfeld (1983) put forward a framework, assummarised below, of conceptual structuren for the unit "10" which children maybe able to draw upon to operate in a numerical context such as this.

It must be emphasised that these manifestations are indicators of meaning,rather than simple behavioural definitions. In Steffe's framework, meaning isindicated by actions or intentions to act, so that behavioural manifestations maycorrespond, in Steffe's view, to significant changes in the numerical meaningstructure of the child. The point is perhaps best illustrated by Steffe's owndescription (Steffe & Cobb, 1988, p.6) of the realisations which cumulativelymay result in the child's construction of abstract unit items:

Before the creation of abstract unit items, the child's counting is stildependent on sensory-motor material, either perceptual orkinesthetic. An important development occurs when the act ofuttering the number words from "one" to, say, "eight" implies thepresence of accompanying tangible items of some kind (countingverbal unit items). Then comes the realisation that an utterance of thenumber word "eight" can, by itself, be taken to imply the numberword sequence "one, two, ....., eight", as well as a collection ofdiscrete unitary items that could be coordinated with that sequence ofutterances. And finally, comes the momentous realisation that anysequence of counting acts can itself be counted.

In the following descriptions of conceptual structures for the unit "10", thesymbol "10" is used rather than its name, to emphasise the distinction betweenthe unit and its name in different languages.

1. A SPECIFIED PERCEPTUAL COLLECTION (SPC) This refers to anycounted collection of 10 perceptual items that cannot subsequently be re-presentedand strategically applied.

2. A PERCEPTUAL UNIT. (PU) Perceptual collections of 10, which are onlytemporarily established in the visual field of the child, and which derive salienceonly from the items of the collection and the patterns in which they are arranged,are abstracted through the ability to re-present counting activity. This may lead to

the realisation that any two perceptual collections of 10 will have a commonfeature, which is that if they were counted, "10" would be the result.

3. A COUNTABLE PERCEPTUAL UNIT. (CPU) If a child can coordinate thenumber word sequence "10, 20, 30, ..." with specific perceptual units whicheach contain 10 items, it is said that, for the child, these units are countable.Children with this conception of 10 probably cannot constitute a collection of 10counted perceptual items into a composite unit, nor move flexibly between thatcomposite unit and its contents.

4. _A FIGURAL PATTERN (CFU) This is a pattere of 10 counted items thatcan be re-presented. It may take the foren of two open hands, or a figural imageof a bundle of 10 or a base 10 long, or may have idiosyncratic significance. If itis co-ordinated with the DNWS, it is called a countable figural unit.

5. A COUNTABLE MOTOR UNIT. (CMU) This type may occur in thecontext of counting perceptual units of 10, when a motor act like putting up afinger or pointing with a finger is used as a substitute for perceptual units of 10that are screened from sight.

6. A NUMBER WORD PATTERN. (NWP) If children are capable of re-presenting and reviewing the results of a counting activity and pulling from it therecurrent results of making intuitive extensions of 10, they are sometimes able toconstruct a number word pattere from a point within a decade, "5, 15, 25, ... ",incrementing by implied 10 counts.

7. A NUMERICAL COMP®SITE UNIT. (NC) This conceptual re-organisation enables the child to take a perceptual collection of 10 items as oneunit, while maintaining its numerosity. This unit refers to any pattere of 10 thatis the result of an integration, the focus being on the elements, and the pattem notbeing taken as one unit.

8. AN ABSTRACT COMPOSITE UNIT. (AC) The ability to co-ordinatecounting by 10's and 1's when counting on, is symptomatic of the use of 10 asan abstract composite unit.

Research PredictionsEach teaching session was videotaped, and a detailed retrospective case

study of the development of each child's conceptual structures for "10" was

Langrage and Counting 9

compiled. The research predictions for the children's conceptual development

were:1. that the BVE children (Hao, Qyen, Ai, and Tony in order of age) wouldexhibit some of the described conceptual structures for 10 earlier than the MEchildren (Kim, Peter, Kylie and Sean).2. that the ME children would not exhibit any of the described conceptualmanifestations for 10 earlier than the BVE children.3. that the BVE children would exhibit conceptual structures for 10 not exhibitedby the ME children.4. that the ME children would not exhibit conceptual structures for 10 not alsoexhibited by the BVE children.

Results

Figure 3 shows the teaching session at which the highest conceptualstructure for 10 was manifested by each child. One BVE child (Qyen) left theschool and the experiment after session 12.

The case studies generally sustained Prediction 1. BVE children Tony, Aiand Hao all evidenced 10 as an Abstract Composite earlier than ME children, andapart from the ME child Kim, the tendency was for BVE children to produce 10as a Perceptual Unit, 10 as a Countable Perceptual Unit, 10 as a CountableFigural Unit, 10 as a Countable Motor Unit, and 10 as a Number Word Pattemearlier than the ME children.

Again, the case studies generally sustained Prediction 2. With theexception of Kim, there was a tendency for BVE children to attain more complexstructures earlier. Certainly the structures attained by Kylie, Sean and Peter (MEchildren) were slower to develop than those of the other children, Peter, forexample, never abstracting 10 as a Countable Perceptual Unit, and Kylie onlyattaining it at the last session.

Prediction 3 was clearly sustained in the case studies. Tony, Ai and Hao(BVE children) all exhibited 10 as an Abstract Composite, a structure notexhibited by any ME child. It will be useful to elaborate on this finding, becauseit may carry certain implications for the treatment of two-digit numbers in theearly childhood mathematics curriculum.

The task which permitted assessment of the availability of the concept "10as an Abstract Composite" involved the child summing two double-digit arrays ofbundling sticks, one of which was hidden. The investigator would put out, forexample, 2 bundles and 3, reach agreement with the child that 23 were present

TEACHING

SESSION END OF

FETIi KYLIE ^^rK19LIE HA0o

Te41

AI

AI

t^+a®AI 1 Kin

Mw Kln

wao

arEn

Kln

Kln

rEAn

PETERKVLIE

IVDAlKIM

WENAlKIM

WilJJHAD

I T iAlKmHAo

SPC PU CPU CFU CMU NWP NC AC

SPC=SPECIFIED PERCEPTUAL COLLECTION PU=PERCEPTUAL UNIT

CPU=COUNTABLE PERCEPTUAL UNIT CFU=COUNTABLE FIGURAL UNIT

CMU=COUNTABLE MOTOR UNIT NWP=NUMBER WORD PATTERNNC=NUMERICAL COMPOSITE AC=ABSTRACT COMPOSITE

(NOTE: Qyen left the experiment after session 12)

Figure 3. Conceptual structuren for °10' (Showing the teaching session atwhich each structure was observed for each child).

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14

13

12

11

10

9

8

7

6

5

4

3

2

1

NOV

OCT

SEPT

AUG

JUL

JUN

Langrage and Counting 11

(by counting if necessary), and then cover the array. A further 1 bundle and 2

might then be placed on the cover, and the child would be asked to fmd the total.No child was found to be unable to attempt the task, but the two types of

responses exhibited by the children were quite distinctive. The ME children

always attempted such a task by unit counting, whereas the BVE children allattempted the task by summing 1O's and l's separately.

Prediction 4 was sustained in the case studies. There were no identifiablestructures for 10 exhibited exclusively by ME children.

The notable exception in these interpretations was Kim, an ME child whoattained 10 as a Countable Figural Unit, 10 as a Number Word Pattem and 10 asa. Numerical Composite before any BVE child. In doing so, she showed that itwas possible for an ME child to attain these structures before BVE children.One explanation of her exceptional performance is that she manifested a clear andearly understanding of the numerical significante of the homonymictransformations in the English SNWS. 7his meant that she was able not only torecite the 2-digit Number Word Sequence quite fluently, but she was also able todecompose a 2-digit number name into a 1O's part and a l's part. She did not,however, exhibit the conceptual structure 10 as an Abstract Composite, preferringto find double-digit rums by unit counting, rather than sequential summation oftens and units.

Discussion

Piagetian conservation theory leads us to consider young children'snumerical knowledge in terras of deficits, and leads us also to make certaindecislons about the kinds of activities such children will need if they are toovercome their deficits. We tend to assume, for example, that there is littie pointin working witti 2-digit numbers until conservation (as exemplified in the abilityto compare small single-digit groups) has been attained. The legact' of Piagetianconservation theory continues in many NSW infants classes, where a great dealof time is spent in establishing and comparing numerosity for small sets ofobjects, while treatment of 2-digit numbers is deferred. The results of this studyseem to show that where young children use a transparent SNWS, they candevelop quite sophisticated concepts for 10, and apply these concepts tooperations on 2-digit numbers.

Teachers of infants in Australia have long recognised the difficulties createdfor young English speaking children by the homonymic transformations of theirSNWS. This recognition is reflected in a general reluctance to expose these

young children to the system of two-digit numbers or numerals too early. But itis important to remember that, given the current price of confectionery, manyyoung children are forced to reckon with 2-digit numbers much more frequentlythan their parents or teachers had to at a similar age. To assist the young child tomake sense of these early encounters, two-digit numbers can be satisfactorilymodelled with base 10 blocks, and reinforced with simultaneous use ofdecimalised number names such as "two tens and three" and the correspondingnumeric symbols. Such representations may be helpful in overcoming thelinguistic disadvantage for children enculturated to a non-transparent SNWS.

The messages for teachers in the present research are that, in contrast toAsian SNWS's, the English SNWS is poorly understood and poorly applied bymany young children, that greater attention needs to be given to the place valueconnotations of the 2-digit SNWS by requiring children to recode in tens andones, and that activities to overcome the inherent linguistic difficulties of theEnglish SNWS need to be developed. In terras of the latten, one strategy mightbe to focus children's attention on number names by exposing them toVietnamese or Japanese names, or by using rhyming slang on English numbernames. So, "What number sounds like STICKY FUN? NIFTY KICKS?DIRTY SHOE?

19

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