Develópmental Processes and Stages in Acquisition of

INTERNAnONAL JOURNAL OF BEHAVIORAL DEVELOPMENT, 1990, 13 (2) 231-250

Develópmental Processes and Stages in the' Acquisition of Cardinality

Vicente Bermejo and Ma Oliva Lago Universidad Complutense, Madrid, Spain

This is a study of the level of children'·s understanding of cardinality, focusing on the difference between a true cardinality response and the .application of a mechanically learned rule. The authors also evaluate and discuss the possible relationship between cardinality and counting., The 'subjects were two groups of 32 preschool children, ranging in age from 4 years 3 moriths to 6 years 3 months. Experimental methodology included two large sets of tests (elements-cardinal vs cardinal-elements), using both numbers and vowels with forward vs backward counting, and visual vs verbal presentation conditions. Results show that cardinality responses, are affected by both the direction and nature of the elements in the counting sequence. Scrutiny of errors committed in the various tests enables us to suggest six stages in the acquisition of cardinality. Although there appears to be a developmental dependency between counting and cardinality, this relationship is not significant in all cases.

INTRODUCTION

D~velopmental Processes and Stages in the Ac'quisition of Cardinality

t A review of current literature on the cardinal meaning of numbers enables one to observe two main lines of resarch. In the first, children indicate the number of objects within a set (Fuson, Pergament, Lyons, & Hall, 1985b;

::' Gelman & Gallistel, 1978; Schaeffer, Egglestori, & Scott, 1974; Wilkinson, , 1984, etc.). In the second, children establish the relationship of equivalence

or inequality between two différent sets. In turn there are different ~

Requests for reprints should be sent to Dr Vicente Bermejo, Departamento Psicología Evolutiva y Educacion, Facljltad de Psicologia, Campus de Somosaguas, 28023 Madrid, Spain.

The authors wish to thank Jeffrey Ring, PhD for his assistance in translating the manuscript and the two anonymous correctors for their suggestions.

© 1990 The Intemational Society for the Study of Behavioral Development

232 BERMEJO ANO LAGO THE ACQUISITION O

approaches: (a) one which is basically concerned with the role of the one- class-inclusion (Bermejo, 1989; Fuson, Lyons, Perga to-one correspondence (Brainerd, 1979; Kingma & Koops, 1984; Michie, 1988), her findings have not been confirmed vis-¡ 1985; Piaget & Szeminska, 1941), and (b) another which stresses the role of Fuson, Pergament, & Lyons, 1985a; Hodges & Fren counting (Clements, 1984; Gelman, 1982; Markman, 1979; Michie, 1984; same orientation, Saxe (1979) has described both Saxe, 1979). : prequantitative method of determining equivalence j

In the former line of research, specification of the cardinal usually ¡ two sets. Children demonstrate a quantitative app presupposes the use of counting. Work carried out by Gelman and Gal- counting to judge the relationship between sets, G

listel (1978) showed that acquisition of cardinality occurred after the approach when they use procedures other than cou acquisition of one-to-one correspondence and stable order. Along this estimation. Similarly, Michie (1984) found that COUI

same line of research, Wilkinson (1984) suggested that counting and 'when the elements of the two sets are placed in sep, cardinality are closely linked to each other during the early and advanced : have be en counted, than when they are placed in phases of counting skills development, but that they may become dissoci- ) children can see. ated during the intermediate periodo Moreover, with regard to Gelman To conclude, Gelman and Gallistel (1978) have a j

and Gallistel's (1978) position, Wilkinson pointed out that elementary ¡I depends on the ability to count, thus highlighting 1

counting skills may be acquired even earlier than cardinality, but that between these two toncepts. Schaeffer et al. (1974) p cardinality reaches functional maturity prior to counting. acquire cardinality via an integration of the earlier F

In the second line of research, when two sets are compared the child may and counting. Wilkinson (1984) found that countin either carry out an item-to-item correspondence between the elements of strongly associated to one another during the early ¡

both sets, or obtain the cardinal value of each, in order to then compare of counting but that they are dissociated during the them. From the first approach (a), Piaget and Szeminska (1941), according Fuson and Hall (1983) have suggested two possible to their logical model, suggested that it is the synthesis between class and sion when a child's response to "How many?" is th asymmetrical relationships, not verbal enumeration, that leads to the The first level of comprehension could be a mechani< operational conservation of the number. In contrast, other authors such as which is not indicative of cardinality. A second and m Clements (1984), Fuson and Hall (1983), Fuson et al. (1985b), Gelman and comprehension indicates the child's reference to the Gallistel (1978), and Saxe (1979), hypothesised that the development of and is considered to be a response of true cardinalit numerical concepts and skills is derived from the integration of count- that all children do not necessarily pass through the ing, subitising, and estimation skills. Michie (1984) integrated these two hension (Fuson, 1988). divergent theoretical conceptions, concluding that the absolute number This brief review reveals the absence of consister ("How many?") appears to develop before the relative concept ("Which about the process of cardinality acquisition. This pa¡ has more?"). Michie argued that those children who used counting to . the cognitive processes that children folIow in acquiri determine the cardinal or absolute value of a set were reluctant to use the fically we focus not only on the steps children foil I

same procedure in certain relational situations. On the other hand, cardinality, but also on the possible relationships h Brainerd (1979) suggested that the development of the concept of number rule of "How many", and the principIe of cardinalit is rooted in ordination, and that ordination is an indispensable prerequisite itself when the child, faced with the question "How m for the child to truly acquire cardination. Michie (1985), however, con- after counting merely and exclusively repeats the las cerned with the developmental sequence of the numerical skills of cardina- sequence given. Cardinality, on the other hand, m tion and order, questioned Brainerd's theory, claiming that children under- response refers to the numeiosity of the whole set o; stand number as an absolute quantity before they can understand ordered though is at times not necessarily the last symb( series. . backwards counting tasks.

With regard to the latter approach ofthis line ofresearch, (b), Markman t We suppose "that there is a certain "cultural" (1979) presented results in which the use of collection terms better facili- ! counting and cardinality, but not necessarily a theoret tated the understanding of cardinality than did class terms. However, while cultural relationship would be due to the fact tbat tbf 1:

sorne studies have supported Markman's position on concepts such as ity is normally associated with the teaching of co

THE ACQUISITION OF CARDINALlTY 233l\GO

:llch is basically concerned with the role of the one: (Brainerd, 1979; Kingma & Koops, 1984; Michie, ¡ka, 1941), and (b) another which stresses the role of 984; Gelman, 1982; Markman, 1979; Michie, 1984;

of research, specification of the cardinal usually f counting. Work carried out by Gelman and Galthat acquisition of cardinaIity occurred after the one correspondence and stable order. Along this 1, Wilkinson (1984) suggested that counting and linked to each other during the early and advanced lIs development, but that they may become dissocinediate periodo Moreúver, with regard to Gelman

position, Wilkinson púinted out that eIementary le acquired even earlier than cardinality, but that Ilctional maturity prior tú counting. [research, when two sets are compared the child may :m-tú-item correspúndence between the elements of ile cardinal value of each, in order to then compare pproach (a), Piaget and Szeminska (1941), according , suggested that it is the synthesis between class and lships, not verbalenumeration, that leads to the iún of the number. In contrast, other authors such as )ll and Hall (1983), Fuson et al. (1985b), Gelman and Saxe (1979), hypothesised that the development úf md skills is derived from the integration of countstimation skills. Michie (1984) integrated these two conceptions, conduding that the absolute number

ears to develúp befúre the relative concept ("Which ¡ argued that those children who used countíng to Uúr absolute value of a set were reluctant to use the certain relational situations. On the other hand, ~sted that the development of the concept of number ~, and that ordination is an indispensable prerequisite , acquire cardination. Michie (1985), however, con.opmental sequence of the numerical skills of cardinaioned Brainerd's theory, claiming that children underIbsolute quantity before they can understand ordered

Iatter approach of this line of research, (b), Markman

class-inclusion (Bermejo, 1989; Fuson, Lyons, Pergament, Hall, & Kwon, 1988), her findings have not been confirmed vis-a-vis cardinality (e.g. Fuson, Pergament, & Lyons, 1985a; Hodges & French, 1988). Within this

l same orientation, Saxe (1979) has described both a quantitative and a prequantitative method of determining equivalence or inequality between two sets. Children demonstrate a quantitatíve approach when they use counting to judge the relationship between sets, and a prequantitative approach when they use procedures other than counting in making their estimation. Similarly, Michie (1984) found that counting is more efficient when the elements of the two sets are pIaced in separate boxes after they have been counted, than when they are placed in two rows which the

: children can see. : To conclude, Gelman and GallisteI (1978) have argued that cardinality I depends on the ability to count, thus highlighting the close relationship

between these two concepts. Schaeffer et al. (1974) proposed that children acquire cardinality via an integration of the earlier processes of subitising and counting. WiIkinson (1984) found that counting and cardinality are strongly associated to one another during fhe early and late development of counting but that they are dissociated during the intermediate periodo FUson and Hall (1983) have suggested two possible levels of comprehension when a child's response to "How many?" is the last counting word. The first level of comprehension could be a mechanically Iearned reaction, which is not indicative of cardinality. A second and more advanced level of comprehension indicates the chiId's reference to the entire set of objects,

i and is considered to be a response of true cardinality. The authors added I that all children do not necessarily pass through the first level of compre

hension (Fuson, 1988). This brief review reveals the absence of consistent and definitive data

about the process of cardinality acquisition. This paper attempts to study the cognitive processes that chiIdren follow in acquiring cardinality. Specifically we focus not onIy on the steps children follow toward achieving cardinality, but also on the possible relationships between countíng, the rule of "How many", and the principIe of cardinality. The rule manifests itself when the child, faced with the question "How many _,_ are there?",

, after counting merely and exclusively repeats the last word or term in the sequence given. Cardinality, on the other hand, means that the child's response refers to the numerosity of the whole set of elements presented, though is at times not necessarily the last symbol pronounced as in

, backwards counting tasks. I We suppose that there is a certain "cultural" relationship between

ults in which the use of collection terms better facili- . counting and cardinality, but not necessarily a theoretical relationship. The ing of cardinality than did class terms. However, while! cultural relationship would be due to the fact that the learning of cardinaliupported Markman's position on concepts such as . ity is normally associated with the teaching of counting, which comes

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234 BERMEJO ANO LAGO

earlier than cardinality, as Fuson (1988), Gelman and GaIlistel (1978), Kingma and Koops (1984), Schaeffer et al. (1974) and, partiaIly, Wilkinson (1984) have argued. However, counting and cardinality could very well be two independent abilities, given that children can count (perhaps perfectly well) without cardin ality , and vice-versa (Fuson, Pergament, Lyons, & Hall, 1985b; Russac, 1983). Furthermore, the cardinal number can be determined not only by means of counting but also by other quantificators such as: subitising, and estimation (Klahr & WaIlace, 1976).

In the current investigation we anaIyse various types of counting and cardinality behaviours. We especiaIly focus on children's mistakes in the foIlowing experimental situations: familiar vs novel tasks, counting forwards vs counting backwards, sequences of number words vs sequences of vowels, and elements-cardinal situation vs cardinal-elements situation (see Fig. 2). We hypothesise that the children's responses in the familiar situations will be more or less influenced by automatised mechanisms, which are very difficult to analyse; the novel situations, on the other hand, will' limit the influence of these mechanisms, and thus will facilitate both the manifestation of the cognitive processes that underlie the acquisition of cardinality, as weIl as our inferences and understanding of those processes. In first block of tests (i.e. Elements-Cardinal) the child determines the cardinality of a given set of elements, while in the second block of tests (i.e. Cardinal-Elements) the process is reversed, and the child determines the elements pertaining to a given cardinal. We expect that these complementary situations wilI differentiate the various leveIs of comprehension in cardinality. FinaIly, the counting backwards situation aIlows us to differentiate empiricaIly the "How many" rule from the principIe of cardinality.

METHOD

Subjects

The subjects- in this study were 64 students in the first and second year of a public preschool in Madrid. They carne from middle-class background s and were chosen at random. Group 1 consisted of 32 children between the ages 4 years 3 months and 5 years 2 months (M = 4 years 7 months). Group II . consisted of the remaining 32 children, whose ages varied between 5 years 4 months and 6 years 3 months (M = 5 years 7 months). Each group was composed of equal numbers of boys and girls.

Materials .

We used up to a maximum of six red chips, each measuring 1 cm ín díameter, to conduct the experimental tasks described below in the empírical procedure section. Secondly, we employed two white cards

THE ACOUISITIOI\

(a) (b)

2 O O 0000

(a) Baekwards eounting praetice card; (b) vowel-c eard; and (e) card indicating vowel-numeral during vowel tasks.

FIG. 1. Empirical material presented during the eXI

Elements-Cardinal

Numbers Vowels Numj

Counting Counting Counting Counting Verbal forwards backwards forwards forwards (2& 5)(2 & 5) (3 & 4) (2 & 5) (3 & 4)

FIG.2. Table oi the empirical desig

(14 X 20 cm) on which were pasted two and three rJ (1.5 cm each), as weIl as the corresponding numeral the last element in the row. These two cards were l

the study in which chiIdren practiced the backwar beginning with these numerals (see Fig. La). (14 x 20 cm), fashioned after the cards already de circles in a row (1.5 cm each), and was used for ve (see Fig. Lb). An additional white card (14 x 20 CI

vowels pasted aboye a correspondíng row ofnuI1 ~inaIIy, we employed four cards (7 x 10 cm) eacl elther a numeral or a vowel printed in Iower ease ti hand in the number of chips indicated (see Empírica] to facilitate the children's approach to the task . they were in a familiar situation, the experimenter "Espinete" (a weIl-known TV puppet in Spain).

Empirical Procedure

Each child took aIl the tests individualIy during scho sessions that lasted approximately 20 minutes e procedure consisted of presenting two blocks of presentation factor) in such a way that the first (elem by asking the child to count the elements of one

\GO

, as Fuson (1988), Gelman and Gallistel (1978), ~4), Schaeffer et al. (1974) and, partially, Wilkinson )wever, counting and cardinality could very well be es, given that children can count (perhaps perfectly ty, and vice-versa (Fuson, Pergament, Lyons, & L983). Furthermore, the cardinal number can be , means of counting but also by other quantificators 1 estimation (Klahr & Wallace, 1976). tigation we analyse various types of counting and We especially focus on children's mistakes in the

1 situations: familiar vs novel tasks, counting for:wards, sequences of number words vs sequences of -cardinal situation vs cardinal-elements situation thesise that the children's responses in the familiar e or less influenced by automatised mechanisms, to analyse; the novel situations, on the other hand, of these mechanisms, and thus will facilitate both

e cognitive processes that underlie the acquisition of lur inferences and understanding of those processes. (i.e. Elements-Cardinal) the child determines the ~t of elements, while in the second block of tests (i.e. le process is reversed, and the child determines the a given cardinal. We expect that these complemen

fferentiate the various levels of comprehension in e counting backwards situation allows us to differenHow many" rule from the principIe of cardinality.

METHOD

ldy were 64 students in the first and second year of a ldrid. They carne from middle-class backgrounds and n. Group 1 consisted of 32 children between the ages 5 years 2 months (M = 4 years 7 months). Group II ning 32 children, whose ages varied between 5 years 3 months (M = 5 years 7 months). Each group was Imbers of boys and girls.

naximum oí six red chips, each measuring 1 cm uct the experimental tasks described below in the section. Secondly, we employed two white cards

THE ACQUISITION OF CARDINALlTY 235

(a) (b) (c)

2 a e i a u O O 0000 123 4 ~

(a) Backwards counting practice card; (b) vowel-counting practice card; and (c) card indicating vowel-numeral correspondence during vowel tasks.

FIG. 1. Empirical material presented during the experimental session.

Elements-Cardinal Cardinal-Elements

Numbers Vowels Numbers Vowels

Counting Counting Counting Counting Verbal Visual Verbal Visual forwards backwards forwards forwards (2 & 5) (3 & 4) (2 & 5) (3 & 4) (2 & 5) (3 & 4) (2 & 5) (3 & 4)

FIG. 2. Table of the empirical designo

(14 X 20 cm) on which were pasted two and three red circles, respectively (1_5 cm each) , as well as the corresponding numeraL '2" or "3" just aboye the last element in the row. These two cards were used during a phase of the study in which children practiced the backwards counting sequence beginning with these numerals (see Fig. _l.a). Another white card (14 x 20 cm), fashioned after the cards already described, had four red circles in a row (l.5 cm each) , and was used for vowel-counting practice (see Fig. l.b). An additional white card (14 x 20 cm) contained a row of vowels pasted aboye a corresponding row of numerals (see Fig. l.c). Finally, we employed four cards (7 x 10 cm) each of which contained either a numeral or a vowel printed in lower ease to request the child to hand in the number of chips indicated (see Empirical Procedure). In order to facilitate the children's approach to the task and make them feel tbey were in a familiar situation, the experimenter introduced a puppet "Espinete" (a well-known TV puppet in.Spain).

Empirical Procedure

Each child took all the tests individually during school hours, in research sessions that lasted approximately 20 minutes each. The empirical procedure consisted of presenting two blocks of tasks (the mode of presentation factor) in such a way that the first (elements-cardinal) began by asking the child to count the elements of one set and to indicate


afterwards the cardinal number of the set; while in the second (cardinaIelements) children were presented a cardinal, and were asked to determine, from a row of six objects, the elements which corresponded to the mentioned cardinal (see Fig. 2). Each block of tasks consisted of two types of tests (counting sequence factor), such that sorne triaIs used numbers and others used the standard voweI sequence for counting. In the first block, using number words, the children carríed out two tasks: one counting forwards and another counting backwards (counting direction factor), both followed by the question: "How many chips are there?". The counting backwards situatíon allowed us to, aboye aH, differentiate between cardinality and the "How many" rule. In both cases we presented two successive sets of chips in a horizontalline consisting of 2 and 5 chips in the forwards counting condition ("Go ahead and count these chips"), and 3 and 4 in the backwards counting condition. In this latter case the children were asked to begin counting backwards where the starting word was one more than the cardinality of the set to be counted ("Go ahead and count these chips backwards, starting from 4 [or 5]"). In the vowel condition, we presented four tests with 2, 3, 4 and 5 chips in a row, and the children were asked to count forwards. We did not inelude a backwards task, due to the difficulty of counting backwards with vowels at these ages. The children were requested: "Go ahead and count the chips using voweIs", and were subsequently asked "How many chips are there?" The child was requested to respond to these questions with voweIs. In this case, as in all the vowel tasks, the children could make use of the voweI-numeraI correspondence card that was placed in front oí them (se e Fig. 1.c). Before introducing the vowel-counting task, however, we checked the subjects' ability to produce the standard vowel sequence, and provided a brief training to those who needed it, such that a11 could produce the sequen ce without problems before proceeding. Likewise, befo re starting the counting backwards test, the experimenter made sure the children understood each task, asking them to count backwards in one or two simple practice situations. , In the second block of tests (cardinal-elements) we asked the children

for a precise set of objects, either using numbers or using vowels. In both cases, the request was made either verbally ("Go ahead and give Espinete 2 [or e] chips"), or visually by means of showing a card which had a voweI or a number written on it ("Go ahead and give Espinete these chips") (request form factor). The inclusion of both a verbal and a visual request factor will allow us more effectively to discriminate Ievels of cardinality acquisition. Our pilot research has pointed our attention to the differences in children's responses when presented with verbal vs visual requests. Furthermore, the visual presentatíon factor is intended to allow us to judge whether children use the visual cardinal as either a symbol oí the entire set of objects, or merely as an indicator of the one object to which it is .

THE ACOUISITI

assigned. In the verbal test we asked for 2 and :. test we asked for 4- and 3. The chíld responded requested by the experimenter from a line of 6 el was onIy one tri al in each experimental situatiOI

The order of presentation of the two bloc balanced, as was that of the tests within e were assigned at random to each of the result presentation of the different tríals within ea, determined at random and was constant for aH ' the counting backwards condition, the set wit befo~e the set of 4 chips in half the subjects, whi tests In the reverse order. Subjects' responses w( correet or incorrecto In the counting forwaJ response was rated as correct when the child 1

word Ol the last counting vowel, whiIe in the dition, the onIy correct response was one that ga1 of the set.

ANALYSIS AND DISCUS

We have analysed our data both quantitatively end we first carried out an are-sin transformat: correet tríals. Using the BMDP2V program, tht Group I1) x 2 (Elements-CardinaI vs CardinaIvs Vowels) with repeated measures as may be significant main effects for age (F [1:62] == 20.39 sequ~nce (F [1,62] = 40.64, P < 0.01), indicatiI obtamed better resuIts than the younger chiIdrel tasks were, globally, easier than the voweIs tasks.

TABLE 1 Average and Standard Oeviations from Transfo

of Correet Responses in Cardina

Elements-Cardinal e

Numbers Vowels Nw

Group I 1.78 1.35 2 (0.35) (0.98) (O

Group JI 2.08 2.20 2 (0.41) (0.81) (O

Maximum possible score is 2.64.

.GO

number of the set; while in the second (cardinale presented a cardinal, and were asked to deter( objects, the elerílents which corresponded to the ! Fig. 2). Each block of tasks consisted of two types uce factor), such that sorne trials used numbers and rd vowel sequence for counting. In the first block, the children carried out two tasks: one counting Dunting backwards (counting direction factor), both ion: "How many chips are there?". The countn allowed us to, aboye a1l, differentiate between Iow many" rule. In both cases we presented two in a horizontalline consisting of 2 and 5 chips in the dition ("Go ahead and count these chips"), and 3 >counting condition. In this latter case the children mnting backwards where the starting word was one lit y of the set to be counted ("Go ahead and count ,starting from 4 [or 5]"). In the vowel condition, we th 2,3,4 and 5 chips in a row, and the children were :Is. We did not indude a backwards task, due to the backwards with vowels at these ages. The children ahead and count the chips using vowels", and were Iow many chips are there?" The child was requested lestions with vowels. In this case, as in aH the vowel lid make use of the vowel-numeral correspondence 1 front of them (see Fig. l.c). Before introducing the .1Owever, we checked the subjects' ability to produce :quence, and provided a brief training to those who aH could produce the sequen ce without problems

lkewise, before starting the counting backwards test, de sure the children understood each task, asking ards in one or two simple practice situations. [( of tests (cardinal-elements) we asked the children ljects, either using numbers or using vowels. In both s made either verbally ("Go ahead and give Espinete sually by means of showing a card which had a vowel on it ("Go ahead and give Espinete these chips")

l. The indusion of both a verbal and a visual request nore effectively to discriminate levels of cardinality : research has pointed our attention to the differences ;es when presented with verbal vs visual requests. lal presentation factor is intended to allow us to judge the visual cardinal as either a symbol of the entire set y as an indicator of the one object to which it is


assigned. In the verbal test we asked for 2 and 5 chips, whiIe in the visual test we asked for 4 and 3. The child responded by selecting the quantity requested by the experimenter from a line of 6 chips in front of him. There was onIy one tri al in each experimental situation.

The order of presentation of the two blocks of tests was counterbalanced, as was that of the tests within each block. The subjects were assigned at random to each of the resulting orders. The order of presentation of the different triaIs within each test was also initiaHy determined at random and was constant for all the subjects. However, in the counting backwards condition, the set with 3 chips was presented before the set oi 4 chips in half the subjects, whilst the other haIf took the tests in the reverse order. Subjects' responses were dichotomised as either correet or ineorrect. In the counting forwards condition cardinality response was rated as correct when the child repeated the last number word or the last counting vowel, while in the counting backwards condition, fue only correct response was one that gave the exact cardinal value of the set.

ANALYSIS AND DISCUSSION

We have analysed our data both quantitatively and qualitatively. To this end we first carried out an are-sin transformation of the proportions oi correct trials. Using the BMDP2V program, the ANOVA 2 (Group 1 vs Group II) x 2 (Elements-Cardinal vs Cardinal-Elements) x 2 (Numbers vs Vowels) with repeated measures, as may be seen in Table 1, showed significant main effects for age (F [1,62] == 20.39, P < 0.01) and counting sequence (F [1,62] == 40.64, P < 0.01), indicating that the older children obtained better results than the younger children, and that the numerical tasks were, global1y, easier than the vowels tasks. Likewise, the interaction

TABLE 1 Average and Standard Deviations from Transformed Proportions

of Correet Responses in Cardinality

Elements-Cardinal Cardinal-Elements

Numbers Vowels Numbers Vowels

Group 1 1.78 (0.35)

1.35 (0.98)

2.33 (0.52)

1.15 (0.88)

Group n 2.08 (0.41)

2.20 (0.81)

2.53 (0.31)

1.70 (1.03)


238 BERMEJO AND LAGO

of these two factors (F [1,62] = 6.08, P < 0.05) was significant as well as the interaction of presentation mode with coimting sequence (F [1,62] = 30.89, P < 0.01). That is to say that when vowels were used as components of the counting sequence, there was a greater difference between the averages of Groups 1 and 11 than when numbers were used. However, the difference in averages between the elements-cardinal and cardinalelements situations was more pronounced whell the children worked with numbers than with vowels, although their performance was greater when faced with numbers than with vowels. The other results were not significant. Thus, there is no significant difference between both blocks of tasks (i.e. Elements-Cardinal and Cardinal-Elements). If in the general analysis we omit counting backwards and visual presentation, we obtain the same results in the ANOVA with respect to the significance of the factors and their interactions. We shall analyse the preceding data in greater detail below, differentiating the two main blocks of tests in order to make our exposition clearer.

Elements-Cardinal Block

With regard to the first block, the overall results with respect to cardinality showed that the children in Group 11 obtained a higher percentage of correct trials than those in Group 1, except when they had to count forwards with numbers, in which case their success was the same .. First we shall examine the results corresponding to the numerical tasks.

Numerical Tests. The ANOVA 2 (Group 1 vs Group II) x 2 (Counting Forwards vs Counting Backwards) showed significant main effects for age (F [1,62] = 10.77, P < 0.01) and counting direction (F [1,62] = 128.44, P < 0.01), as well as the interaction between them (F [1,62] = 7.82, P < 0.01). Thus the task of cardinality is more difficult with a backwards counting sequence, even though we used sets of only 3 or ~ objects (see Table 2). A minimal increase in the number of objects presented to the child (3 vs 4) decreases the success rates of all the subjects, but particularly in the younger group (Group 1). The fact that children's counting success rates are very similar when counting forwards (100%), or when counting backwards with sets of 3 (68%) and 4 (62%) objects alike, strongly suggests that subitising is the main mechanism responsible for the correct cardinal response during backwards counting. Subitising in turo appears to be much easier with 3 than with 4 objects, particularly for the younger subjects (see Fuson, 1988). There are at least two phenomena that might explain the differences we have found between the two conditions (counting forwards and counting backwards). The first is that the children were generally more familiar with the counting forwards situation. The second

THE ACQUISITI

TABLE 2 Pereentages of Correet Trials for the Cardinality é

in the Elements-Cardinal Block Witl

Counting forwards

Counting Cardinality

Group 1 100 97 Group II 100 97

possibility is that the counting backwards tech; criminate between levels of a child's understar while counting backwards allows us to distinguí ~any role" .and the "principIe of cardinality" , it d differences m standard counting. This Iimitation ( in which underIying cognitive operations produce wrong answers in the counting backwards task but in tbe forward task, leading to incorrect inference ing wben only the forward task is given. TI frequent in studies of cardinality (se e Gelman, ~ Wagner & Walters, 1982), and attempts to avoid t useoflan~age aspects (see Fuson, 1988) probabl) tbe expenmental situation.

We bave classified the subjects' mistakes into f with the l~t uumber word of the sequence used; first number word of the sequence; (3) counting ; sequence used in the counting; and (5) responding (see Table 3). A great deal of the wrong answers of

TABLE 3 Pen:entagés of Cardinality Responses Given in the Bac

Three objects

Group 1 Group II

Cardinal 41 72 I.mt number word 41 22 First number word 6 3 Comtting again 6 3 Repeating the sequence 3 Random number word 3

iO

l,62] = 6.08, P < 0.05) was significant as well as ltation mode with counting sequence (F [1,62] = to say that when vowels were used as components

ce, there was a greater difference between the j II than when numbers were used. However, the between the elements-cardinal and cardinal

more pronounced when the children worked with ~ls, although their performance was greater when m with vowels. The other results were not signo significant difference between both blocks of lrdinal and Cardinal-Elements). If in the general ng backwards and visual presentation, we obtain ANOV A with respect to the significance of the

actions. We shall analyse the preceding data in ferentiating the two main blocks of tests in order to

~arer.

Block

st block, the overall results with respect to cardinalldren in Group II obtained a higher percentage oí ¡se in Group 1, except when they had to count , in which case their success was the same. ,First we lts corresponding to the numerical tasks.

he ANOVA 2 (Group 1 vs Group II) x 2 (Counting Backwards) showed significant main effects for age ::: 0.01) and counting direction (F [1,62] = 128.44, , the interaction between them (F [1,62] = 7.82, lsk of cardinality is more difficult with a backwards 'en though we used sets of only 3 or 4 objects (see increase in the number of objects presented to the s the success rates of all the subjects, but particularly (Group 1). The fact that children's counting success when counting forwards (100%), or when counting of 3 (68%) and 4 (62%) objects alike, strongly g is the main mechanism responsible for the correct ing backwards counting. Subitising in turn appears to 3 than with 4 objects, particularly for the younger 1988). There are at least two phenomena that might :s we have found between the two conditions (countnting backwards). The first is that the children were lar with the counting forwards situation. The second


TABLE 2 Pen:entages of Correct Trials for the Cardinality- and Counting, Responses

in the Elements-Cardinal Block With Numbers

Counting forwards Counting backwards

Counting Cardinality Counting Cardinality

Group 1 100 97 48 23 Group II 100 97 83 52

possibility is that the counting backwards technique permits us to discrirninate between levels of a child's understanding of cardinality. So, while counting backwards allows us to distinguish empirically the "how many rule" and the "principie of cardinality", it does not effectively assess differences in standard counting. This limitation could produce a situation in which underlying cognitive operations produce responses considered as wrong answers in the counting backwards task but produce correct answers in the forward task, leading to incorrect inferences of cardinaf understanding when only the forward task is given. This misunderstanding is frequent in studies of cardinality(see Gelman, Meck, & Merking, 1986; Wagner & Walters, 1982), and attempts to avoid tbis problem through the use oflanguage aspects (se e Fuson, 1988) probably inordinately complicate the experimental situation.

We have classified the subjects' mistakes into five types: (1) answering with the last number word of the sequence used; (2) answering with the first numbér word of the sequence; (3) counting again; (4) repeating the sequence used in the counting; and (5) responding with a random number (see Table 3). A great deal of the wrong answers of the older children and a

TABLE 3 Percentages of Cardinality Responses Given in the Backwards Counting Task

Three objects Four objects

Group 1 Group II , Grpup 1 Group II

Cardinal Last number word First number word Counting again Repeating the sequence Random number word

41 41 6 6 3 3

72 22 3 3

6 50 34 6 3

31 19 47 3

L...-... _. _.


considerable percentage of those of the younger ones consist of repeating the number word at which counting had begun. This behaviour may be due either to the fact that the child knew that the first number word used was the largest, and included all the numbers which followed it-although th~y did not notice that it did not really end with "l"-or to a certam understanding of the effects or meaning of the backwards sequence. A relative knowledge of the cardinal meaning of the numbers p~oduced during the backwards counting is reflected in both ~ases. A thlfd and possibly more plausible interpretation suggests that thls sort of error may be produced by the effect of at least two factors: the difficulty of subitising four objects, and the salience (the first and the largest) of the first number word of the sequence employed. This accounts for the fact that almost all the children who make this error with four objects are successful when presented with three objects.

Children who made the first type of error behaved as if it were a standard count, directly using the rule of giving the last nu~ber word of t~e sequence used. This behaviour, typical in the.younger ch!ldr~n aboye all, ~s far from a perfect understanding of the princIpIe of car~ma~lty; al~hou~h lt is normally confused with the correct response of cardmahty m sltuatlOns of standard counting. Perhaps the children who committed type 3 and 4 errors (and, of course, type 5) are even further away from reaching the concept of cardinality. . , ..

Therefore, based on our observations of the chddren s behavlOur m the backwards counting situation, we suggest the following stages (although not in a classic Piagetian sénse) or steps in the acquisition of cardinality: (1) misunderstanding of the task and responding at ran~om; (2) ~ere repetition of the previous counting sequence; (3) countmg the obJects again; (4) giving the final number word of the sequence used (the r~le of "How many?"); (5) suggesting the largest numeral of the countmg sequence; and (6) the response of cardinality. We believe that these are the stages that children in Western cultures normally follow towards t?e acquisition of cardinality in the standard situation, although not every chIld will necessarily pass through each and every stage. In the second stage the child do es not make reference to the objects, while in the third stage a number-object correspondence is. established. The fifth st~ge builds u~on the fourth stage, with the idea of "largest" number used m t~e countmg sequence. In other words, a child in the fourth stage respo~ds wlth the final number in the given sequence, while the typical response m the. fifth stage is to respond with the largest number in the given s~~~ence. ThlS ~ev~lo~mental sequence which we propose for the acqmsltlon of cardmahty IS rather different from that of Gelman and Gallistel (1978), but shares much in common with Fuson's (1988) position. Our sequence differs from Fuson's in 1hat we have found a fifth stage that she did n0t consider, given

THE ACQUISITIOI

the experimental situation she employed. Addi clearly delimited the fourth and sixth stages of thl cardinality proper, respectively.

As for counting in these same tasks, it was ob! attained 100% of correct trials in the forwards ca wards counting percentages were noticeabIy Iower, the fact tbat backwards counting sequences are at sequence of numeraIs (Fuson, Richards, & Briars, errors in tbe backwards counting task were seqw dassiñed ioto three maio types: (1) the omission o decreasing sequence (Group 1 19% vs Group II mixed sequence with both increasing and decrea! 33% -.s Group TI 5%); (3) an increasing sequence

JI is important to note the possible relationship skill. operationalised as the correct sequen ce 01 oonespondences, and the response of cardinality iIl ing backwards. McNemar's test shows that there a ences in any of the groups when the sample consist~ tbe differences are significant when the array consÍ! [1, " = 321= 10.89, P < 0.01 for both groups of seems dear in this lattee case that the child may hav( skiIl befare understanding cardinality, as Gelman al

Sdaaeffec et al. (1974) have pointed out; but it couIe

TABLE 4 Frequency Tables for the Counting Backwal

Three objects

Cardinality C 1 T

Gtunpl Counting 1 C

5 8

12 7

17 15 Connti

T 13 19 32

Cardinality C 1 T

Groupn Counting C

2 21

1 8

3 29 Conntil

T 23 9 32

C. rorred; l. incorrect; T, totals.

j

D

)f those of the younger ones eonsist of repeating 1 eounting had begun. This behaviour may be due : child knew that the first number word used was all the numbers which followed it-although they did not really end with "l"-or to a certain :ects or meaning of the backwards sequence. A he cardinal meaning of the numbers produced ounting is reftected in both cases. A third and lnterpretation suggests that this sort of error may :t of at least two factors: the diffieulty of subitising lence (the first and the largest) of the first number nployed. This aecounts for the fact that almost aH this error with four objects are successful when

jects.le first type of error behaved as if it were a standard he rule of giving the last number word of the laviour, typical in the younger ehildren aboye aH, is .standing of the principie of cardinality, although it .lh the correet response of cardinality in situations 'erhaps the ehildren who committed type 3 and 4 type 5) are even further away from reaching the

our observations of the children's behaviour in the uation, we suggest the foHowing stages (although III sense) or steps in the acquisition of cardinality: )f the task and responding at random; (2) mere lOUS counting sequence; (3) counting the objects lnal number word of the sequence used (the rule ,) suggesting the largest numeral of the counting esponse of cardinality. We believe that these are the n Western cultures normaHy follow towards the ty in the standard situation, although not every child rough each and every stage. In the second stage the :eference to the objects, whiIe in the third stage a pondence is established. The fifth stage builds upon the idea of "largest" number used in the counting

rds, a child in the fourth stage responds with the final equence, while the typical response in the fifth stage largest number in the given sequence. This developeh we propose for the aequisition of cardinality is :hat of Gelman and Gallistel (1978), but shares much son's (1988) position. Our sequen ce differs from ve found a fifth stage that she did n0t consider, given

1..--._. •• _ .••• _.


(he experimental situation she employed. Additionally we have more dearly delimited the fourth and sixth stages of the "How many" rule and canlinality proper, respecti vely .

As for counting in these same tasks, it was observed that both groups attained 100% of correct trials in the forwards counting condition. BackwanIs counting percentages were notieeably lower, which is probably due to (he fact that backwards counting sequences are at times acquired as a new sequenceofnumerals (Fuson, Riehards, & Briars, 1982) (see Table 2). AH enurs in the backwards counting task were sequence errors and can be dassified into three main types: (1) the omission of a number word in the decreasing sequenee (Group 1 19% vs Group II 13%); (2) the use of a mixed sequence with both inereasing and decreasing numbers (Group 1 33% vs Group II 5%); (3) an inereasing sequence (Group 12%).

1t is important to note the possible relationship between the counting skill. operationalised as the correct sequence of objeet-number word correspondences, and the response of eardinality in the situation of counting backwards. McNemar's test shows that there are no significant differeoces in any of the groups when the sample consists of three elements, but tite differences are significant when the array consists of four elements (i [1, n = 32] = 10.89, P < 0.01 for both groups of subjeets). It therefore seems clear in this latter case that the child may have acquired the counting skiD before understanding cardinality, as Gelman and GaHistel (1978) and Scha:effer el al. (1974) have pointed out; but it could also be understood in

TABLE 4 Frequency Tables for the Counting Backwards Task

Three objects Four objects

Cardinality Cardinality C 1 T C 1 T

Group 1 Counting 1 C

5 8

12 7

17 15

Counting 1 C

2 O

14 16

16 16

T 13 19 32 T 2 30 32

Cardinality Cardinality C 1 T C 1 T

GrouplI Counting 1 C

2 21

1 8

3 29 Counting

1 C

2 8

6 16

8 24

T 23 9 32 T 10 22 32

C. corree!; 1, incorrect; T, totals.

L


the first case (3 elements) that these are two distinct phenomena. In fact, counting successes are practically the same in the 3 and 4 objects conditions, especially among the younger children (se e Table 4). However, the children's success changes significantly with respectO to the cardinality response with 3 and 4 objects. This again suggests the importance of subitising to respond correctly to cardinality questions, as can be inferred from the fact that some children count incorrectly and yet still correctly answer the cardinality question (see also Russac, 1983). How could this be if counting is an essential component of cardinality? Might it not be more appropriate to speak of counting merely as one of the quantification proce dures (se e Klahr & Wallace, 1976) that can be used to specify the cardinal of a set, or even the sum of two sets? (se e Bermejo & Rodríguez, 1987). It is for this reason that we suggested in the Introduction the existence of a cultural or situationaI, but not necessarily theoreticaI, relationship between counting and cardinality.

Tasks With Vowels. As for the influence ofthe elements that made up the counting sequence, the use of voweIs instead of numbers significantly reduced the percentage of correct cardinality triaIs in the forwards counting task. We found a significant difference between counting forwards with numbers and vowels (with 2 and 5 objects) (F [1,62] = 44.62, P < 0.01), as well as a significant group difference (F [1,62] = 8.53, P < 0.01). These results cannot be attributed to ignorance of the vowels on the part of the younger chiIdren, since our study procedure ensured a uniform ability to recite the vowels without difficulty. In addition, we know that even very young children can discriminate numbers and letters, although they may not know the structural and functional differences between them.

Regarding cardinality in the vowel condition, the ANOVA 2 (Group I vs Group 11) x 2 (2 and 5 vs 3 and 4 objects) showed significant main effects for age (F [1,62] = 13.17, P < 0.01) and for set size (F [1,62] = 4.2, P < 0.05) (see Table 5). Furthermore, the most frequentIy observed mistakes consisted of counting again and repeating the sequence of vowels employed in the counting, which we have described as stages 3 and 2 (see Table 6). Therefore, the introduction of vowels seems to bring about a return to patterns of behaviour that have already been overcome with regard to numbers (e.g. Markman, 1979; Saxe, Gearhart, & Guberman, 1984; Schaeffer et al., 1974). These children probably consider the vowels as mere labels (Sinclair & Sinclair, 1984), without granting them the cardinal meaning of the numbers. Besides, even when the task entails greater complexity than the standard task with numbers, their errors do not seem to be attributable to deficiencies in coordination or memory, for the percentage of correct counting is higher than the percentage of correct answers of cardinality, and part of the wrong trials are due to the entire repetition of the vowel sequence used in the counting process, particularly

THE ACQUISITIO~

TABLE5 Percentages of Correet Trials tor the Cardinalit'

Responses in the EJements-Cardinal Bloek 1

Two and five objects Thr,

Counting Cardinality Coun

GroopI 72 42 72 GroupII 100 80 98

in Group l. Consequently, these are all signs that atúibution of meaning. The same may be said in n ooronting again, fOl", as Fuson and Hall (1983) have a way of specifying all objects, sueh that tbe child's of tbe elements in the set, but not of the total or J

We aIso found IlUlJkedly higher performance o~ CdJ~ triak (see Table 5). This high success ~ assation ~ we are faced with the <Cassignmenj pmbfem ~ above. As in the counting bacj ~ maioIy ones of sequence, and were onIy 1 ~ düIchen. We can categorise these errors iJ omitone'wowdorrepeat a vowel; (2) they alter the (3) dJey omiJ: a chip ol' count a ehip twice; and ' 'AJIiids for numbers (see Table 6). _ .Regmdiug the relationship between counting w m C8l1finaJity in Group 1 (see Table 7), MeNen

TABLE 6 PeI ...... dages of CardinaJity and Counting Errors With

the Elements-Cardinal Block

Group J

Gmfioatity errors Comning vowels again 39 Rq:Iearing the sequence 31 Sec¡Dence of number words 16 Random. vowel 14

Comning elIOrs Omil vowels of the correct sequence 61 Repeating a vowel 11 Alta tbe order of the sequenceÜI!Jit a chip

11 6

Double-counting of a chip 6 Substitute the vowels for number words 5

AGO

[1ts) that these are two distinct phenomena. In fact, : practically -the same in the 3 and 4 objects condig the younger children (se e Table 4). However, the lIíges significantly with respect' to the cardinality 4 objects. This again suggests the importance of :orrectly to cardinality questions, as can be inferred ne children count incorrectly and yet still correctly question (see also Russac, 1983). How could this be .tial component of cardinality? Might it not be more

of counting merely as one of the quantification r & Wallace, 1976) that can be used to specify the 'en the sum of two sets? (see Bermejo & Rodríguez, reason that we suggested in the Introduction the :al or situational, but not necessarily theoretical, counting and cardinality.

As for the influence of the elements ihat made up ~, the use of vowels instead of numbers significantly ge of correct cardinality trials in the forwards count;ignificant difference between counting forwards with with 2 and 5 objects) (F [1,62] = 44.62, P < 0.01), as ;roup difference (F [1,62] = 8.53, P < 0.01). These ibuted to ignorance of the vowels on the part of the ce our study procedure. ensured a uniform ability to :1out difficulty. In addition, we know that even very iscriminate numbers and letters, although they may ral and functional differences between them. ity in the vowel condition, the ANOVA 2 (Group I vs ,5 vs 3 and 4 objects) showed significant main effects 13.17, P < 0.01) and for set size (E' [1,62] = 4.2,,e 5). Furthermore, the most frequentIy observed , counting again and repeating the sequence of vowels lting, which we have described as stages 3 and 2 (see , the introduction of vowels seems to bring about a f bebaviour tbat have already been overcome witb e.g. Markman, 1979; Saxe, Gearhart, & Guberman, " 1974). Tbese children probably consider the vowels clair & Sinclair, 1984), witbout granting them tbe , tbe numbers. Besides, even :when the task entails han the standard task with numbers, tbeír errors do mtabIe to deficiencies in coordination or memory, for rrect counting is higher than the percentage of correct ty, and part of the wrong tríaIs are due to tbe entire rel sequence used in thecounting process, particularly


TABLE5 Percentages of Correct Trials'for the Cardinality and Counting

Responses in the Elements-Cardinal Block With Vowels

Two and five objects Three and four objects

Counting Cardinality Counting Cardinality

GroupI 72 42 72 38 Groupll 100 80 98 80

in Group L ConsequentIy, these are all signs that tbere may be a lack of attribution of meaning. Tbe same may be said in regard to the response of comding again. for, as Fuson and Hall (1983) have pointed out, it could be a way of specifying all objects, such that tbe child's response refers to each of tbe eJements in the set, but not of the total or cardinal. .

We ü;o found markedly higher performance on counting triaIs tban on cmIinaIity triaIs (see Table 5). This high success rate seems to ratify our assertion that we are faced with the "assignment of cardinal meaning" pmbJem di-;cossed aboye. As in the counting backwards test, tbe errors were mainly ones of sequence. and were onIy found in the group of JODII3eI' dñldren. We can categorise these errors into four types: (1) they amit eme wwd or repeat a vowel; (2) they alter the order of the sequence; (3) tbey omit a dñp or rount a chip twice; and (4) they substitute the WMds Cm numbers (see Table 6).

Rq¡aJding tbe reIationsbip between counting with vowels and success in canfinatity iD Group 1 (see Table 7), McNemar's test is significant

TABLE 6 fUceiltages ofCardinality and Counting Errors With Vowel s With i n

the Elements-Cardinal Block

Group 1 Group II

CmfinaJily errors Coamting vowels again 39 35 Repeating the sequence 31 8 Sequence of nnmber words 16 Random vowel 14 7

Conntingerrors Omit voweIs of the correet sequence 61 Repeating a voweI iI Alter the order of the sequenee 11 ~tachip 6 Doubk-counting of a ehip 6 Substitute the vowels tor number words 5


TABLE 7 Frequency Tables for the Task of Counting With Vowels

Group IIGroup 1

CardinalityCardinality C 1 TC 1 T

1 O O O1 4 6 10 CountingCounting 25 32C 8 14 22 C 7

T 25 7 32T 12 20 32

C, correet; 1, ineorrect; T, totals.

(x: [1, n = 32] = 5.55, P < 0.05), both globally, when three or more correct trials out of four, was considered a correct response, as well as when analysed individually for every set of objects (i.e. for 2, 3, 4, and 5 objects respectively). The children in Group II .routinely count correctly, but sorne respond incorrectly in the cardinality task (see Table 7). Our findings are similar to and partially support the findings of Gelman and Gallistel (1978) and Wilkinson (1984) in that we observed that counting is acquired prior to cardinality, independent of the number of objects or age of subjects. However, sorne children count incorrectly but respond correctly to the cardinality tasks, sustaining the results found in our counting backwards situation. Counting appears to be a quantification procedure closely related to cardinality in certain contexts. However, counting is probably not an essential component of cardinality, despite the position of Schaeffer et al. (1974).

Cardinal-Elements Block

To avoid repeating ourselves we will present a brief summary of our findings in this section. We analysed our data with a repeated measures ANOVA 2 (Group 1 vs Group II) x 2 (Numbers vs Vowels) x 2 (Visual vs Verbal), as shown in Table 8. We found significant main effects for the three factors (F[1,62] = 7.18, P < 0.01; F[1,62] = 74.37, P < 0.01 and

¡:;l' F[1,62] = 21.57, P < 0.01 in this order), and a significant interaction of age with request form (F[1,62] = 5.39, P < 0.05). Once again we observe that t vowels imply a complication of the task and that Group II carried out their task more successfully than Group 1. Additionally we find tasks presented

1I visually are more difficult than those presented verbally, and that the difference of average s between both groups was greater when the cardinal was presented visually. We will now examine these findings in greater

detail.


ch.ildren's success when using vowels, it see' thIS was due to a difficu1ty in attributing quan the vowels presented, as we point out previ01 high number of errors based on random el c~ild's correct answer in this situation impl wlth vowels, or his ability to carrying out coro and .numbers. In both cases the cognitive p solvmg would be longer and more complicate which ouly numbers are used. As for the sign: could be conjectured that it is more difficult . written symbol, whether this be a numeral o tbis same symbol when it is expressed verl children's language development at these agl fundamentally verbal and not written.

There were also differences in the patten between the two groups (se e Table 9). The two types of errors: (a) counting weIl with ve drip to which the vowel suggested as a cardi: group of chips requested, which is typical of tl in all tbe chips without counting, which is ty stage of cardinality -development. On the COI

of errors in the younger sample: the two aIre. a random number of chips; (d) randomly ha fioaIly (e) counting all the chips present at th

TABLE 9 Percentages of Cardinality Responses in the Cardin

Verbal

Group 1 Gr,

Corred answers 38 CountiDg well and giving in an

extra chip Counting well and giving in only

tbe chip corresponding to the Iast counting word 13

Counting once and giving in all the chips 6

GiYiDg in all chips without counting 18

Giving in differen! chips a! random 16 Giving in a single chip at random 9

THE ACaUISITION OF CARDINALlTY 245

TABLE 8 Averages and Standard Deviations from Transformed Proportions ofCorrect Responses in Cardinality Within

the Cardinal-Elements Block

Numbers Vowels

Verbal Visual Verbal Visual

Group 1 2.36 (0.20)

Group II 2.41 (0.00)

1.99 (0.71)

2.25 (0.50)

1.35 (0.80)

1.70 (0.83)

1.09 (0.67)

1.65 (0.84)


Numerical Tasks. We observed that the subjects of both groups carry out the verbally requested tasks more effectively than the visually requested tasks (see Table 8). Likewise, within the visual situation, one can appreciate two patterns of erroneous behaviour that did not arise when the cardinal was presented verbally: (1) randomly giving in any or all chips; and (2) counting the chips correctly and giving in only the chip to which the number corresponding to the cardinal in question is assigned (i.e. giving the fourth chip, when asked to give four chips). The first of these errors appeared in the group of younger children (13% of trials) and corresponds to the aforementioned first stage because this behaviour involves, at least partially, a miscomprehension of the situation and a random choice of the number of elements given. The second type of error was made with approximately equal frequency in both groups (9% and 6% of trials in Groups 1 and II respectively) and would be typical of the fourth stage. In committing this second type of error, the child focuses on the last number word of the counting sequence, as he does in following the "How many" rule, but additionally is able to understand that the last number word repeated represents a particular (the last-counted) object. This response is more frequent in the tests with vowels than in tests with numbers (see Table 9 for vowel task results). It would be quite interesting to analyse how that same number word becomes a label of the whole set in a latter developmental moment, though our current data does not allow us to make this analysis.

Tasks With Vowels. When the cardinal was requested by means of a vowel, we found that on the one hand the number of correct trials was lower, and on the other carrying out of tests was worse with visual than with verbal presentation, just as occurs in the former (numeral presentation) situation (see Table 8). With regard to the drop in the

THE ACaUISJTIO

of them in. The last error mentioned correspond! the two former errors fall under the first stage.

CONCLUSIONS

Our analyses show that the older children better I cardinality than did the younger ones, and that tll consistent over every test we included in tbis stti ences between the two groups increased as thel increased. Likewise, our results allow us to conl tions (counting forwards with numbers) are sim~ (counting backwards and using vowels). In the ~l' success rate was clearly high, whilst in the latt under 25%. This "novel situation" research meth avoid the traditional difficulties inherent in rese~ (2-3 years) and successfully work with preschool development of cardinality, even as it is just unfoll only enable us to specify the levels of cardinalit) analyse those cognitive processes that interven cardinality. More specifically the situation of e example, empirically demonstrates the existence I (the rule,of "how many") prior to cardinality wh investigate with previous counting methodology.

Sorne authors underestimate the possible relatil and cardinality (see, e.g. Piaget & Szerninska, 194 the irnportance of this relationship (see, e.g. Gel Piaget and Szerninska maintain that counting is al and on the contrary Gelrnan and Gallistel clairn tl1 before cardinality. Our data from both nurnber that counting is acquired before cardinality. Thi· between counting and cardinality may be due in p~ rneasured these phenornena, or to how Weste counting to determine a cardinal exactly. Our fin cardinality does not necessarily rule out a cardinal is arrived at by subitising. We see this particularl] ance of children who respond incorrectly in terms ly in terrns of cardinality (see also Russac, 1983)., Lago, & Rodríguez, 1989) confirms that in specifi¡ quantification procedure closely related to, but noi of, cardinality, as Schaeffer et al. (1974) have su~

Error analysis has fundamentally led us to sug stages (though not in a strong Piagetian sense

246 BERMEJO ANO LAGO 'GO

TABLE 7 children's suecess when using vowels, it seems reasonable to suggest that ables for the Task of Counting With Vowels this was due to a diffieulty in attributing quantitative or cardinal meaning to

the vowels presented, as we poiot out previously, and as may be seen in the Group IIGroup 1 bigh number of errors based on random choices. It may also be that a Cardinality child's correct answer in this situation implies either his ability to count

Cardinality C 1 T with vowels, or his ability to carrying out correspondences between vowels C 1 T

and numbers. In both cases the eognitive proeesses involved in problem O O O4 6 10 1 solving would be longer and more complicated in this test than in the one in Counting C 25 7 328 14 22 whieh only numbers are used. As for the significanée of the request form, it

7 32 could be conjectured that it is more difficult to identify the meaning of the T 2512 20 32 written symbol, whether this be a numeral or a vowel, than to understand

lrrect; T, totaIs. Ibis same symbol when it is expressed verbally, given that the level of children's language development at these ages (particularly in Group 1) is fundanlentally verbal and not written. í, P < 0.05), both globally, when three or more

There were also differences in the pattem of errors made with vowels our, was eonsidered a correet response, as well as between the two groups (see Table 9). The older subjects basic~lly madelually for every set of objeets (i.e. for 2., 3, 4, and 5 NO types of errors: (a) counting well with vowels, but handing in only the The children in Group JI .routinely eount correctly, chip to which the vowel suggested as a cardinal is assigned, instead of the :orrectly in the cardinality task (see Table 7). Our group of chips requested, which is typical of the fourth stage; and (b) giving ) and partially support the findings of Gelman and in all the chips without counting, which is typical of behaviour in the first N"ilkinson (1984) in that we observed that counting stage of cardinality-development. On the contrary, we observed five typesardinality, independent of the number of objects or oferroIS in the younger sample: the two already mentioned; (c) handing in lever, sorne children count incorrectly but respond a random number of chips; (d) randomly handiog in any single chip; and iinality tasks, sustaining the results found in our finalIy (e) counting alI the chips present at the outset and then handing all situation. Counting appears to be a quantification

lated to cardinality in certain contexts. However, Got an essential component of cardinality, despite the . TABLE 9

Plen:entages of Cardinality Responses in the Cardinal-Elements Block With Vowels et al. (1974).

Verbal Visual ts Block

Group 1 Group II Group 1 Group IIourselves we will present a brief summary of our m. We analysed our data with a repeated measures Conect answers 38 58 23 . 55 vs Group JI) x 2 (Numbers vs Vowels) x 2 (Visual vs Counting welI and .giving in an

extra chip 2 3 21 Table 8. We found significant main effects for the Counting welI and giving in only~] = 7.18, P < 0.01; F[1,62] = 74.37, P < 0.01 and

the chip corresponding to the : 0.01 in this order), and a significant interaction of age last counting word 13 31 13 31 1[1,62] = 5.39, P < 0.05). Once again we observe that Counting once and giving in all ,lieation of the task and that Group JI carried out their the chips 6 6

Giving in all chips without ly than Group 1. Additionally we find tasks presented counting 18 9 20 12liffieult than those presented verbally, and that the

Giving in different chips at random 16 16 es betweeo both groups was greater when the cardinal Giving in a single chip at random 9 19 llly. We wiU now examine these fiodiogs in greater

, L

THE ACQUISITION OF CARDINALlTV 245

TABLE 8 ard Deviations from Transformed et Responses in Cardinality Within dinal-Elements Block

umbers Vowels

Visual Verbal Visual

1.99 1.35 1.09 (0.71) (0.80) (0.67)

2.25 1.70 1.65 (0.50) (0.83) (0.84)

re is 2.41.

served that the subjects of both groups ted tasks more effectively than the visually . Likewise, within the visual situation, one erroneous behaviour that did not arise when bally: (1) randomly giving in any or all chips; ectly and giving in only the chip to which the cardinal in question is assigned (i.e. giving

:0 give four chips). The first of these errors ger children (13% oftrials) and corresponds 1ge because this behaviour involves, at least of the situation and a random choice of the

rhe second type of error was made with :y in both groups (9% and 6% of trials in and would be typical of the fourth stage. In f error, the child focuses on the last number e, as he does in following the "How many" to understand that the last number word

lf (the last-counted) object. This response is ith vowels than in tests with numbers (see . It would be quite interesting to analyse how omes a label of the whole set in a latter Igh our current data does not allow us to

n the cardinal was requested by means of :he one hand the number of correct trials carrying out of tests was worse with visual n, just as occurs in the former (numeral rabIe 8). With regard to the drop in the

THE ACQUISITION OF CARDINALlTV 247

of them in. The last error mentioned corresponds to the third stage, while the two former errors fall under the first stage.

CONCLUSIONS

Our analyses show that the older children better understood the notion of cardinality than did the younger ones, and that these age differences were consistent over every test we included in this study. Furthermore, differences between the two groups increased as the complexity of the tasks increased. Likewise, our results allow us to conclude that familiar situations (counting forwards with numbers) are simpler than novel situations (counting backwards and using vowels). In the former case the children's success rate was clearly high, whilst in the latter it dropped globally to under 25%. This "novel situation" research methodology has enabled us to avoid the traditional difficulties inherent in research with young children (2~3 years) and successfully work with preschool subjects to measure the development of cardil1ality, even as it is just unfolding. These new tests not only enable us to specify the levels of cardinality acquisition, but also to analyse those cognitive processes that intervene in the acquisition of cardinality. More specifically the situation of counting backwards, for example, empirically demonstrates the existence of a developmental stage (the rule,of "how many") prior to cardinality which had be en difficult to investigate with previous counting methodology.

Some authors underestimate the possible relationship between counting and cardinality (see, e.g. Piaget & Szeminska, 1941); while others insist on the importance of this relationship (see, e.g. Gelman & Gallistel, 1978). Piaget and Szeminska maintain that counting is acquired after cardinality, and on the contrary Gelman and Gallistel c1aim that counting is developed before cardinality. Our data from both number and vowel tasks suggest that counting is acquired before cardinality. This significant relationship between countiog and cardinality may be due in part to how we empirically measured these phenomena, or to how Western cultures usually use counting to determine a cardinal exactly. Our findings of counting befo re cardinality does not necessarily rule out a cardinality without counting that is arrived at by subitising. We see this particularly clearly in the performance of children who respond incorrectly in terms of counting, but correctly in terms of cardinality (se e also Russac, 1983). Our research (Bermejo, Lago, & Rodríguez, 1989) confirms that in specific situations counting is a quantification procedure closely related to, but not an essential component of, cardinality, as Schaeffer et al. (1974) have suggested.

Error analysis has fundamentally led us to suggest the existence of six stages (though not in a strong Piagetian sense) in the acquisition of

I


cardinality in the standard counting situation: (1) incomprehension of the situation and random response; (2) repetition of the number word sequence given in the counting; (3) counting the objects again; (4) giving the last number word of the sequence used (the rule of "how many"); (5) responding with the largest number word of the given sequence; and (6) a true cardinality response. The second and third stages are differentiated basically in that in the former the child does not refer to objects, while in the latter he carries out a strict number word-object correspondence. The fourth stage is an important step towards cardinality, for the child not only knows that each number word in the series represents an object in the set, which is typical of the third stage, but also can correctly answer the question "how many are there?" by giving the last counting word. In the fifth stage, the child knows that the cardinal corresponds to the largest number word of the given sequence. However, it is not until the following (sixth) stage that the child undertands that the last number word given in the forward count isn't only the largest and represents the last object counted, but also represents all the elemeilts counted. This last developmental step is, in our opinion, very interesting, but our data do not allow us to specify how it is acquired. Our developmental sequence, while differing substantially from that of Gelman and Gallistel (1978), shares much in common with that of Fuson (1988), although we believe our findings allow us to propose a sequence better defined and more comprehensive.

Finally, although many authors (Fuson, 1988; Fuson et al., 1985b; Ginsburg & Russell, 1981; Wilkinson, 1984) claim that the size of the sets (from 2 to 19 approximately) does not have an effect on the response to the "how many" rule, our data show that this factor can be relevant to the cardinality response, as appears clearly in the counting backwards test. This is most probably due to the fact that the "How many" rule is related to the counting sequence, while cardinality is related to the set of objects as well.

Manuscript accepted 3 August 1989

REFERENCES

Bermejo, V. (1989). Factores espacio-semánticos y tipicidad en conductas de clasificación e inclusión. Estudios de Psicología, 37, 31-43.

Bermejo, V., Lago, M. O., & Rodríguez, P. (1989). Procedimientos de cuantificación y cardinalidad. Revista de Psicología General y Aplicada, 4, 483-491.

Bermejo, V. & Rodríguez, P. (1987). Estructura semántica y estrategias infantiles en la solución de problemas verbales de adición. Infancia y Aprendizaje, 39-40, 71-81.

Brainerd, C. J. (1979). The origins of the number concepto New York: Praeger. Clements, D. (1984). Training effects on the development and generalization of piagetian

THE ACQUISITION

logical operations and knowledge of number. Jouma/ of 766-776.

Fuson, K. (1988). Children's counting and concepts ofnumber. : ·Fuson, K. & Hall, J. (1983). TIte acquisition of early number w

analysis and review. In H. Ginsburg (Ed.), The developme, New York: Academic Press. (Pp. 49-107.)

Fuson, K., Lyons, B., Pergament, G., Hall, J., & Kwon, Y. terms on class-inc1usion and on number tasks. Cognitive PS)

Fuson, K., Pergament, G., & Lyons, B. (1985a). Collection te the cardinality rule. Cognitive Psychology, 17, 315-323.

Fuson, K., Pergament, G., Lyons, B., & Hall, J. (1985b). ( cardinality rule as a function of set size and counting accur 1429-1436.

Fuson, K., Richards, J., & Btiars, D. (1982). The acquisition al word sequence. In Brainerd, C. J. (Ed.), Children's logical Progress in cognitive development. New York: Springer-Ver

Gelman, R. (1982). Basic numerical abilities. In R. Sternbc psychology of human intelligence. London: Lawrence Erl 181-265.)

Gelman, R. & Gallistel, C. (1978). The child's understanding OJ Harvard University Press.

Gelman, R., Meck, E., & Merking, S. (1986). Young childr Cognitive Development, 1,1-29.

Ginsburg, H. & Russell, R. (1981). Social class and racial infh thinking. Monographs of the Society for Research in Child j

·Hodges, R. & French, L. (1988). The effect of class and col class-inclusion, and nuinber conservation task. Child Devel.

Kingma, J. & Koops, W. (1984). Consequences of task varia1 Genetic Psychology Monographs, 109, 77-94.

.KIahr, D. & Wallace, J. (1976). Cognitive development: An Hillsdale, N.J.: Lawrence Erlbaum Associates Inc.

Markman, E. M. (1979). Classes and collections: Conceptual abilities. Cognitive Psychology, 39, 395-411.

Michie, S. (1984). Number understanding in preschool childrc tional Psychology, 54, 245-253.

Michie, S. (1985). Development of absolute and reIative cone children. Developmental Psychology, 21, 247-252.

Piaget, J. & Szeminska, A. (1941). Le génese du nombre Ci

lachaux et Niestlé. Russac, R. (1983). Early discrimination among small objec1

perimental Child Psychology, 36, 124-138. Saxe, G. B. (1979). Developmental relations between nota

conservation. Child Development, 50, 180-187. Saxe, G. B., Gearhart, M., & Guberman, S. R. (1984). Th.

number development. In B. Rogoff & J. Wertsch (Eds), Ch proximal development. San Francisco: Jossey-Bass. (Pp. 19

Schaeffer, B., Eggleston, V. H .• & Scott, J. L. (1974). Nu children. Cognitive Psychology, 6, 357-379.

Sinclair, A. & Sinclair, H. (1984). Preschool children's inter¡ Human Learning, 3, 173-184.

1counting situation: (1) incomprehension of the ,onse; (2) repetition of the number word sequg; (3) counting the objects again; (4) giving the sequence used (the rule of "how many"); (5) ;t number word of the given sequence; and (6) a ['he second and third stages are differentiated basithe ehild do es not refer to objeets, while in the

riet number word-objeet correspondenee. The 1t step towards cardinality, for the child not only vord in the series represents an objeet in the set, hird stage, but also can eorrectly answer the there?" by giving the last eounting word. In the ws that the cardinal corresponds to the largest sequenee. However, it is not until the following

1undertands that the last number word given in )nly the largest and represents the last object nts all the elements counted. This last developion, very interesting, but our data do not allow us ~d. Our developmental sequenee, while differing f Gelman and Gallistel (1978), shares much in lO (1988), although we believe our findings allow better defined and more comprehensive. authors (Fuson, 1988; Fuson et al., 1985b; Ginskinson, 1984) claim that the size ofthe sets (from es not have an effeet on the response to the "how vthat this factor can be relevant to the cardinality rly in the eounting backwards test. This is most t that the "How many" rule is related to the eardinality is related to the set of objects as well.

Manuscript accepted 3 August 1989

REFERENCES

~spacio·semánticos y tipicidad en conductas de clasificación e 'ogía, 37, 31-43.

Rodríguez, P. (1989). Procedimientos de cuantificación y :ología General y Aplicada, 4, 483-491. (1987). Estructura semántica y estrategias infantiles en la

des de adición. Infancia y Aprendizaje, 39-40, 71-81. igins of the number concepto New York: Praeger. effects on the development and generalization of piagetian


logical operations and kuowledge of number. Joumal of Educational Psychology, 5, 766-776.'

Fuson, K .. (1988). Children's counting and concepts ofnumber. New York: Springer·Yerlag. -Fuson, K. & Hall, J. (1983). The acquisition of early number word meanings: A conceptual

analysis and review. In H. Ginsburg (Ed.), The development of mathematical thinking. New York: Academic Press. (Pp. 49-107.)

Fuson, K., Lyons, B., Pergament, G., Hall, J., & Kwon, Y. (1988). Effects of collection terms on class-inclusion and on number tasks. Cognitive Psychology, 20, 96-120.

Fuson, K., Pergament, G., & Lyons, B. (1985a). Collection terms and preschoolers' use of the cardinality rule. Cognitive Psychology, 17, 315-323.

Fuson, K., Pergament, G., Lyons, B., & Hall, J. (1985b). Children's conformity to the cardinality rule as a function of set size and counting accuracy. Child Development, 56, 1429-1436.

Fuson, K., Richards, J., & Briars, D. (1982). The acquisition and elaboration of the number word sequence. In Brainerd, C. J. (Ed.), Children's logical and mathematical cognition: Progress in cognitive development. New York: Springer-Yerlag. (Pp. 33-92.)

Gelman, R. (1982). Basic numerical abilities. In R. Sternberg (Ed.), Advances in the psychology of human intelligence. London: Lawrence Erlbaum Associates Ltd. (Pp. 181-265.)

Gelman, R. & Gallistel, C. (1978). The child' s understanding ofnumber. Cambridge, Mass.: Harvard University Press.

Gelman, R., Meck, E., & Merking, S. (1986). Young children's numerical competence. Cognitive Development, 1, 1-29.

Ginsburg, H. & Russell, R. (1981). Social class and racial influences on early mathematical thinking. Monographs of the Society for Research in Child Development, 46. .

·Hodges, R. & French, L. (1988). The effect of class and collection labels on cardinality, class-inclusion, and number conservation task. Child Development, 59, 1387-1396.

Kingma, J. &'Koops, W. (1984). Consequences of task variations in cardination research. Genetic PsychoIogy M01Wgraphs, 109, 77-94.

Klahr, D. & Wallace, J. (1976). Cognitive development: An informarion processing view. Hillsdale, N.J.: Lawrence Erlbaum Associates Ine. --

Markman, E. M. (1979). Classes and colIections: Conceptual organization and numerical abilities. Cognitive Psychology, 39, 395-411.

Michie, S. (1984). Number understanding in preschool children. British Journal of Educational Psychology, 54, 245-253.

Michie, S. (1985). Development of absolute and relative concepts of number in preschool children. Developmental Psychology, 21, 247-252.

Piaget, J. & Szeminska, A. (1941). Le génese du nombre chez l'enfam. Neuchatel: De· lachaux et Niestlé.

Russac, R. (1983). Early discrimination among small object collections. Joumal of Ex· perimental Child Psychology, 36, 124-138.

Saxe, G. B. (1979). Developmental relations between notational counting and number conservation. Child Development, 50, 180-187.

Saxe, G. B., Gearhart, M., & Guberman, S. R. (1984). The social organization of early number development. In B. Rogoff & J. Wertsch (Eds), Children's learning in rhe zone of proximal development. San Francisco: Jossey·Bass. (Pp. 19-30.)

Schaeffer, B., Eggleston, Y. H., & Seott, J. L. (1974). Number dcvelopment in young children. Cognitive Psychology, 6, 357-379.

Sinclair, A. & Sinclair, H. (1984). Presehool ehildren's interpretation of written numbers. Human Learning, 3, 173-184.


Wagner, S. & Walters, J. A. (1982). A longitudinal analysis of early number concepts: From numbers to number. In G. Forman (Ed.), Action and thought. New York: Academic Press. ·(Pp. 137-161.)

Wilkinson, A. C. (1984). Children's partial knowledge of the cognitive skills of counting. Cognitive Psychology, 16, 28-64.

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