THE RELATIONSHIP BETWEEN CARDINALITYAND UNDERSTANDING

WITTEN NUMBER

AMANDA E. TESSARO

A thesis submitted to the Faculty of Graduate Studies in partial fulfilment of the

requirement s for the degree of Master of Arts

Graduate Programme in Psychology York University Toronto, Ontario

March 1999

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The Rela t i onsh ip Between Cordinality and Understanding I J r i t t e n Number

by Amanda Tessaro

a thesis submitted to the Faculty of Graduate Studies of York University in partial fulfiltment of the requirements for the degree of

Master of A r t s

Permission has been granted to the LIBRARY OF YORK UNIVERSIN to lend or seIl copies of this thesis, to the NATIONAL LlBRARY OF CANADA to microfilm this thesis and to lend or sel1 copies of the film, and to UNIVERSITY MICROFILMS to publish an abstract of this thesis. The author reserves other publication rights, and neither the thesis nor extensive extracts from it may be printed or otherwise reproduced without the author's written permission.

Abstract

The relationship between cardinality and understanding written numbers was

investigated. The hypothesis was that before a child can understand how Witten

numbers are symbolic he or she must first understand how number itself is symbolic, how

number represents quantity. Thus, cardinality, the understanding that numbers in the

counting sequence describe the quantity of the set, is a prerequisite for understanding the

meaning of written number notations. Children's level of understanding of cardinality

was examined using three subtasks that increased in demands placed on the chiId. It was

reasoned that in order to succeed at these tasks, children had to be able to make the

comection between the last counted word and the total quantity of the set. Their

understanding of written number was determined by success on the moving symbol task

for numbers, analogues, and number words. This task examines whether or not children

understand that number is invariant and that there is a one-to-one correspondence

between the written number and the quantity it represents. This task also examined the

relationship between the size of the nurnbers and symbolic representation. The results

indicated that children did not need to succeed on the cardinality subtasks in order to be

successful on the moving symbol task although results for the analogue condition were in

the predicted direction. The level of dificulty of the numbers affected performance for

the word condition, where representations became hard for both the middle and dificult

levels, and for the analogue condition, where representations became hard with difficult

items only.

Acknowledgements

1 am gratefùl to Prof Ellen Bialystok for her assistance with this study and Judith

Codd for her assistance in data analysis. 1 am also gratefiil to the teachers, parents and

children o f the many daycares in Hamilton and Toronto that took part in this study.

Table of Contents

Ab stract

Acknowledgements

List of Tables

List of Figures

The Relationship Between Cardinality and Understanding Written Number

Stages in the Development of Symbolic Representation of Letters and Numbers

Aiphabetic Knowledge

Written Notations of Number

Differences Between Letters and Numbers

The Relationship between Cardinality and Syrnbolic Representation

The Relationship between Counting and Symbolic Representations

Notational Representations for Number

Hypotheses

Method

Subjects

Tasks

Procedure

Results

Performance on Cardinality Tasks

The Moving Symbol Task

The Moving Symbol Task and Cardinality

vii

Correlation's

Discussion

References

List of Tables

Mean scores for the Visually Cued Recail Task by condition and age.

Mean scores for the "Cive-Me-X' Task by condition and age.

Mean scores for the "Cive-Me-X" Task by condition and question.

Mean scores for the "Are There More X Than Y' Task by condition and age.

Mean scores for the Sharing Task by condition and age.

Mean scores for the first and retum position of the Moving Syrnbol Task by condition and age.

Mean scores for the switch position of the Moving Symbol Task by condition and age.

Mean scores for the switch position of the Moving Symbol Task by level of difflculty, condition and age.

Chi-squares for the Moving Symbol Task and the cardinality subtasks.

Stnngent Chi-squares for the Moving Symbol Task and the cardinality subtasks.

10 Correlation rnatrix for the cardinality subtasks.

List o f Figures

1 Two-way interaction for condition and question on the "Give-Me-X' Task. 76

2 Three-way interaction for level of difficulty, condition and age on the

Moving Symbol Task. 77

3 Two-way interaction for level of difficulty and condition on the Moving

SyrnboI Task. 78

The Relationship Between

Cardinality and Understanding Written Number

Preschool children possess a large amount of detailed knowledge about

number and quantity. They are able to express quantity judgments in the form of

relative size labels, for example, smalI, big or Iittle (Resnick, 1989). They can count and

ofien can recognize written numbers. In order for children to identify a written number,

for example, 3 as a 'three' they do not need to know that number words describe the

quantity of objects. However, if children are to interpret written representations of

numbers, they shouId first have some idea of what is being represented and that the

numbers in the counting sequence refer to quantities and that they describe the quantity

of the set. They should understand cardinality.

Stages in the Development of Symbolic Representation of Letters and Numbers

Bialystok identifies three stages in children's understanding of symbolic

representations for the alphabetic and numerical systems, a gradua1 process, which takes

several years to complete. These stages are characterized by different types of mental

representations. The first stage, conceptual representation, occurs when children learn

the alphabet and numerical sequence as farniliar routines and is based on perceptual

properties of sounds and forms. At this point, they d o not understand the routines as

collections of discrete Ietters and numbers. As the basis for written language, these

representations are inadequate for reading as children are unable to tell you what letter

follows another in the order without going back to the beginning and reciting the whole

sequence. In this stage, Bialystok argues that the child may have only a simple,

unanalyzed representation of these systems.

Formal Representation, the second stage, occurs when children understand the

individual constituents as separate camponents either orally or in their written f o m .

Letters and numbers can be taken out of context of the alphabet and the number

sequence and recognized individually. They can then be understood explicitly. By

producing letters and numbers children begin to see them as separate objects with

specific visual characteristics that have meaning in themselves as objects. However, they

are not yet symbols that stand for meanings. There is probably a parallel development

through the stages in each of the oral and written domains.

At about six years of age, children understand letters and numbers as symbols.

This is the th id stage, symbolic representation. "Symbolic knowledge is represented as

a relation between a symbol (forrn) and an entity (meaning)" (p. 76) (Bialystok, 1991).

Letters and numbers now refer to specific values. Children no longer make the incorrect

assumption that the meaning of the symbol is somehow in the object that is being

represented. They know that there is a difference between the meaning of objects and the

meaning of symbols. They now understand that letters and numbers are not objects but

simply place holders written down to signiQ a parîicular sound or quantity (Bialystok,

1991).

Alphabetic Knowledge

Bialystok (1991) argues that a formal knowledge of letters i s necessary but

not suficient for leamhg to read. Children must understand the correspondence

between oral and written language to reach the symbolic stage where written language

can be represented symbolically. For this, a child must l e m two properties that are not

found in forma1 representation. The first is one-to-one correspondence between a

symbol and its referent. Each symbol stands for a single meaning, for example, each

letter signifies a sound. In an alphabetic writing system, the implication of this is that

more sounds, or longer words, are represented by more letters. The second is

invariance. Since written language stands for meanings, nothing in the presentation of

that wriîten symbol or the context in which it occurs can change the meaning it

represents.

The essential insight children must achieve for literacy is the symbolic

relation by which letters represent sounds and that relation is characterized by the two

properties described above. The names of the entities (letters) achieve symbolic status

once they have been associated with a unique written form and assigned a conventional

meaning (sound). When children realize that alterations in the way a letter is written

does not affect that symbolic representation, they are ready to learn to read. Symbolic

representation presupposes knowledge of both oral and written systems, but builds on

that knowledge by linking the two through a meaning relation @ialystok, 1992).

Using the moving symbol task with words, Bialystok (1991) examined

whether preschool children who have not yet leamed to read understood that

representations of language are invariant and that there is a one-to-one correspondence

between the symbol and its referent. In this task children rnust name a word when it is

placed under a picture that conesponds to the word and then name it again when the

word is placed under an incongment picture. In order to do this task children must first

understand the connection between oral and written language, specifically, the

connection between the letters which together form words and the meanings to which

the words refer. Unless this connection is made explicit, children fail to see written

language as symbolic and they will fail the moving symbol task. Sialystok found that

even prereading children who were able to identie letters and the corresponding sounds

were unable to relate them to their symbolic meaning. They were quick to believe that

once the card shifis to the noncorresponding picture, the representation on that card has

changed to reflect that word (Bialystok, 1997; Tannenbaum, 1997). They believed that

the identity of the printed word changed when it moved to the incongruent picture.

The moving symbol task with words was found to be a good predictor of how

children perfomed on other tasks. Prereading children who were correct on the rnoving

symbol task with words tried to honor the restrictions of letter-sound correspondences.

They understood that the number of letters needed to make a word depends on the

number of sounds in the word. They also understood that taking letters away fkom

words changes them. They were successful on the word task where they were giver, two

words orally and had to determine which was the longer word. They were successful

because they understood that there is a relation between the number of sounds in a word

and the length of the word. Finally, these children were able to use invented spellings to

make words with plastic letters. The children who failed the moving symbol task with

words were unconstrained by the principle of one-to-one correspondence of letter-to-

sound representation. Bialystok (1 99 1) concludes that only when children have passed

the moving symbol task and the word task do they understand the symbolic nature of

pnnt.

Children make errors in interpreting pnnt when they fail to understand this

relationship. Ferreiro and Teborosky (1 982) found that young children counted the iines

and circles that make up letters rather than the individual characters when asked to count

the number of letters in a word. They found that "a letter receives different labels

depending on whether it appears in the context of other letters.. . or whether it appears by

itself' (p. 30). Many prereading children believe that ri written string must contain three

letters to be a word or that no two letters c m be the same within a word (Ferreiro, 1984).

Some preschool children studied believed for example, that one dog should be written

with one letter, while three dogs should be wriîîen with three letters (Tolchinsly

Landsmann & Levin, 1987). A study by LeWn and Tolchinsky-Landsmann (1989),

found that young children ofien preserved the size, shape and colour of objects in

writing the names of objects. Finally, pre-reading children often believe that their

scribbles can be read by others. Bialystok (1995) suggests that these children have

begun to comprehend the 'purpose and structure of print put] have not yet understood

the fùnction of those forms for representing language" @p. 333-334). Thus, these

children have not made the connection between what is d e n and what it represents.

This helps explain their poor performance on the moving word task.

According to Vygotsky (1978), written words are second-order symbols. He

provides the example of a drawing of a dog, which he calls first order symbolism

because the representation directly indicates its meaning. I fa leg is erased the drawing

still shows a dog because the meaning of intention is not a logical surn of its

constituents. With written words, the representation indicates an intermediate system

that in turn signifies its meaning. Therefore the word d-O is not a defonned version of

the word d-O-g. It is a representation of a different word. This requires greater cognitive

complexity to represent, and such a second-order relation may be relatively inaccessible

to younger children. This provides an explanation for why the younger children were

having difficulty, despite being able to recognize the letters and even their sounds.

Written Notations of Number

Before children can l e m the systern of written numerals, they must first l e m

the sequence of number-words and then relate each numeral to a particular number-word

in the sequence (Fuson & Kwon, 1992). However, "the ability to count does not assure

that children will learn to use written numerals either as an aid to counting or as a clue to

quantity" (Bialystok & Codd, 1996, p.4). Children must first leam that the two are

linked. Hughes (1986) asked preschoolers to count a row of magnetic numbers laid out

in their correct sequence fiom 1 to 9. Only a few children answered 'nine' without

counting. They had realized that because the sequence was cornplete (included every

nurnber between 1 and 9), there must be nine numerals in the sequence. Ifthe first few

numerals were concealed, many children had no strategy for finding the answer.

Finally, children were unable to use magnetic numbers to help them count a group of

objects. They did not make use of one-twne correspondence (Hughes, 1991).

The literature suggests that there is a discrepancy between children's ability to

produce numerical notation and their understanding of what those notations represent.

A number of researchers have looked at the beginnings and development of children's

mathematical symbolism. Ahrdice (1977) asked 3 to 6-year-olds to make written

representations of some mathematical ideas. She pIaced small plastic mice on the table

and asked the child to put something on paper to show how many there were. Aimost

half the three-year-olds and three quarters of the four-year-olds used tallies, circles or

pictures to indicate quantity. Otherwise, more global representations that were not

considered adequate were used. Al1 5 and 6-year-olds made reasonably adequate

representations of quantity, which included both analogue representations and written

numerals. Thus, the majority o f the children were capable of using abstract

representations to "preserve on paper the essential information concerning number" (p.

142).

Sinclair, Siegrist, and Sinclair (1982) conducted a similar study with children

aged 4 to 6. The children were asked to represent with paper and pencil collections of

objects (for example, toy cars), which varied in cardinal value from one to eight, and

numerical quantities that were given verbally by the experimenter (for example, three

houses). Al1 except one child were able to produce interpretable notations. Although

most children used several notational types, there was a pattern of increasingly complex

representations across age groups, fiom representations which made use of a one-to-one

correspondance between the quantity to be counted and the number of tallies to written

numbers which are syrnbols for quantity without the one-to-one correspondance. The

youngest children used global representations which did not correspond to the quantity

of objects whiIe older children had moved to analogue representations, for example,

tallies, which were in one-to-one correspondence to the number of objects. The later

types of representation, used by the oldest children, were written numerals, as well as

the number written alphabetically or adding something to specifL the kind of object

being represented, for example, "3 houses".

Preschool children7s interpretation of written numbers was again examined by

Sinclair and Sinclair (1984). They presented the children with various instances of the

use of numerals, for example, a birthday cake with 5 candles, and asked what the

number was and what it meant. Several response types were distinguished and based on

these they found that the majority of children used global representations, for example,

'Tt's a birthday partf', and specific function resporises, for example, "He's 5". Sinclair

and Sinclair believe that global responses occur first in isolation and then are combined

with specific responses. Gradually the global response type should disappear.

Similar results were found in Hughes' (1 986) study. Children aged 3 to 7

were asked to represent the quantities 1 ,2 ,3 , 5, md 6 on paper. Bricks were placed on

the table in front of them and children were asked T a n you put something on paper to

show how many bricks are on the table?". He categorized their responses into four

categories: idiosyncratic (scribbles), iconic responses (one-to-one correspondence, for

example, tallies), pictographic (which represented something of the appearance of what

was in fiont of them as well as its numerosity), and symbolic (conventional numerals).

He found that the preschool children most often used iconic and idiosyncratic methods,

whereas the older children used pictographic and symbolic. For 7-year-olds, the

symbolic method was most common. Older children were more accurate than younger

children in reading their representations in part because idiosyncratic representations

were oflen not recognized.

Bialystok and Codd (1996) Iooked at 3-5 year old children's ability to use and

interpret written representations of quantity. In a method similar to Hughes (1 986),

children were asked to produce or select a representation and use it later to recall the

number of items in a closed box. Responses were categorized as global, analogue

(similar to Hughes pictographic and iconic combined) and numeral. They found an

overall trend in which almost al1 5-year-olds were using numerals, 4-year-olds used

numerals just over half of the time and were inconsistent on the other two possibilities,

and the 3-year-olds showed no preference. Children correctly stated the number of

items for the numeric notations but had more difficulty in interpreting their analogue

representations. They were correct only one fifih of the time when global methods were

used. In general, children's production of symbols became increasingly conventional.

Five-year-olds understood that numerals were the best choice to indicate quantity. The

3-year-olds' use of global representations may have indicated that they did not fùlly

understand the comection between notations and quantity, and that they did not grasp

the cardinal significance of the notations.

In the syrnbol selection task, the 3-year-olds used numerals 60 % of the time.

It was suggested that these children may not have been suficiently skilled in writing

numerals but could use them successfùlly when they were not required to produce them

themselves.

Al1 of these studies have found a developmental trend in children's ability to

produce conventional numerals as representations of quantity. They have progressed

through the use of idiosyncratic methods and analogue representations to the use of

symbolic responses, a deveIopment that takes several years.

Differences Between Letters and Numbers

Like letters, written numbers are also second order symbols. However,

children appear to have more knowledge for representations of written numbers than for

written words at a young age. There are several differences beîween written words and

nurnbers that may explain this discrepancy. Children learn the counting words earlier

than letters and they assign unique values for numbers before letters. Research by Wym

(1990) found that by age 2;4, children were able to identie 'one' and they learn the

meanings of additional number words one at a time for progressively larger numbers .

This may be because individual letters are more abstract than individual written

numbers. For example, the letter 'Wb" on its own does not have a specific meaning

except to refer to a particular phonemic contrast, while the number "2" on its own

represents a particular quantity. Thus, individial numbers have meaning whereas

individial letters do not. This may help children to treat numbers individually

(Bialystok, 1992; Bialystok & Codd, 1997). Letters are not isolated like numbers but

rather they occur in words and sentences, which rnay make the connection between the

symbolic letter and its referent more dificult (Wynn, 1992). This rnay explain results

by Tannenbaum (1997) who found children scored significantly lower on the moving

symbol task with words than with numbers, analogues and pictures.

Therefore, simply knowing how to count rnay help children on the moving

symbol task with numbers in making the connection between a number and the quantity

it represents. This task is similar to the moving symbol task with words except that the

written words are replaced with numbers that correspond to a pile of similar objects. For

example, if one pile of objects has 5 and the other pile has 2 objects and children know

the counting sequence, they know that 5 comes later than 2 in the counting sequence.

Thus, they rnay have an advantage for rnaking the connection between the written

number and the quantity because they rnay realize that the two cards are displaying

different arnounts.

Children know the words in the counting sequence are number words. That

is, they realize that 5 is a cardinal quantity. According to Wynn (1990), if children

know each number word refers to a specific, unique numerosity then they will restrict

the meanings of the number words so that no two refer to the same numerosity. She

found that by the tirne cfiildren understood the cardinal nurnber 'two' they had

determined that counting words refer to specific, unique numerosities. Young children

know very few 7Nfltten words, often only their name that is very personal and familiar on

its own, and brand name logos, for example, McDonalds. Knowing the alphabet rnay

not provide as much information to the child as the number sequence. Therefore, they

have a disadvantage in the moving symbol task with words. According to Resnick

(1 989), children must leam to integrate the number-name sequence with the

protoquantitative cornparison schema, which is one that operates perceptually, without

any measurement process. This is usually accomplished at around 4 years of age.

Children now behave as if the counting sequence forms a kind of "mental number line"

(Resnick, 1983). They can quickly identiQ which of a pair of numbers is 'hiore" by

mentally consulting this number line, without needing to go through the sequence to

determine which number cornes later.

However, the fact that our number system is ideographic, where each written

symbol corresponds to a number word, but there are no sonoric or iconic links between

them, may make the job of grasping the syrnbolic nature of written number dificult

(Sinclair & Sinclair, 1984). While children get much of their experience with letters

fiom books where they are accompanied by the printed form, they do not receive this

experience with written numbers (Bialystok & Codd, 1997).

The Relationship between Cardinatity and Symbolic Representation

To understand how a written numeral represents quantity is a complex task

for children. The hypothesis here is that before a child can understand how written

numbers are symbolic, they must first understand how number itself is symbolic, how

number represents quantity. When children understand that the numbers in the counting

sequence refer to quantities and that they describe the quantity of the set, they have

grasped cardinality. Cardinality, that the last number word used in a count represents

the fiumerosity of the set of items counted, as defined by Wynn, is a prerequisite for

understanding the meaning of written number notations.

Cardinaliiy and the Counting Words

Children as young as two years old understand that the counting words refer

to "a distinct, unique numerosity, though they do not yet know to which numerosity each

word refers" (Wynn, 1992, p. 220). Children must learn that each word's position in the

number word list relates directly to its meaning. When they learn this, they understand

that 5 is greater than 2 because it is fùrther along the word list. Only afier this is

understood, will they be able to determine which number word applies to a particular set

of counted entities, the cardinal quantity.

Children's understanding of cardinality oRen begins by simply recounting

objects already counted when asked how many there are (Fuson & Mierkiewicz, 1980).

Gelman and Gallistel(1978) found children then understand that each object in the

count corresponds to one counting word and that the counting words must follow a

stable pattern after which they can acquire the cardinal principle (that the final tag in the

series has a special significance, that it represents a property of the set as a whole).

Children must learn to map the number concepts ont0 words. This is

particularly difficult in the domain of number as 'Ihe number words do not refer to

individual items or properties of individual items but rather to properties of sets of

items" (Wyqi992, p. 221). Wynn believes that syntactic cues can aid children's

understanding of the counting words. She points out that chifdren can sometimes learn

about the meaning of words when those words are contrasted with known words in the

same domain.

Wynn (1992) points out that these cues are only helpful after a child knows

the meaning of one of the number words. However, research has shswn that the number

"one" is leamed very early (Wynn, 1990). "One" occurs more fiequently than other

numbers and it appears in special contexts, for example, '1 want another oney7. Thus, the

number "oney' may be leamed first as a pronoun that selects a single object. The number

'%O" is then mapped ont0 its corresponding numerosity, etc. Syntax may help to

detennine that number words refer to properties of sets of entities. The nurnber one

could be distinguished in that it refers to an individual entity (no 's' at the end of the

number word) (Wynn, 1992).

Children tend to count discrete, physically separate entities more easily than

they do attached parts of objects or individuaf objects that have been divided into

physically separate parts (Shipley & Shepperson, IWO). Thus, the oneness of discrete

physical objects is believed to be highly salient to children. This may help them in

learning to count.

Cardinality and the Last Word Rule(Cardid Word Principle)

Children acquire considerable skill at counting before understanding that

counting determines the numerosity of a set. Wynn (1990) found that children who

failed to use counting to solve cardinality tasks or to provide the correct cardinal

responses were able to accurately count to four or five. Fuson (1988) also found that

children are capable of counting correctly before understanding the cardinal principle.

In one study, she placed a number of objects in fkont of the child and asksd him to give

her for example, five dinosaurs. Many children counted five and handed the

experimenter the fifih dinosaur rather than al1 five. They could count correctly but

believed that the referent of a number word was the object to which it was assigned.

Thus, being able to segment the number words and assign each one to a unique object is

part of their building symbolic knowledge.

However, Wynn (1992) disagrees with this view. In her study, children were

asked to give a pgpet for example, three animais. If 'children consider the meaning of

the word 'three' to be an object to which it is assigned during a count, then they

presumably should count some objects, and when they get to three, give the puppet the

animal labeled three. Alternatively, they might choose to give the puppet a single

animal while labeling it three" (p. 224). None of the children in the study did this.

Rather, the younger children tended to give the puppet a handfùl of animals oRen

without counting them. Thus, the children attributed a different meaning to number

words than they assigned to objects. A number of children made mistalces in which they

gave an incorrect number of items, for example three, when asked for five. The chifd

would then count, 1-2-5. Wynn suggests that these children h e w that the last number

word in the count should correspond to the word asked for without knowing why,

When teaching children to count, parents and teachers emphasize the last

word Much research has been done on when children begin to appreciate that the last

number in a count represents the quantity of that set. Fuson (1988) found that for small

set sizes, children's counting was superior to their ability to answer the '%ow many"

question, whereas the reverse occumd for large set sizes. This provides firther evidence

that very young children do not understand the relation between counting and

cardinality. Fuson, therefore, suggests that children first learn a last word rule, simply

providing the last counted word without truly understanding its cardinal significance.

Wynn (1990) agrees that children possess a last word rule in which they understand that

the last number word is the answer to the "how rnany7' question but do not understand

"how many" gives the quantity of the set. She calls this the cardinal word principle.

Given that it is the most recent number in their memory, the likelihood of a correct

response increases (Wynn, 1990). Sophian (1 992) believes that initially, children' s

counting and cardinal understanding are separate and that they become integrated only

with development. This development occurs when they discover that the last nurnher in

the count is the result of the count. She agrees with Fuson's last word d e in which

children may answer the "how many" question correctly but fail to indicate that theù

response refers to the entire collections of objects.

A number of other studies have found that children gave more correct

responses to the "how rnany'' question than when asked "are there X' objects (Frye,

Braisby, Lowe, Maroudas, & Nichols, 1989). Children's performance was poorest when

asked to "give me X' objects. These results further support Fuson's last word

hypothesis. The "are there n' and "give me X' place greater demands on children's

understanding of cardinality because they cannot simply respond with the last counted

number word. They need to compare the number given with the actual numerosity of the

set.

Finally, in testing by Schaeffer, Eggleston, and Scott (1974)' most 3-year old

children who were prevented fiom recounting were unable to provide the last counted

word.

Counting and Cardinaiity

Bialystok (1 992) argues that ccchildren initially fail to understand the relation

between the individual nurnbers in the sequences they have learned and the specific

meaning values they stand for" (p. 302). Children begin by representing numbers as

objects with specific visual characteristics rather than as symbols. Thus, they learn the

sequence of numbers (a highly practiced routine) very young but fail to understand that

they stand for meanings.

According to Bialystok and Codd (1997) cardinality grows out of children's

experience with counting in a rnanner similar to that in which a syrnbolic concept of

print grows out of children's experiences with the alphabet. In both cases, routines are

memorized early in childhood and wntexts are learned that elicit their recitation.

Gradually, correspondences are set up between the items in the routines and the abstract

notion that each item symbolizes. Thus, through these routines chi1dren7s concept of

print emerges. But at this point, these labels are simply alternate names or

characteristics. Children still need to understand each item in the sequence is a

placeholder for an actud sound or quantity. If children lack the knowledge that one

numeral must correspond to each object counted, they will not understand how quantity

is represented or the invariant reIation between the symbolic written number and its

referent. It is this relationship which makes written language and number symbolic.

Fuson (1 988) adds that in children's counting situations, the counting words

have reference to the objects to which they are attached by the activity of counting. But,

they must corne to realize that the counting words do not describe the objects. Instead,

counting must unitize the entities to be counted.

Research by Fuson (1988) and Wynn (1990) leads to the conclusion that

counting begins as a purposeless activity without special meaning (Frydman, 1995).

Children learn that the counting words refer to numerosities and only gradually discover

the cardinal meanings of the number words (Wynn, 1990). According to Frydrnan

(1995), "the child has to understand the cardinal meaning of number-names, and nothing

in the counting process itself tells that these names represent cardinal values" (p. 666).

Wynn (1992b) has suggested that it is the ordinal position of each ofthe

number words in the counting sequence that is the means to their representation of

number. There are inherent relationships among the numerosities in that the linguistic

symbol for the numbers have relationships to each other in their ordinality analogous to

the relationships the numerosities have to each other in their cardinality. Thus, the

number sequence that children learn provides them with more information than learning

the alphabet.

Frye et al. (1989) used an error detection method in their study. They

separated trials into judging the validity of the counting procedure (which began: '%et's

count these") and judging the validity of a cardinal response based on the counting

procedure (which began: 'let's see how many there are"). This distinction allowed

them to determine more explicitly whether children can recognize that it is wrong to

give a cardinality response that was the result of a mistaken counting procedure. They

found that children were better on correct trials, but more importantly, they were better

on the counting trials compared with the cardinal trials. The children were unable to

relate cardinality and counting accuracy. Instead, the last word rule suggested by Fuson

(Fuson, Pergament, Lyons, & Hall, 1985) better explains the results. In addition, Frye

found that being a successful counter was not related to success on cardinality (Frye et

al., 1989).

These results are consistent with Wynn's (1990) study. Set size was also

examined and they found that a progression emerged. From easiest to hardest the

progression was: counting a small set, counting a large set, judging the experimenter's

counting of a small set, and judging the experimenter's counting of a large set (Frye et

al., 1 989). The results of the study provide fùrther support for the conclusion that

initially, young children who have learned the counting sequence, and can judge the

accuracy of this procedure, may still lack an understanding of the relationship between

counting and cardinalit y.

Two features of young children's performance have been found consistently

on cardinality tasks. They do not always use counting to determine "how many X's"

there are or to "give X' number of objects. Wynn (1990) found that al1 children under

the age of three and a half sirnply grabbed a number of objects rather than count the

specific number asked for. This occurred despite the fact that d l could count to four or

five accurately. They oflen failed to give even three items, suggesting that children fail

to understand the relation between counting and cardinality. Therefore, Wym argues

that children do not understand the cardinal principle until three and a half. A strong

within-child consistency between the use of counting in this task and correct

performance in answering the "how many" question following a count was also found,

suggesting that these children have learned how to determine the numerosity of the set.

Wynn argues that children must first know the cardinal meanings of the number words

in order to be successfül on these tasks.

There is good evidence for the argument that children's failure in the ''give X'

task is due to a lack of knowledge of the cardinal word principle. Children's

performance on this task predicts 'hhether a child will respond a majority of the time

with the last number word used in a count when asked 'how many' following counting,

whether a child will give the last number word more ofien aRer correct than incorrect

counts when asked 'how many7, and whether a child wiIl tend to count out items aloud

fi-om a pile, when asked for a number that he is generally successfûl at giving" (pp. 187-

188). The 'how many' and 'give X' tasks are conceptually similar so the failure is likely b

due to lack of conceptual cornpetence (Wynn, 1990).

In research by Wynn (1 990) children were asked to count sounds, for

exarnple, an elephant roaring, rather than objects. Children in this condition were unable

to recount because the sounds were presented in sequence and could not be replayed. In

the real object condition children preferred to recount when asked "how rnany". In the

sound condition this strategy was not available and children gave more cardinality

responses (a number word response rather than recounting). However, children were

worse at counting correctly in the sound condition. The older children (3.5 years)

counted comctly on more trials than the younger children (2.5 years). Wym suggests

that children begin to develop an abstract, generalizable representation of the counting

routine very young that can be applied to new counting situations.

A number of researchers have found that children who lack cardinal

understanding are unable to continue counting fkom a partial set they had already

counted in order to determine the amount of the fidl set (Hughes, 1986; Steffe & Cobb,

1988). Bialystok and Codd (1 997) used the çount-on task to study cardinality. In their

task children counted candies in a box after which the lid was placed on the box, and

they were shown some more candies beside the box. They were asked how many

candies there were altogether but were prevented fiom lifting the cover of the box that

contained the counted candies. Many children recounted the items fiom memory,

despite being able to successfully count on fkom a broken sequence. To determine the

total quantity, children must reaiize that &er they count the candies inside the box, they

need only count on the candies outside the box. This suggests that the children were

unabie to translate a counting procedure to a counting outcome. They could not

represent the quantity inside the box fiom memory without recounting.

In a fiirther effort to look at cardinality, Gelman and Gallistel(1978) examined

resuIts of their puppet experiment in which children watched a puppet count (either

correctly or incorrectly) and asked whether the puppet had given the correct answer to

the "holx many" question. Their logic was that it should be easier for the children to

demonstrate their understanding of cardinality if they were fieed fkom the constraints of

having to perform the count themselves. Three and four-year-olds were able to correctly

distiqpish between the correct cardinal answers and the incorrect ones. But, Frydman

(1995) points out that if the aim of this study was to determine whether children

understood more about cardinality than can be demonstrated from their counting ability,

they have failed to do this. The researchers did not compare children's responses on the

puppet task to a task in which they are required to count. The puppet study focuses the

child's attention on the cardinal meaning of counting. In addition, it is possible to detect

the error by a last count-word strategy. Thus the child only has to object to the fact that

the last word used in the counting sequence and the word given in reposes to the "how

rnany" question fail to match (Frye et al., 1989). And as already noted, the ability to

correctly use the last word strategy alone does not demonstrate understanding of

cardinalit y.

Briars and Seigler (1984) conducted a puppet study similar to that by Gelman and

Gallistels and found that children's own counting was more accurate than their ability to

detect errors by the puppet. This suggests that it is counting that leads to the cardinal

principle.

Equivalence and Curdinality

There is also an equivalence aspect of cardinality. Do children understand

that the quantity deterrnined by counting applies as well to an equal set? To fully

understand cardinality, a child must also know that any set with a particular number is

the same in quantity as any other set with that number and different fhom any other set

with a different number (Frydrnan, 1995). Frydman and Bryant (1988) conducted

research to determine whether children were able to make inferences about number on

the basis of sharing. Children four years of age were able to successfilly share blocks in

equai quantities among a different number of dolls on between 71% and 100Y0 of the

trials. In 83% of the correct trials children shared by distributing the blocks one t o each

do11 at a time and only did so for 40% of the incorrect trials. However, children had

little explicit understanding of the numerical significance of sharing. When children

were given a similar sharing task and asked how many blocks they had given to the

second do11 after counting the first doH's blocks, al1 the children started counting the

second doll's blocks. No child realized that counting was unnecessary. Even &er the

blocks were hidden, only 10 children out of 24 gave the correct answer. Frydman and

Bryant suggest that the majority of children were not aware of the relationship that exists

between numerical equivalence and the equivalence of quantities.

Bialystok and Codd (1997) used a task similar to Frydman and Bryant to look

at cardinality. Children were asked to first share a quantity of objects and determine the

quantity of the sets. They were asked whether both piles contained the same amount.

After counting one pile, the children then had to tell the experimenter how many objects

the second pile contained. In order for children to succeed at this task they must realize

that the number counted is the quantity and therefore, two piles of objects with an equal

number of items have the same quantity. Ifthey understand this they should be able to

determine the number of objects in the second pile without counting. Children who

were unsuccessful believed it was impossible to know how many objects were in the

second pile without counting.

Not until children were six years of age were they able to recognize the value

of counting as a basis for numerical cornparison (Sophian, 1995). Younger children did

not use counting to determine the relation between two sets. When asked which of two

rows was more nurnerous, a conservation type task, children typicdly picked the longer

row even when it actually contained fewer objects. But children7s resistance to counting

was not a result of the belief that length was inadequate for comparing rows. Sophian

found that children younger than six were unlikely to count even when the rows were of

equal length.

In addition, even when the counting was done by an experimenter, children

showed limitations in the understanding of counting as a means of comparing sets

(Sophian, 1988). M e r two rows of horses (one with big horses, the other with small

horses) were laid out in front of the children, they were asked '%an every big horse have

its o w little horse?" or 'How many horses are there altogether?". Children had to

decide whether it was better to count al1 the animals together or count the two sets

separately. For the total number problem children typically believed that counting the

horses together was the best strategy to use. But for relational question, three-year-old

children believed both methods were acceptable, whereas the four-year-olds felt it was

best to count the horses together.

The Relations hip between Counting and Sym bolic Represen tations

Fuson (1988) found that the use of counting was strongly and inversely

related to the size of the nurnber word and that for cardinal situations the percentage of

time a child counted increased as the number decreased.

Children are proficient at counting small nurnbers only rnaking the connection

between the written number and quantity more salient for these numbers. However, if

tasks using larger quantities are used, numbers that cannot be detennined visually

(numbers outside the child's counting range), children should have difficulty. They

would need to rely on knowledge of how the written number represents the quantity of

objects.

Wynn (1992) found that infants could discriminate among small numerosities,

as demonstrated by dishabituation. She believes they have some basic knowledge of the

numerosities of two and three. When infants were shown pictures of two and three

objects sirnultaneously with a sound recording of two or three knocks, they displayed

preferential looking at the picture that matched the number of knocks (Wynn, 1990).

According to Resnick (1989), infants' ability to discriminate the numerosity of smal1

sets when they are presented visually is the result of a schema for comparing objects

quantitatively as they make their judgments on the basis of comparative rather than

absolute size.

Sophian (1992) believes that children are using a subitizing strategy in which

they obtain a representation of the nurnber of items in an array by means of a direct

perceptual apprehension mechanism. Each numerosity is grasped, apprehended, taken

in as a whote, and seen as a pattern. According to Sophian, there exist pattern

recognizers that detect oneness, twoness, etc. Therefore, these small numbers afready

have cardinal meaning for the child because they are able to observe that they

correspond to the result obtaïned by counting. This alerts the child to the fact that

counting has the same cardinal meaning. Cooper (1984) also believes that infants learn

about ordinal relations through subitizing, by seeing what happens as objects are added

or subtracted fiom a set.

Fuson (1988) does not believe that this proficiency with small numbers is

necessarily preceded by the ability to subitize these sets. She argues that cornpetence in

cardinal situations that involve large numbers requires an understanding not needed for

smaller numbers. For example, if a child rnust add two numbers, they rnust have some

method of keeping track of the words wunted if the second number is greater than three.

This is consistent with research on children's problem difficulty in addition and

subtraction, which increases as numerosity increases. Children who are unable to solve

large number problems can often do well on smaller number problems (Starkey &

Gelman, 1982).

There is strong evidence that young children understand the difference between

one and the other numbers and that they have substantial knowledge of the smalier

nurnber words. In a study by Hughes (1986), children's ability to represent different

quantities was better with the small numbers, in particdar the number one. And finally,

in Wym7s (1992) puppet study, children who were asked to give one animal to the

puppet were always accurate. ln addition, they never gave one puppet when asked for a

number greater than one. As mentioned earlier, Wynn (1990) found that by 2;4,

children are able to identie "one". In contrast, they did not even use approximation

techniques when asked for two, three, five, or six objects. Thus, it appears that children

learn the meaning of the word "one" very early, followed by the word 'ho".

Children were able to give correct cardinal responses up to a certain numerosity

and unable to give any higher ones. The failures, however, were not a result of an

inability to count that high. Children who failed at some numerosities were able io

successfully count larger set sizes. Although there was an effect of set size, the fact that

older children were able to succeed at larger numerosities fùrther supports the idea that

children's failure on these tasks is due to an inablity to represent the higher numbers, not

an inability to count tliern. Wym (1990) suggests that the children are learning the

meanings of the nurnber words one at a time, for progressively larger numbers. She

argues that children are more ofken correct on cardinality tasks with small set sizes

because they are able to make a simple association between the small number words and

their quantities based of the appearance of the set compared with large sets. Tasks using

larger numbers should therefore, require that children have a full understanding of

cardinality in order to be done successfully.

Notational Representations for Number

Research on developing representations of quantity has found that most children

aged 3-4 use analogue notations to represent quantity despite the ability to recognize

written numerals (Bialystok & Codd, 1996; Hughes, 1986; Allardice, 1977; Sinclair &k

Sinclair, 1984). How does children's knowledge of cardinality affect their ability to

represent it notationally?

Bialystok and Codd (1996) found that when children used analogue notations to

represent quantity they had difficulty interpreting what they had h t t e n at a later time

because they saw an andogue as an object in itself rather than as a representation of

quantity. Thus their symbolic knowledge was incomplete. Tannenbaum (1 997) looked

at preschool children's understanding of the symbolic representation of numbers, words,

analogues, and pictures to determine whether children understood that these forms were

invariant. Using the moving symbol task with analogues she found that children were

ofien successful because they counted the number of circles on the card which would

help them veri@ whether the number of circles matched the number of objects. This

suggests they did not yet have a complete understanding of the invariance of the

representation, particularly because they did not çount again once the card was returned

to it's original position. In contrast, the 5-year-olds did not count. Rather, they were

correct because they understood that the analogue was a symbol for the quantity of

objects presented on the card.

Analogues directly indicate the quantity they represent. Thus, they may be an

intemediate step between pictorial representation, for example, a drawing of the object,

and second order syrnbolic systems like written numbers. The child may first see a

connection between the number of objects and ttle number of dots that make up the

analogue representation. There is a one-to-one correspondence that they can make use

of a more direct relationship than the one found in written numerals. Thus, analogues

are less ideographic than nurnerals as there is a clear link between the object and its

representation.

Children are capabIe of successfiilly assigning one label to each object as they

count before they demonstrate complete understanding of cardinality (Gelman &

Gallistel, 1978). Therefore children who pass the moving symbol task with analogues

may not have complete understanding of cardinality because demonstrating

understanding of analogue representations and the one-to-one correspondence involved

with them is not al1 that is involved in cardinality. Children still have to rnake the

connection between the last counted word and the total quantity of the set.

How do notations come to stand for number? Children first learn the number

words associated with the written numerals without understanding their meaning and

without reference beyond the numeral itself, for example 'That's a six" upon seeing a 6.

Later, children understand the cardinal meaning of the written numerals. They "learn to

label patterns or situations with a cardinal label, for example, there are 5 people in my

family" (Fuson, 1988, p. 127).

By age 5, children appear to have a lot of detailed knowledge about counting and

number. They know how to count and often how to recognize numbers. But they do not

yet fully understand the symbolic fiinction of notations, how they represent number.

Bialystok and Codd (1 996) found that young children were unable to coordinate their

knowledge of the counting and the symbolic function of the numerals. They were

unable to use the number words as symbols for quantity.

Children's acquisition of written numbers and print follow sirnilar paths. In both,

a conventional sequence made up of "arbitrary symbols (numbers, letters) [that] stand

for abstract referents (quantities, sounds) must be learned" @ialystok & Codd, 1997, p.

88). Children zus t cncode the letters and numbers individually so that each has a unique

meaning, a placeholder for actual quantities or sounds. When this relationslip is

understood these symbolic systems c m be used for arithmetic or reading.

In the moving symbol task with words by Bialystok (1991), only children who

understood the one-to-me correspondence between letters and sounds and the invariance

of that relation were able to make the judgrnents necessary to correctly identi& the word

when it was under the noncorresponding picture. Children had coordinated the picture

and the word, but had not yet syrnbolized the word to one meaning. If children lack

cardinality, they should have trouble symbolizing the number word to one meaning.

They must know that a distinct number must correspond to each object counted or they

will not understand how quantity is represented or the invariant relation between the

symbolic written number and its referent.

Hypo th eses

Ifit is true that the establishment of cardinality, the ability to mentally represent

the concept of a numeric value, is necessary before a child is able to understand written

number in a symbolic way, then children should pass the moving number task only d e r

they have demonstrated full understanding of cardinality. It should be more difficult to

pass the moving symbol tasks with larger numbers, as smaller numbers are more

interpretable because they are perceptually discriminable. Thus, it is possible that just as

children who fail the moving symbol task with words lack the alphabetic principle (the

understanding that letters represent the phoneme constituents of words), children who

fail the moving symbol task with numbers may lack cardinality. Children must

understand that "print symbolizes language" (Adams, 1990, p. 33 5) and that written

numbers symbolize quantity.

It is also predicted that children will be more successfùl with written numbers

and analogues than with words. Analogues are not a second order symbolic systern and

should therefore be understood earlier than words. Numbers have individual meaning

whereas individual letters do not. This may help children to treat them individually

compared with letters which are heard in words or sentence making the connection

between the symbolic letter and its referent more difftcult (Wynn, 1992). This is

consistent with previous research that found children were more successful on the

moving symbol task with numbers and analogues than with words (Tannenbaum, 1997).

Children are predicted to be more successfùl on analogues than numbers as numbers are

second-order symbols with nothing to directly indicate their meaning.

Finally, previous research has demonstrated the significant advances that

young chiidren make across the two ages studied in establishing symbolic concepts

(Bialystok & Codd, 1996) and in theu understanding of cardinality (Wynn, 1990, 1992)

and thus, age differences are expected for both the moving syrnbol task and the three

cardindity subtasks. The Visually Cued Recall Task, a measure of memory span, will

be given in order to ensure that children are comparable across conditions.

Method

Subiects

Ninety children consisting of 2 age groups were tested: 45 3-year-olds (mean

age: 3;6), and 45 4-year-olds (mean age: 4;5). There were 52 girls and 3 8 boys. Al1

children participated in three cardinality tasks and the Visudly Cued RecalI Task

(Zelazo, Burack, Jacques, & Frye, 1997). They also received one of three possible

conditions of the Moving Syrnbol Task (Bialystok, 199 1,1997) to which they were

randomly assigned. The three conditions were numeral, word and analogue. For each

age group, 15 children received each condition with the order counterbalanced across

subjects. For each condition there were three levels: easy, middle and difficult. Level

was a within-subjects factor. Al1 tasks were given on the same day in a 30-minute

session. Subjects were selected fiom daycares in Toronto and Hamilton. Children were

seen individually while at school.

Visuallv Cued Recall Task

This task (Zelazo et al., 1997) was used to assess children's memory span and

provide a measure of general cognitive ability to ensure that the children were

comparable across the three conditions. A series of 10 posters (1 4 cm X 1 1 cm) was

shown to the child. Each poster included 12 different pictures of farniliar objects, for

example, a book. The child was told "This is my ffiend Snoopy. He likes certain things

very much. I'm going to show you pictures of things that he likes and when I'm

finished showing you what he likes, I want you to point to them for me". As Snoopy

pointed to the objects they were named by the experirnenter. The child was then asked

"Cm you point to the ones he likes?". Each poster increased the number of items to be

remembered beginning with 2 objects (after a practice trial of 1 object). The order in

which the child selected the items was not important. The task ended afier the child

made an incorrect choice on 2 consecutive posters. Each poster was scored in terms of

the number of items accurately recalled. A child's score was the total number of

pictures correctly recalled including the two posters where a mistake was made.

Additions and deletions were recorded separately.

Cardinaiit~ Task

The cardinality task consisted of three subtasks that increase in the dernands they

placed on the child.

"Give-u-mrrn ber" Task

In this subtask, adapted fiom Wym (1992), a pile of 10 similar objects (Lego) was

placed in front of the child. He or she was asked "can you give me X objects?". Each

child received one trial for each of the quantities three, six, four, and seven objects and a

practice trial of two objects. The score sheet had the numbers up to 10 for each trial.

The child's answer was highlighted.

"Are there more X than Y" task

This task, adapted fiom Sophian (1995), examined understanding of relative

quantity. Two piles of small toys were placed in front of the child. The first pile

consisted of small, identical horses and the second consisted of small, identical spiders.

The use of similar sized objects should prevent any confounding due to differences in

the length of the row due to the size of the object rather than the number of objects. The

child was told Were are some horses and spiders. Are there more horses or more

spiders?'. The child received four problems of this kind in addition to a practice trial

with piles between 4 and 9 and differing by 2 or 3. Which pile was larger alternated

between trials (each child received one trial for each of the pairings: 4-6,7-5,5-8, and 9-

6). For each trial, the score sheet listed the number of horses and the number of

spiders. The answer given by the child was highlighted.

S k i n g Task

This task, adapted fiom Bialystok and Codd (1 997), examines what children

understand about the invariance of number. Children were asked to distribute toys

equally to two d e d anirnals The experimenter helped the children divide the objects

by giving one to each animal in turn. M e r the toys were distributed the children were

asked whether both anirnals had the same arnount. M e r counting one pile and telling

the experirnenter the number, the children were asked to tell the experimenter how many

toys the other animal had. The experimenter kept her hand partially covering the second

pile to prevent the child fiom counting them. This task was repeated four times (each

child received one triai for each of the quantities: four, seven, three and six with an

initial practice trial of two). In order for children to succeed at this task they had to

realize that the number counted was the quantity and therefore, two piles of toys with an

equal number of items have the same quantity. If they understand this they should be

able to determine the number of toys in the second pile without counting. The score

sheet recorded whether children knew that the second pile contained the same amount of

toys as the first pile and the number of toys they beiieved were in the second pile.

Movina S ~ m b o l Task

Numeral Condition

The moving symbol task with numbers examines the child's understanding of the

invariance of the symbolic relation between the reference for the number and the d e n

number itself (Bialystok, 1991). The children were introduced to two toy anirnals and

told that although they are good fnends, they ofien fight with each other. Two piles of

Lego blocks were placed in fiont of the child on a small tray and with the experimenter

each pile was counted. Lego blocks were used as they cany no information or specific

meaning for the child. The experimenter counted the nurnber of Lego with the child

and said 'Here are some Lego. Let's count them together". After counting, the

experimenter told the child, "Three. 1 have three Lego". They were then shown a card

with a numeral that corresponded to the number of Lego in the first pile. The child was

told, "This card has three on it7'. The experimenter then told the child, '2 am going to

place it here", and it was placed under the pile with 3 Lego. The child was then asked

what the card said. At this point in testing the stuffed animals had a fight and the card

was accidentally moved over so that it was now under the noncorresponding pile.

According to Bialystok (1997), using an accident to change the display prevents the

effect found in conservation tasks where children make errors because they expect the

answers to change as a result of the experimenter altering some aspect of the display,

The child was again asked what the card said. M e r the experimenter cleaned up the

animais' mess and ~noved the card back under the correct pile, the child was asked one

final time what the card said.

There were three Iew!s of this task and each child received 4 sequences of

each. In the first level, both piles of Lego had small numbers ranging from 3-6 objects

(3-5,4-6'5-3, and 6-4). In level two, one pile had a small number of Lego, 3-6, and the

other pile a large number of Lego, ranging fiom 9-19. This was based on Gelman and

Galli stel's (1 978) finding that most older preschool children between the ages of 4 and 5

were able to successfilly represent 1 9 objects in a counting study (5- 1 6, 4- 17, 14-6, and

12-3). Finally, in the last level, there were large numbers on both cards (18-14, 17-15,

1 2- 1 6, and 13 - 19). These different levels were separated by the other tasks to ensure

that the chiId did not lose motivation. The order of presentation of the three levels was

counterbalanced across subjects to control for any possible order effects, in particular,

receiving the small-small number order first. Small numbers they are more familiar with

rnay have made it easier for children to make the connection between the quantity of

Lego and the written number on the card.

The research on smaH numbers, in particular the number one, reported in the

introduction demonstrates children's early ability with smalI numbers. As a result, the

number one was excluded fiom the moving symbd task and the additional condition

was included in which larger numbers outside children's usual counting range were also

given. This added difficulty should distinguish between those children who are drawing

on their understanding of the meaning of the small counting words alone and those who

truly understand the symbolic nature of written numerals. The cardinality tasks also

included larger numbers (3 to 9) in order to determine whether children's performance

differs depending on the type of number given.

The use of double digit numbers that adds the difficulty of place value, should

not be a confound as the moving symbol task with words used in previous studies used

words unfafni1ia.r to the child as well. What is important is whether or not the child

understands that number is invariant and so does not then represent the new quantity

after the switch.

Analogue Condition

The objects used in this condition were the same as in the numeral condition

and the card size was the same. The child was told, 'This card has nine on it" when the

analogue representation displayed nine (nine circles).. The experimenter then continued

with the task as in the number condition. In a previous shidy using this task, 3 and 4

year old children counted the circles on the card suggesting that they did not understand

the invariance of the representation (Tannenbaum, 1997). Thus, children were not

allowed to count the dots as a correct answer using this method would not provide any

knowledge of the child' s understanding of the symbolic nature of the representation.

The same three levels and sequences as in the number condition were used with the

numbers being replaced by analogues. Again, the order of presentation was

counterbalanced across subject S.

Word Condition

For this condition the objects were the same as the previous two conditions.

However, the numbers and analogues were replaced with words referring to the number

of objects. For example, if there are 3 Lego, the word 'three" was written on the card.

In previous research with everyday words such as 'car' children found the word

condition more difficult than the number condition (Tannenbaum, 1997). However, they

may not need to have a fùll grasp of cardinality to pass this task. Children may simply

need to know the counting sequence. The wording for this condition was the same

where the experirnenter told the child 'This card has three on it". The same three levels

and sequences were given to subjects in this condition.

Al1 answers were recorded on a prepared sçoresheet that included children's

responses to the three locations (under correct pile, incorrect pile-switch and correct

pile-return). The answer given by the child was simply highlighted as both possible

answers were provided on the score sheet. Which condition the child was in was also

recorded as score sheets were the same for al1 conditions. The children received a

sticker &er completing this task to maintain motivation.

Procedure

Children recieved the Visually Cued Recall Task first, followed by their first

level of the moving symbol task. The c'Give-a-number" task, "Are there more X than

Y" task, and the Sharing task were aven in between the three levels of the moving

symbol task in order to prevent the children from losing motivation for the task.

Results

General cognitive ability was measured using the Visually Cued Recall task,

which assesses memory span. This task was used to ensure that children in the different

conditions had comparable resources for working memory. Children's scores were the

total number of pictures recalled across al1 displays. A two-way ANOVA for age (3,4

years) and condition (numeral, analogue and word) was condücted to detemine whether

there were any differences between children's performance in the three conditions or

between age groups. No significant differences were found for either factor (Condition:

F(2,78) = 0.43, p=.65). Thus, the groups were well matched on general cognitive

ability. A difference between the two age groups approached significance

(F(l,78)=3.52, p.06) with the 4-year-olds scoring higher than the 3-year-olds. The

mean scores for each condition and age are presented in Table 1.

Performance on Cardinalitv Tasks

In the "Give-a-numbef' task children were asked to count out 3,6,4, and 7

objects. A three-way rnixed design ANOVA for the between subjects factors age (3 ,4

years) and condition (numeral, analogue and word) and the within subjects factor

question (the '3' question, the '4' question, the '6' question, the '7'question) was

conducted. Condition was a grouping factor and was included in the analysis to ensure

no diserences between groups. Children received 1 point for each correct answer and

their total score was an average of the 4 trials. There was a significant main effect of

age, F(1,84) = 5.89, F.05 with the 4 years olds (A&0.59,SD.=0.40) scoring higher than

the 3-year-olds 1, w.=0.33). There was not a significant main effect of

condition (F(2,84) = 0.68, p.51). There was a significant main effect of question,

F(3,82) = 18.48, F.001, and interaction of condition and question, F(6,l64) = 2.61,

pc.05. For the andgoue condition there is a drop in the score for '4'question and '7'

question but not for the other two questions. Contrasts were conducted for the repeated

measure question. The '3' question differed significantly fiom the '6' question

(F(1,84)=34.87,fi.001) and the '4' question differed significantly fiom the '7'

question (F(1,84)=3 1.00, F.001). Thus, the srnaller number questions (3 and 4) were

done more successfûlly than the higher number questions (6 and 7). The mean scores

for each question are presented in Table 2a and Table 2b.

In the "'Are there more X than Y' task children's understanding of relative

quantity was examined. They had to determine which of two piles of toys was p a t e r .

There were four trials: the '4-6' question, the '7-5; question, the '9-6' question, and the

'5-8' question. A three-way repeated measures ANOVA for the between subjects

factors age (3,4 yens) and condition (numeral, analogue and word) and the within

subjects factor question (4 trials) was conducted. Again, condition was a grouping

factor and was included in the analysis to ensure no differences between groups. Scores

were calculated as in the "Give-a-number" task. Again, a significant main effect of age

was found, F(1,84) = 10.84, F . 0 1 , with the 4-year-olds (M=O. 86, ==0.2 1) scoring

significantly higher than the 3-year-olds (M=0.69,SD=0.26). There was not a significant

effect of condition (F(2,84) = 0.02, p=.98). A significant main effect of question was

found, F(3,82) = 2.80, F . 0 5 . Contrasts were conducted for the repeated measure

question. The '4-6' question differed significantly fiom the '9-6' question

(F(lJ84)=6.49,p<.01). Both age groups had the most dificulty with the '5-8' question.

The mean scores for each question are presented in Table 3.

Finally, the Sharing task examined children's understanding of number

invariance. M e r distributing toys equally to two stuffed animals they were asked if the

animals had the sarne amount. Children then counted the number of toys in one

animals' pile and were asked if they could tell how many the second animal had without

counting. Trials included pairs of 4,7, 3 and 6 toys. A the-way mixed design ANOVA

for the between subjects factors age (3,4 years) and condition (numeral, analogue and

word) and the within subjects factor question (the 'sharing 3' question, the 'sharing 4'

question, the 'sharing 6' question, the 'sharing 7' question) was conducted. Condition

was a grouping factor and was included in the analysis to ensure no differences between

groups. Scores were calculated as in the previous two cardinality subtasks. There was a

significant main effect of age, F(1,84) = 20.7O,p<.OO 1. Again, 4-year-olds

m=0.62,SD=0.34) scored significantly higher than 3-year-olds (M=0.32,-.28).

There was not a significant effect of condition (F(2,84) = 0.04, p.96). There was a

main effect of question, F(1,82) = 1 1.65, p<. 001. Contrasis were conducted for the

repeated measure question. The 'Sharing 7' question differed significantly Erom The

'sharing 3' question (F(1,84)=6.43, pK.0 1) and theCsharing 3 ' question differed

significantly from the 'sharing 6' question (F(1,84)=ll.44, F . 0 0 1). The 'sharing 3 '

question was the easiest question for both age groups. The mean scores for each

question are presented in Table 4.

The Moving Svmbol Task

In this task each chitd had three scores corresponding to the three times they

were asked what was wriîten on the card (corresponding to the group of Lego, switch

and return). The mean scores for each age group for the first and return positions are

presented in Table 5 and the mean scores for each age group for the switch position are

presented in Table 6. Children always gave the correct answer the first time they were

asked what the card said. A two-way mixed design ANOVA for the factors age (3,4

years), and condition (numeral, analogue and word) was conducted for the retum

position There were no significant effects of age (F(1,78) = 1.98, p=. 16) or condition

(F(2,78) = 3.07, p=.52).

The switch position question was most important in deterrnining whether

children understood the invariance of the symbolic relation between the group of Lego

and written number, word or analog on the card. The task had three levels which

consisted of two small groups of Lego, two large groups, and one small and one large

group. It was hypothesized that because small numbers were more familiar they would

allow the children to make the connection between the quantity of Lego and the number

written on the card more easily. The smaller numbers would also be more salient

because they are more perceptually discriminable compared with the larger numbers.

Comparing children's performance on this task across conditions was important to

examine what connection exists between written words and written numbers and how

children progress through the different orders of syrnbolism.

A three-way mixed design ANOVA for age (3,4 years), and condition

(numeral, analogue, and word) and within subjects factor level of difficulty (small

numbers-easy, small and large numbers-middle, and large numbers-difficult) was

conducted on scores for the switch position question. There was no main effect of age

(F(1,78) = 0.09, p=.76). There was a significant main effeçt of condition, F(2,84) =

9.99, p<.OO 1. Using Scheffes post hoc tests, p . 0 5 , it was found that the scores for the

numeral condition @=.88.SD=.27) were significantly better than those for the analogue

(M=.62,==.35) and word conditions @==.47,==.45) Although not significant, the

scores for the analogue condition are higher than for the word condition. The mean

scores for each condition at each level are presented in Table 7.

There was a significant interaction arnong level, age, and condition, F(4,166)

= 2.56, F . 0 5 (presented in Figure 1). There were no significant differences between 3

and 4-year-olds when examined at each level for each condition. However, by breaking

the analysis down by age a difference emerges. For age 3, there is a main effect of

condition, F(2,42)=6.62, F.0 1, (mean scores: number-. 88, analogue-.66, worcL.43) and

level, F(2.84)=14.29, ~ 6 . 0 0 1 (mean scores: easy-.79, rniddle- .69, dificult-.49) and an

interaction between level and condition, F(4,84)=5.5 1, F . 0 0 1. Compared with the 4-

year-olds, there is a steeper decrease in scores for the 3-year-olds across levels for both

the analogue and word conditions. The level by condition interaction is not present for

the 4-year-olds (F(4.84) = 1.02, r . 4 0 ) . There are only main effects for condition,

F(2,42)=4.09, p<. 05, (mean scores: number-. 87, analogue-. 5 7, word- .48) and level,

F(2,84)=6.29, pc.01 (mean scores: easy-.73, rniddle-.66, dificult--54). In addition, for

age 3 there are significant contrasts for level between the difficult and easy levels

(F(Z,42)=6.97, pK.01) and between the dificult and rniddle levels (F(Z742)=6.O3@. 0 1).

When analysis is broken down by condition, again there are no effects of age and no

level by age interactions.

Although not significant the difference between the two age groups at the

difficult level for the word condition was large (mean score of 0.27 for age 3 and mean

score of 0.45 for age 4). While the 3 -year-olds understand written numbers as well as

the 4-year-olds, the 3-year-olds are behind the 4-year-olds in their understanding of

written words, suggesting a longer period of development for words especially for the

more difficult questions.

More important was the level by condition effect, F(4,166) = 3.1OYp<.05

(presented in Figure 2). Children are consistently successful across levels for the

numeral condition. While the analogue scores are as high as the numeric condition for

the easy and middle level they are significantly lower for the difficult level. For the

word condition scores are the same as the analogue scores at the difficult Ievel but lower

than the other two conditions for the easy and middle level. A main effect of level was

also found, F(2,83) = 14.9 1, F .001 .

Contrasts were conducted to compare the three levels of dificulty. Both the

contrast between the easy (N.=.76,==.42) and difficult @=.52,SI>=.48) (F(1,84) =

27.78, F . 0 0 1) and between middle (M.=.68,==.45) and difficult (F(1,84) = 20.54,

p<.001) were significant. There was not a significant difference between the easy and

middle levels and thus, the difficult level is different fiom the other two levels

suggesting a threshold of dificulty. The size of the numbers affected how successful

ctiildren would be on the moving symbol task where only the largest numbers caused the

task to become more difficult.

SchefXe post hoc, p<.OS, tests were conducted for age and condition at the

three Ievels of difficulty. There were no significant differences between the two age

groups for any level. There were significant differences between the three conditions for

each level. For levels easy and middle the number and analogue conditions were

significantly higher than the word condition while for the dificult level the numeral

condition was significantly higher than both word and analogue condition. Thus,

analogue representations become hard with difficult items only.

The Moving Symbol Task and Cardinality.

In order to determine whether success on the moving symbol task was

preceded by success on the cardinality tasks nine chi-squares were conducted, one for

each condition (numeral, analog and word) for each of the three cardinality subtasks.

Chi-squares were made up of those children who passed and failed the tasks. For the

moving symbol task, a child passed if he or she was successfiil on three of the four t d s

for each Ievel. For the cardinality subtasks, a child passed if he or she was successfùl on

three of the four trials.

Due to the mal1 fkequencies in many of the cells (there were only 30 subjects

per condition) Fisher's Exact tests were used. These tests provide the calculation of

exact probabilities rather than using the continuous chi-square distribution to obtain

approximate probabilities. Results of the tests are presented in Table 8. Only two tests

were significant, the test with the moving symbol task with analogues and the "Give-me-

X' task (1 1.43, pC.01) and the test with word condition and the Sharing task (4.983,

pK.05). For both of these tasks the number of children who failed both tasks was large

in cornparison to al1 other cells and this appears to be the reason for their significance.

In general it suggests that understanding of symbolic representation is independent of

cardinality.

However, the results for the "Are there more X than Y' task, although not

significant, are in the predicted direction. For the number condition(i.22, p . 2 7 ) 18

children passed both tasks, 4 passed the cardinality task but failed the moving symbol

task while only 5 children failed the cardinality subtask and passed the moving symbol

task. For the analogue condition (0.37,p0.54), 7 children passed both tasks, 14 children

passed the cardinality subtask and failed the moving syrnbol task while only 2 children

failed the cardinality subtask and passed the moving symbol task. Finally, for the word

condition(0.27, p=.61), 10 cbildren passed both tasks, 16 children passed the cardinality

subtask and failed the moving symbol task while only 1 child was successfùl on the

moving symbol task aiter failing the cardinality subtask. This is consistent with the

prediction that children should first be successfùl on the cardinality tasks before

understanding written symbolic representations.

In order to conduct the most stringent test of this hypothesis nine more chi-

squares were conducted using the first trial of the difficult level for the moving symbol

task and the trial found most diEcult by children for the cardinality subtasks. These

were the trials that asked for 7 Lego in the "Give-Me-X" task, the trial in which children

had to determine the quantity of an equivalent pile of 7 in the sharing tas& and that had

children compare piles of 9 and 6 Lego in the "Are there more X than Y' task. Only the

chi-square with the analog condition and the ccGive-Me-X' task was significant (5.57,

pc.05). The largest ce11 was for children who failed both tasks. Again, the "Are there

more X than Y' task, although not significant, is in the predicted direction (Numeral -

0.64, p=.43/Anaiogue - 0.10, p=. 76Nord - 0.7 1, p=.40). Results are presented in

Table 9.

Correlation's

A correlation analysis was conducted on the three cardinality subtasks.

Scores for each task were an average of the four trials. The correlations are presented in

Table 10. Only one significant correlation was found; this was between the "Give-Me-

X"taskandthesharingtask,r=0.43,p<.OOl. However,the"'kethereMoreXorYY

task was not significantly related to either the other cardinality subtasks.

Only the "Give-a-number" task was significantly correlated with the moving

symbol task (switch question), r=.Z6, p<.O5. When broken d o m by level of difficulty,

the middle level(r=.22, p<.M) and the difficult level (r=.Z8, f i .01 ) are significantly

correlated with the "Give-a-numbe?' task. In addition, the difficult level of the moving

syrnbol task is significantly correlated with the sharing task, r=.22, f l . 05 .

Discussion

This study examined 3 and 4-year-old children's understanding of written

representations for number and their understanding of cardinality. Scores were not

significantly different for the Visually Cued Recall Task ensuring that children in the

three conditions of the moving symbol task had equivalent working memory. Although

4-year-olds scoring higher than 3-year-olds, the difference was not significant.

The three cardinality tasks placed increasing demands on the child. The "'Are

there more X than Y' task was the task that was least diffrcult for children suggesting

they understand relative quantity by this age. Al1 children could count to 10 and thus,

differences in their ability to solve these tasks cannot be attributed to acquisition of the

counting system.

The "Give-me-X' task was generally difficult for children at both ages. Children

often made mistakes by failing to count consistently in a one-to-one fashion, thus

arriving at an incorrect number of Lego. However, many children simply grabbed a pile

of Lego when asked for different quantities. These results fit with research by Sinclair

and Sinclair (1984) who found that preschool children were not always clear on what

number meant.

Children also had difficulty on the sharing task, in particular the 3-year-olds.

Thus, most children did not have a complete understanding of the invariance or

equivalence of number.

To compare children's pe~ormance across the cardinality subtasks the

number of children to pass (3 out of 4 correct) and fail each task was added across

conditions. It was found that children were most successful on the "'Are there more X

than Y" task where 69 children passed and only 21 children failed. For the "Give-me-

X' task, 36 children passed and 54 children failed. The sharing task proved most

difficult for children with only 3 1 children passing and 59 failing.

Not surprising then was the finding that the ""Are there more X than Y" task

was not correlated with the other two cardinality subtasks, which were correlated with

each other. Children performed better on this task than the other two tasks. Wym

(1990) found that children knew that each number word refers to a specific, unique

numerosity by time they understood the cardinal number 'two'. According to Resnick

(1 989), by age 4 children have Iearned to integrate the number-name sequences with a

protoquantitative cornparison schema, which is one that operates percepîually, without

any measurement process. Children can then identi@ which of a pair of numbers is

"more" by mentally consulting a kind of "mental number Iine". Together, this research

suggests that this task should be relatively easy for children as they had no difficulty

with the counting sequence and were relatively successful for the number 3 in the

"Give-Me-X' task, suggesting they understood its cardinal significance.

Although al1 three cardinality subtasks were expected to require that children

understood the comection between counting and cardinality, it is possible that they were

using an alternative strategy for the "Are there more X than Y' task. Children may have

been successful on this task simply by comparing both sets of toys on a perceptual basis.

This would reduce the overlap between this task and the other two subtasks and rnay

explain why only the "Give-Me-Xy task and the Sharing Task were correlated.

By including three conditions in the moving symbol task, numeral, analogue

and word, it was possible to examine how children build their knowledge of how

number can be represented fiom first order syrnbolism, the analogue condition, to

second order symbolism, the written numerals condition. The hypothesis of the study

was that to get to this last level, children must fust fully understand what numbers

represent. Therefore, they should show understanding of cardinality as assessed by the

three cardinality subtasks, before they are successful on the moving symbol task with

written numbers.

Children in this study did not need to be successful on the cardinality subtasks in

order to pass any condition of the moving symbol task. Thus, in general understanding

of written representations of numbers appears independent of cardinality. The three

cardinality subtasks, as they were distinct fiorn one another in the demands placed on

the child, were expected to be predictive factors for an ability to understand written

numbers. The only task which showed evidence for this when examining the chi-square

analysis was the "'Are there more X than Y' task. The chi-squares for this task and the

numeral condition, although not significant, did appear to follow the predicted pattern

with 18 chiidren passing both tasks and 4 passing the cardinality task and failing the

moving symbol task. Only 5 out of 30 children passed the moving symbol task and

failed the cardinality task. For the analogue condition, 7 children passed both tasks and

14 children passed only the cardinality task. Only 2 children were successfùl only on

the moving symbol task. For the word condition, again a large number of children, 16,

passed only the cardinality task, 10 passed both tasks and only 1 child failed the

cardinality task while passing the moving symbol task. Thus, at least for the "Are there

more X than Y" task, there was evidence that children must understand cardinality, what

the numbers themselves represent, before they can successfùlly understand written

symbols for numbers. However, children were more successful on this task and so were

more likely to pass.

However, the correlations between the cardinality subtasks and the Moving

Symbol Task suggest the two are not independant. The ""Give-a-nurnber" task was

significantly correlated with the moving symbol task (switch question). When broken

down by level of difficulty, the middle level and the difficult level are significantly

comelated with the "Give-a-number" task. In addition, the difficult level of the moving

symbol task is significantly correlated with the sharing task. Thus, when faced with

difficult numbers, children's understanding of written number was associated with the

success on the "give-a-number" and sharing task.

An alternative interpretation of the results is that children have not yet begun

to understand or be able to interpret written representations of number. If this were the

case, they would see no connection at al1 between the two piles of Lego and the written

symbol placed undemeath. Thus, there would be no reason to believe that what was

written on the card would change as it would not be comected with either pile and they

simply repeat the number given to thern by the experimenter. Thus, success on this task

could reflect either advanced knowledge of the representational significance of number

notations or no howledge at al1 that the notations signiQ quantities.

There were a number of subgoals in this study. One was to determine whether

using written nurnber words would affect children's success on the rnoving symbol task

with words, a task generally not well done by children of this age. In previous research

using the moving symbol task with words, the overall proportion of correct responses to

the switch question was .38 (Eiialystok, 1991). In contrast, when easy numbers were

used in the present study the proportion correct was -58. Even when small and large

numbers were used, the middle level, children scored on average .45. Only when large

numbers alone were used did the average score reach that of regular words, .36. Thus,

when small perceptually discriminable numbers are used children's performance on this

task improves.

A second subgoal of the study was to determine whether using different sized

numbers would make the task more dificult. Previous research has shown that children

are able to count much higher than the number to which they can give a correct cardinal

response. Wynn (1990) suggests that children are more oRen correct on cardinality

tasks with small set sizes because they are able to make a simple association between the

small number words and their quantities based of the appearance of the set compared

with large sets. The hypothesis here was that tasks using larger numbers should

therefore, require that children have a fil1 understanding of cardinality in order to pass

the moving syrnbol task and that these larger numbers would distinguished between

those who understood the invariance of symbolic representation and those who were

relying on their knowledge of the smaller counting words. Although the results for the

numeral condition failed to support this hypothesis as children were equally successfbl

at al1 levels of difficulty, the level of difficulty did affect the scores on the analogue and

word conditions. Analogue scores for the easy and middle condition were the same as

for the numeral condition. However, in the numeral condition they continued to be

successfL1 for difficult levels while in the analogue their scores declined in the difficult

level. For the word condition scores were lower for the easy and middle levels with the

same low scores for the difficult level as in the analogue condition. Thus, the difficult

level was important in detennining those who had a complete understanding of the

invariance of written syrnbols and those who relied on knowledge of smaller counting

words.

This study replicates the previous finding by Tannenbaum (1 997) where

children performed much lower in the word condition than the numeral or analogue

condition. As in the present study when small numbers are used there is no difference

between the numeral and analogue conditions. This study extends these findings by

including the dificult nurnber level. When large numbers are used children's

performance on the analogue condition goes down. This may be because when small

nurnbers are used children can discriminate among the nurnbers and rnake the

connection to the quantity of dots on the card. However, when large numbers like 13 or

16 are used children cannot simply look at the pile of Lego or the number of dots and

detemine if they match. They must rely on their knowledge that syrnbolic

representation is invariant and does not change when the card is switched.

Results from the moving syrnbol task show the different levels of

understanding of syrnbolic representation according to the three stages hypothesized by

Bialystok. The children in this study have only reached the stage of forma1

representation for words as they do not yet understand that symbols stand for meanings.

However, in the numerical system children of the same age have reached the third stage,

symbolic representation where they understand that numbers are symbols that signiQ a

particular value. This is evident by the fact that they pass the moving symbol task and

do so even when large numbers outside their counting range are used. They understand

that each number stands for a single meaning, a particular quantity. This occurs despite

the inability of many children to correctly take 6 Lego out of a pile of 10.

Previous research has shown that if a child reaches the syrnbolic representation

stage for words and they understand that each symbol stands for a single meaning they

also understand that more sounds, or longer words, are represented by more letters.

However, children who have reached the symbolic representation stage for numbers do

not show a clear understanding of cardinality. Symbolic representation, according to

Bialystok (1992) requires knowledge of both oral and written systems linking them

through a meaning relation. However, children capable of passing the moving symbol

task with numbers demonstrating knowledge of the written system are not always able to

count out a particular number of objects when given that number orally.

Bialystok and Codd (1996) found that children's understanding of written

number notations, how quantity should best be conveyed and what its symbolic

properties were, was slower to develop compared with their facility in using numbers to

count objects and comment on quantity. However, even the 5-year-olds tested did not

grasp the cardinal significance of the written notations, despite understanding that

numerals were the best choice for indicating quantity. Three-year-olds did not

demonstrate this understanding. Ail these children failed to ''CO-ordinate their

knowledge of the number system with their knowledge of a notational system whose

symbolic hnction it is to represent those numbers". Therefore, although children

demonstrated understanding of written number and its properties, invariance and one-to-

one comespondence between the quantity and the symbol, this does not mean that they

will CO-ordinate this knowledge with their knowledge of cardinality.

Results for the analogue condition are mixed. When smaller numbers are used

children perform well on the moving symbol task suggesting they are at the symbolic

representation stage. This may be because they are very familiar with these small

numbers and their understanding of symbolic representation may not really be tested.

This is similar to prereading children's ability to 'Lead" environmentally familiar words

like McDonalds. If these words are written without any cmtextual cues children are

unable t o read them.

Scores for the analogue condition fa11 to that of the word condition scores when

larger numbers are used. Thus, they do not have a fiill understanding of the relation

between the symbol and its meaning. When the numerosities are not easily recognizable

children become confbsed about the meaning of the written form and fail to see the

analogue representation as a symbol for the number of objects it represented. In

research by T a ~ e n b a u m (1997) the younger children counted the dots on the card.

While children were prevented fiom counting out loud in this study, they may have been

able to count to themselves for the smaller numbers. Numbers in the dificult level were

too large and arranged too randomly on the card for children to count.

These results are surprising given previous research in which children were

asked to provide written representation of different quantities of cookies in a lunchbox

(Bialystok, unpublished). Sixty-five percent of 3-year-old chiIdren chose to represent

the number of objects through an analogue form of representation, usually a picture of a

cookie to match each cookie in the box. None of these children chose to represent the

number of cookies numerically. With the 4-year-old group, over 70% of children used

analogue representations while only about 20% produced numeric representation. These

results fit with the findings that children of this age still have difficulty with tasks that

test their understanding of written numbers.

The predicted effect of age was not found for the moving symbol task. It is

surprising that the 4-year-olds, who have had more experience with number, in

particula. small numbers, would not perform better than the 3-year-olds. The only

exception was the dificult level for the word condition where 4-year-olds scored 45%

correct compared with 27% correct for the 3-year-olds. This is consistent with research

by Bialystok (1991) who did not find a significant age difference on the moving symbol

task with words for 3-5-year-olds although children did appear to improve with age.

This task used dificult words. As mentioned, there appears to be a longer period of

development for the word condition compared with the numeral condition. There is a

steeper decline in scores across levels for the 3-year-olds for both the word and analogue

conditions.

There are a nurnber of limitations in this study. First, the numbers used for the

difficult condition were quite far apart, for example, 1 8 and 14 or '1 2 and 16. Perhaps

children would have had more difficulty if the numbers were closer together and the

piles more perceptually similar. In research by Grunebaum (1998), an effect of level of

difficulty was found for numbers. For the easy number level children were 70% correct

whereas for the difficult nurnber level, children were only 53% correct. She also used

small and large numbers for her easy and difficult Ievels. However, her larger numbers

were closer together. These results may have fit with the hypothesis that children first

must understand what number means, cardinality, before they can perform successfully

on a task that assesses their understanding of written symbolic representation of number.

However, sample size for the present stiidy was smaller as the different conditions for

the moving symbol task were between subjects. Each condition had 30 children

compared with 60 children in the Gmnebaum study where the condition factor was

within subjects. There was an additional 30 5-year-olds who received each condition.

The present study needs replication to determine the exact relationship between large

and small numbers for the moving symbol task compared with the cardinality tasks.

A second limitation was the lack of a 5-year-old age group. Given the large

differences in means for both the different conditions of the moving symbol task and for

the cardinality subtasks and the poor performance of both age groups on a number of

the tasks an additional age group would have been useful to determine more clearly how

development proceeded. In addition, it may have been useful to use more than one task

to assess children's understanding of written symbolic representation as was done for

cardinality. It is possible that the moving symbol task with nurnerals did not require that

children understand cardinality. Knowing what the symbols are called without

understanding that they map ont0 the quantity may make children believe that the

meaning, therefore, would not change when the card is switched to the incongruent pile

of Lego.

Finally, in future research it would be useful to include children's response to

the question "Are there X objects" in addition to "Give me X objects". Both place

greater demands on the child than tasks that ask how many because they cannot sirnply

repfy with the last counting word. However, the "Are there X objects" may provide

more of a cue to count the objects for those children who simply grabbed a handfùl of

objects.

In fiture research detailed observations of how children solved the cardinality

subtasks would be usefiil. It was observed that some children simply grabbed piles of

Lego when asked for a specific number. ûthers knew to cuunt but did not successfùlly

count in a one-to-one fashion for the whole count, often ending one or two Lego short.

These mistakes, however, show dieerences in understanding of cardinality and could be

compared to success on the moving symbol task. Thus, more in-depth observations

would be valuable, for example, in particular the Give-me-X task (for example, some

children simply grabbed piles of Lego, others counted while they picked up pieces but

did not take the right amount and simply did not use a one-to-one method of taking the

pieces out so that the number asked for matched the number in their pile.

There are a number of applications for this study. Children must understand

how letters refer to sounds in order to read and they must understand how numbers refer

to quantity in order to do addition and subtraction. Thus, cardinality is a precursor to

arithrnetic. Children must coordinate their knowledge of the number system, counting,

and its notational system, numerals, to understanding arithmetic, a forma1 study that

usually commences at about 5 or 6 years of age. According to Bialystok and Codd

(1997), children must understand cardinality before they will understand the way in

'khich quantities can be symbolically manipulated through arithmetical procedures" (p.

85). They must realize that there is a relationship between each number in the counting

sequence they have learned and a designated quantity. This is because each number in

the order signifies the cumulative numerosity and the last number counted indicates the

size of the set. They suggest it is because of cardinality that numbers have symbolic

meaning.

One hypothesis of this study was tasks using larger numbers should require that

children have a fùll understanding of cardinality in order to pass the moving symbol task

and that these larger numbers would distinguish between those who understood the

invariance of symbolic representation and those who were relying on their knowledge of

the smaller counting words. Results fiom the analogue and word condition met this

hypothesis. Analogue scores for the easy and middle condition were the same as for the

numeral condition. However, in the numeral condition they continued to be successfùl

for dificult levels while in the analogue their scores declined in the difficult level. For

the word condition scores were lower for the easy and middle levels with the same low

scores for the difficult level as in the analogue condition. This is true for the cardinality

subtasks as well. Children who are successlùl at giving three pieces of Lego may not be

successful at giving higher numbers.

Teachers very often o d y use manipulatives and math questions with numbers 1-

10. They make the assumption that if students can represent and manipulate these

numbers successfùlly they can do so for al1 numbers. However, this research suggests

that even if a child can demonstrate understanding of the invariance of written symbots

for small numbers, they will not necessarily be able to represent higher numbers and if

they know the cardinal value of a particular number they will not necessarily know al1

others. Thus, teachers need to include higher nurnbers when teaching and assessing

children.

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Table 1

Mean Scores for the Visually Cued Recall Task by Condition and Age

Condition n - M - SD - n - M SD

Analogue 15 16.27 10.23 15 22.07 15.08

Word 15 15 6.55 15 18.87 6.78

Table 2a

Mean Scores for the "Give-Me-X" Task bv Aae and Condition

3 years 4 years Total

Quantity - n - M - SD - n M - SD n - M SD

3 question 45 0.60 0.5 45 0.73 0.45 90 0.67 0.47

4 question 45 0.56 0.5 45 0.57 0.48 90 0.61 0.49

6 question 45 0.24 0.44 45 0.53 0.5 1 90 0.39 0.49

7 question 45 0.22 0.42 45 0.44 0.5 90 0.33 0.47

Table 2b

Mean Scores for the "Give-Me-XI' Task bv Condition and Question

Numberal Analogue Word Quantity - n - M - SD - n - M - SD n - - M SD

3 question 30 0.77 0.44 30 0.67 0.48 30 0.57 0.45

4 question 30 0.67 0.48 30 0.7 0.47 30 0.47 0.5

6 question 30 0.43 0.5 30 0.57 0.38 30 0.47 0.5

7 question 30 0.37 0.5 30 0.24 0.38 30 0.4 0.49

Table 3

Mean Scores for the "Are There More X Than Y" Task b~ Aae and Condition

- -

3 years 4 years Tota I

Quantity - n - M - SD - n M - SD n - El - SD

4-6 question 45 0.7 1 0.46 45 0.82 0.39 90 0.77 0.43

7-5 question 45 0.73 0.45 45 0.93 0.25 90 0.83 0.38

5-8 question 45 0.62 0.49 45 0.76 0.44 90 0.69 0.47

9-6 question 45 0.71 0.46 45 0.93 0.25 90 0.82 0.38

Table 4

Mean Scores for the Sharina Task by Age and Condition

3 years 4 years Total

Quantity - n M - SD - n h!! - SD - n M -

3 question 45 0.51 0.5 1 45 0.84 0.37 90 0.68 0.47

4 question 45 0.29 0.46 45 0.5 1 0.005 1 90 0.4 0.49

6 question 45 0.29 0.46 45 0.67 0.48 90 0.48 0.5

7 question 45 0.2 0.4 1 45 0.47 0.51 90 0.33 0.47

Table 5

Mean Scores for the First and Retum Position of the Movinn Symbol Task by Age and Condition

First - 3 years First- 4 years Return - 3 years Return - 4 years Condition - M - SD - n - M - SD n M - SD - n M - SD

Number 15 1.00 0.00 15 1.00 0.00 15 0.92 0.26 15 0.96 0.08

Analogue 15 1.00 0.00 15 1.00 0.00 15 0.92 0.14 15 0.87 0.24

Word 15 1.00 0.00 15 1.00 0.00 15 0.68 0.4 15 0.91 0.24

Table 6

Mean Scores for the Switch Position of the Movina Svrnbol Task bv Age and Condition

Switch -3 years Switch - 4 years Condition - n - M - SD - n - M - SD

Number 15 0.88 0.26 15 0.88 0.27

Analogue 15 0.67 0.3 1 15 0.57 0.38

Word 15 0.44 0.41 15 0.49 0.49

Table 7

Mean Scores for the Switch Position in the Movine. S@ol Task b~ Level of Difficulty. Age. Condition

Easy- 3 years Easy- 4 years Middle-3 years Middle4 years Difficult-3 years SD Condition M - M - SD M - SD M - SD - M - SD

Number 0.87 0.31 0.93 0.26 0.88 0.31 0.93 0.26 0.90 0.26

Analogue 0.87 0.35 0.73 0.46 0.78 0.41 0.57 0.49 0.37 0.41

Wor d 0.63 0.47 0.53 0.52 0.42 0.48 0.48 0.50 0.27 0.46

Difficult-4 years Across Levels SD M - - 3 years 4 years

Table 9

Stringent Chid-squares for the Moving Symbol Task and the Cardinality Subtasks

"Give-Me-X' TAsk 1 Pass 1 Fail

Number 'fi Fail

Analogue -( "Give-Me-X TAsk

Word rx

"Are There More X than Y" Task

"Are There More X than Y' Task

Num ber Pass

1 Pass 1 Fail

Pass 16

"Are There More X than r' Task

Fail 8

Analogue P a s Fail

Word p.UV

Sharing Task

Number

8 11

ShaRng Task

4 7

Analogue Pass Fail

Sharing Task

Table 10

Correlation Matrix for the Cardinality Subtasks

Task 'Give- Me-X "More-X-Than-Y'

"More-X-Than-Y" 0.2

S haring 0.43 * 0.18

o. 1 O

three four six seven Question

Mean Score 0.85 : 0.75 ! 1 1

i I I

E ~ w Middle Difficult

Lewl bt Ufflcultyfor Numeral

Mean Score 0.4

+Age 3

+Age 4

-a- tine 3

O I 1 1 1 I 1

E ~ S Y Middle Difficult

Level of Ufficuity for Analogue

0.7 0.6 + Age 4 0.5

Mean Score 0.4 -- 0.3 -- 0.2 -- 0.1 --

~ S Y Middle Difficult

Lew l of Ufficutty for Word

Mean Score

Level of Dlfficuity