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Research Report

Chinese Children Excel on Novel

Mathematics Problems EvenBefore Elementary SchoolRobert S. Siegler and Yan Mu

Carnegie Mellon University

ABSTRACTKindergartners in China showed greater nu-

merical knowledge than their age peers in the United

States, not only when tested with arithmetic problems,which Chinese parents present to their children more often

than U.S. parents do, but also when tested with number-

line estimation problems, which were novel to the children

in both countries. The Chinese kindergartners number-

line estimates were comparable to those of U.S. children 1

to 2 years more advanced in school. Individual differences

in arithmetic and number-line-estimation performance

were positively correlated within each country. These re-

sults indicate that performance differences between Chi-

nese and U.S. children on both practiced and unpracticed

mathematical tasks are substantial even before the chil-

dren begin elementary school.

Children in China, Japan, and other East Asian countries out-

perform their American age peers on numerous mathematical

tasks, including those involving counting, arithmetic, algebra,

and geometry (Ginsburg Choi, Lopez, Netley, & Chi, 1997;

Stevenson, Chen, & Lee, 1993). This learning gap, evident as

early as kindergarten, persists throughout elementary and high

school, and probably beyond (Stevenson & Stigler, 1992).

Most explanations of the learning gap have focused on

schooling. Children in East Asia spend more time on math in

classrooms, devote more time to doing math homework after

school, and encounter more challenging math problems in each

grade than do their American peers (Chen & Stevenson, 1989;

Stigler & Hiebert, 1999). Compared with American teachers,

East Asian teachers more deeply understand fundamental math-

ematics concepts and use more focused pedagogical practices

(Ma, 1999), provide more substantive explanations of procedures

(Perry, 2000), and more often promote multiple solutions to a

given problem (Geary, 1994).Differing cultural emphases also seem to contribute to the

learning gap. Parents in China place greater emphasis on the

importance of mathematics and are more involved in their

childrens math learning, compared with parents in the United

States (Huntsinger, Jose, Liaw, & Ching, 1997; Zhou et al.,

2006). Even before Chinese children enter school, their skill in

counting and in adding numbers with sums of 10 or less is su-

perior to that of their American peers (Geary, Bow-Thomas, Fan,

& Siegler, 1993; Geary, Bow-Thomas, Liu, & Siegler, 1996).

The basic question that motivated the present study was

whether the superior mathematical knowledge of East Asian

preschoolers is limited to skills that are taught directly byparents or is more general. The skills on which cross-national

differences among preschoolers have been documented

counting and addingare ones for which Chinese parents

provide substantial practice before their children begin ele-

mentary school (Zhou et al., 2006). However, Chinese parents do

not provide instruction on other, less routine mathematical

tasks, such as numerical estimation (Zhou et al., 2006). This

raises the issue of whether the learning gap among young chil-

dren is merely a superficial result of rote learning, or whether the

differences in mathematical knowledge between Chinese and

U.S. children before they begin elementary school extend to the

ability to solve novel problems.In the present study, we examined Chinese and American

kindergartners knowledge of a practiced and an unpracticed

task. The practiced task was arithmetic (addition of single-digit

numbers). The unpracticed task was number-line estimation

(Siegler & Opfer, 2003). For each problem on this task, children

were given a fresh number line with 0 at one end, 100 at the

other, and nothing in between, and the experimenter asked them

to locate the position of a number on the number line. The

logarithmic and linear functions that best fit each childs esti-

Address correspondence to Robert S. Siegler, Department of Psy-

chology, Carnegie Mellon University, Pittsburgh, PA 15213, e-mail:

rs7k@andrew.cmu.edu.

PSYCHOLOGICAL SCIENCE

Volume 19Number 8 759Copyrightr

2008 Association for Psychological Science

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mates were determined, and the success of these functions in

accounting for the estimates of all presented numbers was

compared.

For performance on the number-line task to be optimal, the

best-fitting linear function must account for 100% of variance in

the estimates and have a slope of 1.00. American adults per-

formance closely approximates this ideal (Siegler & Opfer,

2003). In contrast, American children exhibit a consistent age-

related progression from logarithmic estimation patterns to

linear ones. On number lines from 0 to 100, the large majority of

kindergartners generate estimates best fit by a logarithmic

function, the large majority of second graders generate estimates

best fit by a linear function, and first graders are about as likely

to generate one pattern as to generate the other (Geary, Hoard,

Byrd-Craven, Nugent, & Numtee, 2007; Siegler & Booth, 2004).

This means that kindergartners and many first graders esti-

mates of numerical magnitude rise too quickly at the low end of

the scale and are too compressed at the high end, unlike second

graders estimates (see Fig. 1). The developmental pattern re-

peats itself between second and fourth grade in the range from 0to 1,000 (Booth & Siegler, 2006).

Linear representations of numerical magnitudes appear to be

crucial to childrens mathematics performance and learning.

The linearity of individual childrens number-line estimates

correlates strongly with their performance on numerous other

mathematical tasks (Booth & Siegler, 2006; Laski & Siegler,

2007). Linearity of estimates also correlates substantially with

overall scores on math achievement tests from kindergarten

through fourth grade (Booth & Siegler, 2006; Siegler & Booth,

2004). Perhaps most striking is the finding that providing

randomly selected children with visual representations of the

magnitudes of addends and sums along a number line improves

their learning of answers to arithmetic problems (Booth &

Siegler, 2008).

The current study is the first to examine Chinese kinder-

gartners proficiency on the number-line-estimation task. We

hypothesized that Chinese preschoolers practice at arithmetic

and counting would improve their understanding of numerical

magnitudes, even on unpracticed tasks such as number-line

estimation. Consider how the most common early addition

strategy, counting fingers, could contribute to knowledge of

numerical magnitudes. With this strategy, thelarger the sum, the

more fingers children put up, the more fingers they see, the more

numbers they count, and the more time they take to reach the

sum. For example, solving 4 1 4 by counting fingers requires

putting up and seeing twice as many fingers, saying twice as

many numbers, and taking roughly twice as much time as

counting fingers to solve 2 1 2. These kinesthetic, visual, ver-

bal, and temporal cues provide broad-based support for a sense

of numerical magnitudes. Thus, the greater arithmetic and

counting experience of Chinese children, relative to U.S. chil-

dren, was expected to produce superior number-line estimation,despite the fact that neither Chinese nor U.S. children have

experience with this task. The same logic implies that within

each society, individual differences in arithmetic skill and

number-line estimation should correlate positively: Children

who are more experienced and skilled at arithmetic should be

better at the unpracticed number-line task as well.

Thus, this experiment tested three main predictions: (a) that

Chinese kindergartners number-line estimates would be more

accurate, be more linear, and have slopes closer to 1.00 than

those of American peers; (b) that Chinese kindergartners

arithmetic skill would exceed that of their American peers; and

(c)that in both the Chinese andthe American samples,individual

0

20

40

60

80

100

0

Actual Magnitude

Kindergarten

0

20

40

60

80

100

Actual Magnitude

Second Grade

Estim

atedMagnitude

20 40 60 80 100 0 20 40 60 80 100

Fig. 1. Median estimates of U.S. kindergartners and second graders as a function of actual magnitude (data from Siegler &

Booth, 2004, Experiment 1). As the equations show, kindergartners estimates increase logarithmically with numerical

magnitude, whereas second graders estimates increase linearly.

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childrens number-line proficiency would correlate positively

with their arithmetic skill.

METHOD

Participants

The 29 Chinese children (mean age5 68 months, range5 63

73 months; 11 girls, 18 boys) were recruited from a preschoolaffiliated with Tianjin Normal University. The 24 American

children (mean age 5 67 months, range 5 6375 months; 12

girls, 12 boys; 88% Caucasian, 12% Asian) were recruited from

a preschool affiliated with Carnegie Mellon University. Con-

versations with principals and teachers at the two sites indicated

that the socioeconomic status of the Chinese children was no

higher, and was perhaps lower, than that of the American chil-

dren, and that the prestige of the preschool in China, relative to

other Chinese preschools, was lower than the prestige of the

preschool in America, relative to other American preschools.

The principals and teachers descriptions were consistent with

the knowledge of the second author, who had lived in both citiesand was familiar with both preschools.

Procedure

The problems and instructions on the number-line task were like

those in previous studies (e.g., Siegler & Booth, 2004). Among

the 26 numbers whose positions children estimated, 4 were from

each of the first three decades, and 2 were from each subsequent

decade. The reason for oversampling the first three decades was

to better discriminate between linear and logarithmic estimation

patterns. The numbers3, 4, 6, 8, 12, 14, 17, 18, 21, 24, 25, 29,

33, 39, 42, 48, 52, 57, 61, 64, 72, 79, 81, 84, 90, and 96were

ordered randomly.

The addition task was one used previously by Geary et al.

(1996) to compare the arithmetic knowledge of Chineseand U.S.

kindergartners. It consisted of 70 addition problems with sums

between 2 and 10. Children were given 1 min to answer as many

items as possible. Thetwo tasks were presented in random order.

RESULTS

Number-Line Task

To examine the accuracy of the childrens estimates on the

number-line task, we computed their percentage absolute error

(PAE) using the following formula: PAE 5 jestimate esti-

mated quantityj/scale of estimates. Thus, if a child was pre-sented the number 85 and estimated its location as being at the

point that corresponded to 75, the childs PAE on that trial would

be 10% (j75 85j/100).The American childrens mean PAE was 22% (SD5 8.3). The

Chinese children were considerably more accurate, with a mean

PAE of 15% (SD5 7.7), t(51)5 3.05, p< .01, d5 0.82, prep5

.977. The Chinese kindergartners mean PAE was between thePAEs of American first graders and second graders in previous

studies (e.g., Siegler & Booth, 2004).

We next examined the linearity of the childrens estimates. As

in previous studies, American kindergartners median number-

line estimates of the 26 numbers magnitudes were better fit by a

logarithmic function (R25 .90) than by a linear function (R25

.72; see Fig. 2). Unlike the estimates of any previous American

kindergarten sample, the Chinese kindergartners median esti-

mates were better fit by a linear function (R2 5 .95) than by a

logarithmic function (R25 .86; see Fig. 2).

Analyses of individual childrens estimates told a similar

story. For the Chinese sample, the best-fitting linear function

0

20

40

60

80

100

00

20

40

60

80

100

0

Actual Magnitude

ChinaU.S.

Estim

atedMagnitude

20 40 60 80 100 20 40 60 80 100

Actual Magnitude

Fig. 2. Results from the present study: median estimates of U.S. and Chinese kindergartners as a function of actual

magnitude. Also shown are thefunctions with thebest fit to thedata (logarithmic forU.S. kindergartners, linearfor Chinese

kindergartners).

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accounted for greater variance in individualchildrens estimates

than the best-fitting logarithmic function did (means of 63% vs.

58%), t(28)5 2.83, p< .01, d5 0.54, prep5 .959. In contrast,

in the American sample, the best-fitting logarithmic function

accounted for greater variance in individualchildrens estimates

than the best-fitting linear function did (means of 53% vs. 43%),

t(23)5 3.89, p< .01, d5 0.65, prep5 .992. The mean variance

accounted for by the linear function was greater in the Chinese

sample than in the American sample (63% vs. 43%), t(51) 5

2.22, p < .05, d 5 0.65, prep 5 .908. More Chinese than

American students estimation patterns were better fit by a lin-

ear function than by a logarithmic one (62% vs. 17%), w2(1,N5

53)5 11.15, p < .01.

The slopes of the best-fitting linear functions also tended to be

closer to 1.00 for the Chinese than for the American children

(0.53 vs. 0.39), t(51)5 1.85,p< .10, d5 0.50,prep5 .85. Thus,

analyses of individual childrens accuracy, of the linearity of

their estimates, and of the slopes of their best-fitting functions

converged on the conclusion that the Chinese kindergartners

performance on the novel number-line task was considerablymore advanced than that of the American kindergartners.

Arithmetic Performance and Its Relation to Number-Line

Estimation

The Chinese kindergartners also answered more addition

problems correctly than did their American peers (M5 8.4,

SD 5 3.8, vs. M5 5.3, SD 5 4.2), t(51) 5 2.91, p < .01,

d 5 0.88, prep5 .97.

Within both the U.S. and the Chinese samples, individual

differences in the number of correct addition answers were re-

lated to individual differences in the quality of number-line

estimates, both when estimation quality was indexed by PAE,rU.S.(22)5 .64, p < .01, and rChina(27)5 .38, p < .05, and

when estimation quality was indexed by linearity, rU.S.(22) 5

.60, p < .01, and rChina(27)5 .40, p < .05.

DISCUSSION

Results of this study were consistent with our three main hy-

potheses. First, Chinese kindergartners number-line estimates

were more advanced than those of their American peers. Second,

Chinese kindergartners arithmetic performance was also more

advanced. Third, individual differences in number-line esti-

mation and arithmetic proficiency were positively correlatedwithin each country. Thus, even before Chinese children enter

elementary school, their mathematical knowledge is superior on

tasks that parents do not present, as well as on tasks that parents

do present, and individual differences in their performance on a

familiar numerical task are related to individual differences in

their performance on an unfamiliar numerical task.

The literature on intersensory redundancy and learning pro-

vides a useful context for thinking about the present findings.

Providing multiple, synchronous, redundant cues promotes

many types of learning, including numerical learning (Jordan,

Suanda, & Brannon, 2008). For example, Jordan et al. found that

providing redundant visual and auditory cues to numberallowed

6-month-olds to make finer discriminations among sets with

varying ratios of dots than they did when visual or auditory cues

were presented alone. Our predictions that Chinese childrens

numerical estimation would be more advanced than American

childrens and that individual differences in arithmetic profi-

ciency and number-line estimation would be correlated were

based on a similar logic. The well-practiced activities of

counting fingers and other objects to solve arithmetic problems

and determine set sizes convey redundant kinesthetic, visual,

auditory, and temporal information about numerical magni-

tudes. Practice in adding and counting is surely not the only

source of differences between Chinese and U.S. childrens nu-

merical knowledge, but it appears to be one source.

A similar analysis gave rise to a previous prediction that

playing a numerical board game akin to Chutes and Ladders

would improve preschoolers knowledge of numerical magni-

tudes. Such games provide the same type of redundant cues tonumerical magnitude as does counting fingers to solve arith-

metic problems. The greaterthe number in a square of the board

game, the greater the number of movements to reach it, the

greater the number of words the child has said and heard by

that time, the further the distance of the square from the

origin, and the more time it takes the token to reach the square.

In studies consistent with this analysis, preschoolers from low-

income backgrounds who were randomly assigned to play a

linear number board game improved their number-line estima-

tion and magnitude comparison more than did children who

played a similar game on a board with different colors rather

than different numbers in the squares (Ramani & Siegler,2008; Siegler & Ramani, in press). As these studies and

the present study illustrate, analyzing everyday activities in

terms of the cues they provide for inducing desired concepts

may advance understanding of cross-cultural, individual,

developmental, and social-class differences in knowledge and

learning.

AcknowledgmentsThis research was supported by Depart-

ment of Education Grants R305H020060 and R305H050035.

We thank Yajing Zhang, principal of the preschool affiliated with

Tianjin Normal University, and Sharon Carver, principal of the

Carnegie Mellon Childrens School, for their assistance.

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