young american and chinese children's everyday mathematical activity

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Young American andChinese Children's EverydayMathematical ActivityHerbert P. Ginsburg , Chia-ling Lin , Daniel Ness &Kyoung-Hye SeoPublished online: 18 Nov 2009.

To cite this article: Herbert P. Ginsburg , Chia-ling Lin , Daniel Ness & Kyoung-Hye Seo (2003) Young American and Chinese Children's Everyday MathematicalActivity, Mathematical Thinking and Learning, 5:4, 235-258, DOI: 10.1207/S15327833MTL0504_01

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Young American and Chinese Children’sEveryday Mathematical Activity

Herbert P. GinsburgChia-ling Lin

Teachers College, Columbia University

Daniel NessDowling College

Kyoung-Hye SeoUniversity of Wisconsin, Milwaukee

This study examined possible cultural and SES differences in the type, frequency,and complexity of 4- and 5-year-old American and Chinese children’s everydaymathematical activities during free play. 60 American children, 30 each from lower-and middle-socioeconomic status (SES) families, and 24 Chinese children, 12 eachfrom lower- and middle-SES families, participated in the study. Each participant wasvideotaped during free play for 15 min. The results showed that during free playAmerican and Chinese children exhibit similar types of mathematical activity. Yet,Chinese children engage in a considerably greater amount of mathematical activity,particularly pattern and shape, than do American children. However, closer examina-tion revealed that Chinese and American children do not differ in the complexity ofmathematical activities. The results failed to reveal significant differences in the fre-quency and complexity of everyday mathematics between lower- and middle-SESgroups in both cultures.

This article describes a cross-cultural comparison of 4- and 5-year-old Americanand Chinese (from Taiwan) children’s everyday mathematical behavior. We wereinterested in learning whether American and Chinese children exhibit similartypes of mathematical behavior in free play; whether Chinese children show a

MATHEMATICAL THINKING AND LEARNING, 5(4), 235–258Copyright © 2003, Lawrence Erlbaum Associates, Inc.

Requests for reprints should be sent to Herbert P. Ginsburg, Box 185, Teacher’s College,Columbia University, 525 West 120th Street, New York, NY 10027. E-mail: [email protected]

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higher frequency and complexity of early mathematical behavior; and whetherSES is more strongly associated with Chinese children’s everyday mathematicalbehavior than it appears to be with Americans’.

EVERYDAY MATHEMATICS

Everyday mathematics includes perceptions, like “seeing” more and less, that maybe strongly based in biological endowment (Ginsburg, 1989). It includes princi-ples of counting that on one view develop from “skeletal” biological structures(Gelman, 2000). Everyday mathematics also includes “informal” ideas like “add-ing makes more” (Brush, 1978) that may develop from experience in manipulatingobjects or observing events. It also includes culturally derived acquisitions, like thecounting words, and even “formal” skills, like writing conventional numbers.Mostly, everyday mathematics is untaught; adults are usually unaware of its exis-tence (apart from simple counting and recognition of plane shapes). However,sometimes parents do teach young children mathematical ideas and reinforce chil-dren’s spontaneous attempts to deal with mathematical issues (Anderson, 1997;Saxe, Guberman, & Gearhart, 1987). And, of course, some preschools and kinder-gartens may even offer a planned mathematics “curriculum.” In brief, everydaymathematics is whatever mathematics children acquire in their ordinary physicaland social environments. It may be formal and intuitive; it may be based in socialand cultural experience; and, to a small extent, it may even be formalized.

Research has shown that American 4- and 5-year-old children engage in a con-siderable amount of everyday mathematics of various types during free play(Ginsburg, Inoue, & Seo, 1999; Ginsburg, Pappas, & Seo, 2001). Pattern andshape, magnitude, and enumeration activities are frequently observed. Childrenalso show interest in spatial relations, classification, and dynamic change, but to alesser degree.

Little is known about the extent to which behaviors and interests of these typesdevelop in young children from other cultures, particularly in Chinese children.To our knowledge, no observational studies on this specific topic have beenconducted.

What might theory lead us to expect? The acquisition of young children’s ev-eryday mathematics is a constructive process guided by biological endowment,physical environment, and culture. Two factors in this equation make it likely thatbasic aspects of early mathematical behavior are universal, emerging despite wide-spread variation in culture. The human biological underpinning for early mathe-matics learning (Sophian, 1997, p. 345) and universal “supporting environments”for at least some aspects of mathematical development (Gelman, Massey, &McManus, 1991, p. 254) virtually guarantee that children in diverse cultural cir-cumstances develop highly similar early understandings of number, although the

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ages of acquisition may vary. There is a good deal of evidence supporting the no-tion that basic aspects of mathematical thinking are universal (Klein & Starkey,1988). Geary (1996) argued that cultures do not differ in the “biologically pri-mary” mathematical domain—basic cardinality, ordinality, counting, and simpleaddition and subtraction. For example, unschooled African and (schooled) Ameri-can children develop (although not at exactly the same ages) similar methods andconcepts for early addition (Ginsburg, 1982).

Furthermore, in some cases, culture reinforces the contributions of biology andthe supporting physical environment. Nursery schools and kindergartens in manycultures around the world are very similar. One reason may be that they are con-strained by the universal nature of young children, and another that most have beeninfluenced directly or indirectly by Froebel’s ideas. Although differences amongthem clearly exist, organized childcare and educational settings for young childrenoften involve similar activities, social arrangements, and play objects.

In brief, biological endowment, physical environment, and culture all point inthe same direction: Chinese and American children should show similar types ofmathematical behavior in the course of free play.

A CHINESE ADVANTAGE?

Even if our conjecture about universality is correct, children from different cul-tures may vary in the rate of acquisition and use of everyday mathematical con-cepts. For example, presumably because they must calculate in the marketplace,unschooled African children from a commercial tribe learn to perform mental ad-dition earlier than do their peers from a farming tribe (Ginsburg, Posner, & Russell,1981).

The question then is whether Chinese children’s early mathematical behavior ismore frequent or complex than is Americans’behavior. If demonstrated to exist, anAsian precocity in informal mathematical knowledge might help explain the factthat children from China, Japan, and Korea outperform their American peers inmathematics achievement as early as kindergarten (Stevenson, Lee, & Stigler,1986) and certainly by first grade (Stevenson et al., 1990). No doubt other factorscan and probably do contribute to the achievement differences, including the qual-ity of teaching, home environments, and psychological factors like motivation(Stevenson & Stigler, 1992). Yet, superior informal knowledge might help Asianchildren to assimilate the mathematics taught in school.

As previously mentioned, there does not seem to be any research employing ob-servation of behavior in the natural setting to compare everyday interests and be-haviors in American and Chinese children. Available comparative research em-ploys tests, various tasks, and clinical interviews to examine counting and otherbasic mathematical concepts. Several studies show that Chinese, Japanese and Ko-

AMERICAN AND CHINESE MATHEMATICAL ACTIVITY 237

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rean children (Geary, 1994; Miller & Parades, 1996; Miller, Smith, Zhu, & Zhang,1995; Miller & Stigler, 1987) learn counting (particularly between 10 and 20)more easily than do Western children. For example, young Chinese children learnthe first 10 numbers (which are essentially nonsense syllables in Chinese and Eng-lish, with no clear pattern) at the same level as the do American children (Miller etal., 1995). Yet, the Chinese children experience less difficulty than do their Ameri-can peers in learning the numbers from 10 to 20. Why? The teen numbers in Eng-lish are irregular. “Eleven” and “twelve” are strange words with no clear relation tothe base 10 system. “Thirteen” through “nineteen” are “backwards.” If Englishwere consistent, they should be “ten-three, ten-four,” and so on, just like“twenty-three,” and so on. The lack of regularity forces young English speakingchildren to undertake the difficult task of memorizing most of the first 20 numbers.By contrast, Chinese employs a transparent and regular base 10 form of counting(e.g., 13 is said as ten-three and 23 as two-ten-three). The regularity makes learn-ing relatively easy: Chinese children can learn effective rules for generating thenumbers from 10 to 20. In brief, because the structure of the Chinese counting sys-tem (and also the Korean and Japanese systems based on it) is better designed thanis English counting, 4- and 5-year-old East Asian children learn aspects of count-ing with more ease than do Americans.

Some have even argued that the structure of the counting system is a key factorin explaining why Chinese-speaking children in elementary school have an edge incalculation over Western children (Geary, Fan, Bow-Thomas, & Siegler, 1993;Miller & Parades, 1996). Thus, Miura and her colleagues (Miura, 1989; Miura,Okamoto, Kim, Steere, & Fayol, 1993) proposed that the underlying base-10structure inherent in the spoken and written counting numbers of these languagescontributes to East Asian children’s superiority in calculation. For example, al-ways thinking of 13 as 10 and 3 highlights the base 10 structure of the numbers andmakes “carrying” a sensible and easy procedure.

Only a few studies examine aspects of early mathematical thinking other thancounting. American 4-year-old children score slightly (but significantly) betterthan Korean preschoolers do on early mathematical knowledge (Song & Gins-burg, 1987), as measured by the Test of Early Mathematics Ability (TEMA), astandard test (Ginsburg & Baroody, 1983). One reason for the early differencemay be that young Korean children are confused by the need to learn both theKorean informal and formal systems of counting (Song & Ginsburg, 1987). Yet,by first grade the Korean disadvantage disappears and their performance sur-passes Americans’. The writers interpret the result as showing that the Koreansdo not begin with an intellectual “head start”; their later school success must beattributed to other factors, including schooling. Similarly, Chinese children haveno advantage over their American peers in basic number concepts like producinga given number of objects or keeping track of how many have been counted(Miller et al., 1995).

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A somewhat different result was obtained by a study using a series of “birthdayparty” games to investigate key aspects of informal mathematics (including con-cept of more, counting, cardinality, addition, and subtraction) in American, Chi-nese, Japanese, and Korean 4-year-old children (Ginsburg, Choi, Lopez, Netley, &Chi, 1997). The results showed that the Asians generally performed at a slightlyhigher level than did the Americans, although the differences were not large and noqualitative differences in thinking were observed.

A rare study of “spatial structures” shows that 4-year-old Chinese childrenmake more detailed human drawings than their Canadian peers, although bothgroups represented spatial layout at equivalent levels of complexity (Case et al.,1996). The writers note that at this age Chinese children receive more training indrawing than do Western children.

In brief, the Chinese-based counting system may be easier to learn than Eng-lish, and its transparent base 10 structure may provide a sounder conceptual basisfor learning mathematics than English. Yet, on the whole, research shows little dif-ference in the informal mathematics of Asian and American children at the 4- and5-year levels. Some research finds no difference between the two groups on key as-pects of number (Miller et al., 1995) and the complexity of spatial relations (Caseet al., 1996). Other research shows many similarities, and only small quantitativedifferences favoring one group or another (Ginsburg et al., 1997; Song & Gins-burg, 1987). And virtually nothing is known about Asian children’s everydaymathematical interests and behavior.

SES

A second goal of the study is to examine the association of mathematics in the ev-eryday environment with SES. Many studies employing tasks, tests, or clinical in-terviews show that in the United States lower-SES 4- and 5-year-old children tendto achieve less success than do their middle-SES peers on a variety of mathemati-cal tasks (Ginsburg, Klein, & Starkey, 1998). Yet, our previous work shows thatSES differences in U.S. children’s mathematics in free play are virtually nonexis-tent (Ginsburg et al., 1999). One reason may be that when examined in everydayenvironments or with problems relevant to their culture’s expertise, people exhibitfar more competence than may be shown under conditions of standard testing(Cole, 1996). Thus, the street children in Brazil exhibit effective strategies for de-termining the pricing of goods (Carraher, Carraher, & Schliemann, 1985; Saxe,1991); illiterate African merchants perform clever mental calculations (Petitto &Ginsburg, 1982); and the same is true of American adults as they shop for groceries(Lave, 1988). Clearly, some people can express their competence more easily innatural, familiar or comfortable settings than in artificial contexts that present spe-cial demand characteristics, as do tests, tasks, or interviews (Donaldson, 1978).


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Hence we are interested in investigating the issue of SES differences in Chinesechildren’s everyday mathematics. To our knowledge, there is little, if any, researchon the association of SES level with Asian children’s mathematical performance.One of the few studies in this area found that SES was associated with Korean4-year-olds’ performance on mathematical tasks (not mathematics in the every-day environment), with middle-SES children outperforming lower-SES children(Ginsburg et al., 1997).

OUR QUESTIONS

This study focuses on three basic questions. First, do Chinese and American chil-dren display the same types of everyday mathematical behavior in free play?Shared biological endowment, environmental supports, and cultural factors allsuggest that types of everyday mathematical behavior should be similar acrosscultures.

Second, do Chinese children exhibit a higher frequency or complexity of every-day mathematical behavior? Chinese children have the advantage of a more regu-lar counting system than do Americans. Yet, research has uncovered few differ-ences in Chinese and American children’s informal mathematics, and little isknown of Chinese children’s mathematical behavior in the ordinary environment.Hence, we make no predictions concerning cultural differences in the frequencyand complexity of mathematical behavior.

Our third question refers to possible SES differences in frequency and complex-ity of mathematical activity. American data suggest that SES is not strongly associ-ated with these behaviors (Ginsburg et al., 2001), perhaps because people oftendisplay greater intellectual competence in everyday environments than in test situ-ations. We are interested in knowing whether the same is true of Chinese children.

Part A deals with the frequency and types of mathematical activities and Part Bwith their complexity.

PART A: FREQUENCY AND TYPESOF EVERYDAY MATHEMATICAL ACTIVITIES

Method

Participants. A total of 84 children (60 American and 24 Chinese), whoseages range from 4 years to 6 years, participated in the study. Of 60 American chil-dren, 30 were from lower-SES families and 30 from middle-SES families. Fam-ilies were regarded as lower-SES if they qualified for subsidized daycare accord-ing to the local New York City guidelines. If they failed to qualify for such subsidy

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and one or both parents were working, families were considered to be middle-SES.Approximately three quarters of the lower-SES parents had blue-collar jobs, andthe rest were unemployed. About a quarter were single mothers. If they failed toqualify for such subsidy and one or both parents were working, families were con-sidered to be middle-SES. The middle-SES parents engaged in more diverse occu-pations. About 40% had white-collar jobs, 25% blue-collar jobs, and 35% profes-sional jobs. The lower-SES group consisted of 17 boys and 13 girls, and themiddle-SES group, 16 boys and 14 girls.

The Chinese children were from Taipei, Taiwan (Republic of China) and in-cluded 12 lower-SES and 12 middle-SES children. Although not identical to thosefound in the United States, SES distinctions exist in Taiwan. For the purposes ofthis study, we designated as lower-SES those children whose parents were all em-ployees of a municipal company, and were engaged, for the most part, in en-try-level work activities (e.g., clerical and custodial). We classified as middle-SESthose children whose parents held professional positions (e.g., physicians or teach-ers) or were in business. There were 5 boys and 7 girls in each SES group. The av-erage age of the American children was 4.89 (SD = 0.50). The Chinese childrenwere slightly older (M = 5.13, SD = 0.38).

Settings. TheAmericanchildrenwereselected fromfourearlychildhoodcen-ters (referred to as centers A, B, C, and D) located in New York City. Center A served3- and 4-year-old children from lower- and middle-SES families. The majority ofchildren were African-Americans and Latinos. Thirteen children from the4-year-old preschool classroom participated in our study. Their mean age was 4.76.Nine of them were from lower-SES families and four were from middle-SES fami-lies. Center B operated preschool classrooms serving 3-, 4-, and 5-year-old childrenfromlower-andmiddle-SESfamilies.ThemajorityofchildrenwereAfrican-Amer-icans and Latinos. Twenty-one children participated in our study. Their mean agewas 4.83. Center C served 2-, 3-, 4-, and 5-year-old children from lower-SES fami-lies. Most children were African-Americans and Latinos. Seven children from thekindergartenclassroomparticipated inour study.Theirmeanagewas5.29.CenterDoperated separate preschool classrooms for 3- and 4-year olds, as well as a kinder-garten, within a Catholic elementary school. The children were from diverse racialandethnicbackgrounds.Nineteenchildrenparticipated inour study.Theirmeanagewas 4.89. All of the children were from middle-SES families.

Although each of them served children from different socioeconomic and eth-nic backgrounds, these four centers differed little in room arrangement and kind ofplay materials available to children. There was, however, a wide difference in vari-ety and quantity of materials among the centers. A greater variety and quantity ofplay materials were available to children in the more affluent centers.

The Chinese children were selected from two early childhood centers (referredto as centers E and F) in Taipei. Center E served middle-SES children, while center


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F served lower-SES children. Both were similar to the U.S. centers in several ways.All centers were located in urban areas similar in population density. All the set-tings had basically the same types of materials. None was affiliated with a collegeor university. Yet, there were several differences between American and Taiwanesecenters. Taiwanese schools (excepting perhaps upper-SES schools not included inthis study) seem to have less space and hence have room for fewer play materialsthan do American schools. Taiwan is an island country whose area is approxi-mately 13,900 square miles (the size of Massachusetts, Connecticut, and Rhode Is-land combined) and whose population has recently exceeded 21 million persons.This makes Taiwan the second most densely populated country in the world, afterBangladesh, and space is at a premium.

Center E served children from middle-SES families. Twelve children were ran-domly selected from one classroom serving 4- and 5-year-olds. The classroom wasa little more than 600 square feet in area, about two-thirds the size of the Americancenters. Play objects that one finds in centers in Taipei are comparable, for themost part, to those found in centers in New York City. Perhaps most interestingabout center E and many other middle-SES early childhood centers in Taipei is theabsence of continuous quantity play objects like sand boxes, water wheels, and thelike, perhaps because of the need to conserve space. In addition, large woodenblocks were not available for the children at center E. This, too, may result fromspace limitations. The mean age of the twelve children from center E who partici-pated in the study was 4.95.

Center F, located in a residential and commercial area of Taipei, consisted oftwo large rooms in a company building. Thirty-five children ranging in age from 3years to 6 years shared these two rooms, and there was one head teacher for 35children. The rooms in center F were nearly double the size of those in center E. Inmany ways, center F was quite similar to center E in that both offered children arelatively limited number of play objects from which to choose. Again, no continu-ous objects (e.g., water wheels, sand boxes, and clay) were made available duringfree play. Twelve children were randomly selected from the 35 children for partici-pation in the study. Their mean age was 5.31.

None of the Centers in the United States or Taiwan offered children a programof mathematics instruction beyond simple activities like learning counting num-bers or identifying simple shapes.

Data collection. To capture children’s spontaneous mathematical activities,we used a method involving observation in the natural setting. The data collectionbegan by introducing all the children to the graduate student staff, video camera,and cordless microphone. When the children became comfortable with the equip-ment, staff, and procedures, a target child was randomly selected from the class-room for videotaping. After a cordless microphone was placed on the target child,his or her activities were videotaped for 15 min without interruption. In all cases,

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children were videotaped during “free play” time, typically in the morning. AllCenters offered at least an hour a day of free play. The videographer, using ahand-held video camera, was instructed to follow each child wherever he or shewent, except into the bathroom, and to remain at a reasonable distance from thechild (which did not hamper videotaping, because of the camera’s zoomingcapacity).

Data analysis. From pilot observations, we developed a coding system de-signed to capture key aspects of mathematical activity and context. The mathemat-ical codes were intended to analyze the content of mathematical activity thatemerged in play. They involved the following six categories:

• Classification: Systematic arrangement of groups according to clear criteria.For example, the child sorts blocks into groups of cubes and cylinders.

• Magnitude: Description of a magnitude (“There’s a lot here”) or comparisonof two or more items to evaluate relative magnitude. For example, the child claimsthat his tower is “more higher” than his friend’s.

• Enumeration: Numerical judgment or quantification. For example, the childsays that she has “three” blocks.

• Dynamics: Exploration of the process of change or transformation. For ex-ample, the child takes away the buttons on the table one by one and says, “Now Igot two. Now I got one. Now I got none!”

• Pattern and Shape: Exploration of patterns and spatial forms. For example,the child makes a symmetrical tower or identifies an object as “square.”

• Spatial Relations: Exploration of positions, directions, and distances inspace. For example, the child notes that one block is “under” another.

Context codes were intended to describe the environmental conditions associ-ated with children’s everyday mathematical activities. They involved four majorcategories:

• Location: Where the target child plays. For example, the child plays in theblock area.

• Play objects: What kind of object the target child plays with. For example,the child plays with a puzzle.

• Social interaction: Whether and how the target child interacts with peers.Social interaction involves six subcategories including noninteractive (playingalone); semiinteractive (playing side by side with peers but not sharing in the activ-ity); cooperative (e.g., building a block tower with a peer); competitive (e.g., tryingto build a tower bigger than a peer’s); interaction with adults (e.g., talking withteachers and other adults); and others.


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• Play activity: What type of play the target child engages in. Play activity in-volves five subcategories including exploration (e.g., experimenting with soundsthat can be made by a particular instrument); constructive (e.g., building or creat-ing something); dramatic (engaging in a fantasy with dolls); game (e.g., playing aboard game or other game); and others.

In analyzing children’s mathematical activities during play, we employed aminute-based scheme. The videotaped episodes of each child’s play were dividedinto 15 one-min segments. Then each minute-long segment was coded in terms ofmathematical content (whether the target child engages in mathematical activityand, if so, what kind) and context (under what environmental conditions the targetchild engages in mathematical activity). A pair of independent raters coded the en-tire videotapes of 30 American children’s 15-min play. The percent of perfectagreement between coders was 89% on mathematical content, 87% on social inter-action, and 86% on play activity. Another pair of independent raters coded the en-tire videotapes of 24 Chinese children’s play and achieved 96% perfect agreementon mathematical content and contexts.

RESULTS

We begin by comparing the frequency of various types of mathematical activity inthe Chinese and American children from different SES groups.

Frequency of Mathematical Activities

All 24 of the Chinese children (100%) engaged in at least one mathematical ac-tivity during the 15 min of free play. In the American sample, 53 of 60 children(88%) engaged in at least one mathematical activity. The Chinese children wereinvolved in at least one mathematical activity during an average of nearly 69%of the minutes (about 10 of the 15 min) spent in free play. The Chinese mid-dle-SES children exhibited a slightly higher average, 73%, than did the Chineselower-SES children, 64%. Both SES groups showed far more frequent mathe-matical activities during free play than did American lower- and middle-SESgroups. The American lower-SES children’s average percentage of minutesspent in at least one mathematical activity was 44% and middle-SES children’swas 43% (See Figure 1).

To examine whether these group differences were significant, we conducted a 2(culture) × 2 (SES) analysis of covariance (ANCOVA), using age as the covariate,of the percentage of time spent in mathematical activity. The results showed thatwith the effect of age removed, there was a significant main effect of culture, F(1,82) = 7.843, p < .05, but a lack of SES main effect or interaction effect of culture

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and SES. The Chinese children engaged in a significantly greater amount of math-ematical activity than did the Americans. Even though there was a significant cul-tural difference in frequency of mathematical activity, no significant differenceswere found between lower- and middle-SES groups.

In sum, we found a significant difference between the Chinese and Americanchildren in frequency of mathematical activity. The Chinese children engaged in aconsiderably greater amount of mathematical activity than did the American chil-dren during free play. However, we failed to find significant differences in fre-quency of mathematical activity between lower- and middle-SES groups.

Relative frequency of different types of mathematical activity. First, wedetermined the relative frequency of the six different types of mathematical ac-tivity by counting the number of children who were engaged in each. For the Chi-nese group, a large percentage of children engaged in four activities—pattern and


FIGURE 1 Frequency of mathematical activities: Culture and SES differences.

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shape (19 out of 24 children, 79%), magnitude (18 out of 24 children, 75%), enu-meration (18 out of 24 children, 75%), and spatial relations (17 out of 24 children,71%). Classification (4 out of 24 children, 17%) was infrequent, and no one en-gaged in dynamics. For the American group, a large percentage of children en-gaged in three mathematical activities—magnitude (38 out of 60 children, 63%),enumeration (31 out of 60 children, 52%), pattern and shape (25 out of 60 children,42%). Few children, however, engaged in the remaining three activities—spatialrelations (14 out of 60 children, 23%), dynamics (10 out of 60 children, 17%), andclassification (6 out of 60 children, 10%). It is important to consider, however, thatthe previously mentioned percentages indicate the number of children involved inthese activities, and not the percentage of the 15-min total spent in each mathemat-ical activity.

We then examined the average percentage of minutes during which the childrenengaged in each of the six mathematical activities. The percentages and rankingsof the six mathematical categories within the Chinese group are presented in Fig-ure 2. The mathematical activity that occurred most frequently was pattern andshape (an average of 49% of the minutes) followed by enumeration (17%), magni-tude (15%), spatial relations (15%), and classification (3%). As previously noted,dynamics did not occur at all. One major reason for this may be that, as noted pre-viously, Taipei centers do not contain sand boxes, water wheels, and other objectsthat may promote dynamics. The Chinese children exhibited pattern and shape ac-tivity during nearly half of their time in free play. Enumeration activity was a dis-

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FIGURE 2 Relative frequency of six types of mathematical activity: Average percentage ofminutes within a 15-min-episode.

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tant second, but was closely followed by magnitude and spatial relations. Withinthe American group, the percentages and rankings of the six mathematical catego-ries are also shown in Figure 2. The mathematical activity that occurred most fre-quently was pattern and shape (21%), followed by magnitude (15%), enumeration(11%), dynamics (6%), spatial relations (3%), and classification (2%).

Next consider the differences between Chinese and American lower- and mid-dle-SES groups in the relative frequency of different types of mathematical activ-ity. Table 1 presents the average percentages of minutes in which the Chinese andAmerican lower- and middle-SES groups engaged in different types of mathemati-cal activity. We conducted 2 (culture) × 2 (SES) analyses of covariance, using ageas the covariate, on the three most frequent mathematical activities—pattern andshape, magnitude, and enumeration. In regard to pattern and shape, we found a sig-nificant main effect of culture, F(1, 82) = 8.573, p < .05, but failed to find a signifi-cant main effect of SES or a significant interaction effect of culture and SES. TheChinese children engaged in a significantly greater amount of pattern and shapeactivities than did the American children. In regard to magnitude activity, wefound no significant effects of culture and SES, and no interaction between thetwo. Finally, on enumeration, we failed to find significant effects for culture andSES, but there was a significant interaction effect between culture and SES, F (1,82) = 4.467, p < .05. As can be seen in Table 1, the American lower-SES group ex-hibited a lower amount of enumeration activity than did the three other groups.

In brief, while the Chinese children exhibited a substantially greater amount ofpattern and shape activities than did the Americans during free play, the two cul-tural groups did not differ in magnitude and enumeration activities. The SESgroups did not differ in frequencies of pattern and shape, magnitude, and enumera-


TABLE 1Relative Frequency of Six Types of Mathematics Activity:

Culture and SES Differences in the Average Percentage of Minutes

Classification Magnitude Enumeration DynamicsPattern

& ShapeSpatial

Relations

CultureAmericans 2.0 15.0 14.0 6.0 24.0 5.0Chinese 3.0 15.0 17.0 0 49.0 15.0

SESLower-SES

Americans4.0 16.0 9.0 7.0 24.0 0.7

Lower-SESChinese

4.0 18.0 18.0 0 36.0 21.0

Middle-SESAmericans

1.0 18.0 18.0 8.0 24.0 7.0

Middle-SESChinese

2.0 12.0 15.0 0 61.0 10.0

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tion activities. The only exception was that the American lower-SES groupsshowed a considerably lower amount of enumeration activity than did the Chineselower- and middle-SES groups and the American middle-SES group.

In sum, pattern and shape, magnitude, and enumeration activities occurredmore frequently than did other types of mathematical activity in both the Chineseand American groups. Pattern and shape was the most frequently occurring mathe-matical activity in both groups, but the Chinese children exhibited a considerablygreater amount of pattern and shape activities than did the Americans. The Chineseand American children showed no significant differences in frequencies of magni-tude and enumeration activities. There was also a lack of SES differences in thefrequencies of these three types of mathematical activity.

Context

We also examined the environmental conditions in which children’s everydaymathematical activities occurred. To get a rough sense of the association betweencontext variables and frequency of mathematical activities, we conducted correla-tion analyses. In the American group, mathematical activity tended to occur alongwith the use of blocks or Legos (r = .605, p > .01) and constructive play (r = .713, p> .01). The Chinese group showed a similar pattern. Pattern and shape activity waspositively related to use of blocks or Legos (r = .430, p > .05) and constructive play(r = .485, p < .05). No significant relations were found between the frequency ofmathematical activity and any other context variable. In brief, the most strikingfinding is that in both cultures, there is a strong positive association between over-all mathematical activity and constructive play and play with blocks or Legos. Wealso found that in both cultures adults rarely involved themselves in children’s freeplay. The observed mathematical behaviors stem from the children themselves, notfrom adult prompting.

Correlations may fail to reveal whether contextual variables are directly associ-ated with the occurrence of mathematical activity within each minute, whereas con-ditional probabilities can be used to provide this information. We therefore began bycalculating the conditional probability of engaging in any mathematical activity,given that the child is playing with blocks or Legos in a particular minute. The resultwas .88 for the Chinese children and .79 for the American children. To examinewhether these group differences were significant, we conducted a 2 (culture) × 2(SES) ANCOVA, using age as a covariate, of the conditional probability of engagingin any mathematical activity, given play with blocks or Legos. The results showedthat with the effect of age removed, there was a lack of significant main effect of cul-ture. Moreover, there was a lack of SES main effect or interaction effect of cultureand SES. Next, we calculated the separate conditional probabilities of engaging inthe specific mathematical activities, given play with blocks or Legos. The only sub-stantial conditional probability was .83 for pattern and shape. The other figures were

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all .15 and below. Both conditional probabilities and simple correlations show thatfor Chinese children, as for Americans, playing with blocks or Legos is associatedwith mathematical activity in general and pattern and shape activity in particular.

Summary

The results indicated that the frequency of overall and some specific mathematicalactivities was related to culture but not to SES. The Chinese children engaged in aconsiderably greater amount of mathematical activity particularly in pattern andshape activity, than did the American children. However, there was a lack of signif-icant differences in frequency of mathematical activities between lower- and mid-dle-SES groups. Finally, strong positive associations were found between somecontextual variables and mathematical activity regardless of culture. In both theChinese and American groups, mathematical activity in general and pattern andshape activity in particular tended to occur when children played with blocks orLegos and engaged in constructive play.

PART B: DIFFERENCES IN COMPLEXITYOF MATHEMATICAL ACTIVITY

The findings on group differences in frequency of different types of everydaymathematics do not fully capture possible group differences within each type ofmathematical activity. For example, although we found a lack of cultural differ-ences in the overall frequency of enumeration, Chinese and American childrenmight differ substantially in the nature and complexity of enumeration activities,particularly because counting is easier in Chinese than it is in English (Miller et al.,1995). Similarly, we wonder whether the Chinese children not only show a greaterfrequency of pattern and shape activity, but also different and even more complexforms of it than do American children. Therefore, we examined cultural and SESdifferences in the complexity of the most frequently occurring mathematical be-havior–magnitude, enumeration, and pattern and shape. The frequencies of classi-fication, dynamics, and spatial relations were too small to warrant analysis.

Method

To capture complexity differences within each type of mathematical activity, wedeveloped codes based in part on inductive analyses of the videotapes and in parton a review of the research literature. The codes for magnitude involved the fol-lowing categories:

• Saying quantity or magnitude words. This involves describing the globalquantity or magnitude of objects, as in “Oh, this is really big.” The object may or


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may not be present and the child may not compare two objects directly. Thus, thechild may say that she is “faster” than another without adducing direct evidence tosupport the claim.

• Empirical matching. Here the child makes a direct comparison of concreteobjects, as when one child looks at two structures standing side by side and pro-claims, “Mine is more bigger.” The child may or may not use words. Thus, inbuilding a structure, the child may hold up a block next to a space to filled, see thatit is too short, and then get a larger one.

• Comparison without quantification. The child engages in magnitude in anapproximate way, without exact quantification. Thus, one child holds his armsapart to indicate that a picture in a book is “this much scary” and another child dis-agrees, holding his arms even wider apart and saying, “No, it this much scary.”

• Comparison with quantification. The child compares dimensions usingquantitative words. Thus, as two children are building a structure, one says, “Weneed one more,” indicating essentially that the line of blocks was too short by one.The child may estimate the quantity or may measure it exactly.

Codes for enumeration involved the following categories:

• Saying number words. The child simply says a number word, as in “I’m fiveyears old” or “I got it first.”

• Counting. The child overtly counts objects or says the number words withoutcounting objects.

• Subitizing/estimation. Without having counted, the child uses a number wordto designate the cardinal value of a set. The child could have subitized the value—that is, perceived the number without counting—or the child could have estimatedthe cardinal values; there is no way for us to tell. In either case, the context makes itclear that the child is not simply producing a wild guess or randomly producing anumber word.

• Reading/writing numbers. The child reads numbers, for example on a calen-dar, or writes numbers, for example on a piece of paper.

Codes for pattern and shape involved the following categories:

• Symmetry. This involves an exploration of symmetrical relationship, involv-ing a correspondence in size, shape, and relative position of parts on opposite sidesof a dividing line, median plane, or axis. For example, a child draws a picture of abutterfly in which the body serves as a line of symmetry and one of the wings is amirror image of the other. Or a child uses Legos to construct a building in whichthe towers and windows on the left and right sides are identical to one another interms of number, size and shape.

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• Patterning. Objects are arranged in a regular, rule-governed manner. For ex-ample, a child places several rectangular magnets in a row, evenly spaced, and thenplaces another triangular magnet on each.

• Figure Identification. The child’s behavior indicates recognition of particularshapes. For example, during clean-up time, the child places all the cubes in onebin, the rectangular prisms in another, and so on. Or the child consistently calls thecubes “squares” and does not apply this label to cylinders (which might be called“circle things”). The criterion is the child’s consistent ability to identify a shape,not necessarily to label it correctly.

• Shape Matching. The child uses geometric properties of shape to complete atask or solve a problem. For example, to complete part of a puzzle, a child uses aparticular piece because it has a straight edge on one side and a certain contour onanother.

Two independent raters coded magnitude, enumeration, and pattern and shapeactivities in a total of 20 children and achieved satisfactory agreement. The percentof perfect agreement between coders was 98 on magnitude and enumeration, and91 on pattern and shapes.

Results

Magnitude. Thirty-eight American children (out of 60) and 18 Chinese chil-dren (out of 24) engaged in magnitude activity. We calculated the average percent-age of minutes during which each of four types of magnitude occurred when thesechildren engaged in it. The sample did not include those children who did not en-gage in magnitude activity. As shown in Table 2, the Chinese children showed arank order of the relative frequency of the four types similar to that of Americans.


TABLE 2Relative Frequency of Four Types of Magnitude:


Saying Quantity orMagnitude Words

EmpiricalMatching

Comparison WithoutQuantification

Comparison WithQuantification

CultureAmericans 48 15 26 11Chinese 74 12 7 7

SESLower-SES Americans 44 20 23 14Lower-SES Chinese 69 20 0 11Middle-SES

Americans53 9 29 10

Middle-SES Chinese 79 4 14 4

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In both groups, saying quantity or magnitude words was most frequent and com-parison with quantification was least frequent. Yet, the Chinese children showedconsiderably more saying quantity or magnitude words (74%) than did the Ameri-can children (48%). In contrast, the American children exhibited more comparisonwithout quantification (26%) than did the Chinese children (7%).

A similar tendency was found between lower- and middle-SES groups acrossculture (see Table 2). Saying quantity or magnitude words was most frequent in allfour groups. However, the Chinese lower-SES group showed a considerablygreater amount of this type activity than did the American lower-SES group (69%and 44%, respectively). This was also the case for the Chinese and American mid-dle-SES groups (79% and 53%, respectively). In contrast, the Chinese lower-SESgroup exhibited no occurrence of comparison without quantification, whereas theAmerican lower-SES group and the Chinese and American middle-SES groupsshowed similar frequencies (23%, 14%, and 29%, respectively).

To further examine these group differences, we conducted a 2 (culture) × 2(SES) analysis using age as the covariate on the frequency of saying quantity ormagnitude words. The results indicated a significant main effect of culture, F(1,54) = 6.902, p < .05, but a lack of a main effect of SES and a lack of an interactioneffect between culture and SES. The Chinese children engaged in a significantlygreater amount of saying quantity or magnitude words than did the Americans.However, there was no significant difference between lower- and middle-SESgroups. No statistical analyses were conducted with other types of magnitude ac-tivity, because of an insufficient number of cases.

In summary, although the Chinese and American children showed similar fre-quencies of magnitude activity, closer examination revealed significant differencesin types of magnitude. Although saying quantity or magnitude words was the mostfrequent type in both groups, the Chinese children exhibited a significantly morefrequent use of these types of words than did the Americans. However, we failed tofind SES differences in the use of these words across culture.

Enumeration. Thirty-one American children and 19 Chinese children en-gaged in enumeration activity. We examined the average percentage of minutesduring which each of four types of enumeration occurred in those who engaged inenumeration. As presented in Table 3, the Chinese children showed somewhat dif-ferent relative frequencies of four types from those of the Americans. In the Amer-ican group, subitizing or estimation (42%) and saying number words (41%) weremost frequent, and then counting (11%), and reading and writing numbers (5%). Inthe Chinese group, the most frequent type was saying number words (53%), andthen subitizing or estimation (20%), reading and writing numbers (14%), andcounting (13%). The Chinese children exhibited more frequent use of numberwords and written numbers than did the Americans, even though the Americanchildren engaged in subitizing more frequently than did the Chinese children.

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Consider next SES differences. As can be seen in Table 3, saying number wordswas most frequent in all groups. However, the Chinese lower-SES group showedconsiderably more frequent use of number words and written numbers than did theother three groups. The Chinese middle-SES group exhibited a greater amount ofcounting activity than did the other groups. The American lower-SES group en-gaged in subitizing more frequently than did the other groups.

To further examine these group differences, we conducted 2 (culture) × 2 (SES)ANOVAs, using age as the covariate, on the frequencies of saying number wordsand subitizing or estimation. The results indicated a lack of main effects of cultureand SES, and a lack of interaction between the two on the frequency of sayingnumber words activity. Although the Chinese children and the lower-SES group inparticular showed more frequent use of number words than did the Americans,these differences were not significant. This same lack of significant effects charac-terized subitizing or estimation activity. Again, the differences were not significantthough the American children and the lower-SES group in particular showed morefrequent use of subitizing than did the Chinese children. No statistical analyseswere conducted on other types of enumeration activity because of the insufficientnumber of cases.

In sum, a great portion of children’s enumeration activities involved sayingnumber words or subitizing or estimating. The results indicated that the type ofchildren’s enumeration activity as well as its frequency was not related to cultureor SES.

Pattern and shape. Twenty-five American children and 19 Chinese chil-dren engaged in pattern and shape activity. We calculated the average percentageof minutes during which each of four types of pattern and shape activity occurredin those who engaged in pattern and shape. As shown in Table 4, the most frequenttype of pattern and shape activity in the Chinese group was figure identification


TABLE 3Relative Frequency of Four Types of Enumeration:


Saying Number Words CountingSubitizing orEstimation

Reading orWriting Numbers


SESLower-SES Americans 40 13 47 0Lower-SES Chinese 62 3 13 21Middle-SES Americans 43 10 39 9Middle-SES Chinese 43 24 27 6

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(56% of the minutes), and then shape matching (19%), symmetry (16%), and pat-terning. In both Chinese and American groups, figure identification was most fre-quent and patterning was least frequent. Although the Chinese children engaged infigure identification more frequently than did the Americans (56% and 43%, re-spectively), the American children engaged in patterning more frequently than didthe Chinese children (11% and 9%, respectively). Furthermore, the American chil-dren exhibited a considerably greater amount of symmetry than did the Chinesechildren (32% and 16%, respectively).

A similar tendency characterized lower- and middle-SES groups in both cul-tures (see Table 4). Figure identification was most frequent in all four groups. Yet,the Chinese lower- and middle-SES groups showed a more frequent occurrence ofthis type activity than did the American lower- and middle-SES groups. In con-trast, the American lower-SES group exhibited a considerably greater amount ofsymmetry than did the Chinese lower-SES group (35% and 5%, respectively). TheAmerican and Chinese middle-SES groups showed similar frequencies of this typeof activity (25% and 24%).

To further investigate these group differences, we conducted a 2 (culture) × 2(SES) ANCOVA, using age as the covariate, on the frequency of figure identifica-tion. The results indicated a lack of main effects of either culture or SES as well asa lack of a significant interaction effect. No statistical analyses were conducted onother types of pattern and shape activity because of the insufficient number ofcases.

In summary, although the Chinese children exhibited a significantly greateramount of pattern and shape activity than did the American children, a closer ex-amination generally reveals a lack of significant differences in the types of patternand shape activity in which the two groups engaged. In both groups, about half thepattern and shape activities involved figure identification—recognizing, sorting,or naming 2- and 3-dimensional shapes. The Chinese children showed a slightlymore frequent occurrence of figure identification than did the Americans, but the

254 GINSBURG ET AL.

TABLE 4Relative Frequency of Four Types of Pattern and Shape:


Symmetry Patterning Figure Identification Shape Matching


SESLower-SES Americans 35 13 43 9Lower-SES Chinese 5 9 53 33Middle-SES Americans 25 9 43 24Middle-SES Chinese 24 7 58 9

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difference was not significant. There was also a lack of SES differences in the fre-quency of figure identification activity. In contrast, the American children, particu-larly the lower-SES group, exhibited a considerably greater amount of explorationof symmetry than did the Chinese children. However, because of an insufficientnumber of cases, no statistical analyses could be conducted to examine thesedifferences.

Summary. Analyses of the types of three most frequent mathematical activi-ties revealed a similar tendency across culture and SES. In both the Chinese andAmerican groups, magnitude activity involved for the most part saying magnitudewords; enumeration activity was mostly saying number words or subitizing; andpattern and shape activity involved mostly identifying patterns or shapes. In thecase of magnitude, saying magnitude words showed significant effects of culture.The Chinese children exhibited a considerably more frequent use of magnitudeand quantity words than did American children. There was, in general, a lack ofsignificant cultural and SES differences in the various types of enumeration andpattern and shape activities.

DISCUSSION

Our first question was whether Chinese and American children show similar typesof everyday mathematics during free play. The answer is clearly yes. Childrenfrom both cultures show significant amounts of pattern and shape, magnitude, andenumeration. Also, context seems to play the same role in both cultures. In bothcultures, the use of Legos and blocks is strongly associated with mathematical ac-tivity in general and with pattern and shape in particular. Perhaps biological con-straints and environmental supports combine to lead children from different cul-tures to construct a similar everyday mathematics.

Our second question referred to possible cultural differences in the frequencyand complexity of everyday mathematics. The results show first that in general,Chinese children show a greater frequency of everyday mathematical activity infree play, particularly pattern and shape, than do Americans.

Why do Chinese children engage in more everyday mathematics? Some maywish to posit a Chinese or perhaps Asian biological propensity toward mathemati-cal activity. Yet, another possibility is environmental. Because of limited space, theChinese preschools made available to the children relatively few play objects, andprominent among them were Legos and blocks, which our data show tend to affordmathematical activity, particularly pattern and shape. By contrast, the Americanpreschools tended to contain a large number of objects, some of which may fail toafford mathematical activity. Thus, the school environment may have affordedChinese children more opportunities for mathematical play, whereas the American


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children had many alternatives that led them astray (from a mathematical point-of-view). Of course, under these circumstances the American children may haveengaged in more interesting nonmathematical activities than did the Chinese chil-dren. Our results require replication, and further research is necessary to examinethe role of environmental factors—availability of various materials—in facilitat-ing mathematical activity.

Another possibility is that teachers in the two cultures channeled the children’sactivity in different directions. We do not believe that this is likely, because teach-ers in both cultures interacted very little with children engaging in free play and didnot appear to influence it (or even to be aware of the details and significance ofwhat the children were doing).

Whatever its origins, the greater frequency of mathematical activity amongChinese children, particularly if it persists over time, can provide a useful intuitivefoundation in mathematics. Extensive exploration of pattern and shape may resultin visual intuitions that can provide a sound basis for many mathematical ideas.Thus, a picture of a staircase may more vividly capture the idea of a linearly in-creasing number series than may a numerical equation. To the extent that intuitionsof mathematics can usefully be comprised of visual images, experience with pat-tern and shape may enhance mathematical thinking. Further investigations of therole of visual imagery in Chinese children’s later mathematical superiority may beinformative. Furthermore, does the need to learn the intricacies of characters makeChinese children more visually oriented (and hence more mathematically ori-ented) than Americans?

Our second question also referred to the complexity of everyday mathematics inthe two cultural groups. Chinese and American children showed similar subtypesof mathematical behavior within each domain. For example, although Chinesechildren showed a higher frequency of pattern and shape activities than Ameri-cans, both groups showed similar subtypes of pattern and shape. Thus, both groupsexhibited more figure identification than any other type of pattern and shape activ-ity. One exception to the rule of similar subtypes occurred in the case of magni-tude, where Chinese children used more quantity and magnitude words than didAmericans. Nevertheless, the general similarity of subtypes indicates that al-though Chinese children engaged in more mathematical behavior than Americans,and more pattern and shape in particular, the groups did not differ in complexity ofmathematical thinking.

The third question referred to possible SES differences. The story is brief. Aswe found in the American data, there were few if any SES differences. SES makesas little difference in Taiwan as it does in New York for everyday mathematics. Ourinterpretation of this finding is that young children from different SES groups ex-press their competence more easily in natural, familiar, or comfortable settingsthan in artificial contexts like tests, tasks, or interviews. In Taiwan as in New York,lower-SES children are more mathematically competent than often expected.

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young american and chinese children's everyday mathematical activity

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