Running head: INFLUENCES ON EARLY MATHEMATICAL SKILLS
Influence of the complexity of the counting system and the development of cognitive stimulation on early mathematical skills: Differences among kindergarten of children from
Taiwan, the United States, and The Netherlands Loes van Gelderen
University of Amsterdam
Research Master of Educational Sciences Supervisor: Dr. P.F. de Jong
Acknowledgement
First I would like to thank my parents, Gertjan and Ria van Gelderen. I am pleased to thank them for their never ending love, trust, and support. In addition, I am very grateful for their willingness to pay my college fees and study books. I doubt if I would have graduate without them.
I would also like to thank Dr. P.F. de Jong, who has provided me with necessary assistance. He helped me to write my research proposal and to conduct the research. In the last phase of my thesis project, he provided a lot of feedback on my writings. It was great that he always answered my e-mail within a couple of hours. I am also grateful that he always had time for appointments.
I want to express my gratitude to Dr. J. Paik and her research team. Dr. Paik made it possible for me to follow a master course at San Francisco State University. With her enthusiasm for cross-cultural research in cognitive subjects she inspired me for that research area. I also thank Dr. Paik and her research team for the ability to use their data which they collected in Taiwan and the United States. I really appreciate that. Without their help, I would not have been able to make a international comparison.
I thank all the participating schools, parents, and children. Without them, conducting the studies was impossible.
Finally, I would like to express my respect to Vincent, who had to deal with the stress periods I have had in the last months. I thank him for his love, patience, and understanding.
Table of contents Abstract ... 4 Introduction ... 5 Study 1 ... 10 Method ... 10 Participants ... 10 Measurements ... 11 Procedure ... 13
Results and discussion ... 13
Study 2 ... 17
Method ... 17
Participants ... 18
Measurements ... 18
Procedure ... 20
Results and discussion ... 21
General Discussion ... 24
Abstract
From an early age on Asian children are generally found to have better arithmetic abilities than children from other countries. This difference has been attributed to the relative regularity of the Asian counting systems and the larger cognitive stimulation by Asian parents. In the first study 21 Taiwanese, 21 U.S and 24 Dutch 4-year old children were tested on early mathematical abilities. As expected, the Taiwanese children performed better than the U.S. and Dutch children. However, this difference was found on both early mathematical skills that are dependent on the counting system (counting) as on skills that are not
(numerosity). In the second study, parental stimulation was examined in 13 children from Chinese-Dutch and 13 children from Dutch families. Measures for cognitive stimulation by parents and of children’s early arithmetic abilities were administered. Results showed that Chinese-Dutch children had better early mathematical skills than their Dutch peers, but they did not receive more parental cognitive stimulation. The results of the study replicate earlier findings of the better arithmetic abilities of Asian children, but do not provide evidence for the hypotheses that their better performance is due to a less complex counting system and/or more parental stimulation.
Introduction
In the last decades, several studies have been conducted in which the mathematical achievement of children from different countries was compared (e.g., Gonzales et al., 2004; PISA, 2007). These studies show large international differences in arithmetic performance across countries. Asian students tend to perform best, and better than children from other countries like the United States and the Netherlands. These cross-cultural comparisons have mostly involved children from the higher levels of elementary school. However, although limited, there are studies that suggest that Asian children’s advantages may begin much earlier. For example, Miller and Parades (1996) state that there are already differences at preschool and Stevenson, Lee, and Stigler (1986) argue that these differences consist as early as in kindergarten. Because studies have shown that mathematics ability upon entering kindergarten is a strong predictor of later academic success, even stronger than reading abilities, attentional capacity, or socio-emotional functioning (Duncan et al., 2007), it is important to investigate what contributes to the discrepancy between the early mathematical abilities of students in different countries.
Some researchers (e.g., Miller, Smith, Jianjun, & Houcan, 1995; Miura, Kim, Chang, & Okamoto, 1988) suggest that the relative regularity of Asian counting systems is a major contributory factor in the superior performance of Asian children in most aspects of
arithmetic. A counting system is a system in which numbers and arithmetical relationships are expressed in a language (Dowker, Bala, & Lloyd, 2008). Learning number names may be easier in counting systems where new numbers may be inferred rather than having to be learned by rote. So, young children who have to acquire a regular counting system (e.g., Chinese speaking children) will have learned to count to higher numbers at an earlier stage compared to children who have to deal with a more irregular counting system, such as English, in which the structure of the number-naming system limits the induction of the
underlying rules for number formation (Miller & Stigler, 1987). This might lead to a head start in manipulating numbers for children in cultures with a regular counting system
(Dowker, et al., 2008). The influence of counting systems seems to be a possible explanation for different mathematic achievement levels of Asian and American students, but it leads to several implications. The first implication is that as the counting system does not influence all early mathematical skills (Miller et al., 1995), the better performance of Asian children must be limited to skills that depend on language, for example counting, and as a second
implication these differences must show up when the different counting systems begin to differ in complexity (which happens after the number ten).
Miller and Stigler (1987) investigated the role of counting systems on abstract and object counting in American and Chinese children of four, five, and six years old. They found a relation between the characteristics of a language’s counting system (regular or irregular) and children’s acquisition of that system. The Chinese children were able to count (abstract and objects) higher than their American counterparts. As expected, the largest differences were especially found in numbers after ten (‘teens’), where the counting system of English becomes more complex than the Chinese counting system. Miller et al. (1995) also found differences in abstract counting between American and Chinese three, four, and five years olds. Their results showed that language differences were found for counting skills above ten: 94 % percent of the American and 92% of the Chinese children could count to ten while in contrast, only 48% of the American and 74% of the Chinese kids were able to count to 20. With respect to object counting, Miller et al. (1995) did not found any counting system differences in children’s ability to count arrays of 10 or fewer objects and for that reason they concluded that U.S. children do not show any disadvantage in the attentional and conceptual aspects of counting involved in successfully counting of small sets of objects. However, in
comparison with their Chinese peers, they had a substantially greater difficulty in learning the system of number names that their native language employs.
A related implication of complexity of a counting system as contributor to early mathematical performances is that, apart from differences in counting, there must be similar achievements of children from countries with different counting systems on early
mathematical skills in which the complexity of the counting system does not play a role, for example the understanding of numerosity. Numerosity is the basis for the development of arithmetic abilities and it consists of two aspects (Butterworth, 2005). One aspect of numerosity is the ability to judge whether two sets consist of the same number of objects. Some researchers (e.g., Xu & Spelke, 2000) found that for numerosities beyond four, even infants can discriminate between two sets of objects. Adults of an indigenous Amazonian group whose language does not include terms for numbers greater than five, were also able to discriminate between large sets of objects (Pica, Lemer, Izard, & Dehaene, 2004). So, it seems that language is not always necessary for judgments about less and more. Another aspect of numerosity is the rapid and accurate number recognition of small sets of
objects(from 1 to 4) without counting, named subitizing (Hannula, Rasanen, & Lehtinen, 2007). Some researchers believe that this skill builds on a nonverbal pre-attentional ability to individuate small sets of items (e.g., Starkey & Cooper, 1995; Starkey, Spelke, & Gelman, 1990; Trick, Enns, & Brodeur, 1996).
To our knowledge, most cross-cultural studies about the influence of the complexity of a counting system on early mathematical skills have focused on Asian and U.S. children. These studies show that Chinese 4- year-olds outperform their American counterparts. This is in line with the counting system hypothesis: the Chinese counting system is less complex compared to the English counting system. In East Asian languages, such as Chinese, the relationship between units, 10s and higher powers of 10 are very explicit. In English, more
irregular number words are used (e.g., twelve in stead of ten-two). To further examine the influence of complexity of a counting system on early mathematical skills, Dutch children will be included in this study and their mathematical performances will be compared with that of their Taiwanese and U.S. counterparts. The inclusion of Dutch children is interesting because the Dutch counting system is even more complex then the English one. English and Dutch do not differ in complexity until twenty but after twenty, the English system is quite clear: 21 is twenty-one, 25 is twenty-five and so on. As opposed to English, in Dutch 21 is one-and-twenty, 25 is five-and-twenty. That makes the relationship between units, 10s and higher powers of 10 even less clear and therefore Dutch children seems to be at a
disadvantage. So, if the complexity of the counting system explains the difference in
mathematic ability of 4-year-olds from Asian countries and the United States, Dutch children must perform even worse than U.S. children and especially on counting tasks above twenty.
As said, at higher grade levels, Dutch children are outperformed by Asian children. However, despite a more complex counting system Dutch children perform better than U.S. children at a higher age. For example, the Trends in International Mathematics and Science Study (TIMSS) showed that eight-graders from the U.S. were fifteenth in the ranking of average mathematic scores, while Asian eight-graders were number one (Singapore) till five (Japan) and Dutch eight-graders were number seven (Gonzales et al., 2004). The Programme for International Student Assessment (PISA) showed similar differences: the U.S. students were on place thirty-six, while most Asian countries and the Netherlands were in the top five (Chinese students were the best performers, Dutch students were number five) (PISA, 2007). Thus it seems that, despite their experiences with a complex counting system at a younger age, Dutch students are not hindered in their mathematical development. Apparently, apart from the complexity of the counting system, factors in the environment of the children are of
relevance. Another factor that might contribute to the mathematical performances of children is parental mathematical stimulation.
Several studies have been conducted to investigate the influence of the mathematical stimulation by parents. Huntsinger, Jose, Liaw, and Ching (1997) investigated the
mathematical performances of Taiwanese-Chinese, Chinese-American, and Euro-American children and the parental practices and parental attitudes towards mathematics of these children. They found that Chinese-American and Taiwan-Chinese children outperformed Euro-American on mathematical performances. In addition, their results showed that Chinese-American parents gave more formal, direct mathematics instruction, structured their child’s time to a greater degree and reported more encouragement for mathematics-related activities than did Euro-American parents (Huntsinger et al., 1997). This is in line with Siegler and Mu (2008) who state that Chinese parents present arithmetic problems to their children more often than U.S. parents do and they conclude that this practice in adding and counting is one source of differences between Chinese and U.S. children’s numerical knowledge.
But not only explicit practicing or stimulation seems to promote the mathematical development of young children. Siegler and Ramani (2008) studied the effect of playing numerical board games on the numerical knowledge of children from low-income
backgrounds and found that playing these games (in contrast to playing games that substituted colors for numbers) contributes to reducing the gap in numerical knowledge that separates less and more affluent children. In addition, Hong (1999) states that stories and songs are very good tools to stimulate mathematical development, because when mathematical concepts are mentioned, children will informally learn about such concepts. In sum, children of parents who provide information about mathematical problems to their children, play numerical board games with their children and those who offer a lot of stories and songs have an advantage compared to children of parents who do not provide such stimulation.
To examine the influence of complexity of a counting system on early arithmetic development, two studies were conducted. The first study was part of an international project in which it is examined whether the use of a more or less complex counting system will lead to different levels of early mathematical achievement among 4-years-olds from Taiwan, the U.S., and The Netherlands. If counting systems play a role in early mathematical
performances, we expect that Dutch children perform worst on early mathematical
achievement tests followed by the U.S. children, because the Dutch counting system is more complex than the English system. In addition, the Taiwanese children are expected to perform best because their counting system is the least complex. It is also expected that complexity of a counting system will only influence those aspects of early mathematical abilities which depend on counting. Furthermore, it is predicted that their will be no differences between Taiwanese, U.S., and Dutch children in counting skills under ten but differences are expected on counting skills above ten with the Dutch performing worst and the Taiwanese children best.
In the second study, the influence of parent’s mathematical stimulation on early mathematical skills was investigated among two groups of 4-year-olds: children from Chinese-Dutch and Dutch families. It will be studied whether or not Chinese-Dutch children do perform better than Dutch children while both groups use the same Dutch counting system and, if so, whether or not the difference in performance can be explained by differences in cognitive stimulation by the parents. It is expected that children with more early mathematical skills will have received more parental cognitive stimulation.
Study 1 Method Participants
The sample consisted of 21 Taiwanese children, 21 U.S. children, and 24 Dutch children from middle to upper middle class families. The Taiwanese children lived in Taipei, the U.S. children in the San Francisco Bay Area and the Dutch children in a village near Amsterdam. The age of the children ranged from 4 year and 0 months to 4 year and 11 months (M = 4 years and 5 months, SD = 3 months). Fifty percent of the participants were boys.
Preschool/kindergartens around Taipei (Taiwan), San Francisco Bay Area (U.S.), and Amsterdam (The Netherlands) were asked to participate in this international study. In the U.S. and the Netherlands, parental consent forms were mailed to participating schools to be
distributed to their 4-year-olds to take home to their parents. In the U.S. parents signed and returned the consent forms to the school if they chose to give their consent. In the
Netherlands, the parents had to send the letter back if they did not want their child to
participate (passive consent form). In Taiwan, the preschool serves as the legal guardian with regards to children’s in-school activities. Therefore, Taiwanese children were selected and included in the study after permission of the school principal/director’s consent.
Measurements of mathematical skills
Data about the early mathematical skills of 4-year-olds were collected. Next to general early mathematical skills, information was obtained about counting skills (in general, under ten, and above ten) and understanding of numerosity. These skills were measured with the Test of Early Mathematics Ability-Third Edition (TEMA-3) which was developed by
Ginsburg and Baroody (2003). The TEMA-3 assesses numbering skills, number literacy, and understanding of other informal math concepts and has been shown to be free of gender and ethnic biases and shows a strong internal reliability ( α = .92 ~ .96: Bliss, 2006; Hofman & Grialou, 2005).
Originally, the test was developed in English. For this study, the TEMA-3 was translated from English to Dutch and from English to Mandarin. Two experts in early
mathematics, one Dutch and one Mandarin native speaker, translated the English version into Dutch and Mandarin respectively. The translated versions were send to native speakers of English who checked and improved the translations of the test. In every country a native speaker administered the test.
The TEMA-3 consists of 72 items. The items are questions or tasks which must be answered or done by the child. An example of an item is: ‘Can you show me three fingers?’. Whenever there was a new set of questions, the child was given several practice trials. Each item was scored correct or incorrect. After five incorrect answers in a row, administration of the test was stopped. After conducting the items, for each child a raw score was calculated by summing the scores over the items. This raw score was used to measure general early
mathematical skills.
General counting skills. Counting was considered as the ability to say a correct number word sequence (abstract counting) and/or to use this sequence to determine the precise amount of a number of objects (object counting). To measure general counting skills, 22 items were selected. An example of such an item is: ‘Can you count these stars?’ or ‘I would like you to count out loud for me. I will tell you when to stop’ The total score of general counting skills was the proportion of correct answers.
Counting skills under ten. This scale consisted of 11 items. The score of counting skills under ten was the proportion of correct answers.
Counting skills above ten. To obtain data about counting skills above ten, 11 items were selected. The score of counting skills above ten was the proportion of correct answers.
Numerosity skills. Numerosity was considered to be the ability to judge whether two sets consist of the same number of objects, the ability to judge which set has more objects,
and the ability the judge which number is closer to another number. Items of the TEMA-3 (Ginsburg & Baroody, 2003) which measure these abilities were selected. This led to 7 items. An example of an item is ‘Which side has more dots?’ The total score of numerosity skills was the proportion of correct answers on these 7 items.
Procedure
Each child who was willing to participate was escorted from his/her classroom to another room (most of the time a room near the classroom designated by the
preschool/kindergarten) and each child was tested individually. During the test session, the experimenter and the child sat across from each other with the testing material placed evenly between them. The testing materials consisted of a picture book, a Student Worksheet, tokens, blocks, and note cards (used as mats during administration of items using tokens and blocks). The standard procedures as defined in the TEMA-3 manual and examiner’s kit (Ginsburg & Baroody, 2003) were utilized. After the test session, the child received a present (e.g.,
stickers, stamps, puzzle game), even when they did not finish the test, and was escorted back to his/her classroom.
Results
The mean scores on general mathematical skills, counting skills, and numerosity skills of the 4-year-olds from Taiwan, the United States, and the Netherlands are shown in Table 1. General mathematical skills ranged from 70 to 137 (M = 103.70 SD = 17.04, the percentage correct answers on counting skills ranged from 9.09% to 86.36% (M = 42.77%, SD = 18.36%) and for numerosity skills from 0 % to 85.71 % (M = 43.29 %, SD = 23.77%). In addition, the percentage correct answers on counting skills under ten ranged from 18.18 % to 100 % (M = 73.69 %, SD = 22.84% ) and on counting skills above ten from 0 % to 81.82 % (M = 11.85 %, SD = 18.32 %).
The scores of general mathematical skills were converted to norm scores (Ginsburg & Baroody, 2003). The norm score (M = 100, SD =15) was calculated by means of the child’s age (years, months) and his or her raw score. It must be noted that these scores are only available in the U.S. manual and for that reason the Dutch and Taiwanese scores must be seen in light of the U.S. norm scores.
Table 1
Mean scores and standard deviations of early mathematical skills.
General mathematical Counting Counting Counting Numerosity
Country skills under ten above ten
Taiwan 118.24 (11.74) 59.52 (14.51) 93.07 (11.82) 25.97 (22.69) 61.22 (19.77) The United States 95.90 (12.87) 33.77 (11.00) 64.94 (19.56) 2.59 (5.10) 33.33 (18.26) The Netherlands 97.79 (16.49) 35.98 (16.90) 64.39 (22.73) 7.58 (14.10) 36.31 (23.07)
On general mathematical skills, the Taiwanese children performed best and the U.S. children performed worst. An univariate analysis of variance (ANOVA) was done to test if the differences in general early mathematical skills were significant. Results showed that the mathematical skills did significantly differ between the three countries, F (2, 63) = 16.75, p < .01. Deviation contrasts showed that Taiwanese children performed significantly better than American, F (1, 63) = 26.80, p < .01, and Dutch children, F (1, 63) = 23.92, p < .01. The difference between American and Dutch children was not significant, F (1, 63) = 0.20, ns.
To investigate to what extent these differences were specific to counting skills or whether they also extended to numerosity, a repeated measures (RM) ANOVA with
complexity of a counting system as between subjects variables and type of early mathematic ability (counting or numerosity) as a within subjects variable was performed. Results showed that there was a significant main effect of country, F (2, 63) = 17.01, p < .01, and that there was no main effect of type of early mathematical skill, F (1, 63) = 0.11, ns. More importantly, there was no interaction effect between type of early mathematical skill and country, F (2, 63)
= 0.15, ns. To pinpoint the sources of variability between the countries, contrasts on marginal means were performed. These contrast analyses showed that Taiwanese children performed significantly better on both skills, t (63) = 6.44, p < .01 for counting skills and t (63) = 4.85, p <.01 for numerosity skills. The difference in marginal means between American and Dutch children was not significant, t (63) = 0.51, ns for counting skills and t (63) = 0.48, ns for numerosity skills.
The final hypothesis was that there would be no differences between Taiwanese, U.S., and Dutch children in counting skills under ten whereas there would be a difference between countries in counting skills above ten due to differences in complexity of the counting system. Differences between U.S. and Dutch children were expected above twenty. As shown in Figure 1, there was a floor effect on the counting skill above ten. For that reason the
distinction between decided to make no distinction between counting skills between 10 and 20, and counting skills above twenty was dropped. It was also decided to perform two analyses: an ANOVA to test if there are significant differences between the countries on counting skills under ten and a Kruskal-Wallis test for counting skills above ten.
Figure 1. Means of counting skills under and above ten. 0 20 40 60 80 100
Under ten Above ten Counting skills P er cen tag e co rr ect an sw er s Taiwan U.S. The Netherlands
Results showed that there were significant differences in counting skills under ten among Taiwanese, U.S., and Dutch 4-year-olds, F (2, 63) = 16.32, p < .01. Deviation contrasts were performed to study the sources of variability between the groups. Taiwanese students were found to perform significantly better on counting skills under ten, t (63) = 5.71, p < .01. The U.S. and Dutch children did not significantly differ, t (63) = -.10, ns. On counting skills above ten, the countries did also differ significantly, χ2
(63) = 22.53, p < .01. To find the source of that variability, three Wilcoxon-Mann-Whitney tests were performed (Taiwan – The Netherlands, Taiwan – U.S., and The Netherlands – U.S.). Results showed that the Taiwanese children performed significantly better than Dutch and U.S. children, respectively W =
407.00, p < .01 and W = 286.50, p < .01. The difference between Dutch and U.S. children was not significant, W = 456.00, ns.
Discussion
The findings showed that there were differences between Taiwanese, U.S., and Dutch children on general mathematical skills: Taiwanese children performed better than their U.S. and Dutch counterparts. This was in line with our expectations. In contrast with our
expectations was that there were no differences between the U.S. and Dutch children. When looking at the separate mathematical skills, the results were also not as expected. Taiwanese children were found to perform best on counting and numerosity skills and also on counting skills under and above ten. In addition, no differences were found on any of those skills between U.S. and Dutch children.
It could be argued that the findings on the separate mathematical skills (counting skills, numerosity skills and counting under and above ten) are influenced by the fact that administration of the test was stopped after five incorrect items. One could state that children who were stopped early in the test were not able to answer all the items of a scale and
therefore their results could be biased, especially when the items of separate scales were not equally divided over the test. When looking at counting and numerosity skills, the items were equally split up over the test. This had as consequence that, although children were bad in counting skills, they could show their numerosity skills (and the other way round). In contrast, the items for counting skills under and above ten were not equally divided. The counting skills under ten were in the beginning of the test and the counting skills above ten in the end. Consequently, children could be in disadvantage when they are stopped early: they do not get a chance to show their skills on counting above ten. But Ginsburg and Baroody (2003) developed the test in such a way that the easiest items were in de beginning and the harder items in the end. Children, who had to be stopped because of five incorrect items, would most probably not be able to answer the following items correctly. For that reason it seems very unlikely that children, who had to stop the test early, would be able to correctly answer the items about counting skills above ten. In conclusion, the scores on counting and numerosity skills and on counting skills under and above ten are considered to be reliable. Nevertheless, it would be better to use separate tasks to measure the different aspects of counting skills.
In sum the results of Study 1 indicate that, in contrast to what is proposed by other researchers (e.g., Miller et al., 1995), the complexity of a counting system does not explain the better early mathematical skills of Taiwanese children 4-year-olds compared to U.S and Dutch children of the same age. Next to the complexity of a counting system, it is suggested that parental cognitive stimulation might contribute to the better mathematical performance of the young Taiwanese children. In Study 2, the influence of cognitive stimulation by parents was investigated.
Study 2 Method
Participants
The study sample consisted of 26 children of whom 13 were Dutch and 13 Chinese-Dutch. The age range for the Chinese-Dutch children was 4 years and 1 months to 5 year and 4 months (M = 4 year and 8 months, SD = 5 months) and that of the Dutch children was 4 year and 0 months to 4 year and 11 months (M = 4 year and 6 months, SD = 3 months). Sixty-five percent of the children were girls. The children were from three different schools: an elementary school in a place near Amsterdam, a Chinese school in Amsterdam and an elementary school in Rotterdam. For this study, some Dutch children were selected from the sample of Study 1 and others were especially tested for this study. The precise selection procedure of the Dutch children will be discussed in the results section.
Children were considered to be Chinese-Dutch when their grand parents and / or their parents were born in China. Information about the origin of the (grand) parents was obtained by asking the parents or teachers.
Invitation letters and parental consent forms were sent to children’s parents. If the parental consent forms were not sent back, children were allowed to participate (passive consent form). In the letter the parents were also asked to fill in an attached questionnaire and they were asked to send the questionnaire back to the teacher of the child. If the
questionnaires were not send back after the testing period at a school, the parents were called to do the questionnaire by phone.
Measurements of cognitive abilities
Early mathematical skills. To asses the early mathematical skills of the children, the TEMA-3 of Ginsburg and Baroody (2003) was used. See Study 1 for more information about this instrument.
Seriation. Seriation was considered as ordering stimuli along a quantitative dimension such as length. The seriation task (Van Kuyk, 1990) consisted of 14 items. Each item
contained one or several pictures. The child had to answer questions about those pictures. For example, the experimenter showed a picture with three dolls of three different sizes (small, bigger, biggest) in a big box and three small boxes with other figures. The child was asked to point to the box with objects which were placed in the same order. Each item was scored correct or incorrect. To get a score for seriation, the scores were summed over items.
Vocabulary. The vocabulary test of the Revision Amsterdam Child Intelligence Test (RAKIT) developed by Bleichrodt, Drenth, Zaal, and Resing (1984 ) was used. The test has 40 items which are divided in blocks of four items. Each item consists of four pictures. The experimenter had to say a word and the child had to point to the picture (out of four) that fits best with the word. The items were scored correct or incorrect. When there were 4 incorrect answers in a row, administration of the test was stopped after finishing the block of words in which the last incorrect answer was given.
Measurements of cognitive stimulation by the parents
To obtain information about the cognitive stimulation by the parents, a questionnaire was developed. Questions were selected from a questionnaire by Mayo and Leseman (2006) and a questionnaire by Bontekoe (2009). The questionnaire consisted of three scales: general mathematical activities, explicit mathematical instruction, and literacy. To test the reliability of the questionnaire, it was distributed to 130 parents. The response rate was 41.54%: 54 parents returned the questionnaire. These questionnaires were used to determine the reliability of the total questionnaire and separate scales. Intercorrelations among the subscales varied from .40 to .62. Therefore it was decided to use the three separate scales in further analyses.
General mathematical activities. General mathematical activities were considered as activities that are related to mathematical concepts, but not directly meant to learn the child something about mathematics. The scale consisted of 7 items. An example of a item is: ‘When diner is made, does your child weigh ingredients?’ The parents must indicate how often an activity is done: never, monthly (once or twice a month), weekly (once to four times a week), daily (more than four times a week). For each item a score of 1 (never), 2 (monthly), 3
(weekly), and 4 (daily) could be obtained. A total score of general mathematical activities was calculated by summing the scores over the 7 items. For this scale, Cronbach’s alpha was .68.
Explicit mathematical instruction. To obtain a score for explicit instruction by the parents about mathematical concepts, the scores on 12 items were summed. An example of an item from this scale is: ‘Do you tell your child what numbers are?’ Again the parents must indicate how often an activity is done: never, monthly, weekly, or daily. Cronbach’s alpha was .91.
Literacy. This scale consisted of 8 items. The parents were asked to indicate how often (never, monthly, weekly, or daily) several activities with regard to literacy were done (e.g., Does your child watch television programs in which letters are named?). Cronbach’s alpha was .85.
Procedure
Because of the number of tasks and the attention span of a 4-year-old, the test session was split up in two sessions of 20 minutes. In the first session, the TEMA-3 and RAKIT vocabulary were conducted. In the second session the seriation test was administered. The sessions were divided over two days. Children were only tested when they were willing to participate. For each test session, a child was escorted from his/her classroom to another room (most of the time a room near by the classroom designated by the preschool/kindergarten) and
each child was tested individually. During the test session, the experimenter and the child sat across from each other with the testing material placed evenly between them. After the test session, the child received a present (e.g., stickers, stamps, puzzle game) and was escorted back to his/her classroom. All the tests were conducted in Dutch. Despite the TEMA-3, which is translated from English (for more information about the translation, see Study 1), the tasks were already developed in Dutch and therefore no translation was needed.
Results and discussion
Thirteen Chinese-Dutch children were recruited and tested. Next to the 24 Dutch children from Study 1, 15 extra children were tested. These children were from the same school as nine of the Chinese-Dutch children. From the 24 Dutch children, 13 children were selected that could be matched to the Chinese-Dutch children on sex, age, and whether or not the parents had filled in the questionnaire. This led to a sample and a subsample: the full sample consisted of 26 children and the subsample of 16 children of whom the parents returned the questionnaire. Because information about the socio-economic background of the children was obtained by the means of the questionnaire, this was only available for the subsample of children.
Of the subsample, 8 children were Chinese-Dutch and 8 children were Dutch. Of the 8 Dutch children, 2 had an Antillian background. But because the Antilles are part of the
Netherlands, they were considered to be Dutch. The subgroups had a mean economic status of 1.63 (SD = 0.92) for the Chinese-Dutch children and 2.13 (SD = 0.64) for the Dutch children. The difference was not significant, F (1, 33) = 3.56, ns. The scores in both subgroups ranged from 1 (low social economic status) to 3 (high social economic status).
In the subsample, the age of the Chinese-Dutch children ranged from 4 year and 1 month to 4 year and 11 months (M = 4 year and 7 months, SD = 4 months) and the age of the
Dutch children ranged from 4 year and 0 months to 4 year and 11 months (M = 4 year and 6 months, SD = 4 months).
Full sample
The mean scores on the three cognitive abilities for the total sample are shown in Table 2.
Table 2
Mean scores and standard deviations on cognitive abilities for the total sample (N=26)
Dutch Chinese Dutch
Cognitive Abilities Max M SD M SD
Early mathematical skills 143 91.85 18.27 100.38 21.33
Seriation 14 6.31 2.81 8.46 3.26
Vocabulary 40 18.85 6.83 18.15 7.68
Table 2 shows that the Chinese-Dutch children performed best on early mathematical skills and seriation. To test if the differences in mathematical cognitive abilities were
significant, a between subjects multivariate analysis (MANCOVA) was performed with group (Dutch and Chinese-Dutch) as independent variable, general mathematical skills and seriation as dependent variables and vocabulary as covariate. Results showed a significant main effect of group, F (2, 22) = 3.61, p < .05, and of vocabulary, F (2, 22) = 6.86, p < .05. Univariate results showed that, after controlling for vocabulary, the Dutch and Chinese-Dutch groups did not significantly differ in general mathematical skills, F (1, 23) = 3.59, ns, but that the two groups did significantly differ in mean performance on the seriation task, F (1, 23) = 6.93, p < .05. The Chinese-Dutch children performed better than the Dutch children. The results also showed that vocabulary affected both mathematical cognitive abilities: F (1, 23) = 13.93, p < .01 for general mathematical skills and F (1, 23) = 5.18, p < .05 for seriation.
Subsample
In Table 3 the mean scores of Dutch and Chinese-Dutch children on cognitive abilities and cognitive stimulation by the parents are shown.
Table 3
Mean scores and standard deviations on cognitive abilities and cognitive for the stimulation by the parents subsample (N = 16).
Max
Dutch Chinese Dutch
Variables M SD M SD
Cognitive abilities
Early mathematical skills 143 94.24 13.91 112.25 17.35
Seriation 14 6.13 2.75 9.25 3.45
Vocabulary 40 20.38 7.98 22.38 6.55
Cognitive stimulation
General mathematical activities 28 16.63 2.13 16.49 2.39
Explicit mathematical instruction 48 34.00 7.37 28.67 7.66
Literacy 32 22.00 5.90 21.35 5.08
The scores on early mathematical skills and seriation follow the same pattern as in the full sample. The Chinese-Dutch children outperformed the Dutch children. The scores on vocabulary in both samples were reversed: in the full sample the Chinese-Dutch children had a higher score while in the subsample the Dutch children scored best.
To test if the differences in mathematical cognitive abilities in the subsample also were significant, another between subjects multivariate analysis (MANCOVA) was performed with group (Dutch and Chinese-Dutch) as independent variable, general mathematical skills and seriation as dependent variables and vocabulary as covariate. The results showed that the effect of group approached significance, F (2, 12) = 3.43, p = .066. The effect of vocabulary was significant, F (2, 12) = 7.70, p < .01. Univariate results showed that, after controlling for vocabulary, the Dutch and Chinese-Dutch groups did significantly differ in general
groups approached significance in their performances on the seriation task, F (1, 13) = 3.90, p < .070 with the Chinese-Dutch children performing better. In addition, the results showed that vocabulary affected general mathematical skills, F (1, 13) = 16.69, p < .01 and also seriation, F (1, 13) = 6.69, p < .05.
Table 3 also shows that the mean scores on the three scales of cognitive parental stimulation differed between groups (especially on explicit mathematical instruction) with the Dutch children receiving more cognitive stimulation by their parents than their Chinese-Dutch peers. A MANOVA with group (Dutch or Chinese-Dutch) as independent variable and
general mathematical activities, explicit mathematical instruction, and literacy as dependent variables was conducted to test if the differences between the groups were significant. Results showed that the effect of group was not significant, F (3, 13) = 1.31, ns.
As expected, the data showed that the sample of Chinese-Dutch children scored higher on seriation, after controlling for vocabulary than the Dutch children. There was also a trend towards better early mathematical skills of the Chinese-Dutch children but, probably because of the small sample size, this was not significant. In contrast with our expectations, no difference in parental stimulation between Chinese-Dutch and Dutch children was found. It seems therefore that the better mathematical performance of the Chinese-Dutch young children was not due to more cognitive stimulation by their parents.
General Discussion
In the first study we found that kindergarten children from Taiwan had more mathematical skills than children from the U.S. and the Netherlands. This finding is in line with previous studies in which Asian children also performed better than their U.S. peers (Huntsinger et al., 1997; Miller and colleagues, 1995; Miller & Parades 1996; Stevenson et al., 1986). But, in contrast to our expectations, no interaction effect between ethnicity and
type of early mathematical ability was found. That is, Taiwanese children also performed better than U.S. and Dutch children on numerosity skills that were assumed to be more innate. In addition, Taiwanese children performed also best on counting skills under ten that were considered to be independent of the complexity of a counting system. These findings clearly suggest that the better mathematical performance of Taiwanese children is not due to the relative regularity of their counting system.
The finding that Taiwanese students performed best on all aspects of early mathematical skills is not in line with findings of Miller et al. (1995) who found no
differences between Chinese and U.S. children in counting under ten. This different finding might be explained by the fact that Miller et al. (1995) included also children of five years old in their study while we only included children of four years old. Almost all the children (94% of the American children and 92% of the Chinese children) in the study of Miller et al. (1995) were able to count to ten and therefore it is possible that they did not found differences
between Chinese and U.S. children while we did found differences.
In the first study, we did not found, in contrast with our expectations, differences between U.S. and Dutch 4-year-olds on counting above ten. Both groups were not capable to count above ten. Because of this floor effect it can not be concluded that the U.S. and Dutch children do not differ on this skill. This shows an important point in investigating early
mathematical skills: investigators must always keep in mind that some skills are dependent on the age of the child. To obtain reliable information, research must be conducted with children who are in the right developmental phase. So, to obtain more information about the counting skills above ten and twenty, older children must also be included in further studies. Only then it would be possible to test the hypothesis that U.S. and Dutch children differ in counting skills above twenty because of the differences in complexity of their counting system.
As said before, in the first study it was shown that different complexities of counting systems do not explain the better mathematical performance of Asian children. Another suggested explanation for the discrepancy in early mathematical skills between children from Asian countries and other countries, like the U.S. and the Netherlands, is that Asian children receive more parental cognitive stimulation. Therefore in the second study, the influence of parental cognitive stimulation was investigated among two groups of children who used the same counting system but who had a different family background (Chinese-Dutch or Dutch). Results showed that the Chinese-Dutch children performed best on early mathematical skills but they did not receive more cognitive stimulation by their parents. Thus, the results of the second study do not provide support for the hypothesis that Asian children have more early mathematical skills compared to children from other countries because they get more parental stimulation.
Our finding is not in line with the findings Huntsinger et al. (1997), who did found differences in parental stimulation with more mathematical stimulation given by the Chinese-American parents compared to the Euro-Chinese-American parents. An explanation for the
discrepancy in results between our study and the study of Huntsinger et al. (1997) might be that they conducted their study with older children. The mean age in our sample was about 4 year and 7 months, whereas the mean age in the samples of Huntsinger was 5 year and 6 months, that is: the children were one year older. It is possible that parents do not pay a lot of time conducting mathematical activities with their children when their children are just 4 years old. When the children become older, it might be that parents begin to stimulate their children more in cognitive subjects and therefore it might be that the differences will show up when the children become older. To examine this suggestion, a study with older Chinese-Dutch and Chinese-Dutch children and their parents should be conducted.
The different results between our study and the study of Huntsinger et al. (1997) might also be due to different methods of measurements. In this study, only questionnaires were used to obtain information about parental cognitive stimulation. Huntsinger et al. (1997) used both questionnaires and interviews. It might be that interviews provide more precise and correct information about the activities parents do with their children. In an interview it might be that parents report mathematical activities that were not mentioned in the questionnaire. Huntsinger et al. (1997) interviewed the parents and asked more open questions like: ‘How do you facilitate your child’s development in mathematics?’ Such a question will give more precise information and therefore it might be easier to find differences between the two groups. In further studies it would be better to conduct also interviews to obtain more information.
In both studies we found systematic differences between children with an Asian background – in the current studies Taiwanese and Chinese-Dutch children – and children from the U.S. and the Netherlands. But, as in almost all cross-cultural research, it can be questioned whether the founded differences can indeed be explained by the different cultural backgrounds. It is also possible that the differences are caused by uncontrolled factors or that there are problems with controlled factors. In this study, some of these problems might have influenced the results.
The first problem is that the groups in both studies were (roughly) matched on social economic status. In the first study, the children came all from middle class to upper middle class families. However, whether middle class to upper middle class means the same in the three countries is not clear. In the second study, education level of the parents was used to measure the social economic status of the children. This might have leaded to an
underestimation of the social economic status of the Chinese-Dutch children. Because their parents are immigrants, they did not have the same educational chances as the parents of the
Dutch children and therefore it might be that the Chinese-Dutch parents have a lower
education than they could have had. Thus, it is possible that, for example, the Chinese-Dutch parents are more intelligent than the Dutch parents. This might have affected the performance level of the children: the Chinese-Dutch children could be more intelligent than their Dutch peers and therefore they had more early mathematical and seriation skills.
The second problem can be found in the study about parental cognitive stimulation (Study 2). Differences in early mathematical skills were found after controlling for vocabulary of the child. This was done to make sure that the differences in mathematical performance were not due to difficulties in understanding the questions because of the used language. However, it is possible that this manner of controlling was not totally correct. Most Chinese-Dutch children spoke Chinese at home (6 out of 8). This means that if their vocabulary was as high as that of the Dutch children, they had in reality a higher vocabulary (because they also have a Chinese vocabulary). So, there is a chance that the Chinese-Dutch children are underestimated compared to the Dutch children who do not have to deal with another
language: the vocabulary of the Dutch children reaches their ‘true’ vocabulary. This problem means that the Chinese-Dutch children are on a higher developmental level and that might be the reason that they performed better on the mathematical tests. In further studies, a non-verbal intelligence score must be used as a control variable to make a better comparison between the groups.
This is the first cross-cultural study in which the mathematical performances of Dutch 4-year-olds were compared with the mathematical performances of children from other countries. However, Dutch students of higher ages (e.g., children of the 8th grade) have been often included in cross-cultural comparison of mathematical performances. In these studies Dutch students tend to perform better than their American peers (e.g., Gonzales et al., 2004; PISA, 2007). This leads to the suggestion that the differences at a later age might be caused
by differences in the educational systems since results of our study show that U.S. and Dutch children do not differ in mathematical skills at the beginning of their school career. Different educational systems might also be an explanation for the better performance of Asian
children: they might receive education that stimulates their mathematical development. Unfortunately, it is not yet known whether the cultural variations in early education account for the superior math performance of Asian children because most research about cultural variations in curricula and teaching methods have focused on elementary schools.
In conclusion, Taiwanese children had better early mathematical skills, not only skills that are dependent of a counting system, but they performed also better on aspects that were considered to be independent of a counting system (numerosity) or where the counting system did not differ (counting skills under ten). In addition, the Chinese-Dutch children performed best on early mathematical skills and seriation while they did not receive more parental stimulation. So, we did not found evidence for the hypotheses that better performance of older Asian children is due to the two suggested advantages (less complex counting system and/or more parental stimulation) at an earlier age. It might be that we did not found evidence for the hypothesis because of our small sample sizes and the abovementioned problems. But, it is also possible that there are other factors that contribute to the differences in mathematical achievements across countries, such as differences in educational systems. More research with bigger sample sizes is needed to explore the influence of counting systems, parental
stimulation, and other possible contributory factors (e.g., educational systems) in the superior performance of Asian children.
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