6～7岁儿童数字估计能力发展的追踪研究
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摘要
估计能力是数学认知能力的重要组成部分,它反映了人们对数学概念、关系和策略的一般理解和儿童数学认知发展的信息,是高级数学认知能力发展的基础,也是儿童早期数学认知能力发展的重要成果和标志。已有研究表明,儿童的估计能力与其数学认知及其它数学活动(如计算、数字分类及大小比较)密切相关,但5-10岁儿童在估计距离、金钱总额、物体数量、算术以及数字定位时表现不尽如人意。为此,儿童估计能力的培养越来越受到人们的关注。数字估计是一种重要的估计形式,而数字线估计任务具有良好的生态效度,近年来成为估计能力研究中最为流行的研究范式。其他领域的研究推测6-7岁可能是中国儿童数字估计能力发展的重要时期,但已有研究主要集中于成人和小学儿童,而对幼儿园儿童还较少涉及,并且主要采用横向比较研究,这往往考察不到数字估计能力发展的完整过程和转折点。鉴于此,本研究设计了两个追踪子研究,拟就6～7岁儿童数字估计能力的估计精度、估计模式、估计的变异性、上述三者的关系等四个角度进行分析,在这个基础上考察儿童数字估计能力与其数学认知的关系。该研究有利于丰富认知心理学和教育心理学的相关理论,进一步充实国内外关于儿童早期数字估计能力发展的理论和研究。同时本研究还能为教育管理部门、托幼机构和家长等提供儿童数学能力发展的实证资料,为教育工作者制定干预方案提供理论指导和科学依据。
研究Ⅰ随机选取59名大班儿童,通过0～10、0～100和0～1000等3种不同数量范围以及实物、点和数字符号等3种不同情境的数字线估计任务考察大班儿童的数字估计能力及其与数学认知的关系。研究Ⅱ在研究Ⅰ的基础上,追踪考察了32名升入小学一年级的儿童0～100和0～1000数字估计能力的发展及其与数学认知的关系,同时考察大班时期的数字估计能力是否对一年级的数学认知有预测作用。本研究得到以下结论：
1.在各种数字线上,估计精度存在显著差异。0～1000数字线的估计绝对误差率显著高于0～100数字线,0～100数字线的估计绝对误差率又显著高于数字型0-10数字线；而在不同情境0-10数字线中,数字型数字线的估计绝对误差率显著低于点阵型和实物型数字线,但点阵型和实物型数字线的估计绝对误差率并不存在显著差异。
2.大班时期和一年级时期儿童在0～10、0～100和0～1000数字线上都呈线性估计,但仅有大班儿童在0～1000数字线上存在显著的性别差异。
3.在各种数字线上,估计的“等级变异性”并不存在,即估计的变异性不随着估计数量的增大而增大；估计变异性存在显著的年龄差异,一年级时期的估计变异性显著小于大班时期的估计变异性；估计变异性随着数字线的数量范围的增大而增大。
4.估计模式的线性度是估计精度的良好指示器,估计变异性对估计精度的贡献度远低于估计模式的线性度。
5.儿童在大班时期的0～100和0～1000数字估计能力与其数学认知呈显著的正相关；在一年级时期,0～100和0～1000数字估计能力与其数学认知呈显著的正相关；大班时期的0～100和0～1000数字估计能力对一年级数学认知具有预测作用。
As an important part of the mathematical cognition, estimation reflects people's general understanding of mathematical concepts, mathematical relationships and mathematical strategies and the information of children's mathematical cognition development. It's the basis of advanced mathematical cognition, and also the achievement and the symbol of early childhood's development of mathematical cognition.The previous studies has shown that estimation was related to mathematical cognition and other mathematical activities, such as arithmetic, sorting numbers and comparing numbers, but 5 to 10 years old children were not good at arithmetic, location of numbers or estimating distance, amount of money or number of objects. So developing the ability of children's estimation draws more and more attention. Numerical estimation is a significant type of estimations, and number-line is the most popular research paradigm because of the outstanding ecological validity. Researches in other domains has calculated that 6 to 7 years old is the key development period of Chinese children's estimation, but most focused on adult and pupil, and few on kindergarten children. Most of the previous studies couldn't investigate the whole process and key change points of estimation generally. For that reason, the present study designed 2 sub-studies, analyzing the estimation accuracy, pattern and variability of 6 to 7 years old Chinese children, and the relation between them, and the relation between the numerical estimation and mathematical cognition. This survey was conducive to highlight the cognitive psychology and educational psychology theories, and to further enrich the theory and research on numerical estimation development of early childhood. At the same time this study also provided evidence on the mathematical abilities development of children for education authorities, nurseries and parents, and theoretical guidance and scientific basis for educators to develop intervention programs.
59 kindergarten children participated in study I. By three different number ranges number-line(0 to 10,0 to 100 and 0 to 1000) and three different circumstances number-line (numbers, dots and candies), study I examined the numerical estimation and the relation with math achievement. Study II traced the 32 children who had been primary pupil on the numerical estimation, the relation with mathematical cognition and the predictive value of kindergarten children's numerical estimation on primary pupil's mathematical cognition, using 0 to 100 and 0 to 1000 number-line. The conclusions are as follow:
1. Estimation accuracy existed significant difference. The accuracy on 0 to 1000 number-line was higher than 0 to 100 number-line, which was higher than 0 to 10 number-line. For the 0 to 10 number-line, numerical type's accuracy was higher than the dots type and candies type, but there was no significant difference between dots type and candies type.
2. Children in kindergarten and primary school showed the linear estimation on 0 to 10,0 to 100 and 0 to 1000 number-line, and there was no significant gender difference but on 0 to 1000 number-line.
3. There was no "Scale Variability" on all kinds of number-line, in other words, the variability did not rise with the increase of estimation magnitude. The estimation variability existed significant age difference, and the primary's estimation variability was lower than the kindergarteners. At last, the estimation variability rose with the increase of the range of number-line.
4. The degree of linearity was proved to be a good predictor of estimation accuracy, and the degree of linearity could account for more variance of the estimation accuracy than the estimation variability.
5. Both for kindergarten children and primary pupil, the estimation on 0 to 100 and 0 to 1000 number-line was found to have a significantly positive correlation with mathematical cognition.And the estimation on 0 to 100 and 0 to 1000 number-line in kindergarten period also showed a significantly positive correlation with mathematical cognition in primary school.
研究Ⅰ随机选取59名大班儿童,通过0～10、0～100和0～1000等3种不同数量范围以及实物、点和数字符号等3种不同情境的数字线估计任务考察大班儿童的数字估计能力及其与数学认知的关系。研究Ⅱ在研究Ⅰ的基础上,追踪考察了32名升入小学一年级的儿童0～100和0～1000数字估计能力的发展及其与数学认知的关系,同时考察大班时期的数字估计能力是否对一年级的数学认知有预测作用。本研究得到以下结论：
1.在各种数字线上,估计精度存在显著差异。0～1000数字线的估计绝对误差率显著高于0～100数字线,0～100数字线的估计绝对误差率又显著高于数字型0-10数字线；而在不同情境0-10数字线中,数字型数字线的估计绝对误差率显著低于点阵型和实物型数字线,但点阵型和实物型数字线的估计绝对误差率并不存在显著差异。
2.大班时期和一年级时期儿童在0～10、0～100和0～1000数字线上都呈线性估计,但仅有大班儿童在0～1000数字线上存在显著的性别差异。
3.在各种数字线上,估计的“等级变异性”并不存在,即估计的变异性不随着估计数量的增大而增大；估计变异性存在显著的年龄差异,一年级时期的估计变异性显著小于大班时期的估计变异性；估计变异性随着数字线的数量范围的增大而增大。
4.估计模式的线性度是估计精度的良好指示器,估计变异性对估计精度的贡献度远低于估计模式的线性度。
5.儿童在大班时期的0～100和0～1000数字估计能力与其数学认知呈显著的正相关；在一年级时期,0～100和0～1000数字估计能力与其数学认知呈显著的正相关；大班时期的0～100和0～1000数字估计能力对一年级数学认知具有预测作用。
As an important part of the mathematical cognition, estimation reflects people's general understanding of mathematical concepts, mathematical relationships and mathematical strategies and the information of children's mathematical cognition development. It's the basis of advanced mathematical cognition, and also the achievement and the symbol of early childhood's development of mathematical cognition.The previous studies has shown that estimation was related to mathematical cognition and other mathematical activities, such as arithmetic, sorting numbers and comparing numbers, but 5 to 10 years old children were not good at arithmetic, location of numbers or estimating distance, amount of money or number of objects. So developing the ability of children's estimation draws more and more attention. Numerical estimation is a significant type of estimations, and number-line is the most popular research paradigm because of the outstanding ecological validity. Researches in other domains has calculated that 6 to 7 years old is the key development period of Chinese children's estimation, but most focused on adult and pupil, and few on kindergarten children. Most of the previous studies couldn't investigate the whole process and key change points of estimation generally. For that reason, the present study designed 2 sub-studies, analyzing the estimation accuracy, pattern and variability of 6 to 7 years old Chinese children, and the relation between them, and the relation between the numerical estimation and mathematical cognition. This survey was conducive to highlight the cognitive psychology and educational psychology theories, and to further enrich the theory and research on numerical estimation development of early childhood. At the same time this study also provided evidence on the mathematical abilities development of children for education authorities, nurseries and parents, and theoretical guidance and scientific basis for educators to develop intervention programs.
59 kindergarten children participated in study I. By three different number ranges number-line(0 to 10,0 to 100 and 0 to 1000) and three different circumstances number-line (numbers, dots and candies), study I examined the numerical estimation and the relation with math achievement. Study II traced the 32 children who had been primary pupil on the numerical estimation, the relation with mathematical cognition and the predictive value of kindergarten children's numerical estimation on primary pupil's mathematical cognition, using 0 to 100 and 0 to 1000 number-line. The conclusions are as follow:
1. Estimation accuracy existed significant difference. The accuracy on 0 to 1000 number-line was higher than 0 to 100 number-line, which was higher than 0 to 10 number-line. For the 0 to 10 number-line, numerical type's accuracy was higher than the dots type and candies type, but there was no significant difference between dots type and candies type.
2. Children in kindergarten and primary school showed the linear estimation on 0 to 10,0 to 100 and 0 to 1000 number-line, and there was no significant gender difference but on 0 to 1000 number-line.
3. There was no "Scale Variability" on all kinds of number-line, in other words, the variability did not rise with the increase of estimation magnitude. The estimation variability existed significant age difference, and the primary's estimation variability was lower than the kindergarteners. At last, the estimation variability rose with the increase of the range of number-line.
4. The degree of linearity was proved to be a good predictor of estimation accuracy, and the degree of linearity could account for more variance of the estimation accuracy than the estimation variability.
5. Both for kindergarten children and primary pupil, the estimation on 0 to 100 and 0 to 1000 number-line was found to have a significantly positive correlation with mathematical cognition.And the estimation on 0 to 100 and 0 to 1000 number-line in kindergarten period also showed a significantly positive correlation with mathematical cognition in primary school.
引文
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[2]Booth, J. L.,& Siegler, R. S. (2006).Developmental and individual differences in pure numerical estimation.Developmental Psychology,41(6),189-201.
[3]Booth, J.L.,& Siegler, R.S.(2008).Numerical magnitude representations influence arithmetic learning.Child Development,79(4),1016-1031.
[4]Case, R.,& Okamoto, Y.(1990).The role of central conceptual structures in the development of children's though. In C. A. Hauert (Ed.), Developmental psychology:Cognitive, perceptuo-motor, and neuropsychological perspectives, Elsevier Science.,193-230.
[5]Case R., Okamoto, Y. (1996).The role of central conceptual structures in the development of children's thought.Monographs of the Society for Research in Child Development,61,1-2.
[6]Dehaene S. (1997).The number sense:How the mind creates mathematics.New York:Oxford University Press,73-76.
[7]Dehaene,S.,& Izard V. etal (2008).Log or Linear? Distinct intuitions of the number scale in western and amazonian indigene cultures.Science,320,1217-1240.
[8]Ebersbach,M.,& Luwel,K. etal.(2008).The relationship between the shape of the mental number line and familiarity with numbers in 5 to 9 year old children:Evidence for a segmented linear model.Journal of Experimental Child Psychology,99,1-17.
[9]Geary D. C., Mary K. Hoard etal.(2008).Development of number line representations in children with mathematical learning disability[J].Developmental Neuropsychology,33(3):277-299.
[10]Hanson, S.A.,& Hogan T.P.(2000).Computational estimation skill of college students .Joumal for Research in Mathematies Education.31(4),483-499.
[11]Hogan, T.P.,& Brezinski, K.L.(2003).Quantitative estimation:One,two,or three abilities?. Mathmatic Thinking and Learning,5(4),259-280
[12]Huntley-Fenner G. (2001).Children's understanding of number is similar to adults'and rats': numerical estimation by 5-7-year-olds.Cognition,78, B27-B40.
[13]Gibbon, J.,& Church, R.M., (1981).Time left:Linear versus logarithmic subjective time. Journal of the Experimental Analysis of Behavior,7,87-107.
[14]Laski, E.V.,& Siegler, R. S.(2007).Is 27 a big number? Correlational and causal connections among numerical categorization, number line estimation, and numerical magnitude comparison.Child Development,78(6),1723-1743.
[15]Moeller K.,& Pixner,S.(2009).Children's early mental number line:Logarithmic or decomposed linear? Journal of Experimental Child Psychology,103,503-515.
[16]Montague, M.& Garderen,D.(2003). A cross-sectional study of mathematics achievemen, estimationskill, and academic self-perception in students of varying ability Journal of Learning Disabilities,36(5):437.
[17]National council of teachers of mathematics(1980). Agenda for action:Recommendations for school mathematics of the 1980s.Reston, VA:Author.
[18]National council of teachers of mathematics(1989). Currieulum and evaluation standards for school mathematics[S].Reston, VA:National Council of Teachers of Mathematics.
[19]Opfer,J.E.,& Thompson,C.A.(2008).The trouble with transfer:insights from microgenetic changes in the representation of numerical magnitude.Child Development,19(3),788-804.
[20]Opfer,J.E.,& Siegler,R.S. (2007). Representational change and children's numerical estimation.Cognitive Psychology,55,169-195.
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