儿童的分数和小数数量表征及其发展
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摘要
现有的数量发展理论大多基于整数研究,它们对分数或小数的数量能否表征在心理数字线上提出了不同的假设。为数不多的已有的实证研究在这一问题上也得到了相互对立的结果。并且这些研究只考察了分数的数量表征或小数的数量表征,而理解分数和小数数量表征的异同对认知科学和教育科学都非常重要。因此,本研究从发展的视角考察和比较了儿童的分数和小数数量表征以及其与分数小数互化能力的关系。本研究还通过一项干预研究考察了什么样的训练更能提高分数和小数的数量表征水平和互化能力。通过比较中国儿童和美国同龄儿童在数字线估计任务中的表现,本研究也初步考察了分数和小数数量表征的国际差异。
本研究取得了以下主要发现：第一,像整数一样,分数和小数的数量都能表征在心理数字线上。第二,分数和小数的数量表征和互化能力随年龄的增长而发展。第三,分数数量表征和小数数量表征存在较大差异：小数在心理数字线上的表征比分数更准确；小数数量表征比分数数量表征发展地更早；相对于估计小数,儿童甚至成人更容易低估分数的数量。第四,除发展外,强调分数和小数数量意义的短期训练也能提高儿童的分数和小数数量表征水平。第五,分数和小数的数量表征水平与分数小数互化能力之间存在相关甚至因果关系。第六,中国小学四年级儿童的小数数量表征水平高于同龄的美国儿童而分数数量表征水平低于同龄的美国儿童。
本研究的结果对完善数量发展理论、改进分数和小数的教学方式、以及重新审视分数和小数的教学顺序具有重要的价值。
Current numerical development theories, generally based on whole number studies, propose different hypotheses about magnitude representations of common fractions or decimal fractions. Previous studies addressing fraction representations yielded contradicting results. Moreover, these studies explored magnitude representations of these two types of fractions in isolation, while it is particularly important for both cognitive science and educational science to examine and compare magnitude representations of common fractions and decimal fractions. The present study examined and compared, from a developmental perspective, children's magnitude representations of and translation between common fractions and decimal fractions. The present study, using a pretest-training-posttest design, also examined which training is more effective in improving children's fraction representations and performance in fraction translation. The present study also examined international differences in magnitude representations of these two types of fractions.
The present study yielded the following results. First, like whole numbers, both common fractions and decimal fractions could be represented on a mental number line. Second, knowledge of common fractions and decimal fractions and their translation developed with age. Third, substantial differences were observed between magnitude representations of common fractions and those of decimal fractions (e.g. accurate magnitude representations developed more often and earlier for decimal fractions than for common fractions; children were more likely to underestimate the magnitudes of common fractions than those of decimal fractions). Fourth, in addition to development, appropriate trainings could also improve children's magnitude representations of common fractions and decimal fractions. Fifth, correlational, even causal, relationships were observed between children's fraction representations and their performance in fraction translation. Last, Chinese fourth graders gained better magnitude representations for decimal fractions while American peers gained better magnitude representations for common fractions.
These findings have profound implications for revising current numerical development theory, improving fraction instructions, and evaluating the traditional approach of teaching common fractions before teaching decimal fractions.
本研究取得了以下主要发现：第一,像整数一样,分数和小数的数量都能表征在心理数字线上。第二,分数和小数的数量表征和互化能力随年龄的增长而发展。第三,分数数量表征和小数数量表征存在较大差异：小数在心理数字线上的表征比分数更准确；小数数量表征比分数数量表征发展地更早；相对于估计小数,儿童甚至成人更容易低估分数的数量。第四,除发展外,强调分数和小数数量意义的短期训练也能提高儿童的分数和小数数量表征水平。第五,分数和小数的数量表征水平与分数小数互化能力之间存在相关甚至因果关系。第六,中国小学四年级儿童的小数数量表征水平高于同龄的美国儿童而分数数量表征水平低于同龄的美国儿童。
本研究的结果对完善数量发展理论、改进分数和小数的教学方式、以及重新审视分数和小数的教学顺序具有重要的价值。
Current numerical development theories, generally based on whole number studies, propose different hypotheses about magnitude representations of common fractions or decimal fractions. Previous studies addressing fraction representations yielded contradicting results. Moreover, these studies explored magnitude representations of these two types of fractions in isolation, while it is particularly important for both cognitive science and educational science to examine and compare magnitude representations of common fractions and decimal fractions. The present study examined and compared, from a developmental perspective, children's magnitude representations of and translation between common fractions and decimal fractions. The present study, using a pretest-training-posttest design, also examined which training is more effective in improving children's fraction representations and performance in fraction translation. The present study also examined international differences in magnitude representations of these two types of fractions.
The present study yielded the following results. First, like whole numbers, both common fractions and decimal fractions could be represented on a mental number line. Second, knowledge of common fractions and decimal fractions and their translation developed with age. Third, substantial differences were observed between magnitude representations of common fractions and those of decimal fractions (e.g. accurate magnitude representations developed more often and earlier for decimal fractions than for common fractions; children were more likely to underestimate the magnitudes of common fractions than those of decimal fractions). Fourth, in addition to development, appropriate trainings could also improve children's magnitude representations of common fractions and decimal fractions. Fifth, correlational, even causal, relationships were observed between children's fraction representations and their performance in fraction translation. Last, Chinese fourth graders gained better magnitude representations for decimal fractions while American peers gained better magnitude representations for common fractions.
These findings have profound implications for revising current numerical development theory, improving fraction instructions, and evaluating the traditional approach of teaching common fractions before teaching decimal fractions.
引文
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