小学生在非符号材料上的分数表征方式

Primary School Students' Representation of Fractions in Non-symbolic Materials

以小学三到六年级学生117人作为被试,采用非符号性分数材料探讨小学生的分数表征方式,以确定他们能否表征分数值。以心理数字线假设为理论基础,分析被试完成分数比较任务的距离效应和反应编码中的"空间—数字关联"效应(SNARC效应)。结果表明:小学三到六年级学生能够对分数进行整体表征,并且表征效率随年级上升而显著提高。至少从三年级开始,小学生已能够根据分数的值,按从小到大的顺序自左至右地将分数表征在心理数字线上。

How do people represent fractions？ Do they represent the real value of a fraction by accessing the magnitude of the whole numbers composing the fraction （componential） or directly represent the magnitude of whole fraction itself （holistic） ？ Some researchers argued that, in the symbolic fraction comparison tasks, individuals may not directly access the magnitude of the whole fraction, but they process the magnitude of the components and estimate their ratio. However, other studies on symbolic fraction tasks suggested that peo- ple~ fraction representation can be either holistic or componential. We propose, from an evolutionary perspective, that magnitude repre- sentation may be divided into two systems ： physical representation system and symbolic representation system. Compared to the symbolic system, the physical representation system appears earlier, by which people can simulate or represent magnitudes more directly, and a- void some possible interferences of symbolic representation system. Therefore, it is necessary to return to this early magnitude represen- tation system to investigate the representation mode of fractions. Considering the importance of fraction knowledge in the elementary mathematical education, the current study aimed to explore the fraction representation mode of students in primary schools by using non-symbolic fraction materials, which is based on the physical magnitude representation system~ This study sampled 117 third through sixth-grade students in an ordinary primary school, whose par- ents are on the medium level of socioeconomic status. We tested their performance in fraction comparison tasks and recorded the reaction time to examine the distance effect and the effect of the spatial numerical association of response codes （ SNARC）, which are based on the assumption of mental number line. Results of ANOVA and regression analysis showed that： （ 1 ） with grade difference, subjects＇reaction time in the fraction compari- son tasks significantly reduced （ F （ 3, 110） = 28. 50, p 〈 . 05, η2 = . 44 ）, their reaction times were 1060.60ms, 968. 05ms, 813.44ms, and 724. 47ms in each grade. （2） In each grade, there was a significant distance effect between the magnitude of the target fraction and the magnitude of the reference fraction, which were presented in the form of pictures （ the R square were . 91, . 89, . 94 and . 91 respectively, ps 〈 . 05 ） , and the distance effect size remained stable among four grades. Whereas the distance effect between the shadow area of the target fraction and the reference fraction in each grade was not significant （ the R square were. 09, . 00, . 01 and. 08 respectively, the p values were. 39, . 88, . 77 and . 80 respectively）~ （3） In each grade, there was a significant SNARC effect when the magnitude of the target fraction was compared to the magnitude of the reference fraction （the F values were 15.36, 22. 44, 43.17 and 162. 08 respectively, ps 〈 . 05, η2 were. 66, . 74, . 84 and. 95 respectively）. Results of the simple effect test argued that the response of left hand was faster to smaller fractions, and the response of right hand was faster to larger fractions. Therefore, we can conclude that： （1） when comparing the fractions in a non-symbolic form, third to sixth-graders＇fraction repre- sentation are holistic and the efficiency of their representation significantly improves with grade. （2） Third-graders have been able to re- present the magnitude of whole fraction on the mental number line, of which the smaller fractions are represented on the left and the lar- ger fractions on the right. This study enhances our understanding about people~ representation of fractions in non-symbolic fraction mate- rials （pictures）, and suggests educators consider the necessity to change the fraction teaching methods. For instance, teachers can teach fraction knowledge in a non-symbolic form in earlier ages, maybe together with the whole number knowledge, which would assist students to construct fraction knowledge on the basis of their relevant daily life experience and avoid the interferences of whole number symbols.