| Number magnitude representation refers to a psychological representation associated with a particular quantity.The general conclusions about the development of magnitude representation and individual differences come from integers.In fact,the discussion of magnitude representation can not focus only on integers,but also on other types of figures(such as fractional number,decimal).the commonly used method on the study of numerical representation of integer is the number line estimation task,The number line is a line with no mark or division,which has two paradigms;one is the Number-Position Task,that is,the subject is required to mark the position of the specified number on the line segment on a line marked with a starting point;the other is the Position-Number Task,which requires the subject to write a line marked with a starting point on the line to indicate the magnitude of a vertical position.Number line estimation task has many advantages,so it is often used to study the numerical representation of integer.But whether it can be used to explore the representation of the number of other types has been questioned by the researchers until Siegler proposed Integrated Theory of Numerical Development,which argues that the core of magnitude development is to represent the different types of magnitude and magnitude development includes understanding that all real numbers can be represented in a certain order on the mental number line,which can be extended to the number of other types,including fractional number,that is,the number line estimation task can also be used to discuss the magnitude representation of fractional number.Fractional number is in a very central position in math learning,and the understanding of the magnitude of fractional number is the basis for studying the concept of other series of fractional number.At present,there are few researches on the representation of fractional numbers and the research content is mainly about the individual representation pattern and theaccuracy.The representation of an integer is more of an automatic process,where fractional number is not as common as integer and its complex characteristics make it difficult to understand.Its representation process is slow and requires greater control of the will,which determines that fractions need to use strategy and are more prone to errors than numerical representation of integers.It is necessary to pay attention to the use of the number of individual fractional number in the representation of the individual fractional number and to represent the types of errors,and to explore the intrinsic causes of the impact of representation of fractional number from the micro-mechanism level.In addition,the previous study of fractional representations revolves around proper fraction(0~1 Number line),and proper fraction can not represent the characteristics of all fractional number,so the type of fractional number should be expanded in order to understand the problem of the number representation of individual fractional number more comprehensively and to further explore its development and change in representation model,accuracy,strategy usage and representation of error types by expanding the range of number lines.Based on the review of previous magnitude representation research,this study has carried out two studies based on the number line estimation task with 156 students in grade 6~8: study 1: Grade 6~8 students' fractional number representation pattern,accuracy,strategy usage,and representation error type analysis in 0~1 number line.Study 2: Grade 6~8 students' fractional number representation pattern,accuracy,strategy usage,and analysis of representation of error types and their changes in the 0~5 number line.The result indicates:(1)In 0~1 number lines,the fractional number representation of grade 6~8students is linearly represented in NP task or PN task.As for the estimated accuracy: the accuracy of NP task is significantly lower than that of PN.There are significant differences between the grades and the grades,and the interaction between the grade and the task type is significant.In the use of the strategy,the students of grade 6~8 used a certain estimation strategy in the range of 0~1.And with the rising of grade,the strategy is used more flexible with higher frequency.(2)In 0~5 number line,there are differences in the representation pattern of grade 6~8 students.In the NP task,the sixth grade students are logarithmic representation while the students of grade 7~8 are linearly represented.In the PN task,the linear representation of the grade 6 students and the logarithmic representation are matching well,and the students in the 7th and 8th grade are linearly represented.In terms of the estimated accuracy,the accuracy of the NP task is significantly lower than that of the PN task,and there is a significant difference among the grades.The interaction between the grade and the task type is significant;in the use of the strategy,representation strategy is more frequently used in 0~5 number line than 0~1 number line and has some adjustments.(3)The expansion of the number line not only affects the individual representation pattern,accuracy,the selection and use of the representation strategy,but also affects the type of error in fractions.In particular,the error type of proper fraction changes with the number line range.There are four levels of error in the three types of errors,and the types of errors at different levels differ in the range of different numbers and fractional number.The results of this study further reveal the intrinsic mechanism of the magnitude representation of the fractional number and the development characteristics and influencing factors of the fractional number of the students in all ages,which will help to further improve the theoretical research of the number magnitude representation and reveal the cause of the wrong representation to a certain extent which gives some inspiration on how to help the teaching and learning of fractions. |
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