A Study On Quadratic Radical Learning Progression Of Junior High School Students From The Perspective Of APOS Theory

The theory of learning progression first appeared in the United States.It is not only an important tool to describe the cognitive structure of students,but also an important trend in the development of international education.Quadratic radical chapter knowledge not only undertakes the teaching task of quadratic radical knowledge,but also has the task of sorting out the content and method of "number and formula".This paper applies the theory of learning progression to the content research of quadratic radical learning,takes the four stages of APOS theory as the basis of vertical level division,constructs a quadratic radical learning progression model,describes students’ learning performance on quadratic radical learning in detail,and reveals the cognitive development law of students’ learning quadratic radical learning.The research mainly addresses the following three issues: How to construct the quadratic radical learning progression framework? How to test and revise the hypothetical quadratic radical learning progression model? What is the learning status of students in different grades under the quadratic radical learning progression?To this end,three research methods,text analysis method,survey method and statistical analysis method,are adopted to construct the quadratic radical learning progression model.The research process can be divided into four stages as a whole:In the first stage,the relationship between the four stages of APOS theory and the content of the quadratic curriculum is combed through the deep analysis of mathematics curriculum standard and the content of the quadratic curriculum.Combined with the results of expert consultation,the cognitive development process of students on the quadratic curriculum is depicted,and thus the quadratic hypothetical learning progression model is preliminarily constructed.In the second stage,through the analysis of the exercise data related to quadratic root,based on the teacher questionnaire,teacher and student interview,the implementation of the pre-test to develop quadratic root measurement tools;In the third stage,296 students from three classes of Grade 8 and grade 9 were tested.The Rasch model in Winsteps software was used to test and analyze the result data,one-dimensional,project fit,scoring scale,etc.,and the learning progression model was tested and further modified.In the fourth stage,statistical analysis is adopted to further analyze students’ specific learning performance in quadratic radical learning progression model.The main conclusions are as follows.Based on the four stages of APOS theory,the learning progression of quadratic radical is divided into four stages of operation,process,object and diagram on a macro level.In addition,according to specific needs,the model is divided into three advanced dimensions of quadratic radical,namely concept,property and operation,with a total of 14 advanced levels,so as to achieve the purpose of detailed description of learning on a micro level.In the longitudinal analysis of the data,the horizontal difficulty values under the three advanced dimensions gradually increase respectively,and there is a big difference between the operation stage and the schema stage.The difference of understanding of quadratic radical varies with the difficulty of quadratic radical knowledge.On the whole,students of different grades have little difference in learning.Grade 9 students perform better than Grade 8 students in quadratic radical content integration and comprehensive application.There is no obvious difference in ability between different genders,but girls are better than boys in the ability of quadratic radical comprehensive application.Students of different grades have obvious differences in ability in different stages of progression.Only 32% of eighth grade students can reach the schema stage,indicating that eighth grade students’ knowledge of schema stage is weak and still needs to be strengthened.Based on the above discussion,three teaching suggestions are put forward for the deficiencies of students in learning quadratic radical formula.First,pay attention to the construction process of concepts,explain clearly the context of concepts and the difference and connection between similar concepts;Secondly,attach importance to the construction of algebraic knowledge network,guide students to actively incorporate quadratic radical into the existing algebraic knowledge framework,and promote the advancement of mathematical thinking;Thirdly,timely formative evaluation should be carried out to help students at different levels to achieve the improvement of their learning level.