Development And Test Of The Learning Progression Model Of Linear Function For Students Based On APOS Theory

The term "learning progression" originated from the field of science education in the United States and can describe the vertical thinking development process of students' understanding of a certain learning topic in a long time span.In recent years,learning progression has gradually entered the field of education in mathematics and other disciplines,but the research on it is still in its infancy in China.This study takes the "linear function",the first concrete function model built by students,as the core concept,and adopts the top-down method to construct the level of learning progression,which belongs to the study of verification learning progression.By studying mathematics curriculum standards,corresponding mathematics textbooks,and relevant literature on students' cognitive development theory and function concept learning,this paper constructs a hypothetical learning progression model of linear function based on the APOS theory proposed by American mathematics educator Ed Dubinsky and others.Based on the characteristics of each level of progression,the test of the linear function learning progression was prepared,and the three-parameter item response model was used to verify the hypothetical linear function learning progression model,and then the linear function learning progression model was revised by combining the average difficulty value and the analysis of students' progression level from different perspectives.Finally,the learning progression model of the linear function is constructed to describe the learning stage,conceptual understanding level and specific academic performance of students on the learning topic of the linear function,and the actual status of each level in the learning progression model of the linear function is obtained.About a hypothetical learning progression model of the linear function building,the concept of APOS learning theory analyzes the characteristics of four stages,and then on the basis of mathematics curriculum standards of knowledge about linear function requirements,and to the whole 3 edition mathematics textbook contents and chapters content analysis,concluded five learning phase,including: preparation stage,operating stage,process stage,object,scheme,and puts forward 10 expected advanced level,and achieve advanced level of academic performance.Six experts were consulted on whether each level of progression corresponded to the characteristics of the corresponding stage.According to the feedback from experts,the hypothetical learning progression model of the linear function should be modified appropriately.As for the development of the advanced measurement tool for the linear function learning,the compilation of the questions refers to the typical examples and exercises of the corresponding textbooks as well as the senior high school examination questions in recent years.After the prediction,the single-dimensional test was conducted on the data through the principal component analysis in the factor analysis of SPSS19.0,and the test basically met the one-dimensional hypothesis.SPSS19.0 was used to test the reliability of all questions in the whole test paper,and the internal consistency reliability coefficient of the test question was 0.92,indicating that the test question was very ideal.In the process of expert consultation,the test questions are revised and improved according to the feedback of experts.Therefore,it has good expert validity.By using the MML(marginal maximum likelihood estimation),a parameter estimation method,the difficulty coefficient obtained by Matlab self-programming,the two questions in the measurement tool are modified appropriately,so as to develop a better quality measurement tool for the linear function learning.As for the verification of the linear function hypothesis learning progression model,the three-parameter item response model is used to input the formal test data into the Matlab selfprogramming of the parameter estimation method of mle.The results of the formal test show that the internal consistency reliability of the test tool coefficient value is 0.93,the reliability of the whole test question is very high.The learning progression model conforms to the assumption of unidimensionality,has a good fitting with data,and the correlation coefficients of degree of discrimination a and degree of guess c are in the normal range.According to the difficulty correlation coefficient b of each question,the average difficulty of the questions corresponding to each level of learning progression of a function increases in turn.This verifies that the firstorder functional hypothetical learning progression model based on textbooks conforms to the development trajectory of students' linear functional learning.According to the histogram of students' ability distribution,the ability value of most students is between(-2,1.5),mainly concentrated between(-1,1),and the number of students with ability between(2,3)is obviously small,basically no more than 20.Therefore,students have a weak grasp of the knowledge of linear function,and the advanced level still needs to be strengthened.With the gradual improvement of the advanced level,the scoring rate of students is decreasing.With the deepening of learning difficulty,students' cognitive differences in the knowledge related to the first function are also increasing.With the gradual improvement of advanced level,the scoring rate of grade 6,grade 1 and grade 2 students gradually decreased.However,there was a sudden upward trend in grade 3 students at advanced level 4(A1)and advanced level 8(O2).Girls performed better than boys at all levels,especially at level 4(A1),level 6(P1)and level 8(O2).It is difficult for students to understand the three representation methods of linear function,which is mainly affected by the graph method of linear function.For the corresponding problems of process stage and schematic stage,students using the mathematics textbook of Beijing normal university did the best.For the other three stages,there was no significant difference in the scoring rate of students using the mathematics textbook of Beijing normal university.Based on students overall,different grade and different gender students,using a different version of textbooks,students learn advanced levels,students at linear function on different representations of the advanced level,and average difficulty value analysis,understand a student learning from the advanced level of linear function,linear function is verified further study into the order model,and the preparation stage,operating stage,process stage,stage of object to revise some of the content.Finally the linear function learning progression model consists of five learning stages and nine progression levels.This study provides certain ideas for research on learning progression,textbook compilation and teaching design:This study provides some ideas for relevant researches on textbook compilation,teaching design and learning progression:1.The arrangement content and sequence of primary function knowledge should be improved in junior high school mathematics textbooks;2.Teachers should implement the core quality of mathematics into the function teaching;3.In view of the three ways of representation of primary function,teachers should focus on the teaching of image method;4.APOS theory can be used to construct advanced learning models of other mathematical concepts;5.Three-parameter model can be used to verify the learning advanced model of mathematical concepts;6.Subsequent studies need to further test the newly constructed linear function learning model.