Research On Teaching Concepts In Primary School Mathematics From APOS Perspective

At the elementary level,fractions are a central part of the "Number and Algebra" component of elementary mathematics,bridging the gap between whole numbers,decimals,and percents.The APOS theory reveals the psychological development process of students’ learning mathematical concepts,following the cognitive rule of moving from concrete to abstract,and focusing on students’ independent reflection and internalization at each stage.It not only helps students understand and grasp the essence of concepts and build up a comprehensive mental schema,but also provides teachers with effective teaching ideas for the lesson of "Knowing Fractions".At the same time,APOS theory can help to enrich the research of this theory in teaching elementary school mathematics concepts to a certain extent.Based on this,this study takes APOS theory as a perspective and adopts observation,interview and case study methods to investigate the teaching of the lesson of "Knowing Fractions".The paper is divided into the following contents:First,a preliminary analysis of APOS theory and the teaching of the concept of "knowing fractions".First,the four stages of APOS theory are interpreted.The second is an analysis of the lesson of "Knowing Fractions" from the perspective of learning and teaching materials.Thirdly,we analyzed the feasibility of applying APOS theory to the lesson of "Knowing Fractions",and obtained the following findings: the four stages of APOS theory combine concept assimilation and concept formation,which is consistent with the process of students’ concept learning;APOS theory attaches importance to students’ subjectivity and knowledge integrity,and follows the cognitive law of concrete to abstract.The APOS theory focuses on students’ learning process and independent thinking process,which is conducive to the cultivation of students’ core literacy.Based on this,the four stages of APOS theory are used as the main features to analyze the considerations for teaching the lesson of "Knowing Fractions".Second,the research design.The purpose of the observation method and interview method,the research subjects,and the framework of classroom analysis and interview outline under APOS theory are introduced.In order to understand the teaching situation of the lesson "Knowing Fractions",two classrooms of famous teachers and an equal number of daily classrooms of front-line teachers were observed,and three front-line teachers were interviewed with corresponding questions to understand the teachers’ teaching philosophy and attitude.Again,the case studies under APOS theory.First,the four observed classrooms were transformed into actual recordings and analyzed using APOS theory.Second,the results of the analysis were integrated to explore the problems that existed in the four stages of teaching the lesson "Knowing Fractions" : in the operation stage,there were problems of lacking the diversity of visual aids,students’ sense of practical experience was not strong,and the novelty of introduction needed to be improved;in the process stage,there were problems of narrow thinking development,weak induction awareness cultivation,and vague problem design.In the process stage,there are problems of narrow thinking development,weak inductive awareness development,and blurred problem design;in the object stage,there are problems of low inquiry of exercise content,biased guidance of idea elaboration,and lack of significant students’ main position;in the schema stage,there are problems of fragmented concept combing and integration,insufficient connection between multiple schemas,and lack of conceptual understanding and articulation.Finally,corresponding suggestions are made based on APOS theory.Based on the results obtained from the case study,suggestions were made for the corresponding teaching problems,namely: first,designing appropriate activities to enhance knowledge experience;second,using visual backgrounds appropriately to assist teaching with shapes;third,making good use of problem guidance and emphasizing effective reflection;fourth,focusing on variation teaching and adding counterexamples appropriately;fifth,emphasizing expression and generalization to highlight mathematical logic;and sixth,presenting thinking paths to establish diagrammatic connections.