Investigation And Research On The Mastery Of Irrational Number Knowledge In Junior Middle School Students

The irrational number,as the main content of the expansion of the number system,occupies an important position in the history of mathematics.The first formal exposure of students to irrational numbers is at the junior high school stage,which lays an important foundation for the subsequent study of mathematics.Therefore,based on the conceptual framework of Tirosh,this paper explores the mastery of irrational number knowledge by junior middle school students from three aspects: formal knowledge of irrational numbers,intuitive knowledge and operational knowledge.On the basis of literature research and teacher online interview,this paper develops a set of test questionnaires based on the important knowledge of irrational numbers.Through a purposeful sampling,a questionnaire survey and interview were conducted among 280 students in grade 8 and 9 of mathematics in three cities and counties and three different types of middle schools in Hebei Province,and the conclusions were drawn as follows:(1)In the aspect of formal knowledge,most students can accurately say the definition of irrational number,"irrational number is infinite uncyclic decimal number",but little is known about the other forms of definition that appear in history.The concept image of irrational number in students' mind is rich,and attribute class,numerical class and operation class occupy a large proportion,which is closely related to students' definition of irrational number,Kegen number and valuation.(2)In terms of intuitive knowledge,the classification of real numbers and the one-to-one correspondence between real numbers and points on the axis of numbers are still a major difficulty for junior middle school students to learn irrational numbers,and the students lack this idea of classification and combination of numbers and shapes.When students face the score,they tend to turn it into a decimal form and judge whether the number is rational or irrational through whether there is a link or not.Students lack the understanding that irrational numbers cannot be expressed as a ratio of two integers.This leads to cognitive conflict between formal knowledge and intuitive knowledge,which is also an important reason for students to appear cognitive impairment.(3)In the aspect of operation knowledge,junior high school students have a good grasp of the four operations of irrational numbers.However,in the process of operation,there is a lack of grasp of the whole knowledge of the system of numbers,from natural numbers to scores,negative numbers to rational numbers and irrational numbers.The algorithm is always applicable,and it is difficult for students to grasp the relationship between these numbers.(4)Using independent sample T test,the results show that there is a significant difference between students in experimental class and ordinary class in answering unreasonable number knowledge in urban middle school;There are also significant differences between urban students and county students in their answers to irrational knowledge.In terms of different knowledge dimensions,there are significant differences between urban students and county students in responding to irrational number form knowledge,and there is a significant difference between urban and urban areas,between urban and county students in answering irrational number intuition knowledge.There are significant differences between urban students and county students in their answers to irrational number operations.Finally,based on the above-mentioned investigation and research,some teaching suggestions are put forward for teachers,and the innovation of teaching design is carried out in order to better develop students' understanding of the concept of irrational numbers and provide reference for teachers' teaching.