Research On Recognition For Students To Infinity In Mathematics Learning

Mathematics is science of infinity. The infinity of mathematics is the radical motivation,but infinity conceals in the relevant mathematics concepts .The curriculum of high school always keep away from infinity.The destination of infinity don't be offered in the schema of mathematics in high school. Students specially have difficulty in perceiving the concept of limit which become the obstacle of learning calculus. The editor identified the recognition to infinity hierarchly including following two clues:One clue: the status of recogniton for students to every level including such context questions as:(1)How about the actually attainable infinity levels of grade 9, grade 12 and sophomore , the essential factor affecting the recognition for students.(2)How about the steady recognition model and the inevitable inaccurate mental model of students on the basis of the criterion of each level.The other one: understanding to the infinity of the relevant mathematics concepts(1) How about the understanding of students to the infinity of relevant mathematics concepts and what steady recognition model the students showing.(2) what inaccurate mental model the students tend to get during the course of recognizing to the infinity of the mathematics concepts.The editor defined the meaning of the discussed infinity in the paper.Infinity only occurs implicitly in the the contents of mathematics concepts which become the discussed infinity of the paper.In addition,the editor divide recognition for students to infinity into the golden tower structure with five categories,eight hierarchies according to mathematics mental , history of mathematics ,and epistemology as follows: Defining the implication of each level ,The editor analyzed the relationship of each level. In addition, the editor compiled the measuring list of infinity recognition on the basis of which infinity recognition for students can be investigated.According to the idea about infinity recognition of Spiro's flexiable learning theory as well as the theory of APOS of Ed Dubinsky, American mathematics education professor.The editor analyzed the infinity recognition by means of integrating potential infinity and actual infinity .infinity and limit.The editor choose over 500 samples which were tested and interviewed from elementary school, high school and the second grade of university. Some of samples and some of teachers were interviewed . The editor recorded the dialogue of students and teachers,getting the original number resource which the editor analyzed with SPSS software.From such areas as mathematics comprehend and mental custom of mathematics infinity for students ,the editor discussed the reason why the fault of students occur after comparing the status of respective age students,and got some conclusions as follows:1.The general development trend of infinity recognition for students from grade 9 to 12.2. Recognition for students to essence of the infinity.(1) Recognition to abstractation of infinity is the crucial sign of initial intuitive level; (2)The whole recognition is important factor affecting advanced intuitive level; (3) Dynamic analysis to infinity is the main factor to the deductive level;(4) Understanding to the limitary infinity of the ε-δ definition is the key factor. 3 Recognition for students to the mathematics concepts with infinity(1)The sophomore unable to fully recognize the infinity approach idea of the concepts such as continuity ,derivative, integration;(2)The analysis to the main factors affecting understanding to the ε-δ definition. 4 The mental trend of infinity recognition for students(1)Experience to some extent holdbacking the recognition for students to mathematics infinity (2)The mental characteristic trend of infinity thinking style of grade 12. (3) Recognition trend for sophomore to corresponding one by one and transfinite theoryOn the basis of the conclusions.the editor pointed out the didactical suggestion as follows: 1.Foccusing on the cultivating infinity recognition in instruction (1)Seizing on the initial recognition to infinity in instruction (2)Illuminating the recognition as a whole to mathematics concepts . (3)Carrying out consciously instruction means in advantage of infinity thinking (4)Foccusing on the dynamic analysis to limit concept.(5)Foccusing on the illustration to the limitary infinity of the ε-δ definition in instruction (6)Conceming transfinite theory instruction2. Improving the infinity attainment,of mathematics teachers3. Some suggestions to curriculum system arrangment4.The suggestion on the embodied requirements of cultivating infinity idea in the mathematics standard.(1)Definitely ponting out the goals of cultivating infinity idea (2)Adding an example of cultivating infinity idea to the mathematics standard. 5.The suggestion to the evaluation of instruction.