The Study On Junior School Students Function Concept Development

The function plays an important role in the modern mathematics education of middle school, which is the central knowledge of the mathematics and the foundation of studying other mathematics knowledge. Its ideology and method have potential application in solving the mathematics and practical problems. The function concept is one of the most important concepts, which is the cognitive foundation of the function properties and modeling, meanwhile, it is also the basis in studying the advanced mathematics. The understanding of function concept of middle school students can directly affect their learning of other knowledge; the teaching quality can be influenced due to the understanding of the function concept and the mastering of the teaching knowledge with teacher, which influence the mastery of function concept.This text is trying to discuss the understanding of the function concept of the students and cognitive concept. It has the significant meaning to understand cognitive laws of the function concept to middle school students, design of function concept curriculum and instruction and the teaching practice in the classroom in new curriculum.The methods such as literatures, question inquiry and the talking etc are used in the study.It is based on the theory of professor shi about the abstract of number and relation of number and mathematics concept learn with Sfard, There are three levels about understanding of the function concept with students in the junior middle school. Level one: operation level, level two: symbol level, level three: comprehensive level. We select 363 students in the junior middle school as test’s object and use the test paper as instrument, we study the cognitive of junior middle school students, getting the following ideas:1. The cognitive level and laws of the function concept of students in the junior middle school.1) All in all, with grade adding, the cognitive level to function of students in the junior middle school is gradually improved.2) The cognition of students to function concept appears certain characteristics of level, students in grade 7 and 8 are at the same level, students in grade 9 are at a level. The cognition of the students in grade 7 and grade 8 to function concept doesn’t appears notable differences, but students in grade 9 are different from them, the cognition level of students in this grade to function concept is higher than ones in grade 7 and the grade 8.3) The cognition of students to functions presented distinctly different in different patterns. The cognition of students to functions which presented with table representation is the highest, cognition of students to function with symbol representation is the lowest, the cognition of the table representation and the graph representation are higher than the cognition of the symbol representation.4) The cognition of students to functions are at the operation level and symbol level, there are a lot of difficulties to arrive comprehensive level for them.5) On the level 3 of the symbol representation, there is a lot difficulties for students in grade 7 and grade 8 to understand wheter function relation exit in the equation xy =5. The level of grade 9 is the highest.6) The ability which students explain the function relation with the symbol representation is gradually improved. There are notable differences between grade 9 and the others. From the graph representation, there are certain difficulties to explain graphs such as a vertical line, the dot separated from one another, separated graph, the diagram of curves etc for students in grade 7 and the grade 8, It is better to use the function concept in grade 9.7) Most students in junior middle school can’t use dialectical viewpoint such as the motion, the change and the relation to understand the function concept, so their ability of dialectical thinking is very poor.2. The erroneous understanding and reason to the function concept for students in junior middle school.There are many mistaken understanding to the function concept for students in junior middle school, 1) the difficult of understanding special functions; 2) students can’t differentiate nature character of the function; 3) the mistaken understanding of the variables; 4) the mistaken understanding of the function definition; 5) students overly rely on past experiences and memory; 6) students overly rely on basic primary functions; 7) the confusion of the function and the equation.There is erroneous cognition to the function concept for students in the junior middle school. 1)the complexity of the function concept itself; 2) the diversity of the function concept reresenpation; 3) the abstraction of the function notations; 4) the ability of the dialectical thinking of students in the junior middle school is poor; 5) the continuation of the function curriculum is poor; 6) there aren’t any good teaching ways of the function concept.The innovation involved in this research is: the cognitive level of the junior school students to the function concept is evaluated by the theory, The difficulty and the errorous cognition during that the students learn the function concept should be analyzed from the theory of the cognitive psychology and the measures and methods of improving teaching are given, which needs to further explore and research.