The process of obtaining mathematical concepts under the guidance of teachers is generally divided into the following six steps [1]:
(1) Observe a group of examples and extract * * * from them;
(2) Define, analyze the meaning and understand the essential attributes of the concept;
(3) Give two positive and negative examples to clarify the connotation and extension of the concept;
(4) Contact and distinguish this concept from other related concepts;
(5) Re-describe the meaning of the concept;
(6) concretize thinking with concepts.
Through the analysis of the psychological process of the above six steps, the formation of students' mathematical concepts can be summarized into two psychological stages: one is to abstract the definition of the concept from the correct and complete concept image (here, the concept image is the thinking image associated with the concept name to be studied in the students' mind and the nature describing all its characteristics); The second is to make abstract concepts lead to concrete reappearance in the process of thinking. Therefore, teachers mainly grasp the basic requirements of these two stages in concept teaching: how to make students produce correct and complete images of advanced mathematics concepts, and abstract the connotation of advanced mathematics concepts from them, and how to make this concept concrete in students' thinking, that is, visualized concepts.
1. From correct and complete conceptual images to abstract mathematical concepts
The formation of common sense concept needs some experience, which is obtained by summarizing and abstracting examples with certain * * * properties and then classifying them. The concept of mathematics is abstract, but it is still a way to deal with practical thinking. Without practical thinking materials, there is no object of thinking and operation, and without the object of operation, there is no foundation for abstraction. Psychologically speaking, the learning of mathematical concepts should take examples as the starting point, which is the requirement of operational thinking. Therefore, mathematical concepts should be organized, analyzed and summarized, classified and abstracted through appropriate examples for teaching. In fact, these cited examples not only introduce the objective background of basic concepts and their significance in solving practical problems before concept learning, but also help teachers to explain the concepts learned later, such as geometric meaning and physical explanation, so that students can feel that mathematical concepts are not imaginary, but come from reality and are established according to actual needs. More importantly, the conceptual images obtained from these cited examples-these thinking images associated with the name of the concept to be studied in students' minds and the nature of describing all its characteristics are the basic premise for abstracting the concept to be studied.
Here we should emphasize the correctness and integrity of the conceptual image formed in students' minds. Incorrect and incomplete conceptual images will affect the accuracy and comprehensiveness of mathematical concepts formed in students' minds.
When learning the concept of tangent of function graph in differential calculus, we give the image of function. The results show that 80% students correctly think that they can draw a tangent at the origin, but less than 20% students can draw a tangent correctly. The survey shows that more than 90% students report that there is no horizontal tangent at all points except the maximum and minimum points in the function image in their conceptual image of tangent concept.
In addition, inappropriate conceptual images will seriously affect the development of formal theory in students' minds. Taking the concept of limit as an example, Robert( 1982) has analyzed a series of thinking models used by students when dealing with the problem of limit [2], and these models are regarded as good examples of conceptual image. Cornu( 198 1) and Sierpinska( 1985) once regarded the evolution of students' learning limit concept as a process of overcoming obstacles, and put forward five obstacles, the most important of which is the fear of infinity. As a result, many students do not regard infinity as a special mathematical operation, or simply use incomplete induction to get the limit. Wheeler and Martin (1988) have also studied that students' concept of infinity is obviously inconsistent with the conceptual image contained in their minds [2].
2. From abstract regulations to concrete thinking.
Mathematical concepts abstracted from correct and complete conceptual images are the first important psychological process for students to master mathematical concepts. Whether the concept is mastered correctly depends on whether the abstract provisions of the concept can become concrete in students' thinking, that is, the visualization ability of the concept.
For example, when students learn the concepts of derivative and calculus, they often have a strong psychological tendency, that is, to turn these contents into algebraic operations and avoid images and geometric images. The intensive operation of finding functional derivatives and calculus also makes the concepts of derivatives and calculus formed in students' minds lack visualization, which affects the real understanding and application of mathematical concepts.
For example, when discussing the differentiability of f(x, y)=2x+4y+y (+x), more than 90% students directly calculate the partial derivative of f instead of observing the structure of the expression. The reason is that students understand the concept of differential at the level of pure algorithm, but don't understand whether differential is an approximation or a function.
Another example is that when students learn integral, they often take integral calculation as the original function and recite the formula of memorizing integral. They can skillfully write the original function of a function, but when asked to solve the following problem, few students realize that this is a typical integral problem. Examples are as follows: Find the attractive force between a uniform thin rod of a given length placed on a straight line and a particle located on the straight line.
There are two reasons for these results: because of the incomplete understanding of the concept of function, students can't regard differential and integral as functions; Differential and integral are not consistent with the image of functions in their minds. In the final analysis, it is the students' lack of visualization of concepts such as function, differential and integral, which makes the abstract provisions of these concepts unable to be transformed into concrete in their thinking.