(1) visualization
The mastery of mathematical concepts has to go through a process from image intuition to abstract thinking, and then from abstract thinking to practical application, even several iterations. With the intuitive background of concepts, intuitive representation of abstract concepts can improve the effectiveness of concept teaching. Intuition in mathematics is relative, and the objects, teaching aid models, graphics or pictures presented by multimedia belong to concrete and vivid intuition. Well-known concepts, principles and examples are abstract and intuitive.
(2) It is necessary and effective to deepen the understanding of concepts through positive examples and counterexamples.
In fact, in mathematical thinking, concepts and examples often go hand in hand. When a concept is mentioned, the first reaction in mind is often its "example", which shows the importance of examples in concept learning and maintenance. If "function" is mentioned, the concrete analytical expressions and images of linear function, quadratic function, exponential function and logarithmic function may appear in our minds immediately. The counterexample of the concept provides the most useful information for the distinction. It plays a very important role in deepening the understanding of concepts. The application of counterexamples can not only help students understand concepts more accurately, but also eliminate the interference of irrelevant features. It should be noted that counterexamples should be used only after students have a certain understanding of the concept, otherwise, if they are used when they are new to the concept, they may make the wrong concept preconceived and interfere with the understanding of the concept. After revealing the definition of the concept, in order to further highlight the essential characteristics of the concept and prevent conceptual misunderstanding, we can use positive examples or counterexamples of the concept, such as the concept of "non-planar straight line". Through the positive and negative examples of the concept, students can realize that non-planar straight lines can never find a plane to merge them into two straight lines, instead of "straight lines on two different planes"
(3) Use contrast to clarify the concept. Only by comparison can we tell.
Comparing similar concepts, we can sum up the same attributes. Comparing the concept with the species relationship can highlight the unique attributes of the defined concept. Easily confused concepts can clarify vague understanding, reduce intuitive understanding errors, such as "arrangement" and "combination", and avoid confusion; The difference between "maximum value" and "extreme value" can be understood by comparison, that is, the former is holistic, while the latter is local. Of course, the "maximum value" can be obtained, but not necessarily the "extreme value"; Wait a minute.
(4) Using variants to improve conceptual understanding. We can fully understand the concept by studying it from different angles and giving examples.
Variation is to change the non-essential attribute characteristics of an object, change the angle or method of observing things, so as to highlight the essential characteristics of the object and the expression of those hidden essential elements. In short, variation refers to the change of irrelevant characteristics of things. Through variants, students can better grasp the essence and laws of concepts. Because students are used to thinking and remembering in images, we should try our best to guide students to re-understand more abstract mathematical concepts from the perspective of form. In order to obtain the intuitive and image support of concepts, such as "extreme value" and "maximum value", it is worth pointing out that the application of concept variants should serve the understanding of concepts and seize the opportunity. Only in the deepening stage of concept understanding can we get the ideal effect. Otherwise, students can't understand the use of variants, and the complexity of variants will interfere with their understanding of concepts and even cause confusion.
(5) Concept refining can be understood as concept concentration in a certain sense, that is, grasping the essence of the concept!
The concise expression of concepts and chunks occupy less memory space and are easy to extract. The representation of keywords is the representation of the essential attributes of concepts and the height of concept refinement. This also shows that in students' cognitive structure, "concept definition" is inert, even forgotten, and plays a role in refining the essence of concepts. Therefore, concept teaching must go through the process of concept refining, so that students can extract representative features.
(6) Pay attention to the multiple representations of the concept
Mathematical concepts are often represented in a variety of ways, such as using physical objects, models, images or pictures to represent images, and using oral and written symbols to represent symbols. Different representations will lead to different ways of thinking, and multiple representations of concepts can promote students' understanding from multiple angles. Establishing different representation forms of concepts in different representation systems and carrying out transformation training between different representation systems can strengthen students' understanding of concept relations; Establishing a wide range of relationships between different representations of concepts and learning to select, use and transform various mathematical representations are the prerequisites for effectively using concepts to solve complex and comprehensive problems. Therefore, it is the basic strategy of mathematics concept teaching to let students master various representations of concepts and change them flexibly.
(7) the concept of arithmetic.
The purpose of learning concepts is to apply them; On the contrary, application can promote the in-depth understanding of concepts. The application of concepts can be divided into two categories, one is to judge concepts, and the other is to take concepts as attributes. In order to use concepts better, it is necessary to algorithmicize concepts, that is, to transform declarative concept definitions into procedural algorithmic knowledge. The failure of algorithmic definition of declarative concepts is one of the main reasons why students can't apply concepts.