Students' ability to acquire concepts develops with age, intelligence and experience. The research shows that in terms of the influence of intelligence and experience on concept learning, experience plays a more important role, and rich experience background is the premise of understanding the essence of concepts, otherwise it will easily lead to memorizing the literal definition of concepts and failing to understand the connotation of concepts. The "experience" here is not only obtained from school study, but also the experience gained by students from daily life plays a very important role. In fact, many scientific concepts mastered by students are formed and developed from everyday concepts. Therefore, teachers should pay attention to guiding students to accumulate experience beneficial to concept learning from daily life, and at the same time pay attention to using students' daily experience to serve concept teaching.
As far as mathematical concept learning is concerned, the influence of "experience" on new concept learning is more manifested in the expansion of concept system. Some students can find concepts related to new concepts from past experience and establish new concepts on the basis of comparing their similarities and differences, while some students will be disturbed by this experience and have a wrong understanding of concepts. For example, students have been exposed to square operation since primary school. In their experience, the square operation is only associated with "positive"; In addition, they are familiar with linear equations, that is, one equation corresponds to one solution. When learning the concepts of "square root" and "arithmetic square root", it is very difficult to learn the concept of "square root" because the square root of positive numbers involves positive numbers and negative numbers, but in fact these two numbers are the two roots of the equation x2=a, which is very different from their experience. At the same time, we have to learn the concept of "arithmetic square root", so that sometimes we have to take two values and sometimes we can only take a positive number, which leads to confusion in understanding.
In order to prevent the negative influence of experience on new concept learning, we should first work hard on the teaching of basic concepts, put basic concepts in the central position, make them become links of related knowledge, and highlight the internal relations between concepts. Some concepts in mathematics have overall effects, such as "set", "function", "equation" and "distance". These concepts should give students the opportunity to contact again and again, and on this basis, other concepts can be deduced. In Ausubel's words, it is often the most effective to learn the most general concepts first and then gradually differentiate into more specific concepts. For example, the arrangement of algebra textbooks in senior high school is from correspondence to mapping to power function, exponential function and logarithmic function according to the principle of gradual differentiation. Of course, not all the content can be arranged in this way. For example, "number system" cannot be arranged in the order of "complex number, real number, rational number, irrational number, integer number, fraction number and natural number" because this order is contrary to people's daily experience in understanding the concept of "number". For such content, we should pay attention to giving an appropriate number of examples in teaching, so that students have the opportunity to sum up the same characteristics from each specific example, and then abstract the essential characteristics (in fact, we should pay attention to using the teaching strategy of "concept formation" to learn from the shallow to the deep, from the easy to the difficult, and from the known to the unknown. At the same time, we should pay attention to guiding students to explore the relationship between old and new concepts in time and find out their similarities and differences, so that students can have sufficient practical opportunities to establish this sense of connection and difference. Here we emphasize the importance of letting students practice concepts repeatedly. In our opinion, this practice cannot be equated with mechanical repetition, because the mathematical concept is far from the students' reality. If they don't have the opportunity to practice concepts repeatedly, it will be difficult to build up the feeling that they need to understand. For example, in the concepts of "rational number" and "irrational number", students can distinguish, understand and master some numbers that are not cyclic decimals, but finite decimals or cyclic decimals in the process of calculating the square roots of numbers 2, 3 and 5. Of course, this kind of repeated training should be adapted to students' cognitive level, and high-standard understanding should be put forward to students in time. With the growth of students' age and the deepening of mathematics learning, practical training can be gradually carried out under the guidance of abstract concepts, so that the understanding and application of concepts can promote each other, thus speeding up understanding and improving training efficiency.
Second, perceptual materials or perceptual experience.
Concept formation mainly depends on the abstract generalization of perceptual materials, while concept assimilation mainly depends on the abstract generalization of perceptual experience. Therefore, perceptual material or perceptual experience is an important factor affecting concept learning. Specifically, it can be expounded from four aspects: quantity, variation, typicality and counterexample.
1. quantity. The quantity of perceptual materials and perceptual experience is too small, students' perception of the concept is not sufficient, and the experience necessary to master the concept cannot be established, so it is difficult to fully identify the various elements of the concept object, and thus the solid foundation needed to understand the concept cannot be established due to the insufficient comparison of the essential attributes and irrelevant attributes of the concept. Of course, this number should not be too large, otherwise irrelevant attributes will be improperly strengthened and essential attributes will be covered up.
2. variants. Variation is the expression of changing the non-essential attribute of an object, changing the angle or method of observing things to highlight the essential attribute of the object and the hidden essential elements. In short, variation refers to changing the irrelevant characteristics of positive examples of things, so that students can think in variation and better grasp the essence and laws of things.
Variant is put forward in the process of concept transition from concrete to abstract, in order to eliminate some interference caused by the non-essential attributes of concrete objects themselves. Once the specific object changes, those non-essential attributes closely related to the specific object disappear, while the essential attributes appear. Mathematical concepts are established by comparing variants, abandoning non-essential attributes and abstracting essential attributes. For example, when learning the concept of triangle height, we can provide students with different graphic variants, that is, we can provide students with some examples of different triangles changing in shape (acute triangle, right triangle, obtuse triangle) and position, so that students can abstract the definition of "triangle height" through thinking processing of these typical variants.
It is worth noting that variants can be used not only in the process of concept formation, but also in the application of concepts. Therefore, we can not only change the non-essential attributes of the concept, but also change the conditions and conclusions of the problem; It can not only change the form or content of the problem, but also configure various practical application environments. In short, it is to seek the same in change and never break away from your original religion. Here, what changes is the physical properties and spatial expression of things, and what remains unchanged is the essential attributes of things in quantity or shape. The purpose of the change is to give students a chance to experience the process of concept generalization, to make the concepts they master more accurate, stable and easy to migrate, and to avoid taking non-essential attributes as essential attributes.
The application of variants should pay attention to serving teaching purposes. The relationship between mathematical knowledge is the basis of variation, that is, by using the mutual relationship of knowledge, we can systematically get various variations of concepts. In addition, the use of variants should grasp the opportunity. Only when students have a preliminary understanding of the concept, and this understanding needs to be further deepened, can we achieve good results. Otherwise, if they use variants before establishing a preliminary understanding of the concept, they will not understand the purpose of variants, and the complexity of variants will interfere with students' conceptual understanding and lead to confusion in understanding.
3. Typical. Practice shows that the more obvious the essential attribute of a concept is, the easier it is to learn, and the more prominent the non-essential attribute is, the more difficult it is to learn. Therefore, when giving examples of concepts, in order to highlight the essential attributes of concepts and reduce the difficulty of learning, teachers can use the method of expanding relevant features to conduct appropriate classification exercises on the essential attributes of a concept. For example, the concept of "single item" mainly involves the definition, coefficient and number of single items. For the definition, we should emphasize "the product of numbers and letters". Examples are 4x2, -AB and m, let students analyze whether a number is a monomial. For "coefficient", there must be both positive and negative coefficients. In particular, students should point out the coefficients of special monomials such as x, -X and 2, -5. For the "times" of a single item, there should be x2, x3, ab,-7xy3, etc. And let the students tell "how many times is the number monomial". These typical concept examples can help students grasp the essential attributes of concepts and understand all aspects of concepts in concept learning.
4. Counterexample. The counterexample of the concept provides the most favorable information for discrimination, which is impressive and plays a very important role in deepening the understanding of the concept. Proper use of counterexamples can not only make students understand concepts more accurately, but also eliminate the interference of irrelevant attributes. For example, students often confuse the concepts of "modulus of complex number" and "absolute value of real number" and make mistakes. This kind of mistake can be corrected immediately by counterexample. For another example, when students learn the concept of function, they often pay attention to the expression of the function and ignore the definition domain of the function, which shows that students have separated all aspects of the essential attributes of the concept when understanding the concept. At this time, they can also give counterexamples to help students understand. In addition, students often make mistakes in defining concepts, such as "three points define a plane" and "two straight lines with nothing in common are called parallel lines". These problems are not only because of students' carelessness, but also because students have not noticed the relationship between the essential attributes of concepts and have not realized that this relationship is a key feature. Counterexamples can encourage students to enhance their understanding of the importance of this relationship.
It should be noted that the use of "counterexample" is timely. Generally speaking, students can't use counterexamples when they are new to concepts, otherwise, they may make the wrong concepts preconceived and interfere with the understanding of concepts. Counterexamples can only be used on the basis of students' understanding of concepts.
It can be seen from the above that too little or too much experience materials provided by teachers for students will have a negative impact on concept learning, so we should pay attention to giving positive examples in teaching. On the other hand, it is not enough to let students really understand the concept from the front, but also to guide students to understand the concept from the side and back. The so-called "understanding the concept from the side" means understanding the concept with "variants" and describing and understanding the concept with equivalent language; The so-called "understanding the concept from the opposite side" is mainly "giving counterexamples", that is, pulling out one or several key attributes contained in the concept to see what will happen.
Third, students' generalization ability.
Generalization is the direct premise of forming and mastering concepts. The process of students learning and applying knowledge is a generalization process, and the essence of migration is generalization. Generalization is the basis of all thinking qualities, because without generalization, students can't understand and master concepts, so definitions, theorems, laws and formulas derived from concepts can't be mastered by students. Without generalization, it is impossible to make logical reasoning, and it is far from profound and critical thinking; Without generalization, it is impossible to produce flexible migration, and the flexibility and creativity of thinking are impossible; Without generalization, thinking can't be "reduced" or "concentrated", and the agility of thinking can't be reflected. Students' mastery of concepts is directly restricted by their generalization level. To generalize, students must be able to distinguish various attributes of specific cases, and then abstract the same and essential attributes or characteristics through analysis, synthesis and comparison, and then summarize them; On this basis, classification is carried out, that is, the essential attributes obtained by generalization are extended to similar things, which is not only a conceptual application process, but also a higher-level abstract generalization process; Then, it is necessary to incorporate the newly acquired concepts into the concept system, that is, to establish the relationship between new concepts and related concepts that have been mastered, which is the advanced stage of generalization. As can be seen from the above, the differentiation of concept concrete examples is the premise of generalization, and classifying concepts into concept systems has become an important step to deepen concept learning. Therefore, teachers should take teaching students to discriminate and classify specific cases as an important part of concept teaching, so that students can master the skills of discriminating and classifying, thus gradually learning to analyze materials, compare attributes and summarize essential attributes by themselves, and gradually cultivating students' generalization ability. In addition, in the ability of mathematical generalization, it is very important to find the relationship, that is, the ability to find the relationship between various attributes in specific cases of concepts and the relationship between new concepts and related concepts in existing cognitive structures. If you can't find this relationship, it's hard to generalize. For example, when learning the concept of complex number module, it is obtained that the module of complex number z = a+bi is the length of the directed line segment OZ corresponding to the point Z(a, b) in the complex plane, that is, the distance from the point Z(a, b) to the origin O, which is also called the absolute value of complex number a+bi. In order to let students experience the whole process of the popularization of "complex modulus", teachers should guide students to incorporate it into the existing concept system of absolute value of numbers. In practice, students can be guided to compare the similarities and differences between the absolute values of complex numbers and the absolute values of real numbers they have mastered before, and regard the latter as the development of the former and the former as a special case of the latter. Then, in the interpretation of geometric meaning, the real number axis is regarded as a part of the complex plane, the real number A corresponds to the point (a, 0) in the complex plane, and the absolute value of the real number is interpreted as the module of the complex number. Practice shows that in concept learning, only through continuous and in-depth abstract generalization according to the hierarchical structure of mathematical concepts can a concept system with good structure and function be formed, so that students can accurately grasp the essence of concepts and form a relatively perfect mathematical cognitive structure. In fact, the abstraction of mathematical concepts has the characteristics of hierarchy, which brings the hierarchy of generalization activities in concept learning and becomes a spiral process. Concepts with a low degree of abstraction become concrete materials for high-level generalization activities. With the improvement of generalization activities, the abstractness of concepts mastered by students is also improving, and a concept system is gradually formed. Therefore, the study and teaching of mathematical concepts must take care of each other and pay attention to the development of concepts.
Fourth, mathematical language expression ability.
Language names things and expresses their properties and functions. By naming, we can simplify the representation of things in people's minds. Because everything has its own "name", when its expression changes and its essential characteristics are covered up, people can use this "name" to avoid cognitive confusion. The description of the attributes or functions of things can help learners deepen their concept learning, make the relationship between the elements of a concept clearer, and make the connection and difference between a concept and other concepts clearer. Language enables individuals to form concepts directly, without observing things from the beginning or recalling related appearances in the process of understanding concepts. Therefore, language expression is a very important link in the process of concept learning. The conclusion in mathematics depends on logical reasoning, and the expressive ability of mathematical language directly affects the formation of logical reasoning and mathematical concepts. In addition, students can correctly describe concepts in their own language and explain the essential attributes revealed by concepts, which is a sign of students' profound understanding of concepts.
The linguistic expressions of many mathematical concepts represent the conditions for the emergence of concepts, and are abstractions of the occurrence and development process of corresponding things in the form of numbers or quantities. Therefore, the narrative process of a concept actually shows the operating procedures that should be followed when applying a concept. For example, the language expression of the concept of "monotone function" is "Let the domain of function f(x) be E, if the values of any two independent variables belong to an interval in the domain e, x 1, x2, and when X 1 < X2, both have F (X 1) < F (X2 If there is F (x 1) > F (x2) for the values of any two independent variables belonging to an interval in the domain e, when X 1 < X2, it is said that F (x) is a decreasing function in this interval. According to this definition, we can conclude that the judgment function is monotonous.
(1) let x 1, where x2 is the value of any two independent variables in a given interval, and x1< x2;
(2) calculate f(x 1) and f (x2) respectively;
(3) judging the sign of the difference f (x 1)-f (x2);
Therefore, in order to deeply understand and skillfully apply concepts, it is necessary to decompose the language narrative process of concepts and let students master the operating procedures of concept application.