I. Conceptual variants
The variation of mathematical concepts in teaching mainly includes two types: one is to change the presentation of concept extension, that is, the external form of concepts is changing and belongs to the variation of concept extension set; The other is to change the connotation of mathematical concepts, that is, the counterexamples presented in the original concepts have some of the same non-essential attributes and do not belong to the extension set of the original concepts. Concept variation is an important means in primary school mathematics concept teaching, and its function is to help students "get rid of the false and keep the true" and gain a multi-angle understanding and a more comprehensive understanding of the concept.
1. Change the non-essential attribute of the concept
The so-called non-essential attribute of the concept refers to the attribute that has no decisive significance to the concept. The non-essential attribute of changing concept is the most used concept variant in primary school mathematics concept teaching. Its psychological basis is: when the concept variant transforms the non-essential characteristics of things, it shows the diversity of things' representations, enriches students' perceptual experience and makes them know various typical representatives of the concept extension set.
For example, in the teaching of trapezoidal cognition, teachers usually give some "non-standard" trapezoids for students to identify, so as to help students eliminate the negative interference caused by standard graphics and avoid misinterpreting the non-essential attributes such as "long upper bottom, short lower bottom, waist-back (equal waist) and no right angle" as the essential characteristics of trapezoids.
Then, under the background of the new curriculum reform, how can this effective teaching method "keep pace with the times"? I think we can try our best to create conditions and change "teachers play and students watch" into students' own hands-on operation. Taking the teaching of "trapezoidal cognition" as an example, I tried two ways.
First, let the students cut the parallelogram into two quadrilaterals along a straight line, so it is not a parallelogram (as shown in figure 1).
Second, let the students use translucent rectangular and triangular pieces of paper to overlap into a quadrilateral (Figure 2).
It is also a variant figure that observes the non-essential attributes of change, but the object of observation is not provided by the teacher, but constructed by the students themselves. These two methods can enable students to dynamically understand and discover the common characteristics of trapezoid in generative operation and observation activities, and have achieved good results. This also shows that the intuitive teaching effect of variant depends on the initiative and independence of students to some extent.
2. The essential attribute of the concept of change
The so-called essential attribute refers to the unique and inevitable attribute of this kind of thing, so it can also be distinguished from other things. In teaching, it is necessary to change the essential attribute of the concept appropriately, so that students can understand the essential attribute of the concept from counterexamples and mistakes through discrimination, and promote understanding.
In practical teaching, the above two conceptual variants can also be used in combination. For example, the concept of "vertical" is discriminated. Graphics are standard graphics, which are changes of essential attributes and non-essential attributes. They reveal the essential characteristics of the vertical concept from two aspects. Let students make correct judgments by looking at pictures, so as to achieve the teaching goal of understanding concepts from multiple angles and accurately grasping the essential characteristics of concepts.
Second, the process variant
The process of students' mathematics learning is a process of constructing their own understanding of mathematics knowledge. They enter learning activities with their original knowledge background, activity background and understanding, and construct their understanding of mathematics through their own active activities. The implementation of process variant in primary school mathematics teaching aims at optimizing students' learning process, and establishing an inherent and reasonable connection between the learning object and the learners' existing knowledge through variant foreshadowing, so that students can gradually acquire knowledge or solve problems. This is also the embodiment of the concept of mathematics curriculum reform in classroom teaching.
1. Process variables of meaning construction
The process of meaning construction is an experimental connection between new information and long-term memory, accompanied by a process of checking the construction results at any time. In order to realize the meaningful construction of mathematical knowledge, teachers should pay attention to students' nearest development zone, which refers to the distance between learners' actual ability to solve independent problems and the potential development level achieved by using the knowledge of adults or more capable partners. Teachers carry out the variant teaching of meaning construction in teaching, emphasizing that teachers can trigger and promote the formation of students' recent development zone through appropriate and dynamic variants, and finally realize the potential development level. In teaching, the process variant teaching strategy often used by teachers is an effective teaching method to form the meaning construction of mathematical knowledge.
2. The process variables of legal inquiry
Some contents in primary school mathematics are more suitable for students to explore and learn, such as many formulas for calculating the surface and ontology of objects, which are relatively independent, interpenetrating and interrelated.
Take the derivation of trapezoidal area formula as an example. Before this, students have mastered the area calculation formulas of rectangle (including square), parallelogram and triangle, and have a certain understanding of the transformation of graphics and the transformation idea of "transforming the graphics with unknown area calculation formula into the graphics with known area calculation formula". These are the bases that can be used when exploring the trapezoidal area formula.
When teaching, first review the area calculation formulas of rectangle, parallelogram and triangle, and let students describe the derivation process of the area calculation formulas of parallelogram and triangle.
Then put forward the goal of exploration: to find out the calculation formula of trapezoidal area.
Inspire students to think:
(1) Are you going to convert the trapezoid into a figure with the known area formula?
(2) How to transform, is it spelling, cutting, making up, or dividing?
③ Can you calculate the area of the converted graph?
(4) Give it a try and summarize the calculation formula of trapezoidal area.
In the process of exploration and communication, the appearance of various transformation variants is random, and the number of variants that students think of in a class is also quite different. My countermeasure is that students can show and communicate what they can do, but not perfect. Without the three basic ideas of transforming into parallelogram, rectangle and triangle and the three basic methods of spelling, cutting and division, interested students will be inspired to continue their exploration after class. Similarly, students use different methods to get different algorithms, and they don't insist on unifying into the standard form of trapezoidal area calculation formula. Because diversified algorithms are conducive to developing students' ideas, this is also one of the purposes of implementing process variants. In fact, students will eventually agree to the standard form of the trapezoidal area calculation formula:
The differences of different students' mathematics learning are objective. The process variables of conventional inquiry focus on students' inquiry and experience. It is natural and normal for teachers to construct appropriate variation space and lay appropriate potential distance. Different students go through different processes, get different results and realize different things.
Third, training variants.
Mathematics training is an indispensable part of mathematics teaching and an effective means to acquire mathematics knowledge. Training variants include training topics, solutions and training implementation. Mathematical training variants have a long history, and many teachers are designing and implementing variant training intentionally or unintentionally. However, in the past teaching practice, most teachers are most concerned about the variation of problem-solving methods and pursue the diversity of problem-solving methods. This paper focuses on discussing the variants of training questions from the perspective of exercise design.
1. Extensible variant
Expansion variant is to change the structure of mathematical problems from simple to complex (expansion) or from complex to simple (contraction) by changing the conditions or problems in exercise design according to the internal relationship between mathematical knowledge, thus helping students "climb the ladder". "Expanding" embodies the gradual development, change and deepening of cognition and training, and it is a learning and training process from thin to thick. "Shrinkage" embodies the idea of "reduction" in mathematics, which is a learning and training process from coarse to fine.
For example, the comprehensive exercise of "solving equations" can be designed as follows:
This is a design from simple to complex, which aims to highlight that the process of solving equations is the process of simplifying equations by using the properties of equations, and finally the simplest equation x = 2 is obtained, thus helping students to clearly understand the thinking of solving equations and master the methods of solving equations. Practice shows that students can really feel something through practice.
In the teaching and training of practical problem solving in primary school mathematics, extended variables are widely used, which usually show that an actual problem that only needs one or two steps of calculation is turned into an actual problem that needs two or three steps of calculation, or vice versa. This is one of the most commonly used teaching and training methods in problem-solving review class, which can help students see the ins and outs of the development and changes of practical problems and help them form the idea of "controlling complexity with simplicity".
2. Reversible variants
Inversion refers to the substitution of conditions and problems in mathematical problems. It requires teachers to pay attention to the training of reverse thinking while training students' positive thinking, so as to effectively cultivate the flexibility of students' thinking. Inversion is also a common teaching method to solve practical problems. For example, ask students to adapt the topic of finding distance into the topic of finding time or speed. Practice shows that frequent oral practice of adapting such practical questions helps students to master the structure of related questions and grasp the quantitative relations in many aspects.
3. Situational variables
Situational variant is mainly used in the teaching of practical problem solving, which usually keeps the mathematical model of the problem and changes the content of the problem situation. Situational variant is not only helpful for students to understand the close relationship between mathematics and nature and human society, but also helpful for students to understand the value of mathematics. Enhance the understanding of mathematics and confidence in learning mathematics well ",and also help to improve students' ability to analyze and solve practical problems by using the mathematical knowledge they have learned." "
For example, based on the problem of "chickens and rabbits in the same cage", we designed a set of situational variables:
① Assemble 9 tricycles and bicycles, sharing 22 wheels. How many tricycles and bicycles are there?
② 18 students play singles and doubles on six table tennis tables at the same time. How many students are playing singles?
Through practice, students can see the same mathematical essence through different problem situations. If listed as equations, these equations have the same structural form: (1) Suppose there are x tricycles, and according to the meaning of the question, the equation 3x+2 (9-x) = 22; (2) There are X tables in singles. According to the meaning of the question, the equation is 2x+4 (6-x) = 18.
Obviously, it is very beneficial to develop students' abstract generalization ability and cultivate their preliminary mathematical modeling ability.
4. Open variants
Open variant refers to changing the conditions or questions of the topic and diversifying the answers or problem-solving strategies. You can break through the bondage of mindset. Promoting the generation of divergent thinking is an effective way to cultivate students' flexibility in mathematical thinking. Open variants can be divided into three types: conditional openness, conclusion openness and strategy openness.
Conditions are open, for example, "On a straight highway, Xiaoming and Xiaogang ride bicycles of both parties, which are 500 meters apart. Xiaoming travels 200 meters per minute, and Xiaogang travels 300 meters per minute. After a while, they are 5000 meters apart. " If the movement direction of two people is removed here, there will be many situations such as the opposite direction, the opposite direction and the same direction (Xiaoming is in front or Xiaogang is in front).
Conclusion Openness is like "dividing a square into four figures with the same shape and size, how many points can you think of".
The most common strategy opening is the so-called "one question with many solutions" training. There are no more examples here.
Generally speaking, open variant training should be after some basic exercises. According to the needs of teaching and learning, design and implement as appropriate. Appropriate variant training with open conditions, open conclusions and open strategies can stimulate students' interest in participating in mathematical exercises, realize the learning goal of knowledge and skills, and help students cultivate divergent thinking, divergent thinking and intuitive thinking.
In addition, several variant training methods discussed above can also be used synthetically, that is, to form a "comprehensive variant". For example, the solution of the equation given by the above expansion variant is x = 2, which in turn requires students to "write the equation with the solution of x = 2". This is a typical variant training combining reversible variant and open variant.
Variant teaching can make teachers consciously guide students to discover the essence of "invariability" from "invariability" and explore the law of "invariability" from the essence of "invariability", which can help students master what they have learned, let students appreciate the charm of mathematics in endless changes and experience the fun of learning mathematics.
In short, under the new curriculum standards, teachers should constantly update their concepts, teach students in accordance with their aptitude, and constantly improve the "variant" teaching mode, so as to finally achieve the goal of improving teaching quality and lay a good foundation for students to learn and use mathematics well.