As far as its ideological origin is concerned, the discovery teaching method can be traced back to a long time ago. As early as the middle of19th century, Dostoevsky, a famous German educator, put forward that "scientific knowledge should not be taught to students, but should guide students to discover them and master them independently" and "a poor teacher gives the truth, and a good teacher teaches people to discover the truth". Later, the famous British educator Spencer also pointed out that "education should encourage the process of personal development as much as possible, guide students to explore and infer themselves, tell them as little as possible, and guide them to discover as much as possible." These views undoubtedly laid the ideological foundation for the discovery teaching method.
As a strict teaching method, discovery teaching method was first advocated by the famous American psychologist Bruner in the 1950s. He thinks: "When putting forward the basic structure of a subject, we can keep some wonderful parts to guide students to discover for themselves"; "Students can master the basic structure of the subject through discovery, which is easy to understand and remember, and is convenient for the transfer of knowledge and the development of ability"; "Discovery is not limited to seeking things that human beings have not yet known, to be exact, it includes all methods of acquiring knowledge with one's own mind". Because of his advocacy, the discovery teaching method has aroused great concern and attention of educators.
Second, the theoretical basis of discovery teaching method
The main theoretical basis of discovery teaching method is the construction principle and epiphany theory of cognitive constructivism school.
As a teaching method, discovery pays more attention to students' learning, both in the teaching process and in the teaching objectives. In this sense, "discovery learning" is characterized by students' independent exploration and cooperative learning. In the learning process, students can actively and effectively participate in their own metacognition, motivation and behavior on the basis of their original cognition.
The school of cognitive constructivism, represented by fravel, believes that active constructive learning is actually metacognitive monitoring learning, and it is a process in which students actively adjust their learning strategies and efforts according to their learning abilities and learning tasks. Therefore, as a learning method, the essence of discovery method is students' active construction on the basis of original cognition.
The school of cognitive constructivism also believes that learning is a cognitive process, not a blind trial and error, but a sudden "epiphany". People realize from practice that trial and error and epiphany are two complementary processes in learning, which are often carried out alternately. Generally speaking, mastering mathematical skills and trying to solve exercises in mathematics learning often takes the form of epiphany, while understanding mathematical concepts and creatively exploring problems often takes the form of epiphany. Therefore,
Third, the modern interpretation of discovery teaching method
Stepping into the 2 1 century, we are facing a rapidly developing information age. In order to adapt to this rapidly changing situation, people must have the ability of self-learning and lifelong learning. Therefore, an important task of basic education is to help students learn to learn and cultivate their ability of exploration, discovery and innovation.
The high school mathematics curriculum standard points out: "In teaching, students should be encouraged to actively participate in teaching activities, including thinking and behavior. Teachers should not only teach and guide, but also explore and cooperate with students independently. Teachers should create appropriate problem situations, encourage students to discover mathematical laws and problem-solving methods, and let them experience the process of knowledge formation. " This requires our mathematics teachers to change their teaching concepts and update their teaching methods. Carefully design the lesson plan for each class to create a relaxed and harmonious learning atmosphere and environment for students to actively explore problems and acquire knowledge. The concept of discovery teaching method just reflects this demand.
On the basis of the traditional "reception teaching method", the integration of "discovery teaching method", more inspiration in the process of reception and more participation in the process of discovery, and the two teaching forms complement each other to achieve harmony and unity, will become the focus of the new round of curriculum reform.
Fourth, the teaching link of discovery teaching method
Using discovery teaching method to implement mathematics teaching can usually be operated according to the following steps:
The teaching cases of the first n terms and formulas of geometric series are as follows:
1. Create a problem scenario
According to the teaching content and students' learning requirements, by citing examples related to new knowledge, finding out mathematical objects similar to new knowledge from old knowledge, preparing teaching AIDS and materials related to new knowledge, the problem situation is carefully created to guide students' attention and interest in the inquiry activities of mathematical knowledge.
I designed the problem scenario of this lesson like this: SARS virus has brought us infinite panic. Now suppose there is a SARS patient on the first day, he will stop infecting others on the second day, and the other two will infect two people on the third day, and then they will stop infecting others, and so on for 33 days * * * How many people are infected with SARS virus (regardless of the number of deaths).
(The introduction of this topic is based on the following three considerations: (1) Using students' curiosity and taking a real event as the starting point, it is convenient to stimulate students' interest and enthusiasm in learning this lesson. (2) The content of the event is closely related to the theme and focus of the teaching content of this lesson. (3) It is conducive to the transfer of knowledge, so that students can clearly understand the practical application of knowledge. )
2. Organize student activities
Student activities include individual activities such as observation, operation, induction, guessing, verification, reasoning, modeling and proposing methods, as well as group activities such as discussion, cooperation, communication and interaction under the guidance of teachers, with the aim of letting students experience the process of the occurrence and development of mathematical knowledge.
When solving the above problems, students can be guided to compare this problem with the problem of cell division when the textbook talks about the general term of geometric series, and find out the difference: the difference is that cells disappear after splitting into two, but in this problem, SARS patients do not disappear after infecting the other two, so these people should be included in the final calculation of the number of people, so the number of people on 1 day is 1, and the number on the second day is 2.
3. Guide exploration and discovery
On the basis of independent thinking and independent inquiry, students are guided to discover mathematical concepts, mathematical theorems, mathematical formulas and other mathematical knowledge, and find ways to demonstrate mathematical theorems, deduce mathematical formulas and solve mathematical problems, so that students have more opportunities to participate and experience the mathematical process and feel the successful experience like mathematicians. When seeking peace, the author does this:
Teacher: Students, if you want to know whether the data you have guessed is correct or whose error is smaller, you must give the correct solution process of this formula. Let's take a closer look at this formula first. Obviously, 1, 2, …, is a geometric series of 33 terms in * *, so what we need to do now is to find the sum of the first 33 terms in the geometric series. Generally speaking, 2. Do you have any other methods to prove it besides the methods in the textbook?
Give enough time to encourage students to think freely and solve problems actively)
Health 2: I think the formula should discuss q= 1 and classification.
S3: I think we should pay attention to the number of terms in geometric series.
Teacher: Very good. It is true that these two aspects were the places where students were prone to make mistakes in the past, so we should pay attention to the discussion of Q and the number of series terms when using formulas in the future. The proof method in the textbook is called dislocation subtraction. (Teacher demonstrates) (The idea of summation is often used in solving some summation problems, so students should master it) Are there any other proof methods besides the one in the textbook?
Student 4: Get Tong Xiang through geometric series;
Add the two sides of the equal sign of the above n equations respectively to get,,.
When,; When.
Sheng 5: (Piandian) defines: with geometric series, and then, by using the equal ratio theorem, obtains or, or (q = 1).
Health 6: (board performance), and then
So there is, that is, or (q= 1).
4. Construct mathematical theory
Mathematical theory includes concept definition, theorem description, model description, algorithm program and thinking method in solving mathematical problems. After students go through inquiry activities, experience the process, feel the meaning and form appearances, teachers should help to sort out, supplement and improve them in time, standardize them, incorporate them into students' cognitive system, form a complete mathematical theory system, and lay a foundation for mastering the application.
When constructing mathematical theory, the class records are as follows:
Teacher: It is not easy for students to come up with three different methods. Let's carefully study the above three methods: Health 4. According to the definition of geometric series, the first n terms and formulas of geometric series {a n} are derived by superposition method. On the basis of geometric progression's basic concept and geometric progression's definition, Sheng 5 derived the formula by using the equal ratio theorem. Sheng 6, and of course the error subtraction in our textbook is also a very important method, which will appear in a large number of exercises in the future.
The formula for finding the sum of antecedents of geometric series is obtained.
Please think about it. With this formula, what should I do to find the sum of the first few terms of geometric series?
All beings: directly use the formula.
Teacher: What should I pay attention to when using formulas? Inspire students to draw a conclusion that we need to discuss according to whether the common ratio is 1.
Teacher: Besides finding the sum of antecedents of geometric series, is this formula used for other purposes?
(Observe the formula carefully, and guide the students to find out that knowing three and seeking two)
Step 5 try to use math
The application of mathematics mainly refers to the application of mathematical theories discovered through inquiry to solve problems, including distinguishing, explaining, solving simple problems and solving complex problems. Teachers should carefully organize a series of problem groups, guide students to try mathematics application, cultivate students' application consciousness, and test and feedback the effect of students' learning activities.
The classroom record is as follows:
Teacher: We have mastered the summation formula of proportional series. Let's go back to the original question and ask students to accurately calculate the SARS patients after 33 days.
All beings:
Teacher: Calculate the final result.
Sentient beings: 858993459 1.
Teacher: Nearly 8.5 billion people are infected with the SARS virus, but we know that the world population is only over 6 billion. This data can also explain the horror of SARS. Fortunately, under the leadership of the party and the government, we defeated SARS, which also showed the superiority of our party and our socialist country.
Summary, review and reflection
Summary, review and reflection can be described by students first, supplemented and refined by teachers. The purpose is: on the one hand, let students review the activity process, key points and difficulties of this class, as well as the achievements and existing problems in learning activities; On the other hand, it is a re-understanding of the exploration process, a sublimation of the thinking method of studying mathematical problems, and a reflection on mathematical thinking, which provides experience and lessons for students to further study, study and solve problems in the future.
The author asks students to review and reflect on the teaching content of this lesson:
The first n terms and formulas of (1) geometric series;
(2) the derivation method of the formula;
(3) Application of the formula.
Follow-up: What did you learn from this class? This question is left to everyone to think about after class.
Through the reflection of teachers and students, giving full play to students' main role will help students consolidate their knowledge and cultivate their ability of induction and generalization, so as to further realize their cognitive goals and quality goals.
In actual teaching, the above six links don't need to be comprehensive, and they can be chosen flexibly according to the teaching content and teaching environment. The key is to pay attention to the change of students' learning style and put students' exploration and discovery activities in their proper position.
In short, the key to the effectiveness of discovery-based classroom teaching lies in whether students participate, how to participate and the degree of participation. At the same time, only students actively participate in teaching can we change the boring situation of classroom teaching and make the classroom full of vitality. The so-called students' active participation is to give students the right to explore independently, without the teacher setting a frame, and first bind the students' hands and feet. Students are required to operate according to a set designed by the teacher in advance. Every step of inquiry requires students to try first, that is, to push students to the active position and let them learn by themselves. The teaching process is mainly completed by students themselves, so that the discovery-based classroom teaching can enter an ideal realm.