What we want to make clear is that students' mathematical problem-solving ability can not be obtained directly through teaching, but needs to be gradually developed and improved through long-term training. So how to cultivate students' problem-solving ability step by step in mathematics classroom teaching? Combined with my years of teaching practice, I think we can start from the following aspects:
1: Pay attention to the exemplary role of example
The essence of problem-solving teaching is "thinking process". Limited by age and other factors, students' thinking development has its specific laws, which requires that problem-solving teaching should follow students' cognitive characteristics and carry out targeted training. Because students' problem-solving still depends on the problem-solving mode, ideas and steps of examples, so as to realize the classification of problem-solving, I attach great importance to the demonstration role of examples in usual classroom teaching.
I remember that in the review class of trapezoid, I only gave an example:
As shown in the figure, in trapezoidal ABCD, ABCD, with AD and AC as sides, is a parallelogram ACED, which extends the intersection of DC and EB to F, and proves that EF=FB.
Through analysis and discussion, there are many solutions to a problem, and a total of eight solutions are summarized. These eight proof methods include the addition of important auxiliary lines and the knowledge of midline in trapezoidal problems. It can be seen that the teaching of a good example has a positive effect on improving students' thinking quality and problem-solving ability.
2. Pay attention to the infiltration of "mathematical thinking method"
In fact, mathematical thinking method has a higher level and status than basic knowledge of mathematics. It is contained in the process of the occurrence, development and application of mathematical knowledge, and it is a kind of mathematical consciousness, which belongs to the category of thinking and is used to understand, deal with and solve mathematical problems. Mathematical method is the concrete embodiment of mathematical thinking, which has the characteristics of model and operability and can be used as a concrete means to solve problems. Only by summarizing mathematical ideas and methods can we be handy in analyzing and solving problems. Only when I understand the ideas and methods of mathematics can books and other people's knowledge and skills become my own abilities. In the process of teaching, I also persistently cultivate students' mathematical ideas and methods, pay attention to thinking, and achieved good results.
For example, in ABC, AB = AC = 12cm, BC=6 cm, D is the midpoint of BC, and the moving point P starts from point B and moves in the direction of B-A-C at the speed of 1 cm per second. Let the movement time be t, then when t is what value, the straight line passing through point D and point P will divide the circumference of ABC into two parts, so that,
For this kind of dynamic problem, it is difficult, and most students are at a loss. I guide them to think like this, and first determine what kind of topic it is. Students can see that this is a moving point problem. Then ask some questions. What is the focus of consideration? Students can clearly say that it depends on the special position where the moving point moves. Then ask what questions can be determined in a special position? The classification of the situation can be determined. In this way, students are gradually introduced into the thinking of classified discussion, and students can formulate equations according to the meaning of the problem to solve it. After the students finished speaking, I asked again, if I think about adding the whole idea again, will there be a simpler way? In this way, students can gain more through thinking.
Guided by this, the important mathematical thinking methods in mathematics are interspersed in the classroom, and the breadth of their thinking is cultivated imperceptibly and consciously, which not only achieves twice the result with half the effort, but also stimulates students' interest in learning mathematics. Our teachers should pay enough attention to the problem-solving process so that students can improve their problem-solving ability in a subtle way.
3. Pay attention to "general method" teaching.
In the review stage of senior high school entrance examination, we will come into contact with relatively comprehensive topics, and students' ability will be reflected at this time. Students may have many wonderful solutions to the same problem. Most students can only watch others fly with passion on the podium and feel ashamed. At this time, as a teacher, we must give students general methods, because most students can only start with general thinking when facing problems, and there are very few students who can think whimsically. Therefore, in solving problems, we can praise and encourage students who come up with the simplest methods, but we must not forget the most basic ideas and methods.
For example, in the problem of finding the intersection of linear functions in actual situations, there is such a problem: a bus and a taxi commute between A and B every day, and the relationship between the road y(km) from A and the time X (hours) is as shown in the figure. The following question is solved by images: How many times did two cars meet on the way? 2. Find the distance from A when we last met?
When solving this problem, most students can consider using the analytical formula of linear function to construct the equation, get the coordinates of the intersection of images, and then get the results. At that time, some students in the class suggested that there was a simpler way. At that time, I didn't talk to him, but asked the students to write out the process in a conventional way. After the completion, we listened to the students talk about the method of solving by similarity, which is really much simpler than the previous method. The students spontaneously applauded the student at that time. The reason why I didn't let him speak first is because most students have no mind to listen to other methods after hearing the simplest method, but this simple method can't be used for all function problems, and the first method is a general method, and most students' thinking ability can be completed, although it is a little more complicated. After reviewing during this period, I will first emphasize the general methods on the topics of various methods, and then let the students introduce whimsy. Because students are good at expressing and willing to think, they want to compare with teachers in other ways. In this way, an atmosphere of learning and exploration is formed.
Of course, at an appropriate time, I don't mind exposing myself or deliberately guiding students to explore the process of thinking obstruction and failure in the process of solving problems. Even sometimes students are too anxious to explain to me. This situation is intentional on some key issues, in order to let most students have correct ideas and methods. Of course, sometimes it really doesn't work. But I don't think it will make students doubt the authority of teachers. On the contrary, I think it is easier for students to think effectively.
4. Pay attention to the reuse of wrong questions.
For mathematics, it is necessary to do problems. Teachers should guide students to do a certain number of mathematical exercises, accumulate experience in solving problems, sum up ideas for solving problems, form rules for solving problems, stimulate inspiration for solving problems and master learning methods.
Usually teaching is mainly to let students explain the wrong questions in detail. No matter fill in the blanks, choose topics or solve problems, I will set aside some time for wrong questions in class and ask students to write the problem-solving process in red ink. One unit later spent time reviewing the wrong questions. Just discuss and reflect on the wrong questions in the chapter before the exam.
In mathematics teaching, there are endless topics and ever-changing questions. In this context, the purpose of solving problems is not only to satisfy the quantity, process and result of solving problems, but also to guide students to seriously analyze and reflect on the wrong questions after solving problems, pay attention to the radiation effect of the wrong questions and understand other hidden functions of the wrong questions themselves.
5. Pay attention to students' non-intelligence factors and cultivate students' good thinking quality.
Bruner wrote in The Educational Process that students' interest, motivation, attitude, curiosity and emotion play an important role in promoting the development of wisdom. As a teacher, we should understand students' psychological activities, ignite their enthusiasm with our own enthusiasm and care, fully affirm their little progress, let them feel the happiness of success, thus generating upward motivation, fully mobilizing students' initiative, giving full play to their inner motivation, mastering the correct thinking method and forming good thinking quality.
After every exam, I will set aside time to analyze and summarize the exam. No matter whether the grades are good or not, I will tell the students what our advantages are and pass the exam. What are our shortcomings? What is the direction of our future efforts? And praise and encourage them in a targeted manner. Let students know through praise that anyone can learn math well as long as they are diligent, curious and persistent.
In short, the improvement of students' problem-solving ability cannot be achieved overnight, nor can it be achieved by teachers' subtle influence and students' conscious action. Instead, it is necessary to emphasize the word "live" in the guidance of solving mathematical problems, firmly establish the idea of "reading without doing problems, burying one's head in doing problems without summing up and accumulating", and treat mathematical problems with both the ability to get in and out, and the purpose and persistence. Only in this way can students' problem-solving ability be developed and improved!