Modern mathematics curriculum standards put forward: teachers are required to pay full attention to the learning process and guide students to explore new knowledge; Follow the development law of students' cognitive psychology, reasonably organize teaching content and establish a reasonable mathematics training system; Mathematics teaching should not only enable students to acquire basic knowledge and skills of mathematics, but also acquire mathematical ideas and concepts, and form good mathematical thinking quality. Through various channels, students can experience the process of mathematical thinking and creation, enhance their interest and self-confidence, and constantly improve their ability of autonomous learning.
It is only one aspect to make students understand knowledge in mathematics teaching, and it is more important to cultivate students' thinking ability and master mathematics thinking methods. I think strengthening variant training in mathematics teaching is of great help to cultivate students' mathematical thinking ability.
Variation is actually innovation. The implementation of variant training should grasp the main line of thinking training, appropriately change the problem situation or change the thinking angle, cultivate students' adaptability, guide students to seek solutions to problems from different ways, and stimulate students' enthusiasm and profundity of thinking through asking more questions, thinking more and using more.
Of course, variant is not blind. We should grasp the essential characteristics of the problem, follow the development of students' cognitive psychology, and make variant according to actual needs. General types are: multiple questions, multiple questions, multiple solutions, multiple corrections, etc.
First, multiple solutions to one question, through variants, let students sum up the basic laws and cultivate students' thinking ability of seeking common ground while reserving differences.
Many mathematical exercises seem to be different, but their inherent nature (or the same ideas and methods of solving problems) requires teachers to pay attention to the collection and comparison of such topics in teaching, guide students to seek common solutions, let students realize their internal relations and form mathematical thinking methods.
For example: Question 1: Figure A is a point on the CD, ABC and ADE are regular triangles, which proves that CE=BD.
Question 2: As shown in the figure, Abd ABD and ACE are regular triangles. Verify that CD=BE.
Question 3: As shown in the figure, draw a square AEDB and a square ACFG with AB side, AC side and AC side of AB to connect CE and BG, and verify that BG=CE.
Question 4: As shown in the figure, two squares ABCD ABCD with common vertices and BEFG connect AG and EC, which proves that AG=EC.
Question 5: As shown in the figure, P is a point in the square ABCD, and the clockwise rotation energy of ABP around point B coincides with CBP'. If PB=3, find PP'
The above five questions all use the properties of regular triangles and squares to create conditions for proving congruent triangles, and use the properties of congruent triangles for further calculation or proof. Teachers should show such questions to students in groups, so that students can feel their own * * * in comparison.
Second, ask more questions, and cultivate students' innovative consciousness and ability to explore and summarize through variant development, expansion and development of original functions.
In teaching, we should pay special attention to the "modification" or extension of textbook examples and exercises. Mathematical thinking methods are hidden in textbook examples or exercises. In teaching, we should be good at digging such exercises, that is, covering as many knowledge points as possible through a typical example and stringing scattered knowledge points into a line, which often has unexpected effects and is conducive to the construction of knowledge. For example, there is such an exercise in the exercise book of the eighth grade next semester: as shown in figure (1), in ABC, point B=C, point D is a point on the side of DFAB DEAC BC, and the vertical foot is E, F, AB = 10 cm, DE = 5 cm, DF = 3 cm, (/kloc-)
The above problems can be solved by connecting AD: SABC=40 cm2 and dividing it into two triangles with waist as the bottom. With the help of the height CH on AB, it is not difficult to find that the height on AB is 8cm by using the area formula and the conclusion of the first question. I didn't take the conclusion as the ultimate goal of teaching, but continued to ask: 3+5=8. This question explores the internal relationship of DE, DF and CH. Is it a coincidence? (Students guess that CH=DE+DF).
Derive the variant problem (1) as shown in Figure (2). In ABC, B=C, D is any point on the side of BC, DEAC, DFAB and CHAB, and the vertical feet are E, F, H, F and H respectively. Verification: CH=DE+DF.
On the basis of calculating an example, students have the consciousness of connecting vertical line segments with different methods of finding area, and the proof of this problem is easy to solve.
At this time, the enthusiasm of students' thinking was fully mobilized, and I took the opportunity to give the variant (2) as shown in Figure (3). In equilateral ABC, P is any point in the form, PDAB is in D, PEBC is in E and PFAC is in F, which proves that PD+PE+PF is a constant value.
Through this group of variant training, the application of area method in geometric calculation and proof has been well reflected. At the same time, this group of variant training has experienced a process from special to general, which is helpful to deepen and consolidate knowledge, further improve students' guessing and inductive ability, and more importantly, cultivate students' awareness of questions and inquiry.
Mathematics teaching should be designed as a process of "rediscovering and recreating" students' mathematical knowledge, so as to cultivate students' innovative consciousness and problem-exploring ability. Paulia once said, "Before proving a theorem, you must guess the theorem, and before you know the details of the proof, you must guess the leading idea of the proof." Mathematics "Starting from specific problems, establishing guesses through observation and experiments, summarizing laws through analysis and demonstration, and then deepening application and guiding the solution of specific problems".
Third, multiple solutions to one question, through variants, cultivate students' divergent thinking ability and cultivate students' rigorous thinking.
The multiple solutions to a question here have two meanings: one is that there are multiple answers to a question, and the other is that there are multiple solutions to the same question.
For example, when explaining the problem of "solving the distance between the centers of two intersecting circles", students often make the mistake of losing one solution. First of all, I explain to students the formation of the intersection of two circles from the perspective of movement. When two circles are tangent, if the center of one circle continues to approach the center of another circle, when two circles have two common points, it is called the intersection of two circles. Then I draw two intersecting circles on the blackboard, the centers of which are on both sides of the common chord. Let one center continue to approach the other center. When the centers of two circles are on the same side of the common chord, ask the students to calculate the distance between the centers of the two circles. At this time, the students found that the calculation results were not the same under the same known conditions. Therefore, it is concluded that the centers of two circles intersect on both sides or the same side of the common chord.
Maslov's hierarchy of needs theory holds that every student has the need of self-realization and being valued, the desire to attach importance to personal dignity and value, the tendency and unique desire to fully tap and develop their own potential, and to improve themselves and realize themselves through their own creative activities. Therefore, we should attach importance to the exploration of mathematical knowledge in teaching, give play to the variant function of exercises and the diversity of solutions, meet the psychological needs of students, and let students feel the joy of success brought by innovation.
Students' similar "variant" exercises not only help to completely eradicate the problem of missing solutions in multi-valued problems, but also enhance students' awareness of exploration and innovation and cultivate the rigor of mathematical thinking.
Fourth, one topic is changeable, sum up the rules and cultivate the profoundness of students' thinking. Through variant teaching, we can not solve a problem, but solve a class of problems, curb "sea tactics", develop students' problem-solving ideas, cultivate students' exploration consciousness, and achieve "get twice the result with half the effort"
Galileo once said that "science is advancing in the exploration of constantly changing the angle of thinking". Therefore, classroom teaching should constantly bring forth the old and bring forth the new, and more new questions with relevance, similarity and opposition will be introduced through the original questions, so as to deeply explore the educational function of example exercises.
For example, there is a question in the book that proves that the quadrilateral obtained by connecting the midpoints of each side of the quadrilateral in turn is a parallelogram. Teachers can lose no time to make variants. Stimulate students' interest in thinking. Variant (1) What is the graphic variation of the quadrilateral obtained by connecting the midpoints of the sides of the rectangle in turn? (2) What is the graphic variation of the quadrilateral obtained by connecting the midpoints of each side of the rhombus in turn? (3) What is the quadrilateral figure obtained by connecting the midpoints of the sides of a square in turn? After these four exercises, teachers can further guide students to summarize the characteristics of the diagonal of the original quadrangle that affect the shape essence of the figure. Another example is the application problem teaching, which is the difficulty of junior high school teaching. In teaching, we can present the same type of questions to students in a variant way, and gradually guide students' thinking to be profound. For example, two cars, A and B, depart from two cities, A and B, with a distance of 2 10 km. A car is 40 kilometers per hour, faster than B car 10 kilometers, and meet on the road a few hours later. After solving the example, the teacher can make the following changes to this example: (1) change "two cars start at the same time" to "when the A car starts before 1"; (2) Change "two cars driving in opposite directions" to "two cars". Two cars B depart from two cities, A and B, with a distance of 2 10 km. 1 hour later, the B train departed from B city at a slower speed than the A train 10 km, and met on the way 3 hours later. This variant covers both simultaneous and non-simultaneous meetings. Catch up with the basic types of travel problems, such as problems. In this way, through the practice of a problem, we can not only solve a class of problems, but also sum up the most essential things between quantity and quantity. In the future, when students encounter similar problems, their thinking direction will be accurate, which will cultivate the profundity of their thinking. Students don't have to be trapped in the ocean of problems.
To sum up, it is important to guide students to carry out appropriate variant training on the basis of mastering the answers to books and exercises, which can consolidate the foundation and improve their ability. In particular, variant training can cultivate and develop students' divergent thinking, divergent thinking and reverse thinking, thus cultivating students' ability to consider problems from multiple angles and in all directions, which is very helpful for students to improve their ability to analyze and solve problems.
Reference: 1, Mathematics in Primary and Secondary Schools (No.4, 2004)
2,' Mathematics Education Reform and Research' in March 2004
3, Shanghai ordinary primary and secondary school mathematics curriculum standards.
4. "Continuing Education for Primary and Secondary School Teachers in China"