First, the definition of mathematical thinking method
Mathematical thought is the essential understanding of mathematical knowledge, methods and laws; Mathematical method is the strategy and procedure to solve mathematical problems, and it is the concrete embodiment of mathematical thought. Mathematical knowledge is the carrier of mathematical thinking method, and mathematical thinking is at a higher level than basic mathematical knowledge and common mathematical methods. It comes from basic mathematical knowledge and common mathematical methods, and has a guiding position in using basic mathematical knowledge and methods to deal with mathematical problems. For learners, the process of using mathematical methods to solve problems is the process of accumulating perceptual knowledge. When this accumulation reaches a certain level, it will make a leap and rise to mathematical thinking. Once mathematical thinking is formed, it will play a guiding role in mathematical methods. Therefore, people usually regard mathematical thoughts and methods as a whole concept-mathematical thoughts and methods.
Second, the main mathematical thinking methods that should be infiltrated in junior high school.
In junior high school mathematics teaching, at least the following main mathematical thinking methods should be infiltrated into students:
1. Thinking method of classified discussion
Classification is a way of thinking by comparing the similarities and differences of the essential attributes of mathematical objects, and then dividing mathematical objects into different categories according to certain attributes. Classified discussion is not only an important mathematical thought, but also an important mathematical method, which can overcome the one-sidedness of thinking and prevent missing solutions.
2. Analogical thinking method
Analogy is a form of reasoning based on two or two objects having some common properties, which is called the most creative way of thinking.
3. The thinking method of combining numbers and shapes
The thinking method of combining number and shape refers to a thinking strategy that combines number (quantity) and shape to analyze, study and solve problems.
4. Change thinking methods
The so-called "transformation" is to simplify the problem to be solved into another easy problem or solved problem.
5. Thinking methods of equations and functions
Using the thinking method of equation is to transform the problem into solving the equation (group) problem by using the symbolic language of mathematics according to the quantitative relationship between the known quantity in the problem and the unknown quantity in the teaching method.
The so-called functional thinking method is to analyze and study the quantitative relationship in specific problems from the viewpoint of movement and change, and to characterize and study this quantitative relationship through the form of functions, so as to solve problems.
6. Holistic thinking method
The whole thinking method is that when considering a mathematical problem, we should focus on the overall structure of the problem instead of its local characteristics. Through comprehensive and profound observation, we can understand the essence of the problem from a macro perspective and treat some independent but closely related quantities as a whole.
Thirdly, the way to infiltrate mathematical thinking methods in teaching.
1. In the process of knowledge generation, timely infiltrate mathematical thinking methods.
The content of mathematics teaching can be roughly divided into two levels: one is called superficial knowledge, which contains basic contents such as concepts, properties, laws, formulas, axioms and theorems; The other is called deep knowledge, which mainly refers to mathematical thoughts and methods. Surface knowledge is the foundation of deep knowledge and has strong maneuverability. Only by studying textbooks and mastering and understanding certain superficial knowledge can students further learn and understand relevant deep knowledge. Mathematical thinking method is based on mathematical knowledge and contained in surface knowledge. It is the essence of mathematics, which supports and guides superficial knowledge. Therefore, in the teaching process of concepts, properties and formulas, teachers should constantly infiltrate relevant mathematical thinking methods, so that students can grasp the surface knowledge and understand the deep knowledge at the same time, thus making students' thinking leap qualitatively. Only talking about concepts, theorems and formulas without paying attention to the teaching of infiltrating mathematical thinking methods will not be conducive to students' real understanding and mastery of what they have learned, so that students' knowledge level will always remain in the primary stage and it is difficult to improve. In the teaching process, we should guide students to actively participate in the process of exploration, discovery and deduction of conclusions, find out the causal relationship among them, understand its relationship with other knowledge, and let students experience the mathematical ideas and methods they have experienced and applied in creative thinking activities.
Case 1:
Explore:
(1) Ask students to represent the following numbers on the number axis: 0, 1,-1, 4, -4.
(2) 1 and-1, what is the relationship between 4 and -4?
(3) What is the relationship between the distance from 4 to the origin and the distance from -4 to the origin? 1 and-1?
Give the concept of absolute value, and let students discuss and summarize the descriptive definition of absolute value from the relationship between points on the number axis.
(4) How many numbers have absolute values equal to 9? How to use the number axis to illustrate?
You can use the number axis to analyze and solve the problem of absolute value in the future. This method is called "number-shape combination".
In this way, students not only learned the concept of absolute value, but also infiltrated the thinking method of combining numbers and shapes. Here, teachers should properly refine and summarize mathematical thinking methods in teaching to deepen students' impression.
The study of mathematical knowledge can only be mastered and consolidated by listening, reviewing and doing exercises. The formation of mathematical thinking method also needs a step-by-step process, and repeated training can make students really understand. Only through a process of repeated training and continuous improvement can students form an intuitive consciousness of using mathematical thinking methods and establish their own "mathematical thinking method system". In the process of teaching new concepts and knowledge points, students can easily understand and master them if they use the mathematical method of analogy. For example, when learning rational numbers, you can use the "number" in primary school to make an analogy.
Case 2:
Design intention of teaching process in teaching link
Link 2:
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course
study
Xi 1。 Turn a parabola into a general form.
Solution:
=
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2. Panel discussion:
(1) If a parabola is given, can you point out its opening direction, symmetry axis and vertex coordinates?
Whether to discuss it depends on the students here.
(2) Thinking: Given a parabola as or, can you point out their opening direction, symmetry axis and vertex coordinates?
1, this question is to pave the way for the students to have the following discussion.
2. Through discussion, let the students try to find a solution to the problem. When commenting, teachers give positive comments on different solutions to the problems raised by students, thus stimulating students' self-motivation and self-confidence.
At the same time, we should standardize students' writing format.
Through the thinking of two variants, let students know how to deal with the quadratic term when the coefficient is not 1.
After many repetitions and infiltration, students can truly understand and master the analogy method, so as to use it flexibly in future new knowledge learning and problem solving, and at the same time improve their mathematical thinking ability.
2. Reveal the mathematical thinking method in the process of problem exploration and solution.
There has always been such a difficulty in our usual teaching work: there are many topics at ordinary times, but as long as the conditions change slightly, some students will be at a loss and always stay at the level of imitating problem-solving, so it is difficult to form strong problem-solving ability, let alone the formation of innovative ability. Cultivating students' comprehensive ability to solve problems is the core goal of mathematics teaching. In the process of solving problems, teachers should spend the greatest teaching energy on inducing students to think, how to think and where to find solutions, put the application of mathematical thinking methods at the center of solving problems, and give full play to the problem-solving functions of mathematical thinking-orientation function, association function, construction function and fuzzy expansion function. If students can give full play to the problem-solving function of mathematical thinking method in the process of solving problems, they can not only avoid detours, but also greatly improve their mathematical ability and comprehensive quality.
Case 3:
Exercise 1. Know a special angle (or trigonometric function value) and hypotenuse in a known right triangle, and find the right side?
Through several simple variants, the related knowledge is consolidated, geometric thinking is also exercised, and the combination of numbers and shapes is highlighted. )
Exercise 2. Thinking and exploring:
(1) It is known that in Rt△ABC, ∠ c = 90, BC=2 and AC=2. Can you find other corners in △ABC?
(2) It is known that in Rt△DEF, ∠ E = 90, EF=5 and ∠ F = 60. Can you find other edges and corners in △DEF?
(3) It is known that in Rt△ABC, ∠ C = 90, ∠ A = 30 and ∠ B = 60. Can you find other edges in △ABC? If you can find it, write the solution process.
(Show more questions in the exploration, and be precise and thorough; Multi-faceted and multi-angle discussion)
This design gives full play to the main role of students, and students participate in the exploration of problems, which greatly stimulates students' interest in knowledge and makes them feel and understand the charm of mathematical thinking methods while learning knowledge.
3. Refine and summarize mathematical thinking methods in summary and review.
Mathematical thinking method runs through the knowledge points of the whole middle school mathematics textbook and is dissolved in the mathematical knowledge system in a hidden way. In order to make students internalize this idea into their own opinions and apply it to solving problems, we must try our best to externalize the mathematical thinking methods expressed by various knowledge, which is in line with the trend of future mathematical education reform.
As a teacher, first of all, we should make clear the mathematical thinking method embodied in the textbook and its relationship with mathematics-related knowledge, and summarize it in time. In specific teaching activities, we should reveal mathematical thinking methods in an appropriate way, make them superficial, let students really understand and master them, and enhance their awareness of applying mathematical thinking methods.
Case 4:
Summary and thinking of chapter 7 of the seventh edition of Su Ke Edition
(1) Read the "specialization" on page 32 of the textbook. What mathematical thinking methods have you learned from it?
(2) What other important mathematical thinking methods have you learned in the learning process of this chapter? Give examples.
(3) Group cooperation to explore the diagonal number of N-polygon.
Not only in the review of unit knowledge, we should pay attention to guiding students to summarize and summarize the mathematical thinking methods contained in chapter knowledge, but also in the evaluation of exercises, we should not focus on the topics. Teaching "fish" is more important than teaching "fish". Therefore, we should extract this way of thinking from potential exercises, excavate its profound connotation, make it superficial, make students easily grasp the knowledge about mathematical thinking methods, and make this "knowledge" digest and absorb into mathematical thinking with "personality", and gradually form the ability to guide thinking activities with mathematical thinking methods.
Case 5:
As shown in the figure, the ray DE is known to intersect the axis and the axis at points and points, respectively. The moving point starts from the point and moves at a speed of 65,438+0 unit length/second along the axial direction at a constant speed to the left. At the same time, the moving point P starts from point D, and also moves in the direction of ray DE at a speed of 65,438+0 unit length/second. Let the movement time be seconds.
(1) Please use the included algebraic expressions to represent the coordinates of point C and point P respectively;
(2) With point C as the center and a unit length as the radius, the axis intersects with point A and point B (point A is on the left side of point B) to connect PA and Pb.
(1) When it has something in common with ray DE, the value range is;
② When it is an isosceles triangle, the value of.
Train of thought analysis and enlightenment
1. It is the basis of solving the problem to express the coordinates of points A, B, C and P and the length of the line segment with a formula containing T. It is beneficial to solve the problem by listing the coordinates of these points and the length of the line segment one by one.
2.⊙C and ray DE have one thing in common: A and D coincide, and ⊙ C is tangent to ray DE.
3. According to the waist equation, the existence of isosceles triangle PAB is discussed in three cases. When discussing with geometric method, the three cases have their own particularity, and there are two cases of AB=AP.
4. The existence of isosceles triangle PAB is discussed by algebraic method. Be careful when using the coordinates of points A, B and P to represent a three-sided square.
Omitting the process of solving problems;
Reflection:
What important mathematical thinking methods have you learned from solving this problem? (the idea of movement and change, the idea of combining numbers with shapes, the idea of classification, the idea of transformation)
Of course, in order for students to truly have personalized mathematical thinking methods, there must be a process of repeated training and continuous improvement. This requires our teachers to practice boldly and persevere in teaching, and integrate mathematical thinking methods into the usual teaching, so that students can truly form individual thinking activities, thus improving their own mathematical literacy in an all-round way.