The formation process of mathematical concepts is often to provide students with rich perceptual materials through some familiar examples of production and life, objects and models, so that students can observe the similarities and differences of objects, analyze, compare, summarize and abstract the essential attributes of objects, thus forming concepts. Therefore, concept teaching should not simply give a definition, but guide students to feel and understand the mathematical ideas implied in the process of concept formation. For example, when learning the concept of "inverse number" in seventh grade, students are guided to draw the concept of "inverse number" by analyzing the characteristics of 3 and -3: "There are only two numbers with different symbols". In order to deepen the understanding, draw these two numbers on the number axis, or define the reciprocal as: the two numbers represented by two points with the same distance from the origin on both sides of the number axis are opposite. In this way, it is easier for students to understand by using the mathematical thought of combining numbers with shapes to carry out comparative teaching. Another example: eighth grade students learn the definition of "rectangle"
By observing the * * * similarity between rectangle and parallelogram, guide students to get the concept of rectangle through analysis and comparison: "parallelogram with right angle". At the same time, in order to deepen the understanding of the concept, four pieces of wood are used to make a parallelogram movable wooden frame, which stands upright on the ground and gently pushes point D. It can be found that the angle has changed, but the shape of the parallelogram still exists. So you can get parallelogram+a right angle = rectangle.
In the teaching of mathematical concepts, we must find out the regular things from the graphics, so that perceptual knowledge can be abstracted into rational knowledge in mathematical language, so that students can correctly understand and firmly grasp the concepts. Therefore, the mathematical thought of combining numbers and shapes can not only improve students' ability to transform numbers and shapes, but also improve students' ability to transfer their thinking. Hua once said: "If you count less, you will have less intuition, and if you count less, it will be difficult to be nuanced." Through in-depth observation, association, thinking from form to number, thinking from number to shape, intuition is summarized by graphic intuition. Of course, not all mathematical concepts can be understood by graphics, and specific problems are analyzed in detail.
Second, in the teaching of difficult knowledge, cultivate mathematical thinking methods.
As the key and difficult point, their significance and difficulty are self-evident, but how to reduce the learning difficulty and let students master and use it better? Therefore, in the teaching of key and difficult knowledge, we should not draw conclusions too early, but guide students to participate in the exploration, discovery and deduction of knowledge points. Find out the causal relationship, understand its relationship with other knowledge, and let students experience the applied mathematical ideas and methods.
For example, in the ninth stage, in the same circle, the size of any circumferential angle subtended by an arc is equal to half of the central angle subtended by the arc. To test this conjecture, you can fold the circle in half so that the crease passes through the vertex of the center of the circle and the corner of the circle. At this time, three situations may occur:
(1) the crease is the edge of the fillet, (2) the crease is inside the fillet, and (3) the crease is outside the fillet.
When verifying this property, students are guided to analyze, discuss and explore in three situations, so as to master the derivation process of this property and let students know its essence and generality more deeply through the mathematical thought of classification.
Another example is the ninth grade "the root of a quadratic equation", which can be divided into three situations:
(1) equation has two unequal real roots (2) equation has two equal real roots (3) equation without real roots can be generalized to parabola)
(3) The parabola has no intersection with the X axis.
By classifying them, students can easily accept knowledge, so as to better master and use knowledge. Using the mathematical idea of classified discussion can help students think, analyze, discuss and demonstrate problems comprehensively and rigorously, and make the ways and methods to solve problems perfect and reasonable.
Therefore, it is necessary to cultivate the idea of classification in the teaching of some key and difficult knowledge. Hua, a famous mathematician, said: "It's best to look for information in the mathematician's wastebasket when learning mathematics, and don't just read the conclusions in the book." Third, in the teaching of solving problems in mathematics, experience the mathematical thinking method.
There are countless mathematical problems, and the problems are divergent, so the number of exercises is Qian Qian, but the mathematical thinking method contained in the problems is eternal forever. It is the essence of mathematics, an effective means to solve problems and a magic weapon to win. Therefore, in the teaching of mathematical problem solving, we should not just list the solutions directly, but pay attention to summarizing the guiding role of mathematical thinking methods in solving problems.
The thought of conversion is an important mathematical thought. It is a way of thinking that transforms unfamiliar or difficult-to-solve problems into familiar or solved or easy-to-solve problems by some means, so that the original problems can be solved. This mathematical idea is embodied in solving mathematical problems, that is, transforming the original problems into familiar or solved or easily solved problems. At this point, the problem-solving process is a process of continuous transformation.