Graphics and geometry are one of the important modules in junior high school mathematics teaching. In our geometry teaching, there are many ways for teachers to cultivate students' reasoning ability through geometry proof. The revision of the new curriculum standard increases geometric intuition, which makes me feel that it is a long way to go to cultivate students' spatial concept, geometric intuition and reasoning ability in geometry teaching. Let's talk about some ideas:
First, on the cultivation of students' concept of space
1. Mathematics comes from life and serves life. The course content of graphic geometry in junior high school includes intersecting lines, parallel lines, triangles, quadrilaterals and circles, which are common basic graphics in life. Therefore, in the usual teaching, especially in the concept class, students are often asked such questions: Please give examples of triangles, quadrilaterals, circles and other figures you have encountered in your life. In particular, the first chapter of "Understanding Graphics" in grade seven makes students feel close to life, full of curiosity and freshness, which greatly stimulates their interest in learning. At the same time, it takes a long time for students to learn a new graphic, so that they can actively abstract geometric graphics from the real world and improve their perception of geometric graphics.
Starting from the reality of students' life, we should create certain mathematical life situations to guide students to perceive and understand physical objects, to guide students to explore the characteristics of graphics in the process of touching, measuring and discussing, and to make students establish models in their minds. Students' spatial knowledge comes from rich realistic prototypes and is closely related to real life. These rich prototypes in real life are valuable resources for developing students' spatial imagination. Therefore, in teaching, we should link spatial knowledge with real life, guide students to often use the characteristics of graphics to imagine, solve various practical problems in life, develop students' spatial imagination, and thus develop students' spatial concept.
2. Teach students to read pictures, cultivate a sense of pictures, and let students draw from time to time. In teaching, summarize some basic figures, such as adding bisectors between parallel lines to get isosceles triangles. This is especially true for first-year students.
Geometric intuition refers to using graphics to describe geometric or other mathematical problems, explore ways to solve problems, and predict results. Geometric intuitive ability mainly includes spatial imagination, intuitive insight and graphic language thinking ability. Geometric intuition not only plays an irreplaceable role in the study of graphics and geometry, but also runs through the whole mathematics learning process. Let's talk about my superficial views on cultivating students' geometric intuitive ability.
1. Cultivate students' spatial imagination intuitively with geometry.
Pay attention to students' basic life experience in teaching, guide students to connect their feelings about graphics with relevant knowledge in life, and let students actively participate in learning. For example, in the teaching of "straight line and line segment", I give students an intuitive understanding through a set of pictures, which leads to the straight line, so that students can easily find the characteristics of the straight line, especially the straight line is an idealized concept, and the geometric intuitive feeling is more important. Learning intuitive geometry, as the book says, adopts concrete and practical activities that students like, such as taking a look, folding, cutting, spelling, swinging, measuring and drawing. Through personal touch, observation, measurement, production and experiment, we can guide students to cooperate with vision, hearing, touch and kinesthetic sense, cultivate students' spatial imagination, and enable students to master graphic characteristics and form space.
2. Pay attention to the role of models and let students participate in model making.
The new curriculum standard emphasizes the intuition of geometric learning in geometric mathematics and the role of objects and models in geometric learning. After class, students can make their own three-dimensional geometric models and do them themselves, so as to feel the characteristics of spatial geometric figures more directly.
For example, in the section on teaching the nature of parallelogram, I asked students to go home and make parallelogram molds according to the concept of parallelogram. In the process of modeling, students have deepened their understanding of concepts and laid a good intuitive impression for studying parallelogram properties in the future.
3. Make full use of geometric intuition to cultivate students' ability to combine numbers and shapes.
When learning the image of proportional function, students are first guided to draw the image representing proportional function by tracing points. In the process of tracking points, guide students to compare the tracked points with the data in the table, so that students can initially understand the actual meaning of each point on the image. Then, through observation, students can find that the traced points are exactly on a straight line, clearly understand the characteristics of the image of the proportional function, and further understand the changing law of the two quantities expanding or contracting at the same time with the help of intuitive images. After drawing the image, we can further understand the actual meaning of any point on the image and preliminarily understand the practical application of the proportional function image. Through the transformation between the image of proportional function and the relationship of proportional function, we can deepen our understanding of proportional function. .
When it comes to the cultivation of reasoning ability, we often focus on the proof of geometry problems, which is obviously not comprehensive. In fact, reasoning should include rational reasoning and deductive reasoning, and the cultivation of rational reasoning and deductive reasoning ability. Graphics and geometry are very important fields, but they are not the only fields, and are reflected in many fields. The acquisition of regular formulas in algebra can also go through the process from perceptual reasoning to deductive reasoning, including statistical knowledge, and can also cultivate students' reasoning ability.
For the cultivation of rational reasoning, we can set up a good problem scenario, give him a very open space and let him feel the value and significance of rational reasoning. For example, when studying the midline theorem in triangles, we may encounter such a problem: draw an arbitrary quadrilateral and connect the midpoints of the four sides of this quadrilateral to get a figure we call a midpoint quadrilateral. With the same material, if our teacher asks students to prove that the midpoint quadrilateral is a parallelogram, he will soon transition to deductive reasoning; But if we can ask a more open question, students, look at our newly acquired quadrilateral. What characteristics do you think of its shape? What kind of quadrangle might it be? That student may have to think through a series of means such as intuitive observation, measurement and guess. This problem is not as superficial as some problems. There is indeed a certain space for thinking, and it really needs deep thinking. Only through observation, measurement and imagination can we guess that it may be a parallelogram, and this process is more real. With this process, we then ask why it is a parallelogram. By connecting diagonal auxiliary lines and constructing the center line of triangle, this problem is proved step by step. Of course, there is more than one such example, so we should dig more.
In the study of algebra, in fact, reasoning ability can also be cultivated, such as the comparison of algebraic values, that is, if you want to prove ab, you only need to prove a-b0. Through this form of training, students' reasoning ability can also be cultivated.
For the same problem, if it is directly required to prove that the square difference between two odd numbers is a multiple of 8, the result seems to be the same. But if we set questions as before, the student will get more than this conclusion. He will experience observation conjecture and give a case to support his conjecture, and then try to express the law with mathematical symbols and further prove it through algebraic operation. This example inspires us to turn some purely deductive problems into deductive and rational reasoning processes, and cultivate students' ability in this process.
Therefore, in the usual teaching process, we integrate reasoning ability into every field and every class, and cultivate students' reasoning ability from multiple angles and in all directions.