First, beautiful.
This is mainly because mathematical objects are symmetrical, harmonious and concise in form, which brings beauty and beautiful feelings to people's senses.
Geometry often gives people an intuitive aesthetic image. The geometric figure "circle" is an all-round symmetrical figure, which is beautiful and balanced and beyond reproach. Commonly used geometric figures such as regular triangle and pentagram are loved by people because of their symmetry and harmony. There have been many successful experiences in cultivating the aesthetic ability of geometric figures, such as: building a flower bed on a rectangular field, making its area only half of the field, requiring beautiful design. This is a typical subject combining mathematics with art. In the teaching of solid geometry, students are required to make a sports trophy with solid geometric figures such as cylinder, platform, cone, sphere, cylinder and cone, and write the equations of each part. The students' homework is dazzling and beautiful. Some teachers asked students to collect geometric patterns of "panes" in ancient buildings in China, or to show and compare geometric patterns in some famous trademarks, which was very successful. It can be seen that the beauty of mathematics has gained some successful experiences in classroom teaching design. It's not difficult as long as you put your heart into it.
Aesthetic comprehension can be found everywhere in mathematics teaching, not only in geometry, but also in arithmetic and algebra. For example, from n different elements, the total number of all different arrangements of m elements is arbitrarily taken out. This large language is finally condensed into a concise mathematical symbol p, where P→ indicates arrangement, m → indicates the number of elements taken out, and n → indicates the total number of elements. From the structural analysis of the previous symbols themselves, it shows the inherent beauty of harmony. Another example is: triangles A, B and C, marked △ABC, △ "represents the shape characteristics of triangles in form and has formal beauty; The letters A, B and C indicate that it has three vertices, which essentially embodies its inner beauty.
These formulas and rules are very symmetrical and harmonious, and also give people a sense of beauty.
Second, beauty
Many things in mathematics can only be "beautiful" if we realize their correctness. The round structure is extremely beautiful and naturally beautiful. The ratio of the circumference to the diameter of any circle is always a constant, which is neither rational nor algebraic, but transcendental. This inherent mathematical value shows the charm of "circle" and attracts countless heroes. From Zu Chongzhi's calculation to today's computer calculation to 6 billion decimal places, the research on it is not finished.
There are many beautiful mathematical objects. For example, in the teaching of establishing the standard equation of ellipse, it is defined as | MF 1 |+MF 2 | = 2A■+2A ①. This formula is true, but before she started coming to us, we called a thousand times and urged her for a thousand times, but she still hid half of her face behind her guitar from us. Why do you choose "2c" and "2a" instead of "c" and "a" in the process of mathematics? The teacher asked: Can Equation ① be regarded as an elliptic equation? Student A: Of course! Q: Are you satisfied? A: Not satisfied! Q: Why? A: Try to simplify it.
The discovery or creation of mathematical knowledge not only reflects the quantitative relationship and spatial form of the objective world, but also stems from the pursuit of beauty. To measure the success of a theory, there are not only practical standards, logical standards, but also aesthetic standards. When a theory has not reached the realm of beauty, it must continue to improve and "create according to the law of beauty."
Teachers and students get ■ +■ =1(a > c & gt0)②
Teacher: ② It is much simpler in form than ①. Can we continue to simplify? After discussion between teachers and students, introduce B, A2-C2 = B2 (B >; 0) (2) The formula becomes ■ +■ =1(A > B>0) (3) This formula achieves the perfect unity of form, which is pleasing to the eye and wonderful. Equation ③ is also called the standard equation of ellipse. Moreover, based on the standard equation of ellipse, it is convenient to continue to study the image and properties of ellipse.
Third, wonderful.
Good feelings need to be cultivated. Teachers should give students more opportunities to innovate, explore and even discover in class, and experience the joy of discovering truth. For example, the three heights, the three median lines and the three bisectors of the inner angle of a triangle all intersect at one point. This is a beautiful and surprising conclusion. Finding it will make people feel that mathematics, especially geometry, is wonderful. Then, in teaching, don't tell the students the results first, let the students draw by themselves, and let the students discover these invisible "truths". As you can imagine, what a surprise it will be for students to discover a mathematical truth by themselves. Once you realize the beauty of mathematics, you will naturally have a sincere interest in mathematics.
Wonderful feelings often come from "unexpected" but "reasonable" things. The three heights of a triangle intersect at one point, that's all. After the two cylinders intersect vertically, the section expands, and the curve corresponding to the section line is actually a sine curve. Originally guessed that it would be an arc, but the result was "unexpected". After analysis and deduction, it is proved that it is indeed a sine curve. It turns out to be "reasonable" again, and a wonderful feeling arises spontaneously.
Everyone who likes mathematics has felt that moment: an auxiliary line makes the geometric problem that can't be started suddenly clear, a skill makes the puzzling inequality pass, and a specific "relationship-mapping-inversion" method solves the original irrelevant problem. The happiness and excitement at this time is really indescribable, and maybe it can only be summarized in one word "wonderful". This wonderful artistic conception will make people feel the cleverness of mathematics created by heaven and earth, the profundity of mathematics created by mathematicians, and the joy of mathematics learning and understanding. Only by reaching this step can students truly feel the true meaning of mathematical beauty, be attracted by mathematics, like mathematics and love mathematics.
In short, in mathematics teaching, mathematics teachers' reasonable organization, vivid language, standardized blackboard writing, incisive analysis, vivid explanation, ingenious inspiration, appropriate metaphor, strict reasoning and organic connection will surely free students from "learning mathematics is boring" under the influence of beauty. Can't this spiritual satisfaction make students like math? Therefore, teachers should embody the aesthetic principles in mathematics teaching and teaching methods as much as possible, and at the same time explore the aesthetic thoughts and aesthetic values behind them according to mathematical thoughts, so as to cultivate students' aesthetic feeling and aesthetic thinking.