Through mathematical abstraction, human beings obtain mathematical concepts and laws from the objective world and establish mathematical disciplines. Through mathematical reasoning, a large number of conclusions can be further obtained, and mathematical science can be developed. When mathematics is applied to the objective world through mathematical models, it has produced great benefits and further promoted the development of mathematical science. These three points are simply abstraction, reasoning and modeling.
This is the basic idea of mathematics, so there are many mathematical ideas, and there are many mathematical ideas under the basic idea. For example, abstract concepts such as mathematics can only be produced, such as classification concept, set concept, combination of numbers and shapes, symbol concept, symmetry concept, correspondence concept, finite and infinite concept and so on. Many ideas will come from basic ideas.
For example, the idea of mathematical reasoning can also come from inductive thinking, deductive thinking, axiomatic thinking, transformation planning thinking, ideal analogy thinking, gradual approximation thinking, substitution thinking, special general thinking and so on.
For example, ideas such as mathematical modeling can be further derived, such as simplification, quantification, function, equation, optimization, randomness, sampling statistics and so on.
For example, concepts such as classification and geometry can be deduced from mathematical abstract concepts in this way. When people observe the objective world, they analyze association from a certain angle to meet the needs of research, derive these minor non-essential factors, keep these major essential factors, and classify things according to a certain essence in an effective way, and the result of classification will produce a set.
How to distinguish the basic mathematical ideas from what we just said, there are two suggestions for teachers to think seriously. I hope the teacher should first know what is needed for the development of mathematics, because he determines the growth of mathematics. This kind of thing must be basic and important.
Abstraction is a symbolic thing, which constitutes the theme of mathematics. As I said before, we are not satisfied with solving problems one by one. Reasoning includes perceptual reasoning and deductive reasoning. We need these things to establish a scientific system, and mathematics solves practical problems under such guiding ideology. We should turn practical problems into mathematical problems and solve them by mathematical methods, which forms the most basic and important thing to promote the development of mathematics.
The second reason, I also hope that teachers can understand the difference between learning mathematics and not learning mathematics. Mathematics gives us something that other subjects don't. This thing may embody the basic idea of mathematics. What is this unique thing? The three concepts we just mentioned all have such characteristics, which is exactly what we should experience in the teaching of * *. More importantly, we should infiltrate our experience into our regular teaching and gradually help students form such an idea. A good idea can not be established by preaching, but by infiltrating students and guiding them to experience all aspects, so as to achieve such a basic goal. Moreover, this is a long-term process and cannot be achieved overnight. I just want to add that some teachers may ask, is abstraction or reasoning, including models, unique to mathematics? For example, will other disciplines have such characteristics or have the same ideas? We don't rule it out, but it is more fully reflected in mathematics. For example, physics and chemistry are abstract, but there are differences in mathematical abstraction. Including two other characteristics, we regard it as the basic idea, and I think it also reflects the difference between this discipline itself and other disciplines.
The ideological relationship between the three is also something that everyone needs to think about. They are deeply related in essence, but each has its own characteristics, so that we can better understand them.
Our teacher often talks more about mathematical methods, such as method of substitution, etc. But this mathematical method is different from mathematical thought, because it is at a lower level. This kind of mathematical thought can often be described by such adjectives: it is conceptual, comprehensive, universal, profound, general, internal and generalized. Mathematical methods can be described by these adjectives. It is operational, local, special, superficial, concrete, procedural and skillful. However, there is a relationship between the two. Mathematical thoughts should be embodied by mathematical methods, and mathematical methods often embody mathematical thoughts. Therefore, mathematics thought is the essence of mathematics teaching. Teachers must pay attention to embody and embody mathematical thoughts in teaching, so that students can understand mathematical thoughts and improve their mathematical literacy.
Teachers are sometimes a little vague in teaching and will ask questions about this issue. What are the mathematical ideas? Is mathematical method what we call mathematical thought? I hope teachers are aware of this problem. Can have a further understanding. With regard to mathematical thinking methods, it takes a process for teachers to understand and think constantly. Therefore, I also hope that teachers can think more in the teaching after * *: First, in my teaching, how to embody mathematical ideas and how to embody some mathematical ideas behind them through some specific methods, so that we can achieve our goals more smoothly.