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How to abstract mathematical concepts from life examples
Junior high school students' acceptance of some abstract mathematical concepts often needs to start from concrete examples and show their dependence on concrete materials. Although mathematics is abstract, it has extensive concreteness. In teaching, teachers can build a model with very specific materials, list enough examples, let students observe and guess for themselves, then test, draw and do, and finally draw a conclusion. Compulsory education textbooks attach great importance to the teaching function of examples. For example, by enumerating many examples such as temperature, altitude, reservoir water level, object movement, cargo weight and size, on the basis of students' perception of "quantities with opposite meanings", the concepts of positive numbers and negative numbers are introduced. In this way, students are willing to accept and easy to accept. For example, by comparing with fractional operation, students can understand and master fractional operation. Piaget said: "The disadvantage of traditional mathematics is that it is often explained orally, rather than teaching mathematics from the actual operation." Let students practice the operation, which is put forward in view of the limitation of students' thinking of "dependence on specific materials". Experienced teachers will let students cut two corners of a triangular piece of paper by hand and put them together at the vertex of the third corner, thus abstracting the theorem of the sum of the internal angles of a triangle. Even for some abstract problems that we have no definite conclusion to explore, as long as we can "operate" and try, the problem may become concrete and simple. There is a question of "think about it" in the second volume of junior high school geometry in compulsory education: can three matches make a triangle and six matches make four triangles? Using "matches" can easily stimulate students to operate. After the desktop and space operation experiments, students have a real feeling, and it is not difficult to get the experimental results. In mathematics teaching, we will encounter some concepts and laws. These concepts and laws do not necessarily come from specific examples, but are the result of operation and deduction of existing knowledge. Even so, teachers should choose examples as a supplement to understand abstract concepts and laws. It can be seen that starting with concrete examples is the need of students' thinking characteristics, and it also conforms to the basic relationship between abstraction and concreteness, which is conducive to students' understanding of abstract conclusions. Specifically, it is not to accommodate the limitations of students' thinking, but has its positive significance.