? The most basic concept of mathematics is essential and general. It is the navigator for students to learn knowledge and the golden key for students' thinking activities. To cultivate students' mathematical thinking ability, we should take basic concepts as the core, guide students to grasp the connection point between old and new knowledge, grasp the new starting point of logical reasoning, and study the process of problem discovery, knowledge occurrence, concept formation, conclusion derivation and law revelation. This paper studies how the existing knowledge becomes the basis of the subsequent knowledge, so that the knowledge network itself reflects the "order" of knowledge imparting and ability training, so that the previous and subsequent knowledge can be mutually inclusive and naturally deduced, and a logical train of thought from known to unknown can be provided for students in thinking, thus forming a knowledge network with high thinking value. A knowledge network that develops, moves, expands and diverges. ?
? From more than 540 concepts of primary school mathematics, Mr. Ma Xinlan grasped a dozen basic and decisive concepts as the nodes of the knowledge network, and put them at the core of teaching, so as to dominate other mathematical concepts, thus determining the concept subordination in the knowledge network. In the teaching process, the whole knowledge system is commanded by mastering concepts, relationships and structures.
The first mathematical concept and the most important core concept of primary school students is "harmony". Why is this? It can be discussed from two aspects: life experience and the position of "harmony" in the structure of mathematical knowledge:
First, taking the concept of "harmony" as the first basic concept of primary school students conforms to children's existing knowledge, experience and intelligence level.
? Combining the teaching content of primary school mathematics, cultivating students' abstract thinking ability is the primary task of primary school mathematics teaching. The so-called abstract thinking refers to the way and method of thinking that people use concepts, judgments and reasoning processes. The development of primary school students' abstract thinking is in the primary stage. Judging from the overall knowledge of primary school mathematics, children who have just entered school have almost no mathematical concepts necessary for accumulating abstract thinking. So, which concept should we help students master first? It depends on their existing knowledge, experience and intelligence. And the concept of "harmony" meets this requirement.
? The concept of "harmony" essentially embodies the relationship between part and whole, and we can study "harmony" by studying the relationship between part and whole. Because of the experience of "dividing" and "combining" things in daily life, primary school students have accumulated a lot of knowledge and experience about "part" and "whole" before entering school. In teaching, teachers guide children to start with "separation and integration" in life, sort out and improve their own life experiences, and let children understand and master the meaning and relationship of "part and whole". Moreover, life and mathematics correspond. The combination of life into mathematical language is "+",and the division of life into mathematical language is "-". In teaching, teachers should effectively use transformation to transform new knowledge into old knowledge, and the most primitive knowledge is children's life experience. Like, separation and combination in life. From the perspective of children's learning, "part" and "whole" emphasize "relationship" rather than "harmony". In other words, using "whole" and "part" to express "harmony" makes knowledge more concrete, facilitates children to realize the relationship between related knowledge, and can greatly improve the efficiency of learning. For example, combining two parts is a whole, and removing one part from this whole is another part. These well-understood life experiences correspond to a series of mathematical languages-addend+addend = sum, one addend = and-another addend; Negative-negative = difference, negative-difference = negative, negative+difference = negative.
Second, the core position of "harmony" in the knowledge structure of primary school mathematics.
Judging from the overall knowledge structure of primary school mathematics, almost all the problems involved in primary school mathematics are seeking "whole" or "part", which belongs to the concept of "harmony". It's just that the scope of the number studied is expanding, and the method of finding "whole" and "part" is expanding. Let's introduce the order of this part of knowledge according to the "number operation" of each grade:
1. Grade one focuses on the teaching of addition and subtraction and the relationship between "part" and "whole".
When the child reaches a certain abstract thinking, you can give a sketch like this:
2. The second grade mainly focuses on the relationship between size and number, multiplication and division, and the relationship between share and total.
? The so-called relationship between size and number can actually be transformed into first-class knowledge, which can be regarded as the relationship between part and whole when comparing two numbers. The analysis shows that there are two basic situations when comparing two quantities: one is that the two quantities are the same, and the other is that the two quantities are different, resulting in large numbers and decimals. Line charts can be used to represent different situations in this way:
These two quantities seem to be unrelated, but they are not. Compared with decimals, large numbers are naturally divided into two parts: one is as much as decimals, and the other is more than decimals, that is, their differences.
The relationship between size and number is to study the relationship between large numbers, decimals and their differences. At this point, if we hide the decimal, it is easy to see it by observing the large number alone. This line chart shows the relationship between "fractional part", "difference" and "large number".
From this point of view, as long as we use the concept of "as many" to convert decimals into parts with as many decimals in a large number, the relationship between size and number naturally converts the relationship between parts and the whole.
When the whole is divided into several parts on average, and each part is the same, the relationship between the parts and the whole is transformed into the relationship between the parts. Therefore, "share" is a special form in the relationship between "part" and "whole"; The study of multiplication and division is the study of the calculation method of finding the whole and the part under this special situation.
3. The main content of the third grade study is the relationship between "times" and times. If the special forms of "part" and "whole" are transformed into the comparison of two quantities, the concept of "multiple" is formed with the smaller number as the standard. When comparing two quantities, the multiple relation can be regarded as "total relation of shares" or a problem of seeking the whole and the part.
4. An important knowledge of the fourth grade research is the meaning of the score. The difference from "multiple" is that it is based on the larger number, and the smaller number is several times the larger number.
5. The quantitative relationship involved in the mathematics knowledge of Grade 456 is still learned in the first three years, but the range of numbers has expanded from integers to decimals and fractions (percentages).
From the above analysis, we can clearly see that the concept of "harmony" dominates the internal relations of all the one-step application problems in primary schools and is the basis of most knowledge of primary mathematics. Moreover, with the concept of "harmony" as the core, the relationship between knowledge and the expansion of knowledge presents a strict logical relationship, which is a rich resource for cultivating students' abstract thinking ability.