I. Complementarity of content
Some concepts in higher mathematics are abstractions of some quantities in elementary mathematics, and the content of elementary mathematics is an example of abstract concepts in higher mathematics. If we explain the content of primary school mathematics from an abstract height, then the content of primary school mathematics will be orderly and complete. For example, addition, subtraction, multiplication and division is one of the main teaching contents in primary school mathematics, but it is only a few special cases of mapping (algebraic operation) in higher mathematics. Without these examples of elementary school mathematics, it is impossible to understand and abstract the general concept of algebraic operation; If we master the general concepts of algebraic operations and then explain addition, subtraction, multiplication and division, we will fully explain these concepts. Generally speaking, advanced mathematics and elementary mathematics complement each other in the following four aspects.
1. Personal and General
There is the calculation of average in primary school mathematics, and the average in advanced mathematics is a special case of mathematical expectation. If we understand the average from the perspective of mathematical expectation, the teacher will emphasize the difference between the average and each number, and the students will know that the average math score of the whole class and the score of each student are scores, but the meaning is completely different. On the other hand, if students can only calculate the average score, but can't distinguish the average score from each student's score, then students just do some exercises of four operations. This not only can not activate students' thinking, but also is not conducive to improving students' interest in learning. Another example is the problem of finding the positive divisor of natural numbers in primary school mathematics, which is the application of the basic theorem of algebra in higher mathematics, and the calculation formula of finding the divisor of positive integers is also demonstrated in higher mathematics.
2. Finite and infinite
In primary school mathematics, problems are generally discussed in a limited scope, and some problems need to be explained from the perspective of higher mathematics. For example, the understanding of logarithm in primary school mathematics is very simple, but students must understand "number" through "equivalence" and compare the number of elements between two groups. This is because the number method is the basic method to study countable set and uncountable set in higher mathematics. The square rule of "equivalence" is the basic method to compare the number of elements in two sets (finite set and infinite set). For another example, in primary school mathematics, the conclusion that "natural numbers are infinite" can only be explained from the viewpoint of limit, and students can correctly understand this conclusion. On the contrary, if teachers do not have a solid foundation in advanced mathematics, but adopt some incorrect methods to explain it, it will not only help students understand the conclusion that "natural numbers are infinite" accurately, but also affect students' understanding of the concept of limit in the future. For another example, in primary school mathematics, the reciprocity problem of infinite cycle decimal and fraction is the application of series concept in higher mathematics. By explaining the relationship between "0.9", "0.99 … 9" and "1" in teaching, students can experience the concept of limit again.
3. Stationary and moving
Many concepts in primary school mathematics are static if only the results are emphasized. For example, expression 2+3 only discusses how much of its sum is static. If you analyze this expression, it is moving. This is because: if 2 = 3- 1, 3 = 1+2, … then this expression becomes: 3-1+kloc-0/+2, …; If 2 and 3 represent the sum of the number of people in Room 2 and Room 3 respectively, then the meaning of this expression is different. Through this change, students' understanding of mathematical concepts is more complete, and this change is the embryonic form of algebraic thought. Algebraic thinking is one of the most basic ideas in learning mathematics.
4. Calculation and prediction
There is a kind of problem in primary school mathematics, that is, knowing the present value and finding the initial value. For example, the personnel in workshops A, B and C have been adjusted three times. For the first time, workshop C did not move, and one of the two workshops A and B transferred eight people to another workshop; For the second time, workshop B did not move, and eight people moved from one workshop in garages A and C to another. The third time, Workshop A stopped moving, and one of Workshop B and Workshop C transferred seven people to another workshop. After three adjustments, there are 7 people in Workshop A, 2 people in Workshop B/KLOC-0 and 4 people in Workshop C. Ask how many people are in each workshop.
This problem will be complicated if it is calculated in the order of adjustment, but it is very simple to calculate it with a list.
Renshujia workshop b workshop c workshop
7 12 4 after the third adjustment
After the second adjustment 7 5 1 1
After the first adjustment 15 5 3
The result is 7 13 3.
The way to solve this kind of problem is to use list (or drawing), which is generally called backward method. In advanced mathematics, past and present values are more known. Seeking the future, this kind of problem is called prediction, and it is also solved by statistical method through list (or drawing).