Mathematical thinking method is a general method to analyze and deal with mathematical problems.
The basic idea of RMI method is that when problem A is difficult to solve, problem A and it can be transformed through appropriate mapping.
Relational structure R is transformed into problem B and its relational structure R[*] which is easier to solve, and the problem is solved in relational structure R[*].
Problem b, and then inverse the obtained result to r through inverse mapping, thus obtaining the solution of problem a.
The basic content of RMI method: Let R represent the relational structure (or original image system) of a group of original images, which contains uncertainty.
Fix the original image X, let m represent a mapping, and through its function, suppose that the original image structure system R is mapped into a mapping relationship structure.
R[*], which contains the image X[*] of the unknown original image X. If there is a way to determine X[*] in R[*], then through the inverse.
The mapping is the inversion of I=M[- 1], and X is determined accordingly. It is represented by the block diagram as follows:
Attached drawings (drawings)
The steps to solve the problem with RMI method are as follows:
Relation-mapping-mapping-inversion (seeking solution).
Using RMI method, the key is to choose an "appropriate" mapping, that is, the selected mapping M is not only fixable, but also
But also a reversible mapping.
The most typical embodiment of RMI method in primary school mathematics cognition is the mutual transformation of numbers and shapes, which simplifies the complex.
A clear understanding of the infiltration of RMI thinking method in primary school mathematics is not only helpful to cultivate students' problem-solving ability, but also helpful to improve students' problem-solving ability
It is helpful to organize teaching.
(A) the use of RMI ideas to organize teaching
As far as the psychological characteristics of primary school students accepting knowledge are concerned, what they see is more impressive and easier to remember than what they hear.
In this teaching, RMI thinking method is used to transform abstract mathematics into concrete shapes, from which we can find the rules and get them again.
Draw abstract laws.
For example, the concept of multiplication can be taught by straight lines. Take 3×2 as an example, starting from 0 and dividing by vertical lines.
3 units, the demarcation point position is 3. From this point on, divided by 3 units, the vertical bar label falls on point 6 (Figure 1).
Therefore, 3+3=6 and 3×2=6. Of course, 2×3=6 can also be expressed by the same steps (Figure 2). Therefore, it can further
Explain the commutative law of multiplication.
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(B) Examples of using RMI method to solve problems
Although the name "RMI method" does not appear in primary school mathematics (even the name "drawing" does not appear),
But the application of RMI method is always reflected in the problem-solving teaching in the whole primary school stage, which can be summarized into the following three types.
Form:
1. Mapping from atlas (point set) to atlas (point set)
When studying the properties of geometric figures, a certain figure is often regarded as a known and familiar figure.
Transformation (such as symmetry, translation, rotation, expansion, etc. ), geometric transformation is a transformation from a graph set (point set) to a graph.
Mapping of shape sets (point sets).
Its thinking process is:
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Example 1. Find the area of the shaded part composed of two quarter arcs in Figure 3.
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The shaded part ① of the left rectangle can be translated to the unshaded part ② of the middle rectangle; Make the one on the right longer.
The shaded part ③ on the square is translated to the unshaded part ④ of the middle rectangle, that is, from the graph sets (point sets) ① and ③ to.
Equal product mapping of graph sets (point sets) ② and ④. In this way, the required shadow area is obtained, which is equal to the length of the middle block.
Square area:
2×4=8.
2. Mapping from real number set to atlas
With the help of the mapping between positive real numbers and geometric figures (general wired diagram, rectangular diagram, circular diagram, Wayne diagram, etc.). ),
The algebraic (arithmetic) problem is transformed into a geometric problem, and the original problem is solved by using the intuition of geometric figures.
The initial thinking process is:
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Ex. 2: When a car goes from A to B, it goes the whole distance first-the rest of the way-yes.
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Uphill, the rest are downhill. It is known that the downhill slope is 3 kilometers. Find out the distance between Party A and Party B 。
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Analysis: With the one-to-one correspondence (mapping) between positive real numbers and line segments, RMI method can be used to solve this problem.
Convert the inconspicuous quantitative relationship in the problem into a line segment relationship, as shown in Figure 4, and then deduce the original object according to the line segment relationship shown.
The quantitative relationship in the problem, so as to establish the formula.
four
Set the whole process as "1" and the remaining corresponding score as1-; And the corresponding 3 km score.
five
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The score is: (1-) × (1-).
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4 7
Comprehensive formula: 3 ÷ [(1-) × (1-)] = 50 (km)
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Through this example, we can see that in the teaching of application problems in primary schools, it is necessary to strengthen the "translation" of the number of application problems related to graphics.
Practice (such as line segments), so that students can accurately and clearly show the quantitative relationship in the questions and quickly list the calculations.
Type.
3. Mapping from real number set to real number set
In the positive-negative proportional relationship, the expression between two quantities is the mapping from real number set to real number set.
When applying problems, quantity is often solved by transformation and substitution, and it can also be understood as the mapping from real number set to real number set.
Its thinking process is:
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A project can be completed by Party A alone for 63 days, and then by Party B alone for 28 days. If both parties make joint efforts, it will take 48 days.
A few days. Now A does it alone for 42 days, and then B does it alone. So, how many days does B need to do?
As we all know, in a project, A completed 63 days' work and B completed 28 days' work. The sum of them is equivalent to A and B completing 48 days' work.
Therefore, the workload of 63-48= 15 days is equivalent to
20 4 B does 48-28=20 days' work, then A's 1 day's work is equivalent to B-=- days.
15 3
four
Workload (i.e. mapping: A does X days → B does -x days).
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Now Party A will do it for 42 days, and then Party B will complete the problem for a few days alone. Compare the 48 days between Party A and Party B:
48-42=6 (days), the workload completed by Party A in these 6 days will be completed by Party B, and Party B will
4 needs 6×-=8 (days), so B needs to do 48+8=56 (days).
three
To sum up, RMI method is widely used, and when dealing with problems, problems can often be transformed from unknown fields to known fields.
Transformation has the function of changing the difficult to the easy and simplifying the complex, which is very effective in improving students' thinking ability. Therefore,
We should pay enough attention to mathematics teaching in primary schools.
Update 1: I really can't stand it.
Problems with the existing system
Update 2: I don't have a satisfactory answer, but first, I don't w