Improving the ability of primary school students to analyze and solve math application problems has always been the focus of all our math teachers. Although many of our math teachers spend a lot of time and energy on the teaching of application problems, there are still many students whose ability to analyze and solve application problems has not been effectively improved. What is the reason? To this end, I conducted a small survey of students at different levels in my class:
The clarity of students' thinking and mathematical thinking methods when doing application problems.
top student
Solution thinking
The resolution is 99%.
Mathematical thinking method
The clarity is 98%.
solve problems
The accuracy rate is 98%.
Students' learning excitement is 98℅ ordinary students.
Solution thinking
The clarity is 87%.
Mathematical thinking method
The clarity is 85%.
solve problems
The accuracy rate is 86%.
85℅ Learning excitement of students with learning difficulties
Solution thinking
The clarity is 42%.
Mathematical thinking method
The clarity is 39%.
solve problems
The accuracy rate is 32%.
The students' learning excitement is 28%.
As can be seen from the table, students with high clarity in mathematical thinking methods have clear thinking and high accuracy in solving problems. Students are naturally interested in learning. On the other hand, students with low clarity of mathematical thinking methods will have vague or even no problem-solving ideas, and the accuracy of solving application problems will naturally be low, and students' interest in learning will of course be quite low. From this point of view, students' low ability to analyze and solve applied problems has a lot to do with their lack of some mathematical thinking methods. It is important for students to learn mathematics knowledge, but it is precisely because many students have only mastered some explicit knowledge to solve application problems and have not internalized their own mathematical thinking methods that there are always deviations in the process of solving application problems, which reduces the efficiency of our teachers' application problems teaching. Mathematical thinking method is a tacit knowledge system in mathematics teaching. How can our teachers help students internalize some common mathematical thinking methods while teaching students knowledge, so as to improve their ability to analyze and solve application problems? Below, I will talk about my superficial views based on my teaching practice.
First, in the thinking method of combining numbers and shapes.
In daily teaching, we often find that some mathematical problems expounded in language are boring and difficult for students to analyze and understand, especially those with poor spatial concepts. With the help of some line graphs, point graphs, model graphs, tree graphs, rectangular (or square) area graphs, set graphs and direct views, students can correctly understand the quantitative relationship, which will make the questions concise, vivid and intuitive. This idea of making full use of "shape" to express a certain quantitative relationship vividly and thus solve mathematical problems can be called the idea of combining numbers with shapes. Let's experience the benefits of solving problems with the idea of combining numbers with shapes.
Case "Honghong drank a glass of juice, drank half a cup for the first time, drank the remaining half a cup for the second time, drank the remaining half a cup for the third time and drank the remaining half a cup for the fourth time. How many cups of juice did she drink four times? How many minutes are left? "
If students are directly asked to do this problem continuously, most students will be at a loss, but if such a rectangular diagram is cited and combined with graphics, students can easily solve it. (The attached figure is as follows) (Rectangular diagram)
The first time I drank this glass of water was 1/2.
The second time I drank this glass of water was 1/4.
Drinking this glass of water for the third time is 1/8.
The fourth time I drank this glass of water was116.
From this figure, we can quickly calculate that the red wine1/2+1/4+1/8+116 =15/16.
In addition, there are some engineering problems, travel problems, tree planting problems, fractional multiplication and division application problems and so on. We can use the idea of combining numbers and shapes to make problems from difficult to easy, arouse the enthusiasm of primary school students to actively participate in learning, and at the same time give play to their creative thinking potential to improve their ability to analyze and solve applied problems.
Second, in the change of thinking mode.
In mathematics teaching, the idea of transformation is actually to simplify a practical problem into a mathematical problem or a more complicated problem into a simpler one through some transformation. Through transformation, we can communicate the relationship between knowledge and make the solution more flexible. It can be said that reduction is also a common and very important mathematical thinking method when solving mathematical problems.
Case 1 Dad Wang cut a rope with the length of 1/4. If 14m is cut again, the cutting length is 3/5 of the uncut length. How long is this rope?
After reading this problem, we will find that if we use equations to solve it, although the thinking is smooth, it will be very troublesome to solve the equations; If we use arithmetic to solve the problem, we will find that although the unit of "1" in the two conditions representing the score in the problem is the length of the uncut rope, the lengths of the two uncut ropes are not the same. What should we do to solve this problem? We can use the mathematical idea of transformation to transform them into the same standard quantity, that is, "the cutting length is uncut 1/4" into "the cutting length is the full length (11+4) =1/5", and the same is true. At this time, "1/5" and "3/8" both represent the total length of the rope, so the corresponding score 14m can be converted into: (3/8- 1/5). At this point, we can calculate the total length of the rope as: 14. If we students don't establish this transformed mathematical thought in our minds, I'm afraid this problem is really difficult for some students!
Case 2 A choir consists of 36 actors and 30 actresses.
Question: 1. How much do actresses compare with actors?
2. How many actors are there compared with actresses?
3. How many girls are there in the chorus?
4. What is the number of actors in the chorus?
5. How many points is the actress less than the actor?
6. How many more actors are there than actresses?
Although the problem is constantly changing, it can eventually be transformed into a mathematical problem that "a fraction of one number is a fraction of another number", which not only permeates the transformation idea, but also permeates the basic mathematical thinking methods such as comparison and correspondence, making the problem easier.
In addition, the application problems of integer multiplication and division are intrinsically related to the application problems of fractional and percentage multiplication and division, as well as the application problems of fractional and ratio distribution, which can be transformed into each other, making the solution of application problems more flexible and simple, thus better promoting the development of students' thinking ability.
Third, in terms of comparative thinking methods.
We know that students' ability to analyze and solve application problems can be improved by analyzing, comparing and synthesizing various seemingly similar but different questions and then determining their similarities and differences. This mathematical idea of analysis and comparison is often used in the teaching of applied problems, especially in the middle and senior grades of primary schools.
Case 1. There are two kinds of fruit trees in the orchard: apple trees and pear trees, of which 1300 is apple trees, accounting for fruit trees.
65% trees. How many fruit trees are there in the orchard? 2. There are two kinds of fruit trees in the orchard: apple tree and pear tree, of which 1300 is apple tree and 35% is pear tree. How many fruit trees are there in the orchard?
To solve these two problems, we must give full play to the value of comparison, find out their similarities and differences, and deepen them.
Understand different quantitative relationships and solve problems correctly, otherwise the ability to analyze and solve application problems will not be effectively improved.
Fourth, in terms of thinking and modeling methods.
Mathematical modeling refers to the whole process of finding out the mathematical framework to solve this problem, finding the solution of the model and verifying it under certain assumptions. In primary school mathematics, algebraic expressions, relations, equations, charts, graphs and so on established by mathematical symbols such as letters and numbers are all mathematical models. Model thinking plays an important role in compulsory education mathematics teaching. It can not only make students realize that mathematics is not only an abstract subject, but also make students feel the beauty of using mathematical modeling thinking to solve practical problems, which can better improve learning efficiency and make students like mathematics more.
Case 1. Two trains run in opposite directions from both parties at the same time. The local train is 70km | hour, and the express train is 90km | hour. 3.5 hours later, the two cars met. Excuse me, how far is the distance between A and B?
The tallest animal in the world is giraffe. Giraffes are 5 meters tall, two-thirds taller than elephants. How tall is this elephant?
In the first question, our teachers can guide students to solve it easily by using the basic model of encounter problem "speed and x time = total distance". In the second question, we can guide students to build such a mathematical model (that is, the quantitative relationship): the height of elephant × (1+2 ∕ 3) = the height of giraffe, which can be broken through equation method or division, otherwise it is easy for individual students to list an opposite operation.
Above, I mainly introduced the application of the idea of combining numbers and shapes, the idea of transformation, the idea of comparison and the idea of mathematical modeling in improving students' ability to analyze and solve application problems. In fact, in actual teaching, there are many ideas, such as set thought, symbol thought, correspondence thought, classification thought, induction thought, statistics thought and so on. It also plays an important role in our application problem teaching. These valuable mathematical ideas are interrelated, interdependent and intertwined. We math teachers should carefully design all aspects of teaching, persistently and imperceptibly guide students to understand and master mathematical thinking methods in the process of actively exploring mathematical knowledge, and strive to internalize all mathematical thinking methods into their own scientific mathematical thinking methods, which will play a good escort role in improving students' ability to analyze and solve application problems!