How to treat the diversity of mathematical problem-solving methods

"Diversification of problem-solving methods" plays an important guiding role in mathematics teaching. The new version of mathematics curriculum standard advocates a brand-new teaching concept, in which "diversification of problem-solving strategies" is an interpretation of students' problem-solving methods. Advocating multi-strategy problem-solving aims to enable students to develop mathematical thinking, optimize thinking, innovate research, and let teachers implement diversified teaching of problem-solving methods. Teachers should not "teach dead" but students should pay attention to it. "Diversification of problem-solving methods" will rebuild the relationship between teachers and students, the criteria for teachers to evaluate students will become broader, and the forms for students to analyze and solve problems will be diversified, which will strengthen their concepts in the teaching process and truly make mathematics classroom teaching efficient and accurate.

Below I will make a simple exploration of "diversification of problem-solving methods" from two aspects: number and algebra, graphics and geometry.

First, implement "diversification of problem-solving methods" in number and algebra.

I often ask myself: Where does mathematics come from? Why do you want to study math? After listening to many famous teachers' lectures and attending many famous teachers' classes, I don't think mathematics should be unfathomable, especially primary school mathematics, which comes from life. We study mathematics to solve problems in life. Therefore, primary school students also have different knowledge, experience and life accumulation. It is with this or that experience that students will have their own understanding of the problem in the process of solving problems and form their own problem-solving strategies on this basis. Therefore, teachers should provide students with opportunities to explore independently, guide students to explore independently, and encourage students to observe, guess and verify from different angles and in different ways, so as to solve problems and achieve high efficiency in mathematics classroom.

Teaching example 1 When teaching the oral multiplication of a number and a number, use six bundles of sticks to lead out the topic and ask the students: How to calculate the total number of sticks? After a period of independent thinking and group communication, the students finally gave some methods:

(1) Count:

Health 1: I count one by one, ***60.

Health 2: You count too slowly. I count, 10, 20, 30 ... 60 * *.

Health 3: I count to twenty, twenty, forty, sixty, sixty.

② One plus one:10+10+10+10+10 = 60 (root).

③ Multiply by 1:

Health 1: 10× 6 = 60 (root)

Student 2: 20× 3 = 60 (root) The teacher asked: What does this 20 mean? What does 3 stand for?

Student 3: 30× 2 = 60 (root) The teacher asked: What do 30 and 2 in the formula mean?

The teacher presents the students' ideas on the blackboard one by one, so that more students can see different ways to solve this problem and develop their mathematical thinking. Under the traction of these three methods, students will think that such mathematical problems can be solved from two aspects: addition and multiplication. Of course, the teacher will ask: which of the three methods do you think is the easiest? This is also the embodiment of method optimization.

Next, the teacher can ask another question: add six sticks outside the six bundles of sticks. How many sticks are there now? Let the students think. Still solve it in many ways. This problem is actually to add six sticks on the basis of the three methods just now, and consolidate the key content of this lesson again, so that students can learn solid knowledge and achieve efficient classroom.

Teaching Example 2 There is a problem in teaching "Solving Application Problems with Equations": Red Star Primary School organized students to donate books to Hope Primary School, and a class of six students donated 78 books, more than the class of one year 12. How many books do you donate to a class a year? The teacher asked the students to solve this example in different ways. Students calculate in books, and teachers patrol and guide students with learning difficulties. Students report their ideas, and the teacher writes down on the blackboard in time:

Method 1: arithmetic method (78- 12) ÷ 2

Method 2: Calculate and solve by equation: Assuming that X books are donated every year, the equation is as follows:

2x+ 12=78

The teacher guides the students to compare the two methods and asks them to talk about the similarities and differences between the two methods. What should we pay attention to when solving problems with equations? Give students time to fully express their ideas.

The above two teaching examples are the most common examples in teaching. Every time a teacher throws a math problem, students explore independently and form a variety of problem-solving methods. If we subdivide these two cases, the former is a variety of algorithms and the latter is a multi-solution problem. The teaching strategy of algorithm diversification is mainly to let students learn independently, cooperate and explore, and the teaching strategy of solving multiple problems is mainly to encourage students to think from multiple angles.

Whether the algorithm is diversified or there are multiple solutions to a problem, it is a variety of solutions to a problem under the traction of students' flexible thinking. As for the classroom, if students have more ideas to solve problems, teachers should encourage students to express them and give them opportunities to show them. It is precisely because of the constant emergence of children in each class that our classroom activities are full of vitality. Students' thinking is active, teachers' emotions will be driven, teachers' emotions will be high, and scholars will believe in themselves.

Second, "diversification of problem-solving methods" should be implemented in graphics and geometry.

The layout feature of the graphic geometry part of the textbook published by Beijing Normal University is to start from students' real life and enter students' psychology with pictures and examples close to their lives. Simple written expression and bright picture colors are all factors that prompt students to find mathematical information quickly.

In fact, the ultimate goal of mathematics learning is to let students use what they have learned to solve problems in life, so that when facing practical problems, students can actively try to find strategies to solve problems from the perspective of mathematics and according to existing knowledge and experience, and improve their awareness and ability to solve problems. Years of teaching experience in mathematics have taught me that the most effective way is to give students a chance to practice. In teaching, teachers should design realistic and challenging problems in combination with teaching content, so that students can seek solutions.

Teaching Example 3 After teaching the volume of long and cube, the teacher asked the students to bring long and cube objects or containers, as well as irregular objects such as small stones and potatoes, to try to measure and calculate the volume of which objects. On this basis, you can also ask students a challenging question. Can you find a way to measure the volume of small stones with a cubic container, water and a ruler? The students had a heated discussion and exploration in the group, and the teacher went deep into the students' discussion to guide and inspire students to solve such problems with faster, better and more methods. The student representatives gave many wonderful performances during the speech:

Health 1: The method discussed by our group is as follows: Fill the cube container with water and measure the height of the water.

Teacher: Why do you need to measure the height of water?

Health 1: At this time, the height of water is actually the side length of a square. Only by knowing the height of water can the volume of small stones be calculated. Then put the pebbles into this container, and the water will overflow immediately. The volume of spilled water is the volume of small stones.

Teacher: What do you think of this method? Anything to say?

Other students express their ideas.

Health 2: What is the volume of this overflowing water? How to calculate? I think it is necessary to put the overflowing water into a container like this cube, and then measure the height of the water and calculate the volume of the water, that is, the volume of the small stone.

Teacher: That's right! What you said is wonderful! The calculation process of this method is that your two statements are pinched together, which is the solution of the problem. Everyone likes to use their brains to solve problems in life. In your wise expression, Miss Jane suddenly saw the birth of a little scientist! So do other groups have other ways to recite?

Health 3: Our group did this: put a little water in a cubic container and then measure the height of the water. Then put pebbles in, the water surface will rise, and then measure the height of water. The volume of rising water is the volume of pebbles. Finally, the volume of small stones can be calculated by using "the bottom area of cube x the height of rising water".

Teacher: Let's applaud him! Do you understand this second method? Who wants to talk about these two calculation methods?

In the process of communication, teachers showed great interest in each method and fully affirmed it. Finally, ask students to talk about their feelings about these methods: which method do you prefer and why do you like this method? Most students realize that the second one is the simplest one, because its thinking is clear and its operation is not very complicated. The teacher will summarize it again.

When solving exercises in graphics and geometry, the phenomenon in this teaching example often appears. Students should summarize the methods to solve problems through their own research, hands-on operation, practical drills, reports and exchanges. This way of presentation has a warm and active atmosphere and students actively participate. Most students actively strive for the opportunity to speak, and discover the differences and essential connections between different algorithms through communication.

In the above three teaching examples, teachers pay attention to the diversity of methods rather than summarizing which method is good and which method is not good, which is also a puzzle for many teachers, that is to say, do you need to tell students which method is just right? In fact, I think: as long as students can master the convenient method, there is no need to say which method must be used to solve it.

In class, teachers let students learn the basic method of "irregular object volume" through independent inquiry and cooperative communication. This algorithm enables students to understand, master and know why. Therefore, for this kind of special questions, teachers should reasonably grasp the problems in teaching, avoid rushing to give students a correct method, but trigger thinking shocks in students' continuous practice, communication and experience, and truly understand and master the method that suits them best.

"Multiple solutions" have a lot of research value in teaching, and the timeliness of classroom is not groundless. Teachers should grasp the generative problems in the classroom and respond flexibly to all kinds of unexpected problems. When students' answers are in line with the classroom rhythm, teachers should guide them in time and respect students' subjective cognition. Students have great potential and like to solve problems in ways that others have never used before. This is a child's unique desire to explore new things. Teachers should give students enough time to explore in class, let children communicate in groups as much as possible, and let them have a spark of thinking, so that our math class will be active, which is also in line with the concept of "new curriculum standard": "respect students' personality characteristics, pay attention to students' thinking development" and truly "take students as the center". However, it is wrong to pursue multiple methods for the sake of "diversification of methods". Many methods are listed mechanically. If teachers don't sum up in time, find their similarities, improve their thinking, and create an efficient classroom, then it is futile to enumerate more methods. This will only make our class content look too full, but we can't grasp the key points. On the contrary, it will be "counterproductive." Therefore, teachers should grasp this degree and really make "diversified solutions" have guiding significance for teaching, instead of putting on "high-sounding coats".