In recent years, an important reform in American elementary school mathematics is to strengthen the teaching of problem-solving strategies. In the early 1980s, the National Association of Mathematics Teachers of the United States proposed that solving problems is the focus of mathematics teaching in primary and secondary schools, and that basic mathematics skills should include more contents than computing ability, among which there are problems about problem-solving strategies. 1988 also put forward the strategy of making students learn to use problem solving at the sixth international mathematics education conference. At the end of 1980s, the newly drafted Mathematics Curriculum and Evaluation Criteria for Primary and Secondary Schools in the United States further emphasized this aspect. The first criterion of each study period is to learn and use problem-solving strategies. Since then, the content of problem-solving strategies has been compiled into American primary school mathematics textbooks.

Why does the United States attach so much importance to the teaching of problem-solving strategies? This is the need to adapt to the development of modern society. American mathematics educators believe that the United States has entered the information society and needs people who can process information and solve problems by mathematical methods. Therefore, it is very important for students to master problem-solving strategies. This is very different from the past when American primary school mathematics focused on cultivating students' ability to solve practical problems. In the past, the teaching purpose of solving mathematical problems in primary schools was limited to understanding practical problems and solving some simple practical problems themselves. Now, in addition to achieving the above goals, it is also necessary to enable students to master various problem-solving strategies, cultivate their general ability to solve problems and process information, and develop their intelligence, so that students can adapt to the ever-changing society and apply existing problem-solving strategies to solve even if they encounter new problems. Obviously, this is a major reform measure of mathematics teaching in American primary schools.

Second, the content of teaching problem-solving strategies

In American elementary school mathematics, the name "solving application problems" is not used, but called "solving problems". The scope of the problem is wider than the domestic application problems, including both practical problems and some unrealistic text problems and thinking problems. Therefore, the strategies to solve the problem are also extensive. There are both general problem-solving strategies and special problem-solving strategies; In addition, in order to meet the needs of modern information society, some strategies to solve problems by applying modern and modern mathematical methods are put forward. Let's briefly introduce them respectively.

(a) General strategy to solve the problem

In the aspect of general problem-solving strategies, it is mainly the general steps of teaching problem-solving, which are basically the same as the steps of mathematics application problems in primary schools in China. The United States divides the problem-solving steps into the following four steps: 1. Understand the meaning of the problem; 2. Make a plan to solve the problem; 3. Answer as planned; 4. Answer and test. Sometimes examples are given in the textbook for comprehensive explanation, and sometimes individual explanations and exercises are carried out.

1. Regarding the first step, we attach great importance to data collection. In each set of textbooks, more exercises are arranged to collect data from statistical charts. The lower grades mostly appear in the form of image maps, and the upper grades mostly appear in the form of statistical tables. For example, in the fifth grade, the following table will appear:

At (1) temperature 0℃ and wind speed 10 km, what is the air cooling coefficient?

(2) The temperature is-5℃, the air cooling coefficient is-16℃, and what is the wind speed?

The textbook also pays attention to arranging individual exercises on topics with redundant or lack of information. For example, "Tom has four puppies, Sam has three kittens and Bob has five puppies. A * *, how many puppies are there? " "Students go fishing, and half of them have never been there. How many students have never been there?" Through such a topic, students can correctly choose the necessary known number according to the problem, which is helpful to improve their ability to analyze the problem.

2. Regarding the third step, we pay great attention to the training of correct selection of operation methods. For example, given the same known conditions, such as the number of two items, ask questions to find out how many they are, and then ask questions to find out how much they are different. In addition, there is the application of multiplication and division.

Regarding the fourth step, pay great attention to checking the correctness of the answer. On the one hand, it teaches students the methods of testing, such as using subtraction to test addition, using multiplication to test division, and using different operation methods to test whether the calculation results are correct; On the other hand, it teaches students to check whether the high digits of the calculation results are correct through estimation. In addition, pay attention to teaching students to judge whether the answer is reasonable. First, pay attention to what is reasonable. For example, the following question should be divided by 60, but the answer is different: "150 pencils are distributed to 60 students on average. How many pencils does each student have? " (Answer: 2) "150 students, each boat can take 60 students, how many boats do you need?" (Answer: 3) "How small is a movie showing 150 minutes?

Los Angeles is 480 kilometers and the speed of cars is 80 kilometers per hour. How long does it take to get there? Choose the answer: 60 hours, 60 kilometers, 6 hours.

(b) Special strategies to solve problems

Generally speaking, there are the following types:

1. Drawing: Drawing helps to understand the quantitative relationship. For example, "A club member saws wood to make furniture. It takes 5 minutes to saw a piece of wood into 10 pieces. How many minutes does it take once?" By drawing, we can see that it needs to be sawed 9 times, so it is easy to calculate the required time.

2. Simplify the topic: one is to change the complex large number in the original question into a simple smaller number, so that the topic is simple. The other is to change the topic with more complicated description into the topic with simpler description, so as to make the quantitative relationship in the topic clearer.

3. Try to guess: Try to guess, and gradually adjust the results of the try to get the correct answer. For example, "Sonia spent 22.5 yuan on three books. Among them, the mysterious caves are less than the hidden treasures 1 yuan, and the hidden treasures are less than the strange cities 1 yuan. What is the price of each book? " The first attempt: 2 1 close to 22.5, divisible by 3, the average price of each book is 7 yuan; If the treasure is defined as 7 yuan, it is 6+7+8 = 2 1 (yuan), which is close to 22. 5 yuan, but still 1. 5 yuan. The second attempt: If you add 0. 5 yuan in every book, 6 yuan in all. 5+7.5+8.5 = 22.5 (yuan), and the total money is exactly 22. 5 yuan. This shows the price of each book.

4. Backward push: Some topics of reverse thinking can be reversed. For example, "Abbott worked for 3 hours and got the money to buy a bunch of flowers to go to 9.8 yuan, leaving 2.95 yuan. How much does she work per hour? " Drawing helps analysis: push back with the opposite operation.

5. Solving problems with equations: Because we don't talk about simple equations, we take solving problems with equations as part of the problem-solving strategy. Generally limited to one or two steps of calculation.

6. Solve with formulas: for example, find the perimeter or area of a rectangle and the volume of a cuboid.

(c) Strategies for solving problems with modern and modern mathematical methods

This is an important feature of American elementary school mathematics problem-solving strategy. Through teaching, students not only have a preliminary understanding of some modern and modern mathematical thinking methods, but also improve their ability to process information and solve practical problems. Generally speaking, there are the following types:

1. Classification: Pay attention to the practice of classification from the lower grades. For example, circle similar items. In the senior class, students are required to represent related object sets with charts. For example, show the following two pictures:

Then ask the students to combine the two groups of circles and draw the following picture together.

2. Organizing data: infiltrating statistical ideas and methods. For example, a stationery store counts the number of several items as follows, and then calculates them in a list.

3. Thought and method of infiltration statistics. For example, 4000 people want to visit the city, and the city requires them to fill out cards and write down their names and addresses. Without all the cards, I want to know how many people live in each district. What should I do? You can use samples to predict. Randomly draw 100 cards from 4000 cards and distribute them to 5 people, each with 20 cards. The statistics are as follows:

4. Calculate the probability: for example, 6 small cubes, of which 2 are blue and 2 are green.

5. Use paradigm: find out the arrangement rules of numbers or shapes, and then use the rules to calculate or judge. For example, Edward deposited 1 minute in the bank today, 2 points tomorrow, 4 points every day, 7 points on the fourth day, and 1 1 minute on the fifth day ... How much should he deposit on the tenth day? In order to understand this problem, you can make the following table and find out the paradigm.

The pattern found in the table is that the amount of money deposited every day increases 1, 2, 3, 4, 5 ... compared with the previous day, 46 points are deposited on the tenth day, that is, 1+2+3+4+5+6+7+8+9 = 45 (points) more than the first day.

6. Use a tree diagram: For example, there are two kinds of telephones in the store, one is buttons and the other is dials. Each phone has three colors: red, yellow and green. Each color phone has two shapes: square and round. How many kinds can customers choose from? To understand this problem, you can draw a tree diagram as follows.

As can be seen from the figure, there are 12 kinds of a * *. Write the formula as 2 × 3× 2= 12 (species).

7. Open topic: There are generally two situations. One is that a question has different solutions, and the other is that a question has different answers. Examples of the latter are as follows.

Example 1: Draw several items and indicate the unit price respectively, such as shirts 10.99 yuan, pants 13.5 yuan, records 5.98 yuan, toy cars 3.92 yuan, crayons 1.6 yuan. Tade will spend 8- 10 yuan. What can he buy?

Example 2: There are cars and motorcycles with 42 wheels in the parking lot. How many cars can there be in each car? Can be listed as follows:

As you can see from the table, there are 10 answers.

8. Decision-making: This is one of the modern mathematical methods. In primary school, there can only be very simple and specific. For example, "Donna wants to buy a bicycle, which is worth 290 yuan. He saved 225 yuan to earn 40 yuan by working every week. There are three options, which can be decided according to the specific situation.

(1) Save enough for 290 yuan to buy.

(2) Submit it to 90 yuan at that time, and then submit it to 19 yuan every month for one year.

(3) There was no payment at that time, but 28 yuan was paid every month for one year.

Find the total amount paid for each option, and then compare which is favorable and which is unfavorable.

(1) Which option pays the least?

(2) Which option can get the bike immediately?

(3) Can Donna earn enough money to pay for each option?

(4) Which option does Donna choose to pay less, the second or the third?

(5) If you were Donna, which one would you choose?

It can be seen that there is not only one answer to the above question, and the fifth question also varies from person to person.

9. Logical thinking: covering a wide range. Here are just a few representative examples.

Example 1: Qin Na may buy carrots or pears. She doesn't want to buy carrots. What does she want to buy?

Exodus 2: A is not as tall as B, but he is taller than C. Who is the shortest?

Example 3: A, B and C are fitters, electricians and gardeners respectively, but A is neither a fitter nor a gardener, and B is not a fitter. Determine the occupation of each of them.

One way to find out the answer is to build a table, as shown on the right.

Think: A is neither a fitter nor a gardener, so he is an electrician.

B is neither a fitter nor an electrician, so he is a gardener.

Then C is not an electrician gardener, but a fitter.

Example 4: There are 28 students in Grade 4, of whom 14 joined the band, 9 joined the swimming team and 4 participated in these two activities. How many people didn't attend these two activities?

Think: Those who only join the band but not join the swimming team are 14-4= 10 (people). Those who only join the swimming team but not the band are 9-4=5 (people). A * * * joining a band and swimming team is 10+5+4= 19 (person). So those who didn't take part in these two activities were 28- 19=9 (people).

Third, the arrangement of teaching problem-solving strategies

The teaching of problem-solving strategies in American primary school mathematics textbooks, like other contents, also pays great attention to reasonable arrangement. Specifically, it has the following characteristics.

(A) to adapt to the age characteristics of students, starting from the third grade formal teaching. The teaching of problem-solving strategies requires students to have certain mathematical knowledge and accumulate some problem-solving experience appropriately, which is more acceptable. Therefore, it is more appropriate to teach problem-solving strategies from the third grade. However, the first and second grades should also pay attention to some content about problem-solving strategies, such as finding data from graphics, looking at image statistics, choosing operations, initially understanding the steps of problem-solving and opening questions. Only in a more concrete and simple form. For example, the four steps of solving problems appear in the first and second grades: (1) What do you know? Ask for what? (2) How can I solve this problem? (3) do it. (4) inspection. The third grade will be summarized on this basis when it is officially taught.

(2) Decentralized arrangement and appropriate cooperation with other teaching contents. The problem-solving strategies introduced earlier are scattered in each unit of each grade, marked with subtitles, and many problem-solving strategies appear repeatedly in different grades, in which the calculated content matches the teaching content of this grade as much as possible. For example, I learned some decimal addition and subtraction in the third grade, and the estimation content is mainly decimal addition and subtraction; In the fourth grade, I learned some decimal multiplication and division, which is the main part of the estimation content. For another example, the calculation of probability needs to be based on the score, and the probability does not appear until the score is known.

(3) Follow the arrangement principles from easy to difficult, from simple to complex, and from concrete to abstract. For example, the problem-solving strategy of finding a paradigm appears in all grades, but the difficulty and complexity of the topic are different. In the lower grades, it is emphasized to find patterns by looking at pictures, while in the middle grades, in addition to continuing to appear in the form of lower grades, some people also see a list of numbers to find patterns, and then further appear in the list to find patterns. For another example, logical thinking is a problem-solving strategy. In the lower grades, there are "and" or "sentences, in the middle grades, there are rules to solve problems, and in the upper grades, there are set diagrams to solve problems.

Siyiguan

From the brief introduction of problem-solving strategy teaching in American elementary schools, it can be seen that strengthening this teaching is conducive to improving the problem-solving ability of primary school students and promoting the development of their thinking ability. Although there are still some shortcomings in the arrangement, such as the choice of some problem-solving strategies is still worth studying, the practice of multi-step problems is less, and the arrangement of some problem-solving strategies still lacks hierarchy. But the direction of reform is right and meets the needs of the development of modern society.

Strengthening the teaching of problem-solving strategies in the United States has certain enlightenment to the teaching reform of mathematical application problems in primary schools in China. Since the founding of the People's Republic of China, some reforms have been carried out in the teaching of mathematical application problems in primary schools in China, but they are still not enough, especially without jumping out of the box of traditional application problems teaching. The teaching content of application problems is basically limited to the original scope, but some simplified and more reasonable arrangements have been made; Some attention is paid to the problem-solving ideas, which are also reflected in the teaching materials, but there is still a lack of systematic arrangement. Compared with the problem-solving strategy teaching in America, there is a certain gap.

In order to further reform the teaching of applied problems, better improve students' problem-solving ability and develop students' intelligence, I hope our textbook editors, teaching and research personnel and teachers will study how to strengthen the teaching of problem-solving strategies in primary school mathematics. Firstly, the direction of the teaching reform of applied problems, how to determine the content and scope of applied problems teaching and how to arrange the teaching of problem-solving strategies reasonably are clarified. Secondly, vigorously carry out the teaching reform experiment of applied problems, support the directional reform experiment, and concentrate everyone's wisdom, so that the applied problems of mathematics in primary schools in China can go further and make greater contributions to cultivating talents needed by China's modernization drive.