How to Cultivate Students' Critical Thinking in Mathematical Computing Teaching

The so-called critical thinking refers to students' rigorous, comprehensive and self-reflective thinking when judging the content, form and result of classroom teaching activities. Critical thinking is indispensable for learning mathematics well and plays an important role in innovation ability. It is of great significance to cultivate students' critical thinking and make them learn to criticize and innovate. This paper discusses the present situation of primary school students' critical thinking and how to cultivate students' critical thinking in primary school mathematics teaching, hoping to arouse the resonance of peers.

Key words:

Cultivation of Mathematics Teaching for Sustainable Development of Critical Thinking

2 1 century, being diligent in thinking is the foundation of modern people, being brave in questioning is the premise of modern people, and being brave in innovation is the result of modern people. The national conference on education clearly put forward the goal of cultivating students' innovative spirit and practical ability. Professor Yan also clearly pointed out that to cultivate students' innovative thinking, we should "get rid of four superstitions": get rid of superstitions on authority, ancients, foreigners and books. These four breakthroughs are manifested in the classroom teaching of primary school mathematics, which is what we advocate to cultivate students' critical thinking. This paper discusses the present situation of primary school students' critical thinking and how to cultivate students' critical thinking in primary school mathematics teaching, hoping to arouse the resonance of peers.

First, the current situation and analysis of primary school students' critical thinking.

The following two cases are what the author encountered in class. This phenomenon is typical in the current primary school mathematics teaching.

[Case 1] When teaching "Some Fast Calculations of Decimal Addition and Decimal Addition", the teacher showed several exercises after Protestantism, one of which was: More Subtraction ().

Empty: 27.45-20.7=27.45-20+ ()

Obviously, this question is a teacher's slip of the pen. But in the process of completing the exercise, no one raised any objection, and most students gave the following answers: more reduction (0.7).

27.45-20.7=27.45 - 20+(0.7)

After class, I asked several students: Do you think your answer is correct? The answer is roughly as follows:

(1) Correct, it is 0.7 less, it should be 0.7 more, and then it will be 0.7 less.

(2) At the beginning of the exam, I felt that the teacher had made a mistake in this question, but no one mentioned it. I felt that my original idea was wrong, so I filled it in.

[Case 2] When teaching "the volume of a cone", in the process of deriving the formula for calculating the volume of a cone, the teacher asked the students to do the verification experiment with yellow sand in groups after completing the demonstration experiment with water (the experiment was very successful). Because of the relationship between "loose" and "solid" of yellow sand, some groups fell down three times, and the yellow sand in the cylindrical container was not full, so it was filled with sand again; After some teams poured it three times, the sand exceeded the mouth of the cylindrical container and was leveled with a ruler. After the experiment, when the teacher asked everyone what the verification result was, the whole class replied in unison: "The volume of a cone is equal to one-third of the volume of a cylinder with the same height as its bottom."

The above cases show that students always think that teachers can do all the questions they give, teachers are always right, books are always right, and they blindly worship teachers and textbooks. It can be seen that students' thinking is not only influenced by stereotypes, but also has great dependence. They tend to believe conclusions easily, are not good at or can't find their own mistakes in solving problems, lack the necessary habit and ability of independent thinking, judge and question the information of teachers, classmates and textbooks, and lack the quality of critical thinking. In the process of creative activities and creative thinking, critical thinking is an indispensable factor and the core component of the whole psychological system. Without critical thinking, students' creative thinking ability cannot be cultivated and developed. Therefore, the author thinks it is urgent to cultivate students' critical thinking.

Second, the definition of the concept and research ideas

(A) the definition of the concept

The so-called critical thinking refers to students' rigorous, comprehensive and self-reflective thinking when judging the content, form and result of classroom teaching activities. To expound critical thinking from the individual differences of thinking means to be good at judging right and wrong from reality and strictly according to objective standards, which can not only evaluate other people's viewpoints, thinking methods and behaviors, but also conduct self-examination and self-evaluation strictly. Its opposite is blind worship, which is mainly manifested in believing conclusions easily and not being good at reflecting, summing up experience and asking questions. Critical thinking has five characteristics: analytical, strategic, comprehensive, independent and correct. With this kind of thinking, we can get unique methods and answers in solving problems. Therefore, critical thinking should be regarded as an integral part of problem-solving and creative thinking.

(B) Research ideas

In primary school mathematics teaching, we conduct research on cultivating students' critical thinking, which is based on teachers' trust in students, students' talent, and students' ability to acquire all kinds of new knowledge through their own criticism. Therefore, the focus is not mainly on students' criticism of teachers, others, textbooks and themselves, but on students' ideas, behaviors and critical abilities. In the process of cultivating students' critical thinking, on the one hand, teachers can get feedback information about teaching situation in time, understand the gains and losses of teaching and improve the quality of teaching; On the other hand, it can also promote students to learn actively and creatively, which is conducive to the formation of healthy personality and sound personality.

Third, the research process and methods

Research object: Middle and senior students are the main subjects.

It is of great significance to cultivate students' critical thinking and make them learn to criticize and innovate. This is a gradual process, and it is a process of synchronous development of knowledge, ability and quality. In the research, the author mainly tried from the following aspects:

(A) to create a way to cultivate critical consciousness

In primary school mathematics classroom teaching, the cultivation of students' critical thinking should first induce students to have critical consciousness, and the induction of students' critical consciousness depends on teachers creating various opportunities and exploring various methods. In the research, the author established four methods: environment creation method, obstacle elimination method, language encouragement method and role exchange method.

1, environmental protection construction method

Professor Ye Lan pointed out: "A prominent problem in our classroom teaching is the lack of respect for students' life value. "For a long time, the teacher-student relationship characterized by the absolute authority and domination of teachers in the traditional' closed classroom' is a concentrated expression of authoritarian education, which has caused students to be passive, passive and blindly obedient, which is a taboo for cultivating students' critical thinking and innovative ability. To cultivate students' critical thinking, we must establish a new democratic and equal teacher-student relationship, so that students can experience equality, freedom, respect, trust, understanding and tolerance from the teacher-student relationship, encourage students to put forward different opinions in class, doubt the conclusions of books, doubt the opinions of teachers and classmates, encourage students not to stick to ready-made knowledge and be brave in "fantasy", and create a psychological free and safe classroom atmosphere for students. Only in such an atmosphere can students express their opinions boldly, dare to make judgments and question difficult questions. In the study, according to the relationship between teachers and students' classroom activities, the author formulated the index of creating teaching environment (Figure 1). Figure 1:

1, every student has a certain position in the teacher's mind.

Trust students. 2. Be good at finding the bright spot in each student.

3, can guide students to overcome difficulties.

Don't deprive any students of the opportunity to speak.

5. Be able to listen to every student's speech patiently.

Teacher-student relationship respects students. 6. Be able to adopt students' correct opinions.

7, can guide students to supplement and improve their own and others' incomplete opinions.

8. Equal treatment

Treat students sincerely. 9. Encourage students to express different opinions and think from different angles.

10, allowing students to provide suggestions and criticisms to teachers.

In daily classroom teaching, we ask teachers to strive to create a teacher-student relationship of mutual respect, friendship and complementarity according to the requirements of indicators. For example, in class, the author tries to do four things: give students more opportunities for feedback and evaluation; Give students more opportunities to praise; Give students more opportunities to ask questions and ask difficult questions; Create more opportunities to learn from each other. These four "many" laid the psychological foundation for students to dare to criticize.

2. Obstacle elimination method

Although criticism is of great significance to students' personality development, in classroom teaching, teachers may be worried about accepting students' criticism, especially in public observation class, which often makes teachers worried that they will not be able to step down for a while. Therefore, to really cultivate students' critical thinking, teachers must have some courage. Because in the experiment, we constantly strengthened our study and deeply realized that classroom teaching should be a living individual. However, because the old education only taught knowledge, biology turned into death, teachers turned over the "knowledge pot" in their hands, students opened the "bottle mouth" for loading knowledge, and the classroom became a mechanical and emotionless place for transmission and acceptance. Therefore, we believe that to make the classroom full of vitality, teachers should influence students and students should influence teachers. Both sides will have difficulties, pursuits, successes, discoveries and even creations in class.

While eliminating the worries of teachers, we should also eliminate the worries of students. Many students are afraid to give advice to their classmates, especially teachers, thinking that this will cause dissatisfaction among classmates and displeasure of teachers: the friendship between classmates is less, and teachers' love and attention to them is less. In order to make criticism practical, I created an evaluation system of teacher-student interaction. Generally speaking, teachers use star language to evaluate students' critical thinking (table 1). Due to negligence, the teacher did not give affirmation to a student's criticism, and let the students spontaneously remind them with various actions.

Table 1:

One-star system (encouragement) 1, dare to criticize and be enterprising. What you said is reasonable. On second thought, I believe you can answer correctly. What you said is right.

Two-star system (compliment) 4. You can think deeper than before. 5. You are really nice and thoughtful. 6. Your criticism is different.

Three-star system (compliment) 7. Your speech inspired me so much that the teachers didn't think of it. I appreciate your suggestion, and I want to spread it to more teachers.

9. I will give your advice to my uncle and aunt who compiled this book, or you can try to write a letter.

3. Language incentive method

Teachers' affirmation of students' criticism plays an important role in students' minds. However, any form of use is limited. If teachers frequently use praise language, over time, it will lose some meaning in students' minds and students' critical desire will be weakened. So the teacher's praise language must be just right. In the experiment, the author tried to classify the use of various languages and named them stellar systems. The higher the number of stars, the higher the evaluation, which is the evaluation of the star system. Table 1 is the language that we have compiled to evaluate students from shallow to deep. Through these different levels of evaluation language, students can not only experience the teacher's affirmation to them, but also find their own direction of efforts from encouragement and enhance their desire for criticism.

4. Role exchange method

In daily teaching, teachers often blame students who are slow and confused in learning and have difficulties in learning, thinking that they don't work hard and try to change their learning status through pressure, which in most cases has little effect or even fails. The teacher's practice is against the child's psychology: because the teacher also forgot that he had children. We attach great importance to the attempt of dislocation method in research. That is, putting yourself in the students' shoes, understanding, thinking and criticizing problems, characterized by guidance rather than coercion; Experience, not indoctrination.

Similarly, students are also guided to try the dislocation method. For example, when teaching "three-step calculation application problems", there is such a problem: "The school library has brought two kinds of extracurricular reading materials, including 428 literature books, which is 12 more than science and technology books. How many books are there? " I ask students to talk about the equal relationship between quantities in the question. A student said, "Literature and art books plus science and technology books are the sum of the two books." Another student immediately stood up and added, "What you said is not specific enough. You should also explain how to ask for a science book. " "If I speak, first of all ..." Students complement each other and respect each other in their criticism. They talked in Kan Kan, spoke just right and achieved good teaching results.

(B) to consolidate knowledge and strengthen the quality of thinking

Concept is the basic "element" of mathematical knowledge, the "cell" of thinking, and the standard and tool for testing and judging correctness; Formulas, rules, properties and the scope of application of conclusions are often the criteria for making judgments; Mathematical thinking method is a powerful weapon to cultivate students' critical thinking. Therefore, in teaching, students must firmly grasp the basic knowledge and skills of mathematics, attach importance to creating problem situations, fully show the formation process of concepts, the derivation process of formulas, methods, properties and conclusions, and infiltrate mathematical thinking methods into all aspects of teaching to form a complete knowledge network for students. Only in this way can students have the ability to think, judge, question and reflect independently.

In order to ensure that students can criticize better, it is necessary to master some simple judgment methods. Judgment is a form of thinking that affirms or denies the object of thinking. Mathematical judgment is a way of thinking that affirms or denies quantitative relations or spatial forms. To cultivate students' critical thinking, students should master the necessary negative judgment methods. The common methods of negative judgment are counterexample and reduction to absurdity. At the same time, exclusion method and comparison method are also helpful for students to understand and master negative judgments.

1, counterexample method. To prove that a proposition does not hold water or does not have certain attributes, we only need to give a counterexample, which can be found by finding special values, special figures and special conclusions.

[Example 1] True or false.

Natural numbers are either prime numbers or composite numbers. ( )

This question can guide students to find counterexamples by looking for special values. The analysis is as follows: 1 is a natural number, but 1 is neither a prime number nor a composite number, so the original proposition is wrong.

2. Reduce to absurdity. Reduction to absurdity is to study the negative situation of the conclusion of argument, and then solve the problem according to the relationship. The words "at most", "at least" and "must exist" in the title are signals that reduction to absurdity can be used.

[Example 2] Is the following calculation correct?

16700÷200=83…… 1

83

200 16700

16

seven

six

1

If you consider the problem from a positive perspective, it is even more difficult to make it clear. If you use reduction to absurdity, it is quite simple. Guide students to make the following analysis in teaching:

If 16700 ÷ 200 = 83... 1, then 200× 83+1=16700;

And 200× 83+1=16601≠16700, so the original calculation was wrong.

The correct answer should be:16700 ÷ 200 = 83 ...100.

Through the training of reduction to absurdity, students can eliminate the interference of fixed thinking, thus strengthening the criticism of thinking and opening up the breadth and agility of students' thinking.

3. Exclusion method. By eliminating mistakes and drawing correct conclusions, it embodies critical thinking.

[Example 3] Selection: If 5.06× ()

A, less than1; B, equal to1; C, greater than 1

Guide students to analyze: if the multiplicand is greater than 0, if the multiplier is greater than 1, the product must be greater than the multiplicand, that is, the product ratio is 5.06, which is inconsistent with the meaning of the question, then the number in () must not be greater than1; If the multiplier is equal to 1, then 5.06× 1=5.06 is also inconsistent with the meaning of the question; This excludes the possibility of being equal to 1 greater than 1, and the remaining "less than 1" is definitely the correct answer.

4. Comparative method. Negative judgments need comparison, and only comparison can distinguish judgments. Comparative method is usually to compare the existing problems with known concepts, laws, formulas and properties, find differences, make negative judgments and seek correct conclusions.

[Example 4] Right or wrong.

49.56 1 ÷ 5.23=4956 1 ÷ 523………………( )

Teachers can guide students to compare this problem with the quotient invariance of division. Quotient invariance: Divider and divisor expand or shrink by the same multiple (except 0) at the same time, and their quotients remain unchanged. Now the dividend is expanded by 1000 times, and the divisor is expanded by1000 times. If the multiples of two numbers are different, the quotient must have changed, so it is wrong. In order to hold the equation, we can change "4956 1" to "4956. 1" or change "523" to "5230".

In primary school mathematics teaching, there are many methods such as counterexample, counterexample and parallelism. As long as they are often used in mathematics, students will master these methods, improve their negative judgment ability, and then strengthen the quality of critical thinking.

(C) split into two, objective criticism.

Carrying out the idea of dividing into two and criticizing yourself and others appropriately are the signs to measure objective criticism. In the experiment, the author pays attention to the correctness of students' criticism of others, teachers and textbooks, as well as the attitude students take when criticizing others. For example, when teaching "Calculation of Triangle Area", some students suggested that the letter "㎡" was used in the unit name of the calculation result in the textbook, but the Chinese character "square meter" was used in the answer sentence, which was inconsistent and inappropriate. Another student stood up and said, "Not level. We started with rectangular and square areas. What is there to mention for yourself! " "Faced with such a situation, it can not be ignored for teachers to teach students appropriate criticism. So, the author told the class: "This discovery is too level! The uncle and aunt who compiled the book also found this problem and are preparing to revise it. This discovery is too unexpected for me. We need such students, which is the result of being good at thinking. "After listening to the author, the students who just made fun of others immediately stood up and apologized. At the beginning of the experiment, we always find some students laughing at others' immature or incorrect criticism. At this point, we should guide and inspire students to find the bright spots in other people's criticism.

In the whole classroom teaching process, "we should pay attention to students' mathematics learning level, and pay more attention to their emotions and attitudes in mathematics inquiry activities." "In the face of students' critical performance in different degrees, teachers should objectively judge students' critical behavior from the perspective of dichotomy, respect students' unique learning experience, fully consider students' ability to succeed, and give timely guidance and encouragement.

(4) Form a criticism system and improve the quality of criticism.

The improvement of critical thinking quality is not achieved overnight. It requires teachers to make a long-term and purposeful training plan for students.

1, building a multi-level key network

2. Explore various comparison methods.

The purpose of comparison is to let students constantly sum up experience and lessons in the process of criticizing mathematical problems, and to affirm and reflect through comparison is a shortcut to promote students to embark on the road of high-quality criticism.

Modern classroom teaching is based on students' development. Neither books nor teachers should be absolute authorities, both of which have inevitable limitations. Students can accept knowledge and guidance, and they can also challenge books and question teachers. In the experiment, when students come to a key conclusion through thinking, the author will guide students to compare and judge the conclusions in the book. Some students will say, "The language in the book is simpler than mine." Others said, "My summary is simpler and easier to remember than that in the book." There is an example: when learning "the addition of two numbers with different symbols", the book concludes: take the sign of the addend with larger absolute value and subtract the one with smaller absolute value from the one with larger absolute value. But our students' generalization is: the symbol is bigger, and the absolute values of the two numbers are subtracted. It's really exquisite. In the method of solving problems, our students criticized the book again: the example in the book-"There are 120 pear trees and peach trees in the orchard, and the peach trees are twice as many as pear trees. How many kinds of fruit trees are there? " This book is about writing sentences according to the second condition. According to the first conditional sequence equation (X+2X= 120), students think that they can write sentences according to the first condition, and according to the second conditional sequence equation: 2X= 120-X or X=2( 120-X). In this way, the broadness of students' thinking is developed through criticism.

When giving lectures, teachers will inevitably make mistakes, but students always take the mistakes made by teachers for granted. Once teaching practical problems, after the new teaching, the teacher randomly gave the students a topic exercise: two cars left Xin 'anjiang Bus Station and hangzhou east railway station respectively, the speed of Xin 'anjiang Station was 40 kilometers per hour, and that of hangzhou east railway station was 45 kilometers per hour, and they met on the road three hours later. How many kilometers are the two stops apart? After reading the questions, the students quickly answered: (45+40) × 3 = 255 (km); 45×3+40×3=255 (km) was about to go on when a student stood up and said, "No, teacher, why is Xin 'anjiang 255 km away from hangzhou east railway station? My mother works at the station. She told me that Xin 'anjiang is about170km away from hangzhou east railway station. " As soon as the teacher woke up and didn't want to be negligent, he was criticized by the students. The teacher immediately corrected the mistake and affirmed the students' "criticism". Indeed, teachers fully affirm and appreciate students' doubts, which will undoubtedly greatly encourage students to dare to think and express their independent opinions, and help to cultivate the spirit of challenging authority and pursuing truth from an early age.

In addition, the author also asks students to make a comparison on the basis of listening to the criticism of teachers and classmates. Because absorbing critical materials from listening carefully is another effective way to improve the quality of criticism. However, the characteristic of primary school students is that they like self-expression. Sometimes other students have made a certain content very clear, and some students are still struggling to talk about it. Sometimes the topic of criticism deviates from the center. In this regard, I try to carry out comparative training in the following ways: (1) dislocation method, which requires students to listen to others and assume "if I answer, how can I speak"; (2) the difference method, that is, thinking about "what's the difference between other people's thinking criticism and mine", "what part of others can I criticize and how I can't criticize" and "what should I add? For some issues discussed together, students dare to say, "I learned from other people's speeches ... I won't ... I suggested ..." and so on. Students are good at expressing their unique views.

(5) Optimize the form and highlight key benefits.

Pay attention to combining self-criticism with mutual criticism. Self-criticism can fully arouse students' enthusiasm for learning. In self-evaluation, students can reflect on the learning process and promote independent development. Due to the differences in experience and background, students often have different understandings of problems, and various differences themselves constitute a valuable resource. Mutual criticism means that students can communicate with each other, question each other, cause confrontation and self-reflection, get rid of egocentric thinking tendency, deepen their understanding, and use other people's ideas to stimulate their own inspiration and put forward new assumptions, thus giving play to their imagination and creativity.

Pay attention to the combination of immediate criticism and delayed criticism.

Fourth, the research effect.

(A) Dare to take root in people's criticism

Judging from the overall situation of the class, the number of students who speak in class has increased, and those who dare to criticize are also on the rise. As can be seen from Table 2, at the beginning of the experiment, the number of speakers increased from 57.7% to 98. 1%, and basically everyone was willing to speak, among which the number of critical speakers increased from 28.8% to 86.5%, and the percentage of critical speeches increased from 50% to 88.2%, which indicated that students' critical consciousness had been obviously improved.

From the perspective of individual students, an introverted "underachiever" in the class wanted to see him raise his hand at first, but during the experiment, his self-confidence increased and he often raised his hand to answer questions and criticize.

Table 2:

Class Two, Grade Five, 51February 2002-February 2002, 65438+February 2002.

Count the number of people after the preliminary experiment.

Number of speakers (%) 30 (57.7%) 51(98.1%) 21(40.4%).

The number of critics (%) were 15 (28.8%), 45 (86.5%) and 30 (57.7%) respectively.

Times of criticism/speech (%) 50% 88.2% 38.2%

(2) The quality of criticism is becoming more and more mature.

After the experiment, we can see from Table 3 that the quality of students' criticism, the number of students has been greatly improved, not only the criticism of understanding problems and development problems, but also the accuracy of criticism has been improved to 15% and 25% respectively.

The improvement of students' critical quality is also reflected in the emergence of students' innovative thinking. If I encounter this problem: 2537 ÷ 300=8, I ask: the dividend estimate is 2500, which is small. The divisor remains the same. Is the quotient bigger or smaller than the original quotient? Check it with a calculator. The students immediately verified as required, but only one student stood up and said, "You don't need a calculator to verify, just multiply the divisor by the quotient. The result is less than 2537, so 8 is smaller than the original quotient." Students' criticism is of high quality and their thinking is active. It is commendable that he didn't follow the teacher's idea.

Table 3:

Classification of key problems before experiment, critical moment, critical moment, accurate critical moment and accurate critical moment after experiment.

Basic questions 86 75% 25 20 80%

Understand the question 53 60% 65 438+02 9 75%

Developmental problems 2 1 50% 8 6 75%

(c) Significant progress in academic performance.

Fortunately, with the development of students' critical thinking in the experiment, there is no bad situation that students attach importance to written language. On the contrary, students' academic performance has been improved. It can be said that the cultivation of critical thinking not only promotes the development of students' various abilities, but also promotes their academic progress.

Teaching skills are developing vigorously.

In the process of teaching with students' criticism, the formation and improvement of students' critical thinking is also a test of teachers' teaching ability, which urges teachers to concentrate on studying textbooks and not forget to study students, because students' thinking is different, and various "situations" emerge one after another in class, testing teachers' ability to control classroom teaching.

Research experience of verbs (abbreviation of verb)

(A) trust students' potential huge intellectual factors.

The cultivation of critical thinking is based on teachers' full trust in students, believing that students are gifted and that students can acquire all kinds of new knowledge through their own criticism. But in practice, we are sometimes limited by time, and we don't really let students criticize the problem more deeply. Therefore, to fully promote the development of critical thinking of all students, we need to explore and improve in practice.

(B) the study of critical thinking can be extended to various disciplines.

The cultivation of students' critical thinking can be implemented not only in mathematics classroom teaching, but also in other disciplines. In particular, there will be more comprehensive disciplines in the future, and it is more necessary to cultivate and develop critical thinking.

(c) Focus on sustainable human development.

The future society needs people's sustainable development, so our education for students should not only focus on today, but also face the future, so that the education students receive today will become a solid foundation for future sustainable development. Davis, an American psychologist, put forward at the international psychological conference a long time ago that the future creative personality characteristics include strong independence, strong self-confidence and not listening to other people's opinions. The research on cultivating students' critical thinking is just suitable for the cultivation of creative talents in the future cooperative society. This is also to follow the people-oriented principle and pay attention to people's self-improvement.