Reflection on score and division teaching

Part 1: Reflection on Score and Division Teaching

The focus of this lesson is to understand the relationship between scores and divisions. The difficulty is to understand the meaning of fractions in terms of division. Let students learn the relationship between scores and divisions through the study of this lesson. They will use scores to represent the quotient of the two numbers. They can use the relationship between scores and division to solve some simple problems.
Review the old knowledge before introducing the topic. The courseware presents several simple oral calculations to awaken students' memories of integer division and to pave the way for exploring new knowledge. When exploring new knowledge, the courseware presents the story of the pig Bajiehuazhai. From the imagination of 2 cakes per person to a cake, the average cake is divided into 4 people. How many pieces can each get? With the review knowledge and preparation, it is easy to calculate with the formula 1÷4, the students will soon say 1/4, then I will ask again: Why is it 1/4? How did you get it? The student divides the prepared disc into one point; then shows: the pig bastard has three more cakes, and how many sheets per person? The students took out the tools and explored them independently. The student has gone through the process of getting the job step by step, and the meaning of the score is better understood, so I understand why it is 3/4.
When the fraction is used to represent the quotient of the integer division, the divisor is used as the denominator and the dividend is used as the numerator. Conversely, a score can also be seen as dividing two numbers. It can be understood that dividing "1" equally into 4 parts means such 3 parts; it can also be understood that dividing "3" equally into 4 parts means such one part. That is to say, the understanding and establishment process of the relationship between score and division is essentially synchronized with the extension of the meaning of the score.
After teaching, I will reflect on my own teaching and find that in terms of the state in which the mathematics knowledge of the national stage is stored in the minds of the students, in addition to the abstraction, it should be abstract and concrete mathematical knowledge that can be transformed.

Part 2: Rethinking on Score and Division Teaching

The understanding and mastery of the relationship between scores and divisions can not only deepen the understanding of the meaning of scores, but also lay the foundation for the basic nature of learning false scores, scores, scores, and ratios and percentages. Therefore, the relationship between scores and divisions is in the whole textbook. It plays an important role in linking up and down. The new curriculum standard states: "The content of students' teaching and learning should be realistic, meaningful and challenging. These contents should be conducive to students actively conducting teaching activities such as observation, guessing, verification, speculation and communication." Effective learning situations can guide students to carry out learning activities of “independence, exploration and cooperation” and promote students' active participation. Therefore, in the introduction of the new lesson, I deliberately designed two division calculation questions: 8÷9= 4÷7=
When the students saw these two divisions, they were relieved and said, "There are two simple questions!" So I started two men and women in the class, the first question for boys and the second for girls. . Under one command, the boy was immersed in his head, and Hu Wenxin, who was quick-thinking, had already known the answer. He did not write at all. I indicated that she would not say the answer. I turned around and most of the students already had the answer at the tips of the students who were already done. Only the individual boys were still calculating.
After the report, I triggered the students to think: What is the difference between 8÷9=0.88... and 8÷9= 8/9? The most direct answer for students is that it is quick and easy to use fractional notation to express scores. This introduction allows the student to understand that dividing the two numbers can use the score to represent the quotient, laying the groundwork for further learning the relationship between scores and division.
After that, the two figures are divided and the students can quickly use the score to represent the quotient.
Using 1÷3=1/3 in the example to guide the students to find that the dividend in the division is equivalent to the numerator in the score. After the divisor is equivalent to the denominator in the score, let the students replace the numbers with their names: the dividend Divisor = numerator / denominator. At this time, I asked the students to use the letters a and b to indicate the relationship between division and score. Xue Longfeng wrote down on the blackboard seriously: a÷b=a/b, I saw this student writing very seriously, immediately praised her, and asked the students to applaud her. Just as everyone was happy for Xue Longfeng, I played a small "X" behind the formula she wrote. The student immediately said that he was puzzled. The teacher just praised her and now she was given another "X". Or a few thoughts flexible first called, said: "b can not be equal to 0!" I immediately seized this opportunity, asked: "Why can't b be equal to 0?" The class suddenly calmed down, and no one could tell why. This difficulty will soon be broken, and my heart is a little excited. I continue to use the example to divide 1 piece of cake equally to 3 people, and each person gets 1/3 of the piece of cake as an example: "Who said the '3' in this score?" Raise your hand and answer: "Look the cake as a unit '1', '3' means the average number of shares divided into cakes." "If you change '3' to '0'?" The student finally understood: the denominator expressed the unit " 1" The average number of copies divided into "0" is meaningless. As for this "a÷b=a/b" students often forget that the b here must be emphasized as 0. Through this analysis, students can more deeply realize that the divisor cannot be zero in division, and the denominator cannot be zero in the score.
I think that I deal with this link better. I don't directly tell the students that the divisor cannot be 0 in the division. The divisor is equivalent to the denominator in the score, so the denominator cannot be 0. Rather, by analyzing the actual meaning of a score, it is fully understood that the denominator in the score represents the fraction of the average score, and naturally cannot be equally divided into "0" shares.
There are some successes, and there are also some shortcomings. After class, reflect on the understanding of the scores and divisions students, but what are the differences between them, but did not lead students to find and summarize in the classroom. Dividing means dividing two numbers, which is a formula, and the score is a number. This shows that my interpretation of the textbook before the class is not deep enough, and I have not grasped the integrity and coherence of knowledge. In the future teaching, efforts should be made to have a deep understanding of the teaching materials, and at the same time, more information should be consulted in order to expand and extend the knowledge of the teaching materials.

Part 3: Rethinking on Score and Division Teaching

"Mathematics teaching should start from the student's life experience and the existing knowledge background, so that students feel that mathematics is in their own right, and they are in middle school mathematics. Make students understand the importance of learning mathematics and improve their interest in learning mathematics." Division, for primary school students, is a relatively abstract content. The reason why mathematics knowledge can be understood and mastered by students in the national ministry is not only the result of knowledge deduction, but the result of interaction of specific models, graphs and scenarios. So when I was designing the lesson and division, I considered the following two aspects:
1. Start by solving the problem and feel the value of the score.
Introduced from the problem of dividing the cake, let the students in the process of solving the problem, feel that when the business can not be represented by an integer, the score can be used to represent the quotient. This course is mainly carried out from two levels. One is to use the original knowledge of the students to solve the problem of dividing the average score of one cake into several parts, which is represented by commercial scores. The second is to understand the average share of several cakes by means of physical operations. A number of shares can also be expressed in terms of scores. These two levels are developed from the perspective of problem solving.
2. The expansion of the meaning of the score is synchronized with the understanding of the relationship between divisions.
When the fraction is used to represent the quotient of the integer division, the divisor is used as the denominator and the dividend is used as the numerator. Conversely, a score can also be seen as dividing two numbers. It can be understood that dividing "1" equally into 4 parts means such 3 parts; it can also be understood that dividing "3" equally into 4 parts means such one part. That is to say, the understanding and establishment process of the relationship between score and division is essentially synchronized with the extension of the meaning of the score.
After teaching, I will reflect on my own teaching and find that in terms of the state in which the mathematics knowledge of the national stage is stored in the minds of the students, in addition to the abstraction, it should be abstract and concrete mathematical knowledge that can be transformed. The whole course teaching has the following characteristics:
1. Provide rich materials and experience the process of “mathematicalization”.
The understanding of the relationship between scores and divisions is based on the concrete sensible objects and pictures, and the hands-on operation is used to generate mathematical knowledge under the support of rich representations. It is a continuous accumulation of perceptual accumulation, and gradually abstracts and models. process. In this process, we pay attention to the following aspects: First, provide rich mathematics learning materials, and second, on the basis of making full use of these materials, students gradually improve their own conclusions, from text expression to text representation. To the process of using letters, the process from complexity to conciseness, from life language to mathematical language, has also undergone a process of concrete to abstraction.
2. The problem lies in the method, and the content carries the idea.
Mathematical learning is a process of problem solving, and the method is naturally contained in it; the learning content carries mathematical ideas. In other words, mathematics knowledge itself is only one aspect of our study of mathematics. More importantly, we use knowledge as a carrier to infiltrate mathematics thinking methods.
As far as scores and divisions are concerned, the author thinks that if you only teach for a relationship, you just grab the tip of the iceberg. In fact, with the help of this knowledge carrier, we should also pay attention to the methods of inductive and comparative thoughts, and how to use existing knowledge to solve problems, so as to improve students' mathematical literacy.

Part 4: Rethinking on Score and Division Teaching

The relationship between score and division is taught after the students learn the meaning of the score. The purpose is to make the students know that the two integers are divided, whether the dividend is less than, equal to, or greater than the divisor. Their business.
The teaching of this part can not only deepen students' understanding of the meaning of scores, but also the basis of learning the false scores, taking scores, the basic nature of scores, and the ratio and percentage. Therefore, the relationship between scores and division plays a role in the whole textbook. The important role of the link. If you simply learn the relationship between teaching scores and divisions in a formal way, students can learn very solidly, but in this way, the calculation of 3÷4=3/4 is often neglected, in order to let students know what they know and know why. I organized the teaching like this:
1. Sensing new knowledge through practical operations
In teaching, I designed such a teaching situation, and divided a piece of cake on average to four children. How much is each? Ask the students to take a round piece of paper to represent a piece of cake and hand-divide one point to evoke an understanding of the meaning of the score. Then show that you want to divide the three cakes equally into 4 children. How much is each child? A group of four people tried to divide the three round pieces of paper equally into four children. And let the group send representatives to the stage to show the points. Through hands-on operation, students get two different methods, which lead to two meanings, that is, each person gets three quarters of a cake, or can be said to be one quarter of three cakes. In the process, the students fully understood the mathematics of 3÷4=3/4.
2. Make the student clear why the score is used to represent the result of the division formula
After the students understood the relationship between scores and divisions, I consciously designed such exercises. 1÷3= 8÷9= 2÷6= Ask the students to write the calculation results on the exercise book. As a result, some students raised their hands in a second or two, and some students took a long time to write the calculation results. After the report, guide the students to think: 1÷3=0.333...... and 1÷3=1/3 8÷9= 0.88... What is the difference between 8÷9=8/9? The most direct answer for students is that it is too much trouble to use a circular decimal to represent the quotient. It is quick and easy to use without a score. At this time, tell the students that in the future, the quotient of dividing two integers will be used to represent their quotients when there are not enough times or when there are decimals in the quotient. This is simple and fast, and it is not easy to make mistakes.
3, take the opportunity to extend, paving the way for follow-up learning
The first time to introduce students to the difference between the rate and the number. For example, 1 "divide a piece of cake into 4 parts, each part of which is divided into a few parts of the cake? How many pieces of cake are you divided?" 2 "The average length of 2 meters of rope is divided into 7 pieces, each length is How many meters of this rope? How many meters per section "3" divides 4 kilograms of salt equally into 5 parts, each weight is a fraction of the total number of salt / kilograms per part? Let the students understand this The first question asked for the three questions is the "point rate". There is no unit for the rate. The total number is regarded as the unit "1", and the unit 1 is divided into several parts equally. One of them is a fraction of the total. First, they are all divided by the unit "1" divided by the average score. For example, the scores of the first three questions are 1÷4=1/4 1÷7=1/7 1÷5=1/5. The second question is to ask how much each quantity is. Each quantity has a unit. It is obtained by dividing the total quantity by the average number of parts. The number must be the unit name. The first three questions are the second question. The algorithm is 1÷4=1/4 2÷7=2/7 4÷5=4/5
Here, after understanding the rate and the number of each part, the students have made a good foundation for the later learning scores and percentages.
4. Let students construct new knowledge on their own
When the student finds that the dividend in the division is equivalent to the numerator in the score, the divisor is equivalent to the denominator in the score, and the student is led to replace the number with their name: dividend divisor = dividend / divisor. At this time, let the students use the letters a and b on the exercise book to indicate the relationship between division and score. Most students wrote: a÷b=a/b, the teacher took out a slightly poor student's board and deliberately praised the classmate. He was praised but suddenly turned to give the student a big cross at the back of his homework. Just when the classmates are surprised? Why is it wrong? At this time, a few flexible thinking first screamed, saying: "b can't be equal to 0!" I immediately seized the opportunity and asked: "Why can't b be equal to 0?". I continue to use the example in the class to divide an average of 1 cake into 4 people. Each person gets 1/4 of the cake as an example. Let the students say what the '4' in this score means? "If you change '4' to '0'? "Students suddenly realize that the denominator means that the average number of shares divided into "1" is evenly divided into "0". It is meaningless when using letters to indicate the relationship between fractions and division----"a÷b=a/b "Students often forget that b cannot be 0 here. Through this analysis, students can more deeply realize that the divisor cannot be zero in division, so the denominator in the score cannot be 0. The student is not directly told here. In the division, the divisor cannot be 0, and the divisor is equivalent to the denominator in the score, so the denominator cannot be 0. Instead, by analyzing the actual meaning of a score, the students can fully understand the denominator in the score to represent the average score, so The denominator cannot be "0".
The shortcomings of this lesson: Although students have a thorough understanding of the scores and divisions, the differences between them do not lead the students to summarize. Division means that the two numbers are divided, which is an operation, which is an equation, and the score can represent the relationship between the numerator and the denominator, and can represent a value.

Part V: Rethinking on Score and Division Teaching

In this lesson, I am teaching on the basis of the students' learning scores and meanings. When the teaching scores are generated, the average score process often cannot get the integer results. It is expressed by the scores, which has preliminary involved the score and division. Relationship; the meaning of teaching scores, the average division of an object or a whole into several parts, also implies the relationship between scores and division, but they are not explicitly proposed. After the students understand the meaning of the scores, the relationship between teaching scores and division To make the student know initially that the two integers are divided, whether the dividend is less than, equal to, or greater than the divisor, the score can be used to represent the quotient. This will deepen and expand the students' understanding of the meaning of the scores, and also lay the foundation for the basic nature of the scores and scores. Specifically, this lesson has the following characteristics:
First, the visual demonstration is the premise for students to understand the relationship between scores and division.
Since the students are already familiar with the average score of an object when learning the meaning of the score, this lesson teaches the average score of a cake to three people without letting the students operate, but the process of computer demonstration, let the students understand 1 Zhang is the piece of cake. The average of 3 pieces of cake is given to 4 people, and the number of cakes per person is the focus of this lesson. It is also difficult. Teachers provide learning tools to allow students to fully operate and experience the meaning of the two methods. The key point is how to understand the three pieces of cake. Divide 2 pieces of cake to 3 people, how many pieces should each share? Continue to let the students operate, enriching the understanding of 2 or 3 pieces of cake. The accumulation of student operating experience has effectively broken through the difficulties of this lesson.
Second, cultivating students' awareness and ability to ask questions is the key to cultivating students' innovative spirit.
Einstein once said that it is more important to ask a question than to solve it. The ability of students to ask questions is not innate and requires careful and specific guidance from teachers. This lesson has carefully designed a series of thinking and logical questions around the two methods, and “forces” students to think in an orderly manner to further raise valuable questions. For example, after the students show their own points, the teacher inspires the students to ask questions:
a: How many pieces do you have?
b: How many pieces of cake do you get each time?
c: A few times, how many blocks are there?
d: How can I see how many pieces?
The problem is pointed out, which is conducive to students grasping the essence of mathematics.
Third, use the development of thinking to understand the knowledge learned, paying attention to the systematic nature of knowledge.
Mathematical knowledge is not isolated, but closely related. Only by putting knowledge in a complete system, student research is meaningful. For example, when students apply the relationship between score and division, 0.7÷2=, some students will not be able to express the expression method. The teacher explains: This form of score is not common at ordinary times. With the future study, everyone can It translates into a common fractional form.