Case: Teaching the second grade content "Parent-teacher conference tonight. According to statistics, 865,438+0 parents will attend the meeting. The meeting is arranged in the conference room. Tables should be arranged in the conference room, each table can seat six parents (the teacher drew a rectangle to represent the table, and six semicircles around the table represent people). Q: How many tables do you need? "
Student A: According to the teacher's prompt, draw six people around a table, so draw 13 table and draw three people around 14 table. Get the required 14 tables.
Student B: He started to draw tables like A. After drawing two tables, he changed the way. "It's too much trouble." We use the formula to do it, 8 1 ÷ 6 = 13 ............................................................................................ gets the need 14 tables.
Student C: Just use the formula, 8 1 ÷ 6 = 13...3. Draw a conclusion.
Student A only stays at the level of physical operation. Of course, physical operation is an important visualization tool to realize mathematicization, and it is also a necessary stage. However, just staying at this level, students' thinking will not reach a higher level. Without abstraction and symbolization, there would be no mathematics.
Student c is at the level of symbolic operation. It's just a formula, and there's no hands-on process. Problems often arise when seeking the meaning of formulas. For example, some students will answer 13 tables, and there are still three people left. The reason for its error is that it has not gone through a complete mathematical process.
Only student B has gone through the whole process of mathematization. Physical operations are performed first, and then represented by symbols (i.e. formulas). With the foundation of hands-on operation, students naturally understand the practical significance of the formula. So when getting the formula, students will come to the conclusion that 14 tables are needed.
Case: Teaching Grade Two "The Law of Finding Remainder Less than Divider"
The first part:
Teacher: What is the relationship between remainder and divisor? Let's calculate several divisions with the remainder first, and then see what the size of the remainder has to do with the divisor. (student computing)
17÷4=4…… 1,
18÷4=4……2
19÷4=4……3
The teacher inspired the students to draw the conclusion that "the remainder is less than the divisor"
Teacher: Why must the remainder be less than the divisor? Most students say "I don't know".
The second part:
Students experience chopping beans in their brains, and then
Teacher: Now we don't need to divide beans in kind or in our minds. We use the formula to divide the beans, and divide the beans of 17 into four plates on average. How to use the formula? 18? 19?
Think and write the corresponding formula.
Teacher's blackboard: 17÷ 4 = 4... 1,
18÷4=4……2
19÷4=4……3
After writing each formula, ask the students to find the meaning of the formula.
Teacher: Please follow the above formula. What did you find? Can you explain your findings?
After the students reported many discoveries, one student said, "The remainder is less than the divisor, because if the remainder is greater than the divisor, you can continue to divide."
Analysis: Section 1 clearly puts forward the goal of finding the relationship between remainder and divisor. Teachers and students * * * divide with remainder. By observing the formula, the law that the remainder is less than the divisor is obtained. Students get this rule through longitudinal mathematicization. Because students lack the foundation of horizontal mathematicization, they lose the search for meaning and the time and space for exploration. Therefore, students don't really understand the meaning of the remainder, and it is difficult to understand why there is this law.
The second part solves this problem well and helps students to realize mathematicization smoothly. Teachers provide students with time for activities, and their questions are also open to a certain extent. Teachers keep exploring meaning, therefore, teachers inspire and maintain students' high cognitive level. From the perspective of horizontal mathematicization, every time you see an expression, the teacher asks students to explain the meaning of the expression by dividing beans. Therefore, the understanding of the meaning of remainder is deepened in students' minds. The potential line of "the remainder is the undivided beans" is "the number of remaining beans is less than the number of plates", and the further and slightly abstract line is "the remainder is less than the divisor". From the perspective of vertical mathematics, through comparison and incomplete induction, it is found that the remainder 1, 2,3 is less than the divisor, because the number of remaining beans is less than the number of plates, otherwise it can be divided. While seeking the meaning of the remainder, we are also seeking the reason why the remainder is less than the divisor and the law that the remainder is less than the divisor. The meaning of remainder is the foundation and needs to be sublimated into the law of remainder. Back to reality, we must seek to understand the meaning of remainder. (1) Create a situation with mathematical content.
Creating a teaching situation with mathematics content can stimulate students' interest in learning and provide students with a good environment for learning mathematics. Gu Lengyuan pointed out: "The content itself, the expression of the content and the way of learning the content arouse interest, but the teaching activities that blindly pursue interest will also be biased." People have noticed that knowledge is not always interesting or easy, and learning is essentially a process of constantly overcoming difficulties. For students' lifelong learning, long-term core knowledge must be the real pillar of students' learning. " The creation of suitable situations in mathematics classroom should help students learn and understand mathematics knowledge better, serve students in learning mathematics, urge students to pay attention to situations from the perspective of mathematics, provide support for the learning of mathematical knowledge and skills, and provide soil for the development of mathematical thinking. Mathematics class should have the taste of mathematics and reflect the origin of mathematics.
Case: When teaching integer additive commutative law, the teacher created such a situation: the teacher prepared a spoon with a long handle and asked a student to take the soup at the end of it and put it in his mouth. A student tried, but he didn't drink the soup. Then the teacher arranged four students and divided them into two groups to see which group had the soup first. Students gradually understand that only by sending the soup to each other's hungry mouth can they drink it. Then the teacher said, this is the exchange law in life.
This situation is really lively, and students' activities are also very lively, but does it reflect additive commutative law? This is misleading. The exchange law in life is essentially different from that in mathematics. We should not vulgarize the law of exchange into exchange in life. This situation itself lacks the content and essence of mathematics.
Case: How many teaching trees are there in the third grade of Beijing Normal University?
Teachers use multimedia to show teaching situation maps and guide students to observe.
Teacher: Students, do you know the benefits of afforestation to human beings? Our school plants trees every year. Let's take a look at the math problems in tree planting today.
The design aims to create a familiar life situation of tree planting for students, closely link mathematics with real life, and educate students on environmental protection. )
Please observe the picture carefully and tell your deskmate what you see. How many bundles of young trees are there? How many trees are there in each bundle? ) What math questions would you ask?
Guide the students to ask "How many trees are there in a small tree?" .
Students list the formula of 20×3 and try to calculate it.
Teachers combined with the specific situation of life, from the object-problem-formula, let students go through the process of horizontal mathematicization, and at the same time cultivate students' awareness and ability to ask and solve problems. This situation is high in mathematics and not fancy.
(2) Follow the cognitive process from physical operation to symbolic operation.
Bruner's research on children's intelligence development shows that children's cognitive development needs to go through three stages: action cognition, figure cognition and symbol cognition, which corresponds to the three levels of thinking development: operation level, representation level and analysis level. Therefore, for primary school students, there are generally three stages of operation, from physical objects to representations to symbols. But for students with good academic qualifications, they are fully capable of directly entering the formal symbolic operation stage. In addition, for vertical mathematicization, it is naturally the operation from symbol to symbol, but we should also see that its initial symbol operation still depends on intuition and objects.
Case: When students calculate "9+5", students at the level of sports cognition may first count 9 sticks and 5 sticks, and then add them to get 14 sticks. They can't get rid of the concrete operation of counting by using physical schema Students with graphic cognitive level may draw two piles of small circles, a pile of nine and a pile of five, and then draw 1 small circle from the five, and the other pile will become a pile of 10 and a pile of four, and get 14. By using graphic schema, they get rid of actions and can think with the help of appearances. Students with cognitive level of symbols can think abstractly: 9+ 1= 10, 10+4= 14. They use the schema of symbols and have a good feeling for numbers and symbols.
(3) Seeking the meaning of formal symbols
Completing the mathematical process from physical operation to symbolic operation is only one aspect of the cognitive process. On the other hand, in order to seek symbolic meaning, only by returning to reality can formal knowledge realize its own meaning and complete its cognitive mission. This is also the need to maintain a high level of cognition. Only by constantly pursuing the meaning of symbols can we stimulate students' cognitive motivation and improve their cognitive level.