Different people study different mathematics.
"Different people learn different mathematics" is one of the important concepts of the new mathematics curriculum standard, which means to let students receive different levels of mathematics education, adapt to the differences in students' learning and meet the different needs of society. The mathematics expansion questions in the textbook are designed by adhering to this teaching concept, aiming at making students with spare capacity get more development in mathematics. Therefore, teachers should allow all kinds of "differences" when developing problems in teaching.
Different teaching objectives In the process of learning knowledge and skills and formulating teaching objectives at the result level, the basic subject focuses on the foundation, requiring everyone to participate and master together, and the bottom line is the same; The expansion problem focuses on improvement, and the teaching objectives should be formulated at different levels. Different students should seek different learning results, so that students can get as much development and improvement as possible. For example, in the above case, top students are required to solve problems independently in their own way, and middle and lower students only need the help of their peers to solve problems. Top students are required to understand mathematical thinking methods on the basis of understanding the teacher's two solutions, and to solve similar mathematical problems by analogy, so that middle and lower students can understand them. Teachers can flexibly adjust the teaching objectives according to the teaching process and handle them flexibly.
In terms of emotion, attitude and values, there should be no distinction between top students and lower-middle students. Teachers should fully mobilize students' enthusiasm, protect their self-esteem and cultivate their interest in learning. In addition, the expansion questions are generally not included in the content of the customs clearance examination, and can be examined in additional questions or special improvement papers.
There is a big gap in learning ability between top students and lower-middle students with different learning styles. It is very difficult to ask junior and middle school students to explore independently, so the learning methods should be different. Top students are required to explore and solve problems independently and get promoted with the help of teachers. Middle and junior students are more likely to solve problems with the help of their peers and teachers, and acquire knowledge, skills and thinking methods.
Let mathematics take root in students' minds.
Expanding questions has rich mathematical connotation, which can show the unique beauty of rational and logical thinking in mathematics, and will surely win the love of some students with higher mathematical talent and sow the seeds of mathematical research and creation in their hearts.
Sowing with "success", it is difficult to enlarge the problem and it is easy for students to suffer setbacks. If this frustration affects students' interest in mathematics learning, it will not be worth the loss. Therefore, teachers should try their best to make students succeed in the study of expanding questions and sow the seeds with "success".
In the teaching of the above-mentioned problems, the teacher specially arranges peers to help solve the problems on the basis of students' independent problem solving. The communication atmosphere between classmates is relatively relaxed. Therefore, students usually regard the progress made by students through mutual assistance as a kind of self-acquisition. Especially for those students with medium level, their mistakes are often caused by some minor negligence, and after correction, they will still have a sense of success psychologically. In the teaching of the above examples, the classroom atmosphere is always warm, which is based on the successful experience that students have generally gained.
Catalyze with "praise"
Fragment 1: a unique calculation method
Teacher: Is there any other way to calculate the area of red paint?
Health 3: 40 × 40 × 6.
Teacher: What is the reason?
Health 3: The two small pieces in the middle and the invisible ones on the left add up to exactly two 40×40, so there are six in one * * *.
Teacher: Let's have a look. 65-40 = 25cm, 25+10 = 35cm, 5cm is less than 40cm. The height of the leftmost piece is 65 cm, and 65-40 = 15 cm. Does it add up to two squares?
Health: It seems not.
Teacher: Is it too much 10 cm?
Health 3: Yes.
Teacher: What should we do?
Health 3: add 40× 10.
Teacher: Really?
Health: Yes.
Teacher: Let's work out whether the calculation result is/10 000 square centimeters. (The calculation process is omitted)
Teacher: XXX provides us with a brand-new way of thinking. He is really a good boy who can think. Everyone should learn from him and not be satisfied with the general way of thinking.
Psychologist William? James once said: "The deepest need of human nature is to be appreciated and praised by others". Appreciation, praise and encouragement are the wings to help children fly to the other side of success. Although the student who came up with a unique method didn't get the correct result, the teacher gave him a high evaluation, which greatly stimulated his self-confidence and love for mathematics. He doesn't like writing very much after class. He wrote a 500-word math diary to describe the class, with pride between the lines. The teacher's praise has successfully catalyzed the seeds of mathematical research and creation in his heart, which have taken root and sprouted.
Pursuing the Truth of Mathematics Teaching
Part two: Two mathematical thinking methods.
Teacher: The teacher also thought of two calculation methods. Please let the students see if this is correct.
1)(40+40+40)×40+65×40×2
2)(55+40+ 10+40+25+40+40)×40
Method 1
Teacher: Can you understand the above formula? (Students look at the formula and think, and their expressions are blank.)
Teacher: Let's look at the first method first. What does the podium look like from above?
Health: It's rectangular.
Teacher: Can you draw this rectangle? Draw a picture in the exercise book.
Draw a picture on the blackboard and mark the length and width of the rectangle.
Teacher: Can you calculate the area of this rectangle?
Health: Yes. That is (40+40+40) × 40.
Teacher: Then can you understand the first formula? 65×40×2 is the area?
Health: 65×40×2 Calculate the area seen from both sides.
Teacher: Do the students agree with him?
Health: I agree.
Teacher: Seen from the left and right, they are two rectangles 65 cm long and 40 cm wide. So it can be calculated by the method of 65×40×2. Is the red paint area calculated by formula 1 correct?
Health: Correct.
Teacher: Is this a good way?
Health: OK.
Teacher: this is to consider the problem as a whole. This method is often used to find complex surface areas in the future. (The teacher's blackboard: overall consideration)
Method 2
Teacher: Equation 2 also considers the problem as a whole, but it is more "orderly thinking" (blackboard writing: orderly thinking). Please find out where the data of 55+40+ 10+40+25+40+40 (marked in red) is on the podium and what the rules are.
Health: It is the length of the row from left to right. (Teacher, please come up and point)
Teacher: Add them to the total length of the red paint. (The teacher draws a schematic diagram, see Figure 2) The width is 40 cm. This is the area of red paint multiplied by?
Health: Yes.
Teacher: We can imagine that there is a red carpet on the podium, and then we will straighten it. Did it become a rectangle as shown in the picture?
Health: Yes.
Teacher: Do you think this method is good?
Health 1: Good.
Health 2: Teacher Song is really clever.
Teacher: As long as students study hard, think more and pay attention to methods, you will become smarter and smarter.
This clip can be said to be the essence of this lesson. Teachers are striving for the truth of mathematics teaching, which has two main teaching values.
Developing mathematical logical thinking and cultivating students' initial logical thinking ability is one of the important goals of primary school mathematics teaching. Teachers introduce various problem-solving methods to broaden students' thinking and pay attention to cultivating students' divergent thinking; By treating the area of red paint as a rectangle and other mathematical activities, students' abstract generalization ability and profound thinking are cultivated.
Infiltrate mathematical thinking methods Teachers have consciously infiltrated mathematical thinking methods such as "thinking from the whole" and "orderly thinking". Mathematical thinking method is an important basis for students' follow-up study, an important part of students' sustainable academic ability and the essence of mathematics teaching. The mathematical thinking methods of "thinking as a whole" and "thinking in sequence" infiltrated by teachers in the previous example can be internalized into students' thinking mode and manifested as students' thinking process after certain consolidation and strengthening, and the obtained methods can be applied to the calculation of the surface area of combined graphics as shown in Figure 3.
Open up a broad second classroom
The teaching value of expanding the problem is obvious, but it often takes a lot of time. For example, a question has been taught for a class, but there is still a feeling of wanting more. This is really a big problem for teachers, so we can only try our best to arrange the class hours skillfully, extend them after class, and make great efforts to open up a broad second class so that the extended questions can play their due role.
There are three main ways for teachers to do this. 1) Organize a math study group. Assign the task of expanding topic inquiry to the group leader, who will lead the group members to carry out inquiry activities in his spare time, and the teacher will know the situation of inquiry activities from the group leader in time and give appropriate guidance. 2) Offering mathematics activity courses. Mathematics enthusiasts, mainly top students, will be organized to set up activity classes to ensure that there is enough time to start problem teaching. 3) Carry out small math challenge activities. Set up 2-3 extended questions every week, collect correct answers and optimal solutions within the grade range, conduct inter-class competitions, and award prizes once a month.
To sum up, expanding questions have irreplaceable teaching value, so teachers should attach great importance to them, dig deeply, carefully design teaching plans, and organize and implement them carefully, so as to give full play to the teaching value of expanding questions and effectively promote students' development.
(Author: Tongshan Primary School, Shengzhou City, Zhejiang Province)