1, exchange and compare in operation, and realize the necessity of effectively infiltrating mathematical thinking methods.
Let's walk into the "triangle" classroom of two math teachers and experience the different teaching effects deduced from different teaching orientations.
[case A]
The teacher asked each student to prepare two identical triangles before class.
In class, the teacher took out several triangles with squares and asked: Who can calculate their areas? The students worked out the result quickly by counting squares.
Then, the teacher showed several triangles without squares and asked the students to calculate their areas. The students were confused, so the teacher seized the opportunity to let the students discuss the problem together.
So, the teacher asked the students to take out two identical triangles prepared before class, and asked: Can you try to put two identical triangles together into a learned figure?
(Students begin to operate and get the following results. )
1: I spelled parallelogram.
Health 2: I spelled a square.
Health 3: I spelled it into a rectangle.
5. Teacher: What is the relationship between mosaic and original triangle?
6. Ask and answer questions between teachers and students to deduce the area formula of triangle.
[case B]
Before class, the teacher arranged for each student to prepare a pair of scissors, and each group prepared two identical triangles (acute angle, obtuse angle and right angle) and six triangles with different shapes and sizes.
In class, the teacher asked the students to review how we deduced the area formula of parallelogram.
Health: Convert a parallelogram into a rectangle and then deduce it.
Teacher: OK, then can you also convert the triangle into the figure we have learned, and then deduce the formula for calculating the triangle area? (In groups of 4 students, start to combine, cut and fill triangles)
After classroom communication, the students got the following answers.
1: We found that an acute triangle and an obtuse triangle can't be combined into the learned figure. (talk and demonstration)
Health 2: We also found that two different right triangles can't be put together into a learned figure. (talk and demonstration)
Health 3: We made a rectangle from two identical right triangles. (talk and demonstration)
Health 4: We use two identical right triangles to form a square. (talk and demonstration)
Health 5: We make a parallelogram from two identical right triangles. (talk and demonstration)
Then, several students use two identical acute triangles and obtuse triangles to demonstrate that they can also be combined into the learned figures.
Teacher: Did you find anything else?
Health 6: Triangles can also be transformed into learned figures by cutting and filling. (talk and demonstration)
Teacher: You are really something!
Reflection and Enlightenment: What I see from Teacher A is the shadow of "teaching textbooks", just to teach textbooks and organize teaching in the order of textbooks. There is a lack of space for students to explore independently in the whole teaching process. The fundamental reason is the lack of infiltration of mathematical thinking methods and the inability to stimulate students' mathematical thinking. However, through group cooperative inquiry activities, group discussions and classroom communication, Teacher B has fully felt the "transformed" thinking method, and the breadth and depth of mathematical thinking in the classroom are obviously better than the former. Therefore, we think it is necessary to study the infiltration of mathematical thinking methods in primary school mathematics classroom.
2. Experience many times in the situation and gradually refine the mathematical thinking method.
From the formation of students' mathematical thoughts, it is not difficult to find that students' mathematical thoughts can not achieve the goal of mathematical knowledge in one step, and it needs a process of continuous infiltration, step by step, and from shallow to deep. In this process, we teachers need to be a "process" intensifier, constantly using our mathematical ideas to "knock" students' thinking, so that students can accumulate, feel and be clear in the process of "knocking" again and again until the final active application.
Taking the effective penetration of the idea of "turning music into straightness" in the process of understanding the perimeter as an example, this paper probes into how to carry out teaching activities step by step around the idea of "turning music into straightness".
Teaching clip 1: Preview the design and measure the length of the circular sideline, and feel the idea of "turning the curve into a straight line" initially.
Teacher: Please take out a circle from your schoolbag. Question: Can you find a way to know the length of a circle?
1: I can know the length of a circle by rolling along the ruler.
Health 2: I'll surround it with rope first, and then measure the length of the rope to know the length of the circle.
Health 3: I first fold the circle in half twice, then measure the length of the arc with a rope, then measure the length of the rope with a ruler, and finally multiply it by 4 to get the circumference of the circle.
The design intention is to make students feel that the perimeter of a figure is surrounded by a curve like a circle. We can find a way to bend their perimeter into a straight line by folding, rolling, circling and measuring.
Teaching clip 2: Protestant design measures the perimeter of leaves and trunks, and fully understands the idea of "turning curves into straight lines".
Talk: Autumn has come and the leaves have withered. Today, leaves have become a good helper for our study. Can you measure the perimeter of the leaves you prepared with the tools in your hand?
Teacher: The teacher wants to know the circumference of this leaf. Do you have any good ideas?
Health: I can wrap a line around the perimeter of the leaf first, and then measure the length of the line with a ruler to know the perimeter of the leaf.
Teacher: Who can tell us what we should pay attention to when measuring the circumference of leaves with wool?
Raw 1: the amount of wool to be straightened; Health 2: Measure the circumference from the starting point to the end point.
Teacher: Please take out the things prepared before class and start measuring, and record the results, and you will get the answer soon.
Teacher: If you want to measure the width of the trunk of a big tree, what do you want to do? Can you use as many methods as possible? Discuss in groups of four first, and then communicate in groups.
Health 1: rope circumference; Student 2: Soft ruler; Raw 3: 1 tussah; Student 4: Students form a circle hand in hand.
Summary: For the perimeters of figures surrounded by curves like this, we can try to bend their perimeters into straight lines and measure their perimeters.
Design intention In this case, the exploration and measurement methods are divided into two levels, from easy to difficult, closer to the nearest area of students' knowledge development, and fully understand the mathematical thought of "turning joy into straightness". In the process of "turning pleasure into straightness", students not only understand the formation process of knowledge, but also cultivate their interest in exploration, appreciate the mystery in the mathematics kingdom, and further stimulate their exploration spirit and innovation spirit.
Teaching clip 3: homework design calculates the perimeter of bookmarks with different shapes to deepen the understanding of the idea of "turning music into straightness"
Teacher: Look! (Showing bookmarks) What a beautiful bookmark, especially after winding it with gold thread, the bookmark looks even more beautiful. So how long does it take to wrap a bookmark with gold thread? What is the length of the gold thread? Health 1: the circumference of the bookmark.
Teacher: Can you find a way to calculate the circumference of bookmarks? Two people at the same table cooperate to finish. (Hands-on operation by students and guidance by teachers)
Health 1: We study the circumference of a rectangular bookmark. We measured it with a ruler. One of its lengths is 1 1 cm, and the other is 5 cm, so it is 16 cm. Multiplied by 2 is 32 centimeters.
Health 2: We are studying a diamond bookmark. We measured that one side of it was 6 cm with a ruler. Because all four sides are equal, multiplying by 4 is 24 cm, and this is its circumference.
S3: We are studying oval bookmarks. First, we wrap a rope around it, make a mark, and then measure it on a ruler. The circumference is 30 cm.
Health 4: We are studying heart-shaped bookmarks, which are also wrapped with rope first and then measured on a ruler. Its circumference is 36 cm.
Teacher: The students are really amazing. They came up with different methods for bookmarks with different shapes.
The fragment of design intention is designed by creating a question situation: "Put a gold thread around the bookmark and ask: How long does the gold thread need at least?" Arouse students' inquiry. Teachers provide different learning materials for students' learning activities, including bookmarks (rectangles and diamonds) that can directly measure the circumference with a ruler, and bookmarks (ellipses and hearts) that need to be wrapped with a rope first and then measured with a ruler, so as to deepen students' understanding of the mathematical idea of "turning curves into straight lines" and let students combine bookmarks with different shapes to experience the diversity of measurement methods in the process of communication. In class, we are glad to see that students are fully capable of cooperating to solve such practical problems, and their potential has been fully exerted again in the activities.
When reviewing the design of this lesson, I will let the students explore and measure the length of a circle-perimeter straight line by previewing their homework, so that they can initially feel the idea of "turning a curve into a straight line", and then let them try to measure the leaves of irregular figures after getting the definition of perimeter, and explore this type of perimeter measurement method through cooperation and communication. After that, the rules and irregular bookmarks are measured, so it is logical to refine the mathematical idea of "turning joy into straightness" step by step.
3. In the comprehensive application of various mathematical thinking methods, let students at different levels experience mathematical thinking methods.
"Mathematics Curriculum Standard" points out: Mathematics education should be geared to all students, so that everyone can learn valuable mathematics; Everyone can get the necessary mathematics; Different people get different development in mathematics. Therefore, the starting point of students' learning is different, which requires us to treat them differently in teaching. "Infiltrate mathematical thinking methods systematically and step by step, try to present important mathematical thinking methods in a simple form that students can understand, and give vivid and interesting examples." This is also one of the general ideas of the new curriculum standard.
In this paper, taking the review class of rectangular square perimeter calculation as an example, how to help students systematically sort out the knowledge points of mathematics, pay more attention to the comprehensive application of various mathematical thinking methods in each unit, and let students at different levels experience the fun of using different mathematical thinking methods to solve practical problems.
Teaching clips:
1. Through observation, verification and orderly enumeration, let students experience the inner connection of the knowledge of rectangular perimeter.
(1) observation: each of our classmates got two such rectangles (No.1: length 5 and width 4) (No.2: length 7 and width 2). They are different in length and width. What do you think of the perimeters of these two figures?
(2) How can we know the perimeters of these two figures? (Measure the length and width, and then calculate)
(3) The number of students, report: (In order to make us see clearly, the teacher enlarged these two rectangles and pasted them on the blackboard) What is blackboard writing (5+4)? What does 7+2 seek? )
(4) Question: The length and width of these two rectangles are obviously different. Why is their circumference 18cm? (The sum of one length and one width is 9)
2. Orderly listing.
Do you have a rectangle with length and width of 18? How can we find all these rectangles without repetition or omission?
(1) Q: Think about it first, and then discuss it with the children at the same table!
(2) Students' discussion report: (whether there is repetition or omission) (computer demonstration)
(3) What is used to determine the perimeter of a rectangle?
Summary: Yes, when the sum of the widened lengths is determined, the perimeter of this rectangle is also determined.
3. Cut out the largest square from the rectangle, calculate the perimeter of the corresponding figure, and realize the benefits of sketch.
(1) Review the characteristics of a square: What determines the perimeter of a square? Why?
(2) Cutting: Can you cut the largest square from the rectangle 1?
Show: Hold up the square you cut. Who wants to tell everyone what the side length of the square you cut is? Has anyone ever cut a square with sides longer than 4 cm? Why can the side length of a rectangle 1 cut into a square only be 4?
(3) Study the remaining small rectangles: Are there any remaining small rectangles? Can you find its circumference, too? Give it a try.
Report: How did you ask for it? Can anyone calculate its circumference without a ruler? You can know its length and width without a ruler.
(4) Study the No.2 rectangle through sketch.
If you want to cut the largest square from rectangle 2, what should be the side length? What determines the side length of a square?
I won't cut it this time. The teacher drew rectangle No.2 on the blackboard. Can you point out the largest square on the picture?
Look at this sketch. Can you find the perimeter of the remaining rectangle?
Is there a more ingenious way to find the circumference of this small rectangle? The teacher gives you some inspiration: What is the relationship between the widening of the original rectangle and its length?
The design intention is to let students guess whether the perimeters of two rectangles with different shapes are equal, on the one hand, the teacher evokes students' memory of the calculation method of the perimeter of the rectangle; On the other hand: the problem-solving strategy of infiltration observation, conjecture and verification. The teacher's teaching did not end here, but questioned: the length and width of these two rectangles are obviously different. Why is their circumference 18 cm? Do you have a rectangle with a length and width of meters and a circumference of 18 like this? How can we find all these rectangles without repetition or omission? Let the students think for themselves, discuss at the same table, and use the problem-solving strategies listed one by one to find out the answers that are not repeated or omitted, so as to effectively infiltrate the problem-solving strategies listed one by one into the students. Next, the teacher asked the students to cut out the largest square from the rectangle and asked: How to find the circumference of the remaining small rectangle? When the students measured the perimeter of the small rectangle with a ruler, the teacher did not stop exploring, but instructed the students to convert the text into graphics through sketching and calculate the perimeter of the remaining small rectangles. Then he asked: What is the relationship between the circumference of the remaining small rectangle and the length of the original rectangle? At this time, students' thinking is blocked and they don't raise their hands in class. The teacher pointed to the sketch drawn on the blackboard, gently traced it with red chalk, and nudged it at the right time to guide the students to find the law that the circumference of the remaining small rectangle is twice that of the original rectangle, and help them further understand the benefits of drawing to solve problems.
In this teaching clip, the teacher starts from the basic knowledge points of calculating the perimeter of rectangle and square, and comprehensively uses mathematical thinking methods such as observation, guessing, verification, enumeration and drawing, which not only makes students' thinking level reach a new height unconsciously, but also makes students of different levels develop differently.
4. Understand in reflection, apply in understanding and grow in application.
On the one hand, the acquisition of mathematical thinking method needs teachers' conscious infiltration and training in teaching, but it depends more on students' understanding in learning reflection, which is irreplaceable by others. Therefore, in teaching, teachers should guide students to consciously check their thinking activities, reflect on how they found and solved problems, what basic thinking methods, skills and technologies were applied, what detours they took, what mistakes they were prone to make, why, what lessons they should remember and so on. In the process of solving practical problems, it is often necessary to use multiple methods at the same time to be effective.
I often organize some small-scale follow-up surveys in my class, organize students to exchange the optimization strategies of using some mathematical thinking methods to solve problems reasonably, and contribute to the decimal newspaper by publishing some good local laws in the mathematical tabloids, helping students to constantly reflect, use rationally and taste the fun of success. I often practice using mathematical thinking methods to solve practical problems in parallel classes and experimental classes at the same time, constantly reflect on my teaching behavior and improve my understanding of how to effectively infiltrate mathematical thinking methods.