First of all, there are various learning tools-promoting thinking in operation.
The purpose of hands-on operation is to let students realize and reflect their own thinking activities with the help of intuitive activities, but without the participation of thinking, hands-on operation will lose its value. Can we use colorful and varied learning tools to promote students' active thinking? The above idea stems from an inadvertent mistake.
Teaching "Cylinders and Cones", the attached pages of the textbook provide cylinders and cones with equal bottoms and equal heights, because when teaching "Characteristics of Cylinders and Cones", students are arranged to make the learning tools provided in the book. When learning cone volume, many students can't find their own learning tools. "What should I do?" I gave a suggestion to those careless children: "Just imitate other people's learning tools and make a set of cylindrical cones." The students are ready to go. After school, I also began to prepare for this course. I suddenly found that an important link was neglected when arranging tasks: the base and height of cylinder and cone must be equal.
The next day, the lesson of "the volume of a cone" was attended as scheduled. Show cylinders and cones with equal bottom areas and equal heights to help students understand the conditions of "equal bottom areas and equal heights", and then ask students to estimate: What is the volume of this cone? Students have speculated that 1/2, and a few people think it is 1/3.
I took the opportunity to continue the following teaching: "What is the volume of a cone? What method can you use to verify your estimate? " The students shouted, "Do the experiment!" "Fill a conical container with sand, then pour it into a cylindrical container and see how many times it takes to fill it." The students experimented methodically and quickly came to the conclusion that "the volume of a cone is 1/3 of that of a cylinder."
Before I could speak, a student, like discovering a new continent, held up the column representing the math class at the same table and shouted, "No, this column is full of twelve cups of sand, only half full!" " I looked at this group of containers. It turned out that she made a small cone and a very tall cylinder. The students are very happy, and they are all meditating after laughing. I was puzzled myself: "Hey, why are the twelve cups only half full?"? It seems that the volume of a cone is not 1/3 of a cylinder! " Soon, a small hand raised: "Cone and cylinder must have the same bottom area and the same height!" " ""When the bottom area is equal and the height is equal, the cone volume is 65438+ 0/3 of the cylinder volume! "The children are excited about their new discovery. It seems that the mark of equal bottom and equal height has been quietly engraved in the students' hearts.
Looking back at this clip, I thought there would be some twists and turns, but I overcame them inadvertently. I'm glad that I got an unexpected harvest because of a momentary indiscretion. Imagine that if the teacher carefully arranges before class, so that students can prepare cylindrical cones with equal bottom and equal height, and the operation is uniform, then the difficult breakthrough may be dull and the students' understanding may not be profound. It is the huge contrast of students' error resources-"more than a dozen cups have only half a cylinder", which makes students emotional, thinking constantly collide and the enthusiasm for exploring problems is high.
"Unintentionally inserted willows into the shade", but a coincidence may not explain every wonderful time. Chewing this incident has given me a new revelation: it is a good means to arouse mathematical thinking and induce "problems" in operation. If there are no problems, students will not feel the existence of problems and will not think deeply, so the operation activities can only be superficial and formal. How to generate problems in operation and promote thinking? We can start with the change of school tools. Sometimes the tools provided by schools don't need to be unified, but we need to spend some time and deliberately "embarrass" students, thus arousing their demand for exploring and solving problems.
First, seek common ground while reserving differences. That is, organizing learning tools from different angles, changing the non-essential characteristics of things, highlighting the essential characteristics of things in various forms of expression, so that students' understanding of concepts can reach a higher and higher level of generalization. Secondly, find the difference of counterexample. It refers to deliberately changing the essential characteristics of things, transforming them into other things similar to it, comparing and highlighting the essential characteristics of things in comparison and speculation, so as to understand concepts more accurately, discover laws, and constantly think, explore and discover in operation.
Second, learning tools hide and disappear-stimulate imagination in operation
Hands-on practice can't always stay at the level of practical operation, and the goal should be to realize the internalization of activities. One of the ways to internalize activities is imagination. If we can develop mathematical imagination while operating intuitively, we can often better cultivate students' spatial imagination ability and develop spatial concepts. How to cultivate students' habit of consciously integrating imagination in operation? It is a way to make learning tools invisible.
Method 1: Invisibility before operation. For example, in the category of "rotation", we explore "rotating the triangular ruler 90 degrees around the O point". Let the students imagine what the triangle ruler will look like after rotating 90 degrees around the O point. Then draw an imaginary picture and draw a sketch on the draft paper. Finally, the triangular ruler of the operation learning tool is verified. In this way, the proper invisibility of learning tools before operation pushes the implicit imagination to the foreground, making the thoughts in students' minds fully explicit and explicit.
Method 2: Hand-painted "learning tools". When applying the knowledge of "space and graphics" to solve practical problems, a large number of realistic prototypes are often needed as support. Facing every new problem, it is impossible and unnecessary to intuitively perceive and witness every time, which requires students' strong spatial imagination. With the gradual fading out of intuitive learning tools and the gradual strengthening of spatial imagination, we strive to find a bridge for students to achieve a steady leap. The following example may be a glimpse.
After teaching Volume of Cylinders and Cones, I came across a problem: a cylinder and a cone have the same bottom area and the same volume. If the height of the cylinder is 18cm, the height of the cone is () cm. How to make every student deeply understand the relationship between cylinder and cone volume and grasp the change and invariability? I had a brainwave and thought of a stroke. "The bottom areas of these two cylinders and cones are equal. If their heights are equal, what does volume matter? " I made two circular gestures, as if "holding" two figures. "The volume of a cone is 1/3 cylinders with the same height as its bottom surface." The students answered very quickly.
I continued gesturing and slowed down so that every student could follow my gesture: "If their bottom areas are equal, is there any way to make their volumes equal?" The student made a high gesture, "A little longer!" "That is to triple the height of the cone!" "Is there any other way?" I asked the students to stand up and show them to their peers. He made a hard compression action, "reducing the height of the cylinder by three times."
I continued to ask, "Is there any other way to make them equal in volume?" The students thought for a moment and stood up one after another to "demonstrate": "To make the cylinder and the cone equal in height and volume, we can triple the bottom area of the cone or triple the bottom area of the cylinder. Like sign language with explanation, the whole class began to speak and demonstrate, as if they really saw a tall and thin cylindrical cone.
In fact, there are a lot of imagination resources in 3D graphics teaching. With the help of gestures, students draw virtual three-dimensional graphics, which clearly show the vague graphic outline in their minds. Then, facing the graphic representation that I can clearly feel, I will determine the solution to the problem. "Hand-drawn learning tools" fully show what imagination is and help students build the imagination space needed to solve problems. In the long run, strokes promote imagination, express abstract things in an intuitive form, and enhance students' spatial concept.
Third, lack of learning tools-awakening cooperation in operation.
Wei Jie, a special teacher, said in a lecture, what is the value of cooperation? It shows up when cooperation is needed. Therefore, it is necessary to turn the requirement of cooperation into a need of students, and the change of learning tools can turn this requirement into a need. Indeed, in many math classes, students use many learning tools carefully prepared by teachers. What they have to do is to choose-choose their favorite learning tools, not cooperate. How to stimulate students' sense of cooperation in operation?
In the teaching of "area unit", in order to let students experience the practical application of "square decimeter" area unit, I asked students to do an operation exercise: please cooperate at the same table and measure the area of the class desktop with 1 square decimeter paper. Just after operating for a while, the classroom began to stir up. The original 10 1 square decimeter paper in the hands of two people at the same table has been released. Faced with desks that are not full, the students are at a loss. Some looked at each other empty-handed, some looked at me for help, and some simply cried and said, "Teacher, this is not enough ..."
I smiled: "Two people really don't have enough school tools. Can you do something? " Smart students immediately realized that they hurriedly called the students around them: "Come, let's cooperate ..." Inspired by their peers, the students in the classroom were virtually divided into four groups and five groups ... They took their positions in division of labor and cooperation, some along the long and some along the wide, and the desktop of the multimedia classroom was really big enough. Four people didn't have enough learning tools, so they called another classmate to cooperate and worked hard for a while.
Boy, some desktops are dense, and some are only arranged along the length and width. I quietly asked the students to report the calculation results, and the answers were all 64 square decimeters. There is nothing wrong with the complete result. I asked the unfinished group to tell me how they put it. The student replied, "It turns out that the four of us have done the same thing, and 40 yuan is not enough, but there is no paper, so I want to know if there is a good way." Later, we thought that as long as we put 16 blocks along the length and 4 blocks along the width, we need 4 pieces of paper as long as we put them in, so the area is 16 times 464 square decimeters along the long pendulum. " The students said yes again and again, and some quietly revised the plan according to the suggestions. ...
When preparing lessons, I will consider: 1 square decimeter of paper, how many square meters is appropriate? If enough school tools are prepared, it may be time-consuming to operate alone, and the second innovative arrangement may be difficult to show off. Providing fewer learning tools is intended to awaken students' need to actively seek cooperation when there are not enough pieces of paper for two people. When a group finds that many people don't have enough learning tools, they sprout innovative methods, which is precisely the most beautiful voice of cooperative learning.
The core of cooperative learning is whether students have a strong demand for cooperative learning. Psychologist Scott believes that everyone needs others and has psychological requirements for interpersonal communication. Therefore, in cooperative activities, teachers' leading role lies in awakening students' inner psychological needs for cooperation and making students change "I want to cooperate" into "I want to cooperate". We might as well sell the school tools-the prepared school tools are "short of two pounds" and "you have nothing", so that students can naturally have the internal driving force of cooperation that they must cooperate to complete the task, so that they can cooperate from the learning needs of students. Once the desire for cooperation is aroused, business activities will move from formal cooperation to real substantive cooperation.
To sum up, if students can actively think, fully develop their imagination and actively seek cooperation in the operation activities, then we can say that we have developed good mathematical operation habits. A small change in learning tools can start with the change of learning tools, promote students' active thinking, stimulate unlimited imagination, awaken cooperation consciousness and form habits in operational activities.