20 15-03-27

lmzxxwhua

Mr. Ye Shengtao once pointed out that teaching is for not teaching. In order to reach the state of "not teaching" in mathematics teaching, the key is to let students learn "mathematical thinking". "Mathematical thinking" is also called mathematical thinking, which means: (1) looking at the world from the perspective of mathematicians, that is, it has the tendency of mathematicization such as constructing models, symbolizing and abstracting; (2) Have the ability to implement mathematicization successfully.

Crows, an American scholar, believes that learning mathematical thinking is the ability to form mathematical and abstract mathematical views, predict with mathematics and solve problems with mathematical tools. Some people think that mathematical thinking is a way to solve problems from a mathematical perspective when facing various problem situations, especially non-mathematical problems.

China's mathematics curriculum standard regards "mathematical thinking" as one of the goals of mathematics curriculum. The American mathematics education document "Everyone pays attention to the future of mathematics education" points out: "... Americans need to think about mathematics more than ever." The British National Mathematics Curriculum Standard also puts forward that "students have the opportunity to use a series of thinking strategies to carry out activities, consolidate and develop relevant knowledge and skills, and develop mathematical thinking ability."

The essence of "mathematical thinking" is to let students observe and analyze problems from a mathematical perspective when facing various problem situations, find out the mathematical information existing in them, and use mathematical knowledge and methods to solve problems. It can be seen that in order to think mathematically, we must first solve the problem of "what to think"; After abstracting the mathematical problems, we can analyze and study the relevant mathematical information and start the thinking process; Finally, we should improve our original understanding and accumulate thinking experience and strategies in reflection and review. Taking the teaching in cognitive field as an example, this paper discusses how to guide students to learn mathematical thinking.

First, clear observation angle, so that "thinking" has a direction.

(A) in terms of quantity and shape

The premise of thinking is that students should know what to think and how the problem arises. Otherwise, students can only follow the picture. Because "body" is more intuitive to students, I guide students to experience the cognitive process of "body-surface-surface size-revealing the meaning of area" in teaching.

Fragment 1

1, touch and know "the surface of an object".

Teacher: There are many objects in life, and each object has its surface. Can you touch the surface of this tissue box?

Teacher: Who can touch the surface of this ball? (This is a curved object)

Teacher: How many sides does a math book have? This is the cover of the math book. Let's touch it together.

Requirements: Please find some objects, touch their faces and think about their differences while touching them.

2. Compare and know the size of the "object surface".

Teacher: What did you touch? Show it to everyone (three people say one of them is a surface). What's the difference between these surfaces you touch?

Health: Flat, slippery, curved, flat, with different shapes, big and small.

Teacher: From a mathematical point of view, what's the difference between them? (blackboard writing: size)

Teacher: All the surfaces we touch are the surfaces of objects. (blackboard writing: object surface)

3. Tell me about it.

Teacher: Compare the surface of the blackboard with the cover of the math book. Tell me which is bigger and which is smaller.

Teacher: Objects have big faces and small faces. We call "the size of the surface of objects their area". (blackboard writing: the size of an object's surface is called its area) Can you tell me whose area is larger or smaller than whose?

In real life, we will see many objects, but the angles of observation are diverse. From a mathematical point of view, if we pay attention to the "shape" and "size" of its surface, it will help students to make it clear that both the surface of an object and the surface of a figure, whether it is a plane or a surface, can be studied together as an object, and naturally sum up the meaning of the area.

(2) Thinking from the concept extension and counterexample.

We expect students to have a pair of "mathematical eyes", not only to observe from the perspective of "number and shape", but also to be good at communicating the relationship between relevant knowledge and continuing valuable thinking.

the second part

1, draw a face of a physical graph and abstract a plane graph.

Teacher: If we draw along the cover border of the exercise book, what figure will we draw? What does it have to do with the original cover?

Same circumference, same length, same width and same area. (coincidence)

2. Discriminate and clear the area of closed graphics.

Teacher: You said that its area is equal to the area of this cover, but what you want to say is that this rectangle has an area. Smart students will naturally think: will other graphics have areas?

Show all kinds of pictures and let the students talk about them first. Then demonstrate it in the drawing tool.

Communication: Make clear which are closed figures (the last two figures are hidden in the courseware), and the size of closed figures is their area. (blackboard writing)

"Do other plane figures have their areas?" Quickly expand students' thinking, and transition from a special rectangle to a general plane figure. When guessing and demonstrating, demonstrate with the help of drawing tools, so that students can intuitively see that the unclosed figure has no "area". It is worth mentioning that students spontaneously cheered when demonstrating with drawing tools, which is the most eager temperature after mathematical thinking was confirmed.

Second, experience mathematical activities, so that "thinking" has a process.

To teach students to think, they must go through the process of thinking. Therefore, it is particularly important to create a mathematical activity with rich connotation and considerable research value, so that students can actually experience the thinking process and accumulate thinking experience.

(A) concentration of contradictions, so that thinking has a grasp.

Time and space in the classroom are limited, so teachers often need to concentrate on solving contradictions, guide students to sort out research problems in arguments, reflections and collisions, and improve their thinking quality.

the third part

Do the colored parts in the picture below represent the areas of these three figures?

Students think independently.

Teacher: Who do you think can accurately represent this area? What else do you want to say?

The first picture makes students understand that the area of the red part plus the area of the white part is the area of the original big rectangle. (revealing additivity of area)

The second picture asks students to make clear the difference between perimeter and area. Intuitive and profound understanding in the process of using gestures.

In the third picture, let the students know that the irregular plane figure has an area as long as it is closed, which enriches the extension of the area.

(B) Back to the source, so that thinking has a foundation.

If we say that "focusing on contradictions" starts with contradictions related to new knowledge or contradictions that easily confuse old knowledge, so that we can grasp our thinking, then we can go back to the source of life and pick out those blind spots that are not noticed or understood by students, which can make students' thinking more grounded.

When we know the "face", we often pay attention to the "real" face, but not to the "virtual" face, such as the area of the glass mouth, because the glass mouth is empty, which will make students have the illusion that the glass mouth has no area.

When the area is relatively large, the smaller area can directly see the sum, and the larger area can't see the whole at all. How to compare? This kind of practical problem will make students in a real problem state, beyond the understanding of relevant mathematical conclusions in textbooks, and have a more solid thinking foundation.

(C) pay attention to differences, so that thinking is hierarchical.

A successful math activity should attract children's participation to the greatest extent, make the whole class move as much as possible, and let different people get different development in math. Therefore, it is challenging to consider the thinking characteristics of different students as much as possible and meet their psychological needs for successful exploration.

part four

Through the study just now, the students already know the meaning of area, and they also know that there are different sizes of area. Compare any two numbers below to see which area is larger. How do you compare them? Show four pictures. (Figure 1 is a yellow rectangle of 6 cm× 4 cm, Figure 2 is an orange rectangle of 6 cm× 4 cm, Figure 3 is a green rectangle of 10 cm× 2 cm, and Figure 4 is a red square with a side length of 2 cm. The side length of each square is 1 cm. )

1, the size difference is obvious (observation method).

Students will soon compare Figure 1 with Figure 4, Figure 2 with Figure 4, and Figure 3 with Figure 4. (As soon as the students solved the first question, some students asked how to compare the area of figure 1 with that of figure 3. )

Solve the problem in groups of four. The teacher suggested that there were gifts (especially square paper) in the envelope. Please don't use it unless absolutely necessary.

2. Compare similar area sizes (overlapping method, cutting method and spelling method).

In teaching, some students are negatively transferred by one-dimensional length, thinking that two rectangles can only compare one side. When students refute the adjustment again, they must make a right-angle overlap, that is, compare the two dimensions at the same time. Some students folded the figure 1 into two rectangles of 6 cm× 2 cm, and then measured the rectangle of 10 cm× 2 cm, and found that it was only enough to put a rectangle of 6 cm× 2 cm. This method is actually measured by the same standard "6 cm× 2 cm rectangle"; Some students folded the figure in half to make a contrast between 6 cm ×2 cm and 5 cm ×2 cm. In fact, this is also looking for a standard, because the width of the folded figure is the same, as long as it is longer than it, thus turning the two-dimensional problem into a one-dimensional problem; Some directly typeset the figure 1 and figure 3 with a square with a minimum figure side length of 2 cm; Of course, there are also some conclusions that are directly measured with transparent square paper in envelopes. In the meantime, some students said that they knew the calculation method of rectangular and square areas.

3. Similar but not overlapping graphic dimensions must use the same standard (seal, rubber, etc.). ).

4. Compared with irregular polygons, it is most convenient to use squares with the same size.

5. Debate: Is the rectangular area of four squares greater than that of eight squares? Strengthening must unify standards.

In the above math activities, the students are very excited, and everyone is very excited to exchange their ideas, some are rough and imperfect, some are quite incisive, and some are unexpected. To guide students to think, we should not only look at the result of thinking, but also look at the process of thinking. Pay attention to whether to stimulate students' thinking needs, whether it is challenging in thinking, and whether every student can take the initiative to participate as much as possible. Only by letting students go through the process of thinking can students accumulate rich thinking experience, and at the same time, successful exploration experience will also prompt students to further think about the unknown journey.

Third, clear the thread of thinking and let "thinking" have experience.

(A) "Take a step back" to clarify the cognitive path.

We used to think that successful mathematics teaching should be to let students go out of the classroom without "problems". In fact, mathematical thinking is a continuous activity, and good mathematics teaching should be a spiral upward process of cognition from imbalance to balance, and then to a higher level of imbalance, so students should have new feelings and even new problems when they leave the classroom, but this should be our pursuit. To this end, teachers need to "take a step back", just like doing a blackboard newspaper, and feel whether the part and the whole are harmonious from a distance.

The purpose of this lesson is to reveal the meaning of area by paying attention to "the shape and size of the surface or closed figure of different objects", and then to know the size of area through "observation". However, when "observation" can't solve the problem, it is necessary to compare by overlapping, and the essence of overlapping is to find the same comparison standard, thus breeding the thinking method of measuring area by area. Some students' thinking mode of "measurement" not only transforms the two-dimensional "area problem" into the known one-dimensional "length problem", but also lays the foundation for learning the calculation formula of area (indirect measurement) in the future.

After the students have gone through the process of thinking, we need to guide them to "take a step back" and reorganize the cognitive path, so as to understand the thinking method contained in it and accumulate experience for the follow-up study.

(B) "Look further" and refine the essence of mathematics

Some people say that we should learn to be a "lazy teacher". Yes, a "lazy teacher" does not teach a knowledge point, but teaches a kind of thinking mode of problems, and even strings related problems together to teach students snowball learning. In fact, there are many internal similarities between the idea of teaching cognitive area and the understanding of length and volume, which integrates the concepts of one-dimensional, two-dimensional and three-dimensional. For example, for comparison method, one-dimensional lines should have the same starting point; Two-dimensional surfaces overlap, giving consideration to two dimensions, or making one dimension the same, and comparing the other dimension, its essence is to transform into one dimension; A three-dimensional body is naturally more three-dimensional. Although the dimensions have increased, the comparative thinking method is in the same strain. The simplest method is to select a "unit" for direct measurement. This is the inherent logical correlation of measuring length, area and volume.

If we think about the classroom in this way and are willing to make exploratory efforts, then students may continue to find the fun of mathematics learning and feel that mathematics is not that difficult. Slowly, they can gradually learn "mathematical thinking".