What is the importance of teaching "space and figures" in elementary school? How to connect with reality

As one of the four areas of the Mathematics Curriculum Standards (the Standards), "Space and Figures" focuses on the study of the shapes, sizes, positional relationships, and transformations of real-world objects and geometric shapes, and it is an important tool for people to better understand and describe living space and to communicate. The content of "space and figures" is divided into four main areas: understanding of figures, measurement of figures, figures and transformations, figures and position. How to base on the classroom, grasp the teaching practice in this area, we put forward the following suggestions:

I. Understand the concept of the "standard", familiar with the teaching objectives

The concept of the "standard" is the basis of our classroom teaching, the teaching objectives is the direction of the classroom teaching to reach, the importance of the two is self-evident, so we have to achieve " The importance of the two is self-evident, so we must achieve the degree of "comprehension" and "familiarity" in order to achieve a more relevant teaching design, more appropriate teaching strategies, more significant teaching results.

China's mathematics syllabus and teaching materials have gone through several changes, but from the "geometry" of the curriculum content and objectives, elementary school mainly focuses on length, area and volume of the calculation, less involved in three-dimensional space content, the lack of close contact with real life, so that "geometry" intuitive advantages not "The advantage of intuition has not been given full play; over-emphasis on deductive reasoning and "formalization". At the same time, due to the relatively single way of presenting the teaching content, it is also difficult for students' spatial concept and spatial imagination to be developed effectively. Although the "Syllabus" also has a "spatial concept" of the expression, such as "can be imagined by the shape of simple objects geometric shapes, geometric shapes imagined by the shape of objects," and so on. However, there are few explanations and descriptions in the specific teaching contents and teaching requirements. The Standard aims to overcome the tendency of our country's compulsory education curriculum objectives to emphasize too much on basic knowledge and skills, to overcome the drawbacks of focusing on "concepts and skills" and neglecting "emotions and attitudes, experience and reflection, process and independent innovation", and to strive to build a human development-centered We strive to build a human development-centered mathematics curriculum content system: emphasize the real-world background of the content, and connect it with students' life experience and activity experience; increase the content of graphical transformations and position determination; strengthen geometric modeling and the process of inquiry, emphasize geometric intuition, and cultivate spatial concepts; and highlight the cultural value of "space and graphics". For example, the Standard puts forward the requirements of "understanding the golden section through examples in architecture and art" and "through the introduction of Euclid's Principle, feeling the value of the geometric deduction system to the development of mathematics and human civilization", so that students can understand the value of "space and figures". "Space and shapes" have a rich historical origin; emphasize quantity and measurement, and integrate them into the relevant contents to strengthen the practicality of measurement.

The standard pointed out that throughout the primary stage of space and graphics part of the knowledge and skills objectives are: to experience the process of intuitive understanding of simple geometry and plane figures, experience the process of exploring the shape of objects and figures, size, movement and positional relationships, understanding of simple geometry and planar figures and the basic features, feel the phenomenon of translation, rotation, symmetry, can be transformed into a simple figure, can be a preliminary Describe the relative positions of objects, be able to initially determine the position of objects, acquire and gradually develop preliminary skills in measurement (including estimation), graph recognition, and graphing. The goal of mathematical thinking is to develop spatial concepts in the process of exploring the shapes, sizes, positional relationships, and motions of simple objects and figures. The goal of problem solving is to learn to cooperate with others in problem solving activities and to communicate the process and results of thinking with others. Affective and attitudinal goals are: to feel the rationality of the mathematical thinking process to experience the exploratory and challenging nature of mathematical problems, the rationality of the mathematical thinking process and the certainty of mathematical conclusions through mathematical activities such as observation, manipulation, induction, analogy, and inference.

We excerpt these objectives distinctly, on the one hand, it is easy for teachers to comprehend, memorize and familiarize with, on the other hand, it is also a reminder of the need to integrate the teaching of each class into the context of the overall objectives, so that for the teaching of the space and graphics part of the system is not fragmented.

Specifically, the core objective of the Space and Graphics program is to develop students' spatial concepts.

1, how to have spatial concepts? The concept of the standard pointed out that: spatial concept is mainly expressed in the shape of physical objects can imagine geometric shapes, from geometric shapes imagine the shape of physical objects, geometric bodies and their three views, the expansion of the transformation between the figure; can make three-dimensional model or draw a figure according to the conditions; can be decomposed from the more complex figures from the basic shapes and can analyze the basic elements and their relationships; can describe the movement and change of physical objects or geometric figures; can use appropriate methods to develop the spatial concept of students. motion and change; can use appropriate ways to describe positional relationships between objects; can use graphics to describe problems graphically and use intuition to think. This is the direction we are taking in developing students' spatial concepts.

In order to cultivate and develop students' spatial concepts, the Standards not only add some new elements to the reference to "spatial concepts", but also make corresponding arrangements in the content and put forward some new specific objectives.

[e.g.: "Recognize the shapes of simple objects observed from the front, side, and above", "Describe the relative positions of objects in terms of up, down, left, right, in front, and behind", "Read simple road maps", and the concept of transformation. ", as well as visual content about transformations; "can identify the shapes and relative positions of objects seen from different directions" "recognize unfolded diagrams of rectangular, square and cylindrical objects", as well as rich content about transformations and coordinates. These contents are set up to be important learning resources for developing students' spatial concepts, and space and spatial concepts grow with children from the moment they enter school].

2. Developing students' spatial concepts is not isolated. Some teachers think as if only specific contents such as observing objects are developing students' spatial concepts. In fact, the recognition of shapes, shapes and transformations, shapes and position, measurement of shapes, all have an important value in the development of students' spatial concepts, and should be organically integrated in teaching.

Second, the establishment of classroom model, clear teaching ideas

In the grasp of the "standard" concept and teaching objectives, teachers may be more concerned about how to do a good job on a lesson about space and graphic knowledge. The four aspects of "space and graphics" in the "standards" are graphics as a carrier, to develop spatial concepts, reasoning ability, and better understanding and grasp of the reality of our existence as the goal, not only focusing on students to understand and master some of the necessary geometric facts, but also emphasize that the students experience the process of independent exploration and cooperation and communication, the formation of positive learning attitudes and feelings, and the development of a positive attitude towards learning. They also emphasize the process of independent exploration and cooperative communication, and the formation of positive learning attitudes and emotions. The Standards advocate the presentation of content in the basic mode of "problem scenarios-modeling-interpretation, application and extension, and reflection", so that students can experience "mathematization" and "re-creation". "Re-creation" process, do not use the "axiom definition → theorem nature → examples → exercises" structure form.

Here, we provide the corresponding classroom modeling suggestions according to the different content categories of space and figures:

(a) Recognition of shapes

Recognition of shapes is an important content in the field of space and figures. Its content includes: point line surface body awareness rectangular, square, cylinder and ball, rectangle, square , lines and their interrelationships, angles, triangles, quadrilaterals, gardens, cones, three-dimensional views and other shapes. In the understanding of the figure of knowledge teaching, we suggest that the teaching mode, the basic classroom teaching links are as follows: experience the situation, the abstract figure Practical operation, perceive the characteristics Appreciation and expansion, back to life. That is, in the teaching must focus on making students in the real world to accumulate experience on the basis of graphics, recognize the common three-dimensional shapes and plane shapes; in the rich reality of the background, through observation, operation, comparison, generalization and other experience of the nature of the common shapes and use them to solve practical problems; in the observation of the object, put together shapes, design patterns, and other activities, to build a spatial concept; to appreciate the colorful graphic world, and appreciate the extensive existence of shapes in the real world. Specifically described as:

1, so that students experience the whole process of abstracting the figure from the real situation, from three-dimensional figures to planar figures to develop the learning

In teaching, to create a life situation, so that students in the life of the space to discover the figure, to experience the process of abstracting the mathematical model from the source of reality, and to experience the close connection between mathematical figures and the real world. The process is as follows:

Life objects Physical drawings Geometric shapes (models) Back to life

Case 1 As in the knowledge of angles in a lesson, a teacher designed the following teaching steps:

(1), say the life of the angle seen: the students said excitedly: fan, red scarf, books, five-pointed star, desktop, wall, and so on a wide variety of life situations reflecting the introduced.

(2), with multimedia courseware to show the life of physical objects such as fans, red scarves, desktop, and so on, and the corner of the part of the red eye-catching labeled, reflecting the life of physical objects to the initial abstraction of the physical map.

(3), remove the physical part of the courseware, leaving only the red display of the angle of the figure, and then let the students visual observation of the characteristics of the angle. On the completion of the abstraction from the physical to geometric shapes.

Analysis: In this case we can see that the teacher based on the students' life background and knowledge background, and gradually complete the abstract observation from the physical to geometric shapes, very consistent with the students' cognitive law, and the students' understanding of the angle is more three-dimensional.

2, let the students experience practical operation and other activities, in the activities of perception of the basic nature of the figure

"Perception" is based on the corresponding learning materials, through the hands, mouth, brain and the use of the initial feeling and understanding. The development of students' spatial concepts, the accumulation of activity experience, and the experience of the nature of the figure are all carried out in observation, operation, thinking, imagination, communication and other mathematical practical activities. Here, we would like to especially emphasize the importance of hands-on operations. Through folding, cutting and piecing, drawing, measuring, building models, classifying and other activities, students have a first-hand experience of the various properties of shapes, which not only lays the foundation for formally learning the properties of shapes, but also accumulates experience in mathematical activities and develops spatial concepts. Therefore, we advocate that all students take learning aids for operation and practice, which is far more than just letting students look at the teacher's demonstration and classroom demonstration to get far more "insight" and experience of the figure. Especially for rectangles, squares, parallelograms, circles and other shapes of knowledge, we have to let the students look at, touch, fold, fold, fold, put together, cut, cut, measure, draw, trace, compare, share, do a point, do a do, and other basic practical activities, to formally learn the nature of the figure to lay the foundation.

Case 2, such as exploring the characteristics of the rectangle teaching fragment:

(1), create a graphic: the teacher sent a bag of materials to each group before the class, you can use these materials or your own materials around the way to create a rectangle?

(2), show the results: the teacher rounds, name the physical projections placed.

Methods include: arrange the sticks, draw the dot grid, put together the triangle board, put together a small square and so on.

(3), think about the discussion: what **** the same characteristics of these rectangles? What method can you use to prove it? (First think about what you are going to use to verify? And then operate the verification, and your findings and other students to share the discussion, see which group think of more ways).

(4), reporting: rectangle equal sides, four corners are right angles.

Demonstrate one by one: compare, measure, count, fold.

Analysis: In this case we can see that under the guidance of the teacher, the students have carried out sufficient practical activities, such as "compare, measure, count, fold", the characteristics of the rectangle will be more fully perceived.

Case 3: Observing Objects

Observing the Classroom

Teacher: Stand up, observe the front of the classroom, and tell me what you see.

Students: the flag, the blackboard, the schedule ......

Teacher: all turn back and observe the back of the classroom, what do you see?

Students: awards, learning garden ......

Teacher: turn to the left, what do you see?

Student: two doors, a window ......

Teacher: look at the right side of the classroom and say what do you see?

Student: .......

Teacher: Through the observation activity just now, we learned that by observing objects from different positions, we see different results.

Observation of desks

Teacher: students can not study without the desk, the teacher can not lecture without the desk, the teacher asked four students to observe the desk.

Please stand in front of the desk, back, left, back, say what you see?

Student: ......

Teacher: 4 students look at the same podium, why do you see different?

Student: ......

Teacher: because when you look at an object from different positions, you sometimes see different results.

Observing the big rooster

Teacher: Look what the teacher has brought for you.

Student: the big rooster.

Teacher: ask 4 students to come to the front to observe the rooster, you stand in front of the rooster, behind the rooster, on the left side and on the right side. Tell us what you see.

Student: ......

Teacher: Is what you see on the left side and the right side the same?

Follow-up question: different, where is it different?

Student: standing on the left side sees the tail on the left and the head on the right; standing on the right side sees the tail on the right and the head on the left.

Teacher praise: students can be observed very carefully.

Analysis: Again, we can see that in this lesson the teacher let the students experience the process of observation from different directions, from top to bottom, from far to near; let the students in the process of observation, manipulation, imagination, thinking, communication, and continue to find the connection between the physical objects and the shapes they observed, so as to form their views on the three-dimensional space and the two-dimensional plane.

3. Understanding and appreciating some interesting shapes, feeling the richness and colorfulness of the world of shapes

The teaching design of the understanding of shapes should pay attention to providing students with a rich and colorful world of shapes in order to broaden their horizons, stimulate the interest of mathematical learning, and feel the magic of the world of shapes.

Case 4 such as after recognizing the characteristics of axisymmetric shapes, the teacher arranged such a link:

Return to life, appreciate the beauty of symmetry

The teacher provided material themes are: Beijing opera face painting, paper-cutting art, architectural objects, plane shapes, letters and so on.

Analysis: all of a sudden, the students were brought to the wonderful mathematical life, not only once again to experience the characteristics of axisymmetric shapes, but also to fully appreciate the beauty of axisymmetric life, the beauty of mathematics, and to achieve the sublimation of the classroom.

(ii) Measurement of shapes

Compared with the traditional teaching, the Standards in the measurement of shapes part of the strengthening of the understanding of the practical significance of quantity. Combined with the actual life, focusing on hands-on operation, mastering the method of measurement. Attention to the choice of measuring tools and units of measurement, and interpretation of the results of measurement (error). Emphasis on estimation weakens the traditional framework centered on pure calculation (perimeter, area, volume) and unit conversion of pure quantities without practical significance. Accordingly, we propose a teaching model, the basic classroom teaching links are as follows: Combined with the context, understanding the meaning of quantity Operation experience, the establishment of the unit's appearance Explore the method, to solve practical problems. Specifically elaborated as follows:

1. Pay attention to the understanding of the practical significance of the measured quantities in the context of specific problems

For the study of perimeter, area, volume, etc., it is necessary to first understand their meaning. This is not the same as memorizing their definitions, but rather experiencing their practical significance in concrete situations.

Case 5, such as the teaching of "perimeter", the teaching situation is as follows:

(1), create a situation to sense the concept

①. animation to introduce the "week" "first and last" (board book a week).

②. Reveal the "first and last figure" is "closed figure" (closed figure).

(2) Judgment of closed figures as a basis for revealing the concept

①. First judgment, find out the closed graph.

②. Trace the week of these closed figures.

③. Reveal the definition The length of one week of a closed figure is the perimeter of that figure.

(The board book in time to complete)

(3), contact with real life

Touch the perimeter of the figures around you.

Students: desktop math book cover some objects.

Teacher: touch the cover of the blackboard (reflecting the absence of touching the full week).

(4), group work, measuring perimeter

①. Show the problem, discuss and exchange.

Division: What method do you use to measure the perimeter of the following figures?

Teacher: What measurement tools are used for each figure?

②. Ask questions about the method of measurement and the tools used.

③. Measure their perimeter and fill in the report sheet.

④. Physical projection to show the results of the measurements.

(5) Summarize

①. Do you have any gains from this lesson?

②. In real life, there are those places where the perimeter is used?

①.