The learning field of the mathematics course "Space and Graphics" in the compulsory education stage is mainly composed of four contents: cognition of graphics, graphics and transformation, graphics and position (coordinates), and measurement (graphics and proof). The teaching goal of "Figure and Position" is to enable students to describe the positional relationship between objects in an appropriate way. Students learned to distinguish up and down, front and back, left and right in specific situations, and determine the position of objects according to the ranks. In the second volume of the third grade, I learned eight directions, such as southeast, northwest and so on, and personally experienced the knowledge of directions and felt the close connection between the knowledge of directions and daily life; In the study of "Position and Direction" in the second volume of Grade Four, I further realized that the position of an object can be determined by the direction and distance in the plane, and drew a simple road map. In the first volume of the sixth grade, I learned to use number pairs to represent the position of objects in specific situations or to determine the position on grid paper, further improving my existing experience and enhancing the concept of space, laying the foundation for the third phase of learning "graphics and coordinates".

First, the academic status and teaching material analysis

1? Analysis of academic situation.

Before the fourth-grade students in a school learn "orientation and direction", we investigated the cognitive basis and life experience of a class of students in order to understand the knowledge basis of students' "orientation determination". The question of the test is: Please describe the position of your class and your own math representative on the manuscript paper. Several ways for students: (1) the position of the representative of the text narrative department; (2) graphically representing the position of the branch representative; (3) A child said, "The monitor is the second one in my upper right corner."

The students are rich in ideas. So, what is the connection between the students' ideas and the methods in the textbook? This is a puzzle that front-line teachers often encounter in teaching, and it is also a problem worth pondering.

2? Teaching material analysis.

(1) From the comparison of the contents of "figure and position" in primary and secondary schools. Whether it is primary school or junior high school mathematics, the curriculum standard divides the learning field of "space and graphics" into four aspects (as shown in the following table).

As can be seen from the table, the "figure and position" in primary school has been upgraded to "figure and coordinate" in junior high school, and the "figure measurement" in primary school has already mentioned the height of "figure and proof" in junior high school. From the comparison table, it is found that there are two similarities between primary school and junior high school. One is to describe graphics from multiple angles (shape, size, structure, position, relationship and change between objects or graphics, etc.). ) from the past, it simply emphasized the calculation and proof of graphics; Secondly, in the field of "space and graphics", both primary school and junior high school emphasize the cultivation of students' spatial concept, geometric intuition and reasoning ability (rational reasoning and deductive reasoning), but the emphasis is different and the requirements are different.

There are also several differences in learning between primary school and middle school. First, for the study of graphics, primary schools mainly focus on intuitive cognition, and explore and confirm some properties through operation and measurement. In middle school, we should not only confirm it, but also describe it in geometric language and prove it with basic facts (axioms) and theorems. Second, primary schools emphasize some characteristics of a single graph, and middle schools begin to study the status quo, structure and position relationship of one or different types of graphs; Third, there are differences between the strength needed and the way to achieve the goal. Taking "Figure and Position" as an example, the primary school's learning task is to "learn some methods to describe the relative position of objects and establish a preliminary spatial concept", while the junior high school's requirement is to "learn the method to determine the position of objects by using coordinate system and develop the spatial concept". The study of "figure and position" in primary school is mainly based on students' life experience and intuitive feelings, while the study of "figure and coordinate" in middle school is based on students' thinking development level, that is, "only when children can imagine abstract coordinate axes such as vertical axis and horizontal axis, and use this as a means to establish the relative position of surrounding objects, will they begin to express space". (Piaget)

(2) From the characteristics of the concept of space. Since the second phase of primary school, students have begun to describe the positions of figures and their relationships with the help of "number pairs", which involves the expression by algebraic methods, which is a difficult point for primary school students to learn. How to transition from the physical position representing intuitive life to the position of points on more abstract plane graphics? Pupils need rich spatial concepts to learn and master these complex geometric knowledge. On the one hand, it is necessary to explore children's existing geometric cognitive experience before teaching, on the other hand, the curriculum content shows that students' learning of "position" is a spiral and gradual process. Only under the guidance of space concept and space intuition can we simplify the complex.

(3) Determine the position from the perspective of two mathematical principles. In middle school, the position is often expressed in plane rectangular coordinate system and polar coordinate system (see the figure below). In primary school mathematics, the content of determining the position by ranks, directions and distances actually contains the idea of these two coordinate systems, both of which can be expressed by number pairs. As shown in the figure:

The point m in the plane rectangular coordinate system represents m (3 3,3). In polar coordinate system, point M is described by the length (polar diameter) of OM and the angle (polar angle) of OM relative to polar axis, such as M (4 4,45).

From the above analysis, it is found that the representation of students in the above analysis of academic situation is reasonable, which embodies the application of the preliminary ideas of plane rectangular coordinates and polar coordinates in daily life.

Second, "positioning" teaching suggestions

1? Do academic research well. Through the investigation of students' learning before class, we can understand students' original problems and puzzles, their real thinking process and development level, and accurately grasp the cognitive starting point of students' learning. Through preschool research, we can see that children have a better original experience before learning "location", and the key is how teachers dig and use it. In teaching design, we should consider how to establish some connection between students' original ideas and mathematical methods, such as the corresponding relationship between text description, geometric figure description and digital description, and then extend it to the method of determining the position in mathematics with "number pairs".

2? Grasp the knowledge order of figure and position. The knowledge order of primary school textbooks: up and down, front and back, left and right, ranks ―― observing objects from different directions ―― eight directions such as southeast and northwest ―― the position and direction on the plan, the road map ―― the number pair indicates the position. This is presented according to the order of students' cognitive development and a certain logical relationship. When teaching "position", teachers should not only consider students' cognitive order, but also pay attention to the knowledge distribution of this part of the content in the new curriculum, and organically integrate and coordinate them so that students can obtain the perfect unity of "process and result" in their activities.

3? Make full use of the function of dynamic representation. The evolution of children's concept of space is carried out simultaneously at the levels of perception, thinking and imagination, and develops along their own paths. In teaching, we should pay attention to coordinate the development of the two, from perception to image, and then extract the quantitative relationship from image to form the dynamic representation of geometry learning. The formation of representation is a process of thinking and abstraction. Taking the teaching of "position" as an example, the concept of space is first manifested as "imagination", that is, abandoning other attributes of an object, summarizing its geometric forms such as shape, size and position, and abstracting an object as a point to examine its position in space, which is a thinking difficulty for primary school students when they learn to express the position of an object with numbers. When teaching, you can gradually narrow the "character" into a point with the help of courseware demonstration, so that children can understand that this point still represents someone or something; On the other hand, students can also use the function of dynamic representation to understand the shrinking process of gradually abstracting people or things into points.

Third, the case

Subject: Determine the location (1)

1? Guide the class.

Teacher: Students, how many students are there in your class?

Student: There are 40 students.

Teacher: I want to find your monitor among 40 students. How can I find him?

(At this moment, the monitor stood up. )

Teacher: Students, can you tell us where the monitor is?

At the teacher's prompt, the students said: the monitor is in the third row of the first group. )

Teacher: How is the position of the monitor determined? Today, I learn "positioning" with you.

Teacher: To determine the location, we must first define the "group". Now we define the leftmost group as the first group, and the rightmost group as the eighth group. The first row is closest to the podium and the fifth row is closest to the back wall. Is it okay?

Health: OK!

Teacher: Now, please look at your position and tell your deskmate your position.

Then, the teacher organizes the students to play games: please stand up in the third row of the second group! (This classmate stands up) Excuse me, classmates, how can we make people understand this classmate's position in the classroom?

Inspired by the teacher, most students mentioned "several groups and rows". After affirming the students' ideas, the teacher asked the students to quickly record the positions of the students she had read on the draft paper and asked two students to write on the blackboard. The speed of teachers' reading changed from slow to fast, and students' records gradually failed to keep up with the pace of teachers' reading. They talked one after another, and the teacher was also convenient to communicate with the students.

Teacher: Students, is it easy to use "groups and rows" to indicate the position?

Most students shook their heads. )

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Some students mentioned using two numbers "2" and "3" to record two groups and three rows; Some mentioned that it is represented by 2, 3; Some students also thought of using (2, 3) to represent this classmate's position. The teacher skillfully used the generative resources in the classroom to explain several pairs of knowledge in the communication between teachers and students, and affirmed the method of expressing "position" with (2,3).

Then, the teacher called the names of three students on the roster and asked the students to indicate their positions in the classroom with "number pairs".

The design of the course is very creative, which not only makes teachers and students familiar with each other, but also lays a good foundation for introducing teaching content. Students initially establish the corresponding relationship between "position" and "number pair". )

2? Create a situation.

Teachers use multimedia courseware to show the following scenarios:

Teacher: How does the "number pair" indicate the position of Kobayashi in the picture?

After observation, the students thought that Kobayashi's position was (4,3) through "number pairs".

Teacher: If you draw a vertical line along the fourth group and a horizontal line along the third row, what will you find?

Students find that the intersection (4, 3) of two straight lines is Xiao Lin's seat.

The demonstration of multimedia courseware lays a good foundation for students to establish the concept of coordinates, and reserves a generating point for students to learn "rectangular coordinate system" in the future. )

3? Game activities.

(1) The teacher prepared some cards. There are some pairs on the card. The teacher asked the students to go to the stage to pick up the card by name, read the pairs on the card, and then find their new position according to the pairs.

(2) The pairs on the cards that the last two students got were: (3,) and (,1). The two students can't find their new position.

The teacher pointed out that without a piece of information, a pair can't accurately determine the position. For example, the first classmate is in the third group and the second classmate is in row 1. )

4? Introduce the application of "positioning" knowledge. The teacher first introduced the method of determining the position of a point (region) on the earth by "longitude" and "latitude", then asked the students to determine the longitude and latitude of Beijing and Kunming, and finally pointed out that the launch and recovery of Shenzhou VI and VII also needed to determine the position accurately. The successful launch and recovery of Shenzhou VI and Shenzhou VII depend on the "satellite global positioning instrument"-GPS, because there are only longitude and latitude anywhere in the world. The location of a place can be determined by latitude and longitude.

Teaching evaluation:

1. This lesson is close to students' cognitive level and real life from design to teaching, which reveals the connotation of mathematics well: it takes two conditions to determine the position of a point with number pairs. There are two highlights in teaching: one is the use of teaching materials, and the other is the design of games. It is easy for students to learn, because the teaching is targeted, and students can easily understand the content and characteristics of "location", thus placing the whole teaching process in the students' "nearest development zone".

2? The teaching of this course is to introduce the main information through games (let students focus on it)-tell the position (from which several pairs can represent a point on the plane)-create a situation (abstract the mathematical information in the situation as: P(x, y), and infiltrate the idea of plane rectangular coordinate system)-play (strengthen cognition)-introduce knowledge (expand knowledge).

Author unit

Kunming Xishan district xuxiake central school

Sichuan university of arts and sciences mathematics and finance department

Editor: Li Ruilong.

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