Observing Objects (Part Two)

? The function of human eyes is much more powerful than that of light bulbs. We can see a rich and real object, not just a shadow. However, if we assume that our eyes are a light bulb, ignoring other factors and only paying attention to the outline of the object in one direction, it is similar to the projection of the object on the wall under the illumination of the light bulb.

Perspective geometry is closely related to viewpoint, and children generally experience two important nodes: one is to realize that eyes can be used as viewpoint; Second, opinions can be changed. Once the viewpoint of the same object changes-either its own position changes or it becomes another viewpoint (such as a light bulb, or another person, the observer's own real position does not move, just imagining that his observation point has changed)-the result will change.

Children before the age of five only instinctively see everything in the objective world with their own eyes, and can't consciously confirm that their eyes are a viewpoint. For example, if a four-year-old child is asked to add more pieces between two pieces at a distance of one meter on the desktop, so that they form a straight line, then most of the results released by the child are curved curves. They are still in the concept stage of topological space, and they can neither realize the shortest line segment between two points (European space concept) nor consciously use their eyes as sights (perspective space concept), thus effectively correcting unqualified chess pieces (positions) and ensuring them.

Children around the age of seven can use their own eyes as sights, thus successfully completing the task of putting chess pieces on a straight line between two points, which shows that they have formed the concept of views and formally entered the stage of perspective space.

On the whole, a person's concept of space will go through the following three stages: the early concept of topological geometric space, the concepts of European geometric space and projective geometric space in primary and secondary schools (most people stay at this stage), and the highly formalized concept of space after university (formal topological concept, non-European space, projective space, etc.). Among them, the concept of geometric space in Europe is quite special. There is no misunderstanding. We generally call the spatial concept of 6- 12-year-old children as pre-European geometric space concept, while the spatial concept they built in middle school is called Euclidean geometric space concept.

On the whole, the research object of topological geometry is the relationship between elements in the same graph; The research object of Euclidean geometry is the rigid geometric features between different figures (including the same figure, of course). For example, can two triangles be congruent in all cases? The research object of photographic geometry in middle school (perspective geometry is a part of projective geometry) is the change of geometric figure and its changing law after the change of viewpoint.

? In the eyes of adults, these three numbers are completely different, but in the eyes of my three-year-old child, they are exactly the same. It turns out that in the process of children's life growth, there is one of the strangest secrets: the geometric concept of children around three years old is topological, that is to say, the whole substance that constitutes the figure is similar to plasticine, which can be stretched and compressed at will, as long as it does not break or stick, they will always be the same. Moreover, they have exactly the same topological geometric properties, such as: point A and point B always keep close relationship, point A, point B and point C always keep the same order relationship, point A and point C always keep separate relationship, and point B is always closed by point A and point C, and these figures always keep the same continuity ... In European geometry, circles, ellipses, triangles, quadrangles and so on are completely different figures.

Relatively speaking, the concept of European geometry familiar to adults is rigid. When a graph (including a line segment) is stretched or compressed, its length, area or volume will always change, which is the fundamental reason why we adults think that the above three graphs are completely different.

The 23 definitions, 5 postulates and 5 axioms directly given in European geometry, just like Gai Lou, must be laid first. These definitions, postulates and axioms are actually the foundation of the European Geometry Building. On this basis, a point is defined as a figure without size, a straight line is a figure without thickness, which can extend to both ends indefinitely, and a plane is a figure without thickness, which can extend to an infinite straight line around; On the basis of a straight line, a line segment is defined as the part between two points on a straight line (it can be known that a line segment in European geometry can have a length, but not a thickness). So, are the tracks straight? Is the most common drawing tool a triangle? Obviously not, because these common objects in daily life have both thickness and length, and they are all finite, not infinite.

When a seven-year-old takes a ruler to measure the length of a small stick, is this stick a line segment? When a nine-year-old child multiplies the length by the width to find the area of the cover of a math book, is this area the area of a closed plane figure in European geometry? The answer is still no, because no matter how thin the stick is, it is always thick; No matter how thin the cover of a math book is, it is always thick.

These seemingly contradictory questions actually touch on the essence of European geometry that we usually ignore. The concepts of European geometry such as straight line, line segment and triangle do not exist in our limited life world. They may be related to the shape of objects in the objective world, but they do not exist in these objective objects. More importantly, they are not direct and simple abstractions of objective things, they can only exist in the imaginary world of human beings, and they are invisible and intangible formal concepts.

In the long history of human cultural evolution, human beings have created a series of intermediary symbols to express these ideas, including written language symbols, such as straight lines, line segments and triangles, and mathematical graphic symbols, such as-,/,(triangle symbols).

In primary school, no matter the concepts of length and distance in one-dimensional measurement or perimeter and area in two-dimensional measurement, almost all geometric concepts only correspond to specific physical objects in the objective world. Although we can use written language symbols and graphic symbols to express them, they are far from the real concept of European geometry. The concept of European geometry is abstract, infinite and formal. The concept of geometry created by children aged 6- 12 is concrete, limited and physical. Although the concept of geometric space in pre-Europe is completely different from that in Europe, they can use the same name, that is, text symbols and graphic symbols.

? Learners of different ages have different levels of development of ideas generated in their own minds.

When children observe an object, they basically stay in the stage of the blind touching the image-when they watch it in a fixed place, they can only see the local characteristics of an object, and they will mistake the local characteristics for the whole. Today's children can adjust their position according to their needs (in fact, adjust their views) and observe objects more comprehensively.

The concept of children's viewpoint is static, and they can't consciously adjust their viewpoint in all directions, so that the perspective results are holistic, and they always mistake the local visual results for the whole, that is, the blind people touch the image results mentioned above.

Children can't fully coordinate their own concepts of viewpoint and orientation (front, back, left and right), thus accurately distinguishing perspective results.

? Put the teddy bear in the middle of the classroom and let each student observe the teddy bear from four directions and draw a sketch. Then share the outline of each bear and ask the students to tell which direction they are looking from. )

Put the teddy bear on the middle table, surround the table in Pizhou, and students will sketch on the outer ring of the table, which can effectively separate the teddy bear from the children. Then the students raise their hands after drawing one side, and then switch places to draw the other side. )

After drawing, guide the students to tell which direction they are drawing. If you can't see it, you can close your eyes first and imagine standing on the left side of the bear after this moment to see what the observed bear looks like. If you can't imagine what you saw on the left side of the bear when you left the station, you can walk to the left side of the bear and observe.

The second set: show pictures of objects and judge opinions.

(The first part is that students can judge and observe bears by observing them in different directions with their hands, which is more intuitive. The second section will directly show the abstract PPT and judge the perspective. )

These three photos were taken from three different directions in Tiananmen Square. Please judge from which direction this photo was taken.

? From the point of view of two boys, there is no problem for students to observe the new clothes of trucks, but there is a big difference in their observation of marriage in the other two directions. At this time, guide students to imagine first, present a static image in their minds, and then look for a picture of hope from four pictures. For students with difficulties, the teacher made a truck model temporarily, so that students could observe and experience from different directions, and then correct their mistakes.

Section two: discussion and sharing-different views, the same observation?

Close your eyes and imagine what the teaching building would look like if you were standing in front of the Sunbird teaching building now.

Share which direction you saw the picture.

Finally, I reached an understanding that as long as I look at it from different angles, I will see different results.

The teacher is holding a big cuboid box in his hand and observing all directions. It is concluded that the architecture in China pays attention to the beauty of symmetry. The front, back, left and right are usually the same shape and size, but there are some differences in detail decoration. For example, the Sunbird Teaching Building has the same size and shape in front and back, except the doors and numbers.

Part III: Building a three-dimensional school model.

? All the paintings are flat, but the teaching building is three-dimensional. You can find a rectangular box, regard it as this building, and then stick the pictures on it in a certain order and law. Find a student from each of the four groups, regroup into a new group, and paste the corresponding works on the four sides of a rectangular box to make a three-dimensional teaching building model. )

? Part 4: Work Sharing