As we know, in the initial primary school mathematics, the left and right concepts were generally introduced and used as daily life terms or as undefined original names, rather than as mathematical concepts. Since the promulgation of "Mathematics Curriculum Standard" (experimental draft) for compulsory education, the concepts of "left and right" have entered the new textbooks of primary school mathematics in various versions, and almost all of them are treated as a "knowledge point" and as the content of new teaching.
Why should the concept of left and right be introduced into primary school mathematics curriculum? This should start with the reform of mathematics curriculum content.
It is generally believed that mathematical research has two basic categories, one is quantitative relationship, and the other is spatial form. The content of spatial form in primary school mathematics used to be called "basic knowledge of geometry", but now it is called "space and graphics". Is there any substantial change in the name change?
From the content and structure of this field, it can be divided into four parts: graphic cognition, measurement, graphics and transformation, graphics and position, among which "graphic cognition" is basically original content; "Quantity" is to put the knowledge of the length, area and volume of the original quantity and the quantity part together with the content of quadrature calculation; In addition to the original preliminary understanding of axial symmetry, Graphics and Transformation has added two new contents: translation and rotation. "Figure and position" is basically a new addition, including determining the relative position of objects and expressing positions with several pairs, as well as identifying directions and drawing circuit diagrams. Of course, these contents are only a preliminary understanding.
It is not difficult to understand that the most obvious educational value of increasing the learning content of graphics and positions is to help students better understand our human living space. We live and move in this three-dimensional space. In order to make better use of living space and survive and develop better, everyone needs to have relevant common sense and understand and understand three-dimensional space. This should be the main intention of changing the name of the content, that is, changing "preliminary knowledge of geometry" to "space and graphics"
In order to let primary school students understand three-dimensional space, a more suitable starting point is probably to describe the relative position of objects with up and down, back and forth, left and right.
Second, the concept of "left and right" problems in teaching
In previous primary school mathematics textbooks, left and right were all regarded as children's common sense. In the teaching of counting, in the teaching activities of understanding the ordinal meaning of natural numbers, that is, in the teaching activities of learning "which number", it is directly applied and usually passes with a little guidance from the teacher. Although individual students (mainly so-called "left-handed" children) sometimes appear left-right confusion and left-right inversion, it does not constitute the difficulty of teaching. Because for most students, it is not difficult to distinguish between left and right. Why do we regard it as the "knowledge point" of formal teaching now, but it makes both teachers and students "in a dilemma"?
It turns out that the introduction of left and right was mainly due to the need of counting, so students were only required to distinguish their own left and right and count correctly from left to right. This requirement is not difficult for most first-grade children, and they can pass the test. Now it is expanded into a new teaching content of a class, and the teaching requirement is "to describe the relative position of objects with left and right." Therefore, the relativity of left and right is emphasized as a recognized teaching difficulty. Because we don't talk about the relativity of left and right, there is too little content in a class, and almost all the newly compiled textbooks have arranged the content related to the relativity of left and right. Furthermore, when describing the relative position of objects with left and right, a series of new problems arise. For example, in the article Math Teachers in Primary Schools (No.1 No.2, 2005), Teacher Zhou Yanfen discussed three issues.
Question 1:
Are there three triangles or four triangles on the left side of the circle?
Question 2:
Is the hunter holding a gun or a bird in his left hand?
Question 3:
Are there two smiling faces or three smiling faces on the left side of the lion?
Teacher Zhou "consulted experts, studied and discussed with colleagues, and got a deeper understanding of this issue." He thinks that as long as the' standard' is determined, it is not difficult to solve the left and right problems. " So, the answers to these three questions are:
Question 1: Because the observed object is inanimate, and the criterion for judging the left and right is the observer, there are three triangles on the left and four triangles on the right of the diagram.
Question 2: Because "people are observed, what the hunter's left and right hands hold should be based on the hunter, so the hunter holds a bird in his left hand and a gun in his right hand".
Question 3: Because the object of observation is animals, we need to consider the direction. "If the observer is in the same direction as the animal, that is, taking the lion as the standard, the lion has three smiling faces on the left and two smiling faces on the right; If the observer and the animal are in the opposite direction, that is, based on the observer (person), the lion has two smiling faces on the left and three smiling faces on the right. "
According to the comrades in the editorial department of Primary School Mathematics Teachers, they have also received a considerable number of articles about teaching, many of which involve such problems. In the author's teaching and research work, I also constantly meet teachers asking the answers to similar questions. For example:
Question 4:
How many schoolbags are there on the children's left?
There are two opinions, one is the same as the second question; Another view is that, unlike the second question, it is not to ask what the left hand is, so there should be two answers. That is, according to the observer, there are three schoolbags on the left side of the child; Taking children as the standard, she has four schoolbags on her left.
Question 5:
How many blocks are there to the left of the little panda?
There are also two opinions. One thinks that plush toys are lifeless, so it boils down to problem one; Opponents believe that the new curriculum reform requires teachers to pay attention to children's culture. In the eyes of many primary school students, teddy bears are their partners and friends. Don't we allow children to "personify" things?
This really puts the author in a dilemma.
What exactly are we teaching? Is this still math?
Third, psychological research on the formation of children's concept of left and right.
As early as 1960s, Zhu Zhixian, a famous psychologist in China, made a systematic experimental study on the development of the concept of children's left and right, and the results obtained were basically consistent with those of Piaget 1929 in Switzerland and Elgin 196 1 in the United States.
Experiments show that the development of children's concept of left and right has gone through three stages regularly:
The first stage (5-7 years old).
The second stage (7-9 years old).
The third stage (9- 1 1 year).
Children recognize their left and right directions more regularly. Children initially and concretely master the left and right directions. Relativistic children have a more general and flexible grasp of the concept of left and right.
In other words, most of the first-grade primary school students are in the second stage of the development of the concept of left and right, and we should not ask too much of them. At the same time, the experimental results also tell us that with the growth of age, children can naturally enter the third stage of mastering the concept of left and right. Because at that time, the arithmetic teaching in primary schools did not take pains to promote the development of children's concept of left and right as we do today, and imposed all kinds of difficult training.
For comparison, the following excerpts are used to identify two groups of questions that children master the left and right concepts and enter the third stage, namely:
Group 5: (During the exam, the subjects sat opposite the examiner, with pencils, knives and erasers side by side on the table. Tell me: Is the pencil on the left or right side of the eraser? Is the eraser on the left or right side of the pencil? Is the knife on the left or right side of the pencil? Is the knife on the left or right of the eraser? Is the pencil on the left or right side of the knife? Is the eraser on the left or right side of the knife? "
Group 6: (During the test, the subjects sat side by side in the main test place, and the objects remained in the same place. ) Description is the same as group 5. It is obvious about left and right. It is more difficult to discuss the two answers in the above questions 3 to 5 than psychologists to determine the third stage of the development of left and right concepts. In other words, from a psychological point of view, children establish the concept of left and right, and do not need to distinguish whether the observed object is alive or not. This speculative discussion, which can be called "knowledge grindstone", uses the concept of left and right, but it is not necessary to establish the concept of left and right itself.
Fourthly, the attribution analysis of the difficulty of left and right concept practice.
Why do teachers everywhere, without exception, increase the difficulty of practice in the process of teaching related concepts, leading to puzzling problems for adults? In addition to some factors such as the worship of exercise books in the market, I am afraid there are two main reasons.
First, the mindset of "digging deep holes" in teaching material research and exercise design is still having an impact. In the past, all kinds of unified examinations were frequent. In order to avoid similarity and improve the discrimination of students' ability, they are not allowed to go beyond the "outline" and have to dig deeper and deeper artificially. In addition, teachers' practice design is also affected. In order to capture all possible variants, they constantly strive for novelty and flexibility in practice. It leads to the difficulty of teaching materials and the difficulty of exercises. Moreover, because the teaching content of primary schools is particularly "narrow and shallow" in terms of mathematics itself, the result of deep digging is that there are often variations and even deviations from the essence of mathematics. This is the characteristic of exam-oriented teaching, and it is also a practical problem that the new round of curriculum reform tries to "change the current situation of difficult, miscellaneous, biased and old curriculum content". At present, many areas have clearly stipulated that the first and second grades are not unified, but it is difficult to eliminate the long-standing mindset.
Secondly, the goal of "describing the relative position of objects with up, down, left, right, front and back" in mathematics curriculum standards needs to be further concrete and clear. For example, you can add a description of the behavior conditions and performance of the target, or give a case to illustrate it.
Five, how to grasp the teaching requirements of left and right concepts
Since the concept of left and right does not make deliberate teaching efforts, students can naturally form and develop; Because most students in grade one are in the second stage of the development of the concept of left and right, the corresponding teaching goal orientation should be appropriate. We can't make efforts against the law of children's development for other reasons.
So what kind of difficulty is more appropriate? Generally speaking, freshmen can initially understand and master the relativity of left and right. Specifically, as long as they can correctly distinguish the opposite person, which hand is left-handed and which hand is right-handed, it is enough to judge the left and right sides of the opposite person. Don't make a fuss if some students can't correctly distinguish themselves from the opposite person every time. Because children's development is fast and slow, not every normal 7-year-old can reach the level of mastering the concept of left and right. It is also an optional strategy to let children develop themselves to solve the development problem of left and right concepts. Because of the slow development of the concept of left and right, it has little influence on mathematics learning in the second and third grades of primary school.
As for the application of connecting with real life, a situation with appropriate difficulty and practical educational significance is "going up and down the stairs to the right". For example, People's Education Publishing House and Jiangsu Education Publishing House all have this content in their experimental textbooks.
This kind of situation, which has both one's own left and right sides and the other's left and right sides, constitutes a comprehensive application exercise, which is difficult to distinguish. But because students have certain school life experience as the foundation, most students can understand.
In fact, from the teaching purpose of going up, down, front, back, left and right, the main purpose is to help students understand the relative position of objects (such as the relative position of students going up and down stairs) and gradually form the concept of space. Therefore, it is not necessary to introduce the transformation training of judging criteria from people to things, nor to distinguish whether the observed object has "life", let alone discuss various answers in front of the whole class.