Montessori mathematics
Curriculum theory
First, Piaget's concept development theory of children's number
Piaget observed children's psychological development by clinical research in the process of studying human cognition. In the early development of children's number concept, Piaget proposed that the development of children's mathematical ability and the understanding of logarithm depend on the development of their logical concept. For example, the development of the sorting ability of objects and the concept of numbers can be explained by the corresponding characteristics of the logical structure of thinking.
Piaget also believes that the development of children's mathematical concepts is the process of their active construction, while the cultural transmission and acquired learning experience have little influence on the formation and establishment of children's mathematical concepts. Just like, the process of learning a language is a similar feature of different nationalities. However, he believes that teachers are supporters of children's development and their basic task is to promote children's active learning.
Piaget's theory on the development of children's mathematical ability has the following three inspirations for us.
1. He thinks that children have different cognitive operation structures at different ages, which plays an important role in the development and learning of children's mathematical ability. That is, the development of children's number concept is closely related to the development of children's logical thinking ability. The difference between mathematical knowledge and other knowledge is that it contains a certain logical relationship, so the understanding of children's logarithm concept must include the understanding and understanding of logical relationship.
He also believes that the concept of number is the late result of the development of children's logical thinking ability, and children's mathematics learning and understanding took place under the premise of cognitive level at that time. Just like a child's speech is based on the maturity of the pronunciation organs and the development of understanding ability. Piaget pointed out that "the construction of number concept and the development of logical thinking are linked." The conceptual level stage of the former number corresponds to the level stage of the former logical thinking, thus forming a system of mathematical operation and logical thinking. Mathematical operation comes from the generalization and integration of logical thinking. "
2. The acquisition of mathematical knowledge, like other cognitive activities, is not innate, nor a direct response to external stimuli or from one's own experience, but a process in which children actively construct and "re-invent" themselves in activities. In other words, children do not understand and learn mathematics knowledge through oral education and explanation by adults. If the children themselves do not establish this active learning process, even vivid explanations can't make them truly understand. The most fundamental thing is that children's learning mathematics is a process of active construction and exploration on the basis of their own experience. Therefore, for children's mathematics learning, the activity process is more important than the result.
3. Piaget emphasized the important role of action in the development of children's mathematical ability.
He believes that children's application of mathematics is the premise of internalization of cognitive operation. Children's mathematics learning is inseparable from the operation of objects, which is an active and self-disciplined activity.
Piaget pointed out: "It is a great mistake to think that children only get the concept of numbers and other mathematical concepts from teaching." On the contrary, to a considerable extent, children develop these concepts independently and spontaneously. "When children operate school tools, we can't simply think that if children operate, they will be able to achieve the predetermined functions in these school tools. In fact, children's experiences and abilities are closely related to their cognitive development. To let children get the concept of number, they need to accumulate a lot of experience, and there are all kinds of calculation and experience activities related to thinking and action in life. Therefore, teachers should create an environment, make use of various environments in life, and provide children with opportunities to operate and experience exploration; You can also use Montessori's materialized mathematics teaching AIDS to give children the opportunity to discover this quantitative feature of things and related quantitative relations. These processes are of great benefit to children in learning mathematics.
On the basis of Piaget's research, American scholar clements's experimental research also shows that the cultivation of children's logical thinking and the training of counting activities can also effectively promote the development of children's logical operation ability. Moreover, he also found that the cultivation of children's counting ability can better promote the development of children's counting concept. Children's logical thinking level will affect children's learning of symbol system. If children's logical thinking development does not reach a certain level, then they will have difficulty in learning the mathematical symbol system, or they will not understand it after learning it. This proves once again that children's logical thinking ability is closely related to the development of digital concepts.
Second, German and Fu Sen's "number model" theory.
German, an American child psychologist, has long been committed to the study of early children's counting ability. Through research, he believes that children's counting activities play an important role in the development of children's early counting concepts. He found that even a 3-year-old child's counting is not only a simple language ability, but also a complex cognitive ability to abide by the principle of counting.
Children's early counting ability interacts with the development of children's thinking ability. The two complement each other.
Fu Sen, an American child psychologist, also concluded that the establishment of the concept of early childhood counting is closely related to the development of children's counting ability. The development of children's counting concept is helpful to the integration and application of children's counting skills. At the same time, children's learning experience directly affects the establishment and development of the concept of children's counting, and they find that children's counting is also an indispensable tool for them to learn addition and subtraction at first.
Another research view holds that children's counting experience is the result of children's acquired learning. Children's initial counting behavior is an unconscious imitation behavior, and children gradually understand the meaning of counting through counting exercises in real life and specific situations. The development of children's number concept is gradual and based on perceptual experience. At the same time, it is also a process in which social life affects this ability.
Thirdly, Montessori's theory on the development of children's mathematical ability.
1. Grasp the sensitive period of children's mathematics learning and respect the stage of children's mathematics development.
Montessori pointed out in the book Looking for Children that children will not be disturbed by what others may get. On the contrary, a victory will arouse people's praise and happiness, and some people will happily follow suit. Children seem willing to do "what they can do"
She believes that when we think that a child's wish is only to have a piece of knowledge, we repeat it many times. This is very wrong. Because, intellectually, we help children master this knowledge, but in this way, it hinders their self-development.
Montessori put forward an important principle when educating young children, that is, every age and every stage of childhood has its special needs. If these needs are not met at the most prominent time, the development of some abilities of children will be suppressed forever. This is what she thinks is the "critical period" of children's ability.
2. Early mathematics learning should focus on children's operations.
Montessori believes that children's early mathematics learning is characterized by typical perceptual experience learning, rather than abstract and rational learning. She said that we are used to serving children, which is not only for them, but also a dangerous move, because it is easy to kill children's beneficial spontaneous activities, and we don't think about it. Children who don't do it don't know how to do it.
Therefore, she believes that children's mathematics education must guide children to learn mathematics with concrete abstraction. Concrete abstraction is a prominent viewpoint in Montessori's mathematics education. Children need concrete objects to support the exploration of thinking, but at the same time, mathematics is an abstract experience and an abstraction in practice. Therefore, Montessori created a Montessori teaching aid unique to Montessori education, which is the embodiment of her educational thought.
The basic principle of Montessori educational materials is that the actors are coordinated and orderly, so that it is easy for children to judge their success or failure when they engage in activities. The use of these educational materials includes activities that children explore by themselves, as well as activities that groups do and discuss together in a specific environment.
3. Emphasize the influence of "prepared environment" on children's mathematical ability development.
Montessori attaches great importance to providing children with a large number of materials for learning mathematics. This is Montessori's view that it is necessary to set up a "prepared" learning environment for children to gain a real experience and familiarity with the concept of logarithm. She believes that the understanding and perception of logarithm is realized when children are interested in the environment.
Montessori attaches great importance to the influence of preparation environment on children's development. The environment includes not only kindergartens, but also natural environment, social environment and interpersonal environment. She said that the most important thing in education is to provide a "prepared environment" for young children. In traditional education, education includes two factors: teachers and children, teachers teach and children learn. But she thinks that a "prepared environment" should include three elements: teachers, children and the environment, and she lists the environment as the first element of education.
course features
1. Develop children's thinking ability while expanding their knowledge.
2. Provide rich materials and learn through operation and exploration.
3. Pay attention to individual differences and let every child experience success.
4. Close to life and solve life problems by mathematical methods.
Course material composition
Books for teachers: divided into 8 volumes according to the semester of small class, middle class, large class and preschool class. Each volume contains a detailed theoretical overview, rich cases of teaching activities and a systematic teaching evaluation form. Well-designed teaching activities are clear, scientific and effective, which is convenient for teachers to use.
Teaching tools: One-to-one correspondence with the learning tools, the learning tools are enlarged by 4 times, which is convenient for teachers to operate and demonstrate in teaching. Teaching AIDS "materialize" abstract mathematical concepts and are carefully designed according to mathematical goals, involving all aspects of children's mathematics field. They are not only the most essential teaching AIDS selected from Montessori's sensory education and mathematics education, but also the teaching AIDS combining the latest information at home and abroad, which are very suitable for modern children's mathematics learning.
Learning tools are important materials for children to operate and explore in teaching activities. School tools use knife molds. After being stuck, the children can easily take out all the parts of the learning tool with a gentle press of their hands, and they can play with it, combine it and assemble it freely. It is a learning tool and learning tools and similar learning tools of different groups increase step by step, which is very functional, operational and systematic.
Operation book: for teachers to guide children to use in activities in the park, which can be completed in groups or in the corner of the activity area. There are various forms of activities, such as connecting, coloring, cutting and pasting. The activities are rich in content, providing teachers with a lot of materials and injecting rich mathematical factors into the scene. Improve children's direct experience and develop their thinking through operation.
Homework paper: it is the material for parents and children to study in the family. Parents can know the progress of kindergarten teaching, and teachers can also evaluate family activities, so as to realize home education. A special column "Mathematics in Life" is set up in the homework paper, which provides highly operational mathematics parent-child games every semester to guide parents and children to carry out mathematics activities at home, and reminds parents to guide their children to discover mathematics in life and apply mathematics knowledge to life.
Course teaching process
1. Preparation activities
Teachers and children greet each other. Go online and play online games. Concentrate the child's attention, adjust the child's mood or play some small games related to this activity as an introduction.
2. Group activities
Create a certain scene and provide a large number of operating materials for children to explore and achieve the basic goal of this activity.
3. Game activities
Let children participate in the game with rich and interesting games to enhance the experience.
4. Group activities
Respect children's individual differences and provide a lot of materials for children to carry out activities selectively. Generally, it is divided into three groups: physical operation group, learning tool operation group and paper operation group. The three groups of activities either have different forms and difficulties, or use different materials in the same form.
5. Exchange summary and organize school tools.
Teaching methods of this course
First, provide a physical environment to enrich children's math experience.
In the rich physical environment, children's mathematical ability is developed when solving problems in daily life, including the concepts of space, size and quantity. This ability to understand and use concepts when solving problems is the purpose of mathematics education.
Teachers should provide children with sufficient physical environment, guide and encourage children to use these resources and help them accumulate rich and effective mathematics experience.
The activity cases of this course provide a large number of teaching AIDS and learning tools. For example, the activities of small classes last semester were classified, and its "teaching aid preparation" was like this: "supermarket games"; There are red, yellow and blue hula hoops, and there are "supermarket games" with "learning tools to prepare"; Colored fish; One red, yellow and blue round card for each person; Children's clothes, toys, etc. It can be seen that these preparations provide children with sufficient physical environment, and guiding children to use these resources can help them accumulate mathematical experience.
At the end of each activity case, there are extended activities, whether in the park or at home, that is, "Numbers in Life". As mentioned above, in Yuantuo, you can play games such as "Doll's House" and "Shop Opening". Jiatuo can take children to the supermarket and guide them to observe how the items in the supermarket are classified and placed, so that they can classify the items after buying them.
Second, encourage children to explore through operational activities.
From mechanical memory to active construction, from symbols to practical meaning, it is an important way for children to learn mathematics. The new "Outline" also mentions "providing rich and operable materials to provide conditions for each child to explore in a variety of senses and ways."
This course attaches importance to children's operation, and each activity has collective demonstration operation, group operation and individual operation. Children are not passively assigned to operate, but have the space to choose independently. They can choose their own operating materials and modes, which are not only suitable for children's learning characteristics, but also can stimulate children's enthusiasm for learning, so that children can actively explore in repeated operations.
In addition, children's operation is not isolated, it can be organically combined with other methods of learning mathematics, such as games, communication and discussion, in order to obtain better results.
Third, attach importance to children's experience and the process of children's mathematics education.
Paying attention to mathematics learning process is an important direction of modern children's mathematics education reform, and it is an operational process that emphasizes children's active exploration and construction. Piaget pointed out that children's experience of mathematical logic does not come from the object itself, but from the internalization of children's operation and action on the object. Mathematics knowledge is experienced by children through activities, calculations and thinking activities, rather than rote learning. The role of teachers is not only to give children a result, or to ask them for a result, or to be satisfied with the result of children's activities, but to encourage and support children's exploration and learning of mathematics activities and provide them with a learning environment that interacts with materials and people.
The process of children's learning is more important than asking children to get a result. In the process of activities, there are children's happy experiences and children's profound thinking process. Paying attention to children's mathematics learning process requires us to respect and accept each child's concern and interest, respect their exploration and discovery, and respect their explanation and expression. You can discuss or explore with children, so that children can find problems, ask questions and solve problems in the process of activities.
For example, learning addition and subtraction calculation, this course allows children to start with tinkering with small objects and operating the "addition and subtraction board", initially perceive the quantitative relationship of addition and subtraction through touch, then calculate pictures, then calculate ideas, and finally enter the calculation with the help of numbers and symbols.
Fourth, with the help of learning tools, carry out "materialized" mathematics learning.
The learning tools of this course draw a lot of lessons from Montessori's teaching aid function and design a new set of effective tools for learning mathematics. These materialized materials provide children with tools for thinking in images and can help them learn math well.
In the case of teaching activities in this course, colorful mathematical activities and multi-level and multi-level life experience materials are also provided, so that both children and teachers have room for choice and adjustment.
This course has changed the teacher-centered teaching method, taking children as the main body to carry out mathematics education, allowing children to take the initiative to participate and have their own space for operation, rather than being limited to the strict requirements of teachers. At the same time, it also creates opportunities to encourage children to actively communicate and express their feelings of exploration, practice and experience.
Fifth, the theme activities are organically combined with district and corner activities.
Now, more and more kindergartens have carried out theme activities. This course can penetrate into thematic activities, and the goal of mathematical activities can be organically combined with the theme content of activities. If our mathematics education can only carry out mathematics activities on the basis of mathematics activities, it can only show that our mathematics education has not entered the life and real world.
Mathematics education can also permeate daily life and district and corner activities. Children are active and eager to explore, so it is necessary to set up a math area in the classroom. At the same time, we can also use other areas to carry out math activities, such as shops, restaurants and building areas, which are effective ways for children to learn math. In the activity area, children can freely choose activity materials according to their own interests and wishes, decide the content and methods of activities by themselves, and perceive and discover various mathematical phenomena with their senses. Therefore, this course focuses on setting up all kinds of materials that can be used for children's mathematics activities in the activity area according to the recent educational goals, so that children can freely choose and apply them, and let children unconsciously perceive and discover mathematical phenomena and gain mathematical experience.
Six, timely and appropriate conversation and discussion with children, encourage children to express their feelings and findings.
Effective questioning is a form of dialogue and discussion in children's mathematics learning. When children operate materials, teachers naturally get close to children, exchange questions about quantity, and guide children to pay attention to and think about the characteristics related to quantity through questions. For example, encourage children to explore things and give them a name; Encourage children to carry out one-on-one corresponding activities with what they have in their hands; Use outdoor activities to help children explore and describe the characteristics of things; Provide children with many similar materials or very different materials, so that children can easily experience the questions raised by teachers; Teachers can also combine the content of mathematics education with activities in other fields and raise the question of quantity in these activities. If you combine similarities and differences into games or activities, ask your child related questions. In mathematics activities, the questions asked to children should arouse children's interest, and the difficulty should conform to the nearest development zone proposed by Vygotsky, which should not only suit children's existing development level, but also promote the development of children's speech ability and thinking ability.
If the child can't answer some questions, the teacher can talk to the child in a pleasant way and guide the child to try boldly through his own operation.
Curriculum education evaluation
The evaluation of children's mathematics activities should not only pay attention to children's perception and understanding of mathematics knowledge and skills, but also pay attention to children's emotions, attitudes, experiences and development. We should not only pay attention to the results of children's mathematics learning, but also pay attention to their changes and development in the learning process.
Teachers should also consciously use evaluation methods to understand the suitability of education, adjust and improve their work and improve the quality of education. The evaluation of mathematics education activities is also a process for teachers to analyze problems, sum up experiences and reflect on themselves.
I. Five Principles of Evaluation
1. It is clear that the purpose of evaluation is to understand children's development needs, so as to provide more appropriate help and guidance.
2. Fully understand the development of young children, prevent one-sidedness, especially pay attention to knowledge and skills and ignore the evaluation of emotional, social and practical abilities.
3. Adopt natural methods in daily activities and education and teaching. Observing children's typical behavior and their accumulated works at ordinary times is an important basis for evaluation.
4. Acknowledge and pay attention to the individual differences of children and avoid using uniform standards to evaluate different children. Use horizontal comparison carefully in front of children.
5. Pay attention to dynamic evaluation and treat children with a developmental perspective. We should not only understand the current level, but also pay attention to their development speed, characteristics and trends.
Two, the evaluation should pay attention to ten points.
1. Can the objectives, contents, organization and implementation methods and environment of mathematics education activities provide children with a learning experience consistent with the educational objectives required by the new syllabus and meet the needs of children's all-round development?
2. Whether the content and activities of mathematics education are suitable for children's interests and learning characteristics, close to children's lives and attractive to children.
3. Whether the contents and methods of children's mathematics activities and teachers' guidance are suitable for the development level and needs of most children, and also reflect the respect and adaptation to individual differences, so that every child can have a successful experience.
4. Whether the contents, methods and environmental conditions of education are conducive to children's active participation in activities, exploration and creation in activities.
5. Whether the teacher's evaluation and guidance is conducive to children's further exploration and thinking, and whether it is conducive to the expansion, collation and promotion of children's experience.
6. Is the process of mathematical activities a process in which children actively learn and communicate more? Because this is not only a process of understanding, but also a process of communication and cooperation. Children can understand and explain what they have learned through communication, and can help each other, understand and implement operations. At the same time, through communication, let children build mutual trust and communicate effectively.
7. Whether children have curiosity and thirst for knowledge about mathematics activities through mathematics activities. Whether there is a chance to gain a successful experience in mathematics learning activities. Teachers help children to exercise their will to overcome difficulties and build their self-confidence.
8. Do children have a preliminary understanding of the close relationship between mathematics and human life and its role in the development of human history? Experiential mathematics activities are full of fun of exploration and creation.
9. Does the child develop an honest emotional attitude, be good at asking questions and develop the habit of thinking through math activities?
10. Pay attention to the evaluation of children's mathematics learning emotions and attitudes. If children have "math phobia", they should pay attention to their own evaluation of children, encourage, help and encourage their activities from the front, and find their little progress in time.
In a word, the main purpose of children's mathematics education evaluation is to fully understand children's mathematics learning process and encourage children to actively participate in mathematics activities. Therefore, the evaluation of children's mathematical activities should attach importance to the evaluation of the process of children's mathematical activities, rather than overemphasizing the results of the activities; We should pay attention to the level of children's individual development and their emotions and attitudes in mathematics activities.