Mathematics: a way of thinking
In August 2002, during the World Congress of Mathematicians held in Beijing, Mr. Chen Shengshen, a famous mathematician in China, told reporters that each of us had received mathematics education for more than ten years in our lives, but many people only learned to calculate and didn't know what real mathematics was.
The charm of mathematics lies not only in its accurate calculation, but also in its way of thinking ―― it turns concrete problems into abstract mathematical problems, and then applies them to concrete problems by solving abstract mathematical problems. This process is also called "mathematical modeling". Therefore, it has been suggested that mathematical thinking is a way of thinking and mathematics is the science of "mode".
For example, if two people want to share a pile of candy (10) equally, they can adopt different methods: we can divide the candy into two parts by "trial and error", and then compare their quantities and adjust them until we can't see who has more and who has less; We can also give it to two people in turn piece by piece, so as to ensure that the two people get the same amount ... but with the help of mathematics, we can solve an abstract mathematical problem (how much is half of 10) without specific plots, then apply the results to this specific problem and finally solve this practical problem.
It can be seen that mathematics has two characteristics: on the one hand, mathematics is abstract, which is different from concrete things, but abstract from concrete things; On the other hand, mathematics has practical effectiveness and can solve practical problems.
Similarly, mathematics education for children has two values: one is the value of thinking training, because mathematics is an abstract process, and learning mathematics is essentially a method of learning thinking, especially abstract logical thinking; On the other hand, mathematics education can cultivate children's ability to solve problems, especially the ability to solve problems by mathematical methods.
From this perspective, we can't equate mathematics education with pure calculation, and mathematics is not just the result of memory.
Children's Mathematics Learning and Thinking Development
How do children learn math? Is it through memory or understanding? Different answers to this question are directly manifested in different educational methods for young children. Once, the parents of a three-year-old child asked me why their children always count indiscriminately, but he taught me many times to no avail. The parents of another four-year-old asked me, "Why does my child have such a poor memory?"? I have told him many times, but he still can't remember these addition and subtraction problems? " In fact, the most fundamental problem is that children do not learn mathematics through memory!
Let's analyze these simple mathematics for adults: 1. What is a number? The sequence of natural numbers-1,2, 3, 4, 5 ... seems to be a group of sequences that children need to remember, which contains many logical relations in essence. For example, there is an increasing order relationship between the front and back numbers, and each number is larger than the previous number and smaller than the latter number. This order relationship is transitive, that is to say, even non-adjacent numbers can be judged according to their positions in the sequence. For another example, the sequence also contains the inclusion relation, and each number contains the number before it and is also included by the number after it. 5 contains 1, 2,3,4,6 contains 5. ...
Children, 1, 2, 3, 4 ... What they know is by no means the name of some specific things, nor the characteristics of these specific things themselves, but the abstraction of the relationship between things. Even the simplest numbers have abstract meanings. For example, "1" can represent 1 person, 1 dog, 1 car, 1 small disk … any number of objects are "1". Another example is five oranges, which is an abstraction of the quantitative characteristics of a bunch of oranges. It has nothing to do with the size, color, sweetness and sourness of these oranges, nor with their arrangement: whether they are arranged horizontally, vertically or in a circle, there are five. Therefore, children's understanding of logarithm can not be obtained through direct perception like their understanding of size and color, but through an abstract process. None of the five oranges has the attribute of "5". On the contrary, the quantitative attribute "5" does not exist in any orange, but in their mutual relationship-they form a whole with a quantity of "5". Children acquire this knowledge not through direct perception, but through the coordination of a series of actions, specifically the coordination between the actions of "point" and "number". First of all, he must make the finger movements correspond to the mouth movements. Secondly, the coordination of order, the numbers in his mouth should be orderly, and the actions of points should be continuous and orderly, which can neither be omitted nor repeated. Finally, he will put all the actions together and get the total number of objects.
From this point of view, children's counting is only a superficial phenomenon, behind which are the development of children's logical concepts such as correspondence, order and inclusion and the development of abstract thinking ability. Only by understanding these logical concepts can children count correctly. After countless concrete counting experiences, children's understanding of logarithm gradually divorced from concrete things and finally reached abstract understanding.
Let's look at the addition and subtraction of numbers. Similarly, addition and subtraction can not be learned by memory, because it requires children to really understand the logical relationship between three numbers, that is, children should really realize that addition and subtraction are operations that combine two parts into a whole or remove a part from the whole. Four-year-old children can understand the relationship between addition and subtraction with the help of concrete objects and actions, but if they want to add and subtract on the abstract digital level, they must establish the abstract logical relationship contained in the class in their minds. And this will not develop until six or seven years old. So it is not difficult for us to understand why some children can solve specific problems (such as "how much is three sugars plus three sugars") and face abstract problems (such as "3+3=?" ) There is nothing you can do.
In short, children's mathematics learning and thinking development are closely related. On the one hand, children need to be psychologically prepared to learn mathematics, that is to say, children should have certain logical concepts and abstract thinking ability. On the other hand, mathematics education should also point to the development of children's thinking and promote the development of children's thinking through mathematics education. Mathematical knowledge is only the carrier of children's thinking development, not the only purpose we pursue.
Children's Mathematics Education: "Teaching for Thinking"
We put forward the educational principle of "teaching for thinking" in order to fundamentally reverse the memory-based mathematics learning and let children truly feel the charm of mathematics as a way of thinking. Parents are advised to keep the following points in mind:
First, the importance of logical concepts is far more important than the memory of numbers. Don't worry that children can't count and calculate, because they haven't got the corresponding logical concept yet. Parents should provide their children with valuable logical experience, rather than let them memorize those incomprehensible mathematics. For example, matching activities can develop children's corresponding concepts, sorting activities can develop children's order concepts, and classification activities can develop children's inclusive concepts, and so on. These seemingly have nothing to do with mathematics, but they are the necessary foundation for children to learn mathematics.
Second, based on concrete experience, it points to abstract concepts. The essence of mathematics lies in abstraction. However, children's abstract mathematical concepts are not created out of thin air, but must be based on concrete experience. So don't rush to let children perform abstract symbolic mathematical operations, but make full use of concrete objects to let children gain mathematical experience. When children have rich experience in mathematics, even if adults don't teach them, they will draw inferences. For example, children often have the experience of sharing objects equally (cakes, sweets, apples, etc.). ), so they can easily understand the concept of "bisection" in mathematics. When encountering other similar problems, he will also take the initiative to transfer his knowledge. In early childhood, don't force the speed of calculation, but pay attention to giving children rich experience.
Third, life is the source of children's mathematical knowledge. Children's mathematical knowledge comes from their real life. Children encounter real and specific problems in life, which are really their own problems, so they are most easily understood by children and much easier to solve than the problems given to him by adults. At the same time, when children really consciously use mathematical methods to solve problems in life, they will have a more direct experience of the application of mathematics, so as to truly understand the relationship between mathematics and life. For example, what can numbers represent? In the face of abstract digital symbols, it is difficult for children to understand "how significant numbers are". But we can find out with our children: where are the numbers in life? What do they mean? In this way, children will get a lot of specific and rich understanding.