The child enters the fourth grade, and the final mathematics is only 94. Mathematical thinking needs to be improved. In the summer vacation, I saw some valuable math thinking classes in the WeChat group, and the harvest was full.
This article is mainly to let us know how mathematics became Tu Longdao's real problem.
Mathematical literacy
Mathematics is a subject that studies quantitative relations and spatial forms. In the long development of mathematics, several main ideas have emerged. These basic ideas are the fundamental methods to understand the real world and solve practical problems. They are abstraction, reasoning and model respectively.
Abstraction: We use abstract thinking every day. Its thinking process helps us to understand the composition of the world more deeply. When people study the differences of things, they need to separate the objects of study to form a combination, and then define and name the combination, thus forming a concept, which embodies the generality of mathematics.
Reasoning, whose function is to understand the causal relationship between various things, can help us better understand the development trend of things, and also allow us to understand from general to special to general. The rigor of mathematics at this time.
Model thinking: people use creative thinking methods to describe regular things in the real world. If we use models, we can connect the scientific world with the real world. Let's solve practical problems in a scientific way.
People form concepts through abstraction, understand relationships through reasoning, and connect science with the real world through models to solve problems. These three ideas are the basis of looking at the world and solving problems. In addition to mastering basic knowledge, skills and experience in basic activities, it is particularly important for children to acquire basic mathematical ideas. This is the meaning of learning mathematics.
Mathematical thinking: Mathematical literacy is used to discuss mathematical thinking here. Mathematical literacy can be understood as using concepts, ideas and methods in mathematics to observe the world, think, analyze and solve problems, which is both a kind of ability and a kind of consciousness. Mathematical thinking mode is a part of mathematical literacy, which mainly uses mathematical tools and theoretical methods to understand the world, build a knowledge system and optimize and transform the world. Simply put, mathematical literacy means looking at the world from a mathematical perspective, thinking about problems with mathematical thinking, and solving problems with mathematical methods. Look at math, think about math, and do math.
Ten core concepts of curriculum standards: number sense, symbol consciousness, space concept, geometric intuition, data analysis concept, calculation ability, reasoning ability, model thinking, application consciousness and innovation consciousness.
The relationship between mathematical literacy and knowledge and skills. 123rd grade is the first stage of compulsory education. Generally speaking, it is a process of cultivating study habits, accumulating learning experience of basic mathematics and establishing concept system. From the specific knowledge and skills, according to the three main lines of number and algebra, graph and geometry, statistics and probability, the following goals are achieved: a clear goal is a result, and these results are achieved through the learning process.
Learning process is the process of developing students' mathematical literacy. The learning process is properly designed, the learning thinking process is clear, and the cognitive conflict is constant. Children explore independently under the guidance of problems, gradually learn to solve problems with mathematical methods, and their mathematical literacy continues to develop. If it is utilitarian, indoctrinating, spoon-feeding and overemphasizing skill-based learning, this kind of training-based learning enables children to solve problems. They don't know how to ask questions and have no habit of asking questions, which is contrary to the requirements of curriculum standards. Curriculum standards require children to go through the whole process of finding, asking, analyzing and solving problems, and can't let children solve problems in one step. This is not to mention the cultivation of mathematical literacy. Therefore, in the practice of teaching guidance, we resolutely oppose cramming, indoctrination and over-emphasis on skill. We encourage teachers to design learning activities with children as the center in the teaching process and advocate speculative and interactive classroom learning.
Some parents think that doing problems repeatedly can improve their ability. I want to remind you that doing problems repeatedly is to cultivate skills, not skills. Skills alone are not enough and will not work. Because skill is an example, it is very useful for this problem, or it is very useful for this kind of problem, but it will not work with a little change. This is not mathematical ability, let alone mathematical literacy. What we want is to learn a problem in the same way, even analogy.
Let's take operation as an example. How does mathematical literacy penetrate into the process of learning knowledge and skills step by step? Students should master the division of two or three digits divided by one digit three times. The goal of learning is to understand arithmetic, master algorithms and develop operational ability. So in the process of achieving this goal, what kind of learning process should children experience and what kind of mathematical thinking should they master? Children who enter school know numbers first. What are numbers? Understanding it involves two properties, one is number sense, the other is symbol, and the number is symbol (such as 3), which is the abstraction of quantity. Only the concept is not important, and the relationship is very important. Since it is abstracted from quantity, the relationship between numbers comes from the relationship between quantity. What is the essence of quantitative relationship? What is the essence of quantity? Dogs dare to deal with wolves when they come. If a wolf comes, will the dog turn around and run? Do animals know more or less? What animals know is the essence, and what animals know is the essence, which is the most fundamental. The essence of quantity is more and less, abstracting to number is big and small, and losing big and small is the essence and sense of number. If you only let your child count, he doesn't compare logarithms, and he doesn't know the size. It's meaningless to count. Three apples and three chickens can be represented by three squares, and then represented by a number 3, which can represent three tables, three people and three sweets. This process is an abstract process. With the abstract symbol 3, that is, the number 3, children just go through a process from concrete to abstract, and then slowly understand the generality of mathematics. Students begin to experience digital growth from grade one and grade two, from one digit to two digits and three digits. This is the process of cultivating a sense of numbers and symbols. In this process, children have experienced a mathematical abstraction process from understanding numbers to understanding logarithms. We should be curious about their logarithms and symbols, and then have the opportunity to think about the relationship between numbers and symbols, thus producing sensitivity.
Next, children can use multiplication formula to master division in tables, which is also the cultivation of computing ability. What should we experience when we divide two digits into one digit and develop three digits into one digit? At this time, you need to have model thinking and reasoning ability. For example, a student learns the multiplication in the table, and two or three get six. Then he can calculate 6/3=2, and use the multiplication formula to get the quotient. Then 6/3=2. What he understands is that 6 is divided into three parts, and each part is 2, so 60/3=? What does this mean? What does it have to do with 6/3=2? If the student understands that 6/3=2, then we should guide him to divide 60/3 into 6 10 on average, which is the same as 6/3=2, except that the score is 6 10, and the score of 6/3 is 6 1, and then learn 600/3. 6000. So with this foundation, it is convenient for students to understand arithmetic, and in the future study, we will never leave the multiplication and division method in the table, that is to say, we use the multiplication in the table to find the quotient, and every step of our vertical calculation is the division in the table, so why do children propose that the vertical quotient should be aligned with the digits of the dividend? What happens if it is not aligned? This kind of question is beautiful and helps him understand vertical division. Through these curiosity and questions, do children have a chance to see the intelligence and wisdom of human beings behind some unpretentious methods? For example, will they have the desire to create new methods when they see the great decimal counting method? Is this the innovative consciousness we are pursuing? Of course, we can also think the other way around. If you don't go through this process, you can directly teach them arithmetic skills and tell them that the rules are like this. We just need 1, 2, 3, 4, and then teach them skills 5, 6, 7, 8, and we can solve the problem. Then imagine, do children still have the consciousness of asking questions, the process of inquiry and the curious heart? This is worth pondering.
To sum up, firstly, curriculum standards and ten core concepts are a reliable and credible framework system for the cultivation of mathematical literacy. Repeat ten core concepts: number sense, symbol sense, space sense, geometric intuition, data analysis concept, calculation ability, reasoning ability, model thinking, application consciousness and innovation consciousness. The ten core concepts are permeated with the basic ideas of mathematics, and the most essential ones are mathematical abstraction, logical reasoning and mathematical modeling. Mathematical abstraction is mainly manifested in symbol consciousness and number sense in primary school, especially in the first stage, which is an important stage to cultivate students' symbol consciousness and number sense. A little more here, we also see many parents' problems about the completeness and practical operation of this system. At present, we are also doing some experimental work, hoping to fully show this system hidden in the curriculum standards, which is accurate, concrete, popular and evaluable. This work should be shared with you in the near future.
Second, knowledge and skills themselves do not represent mathematical literacy, and the learning process of knowledge and skills is permeated with mathematical literacy. Whether it's math teaching in class or learning in some interest classes outside class, we should inspire children to ask questions as much as possible, reasonably guide independent inquiry, and infiltrate math literacy in the learning process.
Some parents also asked how to cultivate children's mathematical literacy in daily life, which is a good question. Parents are the best teachers for their children, so it is very important to set an example. Parents are curious, children learn to be curious, parents have the desire to explore, and children learn to explore. In fact, I am cultivating children's knowledge, ability, consciousness and personality inside and outside the classroom.
I gave an example of algebra just now. Let me give another example of graphics and geometry. When children see butterflies in the wild, it is easy to observe that the left and right sides of the butterfly's body are very similar, which is to observe the symmetry on both sides of the butterfly's body. This is a mathematical view of the world. If the child is unintentional, we must protect and encourage him. In the future, he will develop a mathematical world outlook, and he will develop a life world with mathematics everywhere.
Children should study this imagination, which can inspire them to do their own research. For example, if you fold it in half, of course you can't use a real butterfly to fold it in half. You can fold it with paper butterflies or imaginary butterflies. We will find that the shapes on both sides can overlap. This folding method is a mathematical thinking method, and virtual folding is the starting point of space concept. Furthermore, since we know the phenomenon of symmetry, how can we describe it? Then we should create some concepts, use some methods and determine some standards to describe them. For example, it is necessary to clarify what is the symmetry axis and study the relationship between the figures on both sides of the symmetry axis. We will calculate the relationship between the straight lines corresponding to two sides and the axis of symmetry, and judge the angular relationship between the corresponding line segments and the axis. Then, after completing these, we will understand the phenomenon of axial symmetry. To study it, judge it, and then create it is the process of learning and applying mathematics. Everyone has noticed that in this process, children should first have the opportunity to observe and simply say it. Secondly, there should be some ways to help children complete the abstract process, such as using paper dragonflies or dragonflies to represent real dragonflies. Think about it and help children feel this phenomenon. After that, it is a very relevant step, heuristic. If children think about this step, they can thoroughly learn the concept of axisymmetric graphics and the corresponding mathematical methods. What can parents do?
Observation creates more observation opportunities. Senior one and senior two should observe more objects, perceive quantity and compare and identify. The third and fourth grades can observe some phenomena, observe some images and feel them in combination with some situations in life. For example, describe, describe and encourage children to talk more. It doesn't matter if it's wrong or incomplete. Make sure you can say it first, such as which is more, which is less, which is bigger, which is smaller, which is on the left, which is on the right, relative to whose left and whose right. If the description is clear, there will be opportunities for identification, induction and deduction. For example, what are the similarities and differences between ab and AB? What has A changed? It can become b, and so on. A particularly important link is to often inspire them to ask questions, such as why we need this thing, what will happen without it, what convenience it brings to our life and research, and whether this method can be used here or elsewhere?
Summarize the thinking process that parents and friends can cultivate with their children, where all kinds of methods come from, to describe, to identify, to induce and to deduce, to practice observation and mathematics step by step in life, to cultivate the consciousness of speculation, and to ask why besides knowing what it is. These are all aspects that parents can help their children in their daily lives.
The second topic is a brief talk about the problem of Olympic mathematics. Olympic mathematics is the process of pursuing the Olympic spirit in mathematics. Pursuing faster, higher and stronger in the field of wisdom. The branch of mathematics, the development of mathematics field and mathematical thinking method. Therefore, it is gradually excavated into a system to expand thinking and cultivate mathematical ability.
It is helpful to expand mathematical thinking reasonably, and children should expand mathematical thinking. The Wide Angle of Mathematics published by Beijing Normal University's Interesting Mathematics People's Education Edition actually embodies these ideas.
Postscript: Teacher Lin Xinhao's Interesting Mathematics also has the foundation of mathematical thinking. Both of them start with thinking and let children understand the essence of mathematics.
One of the gains of this paper: children calculate all kinds of strange mistakes, I will say that you have no sense of numbers. It turns out that I am also a parrot. After listening to teacher Xia's class, I realized that the essence of number sense is the size of number. It turns out that children are 30 × 28 = dozens. I said that you have no sense of numbers, which is the essence.
The second harvest of this paper: both Mr. Xia and Mr. Lin say that mathematics is a study of relationships. I understand that mathematics is a tool to explore the real world, and models are a bridge between reality and mathematics.
The article was revised on 20 19-08-3 1.