What is "mathematical consciousness"? For example, students can calculate "48÷4", which shows that students have the knowledge and skills of division. Students will understand, "there are 48 apples, with an average of 4 apples per person." How many people can you give them? " It shows that students have certain ability to analyze and solve problems, but none of them can show that students have mathematical consciousness. In physical education class, 48 students are jumping long ropes, and the teacher has prepared four long ropes, from which students can think of the formula "48÷4", which shows that students have a certain sense of mathematics.
(1) Understand the meaning of numbers and the relationship between numbers, and cultivate a sense of numbers.
"Sense of number" is the understanding and feeling of logarithmic essence. The essence of numbers is "more and less" or "big and small", thus transitioning to the order of numbers. If people can't count, how much can they say? Experiments show that people can only distinguish 4 or 5. It can be inferred that in mathematics, after the invention of counting, there is an essential difference between human beings and animals. Only with the concept of "quantity" can human beings understand the concepts of "order" and "number". Since l, the natural number system has been formed; The rational number system is formed by four operations of natural numbers; Through algebraic operation of rational numbers, the real number system is finally formed. Therefore, the concept of "how much" and the natural number generated naturally instead of through operation are the most essential concepts of mathematics and the foundation of primary school mathematics. Therefore, cultivating primary school students' "sense of numbers" is the focus of lower grade teaching.
(2) Go through the symbolization process and cultivate the symbolic consciousness.
Russell, a famous British mathematician, said, "What is mathematics? Mathematics is symbols plus logic. " Symbol consciousness mainly refers to the ability to understand and use symbols to express numbers, quantitative relations and changing laws; Knowing symbols can be used for general operations and reasoning.
Of course, the emergence and development of mathematical symbols are not smooth sailing. For example, the birth and use of Arabic numerals is a long process. We can introduce the history of the birth of numbers to students in combination with the teaching of understanding numbers, so that students can understand the development history of digital symbols and feel the infinite charm of mathematical culture.
(C) the combination of practical operation and mathematical thinking to cultivate the concept of space
The concept of space mainly refers to abstracting geometric figures according to the characteristics of objects, and imagining the actual objects described according to the geometric figures; Imagine the orientation of objects and the relationship between them; Describe the movement and change of graphics; Draw graphics according to language description.
In teaching, we should make full use of students' existing life experience and find the fulcrum to develop the concept of space. For example, when I was studying Direction and Position, I took my students to the playground and used their existing life experience of "the sun rises in the east" to determine the east first and then understand the other three directions. In this way, the teaching vision is extended to the living space, and the prototype of life is used to effectively promote the development of students' concept of space.
The development of space concept not only needs rich realistic situations, but also needs a lot of operational activities. When teaching "volume and volume", I use the dynamic process of taking out the chalk from the chalk box and putting it back in its original place to concretize the abstract mathematical concept, so that "the size of the space occupied by an object" can be observed and felt. Here, the teaching process combines students' observation, operation, imagination, thinking, communication and other activities, giving full play to students' spatial imagination, effectively promoting the internalization of activities and the formation of spatial concepts.
Second, mathematical thinking-the cultivation of mathematical thinking ability.
(A) the combination of numbers and shapes to develop students' thinking in images
Primary school students' thinking is in the transitional stage from image thinking to abstract thinking. Number is the abstraction of shape, and shape is the expression of number. The combination of numbers and shapes can help students produce correct mathematical expressions and promote their mathematical understanding. For example, the understanding of "kilogram and gram" belongs to concept teaching, and the content is abstract, which is difficult for students to understand. When I was studying kilograms, I designed a link to find 1 kilogram. I asked the students to weigh 1 kg of washing powder in one hand and the contents in the bag in the other. It is estimated that which bag is also 1 kg. In addition to weighing and estimating, a very important way for people to intuitively perceive the quality of objects is to simply infer according to the number of specific objects. Therefore, when evaluating students' knowledge of grams and kilograms, we should always examine students' "five apples weigh about () kilograms" and "1 box of apples 10 ()". We adults can make simple estimates based on general life experience. The primary school students who have just entered the third grade have little life experience, or have experienced it at ordinary times but didn't pay attention to it. When it comes to solving problems, they can only guess. And objects of the same mass have different sizes and quantities. This requires teachers to awaken students' experience through classroom practice activities and remind them to pay attention to the quality of accumulated experience. For example, after weighing 1kg apples and flour, students can count them and find that 4-6 apples weigh about 1kg, 2 bottles of mineral water weigh about 1kg, and several handfuls of 1kg soybeans (about 4,000 grains). Let students associate the abstract mathematical concept 1kg with the quantity and volume of specific things, which can help students effectively establish the quality concept of 1kg and transform the abstract concept into visible mathematical facts.
(2) Carefully organize mathematical activities to cultivate students' preliminary reasoning ability.
Reasoning is a thinking process of drawing new judgments from one or several known judgments. According to the age characteristics of primary school students, their reasoning ability should be based on reasonable reasoning. Newton, a great scientist, said, "Without bold speculation, there can be no great discovery." Mathematical conjecture is the basis of the development of rational reasoning. "Guess" is an important reasoning strategy. When teaching "the volume of a cone", I asked students to cut cylindrical carrots into cones with the same height and guess the relationship between the volume of cones and the volume of cylinders. Some people think it is cylindrical 1/2, others think it is 1/3, and some people think it is between 1/2 and 1/3. In the above cases, students make bold guesses with the help of observation and experiment; We can also make a guess by analogy. For example, according to "cuboid volume = bottom area × height", it can be analogously inferred that "cylinder volume = bottom area × height".
Because the result of reasonable reasoning is uncertain, we should use case method and deduction method to demonstrate the conclusion, and give priority to case verification. Case verification is mainly carried out through examples, and counterexamples can be cited to overturn the original conclusions or conjectures. You can also cite positive examples and use incomplete induction to verify the conjecture, so that the original conclusion is more reliable. Pupils' reasoning ability is often not "taught", but "understood" in their own reasoning activities. The cultivation of mathematical reasoning ability is not limited to the classroom, and some effective extracurricular activities and games are also good ways to cultivate reasoning ability.
(3) Grasp the whole, break away from convention and cultivate intuitive thinking ability.
Einstein said: "The truly valuable thinking is intuitive thinking." Intuitive thinking is a form of thinking in which the human brain has some direct comprehension and insight into things, problems and phenomena. In teaching, we should cultivate students' intuitive thinking ability. First of all, we should improve students' overall ability to grasp knowledge. For example, Xiaoming is 8 years old and his mother is 36 years old. In six years, how old will her mother be than Xiaoming? According to the general way of thinking, the formulation of this question is "(36+6)-(8+6)", but students with good intuitive thinking will simplify the distance between information and questions and directly express it as "36-8". Secondly, we should choose appropriate questions and forms to train students' intuitive thinking. Such as: In the following time, the closest to your age is (). A.600 hours, b. 600 days, c. 600 weeks, d. 600 months This topic is multiple choice, which only requires choosing a reasonable answer from four options, omitting the problem-solving process, allowing students to use reasonable guesses, which is conducive to the development of intuitive thinking.
Third, solving problems by mathematical methods-the cultivation of problem-solving ability.
I remember the famous Hungarian mathematician Rosa once made a metaphor: If there are gas stoves, faucets, kettles and matches in front of a group of experts, what should I do if I want to boil water? Everyone thinks that we should pour water first, then light the gas stove, and then burn it on the fire. This is the same understanding. But what if the pot is full of water and other conditions remain unchanged? At this time, most experts will directly light the gas stove and then put it on the fire. Only mathematicians will pour out the water, because mathematicians will use mathematical thinking-turning to thinking to think about problems and turn the latter situation into a familiar situation. Although this metaphor is a bit exaggerated, it does show that mathematicians are better at thinking in mathematical ways than other applied scientists. Whether students can use mathematical ideas, methods and strategies to solve mathematical problems or daily problems is an important symbol of students' mathematical literacy.