First, finishing, forming a system
Mathematical knowledge is closely and systematically, and each concept develops vertically and is horizontally related to its adjacent concepts. When reviewing, we should guide students to classify and sort out the concepts on the basis of mastering their meanings, discover and grasp the context of the vertical development and horizontal connection of knowledge, and make it systematic, so as to understand and master the concepts more deeply. For example, the concept of number in primary schools can be reviewed and sorted into the following table:
(attached {Figure})
When reviewing, review natural numbers first. When people count objects, 1, 2, 3 … representing the number of objects are called natural numbers, and the number of natural numbers is infinite. Then review 0 and make it clear that both natural numbers and 0 are integers (there are integers less than 0 to learn later); Then review that the unit of natural number is 1, divide the unit "1" into several parts on average, explain that such a number or several parts lead to a fraction, and further explain that the quotient divided by two numbers can be expressed by a fraction table to show the relationship between a fraction and an integer; Then review the meaning of decimals from the relationship between fractions and decimals; Finally, review the meaning of percentage: it means that one number is the percentage of another. In this way, the ins and outs of the development of numbers are presented to students, and students get a whole piece of relevant knowledge.
For another example, the divisible knowledge of numbers is a closely related concept system. Review, on the basis of understanding the meaning of concepts, grasp the internal relations and development between concepts, and organize them into the following table:
(attached {Figure})
Among them, divisibility is the basis of this piece of knowledge. Based on the divisibility, the characteristics of multiples, divisors and numbers divisible by 2, 5 and 3 are derived. From multiple to common multiple to least common multiple; From divisor to common divisor to greatest common divisor, prime number and composite number are derived from the number and characteristics of divisor, prime factor is derived from prime number, decomposition prime factor is derived from composite number, and prime number is derived from the number and characteristics of two numbers containing common divisor. Derive even and odd numbers from the characteristics of numbers divisible by 2. Finally, use this knowledge to find the greatest common divisor and the least common multiple of two numbers. In this way, all the knowledge about the divisibility of numbers forms a large structure and is stored in the students' cognitive structure.
A certain knowledge or skill of mathematics often includes several aspects. When reviewing, students should also be helped to sort out, recognize the situation one by one and take appropriate measures to deal with it respectively. If you have learned to rewrite multiple numbers in primary school, you can review them one by one: 1. Rewrite the larger multi-digit number into tens or hundreds of millions, such as 43 150 = 435438+05000. 2. Omit the mantissa after a bit from a larger number and take its approximate value, such as 432 150≈430000. 3. Omit the mantissa after a certain decimal place and take its approximate value, such as 3.4 1986≈3.4 (keep one decimal place), 3.4 1986≈3.42 (keep two decimal places), 3.4 1986≈3.420 (keep three. 4. Pseudo-fractions and rewriting of fractions and integers (examples omitted). 5. Interoperability among fractions, decimals and percentages (see the textbook Arrangement and Review). Clearly arrange several rewrites to guide students to analyze and master.
For another example, the comparison of numbers can also be arranged in various situations to study: how to compare the sizes of integers? How to compare the sizes of decimals? How to compare the scores? How to compare with denominator scores? How to compare molecular scores? How to compare scores of different denominators and numerators? How do fractions compare with decimals? In this way, students can master the comparative knowledge of numbers as a whole.
Second, strengthen comparison and communication.
Mathematical concepts are often associated with the essential characteristics of * * * and distinguished by different personality characteristics. By comparison, we can not only seek common ground, but also distinguish differences and curb generalization and confusion. For example, the three concepts of prime number, coprime number and prime factor are all specious literally. By comparison, let the students understand that a prime number is a number, and see if its divisor is only 1 and itself, such as 2, 7, 3 1, is a prime number. The prime number is for two numbers, see if the common divisor of these two numbers is only 1. Although two prime numbers are coprime numbers, the two numbers of coprime are not necessarily prime numbers, such as 8 and 9,6 and 13, 1 and 83. Prime factors cannot exist independently; It must depend on a composite number, which is both a prime number and a factor of this composite number. For example, 2 is the prime factor of 12 and 1 1 is the prime factor of 88. ...
Another example is the reading of integers and decimals, which can be compared as one. For example, in 7645.7645 and 2005.2005, the numbers in the integer part and the decimal part are the same, and they are all read from high places, but they are different: the integer part should not only read the numbers on each digit in turn, but also read them together with the counting unit, and the decimal part only needs to read the numbers on each digit in turn, so it is read as 7645.7645; There are several zeros in the middle of the integer part, just read one zero, and there are several zeros in the middle of the decimal part, so you should read them one by one, so you can't save reading. Therefore, 2005.2005 is read as 2005.205.
Due to the scattered transmission of knowledge, the internal relations between some knowledge cannot be revealed in time. When reviewing, we can connect scattered knowledge together by comparison, so that students can understand it more deeply. For example, the basic properties of fractions and decimals can be linked when reviewing. The basic property of a fraction is that the divisor and denominator of the fraction are multiplied or divided by the same number at the same time (except zero), and the size of the fraction remains unchanged. The basic property of decimal is to keep the size of decimal unchanged by adding or removing 0 at the end of decimal. In fact, the two are consistent. For example, 0.7=0.70=0.700, 7/10 = 70/100 = 700/1000.
For another example, general scores and approximate scores are studied successively, and students can realize that they are both the application of the basic nature of scores by comparison when reviewing. The difference is that the divisor is that the numerator and denominator are divided by the same number (except zero) at the same time, which becomes a fraction with smaller numerator and denominator; A general score is a score that multiplies the scores of different denominators by the same number (except zero) and becomes the same denominator. In this way, the basic nature of the score, approximate score and general score are tied together for review, and knowledge can enter the students' cognitive structure in the form of coding structure, making it a meaningful learning with high generalization ability.
Third, design exercises to deepen understanding
1. Grasp the key points and carry out basic exercises. The basic things are often the most important. For the key points and keys in the textbook, we should strengthen basic exercises. The meaning, separability and nature of numbers must be internalized through practice. All kinds of rewriting and comparison of numbers should be practiced to form skills.
2. Strengthen comprehensive exercises and deeply understand concepts. The general review should make students systematize and integrate concepts, and comprehensively apply what they have learned to solve problems. For example, ()/16=6/( )=( )÷40=0.75=( )% involves the knowledge of decimals and fractions, the reciprocity of percentages, the relationship between fractions and division, the basic properties of fractions and the invariance of division quotient. Another example is a number with the smallest prime number in 10000, the smallest composite number in 100, the smallest odd number in 10, the smallest digit in 1000 and the smallest natural number in 1000. The rest of the digits are all 0, and this number is (), which is pronounced as (). This problem includes the application of concepts such as writing number, reading sum prime number, composite number, odd number and natural number. For another example, A and B are two natural numbers, a÷b=5, the greatest common divisor of A and B is (), and the smallest common multiple is (); According to 4/7×2(5/8)×2/3= 1, the numbers written directly with () are: 4/7×2(5/8)=), 2(5/8)×2/3= (), 4/7× 2/3.
3. By contrast, distinguish easily confused concepts. Comparison questions can be designed in the general review to help students distinguish similar, similar and confusing concepts. For example, 7 ÷ 3 = 2... 1, 0.8÷4=0.2, 18÷6=3, 3÷0.5=6, 40÷8=5 fill in the table as required.
Incomplete division, incomplete division.
Through this comparative exercise, let the students understand that what can be divided must be separable, and what can be divided is not necessarily separable; What is inseparable is sometimes separable, sometimes separable, and what is inseparable must be inseparable.
4. Strengthen targeted exercises and constantly correct the concept of error-prone. For the concepts that students are prone to make mistakes, we should guide them to understand the situation and the original reasons for the mistakes, and then guide them to use the concepts to answer questions and solve problems. For example, judging that "even number is a composite number" and "the prime factor of 42 decomposition is 42=2×3×7 × 1" and "the multiple of a number must be greater than its divisor" is also a process of finding, discussing and correcting errors, so as to strengthen the understanding of concepts from the understanding of errors.
Fourth, inspire students to take the initiative to review
The ultimate goal of general review is to let students master what they have learned. In teaching, we should inspire and guide students to review actively, review and sort out what they have learned, and make it systematic. When recalling and sorting out knowledge, we should let students become the masters of review, let students talk more and make up more, and gradually form a systematic, complete and clear knowledge network. In this way, students not only deepen their understanding of what they have learned, but also feel that they have really improved through review and arrangement, thus stimulating their enthusiasm for review and improving the effect of review.