Keywords: transfer of old and new knowledge in primary school mathematics teaching
Pupils' acquisition of mathematical knowledge, in many cases, follows the process from perceptual to rational, from concrete to abstract. But not all knowledge can be obtained through personal experience. Children often go through the cognitive process from known to unknown and from old knowledge to new knowledge in mathematics learning. This psychological phenomenon is migration.
We can also understand that transfer is the influence of one kind of learning on another, which may be positive or negative. Any previous study that has a positive impact on later study and promotes it is called positive transfer. For example, it is not difficult for a person to learn to ride a bicycle before learning to drive a motorcycle; Learning a foreign language helps to master another foreign language; Children form neat writing habits when doing math exercises, which helps them to keep tidy when finishing their homework.
On the contrary, the existing knowledge and skills will interfere with the newly learned knowledge and skills and have a negative impact, which is called negative transfer. For example, when students begin to learn multiplication, they often confuse addition; Learning a2 is always confused with 2a; When learning integers, I know that "black rabbits are five more than white rabbits" and "white rabbits are five less than black rabbits" have different meanings. When studying scores, "black rabbits are more than white rabbits", then "white rabbits are less than black rabbits" will have some interference, and I mistakenly think that "white rabbits are less than black rabbits". Of course, negative transfer is temporary, and in most cases it is disturbed by superficial phenomena, which can be eliminated through proper practice and knowledge.
For primary school students, effective transfer learning is not an easy task. The research of modern cognitive theory on transfer shows that the greater the positive transfer of students' learning, the stronger their ability to adapt to new learning situations or solve new problems through learning. The essence of this positive transfer is the original cognitive structure of cognitive subjects and the generalization degree of students' mastery of relevant knowledge. Therefore, students' original cognitive structure has become the most critical factor for students' smooth migration.
Generally speaking, in the process of transfer learning, students are mainly influenced by three aspects, that is, can they use the released and fixed concepts in the original cognitive structure? What is the stability and clarity of the original fixed concept? How different is the new potentially meaningful learning task from the original conceptual system that absorbed it? In layman's terms, it is whether there is an internal connection between the old and the new, and how clear this connection is, and whether it can be fully and effectively established and applied.
First, determine the relevant old knowledge.
Determining the old knowledge that can fix the new knowledge from the students' original cognitive structure depends largely on the order in which the textbooks are presented. In the current primary school mathematics textbooks, each "knowledge block" is arranged in the form of cycle, unit and chapter according to the principle of from shallow to deep, from easy to difficult, step by step and spiral. For example, teaching the calculation of integers is from the understanding and calculation of numbers within 20 to those within 100, from those within 10000 to those above 10000; Decimals and fractions are composed of two cyclic segments including preliminary understanding. From the chapter, the addition and subtraction of integers change from non-carry to carry, and from non-abdication to abdication; Fractions with the same denominator are added or subtracted from fractions with different denominators, and so on. The former knowledge is the basis of the latter knowledge, which is the extension and development of the former knowledge. In this way, there are both vertical and horizontal relationships between cycle segments, units and chapters, which is not only a sign of knowledge systematization, but also the starting point and breakthrough point for determining relevant old knowledge when learning transfer teaching. The following is described from two aspects: vertical and horizontal:
1. Grasp the vertical connection and seek the growth point of knowledge.
For example, before learning the addition and subtraction of fractions with different denominators, students have learned the addition and subtraction of integers and decimals, and the addition and subtraction of fractions with the same denominator. In these calculations, it is established that "addition and subtraction can only be done if the counting units are the same".
This highly generalized concept is to migrate the old knowledge base of the method of "addition and subtraction of fractions with different denominators". For another example, using the core principle that "two divided numbers are multiplied or divided by two identical numbers at the same time (except 0) and the basic properties of fractions" established by students, the learning of the basic properties of contrast can be extended.
2. Strengthen horizontal comparison and highlight the connection points of knowledge.
For example, when students learn the reading and writing methods of numbers within 10,000, they master the reading and writing methods of a series, understand the knowledge of number order and counting, and learn the order and reading and writing methods of multiple numbers, and so on. Master the meaning of multiplying numbers by integers and numbers by decimals, and learn the meaning of multiplying numbers by fractions by analogy. In the carry addition within 20, when teaching the calculation of "9 plus several", you can understand the arithmetic of "add 10", and then you can directly transfer the following learning of "8 plus several", "7 plus several" and "6 plus several".
Second, activate cognitive fixation points.
In transfer teaching, we often encounter such a situation: students already have appropriate fixed concepts in their cognitive structure, but they can't make full use of them. This study requires our teachers to try their best to make students wake up these old knowledge before learning new knowledge, make it reappear in the process of students' cognition, and be good at organizing the full interaction between new knowledge and related old knowledge.
When teaching the calculation of remainder division, first organize students to fill in the maximum number in the following formula: 3 × () < 20, 6 × () < 43, 8 × () < 59 ... Then let students think: at 23÷5, 47÷9 ..., the product of the number filled in and this book is the largest. In this way, the thinking process of inequality filling in the blanks is transferred to the vertical calculation of remainder division.
For another example, there are two successive zero subtractions in the teaching minuend. First show two vertical calculation problems: 93-27, 903-27. After group contact, let the students compare: What are the similarities and differences between these two formulas? Through the comparison of the same points, the theory of "which one is not enough to reduce, the previous one will be retired" is highlighted, which runs through the whole teaching process of abdication subtraction; Through the comparison of different points, the summation factor of the second problem is highlighted, that is, "the number of digits is not enough, but the number of digits is 0", which leads to two consecutive reductions. On this basis, the topic is changed to introduce the teaching of "9003-27". This is in line with the original cognitive structure.
Third, the old and the new are connected to realize migration.
Conscientiously identify and activate the related old knowledge that can fix new knowledge in the original cognitive structure, in order to better realize the transition between old and new knowledge, promote the learning of new knowledge and improve the learning effect of new knowledge. However, in order to realize the smooth "integration" of old and new knowledge and complete the migration activities efficiently, it is necessary to choose appropriate migration methods and effectively prevent negative migration.
The following will introduce several common migration methods:
1. Analog migration
The so-called analogical transfer means that when using related old knowledge, we should carefully look for the same factors as new knowledge and assimilate and adapt to new knowledge through classification. For example, the multiplication of one factor with one or two numbers is similar to the multiplication of one factor with three numbers, because the number in which bit of a factor is multiplied by another factor, and the number obtained is the number of counting units in which bit. For another example, students have mastered the derivation method of triangular area, and then when learning trapezoidal area, they can use the same channel of "splicing graphics to derive" to induce students to migrate to the derivation of trapezoidal area themselves.
2. Contrast migration
It can only be identified by comparison. Some new knowledge is often related to and different from old knowledge. In teaching, we can first review the old knowledge we have learned, then compare the new knowledge, and focus on finding out their similarities and differences, and rationally decompose, adjust and reorganize the original knowledge structure to achieve the purpose of "exploring the new with the old". For example, by reviewing the meaning, calculation, unit and function of volume, students can better grasp that their calculation methods and unit names are the same, mainly because of their different meanings. Volume is "the size of the space occupied by an object" and volume is "the volume of other objects that an object can accommodate".
3. Reverse migration
When the old knowledge and the new knowledge are completely opposite, by talking about them and learning together, we can achieve the goal of deeply understanding and mastering knowledge and cultivating the concept of unity of opposites. In teaching, we usually review the original positive questions first, which leads to new knowledge and in-depth research. For example, when teaching fractional division application problems, if we can introduce the meaning of multiplication of fractions and numbers and teach students to solve fractional multiplication application problems by writing relational expressions, then when teaching fractional division application problems, we can organize migration like this: