-Cultivation of Mathematics Core Literacy in the Teaching of "Algorithm"
Mathematics core literacy is the thinking quality and key ability of people who have the basic characteristics of mathematics and meet the needs of students' personal lifelong development and social development. It mainly includes six core qualities: mathematical abstraction, logical reasoning, mathematical modeling, intuitive imagination, mathematical operation and mathematical analysis.
The six core qualities are not separated from each other, but closely linked and integrated. Mathematics comes from life and is practiced in life. The mission of mathematics teaching is to let each of us who participate in learning finally reach and learn to understand the world with mathematical eyes, understand the world with mathematical thinking and express the world with mathematical language.
Understanding the world from a mathematical perspective is inseparable from "mathematical abstraction" and "intuitive imagination"; Understanding the world with mathematical thinking is inseparable from "logical reasoning" and "mathematical operation"; To express the world in mathematical language is inseparable from "mathematical modeling" and "data analysis".
Recently, I have been teaching "Operations Research", including additive commutative law, Multiplicative method of substitution, Additive Association Law, Multiplicative Association Law and Multiplicative Distribution Law. In this paper, how to cultivate students' core literacy in operational research teaching is discussed.
First, the cultivation of mathematical abstraction and intuitive imagination.
The textbook we use is Beijing Normal University Edition. The teaching of algorithm is arranged at the beginning of the second half of the fourth grade. Judging from its position in the textbook, the algorithm is the most important part of this textbook.
"Intuitive imagination" refers to the literacy of perceiving the shape and change of things with the help of geometric intuition and spatial imagination, and understanding and solving mathematical problems with graphics. It mainly includes: understanding the position relationship, morphological change and motion law of things with the help of space; Describe and analyze mathematical problems with graphics; Establish the relationship between number and shape, build an intuitive model of mathematical problems, and explore ways to solve problems. "Mathematical abstraction" refers to giving up all the physical attributes of things and gaining the literacy of mathematical research objects. It mainly includes: abstracting the relationship between mathematical concepts and concepts from the relationship between quantity and shape, abstracting general laws and structures from the specific background of things, and expressing them in mathematical language.
Intuitive imagination is the basis, the primary stage and a kind of mathematical abstraction. It is also an important foundation for constructing and forming other core qualities of mathematics.
Looking at every teaching arrangement of the algorithm in the textbook, it is to provide a situation map or a realistic background map to let students understand the relationship between quantity and quantity in the realistic background. This is to consolidate children's intuitive imagination and to further promote the development of intuitive imagination. Intuitive imagination literacy is the most basic mathematics literacy of children.
For example, in the arrangement of the textbook of additive associative law, two situational diagrams are provided. Take figure 1 as an example. The picture shows monkeys, squirrels and clever dogs picking 30 peaches, 40 pears and 50 apples in the orchard respectively. Formula: What do (30+40)+50 and 30+(40+50) mean respectively? Because children have a certain intuitive imagination and life experience, they can easily describe the meaning expressed by the formula. Even if a few students lack language expression, they can understand the meaning.
The first formula means: How many peaches and pears are there first? So the number of peaches and pears, plus the number of apples, finally * * *, how many? The second formula means: How many pears and apples are there first? Then add the number of peaches to the total number of pears and apples. What is the last * *? Through intuitive imagination, it is not difficult for children to find that although their calculation order is different, they all calculate the number of peaches, pears and apples in the end. So the final result must be equal. In this teaching process, intuitive imagination has played an important role and has been further developed.
On this basis, letters are introduced to represent numbers. The formula: (a+b)+c=a+(b+c) indicates the law of additive combination, which is clear and accurate. Although I never learned to use letters to represent numbers before grade one to grade three, the children have mastered some life experience and are very familiar with the letter abc. Just tell them to use letters to represent numbers, and they will understand the meaning expressed by the formula.
In this teaching process, children's mathematical abstraction ability has been trained twice. First of all, give up the situation, abstract the numbers, then express them with formulas, and express the mathematical operations and activities in this situation with mathematical language. Secondly, letters are abstracted from concrete numbers. Formulas with letters are used to express the operational relationship and order between quantities. This abstraction has been divorced from the concrete number, it can represent another number, and it expresses an unchangeable operational relationship between quantities, that is, some operational law. Children's mathematical abstract literacy has been fully trained and developed here.
Second, the cultivation of logical reasoning and mathematical operation literacy
"Mathematical operation" refers to the completion of solving mathematical problems according to the operation rules on the basis of defining the operation object, which mainly includes: understanding the operation object, mastering the operation rules, exploring the operation ideas, selecting the operation method, designing the operation program and obtaining the operation results. "Logical reasoning" refers to the completion of deducing other propositions from some facts and propositions according to rules. It mainly includes two categories: one is reasoning from special to general, and the main forms of reasoning are induction and analogy; The other is reasoning from general to special, and the form of reasoning is mainly deduction.
Mathematical operation is also a kind of logical reasoning, which provides the carrier and operation mode for logical reasoning. Mathematical operation and logical reasoning are based on intuitive imagination and mathematical abstraction, and carry out higher-level mathematical thinking and mathematical thinking on the pointed object.
In the teaching of additive associative law and multiplicative associative law, mathematical operation ability can be well trained and developed, and students' number sense ability can also be improved. The ultimate goal of learning the algorithm is to improve the computing ability, so let's do simple operations as much as possible. Therefore, in the teaching of this part of the content, mathematical operations can be vividly displayed.
For example, page 52 of the textbook: How to calculate simply? Think about it and do the math.
57+288+43=□
In this question, students are required to keenly find that the sum of 57 and 43 is exactly 100, which is not only the cultivation of students' sense of numbers, but also the training of students' mathematical operation literacy.
Another example is page 54 of the textbook: How to calculate simply? Think about it and do the math.
125×9×8=□
In this question, children are also required to observe the characteristics of operation symbols and numbers in the formula. Through the training of this question, let children understand that 125×8 is equal to 1000, which can be calculated first, which is beneficial to simple operation. In the following training questions, let the children understand that 25×4 is equal to 100, 50×2 is equal to 100, 25×8 is equal to 200, and 125×4 is equal to 500. In these trainings, children's sense of numbers is enhanced and their computing ability is greatly improved.
The first link in the teaching content arrangement of additive commutative law and Multiplicative Commutative Law, Additive Associative Law and Multiplicative Associative Law is "Observe the following formula, can you write another group as it is?" Tell me what you found. " In this teaching process, children's analogy ability has been fully trained and cultivated. This is a logical reasoning from special to special. The second link is to understand the rationality and correctness of the formula with the help of the situation or realistic background. In the third link, letters are introduced to represent numbers, and on the basis of mathematical abstraction, inductive reasoning is used again to implement logical reasoning literacy training from special to general.
The fifth question on page 55 of the textbook: Naughty is calculated by 24×25.
? 24×25
=6×4×25
=6×(4×25)
=6× 100
=600
(1) Can you understand it? Communicate your thoughts with your partner.
(2) Try to calculate the following problems with multiplicative commutative law and multiplicative associative law.
64× 125 125×25×32
This problem is an improvement problem. Not everyone may be able to do it in a short time. But this question is very useful for cultivating students' mathematical literacy. The first small question mainly examines students' sense of number and mathematical operation literacy. The second small problem is to focus on cultivating children's analogical reasoning ability. In the previous research, when children have a sense of number 125×8 equal to 1000 and 25×4 equal to 100, and then under the analogical reasoning of known operation problems, they find the first problem, 64 can be decomposed into 8× 8,8 and 125 multiplication. The second problem, 32 can be decomposed into 4×8, where the multiplication of 4 and 25 is exactly 100 and the multiplication of 8 and 125 is exactly 1000. Problem solved. This is the best opportunity for training in mathematical operation and logical reasoning.
Third, mathematical modeling and mathematical analysis are the same.
"Data analysis" refers to obtaining data from the research object, sorting, analyzing and inferring the data by statistical methods, and forming knowledge literacy about the research object. It mainly includes: collecting data, sorting out data, popularizing information, establishing models and drawing conclusions by inference. "Mathematical modeling" is the result of mathematical abstraction of practical problems, expression of problems with mathematical language, and construction of models to solve problems with mathematical knowledge and summation method. It mainly includes: finding problems, putting forward problems, analyzing problems, establishing models, solving conclusions, verifying results and improving models in actual situations, and finally solving practical problems.
Mathematical analysis is also a kind of mathematical modeling, which provides more basic materials and ways of thinking for mathematical modeling. The basis of mathematical analysis comes from intuitive imagination, mathematical abstraction, mathematical operation and logical reasoning. At the same time, it will lay a better platform for mathematical modeling, and finally let us learn to express the world in mathematical language.
Multiplication and division is the most difficult of the five algorithms to master. The textbook is introduced with real-life examples. From two different observation angles, how many tiles are pasted on a * * *? Let the students understand that different formulas are just different observation angles and bring different ways of thinking. The final result of thinking is to calculate how many tiles are pasted, pointing to the same and equivalent purpose. It is concluded that these two formulas are equal. Conveniently carry out mathematical abstraction and inductive reasoning, and get the multiplication and division method. In order to let this result go deep into the child's heart. The textbook is further verified by citing the realistic background and deduced mathematically. This derivation is actually the derivation of the inverse operation of multiplication and division. Just like the first grade of primary school, learning subtraction is based on addition understanding.
The application of multiplication and division depends more on the literacy of students' data analysis. For example, the textbook is on page 57: Observe and calculate the features of 34×72+34×28.
The focus of this teaching is to cultivate children's data analysis ability and literacy. Let the children find through observation that both multiplication formulas are 34. On this basis, it means 72 34, plus 28 34, and a * * * has100 34. Through data analysis, let students understand that this formula conforms to the characteristics of multiplication and division, and then use the inverse operation of multiplication and division to calculate.
Another example is the fifth question on page 58 of the textbook: Mom ordered a small cabinet that can be assembled freely for naughty. Each small cabinet 18 yuan. How much does it cost to put a picture of 2 yuan on the cupboard door?
There are six small cabinets on the right side of the topic. According to the usual classification idea, children can easily list formulas: 18×6+2×6. The calculation of this formula is very simple and easy for children to accept. In the teaching of this topic, it is necessary to cultivate children's data analysis literacy. First of all, from the perspective of diet, both multiplication formulas have the same number 6. Therefore, children can be guided through multiplication and division to get (18+2)×6. Secondly, children can be guided to observe the picture directly. Through data analysis, it is concluded that each small cabinet 18 yuan, the sticker on the cabinet is 2 yuan, that is, each small cabinet plus the sticker on the door is a ***20 yuan. A * * * has six such cabinets. In this case, the formula can be obtained directly: (18+2)×6.
Among the five algorithms, additive commutative law and multiplicative commutative law, because children have sufficient learning experience in the past. So it is not difficult to master. The skillful application of additive associative law and multiplicative associative law depends on the improvement of operation ability, especially the improvement of number sense ability, which plays an important role. "Rounding" is the core of thinking. Multiplication, division and distribution are difficult to master and require some training.
After training to a certain extent, children can form the idea of mathematical modeling. Of course, children don't need to understand the concept of modeling. When children are faced with an addition to connect two multiplication formulas, they should have a sensitivity. When one of the two multiplication formulas has the same multiplier, the conditions of using multiplication and division can be met, and the inverse operation of multiplication and division can be used for simple operation.
Another example is the sixth question on page 58 of the textbook: We often use vertical calculation to multiply multiple digits.
(1) Can you explain the truth with the law of multiplication and distribution?
? 26
× 2 1
————
? 26
Fifty two
————
546
In fact, 2 1 is divided into 20+ 1 for calculation.
? 26×2 1
=26×( 1+20)
=26× 1+26×20
=26+520
=546
(2) Try to calculate the following problems with the law of multiplication and distribution.
58× 1 1 ? 47× 102
In the teaching of this topic, children can be guided to fully experience, compare, connect and imagine, thus establishing a more advanced model of multiplication and division. Then this model is applied to the multiplication formula of multiplying two numbers. So that the multiplication formula that needs to be solved vertically can be calculated recursively by using the multiplication distribution law directly, and the answer can be written directly, which achieves the effect of simple operation.
Mathematical modeling is the highest level of all kinds of mathematical core literacy. Mathematical literacy and mathematical ability go through intuitive imagination, mathematical abstraction, mathematical operation, logical reasoning and data analysis, and finally form mathematical modeling, reaching the top of the pyramid of mathematical literacy. After the formation of mathematical modeling, we use this accomplishment to reflect on our real life, guide and improve our life, and form a more advanced mathematical modeling when conditions and opportunities permit. So as to form the ability to observe the world with mathematical eyes, understand the world with mathematical thinking and express the world with mathematical language.