In fact, neither additive commutative law nor multiplicative commutative law is a problem. Children in Grade One already know additive commutative law and can use it skillfully. After calculating 8+9= 17, they will immediately say 9+8= 17. Moreover, in the learning process of the first volume of Senior One, there are a lot of additive commutative law's exercises. For example, 8+9 can be divided into 8 and 9, which also contains additive commutative law's thoughts.
Additive commutative law, who can master the multiplication and exchange law skillfully in the first grade, can master it quickly when it appears. How did it become a problem in the fourth grade?
The fundamental reason is that several equations listed in the textbook satisfy additive commutative law's relation. Some teachers asserted that the textbook "proved" additive commutative law with these specific equations. In fact, judging from the appearance of additive commutative law in Grade One, I prefer to believe that these specific equations are quoted in textbooks to awaken additive commutative law's memory.
Of course, if the intention of textbook writers is to "prove" additive commutative law with these equations, there is no problem. After all, besides strict deductive reasoning, there is also reasonable reasoning. Moreover, in my opinion, reasonable reasoning plays a more important role in the development of mathematics. Moreover, the object of study at this time is still the primary school students in the specific operation stage, and the way of giving examples is undoubtedly very suitable (the fractional division operation in grade six can be regarded as the first time that children use deductive reasoning).
Of course, as teachers, we can think more deeply about this problem. Let's find out how additive commutative law should conduct deductive reasoning, which may bring something very different to this part of teaching.
In his article, Mr. Zhang Dianzhou talked about the addition of natural numbers, that is, after two disjoint sets of finite cardinality A and B are merged, the cardinality of A∪B is the sum of the cardinality of A and B. This is probably an attitude towards the definition of addition in our textbooks, at least in the People's Education Edition.
But in my opinion, this is a wrong definition, or at least a definition that is divorced from the essence of addition.
Mr Zhang Dianzhou went on to talk about another kind of understanding, namely "counting". Two piles of stones, A and B, first count A in pile A, then B in pile B, and the final result is A+B.
This is not just a change in understanding, but two completely different definitions. The former takes the cardinal meaning of numbers, while the latter takes the ordinal meaning of numbers. Who is closest to the original meaning of addition? Nature is the latter, that is, the ordinal meaning of numbers. Because the number (shù) comes from the number (shǔ), the ordinal number is the most primitive.
As we know, mathematics is a highly logical subject, and logic is a "chain". After deduction, one link after another, then there must be a starting point, that is, the place where the "chain" begins. For the subject of mathematics, it is mathematical intuition. What is mathematical intuition? That's certain and unprovable. For example, the "shortest line segment between two points" in plane geometry is certain, but I am afraid no one can prove it.
So what is the starting point of the logical "chain" with addition as a link? Or what is the mathematical intuition as the basis of addition?
Piaget told the story of a mathematician when he was a child. The mathematician fiddled with the stones and arranged them in a line. Counting from left to right, 1, 2, 3… finally gets 10. Then, he counted from right to left, 1, 2, 3 ... and finally got 10. He felt amazing and a light rose in his heart.
This is really a wonderful thing (although we are all numb). Whether we start from the left or the right, we get the same number. This process produces an unchangeable property, which is different from the physical properties of stone such as color, smell and hardness. This is the number.
Of course, numbers with mathematical properties are not 1, 2, 3 ... These numbers have always existed, and humans discovered this property long before the number symbols such as "1, 2, 3 ...".
The ancients herded sheep. In the morning, when the sheep came out of the pen, a sheep threw a stone. In the afternoon, the sheep returned to the pen and took another stone. In this way, we can know whether all the sheep are in the sheepfold. In other words, as long as a sheep corresponds to a stone, the sheep and the stone have the same property-number.
We are convinced that as long as they are two kinds of objects in one-to-one correspondence, they have the same attribute-number. As for why? It is really impossible to prove, and it is impossible to queue up in another space or another existence. One-to-one correspondence may not get the same properties.
So the essence of this number is the mathematical intuition of counting, which is the starting point we are looking for.
What is addition? The essence of addition is counting, which is an upgraded version of counting. People's Education Edition defines addition as the union of sets, but when calculating the result of addition, we must go back to counting (counting from the beginning and then counting …). This is inevitable, because the set can still be based on "number".
Addition is an upgraded version of counting because you don't have to go back to one number at a time. For example, 5+4, you don't have to count five before counting four. We can count from five to four and from four to five. Even after the child determines that five sticks and four sticks are nine sticks, he can get five apples and four apples as nine apples at once. All this is because of the fixed nature of objects-numbers.
So, what is additive commutative law? 5+4 and 4+5 are just different in the counting order. No matter the pile counting 5 first or the pile counting 4 first, the two stacked objects have an invariable attribute-$ NUMBER, which can be represented by the operation result "9".
So, what is the multiplicative commutative law? 5×4 and 4×5 are just counted in another order, whether it is the number of rows in a row (5 per row) or the number of columns in a column (4 per column).
In fact, the law of addition and multiplication, even the law of multiplication and distribution, is just that the order of numbers has changed.
Back to the textbook, how do we position ourselves? How to teach? In fact, the fourth grade arrangement exchange method has the flavor of systematic arrangement, which is to name the whole operation method. As the simplest algorithm, commutative law needs no effort. It is important to learn addition before, even to infiltrate in counting.