Some people say that mathematics teaching is a speculative art, with rigorous reasoning, logical argumentation, accurate calculation and step-by-step thinking mode, people can always appreciate the infinite charm of mathematics!
What causes thinking is a fill-in-the-blank question.
In () ÷ 6 = 4 ... (), if the second bracket can at least be filled with (), then the first bracket should be filled with ().
"What's the smallest number to fill in the second bracket" is actually "What's the smallest remainder". This problem has caused controversy among teachers. Some people say that "the minimum remainder is 1", while others say that "the minimum remainder is 0". Which is right and which is wrong? Everyone's enthusiasm for "getting to the bottom of the matter" came up and decided to hold a scientific and rigorous attitude, demonstrate in many ways and find out.
Second, listen to the voice of a line.
Can the remainder be 0? Everyone knows how the front-line teachers look at this problem, so they "interviewed" several teachers at random.
Teacher 1: What is division with remainder? After the average score is not completely divided, what remains is the remainder. If it is total division, there should be no remainder, so I think the division with remainder should not include total division, so the remainder cannot have 0.
Teacher 2: First of all, if 0 is a remainder, then the division of integers is a division with a remainder, and there is no division without a remainder, which is inconsistent with our usual statement. Literally, "remainder" is the remaining number. How can there be a remainder without it?
Teacher 3: I found Modern Chinese Dictionary (6th edition) published on 20 12. In P 1585, the explanation of the word "remainder" is: "In integer division, the dividend is not divisible, and the remainder is greater than 0 but less than the divisor. For example, 27 ÷ 6 = 4...3. That is, the incomplete quotient is 4 and the remainder is 3.
Teacher 4: Any division has a remainder. Of course, 0 is also a remainder. In some versions of textbooks, it is suggested that the remainder 0 may not be written, but it is clear that 0 cannot be used as a divisor, otherwise it is meaningless.
Teacher 5: In division with remainder, the remainder to be divided must be less than the divisor, and 0 is less than the divisor. I think 0 is a remainder.
Third, the theory of reading textbooks.
Teachers' different answers have their own reasons, but no one dares to jump to conclusions, so they decide to use textbooks to find the answers.
Through the Internet and other channels, several widely used primary school mathematics textbooks with different versions have been found. The presentation of "Remainder" related chapters in different versions of textbooks and the interpretation of the textbook "Teacher's Book" have caused some thoughts.
▲ 20 1 1 Year People's Education Press, Volume 1, Grade 3
▲ 20 14 New People's Education Edition, second grade, volume 2
Interpretation of teaching reference
The new textbook Division with Remainder has been moved from the first volume of Grade Three to the second volume of Grade Two. Compared with the experimental teaching materials, the layout of the topic map has changed greatly, from simple formula inquiry to activity inquiry, which highlights the significance of strengthening students' understanding of the remainder through practical operation.
Example 2 of the new textbook, through the operation of putting a square with a stick, on the basis of consolidating the meaning of division with remainder, through observation, comparison, analysis and discovery of the relationship between remainder and divisor, let students understand the reason why remainder is less than divisor. Later, the topic of "making a work" was cleverly designed. Instead of giving the total number of sticks, students are required to use the relationship between remainder and divisor to solve problems, so that students can make it clear that the minimum remainder is 1 and the maximum remainder is smaller than the divisor 1, thus deepening their understanding of the relationship between remainder and divisor.
think
Compared with the old and new textbooks published by People's Education Press, although there are great changes in the concept of arrangement when discussing the relationship between remainder and divisor, there is no direct conclusion about the remainder and a clear answer about how many the remainder may be, which gives us the feeling that "there is still half of her face hidden behind her guitar", and the front-line teachers are still unclear. However, from the interpretation of the faculty, we can vaguely feel that the editor is still more inclined to the view that the remainder cannot be 0.
▲ Old Soviet Education Edition Second Grade Mathematics Volume 2
▲ New Soviet Education Edition, Second Grade Mathematics, Volume II
Interpretation of teaching reference
Because of the geographical location and other reasons, I didn't find the corresponding teacher's teaching books, so I dare not read them rashly. I feel very sorry. I'll fill in the relevant materials later.
think
The presentation form of this part of the old Soviet Education Edition is almost the same as that of Beijing Normal University Edition (3rd Edition). They all put forward a set of formulas and asked students to compare the remainder and divisor of each question. What did you find? There is no clear answer in the textbook. It is estimated that the faculty's statement should be similar to that of Beijing Normal University Edition (Version 3), and the remainder refers to the number of people remaining.
This part of the new Soviet education edition is similar to the new people's education edition. Students observe, compare, analyze and discover the relationship between remainder and divisor by putting a square with a small stick. The difference is that the textbook of People's Education Edition has no conclusion, while the textbook of New Soviet Education Edition clearly tells students that "the remainder may be 1, 2,3, but not 4", and directly tells all the possibilities of the remainder, without saying that there is 0. Personally, I think the editor of Jiangsu Education Edition may prefer the conclusion that the remainder cannot be 0, and think that the minimum remainder is 1.
▲ Beijing Normal University Edition (3rd Edition) Grade Two Mathematics
▲ Beijing Normal University Edition (4th Edition) Grade Two Mathematics
Interpretation of teaching reference
The third edition of the textbook P4 has practiced four questions. By comparing the remainder and divisor of each question, students can explore and discover the relationship between the remainder and divisor independently. In teaching, teachers should make students realize that the remainder is the remainder. The fourth edition of the textbook P4, combined with the activities of building a square, provides intuitive support for students to understand the relationship between remainder and divisor. After the whole activity is completed, observing and filling in the form will help to stimulate students to produce mathematical problems such as "The remainder is getting bigger later, what's going on" and "The remainder is less than the divisor, why", and then naturally guide students to summarize the relationship between the remainder and the divisor.
think
In the fourth edition of Teaching Reference, several paragraphs have caught our attention:
"Teaching Reference" P2 said, "This book is the second time to learn division. The key point is to learn division with remainder by combining average division with operational activities (note: division with remainder in this unit refers to division with remainder, that is, division with remainder not equal to 0), to know the remainder and to explore the relationship between divisor and remainder".
In addition, P 13 also said, "It should be noted that in the above inquiry activities, in order to present the original records of the operation process, students need to fill in the remainder 0 in the formula, but in the usual division operation, the remainder 0 can be omitted."
The keyword "remainder is 0" appeared for the first time in teaching reference. Compared with the third edition of Teaching Reference, from "teachers should make students realize that the remainder is the remainder" to "division with remainder in this unit refers to division with remainder, that is, division with remainder not being 0" to "students need to fill in the formula with remainder being 0 to present the original record of the operation process". The faculty of the fourth edition did not shy away from the topic "the remainder can be 0". I don't think the editor will modify this content at will. It must be after some textual research that the remainder is reasonable and should be intentional.
Fourth, look at the expert's argument.
There is no saying that "the remainder can be 0" in mathematics textbooks, and there is no unified view in teaching reference, but there is evidence that "the remainder cannot be 0" in dictionaries, so it is not difficult for many teachers to insist that the remainder cannot be 0. Is that really the case?
After all, dictionaries can only represent one opinion, so we decided to look for professional books on mathematics and consult relevant experts, hoping to find the answers we want there.
I contacted a friend who is a Ph.D. student and now teaches in the School of Mathematics and Physics of the University, and exchanged my doubts and thoughts with him. He can't jump to conclusions easily. If we really want to find out, we must find relevant evidence.
A few days later, this research-loving friend sent a number of papers about remainder knowledge, and sent him a college textbook-Elementary Number Theory (edited by Pan Chengdong and Pan Chengbiao), which was published by Peking University Press. The first chapter of this book, P 16, is divided by the remainder 3, which is the most important and basic proof of elementary number theory.
[Theorem 1]? Let A and B be two given integers, a≠0, then there must be a unique pair of integers Q and R, and the necessary and sufficient condition for B = QA+R, 0 ≤ R < A, ab is R = 0.
R here refers to the remainder, and the book clearly tells us that the remainder can be 0.
Niu Li Xian, a special math teacher in primary school, has this discussion on the remainder:
Why do some people question "Is 0 a remainder?" ? This may be related to the usually inaccurate language description. For example, "without residue" is called "without residue" and "with residue" is called "with residue". This description of division with or without remainder leads to cognitive conflict: since there is no remainder, how can the remainder be 0?
This passage gave us great enlightenment and made me suddenly enlightened.
Then, I continued to collect a lot of expert papers about the remainder on the Internet. One of them was written by Professor Xiao Jiankeng of Nanchang Normal University, Jiangxi Province. The topic was "Talking about the fact that the remainder can be zero in integer division", and there was a passage in it that "in fact, the formulation of' the remainder is zero' has long been recognized by the mathematical community. The sentence' the remainder is 0' is well documented. " Page 49 of the Handbook for Primary School Mathematics Teachers (People's Education Press, 1982) has the following statement:
To judge whether an integer can be divisible by another positive integer, just divide it. If the remainder is 0, it is exactly divisible; If the remainder is not 0, it cannot be divisible.