Get to the point, as the preface says: this is a collection of books that discuss the core concepts in primary school mathematics. Through comments and suggestions on the current primary school mathematics textbooks, we can promote the reform of mathematics teaching and make contributions to the construction of mathematics education with China characteristics. The researchers of the original textbook are Professor Zhang Dianzhou and Professor Gong Zikun from Hangzhou Normal University. They analyzed and suggested all versions of primary school mathematics textbooks from an expert's point of view. However, the two professors lack practical experience in primary school mathematics teaching, and invited primary school mathematics researchers to discuss and exchange with senior teachers. Finally, Yin Wendi wrote conversation record into words and revised it repeatedly, and finally got such a teaching research book that tried to deeply analyze the core concepts of primary school mathematics from different aspects. This book is divided into four parts: the first part is about numbers, words and equations; The second part is about "division", "fraction" and "ratio"; The third part is about graphics and geometry; The fourth part is others.
I haven't read much about this book. Recently, I only talked about the meaning and nature of fractions, so I focused on the second part: about division, fractions and ratios. This passage impressed me deeply: for the average American student, grades are a very difficult subject, and calculation and application problems are often not done well. But look at the excellent students in America. They have a profound understanding of the essence of mathematics, and feel that the appearance of fractions is a great milestone in the history of mathematics and a manifestation of human civilization. This is the foundation of outstanding students in the United States, which may be more solid than outstanding students in our country. Mr. Qiu Chengtong criticized the foundation of excellent students in China, which means understanding the depth of mathematics. The direct reason why students' understanding is not profound may come from teachers' understanding of knowledge is not profound enough. After reading this book, they also deeply realized the heavy sense of primary school mathematics knowledge and superficial understanding of teaching materials.
For example, there are two division modes: equal division and inclusive division. There are two modes of division: equal division and inclusive division. Whether it is division teaching or fraction teaching, the most common situation is "average division", which involves two kinds of division, commonly known as "equal division" and "inclusive division" to know the total number and number of copies, commonly known as "equal division"; Knowing the total number and each copy, we can find the number of copies by division, that is, how many copies are included in the total number, commonly known as "inclusion and division" It is mentioned in this book that many versions of textbooks are deviant and equally divided, and the number of exercises is so small that children agree to form a one-sided thinking mode, which is very unfavorable for cultivating students' ability to analyze and solve problems, and ignoring "inclusion and division" will emerge one after another. Many students asked when to use division. Divide will be equal to "divide equally". The textbook we used in Qingdao used two red dots in the first volume of Grade Two, which talked about "equal score" and "inclusive score" respectively, which did not belong to that situation. However, after learning the area of rectangles, squares, triangles and other polygons, there are two kinds of questions: after calculating the area, how many seedlings do you need to plant a tree every 8 square meters? How many kilograms of wheat can be produced by producing 0.75 kilograms of wheat per square meter? It is easy to be confused, which is caused by students' incomplete understanding of "inclusion and division" in division.
"Inclusive division" and "average division" are equally important to "twin brothers": the case of average division is more suitable for integer division, and it is evenly distributed to several people, groups and classes. But it is not easy to make sense for fractional division, just as it cannot be said that a piece of cake is evenly distributed to individuals. We learn fractional division according to the rule of inverse multiplication, but it is difficult to explain. Although fractional division is not suitable for equal division, "inclusive division" is easy to use. For example, it can't be said that four biscuits are distributed to individuals equally, but how many biscuits are included in four biscuits? As long as you draw a picture, you can know that there are 2 cookies in 1, so 4 contains 8 cookies. It is 4÷=4×2=8, and it is easy to see it when you multiply it backwards. It is proposed to realize the unity of "equal share" and "inclusive share", make the sub-model more full, and pay attention to the difference between "each share" and "number of shares".
Similarly, our definition of a fraction has always been: "Divide the unit 1 into several parts, and the number representing such one or several parts is called a fraction." Such a definition must be divided into several parts equally, but in many cases it is difficult to do so. Back to the problem of average score, that is, the situation of "equal division" and "inclusive division" 1: Find the known number of copies, divide each moon cake into four parts, and ask how big each piece is. It's a block. This is our common situation. Situation 2: Given the size of a part, ask how much it accounts for the whole. As for the average number of parts to be divided into, it needs to be calculated or measured. For example, a box contains 12 pencils. Now, take three pencils and ask how many pencils you took out. First, it is found that 12 contains four 3s, so after the 12 branch is divided into four parts on average, three are exactly 1, that is. Another example: How many times are four pencils 12 pencils? 1 pencil can be regarded as 1 pencil, and four pencils and 12 pencil include such four pencils and 12 pencil, that is; Consider that two pencils are 1 copy, and four pencils and 12 pencils contain two and six copies, that is; Four pencils can be regarded as 1 copy, and four pencils and 12 pencils include such 1 copy and three copies, that is. What I usually say is regarded as the first case, which is divided into 12 on average, and simplified to get 4 copies. I still don't know enough about the depth of mathematical knowledge.
When talking about the necessity of generating fractions, it can be divided into two aspects from the historical point of view of numbers: the earliest number generated was a natural number (non-negative integer), and later the integer result could not be obtained when measuring the average, so fractions were generated. Scores are generated in the process of actual measurement and average score. However, we usually seem to pay attention to the fact that the average score produces more scores, ignoring the situation that scores are produced in the process of "measurement". When people measure another quantity A with a standard quantity B (measurement unit) and measure it accurately several times, they can use an integer to represent the measurement result. If it can't be measured accurately, it can be divided into two situations: one is to divide the measurement unit into n parts on average, and use one of them as a new measurement unit to measure m times accurately. If the result of measuring A by B cannot be expressed by an integer, a new number fraction must be introduced to express the measurement result. I didn't find the relevant content when I looked for the fifth grade textbook. Maybe I am not familiar with the exercises in the textbook. In short, I still feel unfamiliar with the textbook.
Simplifying the model of division and fraction is not conducive to understanding the essence of mathematics. In order to let students know the ins and outs of knowledge more clearly and improve their understanding of mathematics, teachers need to keep learning and improve their professional quality.
I've read this part once, and I don't know much about some superficial feelings. If there is anything wrong, please criticize and correct me.