In the current primary school mathematics textbooks, the concept of equations has been used as "equations containing unknowns are called equations", and in teaching, students often think that "x=5" is an equation, because it contains both unknowns and equations. At this time, the teacher will tell the students that if they already know that the letters represent specific numbers, such as "x=5", they cannot be called equations, and the purpose of equations is to ask for unknowns.
There is no problem with the student's understanding and the teacher's explanation, and the problem lies precisely in the concept that "an equation with unknowns is called an equation". Regarding the concept of equations in primary school mathematics, Professor Zhang Dianzhou questioned "equations containing unknown numbers" in the article "The teaching of mathematical concepts should be integrated into Chinese culture, and the expression of the concept of "equations" in primary school mathematics should be brought to the fore: Is this so-called definition important?Is this a rigorous logical definition?
Professor Zhang said that in the exercise, the teacher replaced "unknown" with "letter", and the concept of the equation became "an equation containing letters". Letters do not always mean unknowns, and there are many equations that contain letters, not necessarily all of them. For example, the formula a+b=b+a, which describes the commutative law of addition, is also an equation with letters, but it is not an equation. In the letter represents a certain class of numbers, such as the formula for calculating the area of a triangle s=1 2ah, where a is the length of the base edge and h is the height, which has nothing to do with the equation. In addition, among the letters that represent variables, the functions are also equations containing letters s=vt, y=1 x, etc., although they can be regarded as equations for a certain curve, but when studied as functions, they are not the same as equations.
That is, it is okay to say in a textbook that "an equation is an equation with letters", and conversely, it is not correct to think that all "equations with letters are equations". The core of the concept of equations is the requirement of unknowns, therefore, Professor Zhang believes that equations can be defined as "equations that seek unknowns, and establish an equation relationship between unknowns and known numbers." ”
Professor Zhang's redefinition of equations made me feel relieved about the barriers to equations in my heart. Taking the primary school mathematics textbook of the People's Education Edition as an example, there are really phenomena such as the unscientific and unrigorous definition of equation concepts, which makes it difficult for teachers to explain clearly to students.
There is also a question about the meaning and algorithm of multiplying fractions by integers in the first book of the sixth grade. The example question is like this: Xiaoxin, Dad, and Mom eat a cake together, each person eats 2 9, how many do 3 people eat in total?The textbook first presents the addition of "2 9 + 2 9 + 2 9 = 6 9", and then from the addition to the multiplication solution "3 2 9s are added, and the multiplication is 2 9x3 or 3x2 9". The teacher's question is: according to the following textbook "The Meaning of Whole Number Multiplication Fraction", 3x2 9 represents what 2 9 of 3 is, not what 3 2 9 is, and I don't know how to explain the problem of eating cake to students.
This teacher's question, I told the students in class: how much is the sum of 3 2 9, in the sense of multiplication, you can use 2 9x3 to solve;In terms of the commutative law of multiplication, 2 9x3 and 3x2 9 are calculated with the same result, but the meaning of the representation is different. Also, I explicitly told my students that in the sense of the word, I do not agree with the textbook saying that "the sum of three 2 9s is 2 9x3 or 3x2 9 when expressed by multiplication". Later, after learning "multiplying a number by a fraction to find out what fractions of the number are", the students will also agree with my point of view.
Last year, the People's Education Edition of primary school textbooks attracted widespread attention from the society, from the illustrations to the criticism of the content, wave after wave. Last summer, presumably the bad and busy of the Renjiao Society was incomparable. The pressure is too great, and the problems with the textbook are so big that it makes people's jaws drop. As a primary school mathematics teacher, I pay great attention to the textbooks, and have been studying the textbooks in depth, and I often think about and question the presentation of the essence of mathematics in the textbooks. However, the problem of teaching materials that broke the news on the Internet refreshed my cognition of textbooks and my own cognition.
For many years, we have been advocating "teaching with textbooks" instead of blindly "teaching textbooks". Mobike, or superstition, about authority, has blinded the eyes of many teachers and hindered their thinking. Textbooks are made by people, there are no perfect people, and there are no perfect textbooks. As we study the textbook, we might want to ask these questions: "Is that so?"Why is this so?Is this the case?Is there a better explanation or practice?"When we study the material with questioning, we should have a deeper and more comprehensive understanding of the material to determine what to teach our students