This book is divided into the first part and the second part. The first part mainly tells the mathematical thinking method related to primary school mathematics, and the second part tells the case interpretation of the mathematical thinking method in primary school mathematics published by compulsory education. Reading a complete book and feeling to cultivate thinking ability is the core goal of mathematics teaching. Only the teaching of mathematical thinking methods can cultivate students' thinking ability and improve their problem-solving ability.
The book explains in detail some concepts, teaching requirements and problem-solving methods about limit. The idea of limit is to study the changing trend of quantity through infinite approximation. Two key sentences are caught here: one is that the amount of change is infinite, and the other is that the amount of infinite change tends to a constant, and both are indispensable. For example, the natural sequence is infinite, but it tends to infinity, not to a constant, so the natural sequence has no limit. In teaching, on the one hand, students should experience infinity, and more importantly, through specific cases, the amount of infinite change tends to a constant. The limit, derivative and definite integral defined on this basis are powerful tools to solve practical problems expressed by functions. Finite and infinite are a manifestation of dialectical thinking, so we should treat their relationship dialectically, not with the "finite" view of elementary mathematics, but with the idea of limit. Limit method is a powerful tool to deal with the trend of infinite change.
In other words, when we are faced with infinite problems, we should stop thinking from a limited perspective. In order to enter an infinite state, mathematical limit is such a rule and logic, and we just have to follow this rule and logic. In addition, the understanding and expression of cyclic decimal and infinite acyclic decimal also embodies the dialectical relationship between finite and infinite. We know that in middle school mathematics, rational numbers are generally defined by integers and fractions, while irrational numbers are defined by infinite acyclic decimals. Rational numbers and irrational numbers are collectively called real numbers. Rational numbers include integers, finite decimals and cyclic decimals. Integers and finite decimals are familiar to students. So, how to divide the number of components of cyclic decimals? We have introduced before that this problem can be solved by equation method. Next, we use the limit method to solve it.
Case: Convert the cyclic decimal 0.999… into the number of components. Analysis: 0.999… is a cyclic decimal, that is, the number of digits in its decimal part is limited. For primary school students, the acceptable method is the combination of numbers and shapes and the infiltration of extreme thoughts, and the extreme thoughts are described by constructing intuitive geometric figures. First look at the following sequence 0.9, 0.09, 0.009, … With the idea of combining numbers with shapes, this sequence is constructed with line segments as follows: divide a line segment with the length of 1 into 10, and take 9 of them; Then divide the remaining 1 into 10, and take 9 ... The length of all the line segments is 0.9+0.09+0.009+...= 0.999 ... If you take it indefinitely, the length of the remaining line segments tends to 0, and the taken length tends to 1. According to the limit idea, you can get 0.
For teachers, it is not enough to have the infiltration of extreme thoughts, but also need to further understand how to solve it in extreme ways. This is a summation problem of infinite ratio recursive sequence. According to the formula, we can get 0.9+0.09+0.009+… = 0.9 ÷ (1-0.1) =1,so 0.999…= 1.
In short, in my own teaching practice, I contact the theoretical knowledge I have learned and use these theoretical knowledge to guide our teaching.