First, cultivate the rigor of mathematical thinking.
The rigor of thinking refers to the rigor and justification of considering problems. To improve the rigor of students' thinking, we must be strict and strengthen training.
First of all, students are required to think step by step and clearly, that is, to think in a certain logical order. Especially when learning new knowledge and methods, we should start with the basic steps and deepen them step by step.
Secondly, students are required to think comprehensively and carefully, and there must be sufficient reasons for reasoning and argumentation. Use the power of intuition, but don't stop at intuitive understanding; Use analogy, but don't believe the result of analogy; When examining questions, we should not only pay attention to obvious conditions, but also pay attention to discovering those hidden conditions; When applying the conclusion, pay attention to the conditions under which the conclusion is established; Carefully distinguish the differences between concepts, understand the connotation and extension of concepts, and use concepts correctly; Give the answers to all the questions, don't leave them out.
Second, cultivate the depth of mathematical thinking.
Thinking depth refers to the abstraction and logical level of thinking activities, as well as the depth and difficulty of thinking activities. In mathematics learning, students often can't understand the conclusion well. When doing exercises, they can't understand the essence of problem-solving methods at all. Without books and teachers, they can't solve problems independently. This phenomenon is a manifestation of students' lack of deep thinking in long-term study. To overcome this phenomenon, we must consciously and often carry out profound thinking training.
1, looking at the essence of mathematics through phenomena
Whether we can see the essence and connection of mathematical objects through superficial phenomena is the main manifestation of profound thinking. In many mathematical problems, the conditional relationship is relatively hidden. If we only look at the surface of the problem, we can't start. Therefore, in mathematics learning, we should think from the outside to the inside and grasp the essence and law of the problem.
Example 1: There is 17 in Le Ballon Rouge in the store, and 9 yellow balloons of Le Ballon Rouge are missing, and the number of flower balloons is three times that of Le Ballon Rouge. How many flower balloons are there?
Analysis: An application problem contains two unknowns, which are generally unsolvable, but this problem needs the number of balloons. Obviously, this application problem can be transformed into an application problem that contains only one unknown number (the number of balloons). In other words, we can get the serial number of "Red Balloon" through the known conditions, so as to see the essence of the problem through the phenomenon and make clear the direction of transformation.
Solution: (1) How many red barons are there?
17-9=8 (pieces)
(2) How many flower balloons are there?
8×3=24 (piece)
There are 24 flower balloons.
2. Pay attention to the careful examination of the questions to prevent the mindset.
When students encounter similar new problems after solving similar problems many times with a certain thinking mode, they often mechanically apply the previous thinking mode. The more times the same method is used, the more obvious this tendency is.
Example 2: There are 45 myna and 32 orioles in the zoo. The total number of orioles and peacocks is 8 less than that of myna. How many peacocks are there?
Because it is customary to add up the number of orioles and myna to get the total number of two kinds of birds, many students mistake the total number of orioles and peacocks in this question for the total number of orioles and myna, which makes mistakes in solving problems. To overcome this mindset, students can be trained to observe, think and analyze more in their usual homework and practice. In addition, consciously arrange appropriate counterexamples to lure students into being fooled and let them learn from their mistakes.
Third, cultivate the broadness of thinking
The broadness of thinking means that a problem can be considered from many aspects. Specifically, it can explain a fact in many ways, express an object in many ways, and put forward many solutions to a topic. In mathematics learning, paying attention to multi-directional and multi-angle thinking and broadening the thinking of solving problems can promote students' thinking broadening.
For example, to find the circumference of a rectangle, either the four sides are added together, or the lengths of two lengths and two widths are calculated separately and then added together. It is more convenient to add up the length and width first and then multiply by 2 to get the result.
Fourth, cultivate the flexibility of thinking
The flexibility of thinking refers to the timeliness of being able to improvise with the changes of things, not being too affected by the mindset, and being good at getting rid of the old model or the usual constraints. Cultivate students' rigor, profundity and extensiveness in mathematical thinking, but without developing the flexibility of thinking, it is possible to make thinking tend to a specific method and way, unilaterally pursue the stylization or patterning of analyzing and solving problems, and produce thinking inertia.
Flexible thinking is characterized by using knowledge freely, being good at adapting and adjusting ideas, and being good at using differential thinking to analyze specific problems, which is an important manifestation of flexible thinking.
Example 3: Calculating 242-97+55 by Simple Method
Analysis: This is a comprehensive calculation problem of addition and subtraction. If the calculation is simple by conventional methods, the solution is as follows:
242-97+55
=242- 100+3+55
= 142+3+55
= 145+55
=200
It is convenient to show only the first step in the calculation, and other steps do not show much advantage. If we decompose 97 from another angle, there are the following solutions:
242-97+55
=242-42-55+55
=(242-42)-(55-55)
=200
The final result can be obtained simply.
This kind of problem that needs to be solved outside the convention is a good way to train the flexibility of thinking. In addition, the traditional problem-solving method is also a good way to train the flexibility of thinking.
Fifth, cultivate critical thinking.
Critical thinking refers to being good at strictly estimating thinking materials and carefully examining the thinking process in thinking activities. In mathematics teaching, the critical performance of students' thinking is that they are willing to test and reflect in various ways, and can put forward their own views on existing mathematical expressions or arguments, instead of blindly following them. They have completely accepted things ideologically, but also seek improvement and put forward new views and opinions.
Improve the critical consciousness of students' thinking can be carried out from the following aspects:
1, cultivate students' habit of reflection after solving problems.
Cultivating students' habit of reflection after solving problems is to cultivate students to review, think, summarize, evaluate and adjust problem-solving activities, that is, to reflect on experiences and lessons. When solving a problem successfully, we should consider what concepts, methods and conclusions are used in the key steps of the problem-solving process; If you encounter setbacks in the process of solving problems, you should also find out the reasons and what part of the knowledge is unfamiliar. Experience and lessons can strengthen the relevant knowledge of mathematics from two different aspects, which is a prelude to improving the criticality of mathematical thinking; The second is to test and analyze the answer to the question, whether the reasoning is reasonable and whether the argument is sufficient; Finally, consider whether there are other solutions.
2. Error correction training is often carried out in teaching.
The opposite of critical thinking is not critical, which is also the characteristic of many primary and secondary school students. They often believe in conclusions and are not good at or will not find their own mistakes in solving problems. Teachers often make some mistakes in teaching, so that students can discuss and correct them, which is helpful for students to form critical thinking.
In teaching, students are often encouraged not to be superstitious about books and teachers, but to have their own independent thinking and dare to put forward different opinions.
Above, we discussed five main thinking qualities and training methods on how to cultivate students' good teaching thinking habits. In addition to rigor, extensiveness, flexibility and criticism, it is also exploratory, original and purposeful. And these five thinking qualities are the most important. They are interrelated and inseparable. The rigor of thinking is the most basic requirement of learning mathematics and the basis of thinking quality; Profound and extensive thinking is a solid framework based on rigor; The flexibility of thinking affects the level of problem-solving ability to a considerable extent, and it is also the extension and exertion of rigor, profundity and extensiveness of thinking; The criticism of thinking is the comprehensive embodiment of the other four thinking qualities.