Mathematical concepts are the basis of learning mathematics. Learning concepts (including theorems and properties) requires not only knowing why it exists, but also knowing why it exists. Many students only know the concept by rote, but ignore the understanding of the concept, so they will never learn math well or thoroughly.
For every definition, theorem and concept, we should know where it comes from and where it is applied on the basis of keeping in mind the content. Only in this way can we make better use of it to solve problems. Look at some examples.
Students will find that after explaining the basic concepts and theorems, the teacher will always give us some exercises about this concept or theorem, which is to help us understand and consolidate the course content.
Because we have just learned this knowledge, we are not proficient enough to use it. At this time, exercises help us to understand the knowledge and existing concepts in class more deeply and thoroughly. Because of the short class time, the teacher can give very limited exercises in class, so he should find some examples himself.
Second, we should combine thinking with hands-on.
When doing a problem, after reading the problem, we can first think about how to do it, what concepts, properties and theorems are used, and then compare the answers to see what is better than the answers, so as to improve ourselves, or if our ideas and answers are different, we should also find out the reasons and sum up our experience.
When doing exercises, it is a gradual process. The exercises have ready-made answers and clear thinking. As long as we follow its train of thought, we will come to a conclusion. Therefore, you can look at some examples that are highly skilled and difficult to solve by yourself, but not beyond the scope of your study, such as competition questions with moderate difficulty, and do more exercises.
When we say "do more exercises", we don't mean "crowd tactics". When we say "do more exercises", we want everyone to think more about what knowledge it uses, whether it can explain more, whether its conclusion can be strengthened and popularized, and so on.
Three, familiar with all kinds of basic problems and master their solutions
Every exercise in the textbook is aimed at a knowledge point, which is the most basic topic and must be mastered skillfully; Extra-curricular exercises also have many basic questions, with many methods and strong pertinence, which should be done soon. Many comprehensive problems are just the organic combination of several basic problems. If you master the basic problems, you can't worry about solving them.
In the process of solving problems, we should consciously pay attention to the thinking method reflected in the topic in order to form a correct thinking mode. Mathematics is a world of thinking, with many thinking skills.
Therefore, every question will reflect a certain way of thinking in the process of proposition and problem solving. If we consciously pay attention to these thinking methods, after a long time, the "universal" solution of each question type will be formed in our minds, that is, the correct thinking pattern. At this time, this kind of problem is easy to solve; At the same time, I have mastered more thinking methods and laid a certain foundation for doing business comprehensive problems.