First of all, we should pay attention to cultivating students' guessing ability, imagination ability and intuition ability. Students (at least some students) can make good guesses about some problems, or have good intuition. "There are indeed many things in mathematics that can only be understood but cannot be expressed. Sometimes it's hard to explain why. " However, can we come to the conclusion that students should be asked less' why'? A teacher asked this question. )
I don't want to overemphasize the "fallacy" of intuition here; However, if you don't insist on putting forward "why", how can students form a sense of "proof" well? How can we deeply understand the necessity and positive significance of "proof"?
In fact, since the 1980s, the reform practice in the United States with the direction of "solving problems" has provided strong arguments for the above conclusions. Specifically, in the practice of reform, people can often see the phenomenon that students (even teachers) are only satisfied with finding the answer to the question through some methods (including observation, experiment and guess), without further thinking and research, and even without necessary testing and proof of the correctness of the obtained results. This phenomenon has certainly aroused great anxiety among people, especially mathematicians. For example, Professor Wu Hongxi of the University of California once stressed that it is understandable to emphasize intuition and informality, but we can't use it instead of mathematical proof, but only as a necessary supplement to the latter; And "if you are always satisfied with guessing without proof in the process of solving problems, students will soon forget the difference between guessing and proof", which can even be said to be worse than not knowing how to solve problems at all, because "proof is the essence of mathematics".
Of course, it needs to be clear that some teachers emphasize that "mathematics teaching should attach importance to mathematical conjecture and intuitive thinking." Especially in the guessing stage, when you don't know what the conclusion is, ask students' why' as little as possible. "However, it actually involves the following questions, that is, how to cultivate students' ability to guess and intuition. In particular, should these two abilities be regarded as gifts or acquired development processes? In addition, will asking more "why" inhibit the development of students' guessing and intuitive ability?
As for the former question, I think most math workers, including math teachers, will give a clear answer from their own personal experience, that is, to affirm the development and trainability of guessing and intuition. For example, for a professional mathematician, his mathematical intuition is obviously no longer a simple primitive intuition, but a refined intuition, which is gradually developed through years of study and research; In addition, it is based on this understanding that we can talk about "learning to guess" advocated by Polya and how to help students "learn to guess reasonably".
Obviously, if the above position is established, then an inevitable conclusion is that we should not only pay attention to "protecting" students' existing guessing ability and intuitive ability, but also help students learn reasonable guessing methods, so that their intuition can be developed and refined. However, necessary reflection is an important link to cultivate and develop the ability of guessing and intuition. In fact, according to the analysis of many famous scholars (such as Piaget), this is an important feature of mathematical thinking, which is mainly based on the so-called "reflexive abstraction" (which should be clearly distinguished from the "empirical abstraction" in other natural sciences) and is directly related to reflection activities. Special, because often asking yourself "why" is a basic form of "reflective activity". Therefore, in this sense, I think that to cultivate students' guessing and intuition ability, we should not only ask students "why" frequently, but also strive to promote students' transformation from "passive state" to corresponding "conscious state", that is, from passively answering teachers' questions about "why" to constantly asking themselves "why".
It is worth mentioning that the above changes are of great significance not only to cultivate and develop students' guessing and intuition ability, but also to develop students' cognitive ability, which should actually be regarded as an important goal of mathematics education and even general education.
Finally, although some "why" answers are "sometimes difficult", a good guess (or "reasonable" guess) is always meaningful. Of course, the latter cannot be simply equated with strict proof, and we can tell some truth from the perspective of heuristic method. However, any experienced teacher clearly knows that students can often sum up the following "heuristic principles" from their own learning experience: it is often the most effective and reliable to find answers according to the ideas suggested by the teacher.
Of course, in order to cultivate students' ability of guessing and intuition, we should not go too far on the road of "inspiration". Generally speaking, we involve the research object or main characteristics of mathematical methodology in teaching. This means that the main focus of mathematical methodology is the "rational reconstruction" of the process of mathematical discovery and creation, that is, it is hoped that the invention and creation activities in mathematics will be "understandable", "learnable" and "popularized and applied" through the analysis of methodology. In my opinion, this is an important means for us to improve mathematics teaching, that is, to use the analysis of thinking methods to drive the teaching of specific knowledge content.
Because the core of mathematical heuristic is some rigid questions and suggestions, in my opinion, this actually shows this point more clearly: in order to cultivate and develop students' guessing and intuitive ability, we should not only ask fewer questions. And students should be asked more "why". Of course, as Paulia pointed out, the key here is "proper use"; In addition, more importantly, as mentioned above, we should also pay great attention to helping students develop the habit of asking themselves and reflecting.