This article was translated by He Gengxu and Robin Chen, edited by Hong Zhiyang, and originally published in HPM Newsletter. This article is slightly modified and reproduced here (http://highscope.ch.ntu.edu.tw/wordpress/?p=18386)
Polya, who had run a refresher course for secondary school mathematics teachers, learned that secondary school teachers needed a curriculum that was directly helpful for their daily teaching. In this course, he repeatedly emphasizes his personal views on the daily work of teachers, which are finally condensed into ten rules, what he calls the "Ten Commandments":
Interested in the subject you are teaching.
Learn about the subjects you teach.
Try to "read" the students' expressions, understand their expectations and difficulties, put yourself in the student's shoes, and treat yourself as a student.
Know the way to learn: The best way to learn anything is to discover the mysteries for yourself and independently.
It is necessary not only to teach students knowledge, but also to let them know skills, know-how, learn the right mentality and the habit of working systematically.
Let students learn to guess.
Let students learn to prove.
Pay attention to the problem at hand now, and look for some features from it that may be helpful for solving the problem later - try to uncover the universal forms hidden in the current specific situation.
Don't reveal all the secrets at once – let them guess before you tell your students – and let them discover them on their own as much as possible.
Inspire questions; let students dare to publish, do not cram into the students.
Immediately afterwards, he made moral shouts about these "commandments":
❶ It is almost impossible to predict definitively whether a method of teaching will work; yet one thing is certain: if you are bored with the subjects you teach, you will also be bored with your audience. The above should be sufficient to illustrate the first of the Ten Commandments: Interest in the subject you teach.
❷ If the teacher has no interest in the subject being taught, he will not be able to make the student accept the subject, so interest is an indispensable requirement for teaching; but interest alone is not enough, and when you do not know a subject, no amount of interest or teaching methods will allow you to clearly explain an argument or opinion to the student. This should also illustrate the second of the Ten Commandments: Know what you are teaching.
❸ Even after you have an interest and understand the subjects taught, you may still be a poor or rather mediocre teacher. I admit that this situation, though uncommon, is by no means uncommon: most people have met teachers who know the subjects they teach but who are unable to establish a channel of contact with students in the class. The so-called teaching should be that one of the professors can cause learning from others, so there must be some kind of communication channel between teachers and students: teachers should understand the situation of students, support their goals, and reasons. This is the third of the Ten Commandments: Try to "read" the expressions of the students, understand their expectations and difficulties, put yourself in the shoes of the students, and think of yourself as a student.
❹ The first three commandments contain elements of good teaching, and together they form a sufficiently necessary condition- if you are interested in the subject you teach, understand it, and can see the problems of your students, you have become or are about to become a good teacher; all you need is experience.
Experience is required, practical experience enables you to understand the "teaching" and "learning" of teachers and students in the classroom, familiarizing you with the process of acquiring knowledge and skills - including learning, discovery, creation, understanding and many other aspects. Psychologists have done a lot of experiments on the learning process and published some interesting arguments. These experiments and arguments are stimulating for a teacher who is very receptive and understanding; but as far as the main aspect of education which we are discussing here is concerned, they are not yet perfect enough to be of direct benefit to the teacher's teaching, and therefore the teacher must first rely on personal experience and judgment.
Based on nearly half a century of research and teaching experience, and after introspection, I would like to put forward some views on the learning process that I consider to be extremely important for classroom teaching. One thing is repeatedly emphasized: active learning is better than passive, "just accepting" cramming; the more proactive you are, the better: the best way to learn anything is to discover the mysteries for yourself and independently.
In fact, in an ideal teaching program, the teacher is like a "midwife" of the mind – giving students the opportunity to discover on their own what needs to be learned. Often due to the lack of time, this ideal is actually difficult to achieve, but it can lead us in the right direction - just as no one can reach the North Star but can find the right direction by looking at it.
❺ Knowledge includes information and know-how. Skill is a skill, it is the ability to process intellectual messages, to make good use of them to achieve goals; it can be said that a series of appropriate mental activities will eventually make our work systematic. In mathematics, skill is the ability to solve problems, construct proofs, and critically diagnose and prove; skills are much more important than the acquisition of purely intellectual information, so the next fifth commandment is quite important for math teachers: not only to teach students knowledge, but also to let them know the skills, know the tricks, learn the right mentality and the habit of working systematically. It is precisely because skills and know-how are more important than knowledge in mathematics teaching, "how to teach" is more worthy of our attention than "what to teach".
❻ "Guess and prove" - this is how the process of discovery usually begins. From experience, you should know about this, and you should know that math teachers have a great opportunity to show the place of guessing in the discovery process, and therefore to remind students of the importance of thinking activities. The latter is not (although it should be) widely known, and unfortunately, given the limited space, there is no way to discuss it in detail here. Still, I hope you don't lose sight of your students in this regard: let them learn to guess. Careless students are likely to make unfounded guesses. Of course, what we want to teach is not to guess without evidence, but to guess with evidence and reason. Reasonable guessing is based on the wise use of inductive and analogous results, and fundamentally encompasses all the processes of rationalized reasoning, which play an important role in science.
❼ "Mathematics is a good subject for learning plausible reasoning." This sentence briefly describes the implications of the aforementioned law, although it sounds strange and very novel; in fact, I believe it. "Mathematics is also a good subject for learning demonstrative reasoning." The phrase sounds familiar — some of its forms are almost as old as mathematics itself. In fact, what is more real is that mathematics and argumentative reasoning coexist, and argumentative reasoning pervades all scientific disciplines, while elevating their concepts to a sufficiently abstract and clear mathematical level of mathematical logic; under such a high level, for example, in everyday life, there is no room for practical argumentative reasoning (in other words, it is no longer suitable for practical argumentative reasoning), but (there is no need to argue such a widely accepted argument), In addition to the basics, math teachers must still let all students know argumentative reasoning: let students learn to prove.
❽ Tricks and tricks are a more valuable part of mathematical knowledge than just having information. But how do we teach this trick? Students can learn it through imitation and practice. When you ask an answer to a question, appropriately emphasize the instructive features in it. If a feature is worth emulating, then it is educational, that is, it can be used to solve the problem at hand, and it can solve other problems - the more commonly used, the more educational it is. But the emphasis on educational characteristics is not only expressed in praising students (because it will have a counterproductive effect on some students), but also in the behavior of teachers (if you have a talent for acting, it is better to pretend a little). A well-emphasized feature can shift your solution into a "model solution", and by having students imitate an answer that solves more problems, it can also transform it into an impressive form, so the rule is: pay attention to the problem at hand now, and find some features from it that may be helpful for solving the problem later - try to reveal the universal forms lurking in the current specific situation.
❾ I hope to be able to point out some tips here that are easy to learn in the classroom and that teachers should know. When you start discussing a problem, try to get the students to guess the answer. Leave students who speculate or even narrate speculation in a dilemma: they must follow the solution to see if their guess is correct, and they must concentrate on consistency. This is just a peculiar case of the following laws (which themselves are deduced and pieced together from some parts of laws four and six): don't reveal all the secrets at once - before you tell the students, let them check - let them discover as much as possible on their own.
❿ There was a student doing a lengthy calculation line by line, and I saw an error in the last line, but I stayed without correcting him right away. I'd rather take the students and check line by line: "It was pretty good at first, your first line was written correctly, the next line was correct, you did this and that...." It's a good line, but now how are you learning it?" Mistakes happen in this line, and if the student finds out on his own, he has a chance to learn something. However, if I immediately find out what is wrong and say, "Here's wrong!" Students may feel unhappy and will never listen to what I have to say anymore. If I say too often immediately, "You're wrong here!" If the student is likely to hate me, and probably start to hate math, then all the hard work I spent on this student before will be in vain. Try to avoid saying, "You're wrong!" If you do, you are not hypocritical but also humane, and Law Ten implies that you should do this, and we can make it clearer: to enlighten the problem; to make it bolder and not to force it into the student.
* Transferred from Meet Math