There are ten mistakes in mathematical thinking, as long as one is wrong, mathematics will not be learned well!
I gave a simple question to many junior high school students, high school students, and even college students, but no one got it right...... Later, I gave it to many junior high school and high school teachers, but no one did it right.
Moreover, their answers are varied, each has its own ideas, each has its own reason, that is, no one can say, accurate, scientific truth, very strange!
Even if you are in front of the screen, you may not be able to do it right, if you don't believe it, try it, look at the title first:
From this simple question, you can find the second big mistake in mathematical thinking
The second of the four major thinking mistakes in mathematics: thinking superficially
The surface of thinking can be reflected in two ways:
(1) Knowledge learning is superficial
Mathematics is a science, every knowledge, including formulas, theorems, etc., represents a scientific truth, has a profound meaning, must go deep into the essence, in order to understand, however, in order to cope with the exam, teaching is superficial, without in-depth understanding, the real will not know, the knowledge learned is false.
(2) Mathematical problem solving is superficial
Because the knowledge is not good, the solution can not think deeply, just superficial, probably, random, plus the learning is boring, the problem solving is impatient, in a hurry, only seeking the answer, the solution is also fake, the real will not know.
It can be seen from this that the surface of knowledge understanding and problem solving are the huge mistakes of today's mathematics learning, mathematical problem solving, and mathematical thinking, and the main culprit of this mistake is test-oriented education.
In the environment of exam-oriented education, teachers only talk about topics in class, forcing students to memorize questions; Students are anxious to improve their grades and feel that the memorization type improves faster. Schools, teachers, and students only focus on results, ignoring the process of understanding, comprehension, and growth.
It seems to be a set of memorized questions, which meets the needs of the exam, but there are thousands of questions, this question will be tested, that question will be tested, this question seems to have to be done, and that question seems to have to be done...... It is easy to evolve into a sea of questions tactic, students brush so many questions every day, where can they remember them all? In the end, it took effort, bad thinking, and bad exams, and the gains outweighed the losses.
Think deep
How can you learn to avoid superficial thinking? Quite simply, the opposite of the surface of thinking is deep thinking. It is mainly divided into two dimensions: in-depth knowledge understanding and in-depth topic thinking.
(1) In-depth knowledge understanding
Let's move on to this question, first of all, this question is obviously examining the definition of rays.
The essence of the ray is two, one, the starting point, two, the direction, to be represented by two letters, such as ray AB, A represents the starting point, B represents the direction, there is a different ray is different.
Looking at the title, the rays AB and BC, the starting point is different, it is different, and it is counted as 2 rays; In the same way, BC has a different starting point than CD, which is another ray.
However, AB and AC, AD, etc., have the same starting point direction, and they are actually the same ray. Therefore, the only rays that can be represented by letters in the figure are AB, BC, and CD.
When the teacher is teaching, he does not give a full lecture, but allows students to think independently and experience their own feelings, so that learning knowledge is in-depth understanding.
(2) In-depth thinking in problem solving
According to the deep thinking, we need to think about the problem with knowledge, as mentioned in the definition of ray above, judging the ray mainly depends on two points: the starting point and the direction.
Second, let's think step by step:
A is the starting point, AB and AC are the same, and BA is a line segment, not a ray.
B is the starting point, BC and BD are the same.
C is the starting point for a CD, and D is the starting point, because there are no letters, it cannot be represented.
So, the answer is 3 points, think like this, be specific, clear, and must be right.
As you can see, deep thinking is completely different from the surface of thinking, every step seems cumbersome, but it is based on and reasonable, and the final answer is definitely right, and every student can do it.
All knowledge and all topics in primary and secondary schools are the same, superficial thinking is the culprit of the problem, and in-depth thinking is the scientific method to solve the problem.
If you want to learn more deeply and systematically, you can read the book "The Essence of Mathematics · by Zou Huaquan, the founder of Deep Thinking", or learn video method courses and participate in face-to-face ability courses.