"Thinking density" refers to the number and depth of "new problems or problems" that you think about per unit time (such as every hour or every day).

Ordinary people:

Ordinary people, under the pressure of teachers, often only think about a few learning problems every hour; When there is no teacher's compulsion, I almost stop thinking about learning.

Those who are gifted in mathematics:

(1) Flexible math problems (or difficult problems, comprehensive problems), etc.

I can think about dozens to hundreds of flexible problems (or difficult problems, comprehensive problems) in mathematics (and other courses) every hour.

(2) textbook intensive reading and intensive reading

In intensive reading, you can come up with dozens to hundreds of new ideas every hour.

(3) Take the bus, sit still and listen to others' nonsense.

New problems and new methods are constantly appearing in my mind, and I can think of dozens to hundreds of methods every hour.

(4) misunderstanding

In the face of movies, TV, novels, newspapers and online articles, at least dozens of thinking loopholes and "weak thinking" can be seen every hour.

Second, thinking compression.

"Thinking compression" means that you think about some problems faster and faster, do problems faster and faster, and read books faster and faster. From reading a book for the first time in a month to reading it for the twentieth time and a half hours, this process is often the process of thinking compression.

Those who are gifted in mathematics:

1) Flexible math problems (or difficult problems, comprehensive problems), etc.

For example, it takes 5 hours to do a selected 100 elastic math problem (or a difficult and comprehensive problem) every day. One month for the first time, three days for the second time and three hours for the third time.

2) Intensive reading of textbooks

For example, it takes 10 hours a day to read the textbooks to be learned next semester, one month for the first time, three days for the second time and three hours for the third time.

Third, "thinking feet"

Generally speaking, "thinking feet" are your opinions, judgments, preferences and so on about something. "Thinking feet" are mainly measured by "size". Generally speaking, the bigger a person's "thinking feet", the worse his thinking ability.

For example, children simply divide people into "good guys" and "bad guys". But adults divide people into more categories. Adults are smaller than children on this "thinking leg" of the classification of "people".

Those who are gifted in mathematics:

(1) The classification of mathematics is fine and varied.

For a certain part of the knowledge or method in the textbook, the book says it is divided into 10, but you can divide it into 100.

For example, people ask you how many kinds of apples are there? Most people may only think of "big apple" and "small apple" according to size, color and brand. As for you, you can think of dozens or hundreds of categories, such as poisonous apples and nontoxic apples, edible apples and inedible apples, flat apples and three-dimensional apples.

Generally speaking, for a certain part of knowledge or method, if the number you can classify is more than 10 times that of others, you can think that your thinking feet are "extremely small". If it is more than five times that of others, it is "very small".

(2) There are many mathematical methods and skills.

For example, if you ask ordinary people how to solve math problems, others may only come up with a few to a dozen methods such as "using formulas", "special values", "reduction to absurdity" and "reverse deduction". But you can name dozens or even hundreds of math problem-solving methods (of course, these problem-solving methods are not compiled by you, but summed up by yourself during the problem-solving process and can be applied to various topics).

Generally speaking, for a certain course or a certain part of knowledge, if you master more than 10 times that of others, you can think that your thinking feet are "minimal", and if it is more than 5 times that of others, it is "minimal".

(3) There are many changes in thinking direction.

When considering a problem, you should not just think in the direction that others (such as the media) make him think, like ordinary people. You have to think backwards, backwards, sideways, sideways.

(4) contact is much more.

Any seemingly unrelated question is related.

(5) There are many differences.

For example, two "concepts", others can think of a difference, but you can think of the difference between 10 and 100.

For example, if you ask someone the difference between "going up" and "going down", the person says, "Of course there is a difference. The top is the top and the bottom is the bottom. " But you tell this person the difference between "up" and "down" from mathematics, physics, language, philosophy and other aspects until this person thinks that you have mental problems and drives you away.

The fourth "thinking hook"

"Thinking hook" refers to other things that someone can think of when thinking about a problem. Just as you hook an object with a hook, it is also like an octopus's tentacle. Therefore, it is called "thinking hook". Generally speaking, people usually say "connection", "interrelation" and so on, which means "thinking hook".

A man with a gift for mathematics.

(1) concept ".

For the "concepts" of different parts of knowledge in textbooks, you can think of many connections, far exceeding ordinary students.

For example, if someone asks you the difference between "apple" and "watermelon", you will think of many differences.

Generally speaking, for every 10 "concept" in a course, the number of "thinking hooks" you can find is more than 10 times that of others, so you can think that your thinking hooks are "extremely many", and if they are more than 5 times that of others, it is "many".

(2) the connection of different problem-solving skills (problem-solving methods).

For example, for a math problem, you can use "formula", "special value", "reduction to absurdity" and "reverse deduction" to solve it, and at the same time you can find out the relationship between these different problem-solving skills. For example, "when using the reduction to absurdity, you can bring in special values, so try it first."

Generally speaking, for a certain course or a certain part of knowledge, if you master more than 10 times that of others, you can think that your thinking feet are "minimal", and if it is more than 5 times that of others, it is "minimal".

(3) contact.

Any seemingly unrelated question is related. The connection with oneself is often related.