1. Functional concept:
A mathematical problem is expressed by function, and the general law of this problem is explored by function. This is the most basic and commonly used mathematical method.
2. The combination of numbers and shapes:
"Numbers are invisible, not intuitive, and numerous shapes make it difficult to be nuanced", and the application of "combination of numbers and shapes" can make the problem to be studied difficult and simple. Combining algebra with geometry, such as solving geometric problems by algebraic method and solving algebraic problems by geometric method, is the most commonly used method in analytic geometry. For example, find the root number ((A- 1)2+(B- 1)2)+ root number (A 2+(B- 1)2)+ root number ((A- 1) 2+B).
3. Discuss ideas by category:
When a problem may lead to different results because of different situations of a certain quantity, it is necessary to classify and discuss the various situations of this quantity. Such as solving inequality | a-1| >; 4. It is necessary to discuss the value of A.
4. Equation concept:
When a problem may be related to an equation, we can solve it by constructing the equation and studying its properties. For example, when proving Cauchy inequality, Cauchy inequality can be transformed into a discriminant of quadratic equation.
5. General idea:
Starting from the overall nature of the problem, we emphasize the analysis and transformation of the overall structure of the problem, find out the overall structural characteristics of the problem, and be good at treating some formulas or figures as a whole with the "overall" vision, grasping the relationship between them, and carrying out purposeful and conscious overall treatment. The holistic thinking method is widely used in simplification and evaluation of algebraic expressions, solving equations (groups), geometric proof and so on. Integral substitution, superposition multiplication, integral operation, integral demonstration, integral processing and geometric complement are all concrete applications of integral thinking method in solving mathematical problems.
6. Change ideas:
It is through deduction and induction that unknown, unfamiliar and complex problems are transformed into known, familiar and simple problems. Mathematical theories such as trigonometric function, geometric transformation, factorization, analytic geometry, calculus, and even rulers and rulers of ancient mathematics are permeated with the idea of transformation. Common transformation methods include: general special transformation, equivalent transformation, complex and simple transformation, number-shape transformation, structural transformation, association transformation, analogy transformation and so on.
7. Implicit conditional thinking:
Conditions that are not explicitly stated but can be inferred from existing explicit expressions, or conditions that are not explicitly stated but are routines or truths.
8. Analogy:
Comparing two (or two) different mathematical objects, if they are found to have similarities or similarities in some aspects, it is inferred that they may also have similarities or similarities in other aspects.
9. Modeling ideas:
In order to describe an actual phenomenon more scientifically, logically, objectively and repeatedly, people use a language that is generally regarded as strict to describe various phenomena, which is mathematics. What is described in mathematical language is called a mathematical model. Sometimes we need to do some experiments, but these experiments often use abstract mathematical models as substitutes for actual objects and carry out corresponding experiments. The experiment itself is also a theoretical substitute for the actual operation.
10. Back to thinking:
The idea of transformation is to turn the unknown into the known, the complex into the simple and the difficult into the easy. For example, fractional equations are transformed into integral equations, algebraic problems are transformed into geometric problems, and quadrilateral problems are transformed into triangular problems. The methods to realize this transformation are: undetermined coefficient method, collocation method, whole generation method and the transformation idea of turning dynamic into static and abstract into concrete.
1 1. inductive reasoning thought:
An object has some special characteristics ... >; & gt
Question 2: What are the thinking methods for solving mathematical problems?
Introduction to mathematical thinking methods
High school mathematics is a line, algebra and geometry are two beads;
Keep in mind the three basics, four can not be idle.
Five routines are practiced every day, and six strategies change from time to time.
Intensive reading of seven thoughts on mathematics is fun to learn.
A line: the main line of function (running through the textbook)
Two beads: algebra and geometry (focusing on the intersection of knowledge)
Three foundations: method (familiarity), knowledge (firmness) and skill (dexterity)
Four abilities: conceptual operation (accurate), logical reasoning (rigorous),
Spatial imagination (rich), problem decomposition (flexible)
Five methods: method of substitution, collocation method, undetermined coefficient method, analysis method and induction method.
Six strategies: simply control complexity, deal with difficulties, retreat for progress, change differences into similarities, replace trees with flowers, calm down and move.
Seven thoughts: Functional equations are the most important, and classified integrals are often used.
The combination of numbers and shapes is as good as ever, and the transformation is inseparable;
The limited self will be described infinitely, or will be expressed inevitably.
Special and general dialectical, knowledge gradually cross.
2. On mathematical knowledge and methods;
* * * and logic
* * * In the logical mutual table, children intersect to form a complete set.
Distinguishing right from wrong is difficult, and distinguishing right from wrong is a proposition;
Whether the crisscross is primitive or not, there are four relations that are necessary and sufficient.
When it is true or false, the false is not true, or the operation of true and false is strange.
Function and sequence
Mother and son of sequence function, arithmetic difference ratio is self-contained
What is the sum of series? The idea of general recursion is open;
There is no difference between the separation of variables and the synthesis of functions.
The same increase and different decrease monotony, and the interval digs the maximum.
trigonometric function
The triangle defines the birth ratio, and the radians are fused with each other.
From the same point of view, the flexibility of three kinds of good induction is twice as different.
If there are three equilibria before the solution, there will be a pulse after the solution;
The calculation of angle value is large or small, and the chord tangent is also different.
Equality and inequality
Unequal roots of functional equations often lead to parameter ranges;
One positive, two definite and three equal phases, the mean value theorem is the best.
The parameter ratio is uncertain, and the two formulas are different.
Equality and inequality are not absolute, but variable separation is constant.
Analytic geometry
The intersection setting of simultaneous equations does not require clever discrimination;
Vieta's theorem indicates the chord length, and the slope is converted into the midpoint.
Select parameters to model the trajectory and calculate the distance symmetrically;
The moving point is related to the definition, and the static auxiliary analysis is obtained during the moving process.
solid geometry
Multi-point * * * lines cross on both sides, and the multi-line * * * surface is ingenious;
The vertical chord in space is large and the spherical arc is small.
Line-to-line relationship line opposite search, face-to-face angle line table;
Equal product transformation is continuous projection and can cut bridges.
Arrangement and combination
Step by step, classification, multiplication and addition, if you want to be adjacent, you need to tie it up and insert it once in a while;
Orderliness leads to disorderly groups, and difficulty leads to exclusion.
Repeated multiplication of elements, you first take special elements;
Average grouping factorial division, I am a master of diversity.
binomial theorem
How much binomial power knows, the source of Wan Li is the general term;
Expand the trinomial index system, and the combination coefficient is Yang Hui angle.
Divisibility proves that the bottom is wonderful, binomial and unique;
Who is the most symmetrical at both ends? The main peak is all the other peaks that look short in the sky. ..
Probability and Statistics
Probability and statistics have the same root, random occurrence and so on;
Mutually exclusive events is a show, fighting for independence at the same time.
The whole sample sampling test, binary point independent repetition;
Distribution table, a random variable, false expectation variance theory.
Question 3: What are the basic mathematical thinking methods in primary school mathematics? 1, the corresponding thinking method.
Correspondence is a way of thinking about the relationship between two * * * factors, while primary school mathematics is generally an intuitive chart with one-to-one correspondence, which breeds the idea of function. For example, there is a one-to-one correspondence between points (number axes) on a straight line and specific numbers.
2. Hypothetical thinking method
Hypothesis is a way of thinking that first makes some assumptions about the known conditions or problems in the topic, then calculates according to the known conditions in the topic, makes appropriate adjustments according to the contradiction in quantity, and finally finds the correct answer. Hypothetical thinking is a meaningful imaginative thinking, which can make the problem to be solved more vivid and concrete after mastering it, thus enriching the thinking of solving problems.
3. Comparative thinking method
Comparative thinking is one of the common thinking methods in mathematics, and it is also a means to promote the development of students' thinking. In the application of teaching scores, teachers are good at guiding students to compare the situation before and after the change of known quantity and unknown quantity, which can help students find solutions quickly.
4. Symbolic thinking method
Symbolic thinking is to use symbolic language (including letters, numbers, graphics and various specific symbols) to describe mathematical content. For example, in mathematics, all kinds of quantitative relations, quantitative changes and deduction and calculation between quantities all use lowercase letters to represent numbers, and use condensed forms of symbols to express a large amount of information. Such as laws, formulas, etc.
5. Analogical thinking method
Analogy means that based on the similarity between two types of mathematical objects, the known attributes of one type of mathematical object can be transferred to another type of mathematical object. Such as additive commutative law's sum-multiplication commutative law, rectangular area formula, parallelogram area formula, triangle area formula, etc. The idea of analogy not only makes mathematical knowledge easy to understand, but also makes the memory of formulas natural and concise.
Step 6 change your way of thinking
Changing ideas is a way of thinking from one form to another, and its own size is unchanged. Such as geometric equal product transformation, homotopy transformation for solving equations, formula deformation, etc. A-B = A × 1/ B is also commonly used in calculation.
7. Classified thinking method
The thinking method of classification is not unique to mathematics, but embodies the classification of mathematical objects and its classification standards. For example, the classification of natural numbers can be divided into odd and even numbers according to whether they can be divisible by 2; Divide prime numbers and composite numbers according to the number of divisors. Another example is a triangle that can be divided by edges or angles. Different classification standards will have different classification results and produce new concepts. The correct and reasonable classification of mathematical objects depends on the correct and reasonable classification standards, and the classification of mathematical knowledge is helpful for students to sort out and construct their knowledge.
8, * * * way of thinking
The idea of * * * is a way of thinking that uses the concepts of * * *, logical language, operation and graphics to solve mathematical problems or impure mathematical problems. Primary schools use intuitive means, graphics and objects to infiltrate the idea of * * *. When talking about common divisor and common multiple, we adopt the thinking method of intersection.
9. The thinking method of combining numbers and shapes
Numbers and shapes are two main objects of mathematical research. Numbers are inseparable from shapes, and shapes are inseparable from numbers. On the one hand, abstract mathematical concepts and complex quantitative relations are visualized, visualized and simplified through graphics. On the other hand, complex shapes can be expressed by simple quantitative relations. When solving application problems, we often use the intuitive help of line segment diagram to analyze the quantitative relationship.
10, statistical thinking method:
Statistical charts in primary school mathematics are some basic statistical methods, and finding the average application problem is the thinking method of data processing.
1 1, extreme thinking method:
From quantitative change to qualitative change, the essence of limit method is to achieve qualitative change through the infinite process of quantitative change. When talking about "the area and perimeter of a circle", the idea of limit division of "turning a circle into a square" and "turning a curve into a straight line" is to imagine their limit states on the basis of observing the limit division, which not only enables students to master the formula, but also germinates the limit idea of infinite approximation from the contradictory transformation of curves and straight lines.
12, alternative thinking method:
He is an important principle of solving equations, and one condition can be replaced by other conditions when solving problems. If the school buys four tables and nine chairs, it will cost 504 yuan. The price of a table and three chairs is exactly the same. What is the unit price of each desk and chair?
13, reversible thinking method:
It is the basic idea in logical thinking. When the positive thinking is difficult to solve, we can seek the way to solve the problem from the condition or problem thinking, and sometimes we can use the line segment diagram to push back. For example, if a car goes from A to B, it can run the whole course in the first hour ... >; & gt
Question 4: What are the general mathematical thinking methods? What are the thinking methods of primary school mathematics?
1
Corresponding thinking method
Correspondence is a way to think about the relationship between two factors.
Elementary school mathematics is general
It is a one-to-one intuitive chart and the idea of pregnancy function. For example, a point on a straight line (number axis)
There is a one-to-one correspondence with the specific figures.
2
Hypothetical thinking method
Hypothesis is to make some assumptions about the known conditions or problems in the topic first.
Then follow the already in the question.
Know the conditions to be calculated, make appropriate adjustments according to the contradiction in quantity, and finally find the right one.
A way of thinking about the answer. Hypothetical thinking is a meaningful imaginative thinking that can be mastered.
In order to make the problem to be solved more vivid and concrete, thus enriching the thinking of solving problems.
three
, comparative thinking method
Comparative thinking is one of the common thinking methods in mathematics, and it is also the starting point to promote the development of students' thinking.
Duan. In the application problem of teaching scores, teachers are good at guiding students to compare the known and unknown quantities in the problem.
Changing the situation before and after can help students find a solution to the problem quickly.
four
Symbolic thinking method
Use symbolic language (including letters, numbers, graphics and various specific symbols) to describe numbers.
Learning content, this is symbolic thinking. For example, various quantitative relations in mathematics, the change of quantity and quantity and quantity.
The deduction and calculus between them are expressed in lowercase letters and expressed in the condensed form of symbols.
Have access to a lot of information. Such as laws, formulas, etc.
five
Analogical thinking method
Analogy means that according to the similarity of two mathematical objects,
It is possible to compare a known math class.
The idea that the essence of an image is transferred to another mathematical object.
Additive commutative law, for example, multiplicative exchange.
Summary of teaching plan exercises in primary school courseware.
First, second, third, fourth and fifth grade.
Law, rectangular area formula, parallelogram area formula, triangle area formula. similar
Thought not only makes mathematical knowledge easy to understand,
But also makes the memory of the formula natural.
And simplicity.
six
Change the way of thinking
Changing ideas is a way of thinking from one form to another.
And its own size
It is the same. Such as equal product transformation of geometry, homotopy transformation of solving equations, deformation of formulas, etc.
A-B is also often used in calculation.
=
A×
1/
B.
seven
Classified thinking method
The method of classified thinking is not unique to mathematics,
The classified thinking method of mathematics embodies the correct understanding of mathematics.
Classification of elephants and its classification standards. Such as the classification of natural numbers, if possible.
2
Divide evenly into odd numbers
And even numbers; Divide prime numbers and composite numbers according to the number of divisors. Another example is that triangles can be divided by edges, and can also be divided by
Divided by angle. Different classification standards will have different classification results and produce new concepts.
Correct and reasonable classification of mathematical objects depends on the classification standard and the correctness and rationality of mathematical knowledge.
The classification of knowledge helps students organize and construct knowledge.
eight
* * * mode of thinking
The idea of * * * is to use the concepts of * * *, logical language, operation and graphics to solve mathematical problems.
Thinking methods of problems or non-pure mathematical problems. Primary schools use intuitive means, graphics and physical infiltration.
Through * * * thoughts. When talking about common divisor and common multiple, we adopt the thinking method of intersection.
nine
Thinking method of combining number and shape
Number and shape are two main objects of mathematical research. Number cannot be separated from form, and form cannot be separated from number. On the one hand,
Abstract mathematical concepts and complex quantitative relations are visualized, visualized and simplified by graphics.
Simplicity On the other hand, complex shapes can be expressed by simple quantitative relations. When solving application problems
Visualization of line graph is often used to help analyze quantitative relations.
10
, statistical thinking method:
Statistical charts in primary school mathematics are some basic statistical methods.
General application problems are reflected.
The idea and method of data processing are put forward.
1 1
, extreme thinking method:
Things change from quantitative change to qualitative change,
The essence of limit method is realized through the infinite process of quantitative change.
Qualitative change. When it comes to "the area and perimeter of a circle",
"Turn a circle into a square"
Limit score of "turning curve into straight line"
Cut off the train of thought and imagine their limit states on the basis of observing finite division, which not only makes learning
The formula mastered by students can also sprout the limit idea of infinite approximation from the contradiction transformation between curve and straight line.
12
Alternative thinking method:
He is an important principle of solving equations, and one condition can be replaced by other conditions when solving problems.
If the school buys it.
four
A table and
nine
Use the chair, * * * *
504
Yuan, a banquet.
three
a chair
The price is the same, table ... >>
Question 5: What are the common ways of thinking in mathematics? 1. The idea of using letters to represent numbers
This is one of the basic mathematical ideas, which is mainly reflected in the second chapter of the first volume of Algebra, "Basic Knowledge of Algebra".
For example, let the number A be a and the number B be represented by an algebraic expression: (1) 2 times the sum of two numbers A and B: 2 (a+b) (2) the difference between 2 times A and 5 times B: 2a-5b.
Second, the idea of combining numbers with shapes.
The combination of numbers and shapes is one of the most important and basic thinking methods in mathematics, and it is an effective idea to solve many mathematical problems. "Less is not intuitive, but more is difficult to be nuanced" is a famous saying of Professor Hua, a famous mathematician in China, which highly summarizes the role of the combination of numbers and shapes. The following contents in mathematics textbooks reflect this idea.
1, the one-to-one correspondence between points on the number axis and real numbers.
2. One-to-one correspondence between points on the plane and ordered real number pairs.
3. The relationship between function and image.
4. The sum, difference, multiplication and division of line segments (angles) should make full use of numbers to reflect shapes.
5. Solving triangles, finding angles and side lengths, and introducing trigonometric functions are how to solve problems by algebraic methods.
6. In the chapter "Circle", the definition of circle, the positional relationships between points and circles, straight lines and circles, and circles and circles are all treated as quantitative relationships.
7. The second statistical method in preliminary statistics is to draw statistical charts, which are used to reflect the distribution and development trend of data. In fact, it is through the "shape" to reflect the data dressing situation, development trend and so on. In fact, it is to embody the characteristics of numbers through "shape", which is a direct application of the idea of combining numbers and shapes in practice.
Third, change ideas (return to ideas)
In the whole junior middle school mathematics, the idea of transformation has been running through it. Transforming thinking is to transform an unknown (to be solved) problem into a solved or easy-to-solve problem, such as simplifying the complex, changing the difficult to the easy, changing the unknown to the known, and changing the high order to the low order. It is one of the most basic problem-solving ideas and one of the basic thinking methods of mathematics. The following contents reflect this idea:
1, the solution of the fractional equation is to transform the fractional equation into the quadratic equation that I learned before. Here, the new problem to be solved has become a solved problem, which embodies the transformation idea.
2. Solve the right triangle; Turn the non-right triangle problem into a right triangle problem; Turn practical problems into mathematical problems.
3. It is proved that the sum of the internal angles of a quadrilateral is 360 degrees, that is, a quadrilateral is transformed into two triangles. At the same time, the idea of transformation is also used to discuss the sum of the inner angles of polygons.
Fourth, the idea of classification.
The classification of rational numbers, algebraic expressions, real numbers, angles, triangles and quadrangles, the positional relationship between points and circles, straight lines and circles, and the positional relationship between circles are all discussed through classification.
Question 6: What are the common mathematical thinking methods in mathematics? 1. Common mathematical thinking (four major ideas in mathematics)
1. Concepts of functions and equations
The way to think about problems with variables and functions is function thought, which is a higher-level refinement and generalization of knowledge such as function concepts, images and properties, and a method to guide with ideas abstracted from repeated learning of knowledge and methods.
A deep understanding of the images and properties of functions is the basis of solving problems by applying the thought of functions, which can be summarized into three steps: ① transforming the problems faced into equation problems; ② Solve or discuss this equation, and draw relevant conclusions; ③ Return the conclusion to the original question.
2. Combination of numbers and shapes
In middle school mathematics, we can't completely separate "number" from "shape", that is to say, algebraic problems can be geometric problems, geometric problems can also be algebraic problems, and "number" and "shape" can be transformed and infiltrated with each other under certain conditions.
Step 3 discuss ideas by category
In mathematics, we often need to investigate the research object in different situations according to its different nature. This is an important mathematical thinking method and an important problem-solving strategy. There are many factors that cause the discussion of classification, which can be summarized as follows: (1) the discussion caused by the restrictive conditions of mathematical concepts, properties, theorems and formulas; (2) Classification discussion caused by the restrictive conditions required for mathematical deformation; (3) Discussion caused by the uncertainty of graphics; (4) Discussion caused by topics containing letters.
Generally, the solving steps of classified discussion are: (1) determine the objects to be discussed and all the objects to be discussed; (2) The classification is reasonable and the standard is unified, so as not to omit or repeat; (3) Step by step and discuss at different levels; (4) Summarize the whole topic.
4. The idea of equivalent transformation
Equivalent transformation refers to the equivalent form of the same proposition, which can be realized by the conditions and conclusions of variable problems, or by substituting it into the form of transformation problems appropriately, or by using the equivalent relationship of mutually negative propositions.
Commonly used transformation strategies are: known and unknown transformation; Forward and reverse conversion; Conversion between number and shape; Generally in special transformation; The transition between complexity and simplicity.