1 Mathematical thinking method from general to special
Function is the backbone of senior high school algebra, and the idea of function runs through all the contents of senior high school algebra. Functional thought is the abstraction, generalization and refinement of functional content at a higher level, and it is to consider, study and solve problems from the internal relationship and overall perspective of functional parts.
The idea of equation is to highlight the equal relationship between known quantity and unknown quantity, and to achieve the purpose of evaluation by setting unknown quantity, listing equation or equation, solving equation or equation, etc. It is the basic idea to solve all kinds of calculation problems and the foundation of calculation ability.
2 Introduction to Mathematical Thinking Methods
Mathematics is the gymnastics of thinking, and learning mathematics is more about cultivating students' good mathematical literacy. Students' good mathematical literacy is an organic combination of mathematical resources and mathematical ideas. The former belongs to "tangible" resources and has become a necessary tool and key object for mathematics learners. The understanding of "intangible" mathematics thought must be guided by teachers in the classroom, which can effectively penetrate into students' mathematics learning and will also become an inevitable way to improve students' comprehensive mathematics literacy.
The thoughts of "from special to general" and "from general to special" reflect the mutual characteristics in many mathematical thoughts. For example, in the first section of Pythagoras theorem, the process of revealing the origin and proof of the theorem is only an interpretation of the idea of "from special to general". From the introduction of Pythagoras floor tiles in teaching, the exploration of special triangle-isosceles right triangle (as shown in Figure A), and the discussion of general right triangle (as shown in Figure B), it is understandable that from incomplete inductive conjecture to strict deductive reasoning conclusion, from special to general mathematical thinking has played a role in attracting jade.
For another example, the parallelogram in the next chapter reveals the idea of "from general to special". In teaching, we start with the general quadrangle, then turn to the special quadrangle-parallelogram, and start with the parallelogram from three special angles: (1) the parallelogram with special angle-rectangle; (2) A parallelogram with special sides-a rhombus; (3) A parallelogram with special sides and angles-a square.
The thought of "from special to general" and "from general to special" seems unremarkable but ubiquitous, which better embodies that mathematics comes from life and is applied to life. Teachers may wish to set up step-by-step questions in the teaching process to inspire students to think better, so as to better exercise their thinking.
3 Introduction to Mathematical Thinking Methods
Hypothetical thinking: Hypothesis is a thinking method 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.
Symbolic thinking: using symbolic language (including letters, numbers, graphics and various specific symbols) to describe mathematical content, which is symbolic thinking. 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.
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 problem of teaching scores, teachers should be good at guiding students to compare the known quantity and the unknown quantity before and after the change of the problem, which can help students find the solution quickly.
Analogical thinking: Analogical thinking refers to the idea that based on the similarity between two types of mathematical objects, it is possible to transfer the known properties of one type of mathematical object to another. 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 as natural and concise as logical conclusions.
4 Introduction to Mathematical Thinking Methods
Concepts of functions and equations
Klein, a famous mathematician, said: "The important thing that ordinary educators should learn in math class is to think with variables and functions." . A student who only studies the knowledge of functions is often passive in solving problems. Only by establishing the thought of function can he actively think about some problems.
Function is the backbone of senior high school algebra, and the idea of function runs through all the contents of senior high school algebra. Functional thought is the abstraction, generalization and refinement of functional content at a higher level, and it is to consider, study and solve problems from the internal relationship and overall perspective of functional parts.
The idea of equation is to highlight the equal relationship between known quantity and unknown quantity, and to achieve the purpose of evaluation by setting unknown quantity, listing equation or equation, solving equation or equation, etc. It is the basic idea to solve all kinds of calculation problems and the foundation of calculation ability.
Functions, equations and inequalities are related to each other through the function value being equal to zero, greater than zero or less than zero, and there are both differences and connections between them. The thought of function and equation is not only the embodiment of function thought and equation thought, but also the embodiment of their comprehensive application, and it is the basic mathematical thought in the process of studying variables and functions, equality and inequality.
The college entrance examination focuses on the thought of function and equation, and examines the basic application of function and equation with multiple-choice questions and fill-in-the-blank questions. In solving problems, it comprehensively examines the relationship between thinking methods and related abilities from a deeper level at the intersection of knowledge networks.
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