On trigonometric functions and mathematical thinking

Modern Education emphasizes "knowledge structure" and "learning process", aims to develop students 'thinking skills, and knowledge as the thinking process of the material and media. Only to acquire knowledge, skills as an intermediary to develop students' quality of thinking is consistent with quality Education the basic requirements. mathematical knowledge may be forgotten in the future, but thought the quality of training will affect the student's life, thinking the quality of training is the value of Mathematics Education to be truly the ideal way.

Educational psychology theory: thinking is the human brain the nature of things and things of the relationship between the regularity of general indirect reflection. The core components of cognitive thinking, thinking, knowledge and level of development determines the structure and function of the system, so development potential of high school students thinking, to improve the quality of thinking, of great significance.

The quality of thinking including flexibility of thinking, a broad, agility, depth, originality and critical of several aspects. Flexibility of thinking is based on a broad and profound thinking on the basis of and for the quick thinking of , originality and critical to provide a guarantee of good quality in the people's work, life, easy to play by the rules, innovation difficult, the challenge in the absence of flexible thinking, so thinking, flexibility training is particularly important.

Thinking, flexibility refers to flexibility of thinking activity, the development means good things according to changes in a timely manner with a new perspective on things has changed, and proposed realistic solution to the problem of new ideas, new programs and methods, students flexibility of thinking mainly to: (1) a flexible starting point for thinking: from different angles and at different levels and in different ways according to the new conditions quickly determine the direction of thinking (2) flexible thinking process: the flexibility to use various rules, axioms, theorems, laws, formulas and other ways to move from a problem-solving is another way. (3) Flexible migration of thinking: to learn by analogy, analogy.

Thinking how to make more students with flexible features? I made some in the teaching trigonometry to explore:
1. To "divergent thinking" training to improve flexibility of thinking
American psychologist Guilford's "divergent thinking" is thinking of training flexibility training. "Divergent thinking" means "from the definition of information to produce the information, its focus is produced from the same source in a variety of large number of the output, conversion effect is likely to occur."
In the current Mathematics teaching, more emphasis on the prevalence of concentrated thinking, training, and the relative neglect of the divergent thinking ability. Divergent thinking is to understand the material, flexible use of knowledge required, but also to meet the information age, life should have to adapt to the future ability.

1.1 Solution of the problem to guide students to divergent, in the teaching process, using a variety of methods, from different angles and different ways to seek answers to questions, with a given problem to develop student's thinking process flexibility.


Through a given problem to guide the students to summarize that the trigonometric identities of the basic methods: (1) unified function type, (2) unified point of view, (3) unified operation. A given problem can be open-minded, and enhance knowledge linkages, learn to multi-angle think about problem-solving approach and flexible way of thinking.

1.2 to guide students to the problem of divergent conclusions, the conclusions of the divergence is known to determine the conditions no ready conclusions. Allow students to explore as much as possible to find their own conclusions, and solve it.


The introduction of open questions that can guide students to think from different angles, not just think about the conditions themselves, but the conditions need to think about the relationship between the use of a variety of conditions, according to comprehensive transformation means to process information, explore the findings will help the starting point of thinking flexibility training, but also conducive to the spirit of tireless study and cultivation of creativity.

2. Increased flexibility of thinking led to the other to improve the quality of thinking, to thinking of other quality of training to promote the cultivation of flexibility of thinking
As the thinking of various quality is with each other, inseparable, in an organic unity, so, thinking of other quality of training can effectively promote the increased flexibility of thinking, following on the quality of thinking and talk a bit about the nature of some sentiment.

2.1 refers to the depth of thinking, the thinking process of the level of abstraction, refers to the phenomenon are good at things found in nature, is good at relationships and linkages between things, reveal the law.

<Example> equation sinx = lgx solution with () one. (A1 (B2 (C3 (D4
Students are accustomed to solve equations, and solving this equation can not often that students know what to do. If the use of flexible thinking for a perspective: the nature of this issue for the sake of equations The common solution. For the sake of using Shuxingjiege thinking into question the function diagram intersection of family, nature and geometry of algebraic equations for the intrinsic link between the through knowledge of the series, horizontal communication and firmly grasp the essence, the basis of the depth of thinking on the flexibility of thinking only useless.

2.2 thinking agility refers to the speed of thinking activity and it has two indicators: first, speed second, the correct rate. Students with the quality and reduce the operational aspects of the reasoning process. Flexibility of thinking for thinking improve the speed and accuracy play a decisive role.

<Example> adjacent side of the parallelogram a and b, respectively, from the geometry around both sides of the rotary volume Va (around a while) and Vb (around the b-side, the Va: Vb = ()
(Aa: b (Bb: a (Ca2: b2 (D) b2: a2
Solved by direct method: a general parallelogram, for example Figure, rectifiable:

Then Va: Vb = b: a, due to the introduction of both sides of the angle to solve, students often can not start. If the special parallelogram (a rectangle to handle, it is quite simple.

This problem solution fully reflects the flexibility of thinking, to Jane Yu Fan, with a specialization thought to solve, solve problems quickly and correctly. Links to free download http://www.hi138.com 2.3 creative thinking means thinking The original level of activity, with new resilience characteristics. thinking, creative thinking, flexibility is provided fertile ground for problem-solving "inspiration" of the flash to provide the fuel.


2.4 Critical thinking refers to thinking, independent analysis of the degree of activity, whether it be good at thinking strictly estimates carefully check the materials and thought processes in teaching, encouraging students to present different views or even suspect, pay attention to guide and inspire, to promote independent thinking ability.


Students to doubt the reliability of conclusions in an independent analysis, based on the flexible use of trigonometric functions to determine the angles of a triangle of the monotony of the range, the value of rigorous argument of the trigonometric values ​​of the possibilities.

Ingenious idea of ​​a unique flexible, Shuxingjiege thought to be fully reflected. teaching students to pay attention to the unique nature of problem-solving ideas, novelty recognition and promotion of the full grant attempt to explore opportunities to active thinking, personality development.

In recent years, with the purpose of teaching students after the training, the quality of thinking has been greatly improved. Accordingly, the quality of student learning has been greatly improved and many students enter university, or even go to work, often Although the letter mentioned there are many mathematical knowledge has been forgotten, but the teacher has to teach mathematical way of thinking often makes them work, study and life to benefit a lot. With the advance of reform curriculum materials, highlighting the quality of training of thinking has become the the consensus of teachers and educators, I will continue to explore down, in order to get more Educational theory and methods.

References:
[1] <<high ​​school students studying psychology, "Writing Group of the Guangdong Higher Education Press.

[2] <<Mathematics Education>> Tian Wanhai book Zhejiang Education Publishing House.

[3] <<students of quality education>> Xu Zhongan the Shanghai Science and Technology Press. Links to free download http://www.hi138.com