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Abstract：Basing on the nature and special characters of normal college， talk about mathematical analysis teaching in several aspects.First， mathematical analysis should be taught as a foundation course because it will be used as one of the basic tools for further mathematics’study， and in the succeed course.The second， it being strictly relative with the primary mathematic course， so it should be taught as a guidance to pre-middle school mathematics teachers.The third， as any subject， getting the general structure of it is useful to let the students get the whole sensation， and help them to stimulate them to explore. Especially，for the future teacher， the course should build research interest and make the students know how to excite creation emotion from any subject as we do in mathematical analysis.
Key words： normal college， mathematical analysis teaching， normal college teaching nature and character， basic course， guidance， creation
1.Introductory
As we all known， mathematical analysis is one of the basic courses for mathematic major， and so the teaching should be considered as the basis of advanced mathematics teaching and suitable to the characters of the course itself. However，the course is a little special to normal college students and so the corresponding teaching methods should be adapted to be fit to its characters and show some information to the future teachers， and guide the middle school mathematics teaching and mathematic study.
2.Accuracy Concept and Useful Ideas
Mathematical analysis is one of the elementary courses for normal college students whose major is mathematics， as the same as any other common mathematic major college students. But it has some normal characters which other mathematic major don’t have. So its teaching should be thought over the effect which will affect on both the normal college students themselves and the future students who will be taught by the normal college students. Since the concepts are the key to any subject， the same to
mathematical analysis， and we can not image what the students will do if they were shown the error concepts about some topics， the only thing they would do is to hand down the mistakes generation by generation.
Since accuracy of concept is very important， how to explain the concept exactly？ The right choice of tools and the suitable formula are both useful to get it. The fit tool will make it easier to reach the quiddity of the topic which is talked about. For example， the definition of limitation of sequence is one of the most important concepts in mathematical analysis， we should not only give the definition： a number A is called the limit of the numeral sequence ， if for any ， there exist an index N such that for all，but also explain the real meaning of the definition， what the formula of the definition show and the reason why the formula can express the right idea with it. At the same time we should talk about the choice of the tool which was used to define the concept， and whether there was another way can be used to define， what’s more， why the absolute value had to be selected to help define the limit. As we know， if we want to get the limit situation of a sequence， we need to show almost numbers of the sequence are sufficient close to some fixed point. The only way to examine the closure between two real numbers is to use absolute value of two numbers. Therefore the formula for the formula is not strange. In another way， the useful idea is also the resource which the subject will bring to the reader and the learner should get from. In the same problem of limitation， use infinite items of a sequence to show the tendency but not some finite item，for any finite items can only get finite distance to some fixed point， and finite distance can not be filled with some items， but infinite items can show the tendency which is not the same as the finite distance to some fixed point， hence the limit can only be expressed by infinite items， and in the course， about the concept of limit must be emphasized on the number n which means the sequence items and through n>N to show the items being infinite clearly with the properties of nature numbers. By the means， in many situations， we present the whole item with all most of them except finite tems of it. This idea is continuously used in integral and most of the concept of mathematical analysis.
3.A Guidance
Almost normal college students should be teachers， and what they learn from the college would have some important effect on their future teaching. So the teaching to them may continuously be followed by several gaps and affect many successor， in this way， we should keep more attention on the normal college mathematic analysis teaching， and try our best to let the teaching be a guidance to the future teachers.
At first， the knowledge itself should be taught as a guidance to the future teachers.As what F.Klein had said， the advanced mathematics should be the basis of elementary mathematics.Many mathematic topics of primary school are relative with mathematical analysis， and some problems can not be expressed clearly with precise terminology for the students lack of enough professional knowledge， and only depending on the teacher， the students can really grasp the problems. But what the teacher can do is to turn to what they had learned in college.
In another aspect， the teaching should give a guidance way to study， generally speaking， any subject has its own knowledge structure and logical order. The structure and logic come from the subject’s characters. Getting the subject’s logic and knowledge structure will show learners the right way to master it， what’s more， the utilization of some subject’s logic， the learner will get some unknowns. For example， how to use an axiom system to study some problem， and how to build the axiom system with experience， even find some thing new with the axiom system. It is especially important for the people who will be middle school teachers， know how to use several axiom to induce the theory system of some item，and lead the students to share the whole knowledge system. 4.Exciting More Creation
Many noted mathematician had pointed out that the core of mathematics is creation， and creation is the inexhaustible resource of a subject.Therefore，the core of mathematic teaching is exciting students’ creation. How to show the creation in mathematics class and try to explain clearly how to obtain the idea of creation becomes the first thing in class and all of this should be found by the teachers who give the lessons of mathematic analysis.
In one way， the teacher should show the creation point in the course， and give the background of the problem such as when， where and who are interested in， and what condition and goal lead to the creation， and how they achieved at that time. In this way， the teacher will interest the students and give them the bricks of further research on creation.After giving the right place where the creation points are in the subject， then the teacher should analyze the condition which can be used to acquire the creation idea and induce the students to examine the original source of the creation is another key step. The typical example is to compute the square of any irregular plane graph with integral of one-element function. we should guide the students to not keep eyes only on the regular graph whose area can be computed， but some similar and relatively regular one to take place of the computed one， and warn that any finite analog can not absolutely replace the right one. Then hint to consider infinite shades to approach. As we know， if the sum of infinite shades has no limitation， the sum is not unique， and the sum can not really reflect the fact， otherwise， they will get the unique square， and the integral can solve this problem， and with the same thought， people can create all kinds of accumulate the sum of some thing in different way， and get the formulas of all suitable kinds of sum with the formula
In another way， the teachers should also mention that mathematical analysis can be used in other several subjects such as ODE， OR， real analysis， complex analysis， functional analysis and probability theory etc. Finding the reason why one thing can be applied to solve another matter， and only under the help of creation， a new method can be obtained. So the teacher should train the student to develop ability in using one viewpoint to solve another problem in mathematical analysis teaching.
In conclusion， normal college mathematical analysis teaching should tell the right information of concept， show the useful guidance to learn and research， and cultivate the creation ability.
References：
[1]Vladimir A.Zorich， Mathematical analysis，Hong Kong，Springer，2004.
[2]F.Klein，Elementary mathematics from advanced standpoint，London，Springer，1936.
[3]James L.Adams，Breaking out the barrier of though，New York， Springer，1985.
[4]JIN Ling-yu，FANG Shao-mei，LIU Wen-yan，Someopinions and understandings on teaching reform of mathematical analysis，college mathematics，Vol.28，No.4，Aug.2012，25-30.
[5]XIANG Zhao-yin， HUANG Ting-zhu， The Ditactical Principle of the Research Teaching in the Exercises Lecture of Mathematic Analysis，college mathematics，Vol.28，No.4，Aug.2012，151-154.
[6]Boris Handal，Teachers’Mathematics Beliefs：A Revie，the mathematics educator，2003，Vol.13.No.2，47-57.
基金项目：
黔教高发[2013]446号
Key words： normal college， mathematical analysis teaching， normal college teaching nature and character， basic course， guidance， creation
1.Introductory
As we all known， mathematical analysis is one of the basic courses for mathematic major， and so the teaching should be considered as the basis of advanced mathematics teaching and suitable to the characters of the course itself. However，the course is a little special to normal college students and so the corresponding teaching methods should be adapted to be fit to its characters and show some information to the future teachers， and guide the middle school mathematics teaching and mathematic study.
2.Accuracy Concept and Useful Ideas
Mathematical analysis is one of the elementary courses for normal college students whose major is mathematics， as the same as any other common mathematic major college students. But it has some normal characters which other mathematic major don’t have. So its teaching should be thought over the effect which will affect on both the normal college students themselves and the future students who will be taught by the normal college students. Since the concepts are the key to any subject， the same to
mathematical analysis， and we can not image what the students will do if they were shown the error concepts about some topics， the only thing they would do is to hand down the mistakes generation by generation.
Since accuracy of concept is very important， how to explain the concept exactly？ The right choice of tools and the suitable formula are both useful to get it. The fit tool will make it easier to reach the quiddity of the topic which is talked about. For example， the definition of limitation of sequence is one of the most important concepts in mathematical analysis， we should not only give the definition： a number A is called the limit of the numeral sequence ， if for any ， there exist an index N such that for all，but also explain the real meaning of the definition， what the formula of the definition show and the reason why the formula can express the right idea with it. At the same time we should talk about the choice of the tool which was used to define the concept， and whether there was another way can be used to define， what’s more， why the absolute value had to be selected to help define the limit. As we know， if we want to get the limit situation of a sequence， we need to show almost numbers of the sequence are sufficient close to some fixed point. The only way to examine the closure between two real numbers is to use absolute value of two numbers. Therefore the formula for the formula is not strange. In another way， the useful idea is also the resource which the subject will bring to the reader and the learner should get from. In the same problem of limitation， use infinite items of a sequence to show the tendency but not some finite item，for any finite items can only get finite distance to some fixed point， and finite distance can not be filled with some items， but infinite items can show the tendency which is not the same as the finite distance to some fixed point， hence the limit can only be expressed by infinite items， and in the course， about the concept of limit must be emphasized on the number n which means the sequence items and through n>N to show the items being infinite clearly with the properties of nature numbers. By the means， in many situations， we present the whole item with all most of them except finite tems of it. This idea is continuously used in integral and most of the concept of mathematical analysis.
3.A Guidance
Almost normal college students should be teachers， and what they learn from the college would have some important effect on their future teaching. So the teaching to them may continuously be followed by several gaps and affect many successor， in this way， we should keep more attention on the normal college mathematic analysis teaching， and try our best to let the teaching be a guidance to the future teachers.
At first， the knowledge itself should be taught as a guidance to the future teachers.As what F.Klein had said， the advanced mathematics should be the basis of elementary mathematics.Many mathematic topics of primary school are relative with mathematical analysis， and some problems can not be expressed clearly with precise terminology for the students lack of enough professional knowledge， and only depending on the teacher， the students can really grasp the problems. But what the teacher can do is to turn to what they had learned in college.
In another aspect， the teaching should give a guidance way to study， generally speaking， any subject has its own knowledge structure and logical order. The structure and logic come from the subject’s characters. Getting the subject’s logic and knowledge structure will show learners the right way to master it， what’s more， the utilization of some subject’s logic， the learner will get some unknowns. For example， how to use an axiom system to study some problem， and how to build the axiom system with experience， even find some thing new with the axiom system. It is especially important for the people who will be middle school teachers， know how to use several axiom to induce the theory system of some item，and lead the students to share the whole knowledge system. 4.Exciting More Creation
Many noted mathematician had pointed out that the core of mathematics is creation， and creation is the inexhaustible resource of a subject.Therefore，the core of mathematic teaching is exciting students’ creation. How to show the creation in mathematics class and try to explain clearly how to obtain the idea of creation becomes the first thing in class and all of this should be found by the teachers who give the lessons of mathematic analysis.
In one way， the teacher should show the creation point in the course， and give the background of the problem such as when， where and who are interested in， and what condition and goal lead to the creation， and how they achieved at that time. In this way， the teacher will interest the students and give them the bricks of further research on creation.After giving the right place where the creation points are in the subject， then the teacher should analyze the condition which can be used to acquire the creation idea and induce the students to examine the original source of the creation is another key step. The typical example is to compute the square of any irregular plane graph with integral of one-element function. we should guide the students to not keep eyes only on the regular graph whose area can be computed， but some similar and relatively regular one to take place of the computed one， and warn that any finite analog can not absolutely replace the right one. Then hint to consider infinite shades to approach. As we know， if the sum of infinite shades has no limitation， the sum is not unique， and the sum can not really reflect the fact， otherwise， they will get the unique square， and the integral can solve this problem， and with the same thought， people can create all kinds of accumulate the sum of some thing in different way， and get the formulas of all suitable kinds of sum with the formula
In another way， the teachers should also mention that mathematical analysis can be used in other several subjects such as ODE， OR， real analysis， complex analysis， functional analysis and probability theory etc. Finding the reason why one thing can be applied to solve another matter， and only under the help of creation， a new method can be obtained. So the teacher should train the student to develop ability in using one viewpoint to solve another problem in mathematical analysis teaching.
In conclusion， normal college mathematical analysis teaching should tell the right information of concept， show the useful guidance to learn and research， and cultivate the creation ability.
References：
[1]Vladimir A.Zorich， Mathematical analysis，Hong Kong，Springer，2004.
[2]F.Klein，Elementary mathematics from advanced standpoint，London，Springer，1936.
[3]James L.Adams，Breaking out the barrier of though，New York， Springer，1985.
[4]JIN Ling-yu，FANG Shao-mei，LIU Wen-yan，Someopinions and understandings on teaching reform of mathematical analysis，college mathematics，Vol.28，No.4，Aug.2012，25-30.
[5]XIANG Zhao-yin， HUANG Ting-zhu， The Ditactical Principle of the Research Teaching in the Exercises Lecture of Mathematic Analysis，college mathematics，Vol.28，No.4，Aug.2012，151-154.
[6]Boris Handal，Teachers’Mathematics Beliefs：A Revie，the mathematics educator，2003，Vol.13.No.2，47-57.
基金项目：
黔教高发[2013]446号