Discussion on the Application of Discovery Learning Theory to the Mathematical M

Abstract. The learning model and mental mechanism of the mathematical modeling in colleges and universities are analyzed with the theory of discovery learning. Not only the process and ways for freshmen studying in mathematical program to change their original thinking to higher mathematical thinking， but also the practicing methods provided by the mathematical modeling teachers for students to necessarily develop the skills of higher mathematics are discussed. In this paper， by taking the mathematical model laid in oil refinery for the pipelines of refined oil products as an example， an optimization model targeted at total cost is established， and also an optimal designing scheme is concluded through comparing the solving results of the model.

Keywords： Discovery Learning Theory； Mathematical Modeling in Colleges and Universities； Higher Mathematical Thinking； Mathematical Optimization

1. Introduction

After entering colleges and universities， students and especially those in the department of mathematics are primarily necessary to change their original thinking to higher mathematical thinking. Especially in the first semester， the students studying in mathematical program will encounter all kinds of cognitive conflicts. Therefore， developing the higher-level mathematical skills of students is one of the problems necessary for higher education to solve. The mathematical modeling in colleges and universities is a required course for students studying in mathematical program. How to realize the change of the mathematical thinking of students is an issued necessary for each mathematical modeling teacher to solve. However， can the higher-level mathematical skills of students be embodied in the concrete practices？ The key lies in whether the practicing methods of developing necessary skills can be provided by teachers for students. In this paper， by taking the mathematical model laid in oil refinery for the pipelines of refined oil products as an example， the change of the mathematical thinking of freshmen is discussed with the theory of discovery learning.

2. Overview of the Theory of Discovery Learning

2.1 Brief introduction to the theory of discovery learning

The theory of discovery learning， as a teaching approach in a strict sense， was first proposed by American cognitive psychologist Jerome S. Bruner （1915- ） in the book The Process of Education [1]. This method requires the students to "discover" the causality of the changes of things as well as the internal relationship through their own exploration and learning under the guidance of teachers， and thus concepts and principles can be produced. In this process of cognition， the excitement of "discovering" knowledge and the confidence of finishing tasks can be experienced by students at the same time. With these excitement and self-confidence， the intrinsic motivations of the students can be stimulated. Bruner said that the discovering included all forms of acquiring knowledge with their own heads. Besides， the theory of discovery learning [2] can accurately and fully give an expression to the relationship between teaching and learning as a pair of contradictions in development. Bruner thinks that the theory of discovery learning possesses the advantages as follows.

（1） Improving the wisdom of students and playing the potentials of students

（2） Promoting students to produce the intrinsic motivations in learning and enhance their confidence

（3） Promoting students to learn the exploring ways of discovery， and also training their abilities in raising and solving problems and their attitudes toward creation and invention

（4） Students can better comprehend and consolidate what they have learnt and also can better make use of the learnt because they make their knowledge systemic and structural

In mathematics learning and mathematics teaching， great numbers of mathematics learning experiments were made by Bruner， and also four major principle-oriented theorems were concluded [3] as follows. These are of positive significance for guiding the mathematics teaching.

（1） Structural theorem： The best method for students to learn mathematical concept， principle or rule at the beginning stage is constructing an expressive form for it.

（2） Mark theorem： If the structure and expressive form applied in the early days are marks are suitable for the intelligence development levels of students， it will be easier for students to get cognition and comprehension

（3） Comparative and changing theorem： From the concrete expression of a concept to abstract expression， the comparative and changing theorem is also applied

（4） Connection theorem： Each concept， principle， or skill in mathematics has a close connection with other concepts， principles， or skills

2.2 Learning model and mental mechanism

From the introduction to the theory of discovery learning， it can be known that it is necessary for students to "discover" the causality of the changes of things as well as the internal relationship through their own exploration and learning， and thus concepts and principles can be produced. However， in education psychology， the learning of students is classified into accepting learning and discovery learning. In the process of accepting learning， learning contents are directly presented with the form of final conclusion. Thus， it is a copy process in essence. However， in the process of discovering learning， learning contents can be generated. It is a highly cognitive process including creation， reflection and critical behaviors in essence.

2.3 Operational steps of discovery learning

Step 1： Proposing equipments and promoting students to know clear purpose

Step 2： Making assumption and promoting students to know clear thinking direction

Step 3： Setting up situation and promoting students to confront with conflicts

Step 4： Guiding students to classify information， list evidences， and discover and draw up conclusion according to real cases

Step 5： Combining the discovered conclusion with the real information and enhancing comprehension

Step 6： Applying actually-acquired knowledge to practice

3. Taking the Mathematical Model Laid in Oil Refinery for the Pipelines of Refined Oil Products as an Example

3.1 Proposal of problem

An oil field plans to construct two oil refineries on one side of railway， and also add a station along the railway for transporting oil products. Because this model is of certain universality， oil field design institute hopes to use common mathematical model and method， which can save the cost of pipeline laying to the maximum.

According to the different distances of two oil refineries from the railway and the distance between two oil refineries， the following design scheme is proposed. In designing scheme， if pipeline is shared， it is necessary to consider the common or different points of shared pipeline cost and non-sharing pipeline cost. Then， the pipeline laying plan and corresponding cost should be provided by the design institute.

3.2 Model assumption

（1） Assuming the railway line segment of oil delivery station to construct is straight

（2） Assuming two pipelines are laid at the same plane with no high and low change of terrain

（3） Neglecting the shared part of two oil pipelines because of connection

（4） Seeing oil refineries and the railway station to construct as a point， and leaving out the effect of its size on the laying of oil pipelines

3.3 Problem analysis

It is required to propose design scheme according to the different distances of two oil refineries from the railway line and the distances between two refineries， aiming at making the total cost of pipeline laying saved to the maximum. In fact， this is a mathematical optimization question. Therefore， a rectangular plane coordinate system is established by using refineries A and B and the railway and also a pipeline trend with universality is designed， and then an optimization model targeting at total cost is set up according to pipeline map [5] [6]. However， no specific data is provided in question， and thus the optimal design scheme is solved by us only according to assumption.

3.4 Model establishment and solving

Two oil refineries are expressed with A and B， respectively； the vertical distances of A and B from the railway line are a km and b km， respectively； the horizontal distance of A and B along the railway line is L km； the laying costs of non-shared pipeline and sharing pipeline is （10000RMB/km） and （10000RMB/km） ， respectively. Rectangular coordinate system is established by using the straight line of the railway as x-axis and the straight line passing through the center of A and vertical to the railway line as y-axis， as shown in figure 1.

Special circumstances： When points D and A are coincided （i.e. two pipelines intersects at point A）， the line of oil pipeline is ， and thus total laying cost is .

When points D and B are coincided （i.e. two pipelines intersects at point B）， the line of oil pipeline is ， and total pipeline laying cost of two oil refineries is .

Then， the three costs ， and are compared， and the scheme with the lowest cost as the optimal， namely optimal model.

In this paper， through establishing a rectangular plane coordinate system， an optimization model targeting at total cost is set up. This model own simple principles and is easy-to-understand， and can be greatly promoted in solving similar problems. Also， the numerical values of it are accurate， reliable， and easy-to-promote.

4. Conclusion

In this paper， by using discovery learning theory and taking the mathematical model laid in oil refinery for the pipelines of refined oil products as an example， whether the learning of a new mathematical conclusion is element for teachers to directly provide introduction or explore new conclusion with the existing knowledge and experience of students is discussed， and subsequently the introduction to new conclusion is obtained. Meanwhile， how to promote students to smoothly transition from original thinking to higher mathematical thinking is discussed. The products of higher mathematical thinking should not be only imparted by teachers to students， but more importantly students should be promoted to obtain higher mathematical thinking.

5. Acknowledgements

Fund Projects： New-century Higher Educational Reform Project of Guangxi Educational Department in 2010 （No.2010JGB115）； New-century Higher Educational Reform Project of Guangxi Educational Department in 2012 （No.2012JGA267）.

References

[1] [U.S.] Jerome S. Bruner. Translated by Ruizhen Shao. The Process of Education. Cultural Education Press， 1982， June， Edition 1.

[2] Aimin Ji. An Analysis of Bruner's Theory of Discovery Learning. Journal of Higher Correspondence Education （Philosophy and Social Sciences）， 1998 （06）.

[3] Toward a Theory of Instruction by Jerome S. Bruner.1966.

[4] Liansheng PI. Educational Psychology. Shanghai Education Press， 2004， April， Edition 3.