Reconstruction Of "Conic Section And Equation" Teaching Based On Problem-driven Theory

The problem-driven theory is the further extension of H.Freudenthal's mathematics view of education and the embodiment of his thought of recreation.It advocates that teachers make use of mathematical history to deeply analyze teaching content within mathematics discipline,excavate background and value of knowledge generation,explore the formation process of mathematical thought and method and combine the requirements of mathematical curriculum standards with students' reality,thus to create real and effective problem situation to drive mathematical teaching.To reveal the essence of mathematics by problem-driven teaching is the trend of mathematics classroom teaching research of middle school and in the meantime the requirement of mathematical teaching.Taking the unit of "conic curve and equation" in senior high school as an example,this study reconstructs teaching materials and teaching process based on the problem-driven method and explores how to integrate the problem-driven teaching theory with teaching practice.This paper mainly explains the following four aspects.(1)Summarizing relevant research literature of "conic curve and equation" unit to obtain research enlightenment.Introducing problem-driven teaching theory and pointing out the connotation of "problem" and essence of "real effective problem situation",which lays theoretical basis for latter study.(2)Historical development of conic curve is analyzed.To clarify following two important questions by sorting out relevant mathematical history: 1)what is the problem that gives rise to conic curve? 2)Why do people study conics curve? The history also shows a close relationship between conic curve and branches of natural science and mathematics.Teaching enlightenment is obtained from history,which can provide strong support for teaching reconstruction of "conic curve and equation" unit.(3)Three different versions of "conic curve and equation" in senior high school mathematical textbooks are compared and analyzed.From knowledge system and content arrangement,column setting,chapter introduction,presentation of concept and nature,and review of the end of chapter these five dimensions to analyze characteristics of compilation in different textbooks and their respective shortcomings.Thereby,necessity of teaching reconstruction of "conic curve and equation" unit is demonstrated.(4)Based on problem-driven teaching theory,historical enlightenment of conic curve,analysis of corresponding textbooks and overall grasp of knowledge,"conic curve and equation" was reconstructed according to students' reality.Teaching reconstruction emphasizes overall design with unit as the main body,and uses problem-driven teaching to organize teaching process.Teaching designation and teaching practice are committed to solving four key issues of "conic curve and equation" unit teaching.That is,realizing natural transition from original definition in space to first definition on the plane;highlighting the characteristics of ellipse,hyperbola and parabola and at the same time revealing inherent unity of the three;understanding the consistency of "centrifugal rate" in different expression;using proper conic optical principle to organize teaching.The main results of this paper are as follows.(1)Unit teaching reconstruction of "conic curve and equation" based on problem-driving theory is realized.According to problem-driven theory,conic curve history is combed and teaching enlightenment is obtained.Besides,teaching content is deeply analyzed from the perspective of mathematics subject structure.Furthermore,requirements of mathematics curriculum standards and students' reality is combined to reconstruct teaching content.Teaching reconstruction follows three main lines to carry out problem-driven teaching.That is,the Dandelin double sphere model,the optical properties of conic curve and the knowledge points' close relation inside the conic curve.Through the whole design of teaching unit and problem-driven teaching,it can help promote students to understand deeply learning content,obtain whole structure with rich connection and corresponding mathematical thought.(2)A set of complete teaching designs of "conic curve and equation" unit has been formed,which can provide teaching research paradigm for middle school mathematics teachers.According to the organizational order of "conic curve and equation" unit teaching reconstruction,a set of complete teaching designs is given.Teaching designs are divided into three parts: teaching of unit initiation,teaching of specific concepts and propertiesand teaching of unit review.They are linked with each other,gradually foregrounding and echoingand finally forming an organic entity.Four key teaching problems of "conic curve and equation" unit which have been mentioned above can be solved through teaching reconstruction.Through this kind of study,students can form comprehensive understanding of conic curve content,fully realize the interrelation between knowledge as well as develop mathematical thought and method in knowledge.Then how to apply problem-driven theory into mathematical teaching? Teaching reconstruction of mathematics unit in middle school by problem-driven teaching emphasizes the overall interpretation of teaching content with effective teaching organization and design.The exploration process of this study provides research paradigm of using problem-driven theory to analyze textbooks and organizational teaching for front-line teachersand also provides direction of thinkingfor overall grasp of mathematics teaching structure and teaching organization of specific class hour.(3)This research enriches problem-driven theory.From the perspective of educational philosophy,problem-driven teaching penetrates intomathematics to analyze background and value of knowledge generation,so as to understand the value of mathematics education and create problems which can reflect the essence to drive mathematics teaching.The focus of this study changes from "why to teach" to "how to teach".Teaching reconstruction and teaching designs of "conic curve and equation" in this study are practical exploration and reflection on problem-driven theory and therefore is a beneficial supplement to existing theoretical system.From holistic perspective,this research interprets teaching content,reveals the essence of mathematics and traces the origin of knowledge according to mathematical history and mathematical subject structure.Furthermore,requirements of mathematics curriculum standards and students' reality is combined to recreate teaching content and design real effective problem situation to drive mathematics teaching.Problem-driven mathematics teaching can reproduce the generation process of knowledge.Therefore,this research is conducive to facilitate the transformation of teachers' teaching view and also helps students to learn to "think mathematically".