Research On Problem-driven High School Mathematics Vector Teaching

The vector in high school mathematics,which is different from the traditional high school mathematics knowledge,is an important concept in modern mathematics.It is originally the curriculum content of college mathematics,which contains rich mathematical connotations and is incorporated into the high school mathematics curriculum in the new curriculum reform.Vectors play an important role in middle school mathematics.They can not only help students understand middle school mathematics knowledge,clarify the essential relationship between algebra and geometry,and obtain mathematical learning methods,but also enable students to understand the abstract structure system of modern mathematics,and effectively establish the connection between middle school mathematics and college mathematics curriculum.However,the extremely simple formal definition and rich physical background of vectors make them easy to misunderstand that the learning of vectors is simple and easy,while ignoring the complexity of vector structures.This is bound to the fact that the high school vector teaching only stays in the surface of knowledge,and cannot reveal the essence of vectors and contains important modern mathematical thought.This research takes what and how to teach vectors in high school mathematics as the main purpose,analyzes the research on vector teaching in high school mathematics at home and abroad,and finds out the shortcomings and innovations of the vector teaching in high school,based on the problem-driven mathematics teaching theory.The teaching and learning of vectors in high school are deeply studied from the perspective of mathematics.First of all,according to the setting and teaching requirements of vectors in high school mathematics according to the mathematics curriculum standards,the vector arrangement content and classroom teaching status in high school mathematics textbooks are investigated and analyzed to find out the problems existing in vector teaching:(1)The introduction of vector concepts is too dependent on the physical background,and there is no clear motivation,meaning and value of vector generation;(2)There is a deviation in the position of the goal of vector teaching,which takes the vector as a tool to solve problems,and does not focus on revealing the essence of the vector method and the mathematical thought involved in vectors;(3)There is a phenomenon of fragmented knowledge accumulation in vector teaching,which emphasizes too much on the geometric meaning of vectors and lacks the construction of vector theoretical systems from an algebraic perspective.Secondly,aiming at the problems existing in high school mathematics vector teaching,this paper combs the historical process of vector generations,formations and development in mathematics,and analyzes the important influence of vectors on mathematics development.The original problem of mining vector generations and its mathematical thought are how to use algebra’s quantitative operation to simply quantify the properties of geometry,and how to use geometry’s intuitive figures to illustrate the abstract quantitative relations in algebra.Again,based on the current investigate on the vector teach in high school mathematics,we give new ideas of the vector teaching in high school mathematics.From the geometric intuitive vector,as an algebraic operation of the object,constitute the mathematical structure of the vector three levels gradually expand.From the perspective of modern mathematics,high school mathematics vector,from the perspective of modern mathematics,reveals the essence of vector and its operation,and explores the rich mathematical connotation contained in the content of high school mathematics vector.This paper analyzes the teaching value of high school mathematics vectors from two aspects of horizontal connection and vertical development of mathematics knowledge,and points out that vector can not only clarify the nature of geometric qualitative property,depict the initial form and research method of trigonometric knowledge,clarify the essence of mathematical knowledge such as the concept and operation of complex numbers,and has wide application in physics.It can also be used as a bridge to effectively connect middle school mathematics and university mathematics knowledge,and help students to deepen their mathematical understanding from algebraic operation to algebraic structure,from plane to high-dimensional space,from intuitive geometric space to abstract space described by algebra.Finally,based on the pattern of problem-driven mathematics teaching,in order to reveal the essence of vector content,penetrate the idea of modern mathematics,and improve students’ mathematics core literacy,the teaching design of each chapter of high school mathematics plane vector is given.In the analysis of the essence of the vector concept,the cognitive transformation from the geometric intuitive vector to the vector that can establish the operational relationship is realized.It deepens students ’ understanding of algebraic operation in vector linear operation,realizes the transformation from arithmetic operation of number to operation of general algebraic object,from algebraic operation to understanding of algebraic structure,and penetrates the idea of axiomatic vector concept.In revealing the essence of the basic theorem of number multiplication and plane vector,the transformation from qualitative research on geometric space to algebraic quantitative description is realized.In the construction analysis of vector quantity product,it is further clarified that vector constructs a method to describe geometric properties directly by algebraic quantitative operation.Finally,the students’ perception vector theory can form an algebraic theory system which is completely equivalent to the Euclidean geometry system and can describe the geometric properties,namely Euclidean space.