本文關鍵詞： 問題驅(qū)動 復數(shù)概念 復數(shù)運算 教學內(nèi)容重構　出處：《廣州大學》2016年博士論文　論文類型：學位論文

【摘要】：復數(shù)不僅在數(shù)學發(fā)展史上有著重大的歷史意義,而且對科學的認識論和方法論產(chǎn)生了深遠的影響,同時它在現(xiàn)代科技中也有著廣泛的應用.在高中數(shù)學課堂教學中,復數(shù)章節(jié)的教學是“容易教”的難點課.一方面教材內(nèi)容代數(shù)形式化的敘述,使學生對復數(shù)概念及其運算可以較輕松地獲得機械的識記和應用,這就是復數(shù)內(nèi)容表面上相對易教的原因;另一方面,正是教科書以復數(shù)產(chǎn)生的歷史結果為出發(fā)點,以代數(shù)形式表示復數(shù)及形式化定義復數(shù)運算,它們掩蓋了復數(shù)概念及復數(shù)運算產(chǎn)生的本原性問題,不利于復數(shù)及其運算的數(shù)學本質(zhì)和思想方法的揭示.本研究基于問題驅(qū)動的數(shù)學教學理念,審視了高中復數(shù)教學的基本現(xiàn)狀:首先,剖析了現(xiàn)行中學數(shù)學教科書中有關虛數(shù)引入、數(shù)系擴充和復數(shù)運算等知識點的敘述脈絡,分析了教材編寫中的存在問題;比較了主要發(fā)達國家及地區(qū)和我國有關中學復數(shù)知識的教學要求和教學內(nèi)容的差異,指出我國高中復數(shù)教學要求的差距.其次,為了掌握中學復數(shù)教學的實際現(xiàn)狀,通過聽課觀摩、教學研討,對復數(shù)章節(jié)的教學進行調(diào)查研究,分析和尋找高中復數(shù)教學存在的問題及其根源;研究了中學一線教師有關復數(shù)教學方面的研究文獻,考察數(shù)學教師對復數(shù)教學問題的相關認識.論文進一步介紹了復數(shù)產(chǎn)生和發(fā)展的脈絡,闡述推動復數(shù)理論發(fā)展的深刻的物理背景,探討了復數(shù)及其運算的重要思想和方法,并從數(shù)學思想性角度審視了高中復數(shù)內(nèi)容的教學問題,提出高中復數(shù)教學內(nèi)容重構的建議.最后,本研究在調(diào)查和分析的基礎上,重構和編寫了基于問題驅(qū)動的高中復數(shù)教學內(nèi)容,希望能夠準確把握復數(shù)產(chǎn)生與發(fā)展的本原性問題,希望能夠深度挖掘在復數(shù)認識中起統(tǒng)帥作用的、觸及與揭示復數(shù)本質(zhì)的問題,并貫穿于問題驅(qū)動的復數(shù)教學內(nèi)容的設計與重構.重構方案進行了教學試驗和評價.新教學內(nèi)容摒棄了代數(shù)視角,確立了幾何與物理視角介紹復數(shù),它能夠很好地回答:為什么要提出和引入虛數(shù)概念?如何適當?shù)囟x復數(shù)概念和復數(shù)運算?這些概念和運算給我們帶來了什么?復數(shù)的重大意義和科學價值體現(xiàn)在哪里?
[Abstract]:Plural is not only of great historical significance in the history of mathematics development, but also has a profound influence on the epistemology and methodology of science. At the same time, it is also widely used in modern science and technology. In the teaching of mathematics in senior high school, the teaching of plural chapters is a difficult course which is easy to teach. On the one hand, the formal description of the content algebra of the teaching material. It is easy for students to obtain mechanical memorization and application of complex number concept and its operation, which is the reason why the complex number content is relatively easy to teach on the surface; On the other hand, the textbook takes the historical result of the complex number as the starting point, expresses the complex number and formally defines the complex number operation in the algebraic form, which cover up the original problem of the complex number concept and complex number operation. This research is based on the problem-driven mathematics teaching concept, examines the basic situation of the complex number teaching in senior high school: first of all. This paper analyzes the narrative context of the knowledge points in the current middle school mathematics textbooks, such as the introduction of imaginary numbers, the expansion of the number system and the complex number operation, and analyzes the problems existing in the compilation of the textbooks. This paper compares the differences of teaching requirements and contents of plural knowledge in middle schools between the main developed countries and regions and our country, and points out the differences in the requirements of plural teaching in senior high schools in our country. In order to grasp the actual situation of plural teaching in middle schools, through the observation of lectures, teaching discussion, the teaching of plural chapters of the investigation and research, analysis and search for the problems and root causes of plural teaching in senior high school; This paper studies the literature on the plural teaching of the first-line teachers in middle schools, and investigates the mathematics teachers' understanding of the plural teaching. The paper further introduces the context of the emergence and development of the plural. This paper expounds the profound physical background of promoting the development of plural theory, probes into the important ideas and methods of plural number and its operation, and examines the teaching problems of plural content in senior high school from the angle of mathematical thought. Finally, on the basis of investigation and analysis, this study reconstructs and compiles the plural teaching content based on problem driven. Hope to accurately grasp the origin of the emergence and development of the plural, hope to be able to dig in depth in the plural understanding of the role, touch and reveal the essence of plural problems. And the design and reconstruction of the complex teaching content which is driven by the problem. The reconstruction scheme has carried on the teaching experiment and the appraisal. The new teaching content has abandoned the algebra angle, established the geometry and the physics angle to introduce the complex number. It can answer well: why put forward and introduce the concept of imaginary number? How to define plural concepts and complex operations properly? What do these concepts and operations bring us? Where is the significance and scientific value of the plural?
【學位授予單位】：廣州大學
【學位級別】：博士
【學位授予年份】：2016
【分類號】：G633.6

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