【摘要】：數(shù)與形是數(shù)學研究的重要主題,數(shù)與形的結合有著悠久的歷史。從畢達哥拉斯“萬物皆數(shù)”開始就將數(shù)和形緊密的聯(lián)系在一起。笛卡爾創(chuàng)立的解析幾何是數(shù)形結合模式的典范,極大地推動了數(shù)形結合的發(fā)展。數(shù)和形也是數(shù)學中的兩個最基本的概念,他們相互聯(lián)系并相互論證。在《義務教育數(shù)學新課程(2011年版)》中明確指出：數(shù)形結合是探索數(shù)學新知識的重要方法之一。因此,數(shù)形結合一向是教學的重點,也是歷年高考的必考內(nèi)容。新一輪課改更加注重學生探索和創(chuàng)新能力的培養(yǎng),這對學生數(shù)形結合思想的理解以及運用數(shù)形結合解題的能力,提出了更高的要求,也對教師的教學提出了新的挑戰(zhàn)。因此,有必要對數(shù)形結合思想在解題中的應用以及學生數(shù)形結合解題能力現(xiàn)狀進行研究分析,從而給出一些教學上的策略建議。本文試圖在總結前研究成果的基礎上,以教材為依據(jù),以教育實踐經(jīng)驗為參考,深入研究數(shù)形結合在高中數(shù)學中的應用方面,并通過問卷調查、口頭訪談和課堂實踐的方式,了解學生數(shù)形結合解題能力現(xiàn)狀。并在充分分析結果的基礎上,探討教師應如何培養(yǎng)學生數(shù)形結合能力。使其不單成為學生快速有效解題的方法,而且更高層次的,使其上升為一種數(shù)學思想,并內(nèi)化為學生的數(shù)學素養(yǎng)。具體里來說,本文共分5章內(nèi)容：第一章緒論,主要闡述問題研究的背景和意義；第二章文獻綜述,主要總結前人的研究成果,包括數(shù)形結合的演變簡史、數(shù)形結合的理論依據(jù)和數(shù)形結合的教育價值。第三章數(shù)形結合在中學數(shù)學解題中的應用,總結并用實例說明了數(shù)形結合應用的三種類型：以形輔數(shù)、以數(shù)助形和數(shù)形并重。而在運用數(shù)形結合解題中應遵循等價性原則、雙向性原則和等價性原則,最后根據(jù)對近幾年高考統(tǒng)計分析,總結高考對數(shù)形結合思想考察的特點；第四章對數(shù)形結合思想應用的現(xiàn)狀做實證調查,發(fā)現(xiàn)學生對數(shù)形結合思想的理解比較狹隘,學生對數(shù)形結合思想的應用能力不強。主要原因是學生構圖能力不強、學生對同一知識點的數(shù)表征和形表征的對應關系較弱。但相對來說學生“以形思數(shù)”的能力要強于“以數(shù)思形”的能力。第五章根據(jù)第四章調查研究所發(fā)現(xiàn)的問題給出一些策略性建議,包括轉變教師觀念、用好教材中的素材、注重數(shù)學語言的教學和合理利用信息技術加強數(shù)與形的對應。
[Abstract]:Number and shape are important topics in mathematical research, and the combination of number and shape has a long history. Since Pythagoras, everything has been closely linked to form. The analytic geometry created by Descartes is a model of numerical combination model, which greatly promotes the development of numerical combination. Number and form are also the two most basic concepts in mathematics, which are related to each other and demonstrated to each other. In the New Mathematics Curriculum of compulsory Education (2011 Edition), it is clearly pointed out that the combination of numbers and shapes is one of the important methods to explore the new knowledge of mathematics. Therefore, the combination of numbers and shapes has always been the focus of teaching, but also the required content of the college entrance examination over the years. The new round of curriculum reform pays more attention to the cultivation of students' exploration and innovation ability, which puts forward higher requirements for the understanding of students' thought of combining number and shape and the ability to use the combination of number and form to solve problems, and also puts forward new challenges to teachers' teaching. Therefore, it is necessary to study and analyze the application of the thought of combination of numbers and shapes in problem solving and the present situation of students' ability to solve problems, so as to give some strategic suggestions in teaching. On the basis of summing up the previous research results, taking the teaching materials as the basis, taking the educational practice experience as the reference, this paper attempts to deeply study the application of the combination of numbers and shapes in senior high school mathematics, and through questionnaires, oral interviews and classroom practice. To understand the present situation of students' problem-solving ability combined with number and shape. On the basis of full analysis of the results, this paper probes into how teachers should cultivate students' ability of combining numbers and shapes. So that it not only become a fast and effective method for students to solve problems, but also a higher level, so that it rises to a kind of mathematical thought, and internalizes it into students' mathematical literacy. Specifically, this paper is divided into five chapters: the first chapter is the introduction, which mainly expounds the background and significance of the problem research; The second chapter is a literature review, which mainly summarizes the previous research results, including the brief history of the evolution of the combination of numbers and forms, the theoretical basis of the combination of numbers and forms and the educational value of the combination of numbers and forms. In the third chapter, the application of number-form combination in middle school mathematics problem solving is summarized and illustrated with an example: the auxiliary number of the number, the weight of the auxiliary number and the number of the number. In using the combination of number and form to solve the problem, we should follow the principle of equivalence, the principle of two directions and the principle of equivalence. Finally, according to the statistical analysis of the college entrance examination in recent years, this paper summarizes the characteristics of the thought of the combination of logarithm and form in the college entrance examination. In the fourth chapter, we make an empirical investigation on the present situation of the application of logarithmic combination thought, and find that students have a narrow understanding of the thought of numerical combination, and the application ability of students' thought of logarithmic combination is not strong. The main reason is that the students' composition ability is not strong, and the corresponding relationship between the number representation and the shape representation of the same knowledge point is weak. But relatively speaking, students' ability of thinking in shape is stronger than that of thinking in shape. The fifth chapter gives some strategic suggestions according to the problems found in the fourth chapter, including changing the teachers' concept, making good use of the materials in the teaching materials, paying attention to the teaching of mathematical language and making rational use of information technology to strengthen the correspondence between numbers and shapes.
【學位授予單位】：華中師范大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：G633.6

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