發(fā)布時(shí)間：2019-02-21 17:58

【摘要】：三角學(xué)是在解決天文學(xué)、物理學(xué)實(shí)際問(wèn)題過(guò)程中而產(chǎn)生與發(fā)展起來(lái)的,對(duì)數(shù)學(xué)與自然科學(xué)產(chǎn)生了深遠(yuǎn)的影響.中學(xué)階段主要包括三角函數(shù)、正弦定理與余弦定理等內(nèi)容,三角函數(shù)蘊(yùn)含著深刻的數(shù)學(xué)思想,研究三角函數(shù)的教材對(duì)教材編寫(xiě)和三角函數(shù)的教學(xué)實(shí)踐與教學(xué)理論均有重要意義.本文以北京師范大學(xué)出版社、江蘇科學(xué)技術(shù)出版社、人民教育出版社出版的教材為參考,以初中銳角三角函數(shù)、高中三角函數(shù)為研究對(duì)象.重點(diǎn)研究三角函數(shù)概念的產(chǎn)生與發(fā)展.本研究以質(zhì)性研究方法為主.通過(guò)消化吸收已有的數(shù)學(xué)史知識(shí)、數(shù)學(xué)教學(xué)理論知識(shí)、數(shù)學(xué)學(xué)習(xí)心理知識(shí)、數(shù)學(xué)方法論知識(shí)等,結(jié)合數(shù)學(xué)教學(xué)的課堂觀察與研課,對(duì)教材內(nèi)容作出解讀,在此基礎(chǔ)上對(duì)教學(xué)內(nèi)容進(jìn)行重構(gòu).研究成果主要有:●初中部分三個(gè)版本教材為形成銳角三角函數(shù)概念都創(chuàng)設(shè)了問(wèn)題情境,北師版教材從梯子陡坡程度、蘇科版教材從臺(tái)階的坡度、人教版教材從綠化荒山的角度創(chuàng)設(shè)問(wèn)題情境,這些問(wèn)題并沒(méi)有真正揭示出銳角三角形相似比的不變性這一深刻的數(shù)學(xué)思想,事實(shí)上,將之作為相似三角形的問(wèn)題情境更合適.如何通過(guò)相似三角形引導(dǎo)學(xué)生發(fā)現(xiàn)直角三角形的相似不變量是問(wèn)題的關(guān)鍵.●高中部分三個(gè)版本教材在任意角、弧度制、任意角的三角函數(shù)、三角函數(shù)的誘導(dǎo)公式等內(nèi)容的編寫(xiě)風(fēng)格不同.北師版教材直接給出相關(guān)知識(shí).在弧度制、任意角的三角函數(shù)、三角函數(shù)的誘導(dǎo)公式設(shè)計(jì)上均以單位圓為載體,教材希望以單位圓為載體給出上述知識(shí),但是知識(shí)間的關(guān)系不清晰.同時(shí),有些內(nèi)容的編寫(xiě)出現(xiàn)疏漏.沒(méi)有體現(xiàn)三角函數(shù)的科學(xué)價(jià)值與應(yīng)用價(jià)值,也沒(méi)能體現(xiàn)出三角函數(shù)蘊(yùn)含的數(shù)學(xué)思想.蘇科版教材通過(guò)問(wèn)題情境引發(fā)概念的生成,并以引言中的相關(guān)問(wèn)題統(tǒng)領(lǐng)知識(shí)間的關(guān)系,知識(shí)之間的關(guān)系清晰.但在編寫(xiě)方面也存在一些瑕疵,例如,在靜態(tài)直角三角形中標(biāo)記動(dòng)態(tài)的角的旋轉(zhuǎn)符號(hào)等.三角函數(shù)緣何產(chǎn)生以及它真正的科學(xué)價(jià)值沒(méi)有在教材中體現(xiàn)出來(lái).表現(xiàn)出的主要是這些知識(shí)之間的數(shù)學(xué)關(guān)系,但是有些關(guān)系比較牽強(qiáng).人教版教材在三角函數(shù)章引言上給出了宇宙天體運(yùn)行圖,但是在各節(jié)中卻沒(méi)有運(yùn)用這些有價(jià)值的天文學(xué)背景為三角函數(shù)相關(guān)知識(shí)的形成提供基礎(chǔ).人教版教材在與讀者互動(dòng)方面設(shè)計(jì)較好,但是各節(jié)的問(wèn)題之間缺乏聯(lián)系,且有些是無(wú)效問(wèn)題.三角函數(shù)揭示的天文學(xué)或物理學(xué)中旋轉(zhuǎn)運(yùn)動(dòng)與直線運(yùn)動(dòng)間的關(guān)系等很少在教材中體現(xiàn).本文以真實(shí)且歷史上具有重要影響的問(wèn)題情境——擺線統(tǒng)領(lǐng)任意角、弧度制、任意角的三角函數(shù)、三角函數(shù)的誘導(dǎo)公式.這一問(wèn)題情境既是學(xué)生生活中常見(jiàn)的現(xiàn)象又滿(mǎn)足學(xué)生的數(shù)學(xué)現(xiàn)實(shí).以啟發(fā)性問(wèn)題引領(lǐng)學(xué)生解決物理問(wèn)題的同時(shí)關(guān)注從數(shù)學(xué)內(nèi)部建構(gòu)數(shù)學(xué)知識(shí)等,引導(dǎo)學(xué)生在解決真實(shí)的物理問(wèn)題過(guò)程中建構(gòu)三角函數(shù)知識(shí),并揭示三角函數(shù)形成的根源及其應(yīng)用價(jià)值.
[Abstract]:Trigonometry is developed and developed in the process of solving the practical problems of astronomy and physics, and has a far-reaching influence on the mathematics and the natural science. The middle school stage mainly includes the trigonometric function, the sine theorem and the cosine law and so on, the trigonometric function contains the deep mathematics thought, the teaching material of the study of the trigonometric function is of great significance to the teaching practice and the teaching theory of the preparation of the teaching material and the trigonometric function. This paper is based on the teaching materials published by the Beijing Normal University Press, the Jiangsu Science and Technology Press, the People's Education Press, and the high school triangle function and the high school triangle function as the research object. This paper focuses on the generation and development of the concept of the trigonometric function. This study is based on the qualitative research method. Through the digestion and absorption of the existing knowledge of the mathematical history, the theoretical knowledge of the mathematical teaching, the psychological knowledge of mathematics learning, the knowledge of the mathematical methodology, etc., the content of the teaching materials is interpreted in combination with the classroom observation and the research of the mathematics teaching, and the content of the teaching is reconstructed. The research results are as follows: the three editions of the teaching materials in the junior middle school form the problem context for forming the concept of the sharp-angle triangle function, the teaching material of the north division is from the slope of the ladder, the teaching material of the Su-ke edition is changed from the slope of the step, the teaching material of the human version creates the problem context from the angle of the green barren hill, These problems do not really reveal the invariance of a sharp-angle triangle-like ratio, which is a profound mathematical thought. In fact, it is more appropriate to use it as a similar triangle. How to guide students through similar triangles is the key to the problem. The writing style of the three versions of the textbook in the high school is different from the writing style of any angle, the radian system, the trigonometric function of any angle, the induction formula of the trigonometric function, and the like. The North Division teaching material directly gives the relevant knowledge. In the radian system, the trigonometric function of any angle and the induction formula of the trigonometric function are designed with the unit circle as the carrier, and the teaching material hopes to give the above-mentioned knowledge in the unit circle as the carrier, but the relation between the knowledge is not clear. At the same time, there is an omission in the preparation of some of the content. The scientific value and application value of the trigonometric function are not reflected, and the mathematical idea contained in the trigonometric function is not reflected. The teaching material of the Su Ke version is the generation of the concept through the situation of the problem, and the relationship between the knowledge and the knowledge is the clear relationship between the knowledge and the knowledge in the introduction. however, there are also some blemish in that preparation of, for example, the rotation symbol of the dynamic angle in a static right-angled triangle, and the like. The origin of the trigonometric function and its real scientific value are not reflected in the teaching material. The main thing to show is the mathematical relationship between these knowledge, but some of them are far-fetched. In the introduction of the chapter of the trigonometric function, the teaching material of human teaching has given the operation diagram of the universe, but it is not used in each section to provide the basis for the formation of the related knowledge of the trigonometric function. The teaching material of the teaching version is well designed in the interaction with the reader, but there is a lack of contact between the problems of each section, and some are invalid. The relation between the rotational movement and the linear motion in the astronomy or physics revealed by the trigonometric function is seldom reflected in the teaching material. In this paper, the induction formula of any angle, radian system, trigonometric function of arbitrary angle, and trigonometric function is given in the real and historical context of the problem with the important influence. This problem is not only the common phenomenon in the life of the students, but also the students' mathematical reality. In order to lead the students to solve the physical problems with the heuristic problems, the paper focuses on the construction of the mathematical knowledge from the interior of the mathematics, and guides the students to construct the knowledge of the trigonometric function in the process of solving the real physical problems, and reveals the root causes of the formation of the trigonometric function and its application value.
【學(xué)位授予單位】：廣州大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：G633.6
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本文編號(hào)：2427751