注本文为 “基于微积分思维的数学分析教学” 相关译文机翻未校。略作重排如有内容异常请看原文。基于微积分思维的数学分析教学Teaching of Mathematical Analysis or Calculus Based on the Thinking of Calculus收录于《社会科学、教育与人文研究进展》第 551 卷第六届教育改革与现代管理国际会议论文集ERMM 2021王伟*中国人民大学 数学学院*通讯作者邮箱wweiruc.edu.cnABSTRACTMathematical analysis or calculus is one of the core courses of mathematics and many related disciplines. For students to lay solid foundations and to have better developments in their future careers, teaching of mathematical analysis plays an important role in mathematics and related disciplines. Unfortunately, many students feel that it is difficult in the understanding and learning of the subject. Based on many years of the teaching experiences, in this paper, we put forward the new idea that the teaching of mathematical analysis can be carried out by using the thinking of calculus. The essence of the method is to carry out the teaching and learning of mathematical analysis or calculus with a simple and efficient way. Some suggestions on the teaching and learning of the subject are also provided.数学分析微积分是数学及众多相关专业的核心课程。为帮助学生夯实专业基础、助力未来职业发展数学分析的教学工作在数理类专业教育中占有举足轻重的地位。但现实中大量学生在理解、学习这门课程时倍感吃力。本文结合多年一线教学经验提出依托微积分思维开展数学分析教学的新思路。该教学思路的核心是以简洁高效的方式组织数学分析的教与学同时本文针对该课程的教、学两端给出若干实操建议。KeywordsTeaching reformation, Mathematical analysis, Calculus thinking, Automatic method.教学改革数学分析微积分思维自动化计算方法1. INTRODUCTION引言Calculus or mathematical analysis is one of the basic courses of mathematics, and it is also a basic course for many other majors. It plays an important role in cultivating students’ mathematical professionalism and mastering scientific and efficient calculation methods.微积分数学分析是数学专业基础课同时也是绝大多数其他专业的必修基础课对培养学生数学素养、掌握科学高效的计算工具至关重要。However, in the teaching procedures of calculus or mathematical analysis, it is generally believed that learning of mathematical analysis or calculus is a very difficult thing. I was very impressed by a very serious student who had studied calculus for a year. When she was asked what she felt about the course, her answer was that it is nothing but difficulty, or it is more difficulty than difficulty. In addition, a considerable number of students fail in this course each year.但在实际教学过程中普遍存在学生认为微积分、数学分析晦涩难学的现象。我印象很深一名学习态度十分端正的学生在学完一年微积分后被问及课程感受时直言这门课除了难还是难难上加难。每年都有不少学生该课程考试不及格。During the mid-autumn festival in 2020, the mooncakes sold by the University of Science and Technology of China (USTC) take some of the courses which are difficulty to learn, such as mathematical analysis, as the object of moon cakes. The moon cake packaging sent to students by USTC the year 2020 is full of creativity. The packaging box of mooncake is covered with the textbooks of Astrophysics, the Course of Mathematical Analysis, Introduction to Quantum Mechanics and other textbooks, which are marked with ‘Excellent Textbooks of China University of Science and Technology’. The small package also has the words ‘for you’. The official micro-blog of USTC also wishes that the students will never fail after eating the cake. It indicates that mathematical analysis is also a difficult course for students of USTC.2020 年中秋中国科学技术大学推出的月饼设计极具创意将数学分析等难度较高的课程印在月饼包装上。月饼礼盒印满《天体物理》《数学分析教程》《量子力学导论》等教材封面并标注“中科大精品教材”独立小包装印有“送你一份”字样。中科大官方微博还打趣祝愿学生吃完月饼该科永不挂科。这件事也侧面说明即便是中科大的学生同样觉得数学分析学习难度很大。A foreign colleague lamented that it is not easy to teach multiple variable calculus, because students do not have enough motivation. And it will be easier to teach the multi-variable analysis only when students knowing the applications.一位国外同行也曾感慨多元微积分授课难度高学生学习动力不足只有让学生充分了解其实际应用多元分析的教学才能顺利推进。Then, why should the students learn this course well? How can we carry out the teaching of calculus and mathematical analysis, and how can we simplify the problem of teaching and learning? Combining with ourselves experiences in teaching of calculus or mathematical analysis for many years, in this paper, we put forward the viewpoint of using the thinking of calculus to carry out the teaching of calculus or mathematical analysis. First of all, our viewpoint is that mathematical analysis is necessary for almost all majors, and it should be taught or learned seriously. At the same time, our teaching should adopt the idea of calculus, i.e., we should try our best to translate some of the complex calculations into simple ones, and to turn the problem of complex learning processes into easy learning processes.那么学生为何必须学好这门课我们该如何开展微积分、数学分析教学简化教与学的难点本文结合多年微积分与数学分析授课经验提出以微积分思维统领课程教学的观点。核心观点分为两点第一几乎所有专业都离不开数学分析必须认真教、认真学第二教学全过程贯穿微积分核心思想尽可能把复杂计算转化为简单运算把繁琐的学习流程简化。The organizing of the paper is as follows. In section 2, we will concern about the significances and the roles of mathematical analysis; In section 3, the viewpoints on the contents of mathematical analysis is discussed; In section 4, the reasons on the difficulty of learning calculus or mathematical analysis are pointed out; In section 5, to learn mathematics well, some suggestions are provided. Followed by some conclusions.本文行文结构安排如下第 2 节论述数学分析的价值与必要性第 3 节剖析数学分析的核心内涵第 4 节梳理学生学习微积分、数学分析感到困难的根源第 5 节给出学好该课程的相关建议最后为全文总结。2. THE VIEWPOINTS ON THE FUNCTIONS AND NECESSITIES OF CALCULUS OR MATHEMATICAL ANALYSIS关于微积分或数学分析的功能和必要性的观点From the basic viewpoint on the division of higher mathematics and elementary mathematics, mathematical analysis belongs to the higher mathematics. It is the development of elementary mathematics. Elementary mathematics concern about the mathematics of studying constants, and mathematical analysis concerns about the science of variables. Therefore, mathematical analysis provides the mathematical or scientific methods to describe the characteristics on changing of variables [1].从初等数学与高等数学的划分标准来看数学分析属于高等数学是初等数学的延伸拓展。初等数学以常量为研究对象而数学分析是专门研究变量的学科。因此数学分析提供了一套刻画变量变化规律的科学数学工具[1]。It is precisely because of such viewpoint that the subject of mathematical analysis is a foundation of various disciplines. Because dialectics tells us that all things or materials are always in changing, change is absolutely, and invariance is relatively. Any subject is related with the problem of changing, not only scientific subjects, but also the subjects on society, humanity and culture. How can we describe the laws or rules of changing? It is inseparable from the methods provided by mathematical analysis. In other words, mathematical analysis solves the important problems in philosophy. That is to say, it provides the measurement methods of changing. Especially, the differential or derivative solves the problem of changing speeds, and the integral solves the problem of cumulative effects on changing.正因如此数学分析成为各类学科的底层基础。辩证法指出万事万物始终处于运动变化之中运动是绝对的静止是相对。不管是理工科还是社科、人文类学科都会涉及变化规律的研究而描述变化规律离不开数学分析的工具。从这个角度来说数学分析回应了哲学层面的核心问题它给出了量化变化的手段微分导数刻画变化速率积分刻画变化的累积效果。It’s not only the students from mathematical majors but also for students from many other majors, who need to learn this course well. For example, in computer applications, information processing is a very important aspect. The basis methods of information processing come from mathematical analysis. If you are engaged in CAD or 3D modelling, the methods in multivariate calculus are necessary and more useful. If you are engaged in information management, the characteristics of information flow can be illustrated by the curve area, and if you concern about the economic management and resource management, you are also inseparable from mathematical analysis [5]. And, if you want to be effective in financial analysis in the future, you’ll have to learn stochastic analysis, which is the extension of the methods or thinking of calculus. And mathematical analysis is also the basis of the follow-up courses of mathematics, such as, ordinary differential equations, real analysis, complex analysis, probability and statistics, which are all based on the knowledge of calculus.不仅数学专业学生其他各专业学生都需要扎实掌握这门课程。举例计算机领域中信息处理的核心方法根植于数学分析从事计算机辅助设计、三维建模多元微积分是必备工具信息管理领域可用曲线面积刻画信息流特征经济、资源管理专业也离不开数学分析工具[5]。若未来从事金融分析还要学习随机分析而随机分析正是微积分思想与方法的延伸。同时数学分析也是数学系后续专业课的基础常微分方程、实变函数、复变函数、概率论与数理统计等课程全部建立在微积分知识之上。3. THE ESSENTIAL IDEAS OF CALCULUS OR MATHEMATICAL ANALYSIS微积分或数学分析的基本思想Calculus is considered to be one of the greatest inventions in the history of human science. It is the discovery on the idea of calculus that make the scientific calculation effectively and simply [2]. As mentioned above, the basic problem of calculus is to study the methodology of solving the change of things. Therefore, it takes a series of functions as its objects. It studies the change rate or cumulative effect of the change of one variable functions at first, and then, they learn multiple variable functions. So, the subject mainly includes elementary functions and corresponding functions.微积分是人类科学史上最伟大的发明之一微积分核心思想的诞生让复杂科学计算变得简洁高效[2]。前文提到微积分的核心任务是研究刻画事物变化的方法因此课程以各类函数为研究载体先讨论一元函数的变化速率与累积效应再拓展至多元函数课程主体围绕初等函数及其衍生函数展开。In order to solve the problem of calculus, the method of the limit theory was introduced [3]. It is the introduction of limit method that we can understand ‘infinity’ from ‘finiteness’, understand ‘variable or change’ from ‘constant’, and understand ‘qualitative change’ from ‘quantitative change’, and so on. That is the essential features of mathematical analysis, i.e., to realize the complex calculation from simple calculation.为解决变量分析问题微积分引入极限理论体系[3]。借助极限思想我们可以通过有限认识无限、通过静态常量理解动态变量、通过量变推导质变这也是数学分析的核心特质把复杂运算拆解为简单运算。It is the introduction of differential or derivative that makes the calculation of the increment of a complex function becomes a linear function of independent variable. In particular, differential is to solve the complex nonlinear calculation by linear ‘approximation’ (main component) plus infinitesimal quantity. It is because of the characteristics that make the calculation for the area or volume of unusual complex geometry simplify with efficient calculation of calculus [4].微分导数的出现将复杂函数的增量转化为自变量的线性表达式。微分的核心思路是用线性主部做近似搭配无穷小误差项处理复杂非线性运算。依托这一特性各类不规则复杂几何体的面积、体积计算得以简化运算效率大幅提升[4]。It is because of the functions of limit that determine the structure of mathematical analysis. The whole course is composed by the concepts and properties of the limit, continuity, differentiability and integrable of a series of functions. And the fundamental part of limit theory is the theory of real numbers.极限理论搭建起数学分析的整体框架整门课程围绕函数极限、连续、可微、可积四大核心概念与对应性质展开而实数理论又是极限理论的底层根基。It is the establishment of the relationship between derivative and integral, i.e., Newton-Leibniz formula, that makes the analysis of the boundary situation can be reflected by the internal situation.牛顿-莱布尼茨公式搭建起导数与积分的桥梁实现了通过区间内部变化刻画区间边界特征。So, from the perspective for the majors of mathematics, it is based on real number theory, limit theory, continuity, differentiability and integrable. Each section has its corresponding concepts, properties and algorithms. From the perspective of applications, the theory on real numbers can be ignored, that is to say, the students can pay more attention to the concepts or methods of limit, continuity, differentiability and integrable without the proof of the main theories [6].因此站在数学专业视角课程体系以实数理论、极限、连续、微分、积分为主线每个板块配套专属概念、性质与计算方法而面向应用型专业学习时可弱化实数理论学生重点掌握极限、连续、微分、积分的概念与实操方法不必深究核心定理的完整证明[6]。4. WHY DO STUDENTS FEEL DIFFICULTIES FOR THE LEARNING OF MATHEMATICAL ANALYSIS?为什么学生对数学分析的学习感到困难Why do students feel the difficulty on the learning of mathematical analysis or calculus? From the viewpoint of ourselves, there are some aspects affecting the performance of teaching and learning.学生学习微积分、数学分析倍感吃力的根源是什么结合教学实践总结出几方面影响教、学效果的关键因素。First of all, it relies on the serious lack of the understanding to the necessary or/and the backgrounds of mathematical analysis. The reasons why we usually feel difficult are that we don’t know why to learn. Usually, the mathematical analysis is presented to the students not by the usefulness, but by the abstraction and a lot of mathematical skills. Meanwhile, the teaching of mathematical analysis usually performed by the scholars or professors who are with deep basics of mathematics, they are familiar with the way of abstract thinking of mathematics. They may or may not also understand the idea or the backgrounds behind the concepts. They cannot tell the real backgrounds on some of the mathematical thinking, or what are the reasons which we can learn from the mathematics.第一学生对数学分析的学习必要性、应用背景认知严重缺失。学生觉得难的首要原因是不清楚学习这门课的意义。课堂讲授往往优先抛出抽象定义、繁杂技巧很少结合实际用途展开。同时授课教师大多具备深厚数学功底习惯抽象思维模式有时未能深挖概念背后的思想起源无法清晰向学生讲解各类数学方法的现实来源与学习价值。Secondly, the difficulty of mathematic analysis lies in its abstractness. Of course, this does not mean that mathematical problems and methods are purely mathematicians’ business. They come from real life and production. On the other hand, textbooks often contain mainly a lot of abstract concepts, signs and skills without much more materials on the applications.第二课程高度抽象。但数学问题、数学方法并非脱离现实全部源自生产生活而教材大多堆砌抽象定义、符号与解题技巧应用案例篇幅严重不足。Many abstract concepts in mathematics come from real phenomena, and they are the embodiment of the most essential characteristics. For example, particles suspended in the liquid to do irregular motion is called Brownian motion. The mathematical abstraction of this motion is called Wiener process. Now, people use Brownian motion to approximately describe the changing law of the stock markets. This is one example of the important roles of mathematics. Without mathematical abstraction, it may be difficult to relate the motion of particles in liquid with the fluctuation law of stock markets. This is mathematics, i.e., it is a highly abstract science. It is this abstraction that enables people to see the essence of the problem through phenomena. It is precisely such kind of features that we can equate the oil exploration of the earth with the medical diagnosis of the disease source. Maybe the two aspects have the characteristics of ‘isomorphism’. It is precisely because of this feature that we can get a breakthrough in another field by ‘transplanting’ under the inspiration of research methods in other fields [8].大量抽象数学概念都源于现实现象是事物核心特征的凝练。举个例子液体中悬浮微粒的无规则运动称为布朗运动其数学抽象模型为维纳过程如今人们借助布朗运动拟合股市波动规律这正是数学价值的典型体现。若没有数学抽象工具很难建立液体微粒运动与股市涨跌之间的关联。数学是高度抽象的学科抽象性让人们透过表象抓住事物本质。依托抽象带来的同构特性石油勘探、病灶诊断两类完全不同的领域可以建立对应关系我们能够将一个领域的研究方法迁移到另一领域实现研究突破[8]。Thirdly, the difficulty also depends on whether the basics of mathematics is solid or not. Mathematics is the subject needs strong ability of logical reasoning. The former knowledge will be important for the studying of follow-up contents. For example, many of the ideas and methods related with multi-variable calculus are based on the ones from single variable calculus. So, if students don’t have a firmly grasp of the methods at the beginning, the consequences will be the ones that the students cannot understand what the teachers said later, or that the students have no idea on what we will discuss.第三数学基础扎实程度决定学习难度。数学高度依赖逻辑推导前后知识点层层递进。多元微积分的思想、方法全部建立在一元微积分之上若前期基础掌握不牢后续课堂内容会完全跟不上听不懂授课内容。Maybe all of these reasons lead to students the impression of mathematical analysis that it is difficulty or more difficulty than nothing.以上多重因素叠加最终让学生形成“数学分析难、难上加难”的固有印象。However, what is the inspiration for students that mathematical analysis bringing about? We really hope that mathematical analysis can tell them that calculus can make the complex calculating more easily, for example, the calculating of the area or volume for some unusual objects. And from the relationship between differential and integration, i.e., Newton-Leibniz formula, we can understand that the cultivation of small things can bring remarkable/serious results, not only for our behaviours on the learning or actions for doing things, but also for our healthy. It is almost the same that we can learn from the aphorism, i.e., don’t do anything small if it is evil, but do something small if it is good. The methods in calculus or mathematical analysis are also the scientific expressions of some well-known aphorism, such as, Rome is not built in a day.但数学分析本可以给学生带来正向启发微积分的核心作用是简化复杂运算比如各类不规则几何体的面积、体积求解。同时从牛顿-莱布尼茨公式体现的微分积分关系中学生能领悟点滴积累终将收获巨大成效这条道理适用于学习、处事、养生和“勿以善小而不为勿以恶小而为之”的古训相通。微积分思想也科学诠释了“罗马不是一天建成的”这类俗语。5. HOW CAN WE LEARN MATHEMATICAL ANALYSIS WELL?我们怎样才能学好数学分析Mathematics is an integration composed by some abstract concepts, properties and algorithms. For example, geometry is a subject based on an axiomatic system. Similarly, probability theory is based on certain axiomatic system, and topology is embodied by some axiomatic system. Therefore, to learn mathematics well, we need to understand these axioms and the deduces based on them. Just as we need to learn the rules of the game early when playing chess or playing games. It will be easier to learn mathematical analysis or calculus by remembering the concepts and the related rules at first.数学由抽象概念、性质、运算体系构成几何、概率论、拓扑学全部建立在公理体系之上。想要学好数学必须吃透公理与公理推导的全部结论如同下棋、玩游戏要先弄懂规则。学习数学分析、微积分可以先熟记核心概念与对应规则降低入门难度。According to the analysis above, the core of mathematical analysis has its basic framework. Each part of the framework has some concepts, properties and algorithms. So, as a student, mastering the concepts of limit, continuity, derivative and integral for elementary functions, and understand them better. Even if you don’t understand them, you should recite them at first, and then try to use them by doing some exercises. Only in this way, you can cultivate the self-confidence on the learning.结合前文分析数学分析拥有清晰核心框架每个模块配套专属概念、性质、计算方法。学生首先要吃透初等函数的极限、连续、导数、积分核心概念暂时无法完全理解时可先背诵熟记再通过习题实操巩固运用以此建立学习信心。Therefore, to learn mathematics well, we should understand the original and development of these ‘rules’. To speak back, just as the learning of English we must remember the grammar and the necessary number of words, we should remember the concepts and some algorithms at least, including some necessary skills in mathematical analysis. That is to say, we can learn by grasping the main algorithms of the subject, getting rid of some details and proofs. Then we do not be intimidated by some unusual skills.想要学好数学还要理清各类“规则”的起源与演变。类比英语学习需要熟记语法、核心词汇学习数学分析至少要牢记核心概念、计算流程与必备解题技巧。学习时优先掌握主干计算方法暂时搁置细碎细节与复杂证明不必被偏难怪解题技巧打击信心。I usually teach the third part of mathematical analysis. Many contents in the third part are parallel to the first part of mathematical analysis. The main considerations of my first class are to solve the problem of why and how to learn, because some students don’t know what should they learn in mathematical analysis even having one year studying. Therefore, for each new problem or concept, we should always remind the students of the original problems and methods, and what is the difference in the multi-variable analysis. Once again, many calculations in the section three are based on the knowledge and conclusions of that in the section one. Therefore, the conclusions and methods of the section three are divided into the ones that should be memorized as common sense. Of course, it is the best to do so based on understanding.我常年承担数学分析下册授课工作下册大量内容和上册呈平行对应关系。我的第一堂课重点解决“为何学、怎么学”的问题不少学生学完一年依旧不清楚数学分析的学习重点。因此每引入新问题、新概念都会带领学生回顾一元函数对应的原型与方法对比多元场景下的差异。下册绝大多数计算依托上册结论因此下册的公式、方法可以当作常识记忆当然在理解基础上记忆效果最佳。To learn mathematical analysis well, we must overcome the fear of difficulties. Some students say that their foundations in middle school are not solid. As long as we understand the ‘rules’ in mathematical analysis, we can learn them well. We wish all the students can learn mathematical analysis well and lay a good foundation for their future development.学好数学分析首先要克服畏难情绪。不少学生认为自己中学数学基础薄弱但只要吃透数学分析的核心规则完全可以学好这门课。希望所有学生都能扎实掌握数学分析为未来发展筑牢根基。In addition, when learning mathematical analysis, it should be noted that some of the analysis and proof methods, such as constructing auxiliary functions, and proving convergence, were accumulated for several centuries of exploration. Therefore, we should recite the common analysis and proof methods or skills at first. This is the suggestion we got when consulting some great mathematicians from the Chinese Academy of Sciences. After that, we can solve the problem with proper imitation.另外学习过程需要注意构造辅助函数、收敛性证明等分析技巧是数百年数学家逐步摸索积累形成的。笔者咨询中科院资深数学家后得到建议先熟记通用分析方法、证明套路再模仿例题思路独立解题。Finally, we must recognize that learning is a hard work, certain time paid is necessary. According to the statistics of foreign scholars, the ratio used for learning of mathematical analysis is 4:1. That is to say, if we learn one hour in class, and three times more time is needed for previewing, reviewing and doing homework.最后必须认清学习没有捷径需要投入足量时间。国外学者统计得出数学分析课内外时间配比为 4:1课堂学习 1 小时课前预习、课后复习、习题训练合计需要 3 小时。The viewpoints above are some of our experiences, just for your reference only. We hope that all students can learn mathematical analysis or calculus easily, and they can grasp the idea effectively, and they will have a better future in their careers.以上观点均来自一线教学实践仅供同行参考。祝愿所有学生轻松掌握数学分析、微积分吃透核心思维为职业发展铺路。From the viewpoint of applications, the automatic method for calculating the derivatives can be realized by using tracking-differentiators [7]. With the development of artificial intelligence, we can have more chances to simplify the method for complex calculations.从应用层面来说跟踪微分器可以实现导数自动求解[7]随着人工智能技术发展复杂计算的简化工具会越来越丰富。6. CONCLUSIONS结论Mathematical analysis or calculus is one of the key courses of mathematics and many related subjects. Teaching of mathematical analysis plays an important role in mathematics and related disciplines, but many students feel that it is difficult for the learning. On the basis of the author’s teaching experience for many years, this paper puts forward the viewpoints that we can teach of mathematical analysis based on the idea of calculus. Its essence is to carry out the teaching and learning of mathematical analysis with a simple and efficient way.数学分析微积分是数学及众多相关专业的核心必修课在数理类人才培养中地位关键但普遍存在学生学习困难的问题。本文结合作者多年授课经验提出依托微积分核心思维开展数学分析教学的思路该思路的核心是用简洁高效的模式组织课程教与学。ACKNOWLEDGMENTS致谢The authors would like to thank the financial support from the ‘Financial Calculation and Digital Engineering’-the Engineering Research Centre of Ministry of Education of China.本文受教育部金融计算与数字工程工程研究中心项目资助在此致以谢意。REFERENCES参考文献[1] A. D. Alexander, et al., Mathematics: Its Contents, Methods and Significance, vol. I, CCCP, 1956. (Translated into Chinese by Xiaoli Sun, et al., published by Science Press of China, 1984.)A.D.亚历山大等. 数学——它的内容、方法和意义第一卷[M]. 苏联1956孙小礼等译科学出版社1984.[2] M. Kline, Mathematical Thought: From Ancient to Modern Times, vol. I, Oxford Univ. Press, New York, 1972. (Translated into Chinese by Lijing Zhang, et al., published by Shanghai Science and Technology Press, 1981.)M.克莱因. 古今数学思想第一卷[M]. 纽约牛津大学出版社1972张理京等译上海科学技术出版社1981.[3] L. 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Wang, Nonlinear trackingdifferentiators, Systems Science and Mathematical Science, vol. 14, no.2, pp. 177-183, 1994.韩京清王伟. 非线性跟踪微分器[J]. 系统科学与数学199414(2):177-183.[8] W. Wang, Mathematics is not just an abstract subject but an economic and effective methodology: A viewpoint for education reformation, in: Deyao Tan (Eds.), Engineering Technology, Engineering Education and Engineering Management, CRC Press, pp. 151-154, 2015.王伟. 数学绝非纯粹抽象学科而是高效实用的方法论——教改观点[C]//谭德耀主编. 工程技术、工程教育与工程管理CRC出版社151-1542015.ReferenceTeaching of Mathematical Analysis or Calculus Based on the Thinking of Calculushttps://www.atlantis-press.com/article/125956520.pdf(PDF) Teaching of Mathematical Analysis or Calculus Based on the Thinking of Calculushttps://www.researchgate.net/publication/351819287_Teaching_of_Mathematical_Analysis_or_Calculus_Based_on_the_Thinking_of_Calculus