曼士先生的教育理论

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漫士是一名清华大学在读博士生，研究人工智能和理论机器学习，同时也是科普视频创作者。在其分享中，包含了自身考试经历、对教育的反思、费曼的相关例子以及黄金圈理论：

作者考试经历：漫士初三数学考试时，遇到一道求两个抛物线中间阴影区域面积的题目。由于出题人阴影填充方式的问题，这道题并非常规的长乘宽计算，而是要计算抛物弓形面积。其他同学因不会便默认按平时练习的长乘宽来算，但漫士觉得题目有趣。他依据听说的微积分思想，将横坐标0 - 1分成N份，每份宽度为N分之一，用小矩形面积代替该部分面积，宽是N分之一，长是横坐标的平方，把整个面积表示成N个小矩形求和的形式。在计算过程中，由平方求和联想到正方形，进而想到小立方体体积、金字塔，最终联系到四棱锥体积公式（四棱锥体积是1/3底面积乘高，此处底面积为1×1，高为1 ，体积是1/3），由此算出抛物弓形面积为1/3，进而得出整个阴影面积。然而，当他自信交卷后，老师只是打了个叉，未作任何解释。后来漫士学习了微积分，才意识到自己当时在考场上的做法，实际上走过了先贤探索黎曼积分的道路，可自己的探索就这样被扼杀了。
教育反思：漫士认为应试教育存在很大问题。在学习过程中，学生们会产生很多有趣的想法，比如为什么尺规作图不能三等分角、二元一次方程组无解的条件与平行四边形面积公式有什么联系、无限大加一等于什么、自然数的和为何等于 -1/12等。这些本可以成为理解深邃数学知识和思想的窗口，但应试教育却以“考试不考”为由将其扼杀，让学生在重复计算中打转，导致很多人讨厌数学，而有兴趣深入学习的同学也找不到有价值的学习资料。同时，课堂教学方式也不佳，像“孔乙己排开几文大钱”一样，直接将大量定义、定理、证明、技术名词和方法一股脑地抛给学生。学生只能疲于记忆，机械地重复课本上的计算和基本方法，根本不理解这些知识为何存在，以及方法背后的核心精神和原理。这样的学习方式，就如同只知道鸟的各种名字，却对鸟的生活习性、行为本质一无所知，学生学到的只是一些机械的信息和概念，无法转化为有生命力和创造力的创新。
费曼的例子
- 费曼父亲对鸟的讲解：费曼是美国著名物理学家，参与过曼哈顿计划并获得诺贝尔奖。他的父亲曾教导他knowing和knowing the name的区别。比如看到一只鸟，知道它叫brown throated thrush，在德国叫HEYSONFUGA，中文叫棕景东或棕猴冬，但这些只是人类对鸟的称呼。鸟会唱歌、教幼鸟飞翔、夏天能飞越全国好几英里，而人们并不知道它是如何做到这些的。所以，仅仅知道鸟的名字，并不代表真正了解这种鸟。
- 关于“什么让它动起来”的回答：有一本书展示了发条玩具、汽车、骑自行车的男孩的图片，并提出“什么让它动起来”的问题。常规教育方式会从机械学角度讲发条玩具内部弹簧如何工作、有什么弹簧，从化学角度讲内燃机、发动机的运作和部件名字，从生物学角度说肌肉如何工作。但费曼的父亲回答是因为太阳在照耀。费曼不理解，他父亲通过一步步引导，问他发条怎么上紧、他自己如何动起来，最终让费曼明白吃的东西能生长是因为太阳照耀，传达出能量转化的思想。同时，费曼父亲还指出，书里用“能量”解释和用“挖ALAXIS”等随意的词解释没区别，因为只知道这些名字，并不包含有意义的知识或理解。这就像平时上课，老师讲了一堆名字和机械联系让学生背诵，却没讲清楚名字背后真正的原理和知识，所以学生考完试就容易忘。
黄金圈理论：由著名学者SIMONSENIC克提出，该理论指出想要完全理解一件事情，有三个层面。最表层是“what”，指具体的定义、概念、方法、标准；中间层是“how”，即这些具体的概念和方法是如何产生的；核心层是“why”，比如为什么会有对偶空间、编程中为何提出继承和多态的概念、为什么要研究随机变量的特征函数、函数极限的定义为何是这样等。只有能很好地回答这些“why”的问题，才算真正理解一个学科，并且能够将这种知识转化为未来可以运用的素养。可惜的是，现在的课堂和辅导机构在这方面做得远远不够，大多只注重“what”，很少涉及“how”，几乎没有“why”。漫士希望自己的视频能回答这些“why”的问题，填补空白，讲清各学科知识体系背后的精神思想和联系。

Man Shi is a Ph.D. candidate at Tsinghua University, researching artificial intelligence and theoretical machine learning. He is also a science popularization video creator. In his sharing, he included his own exam experience, reflections on education, examples related to Feynman, and the Golden Circle Theory.

The Author’s Exam Experience: During Man Shi’s junior high school math exam in grade three, he encountered a question asking for the area of the shaded region between two parabolas. Due to the way the examiner filled in the shaded area, this question was not a regular length-times-width calculation but required calculating the area of a parabolic segment. Other classmates, not knowing how to solve it, defaulted to using the length-times-width method as they did in usual exercises. However, Man Shi found the question interesting. Based on the calculus ideas he had heard about, he divided the abscissa from 0 to 1 into N equal parts, with each part having a width of 1/N. He used the area of a small rectangle to approximate the area of that part, where the width of the rectangle was 1/N and the length was the square of the abscissa. In this way, he expressed the entire area as the sum of N small rectangles. During the calculation, from the sum of squares, he thought of a square, then the volume of small cubes, a pyramid, and finally related it to the volume formula of a square pyramid (the volume of a square pyramid is 1/3 of the base area times the height. Here, the base area is 1×1 and the height is 1, so the volume is 1/3). Thus, he calculated that the area of the parabolic segment was 1/3, and then obtained the area of the entire shaded region. However, when he confidently handed in the test paper, the teacher only marked it wrong with an “X” and did not give any explanation. Later, after learning calculus, Man Shi realized that what he did in the exam actually retraced the path of the predecessors’ exploration of Riemann integration, but his exploration was stifled like this.
Reflections on Education: Man Shi believes that there are significant problems with the exam-oriented education system. During the learning process, students come up with many interesting ideas, such as why it is impossible to trisect an angle using only a compass and straightedge, what the connection is between the condition for a system of binary linear equations to have no solution and the area formula of a parallelogram, what is the result of infinity plus one, and why the sum of natural numbers is equal to -1/12. These could have been windows to understanding profound mathematical knowledge and ideas. However, the exam-oriented education system stifles these explorations with the excuse of “not being tested in the exam,” making students get stuck in repetitive calculations. This causes many people to dislike mathematics, and students who are interested in in-depth study cannot find valuable learning materials. At the same time, the classroom teaching method is also unsatisfactory. It’s like “Kong Yiji laying out a few big coins,” directly bombarding students with a large number of definitions, theorems, proofs, technical terms, and methods. Students can only struggle to memorize and mechanically repeat the calculations and basic methods in the textbooks. They根本 do not understand why such knowledge exists or the core spirit and principles behind these methods. This learning method is like only knowing the various names of a bird but having no idea about its living habits and behavioral essence. Students only learn some mechanical information and concepts and cannot transform this knowledge into creative and innovative power.
Examples of Feynman
- Feynman’s Father’s Explanation of the Bird: Feynman was a famous American physicist who participated in the Manhattan Project and won the Nobel Prize. His father once taught him the difference between “knowing” and “knowing the name.” For example, when seeing a bird, knowing that it is called “brown throated thrush,” “HEYSONFUGA” in German, and “Zong Jingdong” or “Brown Monkey Winter” in Chinese, these are just human names for the bird. The bird can sing, teach its chicks to fly, and fly across the country for several miles in summer, but people don’t know how it does all these things. So, just knowing the name of the bird does not mean truly understanding this kind of bird.
- The Answer to “What Makes It Move?”: There was a book that showed pictures of a wind-up toy, a car, and a boy riding a bicycle and asked the question “What makes it move?” The conventional educational approach would explain from the perspective of mechanics how the internal spring of the wind-up toy works and what kind of spring it is, from the perspective of chemistry about the operation of internal combustion engines and engines and the names of their components, and from the perspective of biology about how muscles work. However, Feynman’s father answered that it was because the sun was shining. Feynman didn’t understand. His father guided him step by step, asking him how the spring was wound up and how he himself moved. Eventually, Feynman understood that the food he ate could grow because of the sun’s irradiation, thus conveying the idea of energy conversion. At the same time, Feynman’s father also pointed out that explaining with the word “energy” was no different from using random words like “waALAXIS” because just knowing these names did not contain meaningful knowledge or understanding. This is just like in normal classes. Teachers teach a bunch of names and mechanical connections for students to memorize but do not explain clearly the true principles and knowledge behind these names. Therefore, students tend to forget quickly after the exam.
The Golden Circle Theory: Proposed by the famous scholar SIMONSENIC, this theory points out that to fully understand something, there are three levels. The outermost layer is “what,” referring to specific definitions, concepts, methods, and standards. The middle layer is “how,” that is, how these specific concepts and methods came into being. The core layer is “why,” such as why there is a dual space, why the concepts of inheritance and polymorphism are proposed in programming, why we study the characteristic function of random variables, and why the definition of the function limit is like this. Only by being able to answer these “why” questions well can one truly understand a subject and transform this knowledge into qualities that can be applied in the future. Unfortunately, current classrooms and tutoring institutions do not do well enough in this regard. Most of them only focus on “what,” rarely touch on “how,” and almost never address “why.” Man Shi hopes that his videos can answer these “why” questions, fill in the gaps, and clarify the spiritual ideas and connections behind the complex knowledge systems of various disciplines.