每天晚上，雷打不动有1个多小时陪孩子学习数学和编程。昨天晚上，小树遇到一个数学题不会做：
Every evening, without fail, there is more than an hour spent accompanying the child in learning mathematics and programming. Last night, Xiaoshu encountered a math problem he couldn't solve:

思考的阻塞，来自知识的漏洞 The blockage in thinking comes from gaps in knowledge.

学校里还没学过 ，孩子不知道这个（平方差）公式可以（因式分解）/拆解为 。这个题不会做是正常的。
This hasn't been taught in school yet, and the child doesn't know that this (difference of squares) formula can be (factored) / broken down into. It's normal not to be able to solve this problem.

毕竟，思考的阻碍，几乎都来自知识的漏洞。这是思考的第一只拦路虎：知识漏洞。
After all, the obstacles to thinking almost always come from gaps in knowledge. This is the first roadblock to thinking: knowledge gaps.

于是，我现场给她推导了这个公式（用两个正方形相减的方式，直观，具体，还能得出通用结论）。我说，这个公式不要死记硬背，一定要自己推导。以后遇到所有的公式，都必须会自己推导。如果不能推导，说明我们不理解 why。你可以记住 ，但这是不对的，记住不等于理解，所以记住不是真的掌握，必须理解知识背后的 why。
So, I derived this formula for her on the spot (using the method of subtracting two squares, which is intuitive, specific, and can lead to a general conclusion). I said, do not memorize this formula by rote, you must derive it yourself. In the future, for all formulas you encounter, you must be able to derive them yourself. If you cannot derive them, it means we do not understand why. You can remember it, but that is not correct, remembering is not the same as understanding, so remembering is not truly mastering; you must understand the why behind the knowledge.

解题思路哪里来？ Where does the problem-solving approach come from?

然后，小树发现，这等于 ，每个数字之间差 4，是个等差数列。然后又呆住了：这下可怎么办？
Then, Little Tree discovered that this equals, each number has a difference of 4, which is an arithmetic sequence. Then he was stunned: what should he do now?

等差数列求和又是一个公式。和平方差因式分解一样，这两个知识点都是初中二年级才学的。但是，它确确实实出现在小学四年级的课外练习册里面了（这方面，教培集团们真的是“功在当代，利在千秋”啊！）。
The sum of an arithmetic sequence is another formula. Like the factorization of the difference of squares, these two knowledge points are only taught in the second year of middle school. However, it indeed appeared in the extracurricular exercise book for the fourth grade of elementary school (in this regard, the education and training groups are truly "beneficial to the present and advantageous for future generations" indeed!).

解题时，我们会有好几个思路。小树尝试一个思路（孩子想快速解题，顺着本能会一路狂奔），然后遇到了问题，然后就卡在那里了。
When solving problems, we have several approaches. Little Tree tried one approach (the child wanted to solve the problem quickly and instinctively rushed forward), then encountered a problem and got stuck there.

于是，孩子遇到了思考的第二只拦路虎：思路阻塞。
So, the child encountered the second roadblock in thinking: thought blockage.

如何得到思路？ How to get the thought process?

我让她不要思考的太快。思考太快，卡到弯路里就回不来了。数学和编程一样，这些学科的知识非常少，所以不求快，不要着急，关键是思考的过程。
I told her not to think too fast. Thinking too fast, if you get stuck in a detour, you can't come back. Like mathematics and programming, these disciplines have very little knowledge, so don't rush, don't be anxious, the key is the process of thinking.

现在，我们一步一步思考，think step by step。你先把解题过程一步一步写下来。
Now, we think step by step, think step by step. First, write down the problem-solving process step by step.

小树开始在草稿纸上写步骤。 The little tree began to write the steps on the draft paper.

• 第一步： Step one:
• 第二步： Step two:

我说停！现在，放下笔。坐好了，不着急往下赶，对着这个题目，盯着它看，一直看，直到看懂这个题目，看出这个题目中的规律。
I say stop! Now, put down the pen. Sit properly, don't rush to move on, look at this problem, stare at it, keep looking until you understand this problem, and see the pattern in this problem.

然后，娃就乖乖盯着题目看。看了一会，她就笑了：不过就是从 1 加到 100,一共 100 个数，答案是 5050 嘛。
Then, the child obediently stared at the problem. After a while, she laughed: It's just adding from 1 to 100, a total of 100 numbers, the answer is 5050.

本以为这个题目很难，是个吃人的大老虎，原来不过是一只纸老虎啊。
I thought this problem was difficult, like a man-eating tiger, but it turned out to be just a paper tiger.

关于解题，关键是学习、练习并掌握解决一切问题的“解题元技能”。之前用万字长文写过这个问题，此处不赘述。有需求移步：《解题元技能》。（这个文章写的时候设置了付费。现在以《费曼学习法训练营》的形式集体学习中，有需求的话，可以点击“阅读原文”加入。）
Regarding problem-solving, the key is to learn, practice, and master the "meta-skill of problem-solving" to tackle any issue. I have previously written a lengthy article on this topic, so I won't elaborate here. If you're interested, please refer to: "The Meta-Skill of Problem-Solving". (This article was set as paid content when written. It is now being collectively studied in the form of the "Feynman Learning Method Workshop." If interested, you can click "Read Original" to join.)

思考的乐趣 The Joy of Thinking

费曼在小时候和爸爸一起去树林里看鸟。费曼爸爸说，你知道用各种语言称呼这只鸟的名字，但是，知道名字并不等于理解。现在，忘掉这些名词，真正观察一下这只鸟。
When Feynman was a child, he went to the woods with his father to watch birds. Feynman's father said, you know the names of this bird in various languages, but knowing the name does not mean understanding. Now, forget these nouns and truly observe this bird.

我经常对小树说，读题，读题，再读题！我们不要觉得读题目浪费时间，觉得自己题目读懂了。不要看一眼就以为自己读懂题目了。解题只分四步，而第一步就是读题，真正读懂题目，读到不看题目能讲出来，讲的很清晰的程度。
I often tell the little tree, read the question, read the question, and read the question again! We shouldn't feel that reading the question is a waste of time or think that we have understood the question. Don't assume you understand the question after just a glance. Problem-solving is divided into four steps, and the first step is to read the question, truly understand it, to the point where you can explain it clearly without looking at the question.

现在，放下笔，看看这个题目，盯着这个题目看。这个题目到底在问什么？它告诉了我们什么信息？题目里包含了几条信息？每一条信息都要用自己的话说出来。每一条信息，到底告诉了我们什么？
Now, put down your pen and look at this question, stare at it. What exactly is this question asking? What information does it provide us? How many pieces of information are contained in the question? Each piece of information should be expressed in your own words. What exactly does each piece of information tell us?

解题时，当我们真正把题目读进去了，读懂了，后面的解题过程就容易了。我们会遇到知识欠缺、思路僵化导致的思考阻塞，但是，这都是可以通过知识学习和解题练习来稳步解决。
When solving problems, once we truly understand the problem by reading it thoroughly, the subsequent solving process becomes easier. We may encounter thinking blockages caused by lack of knowledge and rigid thinking, but these can be steadily resolved through learning and practice.

聊完了这些。然后，我们又一起聊了一个衍生的题目：  算出来之后，再算一下：
After discussing these, we then talked about a related problem: After calculating it, calculate it again:

因为题目比较简单，我们就玩心算加口算。我们坐在一起，用手操作空气中的假想的数组，用嘴说出思考的过程，嘻嘻哈哈地在扯淡中就把题目给聊了，一起快乐地度过了今天的解题时光。
Because the problem was relatively simple, we played with mental arithmetic and oral calculation. We sat together, using our hands to manipulate imaginary arrays in the air, and spoke out our thought processes, chatting and laughing as we discussed the problem, happily spending today's problem-solving time together.

然后，她又写了两个 C++程序，还是两个解题，只不过用编程语言这种形式简洁、强调秩序的人工语言。还练习了一下盲打，字母模块已经达到熟练水平，编程打字模块已经达到入门一级。
Then, she wrote two C++ programs, both solving problems, but using the concise and orderly form of a programming language. She also practiced touch typing, reaching proficiency with the alphabet module, and achieving beginner level in programming typing.

然后，娃心满意足，快快乐乐地结束了今天的亲子学习。
Then, the child was satisfied and happily concluded today's parent-child learning session.

来自实例的几条启发 A few insights from the example

不要教，要prompt Don't teach, prompt

父母陪孩子学习，不要走教培模式：不要代替孩子思考，不要帮孩子思考。父母只应该做一件事：prompt（提示）。
When parents accompany their children in learning, they should not adopt the teaching and training model: do not think for the child, do not help the child think. Parents should only do one thing: prompt.

父母可以提供脚手架，把思考问题转化成孩子稍微努力一下就能够得着的程度，转化为孩子在最近学习区范围的思考。整个过程中，父母只是提供 prompt，而非灌输答案、教授僵化孩子思维的解题思路。
Parents can provide scaffolding, turning problem-solving into something the child can reach with a little effort, converting it into thinking within the child's zone of proximal development. Throughout the process, parents only provide prompts, rather than instilling answers or teaching rigid problem-solving methods that stifle the child's thinking.

你提供 prompt，孩子就可以主动思考；你提供答案，孩子就停止思考，而且记不住你的答案；你提供解题思路，孩子继续停止思考，她会以为自己思考了，以为自己会做了，然后下次遇到时仍然不会。因为，没有经历自己的主动思考过程，她不知道背后的 why，于是连带着 what 和 how 都记不住。
You provide a prompt, and the child can think actively; you provide an answer, and the child stops thinking and cannot remember your answer; you provide a problem-solving method, and the child continues to stop thinking, believing they have thought it through and can do it, but when faced with it again, they still cannot. Because they have not gone through their own active thinking process, they do not know the underlying why, and thus cannot remember the what and how either.

父母不要用成年人的知识积累和抽象思考能力去要求孩子，去灌输你觉得很简单很明显的东西。父母不要以为自己会教，去扮演老师或教培的角色。我个人从来不敢、也不屑于去做”教“的工作。如果父母有这样的习惯和行为模式，只是因为每个人都是有局限的，而父母没有认识到自己的局限性。
Parents should not use the knowledge accumulation and abstract thinking ability of adults to demand from children, to instill things that you think are simple and obvious. Parents should not assume they can teach and play the role of a teacher or educator. Personally, I never dare, nor do I care to do the work of "teaching." If parents have such habits and behavior patterns, it is only because everyone has limitations, and parents have not recognized their own limitations.

清晰思考之难 The difficulty of clear thinking

写作等高级认知活动之难，难在清晰思考是困难的。而思考之难，在于思考经常被阻塞。要么是因为背景知识有漏洞，要么是解题思路僵化。
The difficulty of advanced cognitive activities such as writing lies in the fact that clear thinking is difficult. And the difficulty of thinking is that it is often blocked. Either because there are gaps in background knowledge or because the problem-solving approach is rigid.

不论成年人还是孩子，因为当前个人知识体系的局限性，一定会遇到阻塞。所以，要想清晰思考，一定要基础牢靠。在稳固的知识基础上，概念清晰准确，逻辑推理、复杂思考才有可能发生。
Whether adults or children, due to the limitations of their current personal knowledge system, they will inevitably encounter blockages. Therefore, to think clearly, a solid foundation is essential. On a solid knowledge foundation, clear and accurate concepts, logical reasoning, and complex thinking can occur.

孩子在解题中，在思考上遇到问题，父母光喊口号（你要思考啊！你要深度思考独立思考清晰思考！你要努力思考啊！）（你怎么不动脑子呀！动呀，动呀！），越这样，孩子越会对着题目发呆。
When children encounter problems in solving questions or in thinking, if parents just shout slogans (You need to think! You need to think deeply, independently, and clearly! You need to think hard!) (Why aren't you using your brain? Use it, use it!), the more they do this, the more the child will stare blankly at the problem.

当然，还有的父母喜欢拍桌子，喜欢生气。这会导致孩子的应激反应，长期下去，不但无法找到思考的乐趣，反而会陷入抑郁。
Of course, some parents like to slam the table and get angry. This can trigger a stress response in children, and over time, not only will they fail to find the joy of thinking, but they may also fall into depression.

孩子解题时遇到思考阻塞，唯一的办法是让孩子给父母讲题（费曼学习法）：讲自己对题目本身的理解，讲自己的解题思路。父母听孩子讲解，一方面自己可以借机了解下题目（孩子做题时，父母应该在旁边读自己的书），听的过程中父母提出一些问题（这个题目什么意思，我怎么还没看懂？你这一步为什么要这么做，我还没想明白？），借机组织自己的解题思路；这个过程中，父母会发现孩子解题时遇到阻塞的根本原因，到底是哪个知识点欠缺，那个环节把自己绕进去了。
When a child encounters a mental block while solving a problem, the only solution is to have the child explain the problem to the parents (Feynman learning method): explain their understanding of the problem itself and their problem-solving approach. As parents listen to the child's explanation, they can take the opportunity to understand the problem (while the child is solving the problem, parents should read their own book nearby). During the listening process, parents can ask some questions (What does this problem mean? I still don't understand it. Why did you do this step? I haven't figured it out yet?), and organize their own problem-solving thoughts. In this process, parents will discover the root cause of the child's mental block in problem-solving, which knowledge point is lacking, and which part got them stuck.

就像医生看病一样，诊断了病因，然后对症下药就容易了：一步步引导孩子主动思考即可。
Just like a doctor diagnosing an illness, once the cause is identified, prescribing the right remedy becomes easier: guide the child to think actively step by step.

ChatGPT作为思考教练 ChatGPT as a thinking coach

有父母说，我不会啊，小学题目我也不会做。 Some parents say, I can't do it either, I can't solve elementary school problems.

这不是借口，两年前你可以拿“不会”当借口，现在不行了，因为 ChatGPT 发布快两年了！如果父母不会引导孩子思考，甚至也无法识别孩子在解题中在什么环节遇到了什么阻塞，那么，让 chatgpt 来引导。
This is not an excuse, two years ago you could use "I don't know how" as an excuse, but not anymore, because ChatGPT has been out for almost two years! If parents cannot guide their children in thinking, or even cannot identify where their children encounter obstacles in problem-solving, then let ChatGPT guide them.

以刚才那道题为例，事实上，chatgpt 提供了比更好的解题思路，更好版本的引导。普通数学题，用 gpt-4o 模型足够；上难度的题目，用国际奥数铜牌水平的 o1 模型也足够了。
Taking the previous problem as an example, in fact, ChatGPT provides a better problem-solving approach and a better version of guidance. For ordinary math problems, the GPT-4o model is sufficient; for more difficult problems, the o1 model at the level of an international math olympiad bronze medalist is also sufficient.

AGI时代的孩子，学习到底是学什么？ In the era of AGI, what exactly are children learning?

在 AGI 时代，学习的目标不是某个具体的知识和技能（平方差公式、因式分解、等差数列），而是思考的习惯和能力。这个世界，愈发分裂为思考者与不思考者这两个阶层。
In the AGI era, the goal of learning is not specific knowledge and skills (such as the difference of squares formula, factorization, arithmetic sequences), but the habit and ability to think. This world is increasingly divided into two classes: thinkers and non-thinkers.

父母与孩子一起学习，孩子到底在学什么？
Parents and children learning together, what exactly are children learning?

只学习一件事：主动思考，独立思考，深度思考，清晰思考。
Learn only one thing: proactive thinking, independent thinking, deep thinking, clear thinking.

父母与孩子一起学习，父母到底在干什么？
Parents learn with their children, what exactly are the parents doing?

父母不是化身教培老师，给孩子灌输死的知识，灌输惰化孩子思维的解题套路。
Parents are not substitute teachers, instilling dead knowledge into children, or teaching problem-solving routines that dull their thinking.

父母只要做一件事，只应该做一件事：prompt。 Parents only need to do one thing, and should only do one thing: prompt.