(This is Chapter 4 of the book, in draft form, about teacher Xu. I wrote my recent book "Unbalanced: Memoir of an Immigrant Math Teacher" to tell a story that is similar to his, but in the post-pandemic era. As a math teacher, I was unable to accomplish what Mr. Xu had, but I want to explore what we can learn from a unique and successful teacher, especially on how to teach math effectively in this country.

This chapter elaborate on an approach that Mr. Xu often uses in his class, to say, "I don't know" and encourage students to say it. For a teacher to say, "I don't know", in fact for anyone to say it, is a sign of humbleness and openness that often lead to deeper learning. I hope people would do it more often.)

(travel photo from Dolomites, 2023)

Chapter 4: The "I Don't Know (IDK)" Method

Not long after the first state standardized test, the school district organized a math competition with two divisions: honors and regular. Each division had both individual and team awards. The honors classes were made up of the strongest math students in each school, and their curriculum was far ahead of the regular classes. I was teaching a seventh-grade regular class. Although we had not prepared specifically for the competition, my students outperformed the honors students by a wide margin. To make the comparison less embarrassing, the organizers combined the two divisions into one. My students won first place as a team, as well as the individual gold and silver medals. The top honors student received the bronze medal. I did not mind, but I suspected that bronze medal was awarded partly to save face for the honors class, because if all the awards had gone to the regular class, it would have been too humiliating.

After my students did well on both the state test and the competition, other schools gradually began approaching me and saying, "Come teach at our school." At that time, I was teaching only regular classes. To attract me, some schools promised that I could teach honors classes and the Gifted Program, which was then being established. Gifted programs are common now, but at the time Arizona was just beginning to create them, and students had to pass an entrance exam. Not many qualified, and some who did qualify did not want to go, because they would have had to leave their circle of friends and take a bus to another school, which was inconvenient. In the end, the school that recruited me was able to open only one gifted class. After I transferred, I taught students across a wide range of levels, in gifted, honor, and regular classes.

The head of the gifted program at the new school did me a favor. Teaching gifted classes required a special certificate. According to the rules, I still needed 15 additional credits, but teachers who had taught gifted classes for more than three years and whose students had performed satisfactorily could use their practical experience to prove their qualifications. At the time, I had already taught regular classes for more than three years, and my students' performance fully met the graduation standards for the gifted program. He wrote a special letter for me stating that I already had three years of qualified teaching experience. Because of that, I did not need to earn the extra 15 credits and was able to obtain the gifted teaching certificate directly.

Later, when I retired, that same program director said to me, "At the time, only half the students in your gifted class were actually qualified. The other half were put into your class at the last minute because we could not recruit enough qualified students. I'm sorry I never told you. But it turned out all right — by the time they graduated, every one of them had reached the level expected of gifted students." No matter what school I taught in, or what level students were at when they entered my class, once they were in my class, by the time they graduated they usually reached the level of gifted students.

***

The principal at the new school did not understand math, so he mostly let me teach as I saw fit and did not interfere much with my unusual teaching methods. Two years after I joined the school, a new assistant principal was assigned there. He had previously been the district's top person in charge of training math teachers, and he believed that no one understood math — or how to teach it — better than he did. He believed that 80% of K-12 teachers did not understand math at all, and that only 20% understood at least a little. When the principal introduced me to him, he said half-jokingly, "This one doesn't like writing lesson plans, doesn't do things by the book, and isn't easy to manage." So, from the very beginning, the assistant principal placed me among that 80%. He thought I was disobedient, unreasonable, and the sort of troublemaker who did not understand what he was doing yet still refused to listen.

As for lesson plans, I always looked at my students' interest and level during the class, then decided what to teach next and how to teach it. Each lesson was adjusted on the spot according to how the previous one had gone. I had two kinds of lesson plans: the formal kind the school required, and my own informal lesson plan that followed the students' progress. I wrote my real lesson plan only after the previous class ended. Every day after class, I would jot down three kinds of notes, e.g., this student now understood 7/21; that student had reached a certain point in multiplication; another student was stuck at a particular point in understanding.

I could not remember the students' English names, so I used their seating chart. Under each seat I drew ten little boxes, and every day I recorded progress in those boxes. One seating chart lasted about two weeks. That way, what I needed to review in the next class, and how I needed to review it, all came from what had actually happened that day. Even when the content was exactly the same, each class moved at a different pace. Students in the first class in the morning and the last class of the day were never in the same mental state, and their learning was different too. Some examples worked wonderfully in Class A but got no response at all in Class B, in which case I had to switch examples. So my real lesson plans were not something I could write one or two weeks in advance.

The assistant principal wanted everyone to follow the rules and write lesson plans in a fixed format for every class. Each lesson plan had to include 13 categories, all built around a structure like "I do / We do / You do," and required every day. It reminded me of the old rigid eight-legged essay in China: formal, formulaic. But while many teachers taught only one subject, I taught many different classes — seventh grade, eighth grade, gifted, honors, regular — a total of six different kinds of classes. If I had to write six such elaborate lesson plans every day, I would have had no time left to do anything except produce reports for administrators showing what I was supposedly doing each day! That was why I had no enthusiasm for writing them.

The assistant principal was very dissatisfied with my lesson plans. He sent me an official letter of direction saying that if I did not improve, he would fire me. Although dismissing a public-school teacher is not actually easy, the letter still put me under a great deal of pressure. We argued for a long time, and I never gave in. I kept trying to reason with him. In my reply, I wrote: "I understand your reasoning, but you may not understand mine. I believe my way of teaching is better for students; the results speak for themselves. If you can prove that I have harmed students or the district in any way, I will do as you say. Otherwise, I will continue teaching in my own way."

In the end, he decided to come observe my class himself to see exactly how I taught, hoping to gather firsthand evidence against me. During class, when a student asked, "How do you do this problem?" I said, "I don't know. What do you think?" I treated every student that way, asking them again and again. Some gave answers that were all over the place. Some said things that had nothing to do with the problem. Others came close. In all these cases, I did not rush to correct them. I refused to give them the answer directly.

Then one student would ask another, that one would ask a third, and the students would argue back and forth until, amazingly, they figured the problem out. The assistant principal sat there watching, increasingly anxious. According to his old habit, when students asked a question, the teacher should answer immediately, provide the method, and lead students through it step by step. But what he saw me doing was refusing to answer, refusing even to correct mistakes right away, allowing students to be wrong, allowing them to argue, and letting them reason their way to the result themselves. Later he told me he had been so anxious that he nearly jumped in and taught the lesson for me. But he held himself back, because he wanted to see just how bad I really was, and why I would not answer students' questions. What he saw in the end was this: the students not only figured out the answer themselves, they actually understood it.

After that class, he said to me, "I find it hard to believe you could hold back that long without helping the students." He had to admit that I had a point. My method opened his eyes. Although our conflicts would continue, he let me off for the time being and stopped making my life difficult over lesson plans.

I told the assistant principal that the core of my teaching method could be summed up in one phrase: "I don't know (IDK)." I treat "IDK" as a badge of honor. When a student asks me a question, I typically answer "IDK" and asked the student, "What do you think?" As long as a student gave an honest answer and a reason — even "because today is Friday" — I gave him credit, because I believed that no one is born wanting to be foolish. Everyone wants his or her brain to work. In my classroom, I graded only one thing: whether the student had really engaged in thinking.

***

In most classrooms, teachers never say "I don't know" (IDK), because they would feel it's embarrassing. IDK is something only students say — usually the weaker students — and they say it with a sense of apology and self-blame. In my classroom, not only do I say IDK, I actively encourage students to say it often. When a student says IDK, there is no need for shame — on the contrary, it should be said with the pride of someone who is being honest, responsible, and genuinely eager to learn. IDK is the prerequisite for wanting to find out, the first step toward finding an answer and mastering a skill.

After a student says IDK, I will ask for additional thoughts, inferences, questions, estimates, and guesses — even wild guesses are welcomed and encouraged. I'll ask the student to share what the best guess or estimate might be, what the second most likely possibility is, and even what a completely groundless guess might be. For all of these answers, even the wildest guesses, I give sincere encouragement and appreciation — praising the effort they're putting into thinking. After the students raise their question, I'll ask the whole class: who can help, or who has a better inference or guess? The ideal classroom situation is one where nobody is completely sure, but through lively back-and-forth discussion, the answer gradually becomes clearer and more accurate — sometimes arrived at through several different methods.

Even if the class collectively arrives at a wrong conclusion, I still won't step in and give the correct answer. Instead, I guide them through related problems — what's called "scaffolding." I am very patient; sometimes I'll even wait until the next class, letting students discover the issue themselves while working on similar exercises that give contrasting results. Then I'll say, "Is that so? Why?" and have the class discuss together whether the answer is 7 or 8. In the vast majority of cases, when the right and wrong, the good and the flawed are placed side by side, students are capable of making the right judgment themselves.

Finally, I ask students to explain their reasoning. If they can't explain it clearly, I'll act as though I still don't understand — until they've explained it completely and clearly, at which point I say, "Wow, so that's really how it works!" This way, what students feel is not frustration but a sense of achievement — delight, not discouragement, and not hollow praise either. When students work through things themselves, find their own answers, and experience the joy of their own success, they come to love mathematics and their confidence grows: I can actually can learn math — I am smart enough!

Most math problems have some connection to things previously learned, or can be guided toward through what's already known to students. If everyone is completely lost, then the problem itself must have been poorly chosen — it lies beyond the zone students are able to explore. In that case I will say, "This is something entirely new, so nobody knows it yet." Even then, I won't give away the answer or have students look up the standard answer in a textbook. I'll send them home to find out for themselves, and we'll discuss it the next day.

***

The ultimate sign of success in my classroom is this: did the students enjoy the process of thinking? Discussion is graded on effort alone, not on right or wrong. A student who offers any possible answer passes; a second attempt earns a B; a third attempt earns an A — none of it depends on whether the answer is correct. The classroom norm has been discussed and are well known to all students. There are three things that result in a failing mark: 1. Waiting, 2. Hiding, 3. Pretending — waiting for someone else (including the teacher) to give the answer and then quickly copying it down; hiding to avoid being called on; pretending to know when you actually don't. When students know they can say IDK without shame and that they must not wait, hide, or pretend, their learning becomes active, genuine, and deep.

In the age of AI, the primary purpose of math education is not to have students master a body of knowledge or memorize standard problem-solving procedures — nor even to train them in a technical skill for scientific research and engineering innovation. It is to help them learn how to think, and in that process build their confidence and bring them joy. Confucius said this two thousand years ago: "Those who know it are not as good as those who love it; those who love it are not as good as those who delight in it." The teacher's goal is not merely to help students acquire knowledge, but to help them learn as independently as possible — and more than that, to make them feel happy, to believe they are capable of learning, to make them feel that math really suits them. The love of learning will grow from that feeling.

Teachers and parents are often far too eager, desperate to pour all their knowledge into students as fast as possible, while relishing how knowledgeable and helpful they are. But this robs students of the pleasure and sense of achievement that comes from the learning process. It misses the central purpose of mathematics education in the AI age: the development of the intellect, not the acquisition of knowledge or skills. Under the largely compulsory, top-down methods used in most schools today, students are forced to learn with little interest or love, perpetually in a mode of simply fulfilling some duties. In such a state, there is little cultivation of creativity, because creativity requires relentless questioning, probing, and deep investigation. Without love for the subject, it is very hard to enter that kind of state.

***

Many teachers are skeptical of my approach. They believe that letting students write three letters — IDK — and call it done will encourage students not to try, just to coast. Writing three letters takes almost no effort, and it makes it easy for unwilling students to slack off. This concern has some merit in the context of literary composition — even if you can't write, you push yourself, and you can usually squeeze something out. But in mathematics, especially in elementary arithmetic, if you don't understand, you simply can't squeeze out anything useful. If you do understand, writing it down is easy. Often IDK is the most honest answer. When students are willing to say so — and receive the teacher's respect and trust in return — they would be more motivated to learn.

I do have one requirement: for every assignment where a student writes IDK, they must ask someone else until they understand at least one of those problems. I tell students: "Not understanding this" is the first step toward learning it. I help students accept that IDK is nothing shameful, and "I know" is nothing to be smug about. We are all students, because we came here precisely because there are things we don't yet know. What students do every day is discover what they don't understand and then come to understand it.

I also say that those who understand sooner learn faster, those who understand later learn more slowly — it's just a difference in how individuals learn, but understanding is understanding. I invite students to look at it from another angle: those who learn quickly may also forget quickly, while those who learn slowly often forget slowly too. Compared to someone who looks bright and gets it right away, someone who looks slow, who tries three different approaches before finally getting it, knows two paths that don't work. Therefore, their knowledge is richer and more solidly built.

Any test or quiz where students are only doing problems and getting marked right or wrong is basically a waste of time. Every quiz and test I give is an opportunity for students to discover where they don't understand, and then come to understand it, often through discussion. For a student to discover what they've figured out, what they've learned, how much they've grown: that is an extraordinarily joyful experience, and one of the goals of my teaching. I don't focus on whether homework answers are right or wrong. I won't leave comments like "This is wrong", I'll put a question mark and say, "That is interesting. Tell me how you got it." This way, the teacher doesn't need to go through every problem marking right and crossing out wrong, because every cross is a slap to the student, deeply damaging their interest and confidence, and silently eroding their motivation to learn.

***

From my very first year of teaching, I have had a particular love of observing other teacher's classes. When I began teaching eighth-grade physics in China, I often sat in other teachers' lessons, including lessons in physics and other subjects. Later, when I majored in language and literature at university, I decided I would study education afterwards. I made a point of attending almost every teacher's classes — well-known and unknown, full professors and junior instructors alike. Every lesson I attended, I made sure to identify several strengths and several weaknesses. Even for the best teachers, I had to be able to point out a flaw. If you can't do that, it means you haven't truly understood it. Having observed so many classes, I became better able to see the classroom from a student's perspective, because I observed as if I was a student.

In the United States, opportunities to observe classes are rarer, perhaps because everyone is busy. Aside from the assistant principal, only a very small number of teachers have come to observe my class. What follows is a report written by one of those teachers after observing my class:

Teaching Approach:

Mr. Xu designed the lesson by giving students a set of problems with a wide range of difficulty levels. Each student marked IK ("I Know") and IDK ("I Don't Know") for each problem. The lesson began by focusing on what students didn't know, while also reviewing what they thought they already knew. Interestingly, some students discovered mistakes in problems they believed they could solve, leading to deeper learning moments.

Rather than explaining answers directly, Mr. Xu guided students through questioning, prompting them to reread, reflect, and think critically until they reached their own solutions. He encouraged students to raise their hands, share their thinking, and feel comfortable expressing uncertainty. Wrong answers were welcomed as learning opportunities, creating an open and respectful classroom environment.

What I Found Inspiring:

This was a teaching style I had never observed before. It encouraged students to be proud of both what they knew and what they didn't know. Every student was empowered to share ideas without fear of being "wrong." Students learned by brainstorming individually and collaboratively, while the teacher talked less and the students talked more — keeping the entire class engaged and active. No one was left behind.

I especially liked the system where IK or IDK responses still allow students to earn an A, because IDK becomes the starting point for individualized instruction. This ensures lessons meet students where they are and helps every student progress. I also appreciated the idea that students "fail" only if they disengage, hide, or pretend to understand — rather than based on getting wrong answers.

Classroom Impact: Students were visibly excited during "aha!" moments and stayed fully engaged from start to finish. Mr. Xu praised both correct and incorrect answers, then let the class collectively determine which was correct. This kept the entire group thinking, not just the students who solved the problems first.

Questions & Considerations:

· How might this approach be adapted for English Learners who may not fully understand the problem statements? Could peer explanation in small groups help?

· Since there were many questions but limited class time, how could students continue working on problems they are still struggling with after class?

· When worksheets are collected and replaced with new ones the next day, how can students revisit unresolved questions to deepen their understanding?

Overall Impression:

The lesson was engaging, empowering, and student-centered. I left feeling inspired by the way Mr. Xu created a classroom where mistakes were part of learning, and participation was valued over perfection.

中文版

第四章 "我不知道（IDK）"教学法

第一次州统考之后不久，学区组织了一次数学竞赛，分荣誉班和普通班两个级别，每个级别有个人奖和集体奖。荣誉班集中了全校数学成绩最好的学生，学习进度比普通班快。我教的是七年级普通班。虽然没有专门为竞赛准备，但我的学生的结果远超荣誉班。为了不让这个对比过于刺眼，主办者把两个级别混为一个，我的学生得了集体第一名，以及个人金奖和银奖，荣誉班的第一名得了个人铜奖。虽然我并不介意，但我怀疑这个铜牌也是为了照顾荣誉班才给的，因为如果全部奖都被普通班拿了，实在是太不给面子。

自从我的学生在统考和竞赛中取得好成绩，慢慢就开始有一些别的学校找我，说："你到我们这儿来教吧。"我那时只教普通班，为了吸引我，有的学校承诺我可以教荣誉班，以及正在筹建中的天才班（Gifted Program）。现在天才班比较普遍了，但那时亚利桑那州才刚开始筹建天才班，学生要经过考试选拔。能通过选拔的学生不多，有些通过的孩子也不愿意去，因为要离开朋友圈，还要坐车去别的学校，很不方便。所以，最后把我挖去的那个学校，只开出一个天才班。我换了学校以后，教的学生水平跨度很大，有天才班的，荣誉班的，也有普通班的。

新学校的天才班负责人帮了我一个大忙。教天才班需要特殊的证书，按规定，我还需要进修15个学分，但教了天才班三年以上、学生表现合格的老师，可以用实际经历证明自己有资历。当时我已经教了三年以上普通班，而且学生成绩完全达到了天才班的毕业要求。他特地给我写了证明，说我已经有三年的合格教学经历。所以我不用补学分，直接就拿到了教天才班的证书。

后来等到我退休的时候，那位负责人对我说："当时你的天才班里，只有一半是合格的学生，另一半是因为招不到足够多合格的学生，才被临时塞给你的。对不起，我一直没有告诉你。不过没关系，你的学生毕业时全部达到了天才班的水平。"不管在哪个学校，学生进入我的班上时的水平如何，只要到了我的班上，毕业时基本都能达到天才班的水平。

***

新学校的校长不懂数学，所以基本随我怎么教，对我不同寻常的教学方法也不多过问。我加入两年之后，学校调来了一位副校长，他原来是学区数学老师培训的总负责人，觉得不会有人比他更懂数学、以及怎么教数学。他认为中小学老师里有80%根本不懂数学，只有20%或多或少懂一点。校长介绍我时，半开玩笑地对他说：这个人不想写教案，做事不按规则来，不太好管。所以一开始，副校长就把我归到那80%里，觉得我不听话、没道理，明明不懂还不听他指挥，是个刺头。

关于教案，我是看学生的兴趣和水平，然后决定接下来教什么、怎么教，每堂课会根据上一堂课的反应临时调整。我有两种教案：一种是学校要求的那种正式报告，另一种是我自己非正式、完全跟着学生走的教案。我自己的教案是上节课结束后才写的。每天一下课，我会记下三个方面，比如，这个学生7 ÷ 21已经懂了；这个学生乘法练到几了；那个学生的理解卡在哪儿。我记不住学生的英文名字，就拿着他们的座位表，每个座位下面有十个格子，每天我就在格子里写进度，两周用掉一张座位表。这样，我下一节课要补什么、怎么补，都是根据当天课堂实际情况决定的。即使教的内容一模一样，每个班进度也会不同，比如早上第一节课和最后一节课学生完全不是一个状态，学习效果也不一样。有的例子对A班特别有效，B班却一点反应都没有，那就必须换例子。所以，我真正的教案，不是提前一两周能写好的。

副校长要求所有人按规矩，给每门课写固定格式的教案。每份教案要填13个类别，都是"你干什么 / 我干什么 / 我们一起干什么"这种结构，像中国的八股文，每天都要填。可是你想，别的老师经常只教一门课，我却教很多门，有七年级、八年级、天才班、荣誉班、普通班，加起来有六种不同班。要我每天写六份这么复杂的教案，我除了花时间写报告给上级看自己每天在做什么，根本没有时间干别的活。所以我写起来就很不带劲。

副校长对我写的教案很不满意，给我发了一个官方指令文件（letter of direction），说我如果不改进，他就要解雇我。虽然真要解雇一个公立学校教师也不容易，但这件事还是给了我很大的压力。饶是如此，我们争了很久，我也没有投降，一直都在讲理。我回信说：你的道理我懂，但我的道理你可能没懂。我这样的教法对学生更好。我的教学法效果是什么，大家都看得到。如果你能证明我给学生、给学区带来了伤害，我就听你的。否则我就要坚持自己的教学方式。

最后，他决定亲自来听我的课，看我到底在课堂上怎么教，来收集对我不利的第一手证据。在课堂上，有学生问我："这道题怎么做？"我就会说："我不知道。你怎么想？" 我就是这样对待每个学生，反复地问他们。有人回答得乱七八糟，有人说得完全不沾边，也有人说得有一点接近。这些我都不急着纠正，我始终不肯直接告诉他们答案。

于是，张三问李四，李四问王五，学生们争来争去、吵来吵去，最后居然把问题研究出来了。副校长在旁边看得很着急。照他以前的习惯，学生问问题，老师应该有问必答，立刻给出解法，一步一步带着学生去做。他看到的我却是一直不答、也不改错，就让学生错、让他们争，争到最后自己推出结果。他后来说，他当时急得都快冲上来替我讲课了。但他终于忍住了，因为他想看看，我到底能差到什么程度，为什么学生问问题我不回答。结果他看到的是：学生不仅自己研究出了答案，而且真的理解了。

那节课下课后，他对我说："我很难相信你能忍那么久，不给学生帮助。"他不得不承认我有道理，我的方法让他开了眼界。虽然我们的冲突还将继续下去，但他暂时放了我一马，不再为了教案为难我。

我告诉副校长，我的教学法核心就是一句话："I don't know. What do you think?" 我把"I don't know（IDK）"当成一种荣耀：学生问我，我回答"我不知道"，然后让他自己去想。只要学生说出任何理由 — — 哪怕说"今天是星期五"，我都会给分，因为我相信：没有人天生愿意傻，人人都希望自己的大脑能运转。我的课堂评分就只看一件事：你动脑筋了没有。

***

在多数的课堂上，老师从来不会说"我不知道（IDK）"，因为会觉得丢面子。IDK 只有学生说，通常是差的学生说，说的时候带着一些抱歉和自责。在我课堂上，不仅我会说，我也鼓励学生多说"我不知道"。学生说"我不知道"，不仅无需自责，而且应该带有诚实、负责任和热心求知的自豪感。因为"我不知道"是想要努力去知道的前提，是找到答案、学会技能的第一步。

在学生回答"我不知道"后，我会问他的想法、推断、疑问、估计和猜想，哪怕是胡猜也都是允许和鼓励的。我会让学生说出他认为的最佳猜想或者估计是什么，其次的可能是什么，甚至没有根据的猜想是什么。对他这些哪怕胡猜的答案，我都会给以真心的鼓励和赞赏，赞赏他在动脑筋，以及他付出的努力。被问的学生提出疑问之后，我会问全班同学，谁可以帮助他，或者谁有更好的推断或猜想。最佳的课堂状态是没有人完全清楚，但通过七嘴八舌的讨论，答案越来越清晰，越来越正确，甚至可以通过几种不同的方法获得。

即使学生集体得出了一个错误结论，我仍然不会站出来给出正确答案，而是引导他们去做一些与这个问题有关联的题，即所谓的"脚手架"（scaffolding）。我会很有耐心，有时甚至等到下一堂课，让他们在做类似的题时，自己发现问题。然后我会说："是吗？为什么？"让同学一起讨论，到底答案是 7 还是 8。在绝大多数情况下，把对的和错的、好的和差的摆在一起，学生是能做出判断的。

最后，我会让学生解释，如果解释不清楚，我还会表现出不懂。直到他们解释得完全清晰明白了，我才说："哇，真的是这样啊！"这样，学生得到的不是挫折感，而是成就感，是高兴，而不是打击，或者虚假的表扬。让学生自己努力、自己找到答案、自己感受成功的喜悦，学生就会喜欢上数学，就会生出自信：原来我可以学会数学，我是足够聪明的。

绝大部分的数学问题与以前学过的东西都有某种关联性，或者可以引导出来。如果所有学生都摸不着头脑，那么一定是给出的这个问题不好：它超出了学生能够探究的区域。此时我会说，这是一个崭新的东西，所以谁也不知道。就算这个时候，我也不会告知答案，或者让学生去找书中的标准答案。我会让他们回家去自己查，明天再来讨论。

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我的课堂成功只看一件事：学生有没有享受思考过程。关于讨论的评分标准是只看努力，不看对错。学生只要给出一个可能的答案就及格，再次尝试就是 B，三次尝试就是 A，和结果的对错无关。课堂的规则大家都已经讨论过，学生都熟知，有三种情况会不及格：1、等待（Waiting），2、躲避（Hiding），3、假装（Pretending）：等待别人（包括老师）给出答案，然后赶紧记下来；躲避被问到；假装知道，实际上却不知道。学生知道可以大胆说"我不知道"，不能等待、躲避和假装，他们的学习会更主动、真实且深入。

在 AI 时代，数学教学的首要目的，不是要让学生懂得一门高深的知识，记住一些标准的解题步骤，也不仅仅是训练学生掌握一门研究科学和工程创新的技术，而是要锻炼他的思维能力，并在这个过程中让他提升自信，得到乐趣。两千多年前孔子在《论语》里就讲过："知之者不如好之者，好之者不如乐之者。"老师教学生的目的，不只是让学生学会知识，而要尽可能让他自己学会，更要让他感到高兴，相信他自己能学会，让他发现这门课很适合他。对学习的爱和渴望就会从这种感觉里生长出来。

我们的老师和家长，总是太过热心，恨不得把自己所有的知识尽可能地灌输给学生，同时让学生觉得他们超级有知识，并且乐于助人。但这样做会剥夺学习的乐趣和成就感，也会错失 AI 时代数学学习的主要目的：智力的提升，而不是知识或技能的获得。在大多数学校现行的、带强制性的灌输方法之下，学生们被迫学习，没有多少兴趣和爱，始终处于一种应付状态。这种状态中自然缺少创新性的培养，因为创新需要契而不舍的质疑、钻研、深究。没有对这门学科的爱，很难进入到这种状态。

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很多老师对我的这个做法不以为然。他们认为让学生写 IDK 三个字母就可以交差，这样会鼓励学生不去努力，试图蒙混过关。写三个字母不用费什么劲，这显然容易让不想努力的学生偷懒。这个观点在文学创作方面可能有些道理；你写不出来，使点劲儿，总能挤出点东西来。但对于数学，特别是中小学生的基础算术，不懂就是不懂，是挤不出来的；懂的要写出来也很简单。"我不知道"是最诚实的回答。学生愿意这么说，还能得到老师的尊重和信任，对他们也是一种激励。

当然，我有一个要求：所有写 IDK 的地方，学生都必须问老师或者别人，至少搞懂一道题。我对学生讲，"我不懂这个"是学习"这个"的第一步。我让学生接受，"我不懂"不是什么耻辱的事，"我懂"也不是什么神气的事。我们都是学生，因为不懂才来学习的。学生每天做的事就是发现不懂的地方，然后把它搞懂。

我会跟学生讨论：先懂得人学得快，后懂的人学得慢，是个人的学习方式不同，但懂了就是懂了。我会让学生从另一个角度看这个问题：学得快的可能忘得也快，学得慢的常常忘得也慢。那些看起来很笨、试了三种方法才搞懂的人，和那些看起来很聪明、一下就搞懂的人相比，前者知道两条走不通的路，他的知识更丰富，学得更扎实。

任何考试或测验，如果学生只是在做题、只有对错，那么它基本就是在浪费时间。我的每一次测验和考试，对学生都是一次寻找哪里不懂、然后通过讨论把它搞懂的过程。学生发现自己搞懂了什么、学会了什么、进步了多少，是一个极其愉快的经历，这也是我的教学目的之一。我不那么在意作业的对错，不会给哪道题一个"很差"的评价，最多画个问号，说："That is interesting. Tell me how you got it." 这样，老师不需要每道题都在那儿改对错、打叉打勾，因为每一个叉对学生来说都像打一巴掌 — — "啪、啪、啪"，非常伤害他们的兴趣和信心，无形中减少他们的学习动力。

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从第一年教书开始，我就特别喜欢去听课。我在国内开始教的是初二物理，只要有机会，别人不介意，我就会去听同校老师的课，包括物理和其它学科。后来我在大学主修语言和文学，但决定以后一定要学教育。在大学里，我几乎把所有老师的课都听过 — — 有名的、无名的、大教授、小教授，我都去听。我每听一节课，一定要找出几个优点，也要找出几个缺点。哪怕是再好的老师，我也要能指出他的缺点，因为指不出缺点，说明你其实没有真正理解，只是囫囵吞枣。因为听过很多课，我能更站在学生的角度去看课堂 — — 我听课的时候就是以学生身份去听的。

在美国，听课的机会比较少，可能因为大家都比较忙。除了副校长之外，只有极少数老师来听过我的课。这是其中一位老师听了我的课之后写的报告：

教学方法：

徐老师设计课程时，会给学生一组难度跨度很大的题目。每个学生对每道题标注 "IK"（我会）或 "IDK"（我不会）。课程从学生不会的内容开始，同时也复习他们认为自己已经掌握的内容。有趣的是，有些学生发现自己原以为会做的题目其实做错了，这反而促成了更深层的学习。

徐老师不直接讲解答案，而是通过提问引导学生重新阅读、反思和批判性思考，直到他们自己找到解决方法。他鼓励学生举手、分享想法，并且让他们能够自在地表达不确定。错误的答案被视为学习的机会，营造出一个开放且相互尊重的课堂氛围。

我受启发的地方：

这是我从未见过的教学方式。它鼓励学生既为自己懂的知识感到自豪，也为自己不懂的地方感到自豪。每个学生都能放心地分享想法，不怕"答错"。学生通过独立思考和合作讨论来学习，老师讲得少，学生讲得多 — — 整堂课保持参与度和活跃度。没有人被落下。

我特别喜欢这个系统：无论标注 IK 还是 IDK，学生都有机会拿 A，因为 IDK 成为个性化教学的起点。这确保了课程能从学生的实际水平出发，帮助每个学生进步。我也很欣赏这个理念：学生只有在逃避、隐藏或假装理解时才算"失败" — — 而不是因为答错了。

课堂效果：

学生在"恍然大悟"的时刻明显很兴奋，从头到尾都全神贯注。徐老师对正确和错误的答案都给予肯定，然后让全班一起判断哪个是对的。这让所有学生都在思考，而不只是最先解出题目的那几个。

问题与思考：

• 对于可能无法完全理解题目表述的英语学习者，这种方法可以如何调整？小组内的同伴互助讲解是否会有帮助？

• 由于题目很多，但课堂时间有限，学生课后如何继续练习他们仍然不会的题目？