Note: I am continuing to post stories of the teachers I have encountered on my teaching journey. More and more I am transitioning to my colleagues, some of whom didn’t fit the roles of “inspiring teachers” that I have depicted so far. This one was a senior teacher in our department and a former department head.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you enjoy what you are reading, you can purchase the eBook for $2.99; or you can email me for a copy at yellowheight@hotmail.com)

CHAPTER 30 THE CAREER TOTEM POLE

Coming into the school, I had expected that I would be the lowest member on the career totem pole. But I was so busy adapting to a new career, and the school leaders and the department head all projected an image of an open environment that welcomed fresh perspectives and input from new teachers. For a while, I believed in them and forgot about the totem pole.

Tanya was the co-teacher for the first course I taught at the school. During the summer before the fall semester started, I came to the school to prepare classes with all my co-teachers, but Tanya didn’t come, so we missed the opportunity to get to know each other. Tanya had a severe expression and strong opinions on many topics. She was not physically imposing, but there was something distinctive about her tone, which I later recognized as a form of supreme confidence that quietly demanded compliance. At that time, I didn’t know that she was a 22-year veteran teacher and the former department head who had hired the current head, Darren.

Most math courses in our school, including this one, did not rely on any existing textbook and had a repository of teaching notes accumulated from previous years. As a new teacher, I was glad to use the material from past years complied by Tanya. When I noticed a mathematical concept that was repeatedly used and tested but not well explained in our materials, I asked her about it. I suggested we either fully explain it or drop it altogether. Our colleague Ed, who was sitting next to Tanya, joined the discussion casually.

When we started to have differences in our interpretations of the concept, I looked at Ed, hoping that he would share his opinion. Tanya quickly got upset and said: “Why do you look at Ed? He doesn’t know statistics. You should talk to me! Even if you are right, this unit is almost finished. Why are you raising this issue only now?” Her loud voice turned everyone’s heads in our direction, and I quickly ended the conversation in embarrassment.

Later, when I made the mistake of giving out a proposed quiz as homework, Tanya called me into the hall and lashed out at me for my mistakes as a new teacher, which cost her extra time to make a new quiz. I took the blame in silence. It was my mistake, although an understandable one for a new teacher and not a particularly costly one. Maybe this was the price a new teacher needed to pay to be accepted into his rightful place. After another similar incident, I tried to talk to her in private to bridge our differences. But it didn’t bring us any closer. When I said I might have different ways of thinking, she didn’t acknowledge me and simply walked away in silence.

In another course we co-taught during the following semester, I noticed that the last unit was designed to be short, but the content was more difficult than the other units. When my students complained, I proposed extending the teaching time or reducing the length of the test. At first, Tanya simply didn’t acknowledge any of my points. When I pointed out that the inconsistency was unfair to students, she dug in and questioned why I was raising the issue so close to the end of the semester. I sent another email apologizing for the inconvenience but still pushing for changes. At last, another co-teacher jumped in and produced a version with reductions, and Tanya finally agreed. Although I got what I wanted, it was an exhausting process that I didn’t want to repeat.

Whether or not to allow note cards in tests was a confusing matter to me. In the pre-calculus course, it was always allowed, but in this course, Tanya insisted we should not allow note cards. As a new teacher, I had no choice but to follow what others did. However, on the last day of teaching, some of my students said that the other teachers were allowing note cards in the final. I checked with Tanya, and she said, “Oh, I forgot to tell you that we allow note cards only in finals.” I thought silently: “You didn’t think it necessary to notify me as your co-teacher? And how does this sudden change in rules make logical sense?”

On the final exam, there were a few borderline cases where I struggled with grading: I was trying to send the right message through grades while maintaining objectivity, and it was so hard to do. Frustrated, I emailed the department head, Darren, about the situation and said some of my students might have been unfairly disadvantaged due to the last-minute notice. They might not have had enough time to write their note cards. Darren said he would take care of the issue.

I didn’t know what Darren did, but the next day, I got an email from Tanya with a tone of humility and sincerity that surprised me. She wanted to meet me to apologize and talk about the issue. I was impressed by her open-mindedness and imagined a scenario where we could iron out our differences and have a fresh start. But when I woke up at 5 a.m. thinking about what it would take to bridge our gaps, my tense body told me that it would not be easy.

Finally, we had our meeting. Tanya suggested that we talk privately outside the office. We chose a spot in the park next to our school building. The summer air was crisp and clear, and I had hoped that our conversation would similarly clear the air between us. She apologized for not notifying me of the change earlier, and I expressed my appreciation for her help as a lead teacher, acknowledged her long experience, and apologized for the mistakes I had made as a new teacher. There was a moment of mutual appreciation as she acknowledged that my math skills were very strong. To show my respect, I asked about her teaching career and the advice she might have for a new teacher like me.

Tanya went to Georgia Tech for her undergraduate education in the same year I attended college in China. She didn’t graduate from Georgia Tech and didn’t discuss the specific circumstances with me. She only explained that she wanted to be a teacher, and since Georgia Tech didn’t have a teaching program, she earned her teaching degree from a nearby state college. In her first year of teaching, she taught matrix inversion totally wrong. She was always a tough person, but during her first year, she cried quite a few times.

She taught summer school to make extra money. A high school student was there because he didn’t pass algebra at his school. She failed him, and he did not get a high school diploma. After all these years, she still wondered what had happened to him. She could have destroyed his life. She regretted the decision and thought about it whenever she was about to give a failing grade.

Tanya was strict with students and often exhorted other teachers against treating students too leniently. I had never seen the soft side of her before. Obviously, she didn’t like disagreements, and I wanted to say something complimentary, but for reasons I couldn’t fully explain, I still sounded skeptical: “Thanks for sharing your experiences. I see how much you cared about the student. It was a tough decision, but I am not sure it was a bad one. If the student had cared and worked hard, I might have passed him even if his score was slightly below the guideline. If he simply didn’t care, failing may not be so bad. It was his choice. We had to do our job as teachers and hold certain standards.”

Tanya didn’t look pleased with my comments and didn’t chime in. Sensing my mistake, I tried to find a better way to communicate, saying, “It would be great if you could understand my learning style. It often takes time for me to think through an issue, and initially, I may not completely understand or agree with your points. It really helps if you explain the logic behind certain teaching practices to demonstrate their effectiveness; this way, I will get on board faster.”

In my previous careers, I had never encountered anyone who would so flatly turn down such a request, but Tanya did it casually and naturally: “It will be hard for me to do that. I can’t do it without assuming whether you know something or don’t know something, which is difficult to find out. And frankly, I don’t have the time. I am not paid to be anyone’s mentor. Nothing in my teaching contract says I should be.”

Not ready to give up, I said: “I am new to the teaching profession, but I have other perspectives from years of working experience using math. Sometimes, I might have ideas that could be useful for improving teaching. Do you consider that valuable sometimes?”

Tanya didn’t relent: “Contractually, we are equals as math teachers, although you just said you have no experience teaching and don’t know the sequencing of the course. And I have done this for twenty-two years. You will see my point of view when you get more experience.”

I felt I had met a stone wall and changed the topic: “How do you stick to the grading guideline while exercising teacher discretion fairly? Where is the boundary?” Tanya answered: “I know it’s hard for a new teacher to get it. You will have to use all the information you have. It’s normal that you struggle with it now. You will get better next year. For this course, I think it’s OK for you to override the final exam results, since you feel your teaching has caused your students to fail. You can raise their grades this time because it was not their fault.”

I swallowed back my questions: “Was this my fault? I didn’t think so and never said it. Was it caused by my teaching, the way it was designed, or your failure to notify me of the change in time? I don’t think this is a fair statement.”

Finally, I asked: “How can we tell if a teacher is doing a good job? What are some measures?” Tanya said: “It’s hard. Ed and I joined the department at the same time; we both had eleven years of teaching experience before that. We both went through formal educational training and had undergraduate degrees and certificates in teaching. We know what we are doing. You will get the hang of it after you have been doing it longer.”

I asked more questions, but this time only silently in my mind: “Do experiences automatically lead to better teaching? You have run this department for years and are the most senior teacher, yet our students are so weak in mathematical thinking. You are the one who told me not to teach students why, only to show them how to do the steps. How can I believe that you are as good as you say you are, even if you have done this for twenty-two years?”

It was only with the passage of time that I started to understand why Tanya treated me this way. The status of teachers, as well as their salaries, primarily depended on their tenure. As the former department head with 22 years of teaching experience, Tanya never considered me anywhere near an equal, someone dared to raise different opinions on issues of math and teaching. Coming from the tech and investment industry, I was used to challenging authority opinions, often with large stakes (like hundreds of millions of dollars), when I knew that I was correct. It was the collision of two vastly different worlds.

I was not alone in receiving this treatment. In an official gathering of new teachers at our school, nearly everyone said they didn’t feel heard or respected by their more experienced colleagues. They complained about the school’s underlying culture where experience meant authority. To the experienced teachers, the new teachers had a lot to learn and little to contribute to the practice of teaching. The promise was that new ideas would be heard and valued, but the reception was often intentional neglect or occasional outbursts of hostility, like the ones I experienced with Tanya.

After the long conversation, Tanya and I entered a state of truce, although we kept a wary distance from each other.

(If you like what you are reading, please support my work by clicking on the image below)

Note: I am continuing to post stories of the teachers I have encountered on my teaching journey. Mr. Felix has an unusual life story, from the LA street gang to then No.1 private school in the U.S., onto a renowned professor at a top university.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you enjoy what you are reading, you can purchase the eBook for $2.99; or you can email me for a copy at yellowheight@hotmail.com)

CHAPTER 14 THE SOCIAL JUSTICE CHAMPION

As a graduate student, I didn’t know that Mr. Felix was a legend in the ULW education department initially. During the first few online classes, I simply noticed a short, plain-looking Latino who was about my age. Beneath his reserved manner, though, there was a trace of deep passion and sadness in his eyes, which had surfaced only briefly. He spoke in such an unemotional tone and even pace that classmates soon began mocking him in private chats. However, I quickly changed my mind.

Mr. Felix taught the math instruction method class in our program. His first assignment was to write an autobiography on our math learning. I suggested that we should share our individual relationships with math in class to get to know each other more deeply. I was surprised that most of the class time was indeed dedicated to exactly what I had suggested. For the first time, I felt someone in the program actually listened to me.

His assigned reading stoked my curiosity in teaching methodology. One was about the difference between the teaching styles of Japanese and American teachers, which I found striking and insightful. I said to myself this guy had real substance. In addition, he was the only teacher who actively allocated time for student appointments and encouraged—even enforced—1:1 meeting through Zoom. I signed up twice and got to know him better as a person.

Felix moved to the U.S. from Mexico with his family in third grade. For several years, all six of them lived together in a single garage. More than half the students at his school were Latino. The school bullies quickly learned that this quiet boy was a fierce fighter when provoked. Before long, he developed a tough reputation and became involved with a street gang.

He was good at math and once earned the highest score in a math competition—but the teacher didn’t believe him and made him retake it. (He would later have a similar experience in college.) Experiences like these left him disillusioned with school, and for a long time, he simply wasn’t interested in academics.

His life changed dramatically when a school counselor took a keen interest in him. To channel his fighting spirit, he started training for boxing, played on the school soccer team, and ran a full marathon as a thirteen-year-old. He became more motivated to learn and caught up academically. As a well-rounded student, he was admitted to Phillips Exeter (the No. 1 private school in the U.S.) on a scholarship, while many students with better academics were turned down.

Life at Phillips Exeter was daunting. Beyond adapting to a new climate—from sun-drenched Los Angeles to the cold, damp winters of New Hampshire—he had to adjust to an entirely different academic culture and class of students. He told me that Phillips Exeter differed from local private schools in a fundamental way: while local parents often sought entry into elite social circles, Exeter families had already arrived. Its graduates included a U.S. president, three Nobel laureates, and billionaires such as Facebook founder Mark Zuckerberg. His roommates’ parents included a neurosurgeon, a Saudi prince, and a cabinet minister from an Asian country.

His time in the big public school system did not prepare him for the academic rigor at Phillips Exeter. He had to work extra hard to feel adequate. It was only in the last year that he started to feel on par with his peers. For college, he went to Occidental College and found the academics completely unchallenging there.

After graduating from college, he became a math teacher at his alma mater, Garfield High School in LA, which was depicted in the inspiring movie Stand and Deliver, a film he recommended to me. He admired the main character, Jaime Escalante, a recipient of the Presidential Medal for Excellence in Education, who had taught him in real life. However, he believed Escalante’s focus on bringing students to academic success was too centered on standardized tests, which are inherently flawed.

Garfield did not work out for him: he was neither trusted nor respected by the more experienced teachers, was often overlooked, and was even mistaken for a student. Later, he moved to a middle school where he finally flourished as a teacher, connecting deeply with students and the community and teaching fantastic math. By the time he started his PhD at UCLA, he was already sought after as a consultant. After that, he came to teach at ULW and became an inspiration for the Latino students and teachers here.

He asked students to prepare a lesson demonstrating that math was invented not just by Westerners but by people of color, but their inventions were often stolen by Westerners. It made sense that when students of color feel proud, and relate to the inventors of math, they tend to connect to math naturally, feel proud and are motivated to learn.

Growing up in China, I was always told that Chinese people had invented math (and many other great things) before the West. The Mathematical Book of Sun Zi, written around the sixth century, provided vivid examples. One of those was: “In a cage with certain numbers of chickens and rabbits, one can count 35 heads and 94 feet. How many chickens and how many rabbits are there?” I created a lesson on linear systems, referring to this example and the Chinese achievements in math, and delivered it to the applause of my classmates.

Listening to my classmates’ presentations, I was amazed to learn that so many cultures could claim to have invented math on their own. However, when I learned of a proposed law in our state requiring every class to include examples like this, I began to worry that I would run out of ideas. Although I wanted my ancestors to take more credit for inventing math, I could only do so much without bending the facts to serve my own cultural pride or a social purpose.

Our final class in the summer of 2020 coincided with the escalating protests over the killing of George Floyd. Mr. Felix came to class with bloodshot eyes, his typically hidden emotions boiling to the surface. He was upfront that he hadn’t gotten much sleep the night before. He watched the George Floyd video several times and couldn’t stay calm. His mind flashed back to the LA streets, where some of his young gang friends were killed by cops in similar ways. To this day, he and his older son could be stopped by the police for questioning for no reasons other than how they looked, even on the ULW campus.

After all these years, the reticent, poor Latino immigrant, known for his fighting spirit and untrusted math brilliance, has finally grown into a staunch champion for social justice.

Disenchanted with his local public schools, Mr. Felix pulled his youngest son out to be homeschooled. His son joined 3.7 million other homeschooled children, representing 6.7% of all school-aged children in the U.S., up from 3.3% in 2016. Despite this, Mr. Felix remained actively involved in public schools, volunteering for many years to lead a social justice club for the indigenous students.

Most instructors in our program were PhD students or lecturers who preferred to be called professors. Mr. Felix, one of the few tenured faculty members, was more low-key and advocated for social justice in a natural and open-minded way. He once told me that while teachers should care deeply about social justice, not everyone needs to be an activist. He was the first instructor to present this as a morally acceptable position.

I wondered why he told me that specifically. Perhaps he had noticed that many instructors, few of whom had faced the kinds of challenges he had, were enthusiastically urging all teacher candidates to become activists for social justice.

Mr. Felix was supposed to teach us an advanced class the following semester, but he never showed up. I later heard from classmates that he had been undergoing cancer treatment for some time. From time to time, I would think of this plain-looking man and the blend of passion, sadness, and combativeness that seemed to lie just beneath the surface. Did it come from his battle with cancer, from the teachers who didn’t believe he could achieve the highest score in math, or from the police on the LA streets who took the lives of his young friends?

(If you like what you are reading, please support my work by clicking on the image below)

I am continuing to post stories of the teachers I have encountered on my teaching journey. This is a special one, because he is my father.

I am at an airport returning to the U.S. from China. Finally free Internet. I saw friends, and more importantly, my parents there. My father has certainly aged; his mind is somewhat sharp for his age, but his memory and attention has deteriorated so that it is hard for us to carry on long conversations. Unfortunately, he has strong symptom of Parkinsons and refused to see doctors. He has suffered from health problems since his forties and doctors haven’t been able to help him much, so he thought he would take on them all by himself. No one can convince him otherwise.

This chapter is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you like what you read, please click on the link.

CHAPTER 7 MY FATHER’S PATH

The weekend I was accepted into the ULW teacher’s program, I called my father.

My father was of the silent generation in China. He didn’t like talking about his childhood or adolescence. After piecing his life together, I started to understand his reluctance. His father died of pneumonia when he was five, a few years before China went through the famine of 1959-1961, following the disastrous Great Leap Forward movement. An estimated 16 million to 43 million people died in the famine, although the true number was suppressed by the government.

My father’s early memories were filled with hunger and tragedies: his grandfather died of poor nutrition, strangers collapsed on the road from starvation, and he had to forage suspicious plants to survive. I remember him talking about the strange fruits and tree bark he ate at that time, which resulted in a bloated stomach and prolonged constipation. I was young when he told me this in an attempt to instill in me the spirit of “eating bitterness,” the Chinese version of grit.

To keep him in high school required huge sacrifices from his widowed mother, who was constantly pressured to pull him out of school to work as a farmhand. During summer breaks, he did back-breaking work like any farmhand, often spending twelve hours a day in the rice fields. At one point, the labor was so grueling and there was so little food to eat that he contracted severe hepatitis. He couldn’t get any treatment and nearly died as a result.

Starting at the age of twelve, he had to travel ten kilometers every weekend, carrying heavy bundles of wood on a shoulder pole to exchange for food at school where he stayed during the week. The forty-pound load bore down on his small shoulders and frail frame. Sometimes he left late, only after finishing work in the fields. As darkness fell, he shivered with fright at the howling of wild animals and had to ask villagers to stay the night under their roofs.

He attended Wuhan University in the last cohort of high school graduates to go to college. The following year, Chairman Mao started the Cultural Revolution, stopping college education for an entire decade. In college, he witnessed classmates being killed in the struggles among different factions of the Red Guards vying to show higher loyalty to Mao and the cause of the revolution.

After graduating, he was assigned a teaching job in our hometown, gradually working his way up to a position in the city. With the social immobility in China at the time, settling in a city as a country boy was a challenge that was not unlike my immigrant experience in the U.S. We couldn’t afford rent in the city, so until elementary school, I slept in a small room in my mother’s assigned factory dormitory, sharing a bed with my parents and younger sister.

My father didn’t talk much about his life as a teacher, and I only knew years later from other people that for fifteen years, he had been the best physics teacher in our city before being promoted to an administrative role. My mother told me that my father worked so hard as a teacher that he returned to the classroom only two days after their wedding, and that he began losing his hair and became nearly bald before the age of thirty. She joked that she had been cheated, as the handsome young man she had married was soon left with only a few tufts of hair.

When I was going through school, almost every teacher and principal knew him and gave me some special treatment (this often meant more challenges or back-channel reports, which I resented). His students were so widespread that after I went to college a thousand miles away, some welcomed me into their homes there. They told me that my father’s teaching was so effective that his classes consistently ranked at the top in our city, and his students were disproportionately admitted to the country’s most prestigious universities. At that time, such outcomes meant entirely different life opportunities for those students.

Growing up, I was not close to my father, and we had our share of father-son struggles. I felt he was ill-tempered and treated me with insufficient love and respect. But my father grew up without a father or the warmth of a normal home. There were several occasions when he could have died, and his mere survival was a miracle. His instinct was for me to acquire the skills to survive the harsh environments he had experienced. Having lived through numerous revolutions that promised to rescue the oppressed but only made things worse, he knew that my only path to a better life was through education, and he did everything he could to help me pursue it.

My father never talked about his struggles and sacrifices, and I was the teenage rebel who left home at the age of thirteen, never having the faintest idea to understand what he had endured. Years later, as a father, when I began to hear the same complaints and resistance from my own sons, I realized that I was him, and they were me, and such was the cycle of life.

After I came to the U.S., he was always worried about me from thousands of miles away: Did I make the right choice to quit the PhD program? Why did I leave the stable “iron rice bowl” job at Microsoft? Was running marathons too much for my body? How could I make enough money if I left my job to start a company? Would there be enough saved for my children’s college education? Although I wanted him to acknowledge that I made the right choices, he was always worrying, as if the act of worrying would somehow protect me from the very things he was worried about. Maybe it was his generation’s way of showing love. Maybe, it blessed me.

When I told him I planned to go to graduate school to study and become a teacher, he was silent for a moment. Then he asked, “Are you OK? Did you think this through? What did your wife and children think? How would you know if you were a good teacher?” He had never asked me so many questions at once, and he seemed a bit unsettled by it. After listening to my answers, he ended the call quickly. Before hanging up, he said, “I think you made a not‑too‑bad choice. I am proud of you.”

Two surprising things happened: For once, my father did not worry about me and approved of my life choice. “Not-too-bad” was probably as much a compliment anyone could ever extract from him. And, he said, “I am proud of you.” I knew he was proud of me before, but this was the first time he said it to me directly.

Yes, my father was proud of me for wanting to be a teacher. It must have meant something important for him to break his old habit and say it. To me, hearing him say, “I am proud of you” meant a great deal. I struggled to hold back tears, as though afraid that he might see them at the other end of the line. At that moment, I suddenly realized that during all these years of struggling with my father, I was proud of him too and wanted to be like him in some important ways. I just didn’t know it.

After the call to my father, I accepted the offer.

A few weeks before my program began, cases of a new breed of virus called COVID-19 were reported in the United States. Within a few months, the pandemic brought the world to a standstill.

Note: I am continuing to post stories of the teachers I have encountered on my teaching journey. I will be traveling for two weeks and may not have access to Substack, so this one comes out a bit earlier.

The third teacher is Sensei Sato, who shared some similar characteristics with me. Our work time at Microsoft overlapped. He gave me inspiration, guidance and resources when I was looking for a teacher job. Later, he became my son’s teacher and a friend. His story is mixed one, coming a long way to show the sacrifices some teachers make to pursue their dreams.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. )

CHAPTER 11 SENSEI SATO

When considering my third career in teaching, I heard about Mr. Sato, an engineer who had left Microsoft to become a teacher at my son’s school. He had been both a manager and a principal engineer, one of the more senior technical roles at the company. I invited him to a chat over coffee.

Sato was about my age, quiet and earnest looking, with large and clear eyes that reflected the calm serenity and self-assuredness of a “sensei” (“master teacher” in Japanese). He preferred that students call him “Sensei Sato.” A second-generation Japanese American, he went to Williams College and double-majored in computer science and educational psychology. After graduation, he joined Microsoft in 2001, the same year I did.

While his work at Microsoft was financially rewarding, Sato had always felt a strong desire to work closely with students. He volunteered as a STEM teacher at local schools for years. After helping middle and high school students successfully build and launch a Lego Mindstorm science experiment into space on the International Space Station and seeing their incredible growth, he decided that he could no longer put off his teaching dream.

His college roommate was on the senior leadership team of my son’s school and told him about his work, which sounded exciting and deeply impactful. Sato had no formal teacher training or classroom experience, but the school leadership was willing to take a chance on an inexperienced teacher. With his roommate’s endorsement, he moved smoothly through the interview process and was hired as a computer science teacher.

When he made the announcement, Sato received hundreds of encouraging comments on his LinkedIn page, but it was a tough personal decision. He was the sole income earner in the family, with two kids heading to college soon. The transition would reduce his family’s income by three-quarters. What impressed me was not only the immense economic sacrifices but also the courage and humility to start an entirely new career at the lowest level of the career totem pole. Perhaps because of our shared Asian heritage, we seemed to understand each other deeply.

Sato was upbeat in the first year and found his work fulfilling. Everyone was nice and cooperative at the school, and he had few complaints as a new teacher. He was given complete freedom to design a new course curriculum and put his heart into it. He enjoyed the interactions with students and felt that his work as an advisor gave him a closer understanding of students’ lives, especially those from challenging family backgrounds. The only drawback, he noted, was the lack of feedback on his performance. Still, he believed he was doing well.

Sato became my son Matt’s teacher a year later, and we met at a parent-teacher conference. Matt and his friends loved Mr. Sato’s teaching style, the breadth and depth of his knowledge, and the respectful, encouraging way he interacted with students. They appreciated the real-world insights he brought from his previous career, which made his teaching more grounded and engaging, offering a refreshingly new perspective. One of his practices, similar to Ms. N’s, was offering unlimited office hours. During those sessions, he stayed as long as students needed and was always patient and encouraging.

Later, after the COVID lockdown ended, I ran into Mr. Sato at a school social event. It had only been two years since we last met, but he looked ten years older. His hair was grayer than I remembered, and he had begun growing a long beard that was turning salt‑and‑pepper. As we sat at a small table on campus eating catered gourmet food, I noticed the dark circles under his eyes. There was a tiredness in his furrowed forehead, dimming the optimism I had always been used to seeing on his face.

He told me that teaching was easy to do but hard to do well, at least by his own standards. Although he believed he was a good teacher, there was no real acknowledgment of the quality of his work. Everyone was so nice to one another all the time that it almost felt as if no one cared whether the teaching was actually good. He also began to doubt his ability to influence students’ lives as their advisor.

When Mr. Sato learned that I was looking for a teaching position in 2021, he referred me to a job fair for local independent schools. He said he would go there to explore other opportunities too, but he didn’t show up. He was too busy to take even one day off. I attended the fair and received an offer to teach high school math at a local private school.

The offer I received as a high school teacher was $70,000 a year, hardly enough to support a comfortable living in 2021, even with the benefit of two master’s degrees. Without those advanced degrees, and despite more than twenty years of work experience in other industries, my salary would have been under $60,000 a year, while the school’s annual tuition was $48,000. I could manage because of other income sources, but not everyone has that option.

I wondered how Mr. Sato was getting by with a similar salary while supporting his family of four. He was a pure person, idealistic and totally indifferent to fame and wealth, who felt that as long as he had enough money to get by, that was enough. But relying on an income of less than sixty thousand dollars a year made it challenging to live in our area. His family did not travel much, nor did they eat out or buy big-ticket items. Even so, the mortgage, car loan, various insurance premiums, phone and computer bills, extracurricular activity fees for his children, and the daily necessities of life all required his wife to budget family expenses with great precision.

Since COVID, inflation in the U.S. had reached 9.1%, and the price increases of everyday goods far exceeded that figure. It was hard for them to avoid running a deficit. After working at Microsoft for eighteen years, Mr. Sato’s family had some savings. But when the U.S. stock market plummeted in 2022, his retirement account, like those of many of his friends, lost half its value.

At the same time, he faced a difficult choice. His older son had just received admission offers from the University of Lake Wobegon (ULW) and the University of California, Berkeley, to study math. While his son loved Berkeley, the academically stronger option, it was three times more expensive than ULW for an out-of-state student. He was struggling mightily over the decision because of his much lower earnings as a teacher. It pained me to see his kind face filled with anxiety and torment.

In the end, his son chose Berkeley, and Mr. Sato started online tutoring in his spare time, which paid five times his hourly rate as a teacher at the school.

Note: I am continuing to post stories of the teachers I have encountered on my teaching journey. The third teacher is Ms. N, who is a kind of role model for me. After her father died after being a teacher for forty years, she quit her Wall Street lawyer job to become a math teacher in the poorest schools in our region. She was my mentor teacher when I was getting my master’s degree in teaching.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you enjoy what you are reading, you can purchase the eBook for $2.99; or you can email me for a copy at yellowheight@hotmail.com)

Chapter 10 The Guilt She Carried

When Ms. N was assigned to be my mentor teacher during the ULW teaching program, I did some research on her. From what I could find online, she cared a lot about students and was always available to them. Students commented, “Ms. N has a math camp open before school, during lunch break, and after class.”

In our first conversation, we introduced ourselves and established how we would work together. I heard from classmates that some mentors treated teacher candidates like free labor and subordinates, but Ms. N treated me as an equal. She readily admitted that during COVID, she was learning everything from scratch and hoped we could learn from each other.

Ms. N was in her late fifties, wearing thick-rimmed glasses that made her look serious and a little intimidating. She was Jewish and deeply religious. Career-wise, we had similar backgrounds: she had worked in the financial industry before becoming a teacher, serving as a tax lawyer at a major Wall Street bank. She began her teaching career in her forties, around the same age as I did. She worked from her garage, while I worked from my basement—both of us dealing with unstable internet connections at times.

Ms. N spearheaded the accelerated math pathway program, a very popular math program in this Title-1 school, which receives federal funding for having a large proportion of low-income students. As the only qualified teacher to teach the advanced classes, she was teaching six classes with over two hundred students.

The program had its own unique challenges, as the students joined with very different levels of prior knowledge. She knew well how to teach the courses and engage students before COVID, but for remote learning, she had to find a new approach. She was working eighteen-hour days to get everything ready and spending a lot of time adapting her teaching to the new learning environment.

Besides that, she had to replace her school-assigned laptop five times to get it working and wasted an enormous amount of time. Students were having all kinds of problems, from online learning logistics to troubles at home to lack of motivation, while she had to wait six weeks for her assistant teacher (me) to be on board because of bureaucratic delays. She always maintained a calm and caring attitude, no matter how discouraging things were. However, when she needed to share her account password with me, I learned that it was “I-hate-LWPS***” (LWPS stood for her employer, Lake Wobegon Public Schools).

During the first few sessions, I observed and took notes as she struggled with engaging the students. After the class, I gave her a few suggestions and provided the supporting material. She used them and got a much better reception from students. For the next class, she suggested that I teach the class on my own while she would stand by to help with anything I needed. From then on, I started teaching some of her classes independently. She was always supportive and let me drive my own teaching but gave me detailed feedback when the classes ended.

Whenever I asked for help, she immediately got on it. She didn’t talk much about how she helped, but she was actively helping me behind the scenes. In mid-semester, an African American boy was transferred to my class and found it challenging. After a few attempts in class, he gave up. When I asked, he sent me a message full of desperation. I kept encouraging him and sent a message to Ms. N, who was in a better position to offer more assistance. We were both busy, and I didn’t hear from her about the issue.

After a week or so, the student suddenly showed more enthusiasm and initiative in class.. He was trying to answer questions, and his tone was noticeably more upbeat. He even accepted my offer for 1:1 help after class. Out of curiosity, I asked Ms. N: “Did you do anything?” She replied, “Yeah. I talked to another teacher who could help him.”

First, she invited the student to talk, encouraging him and offering him time to work together, but she got no reply. She then thought of a colleague, a young African American male teacher with a body full of tattoos. Maybe a teacher like him could connect with the student better than an old white lady like her. She asked that colleague to talk to the student, and it apparently worked. That was why the student became motivated again and started to reengage in learning. If I hadn’t asked, Ms. N wouldn’t have mentioned it.

During one session, a student struggled with a practice problem and forgot to mute her microphone. The entire class could hear the parents in the background explaining the problem to the student. The explanation was incorrect, and the whole class could hear it. Ms. N did not ignore it but told the student: “I think the comment got the right answers, even though it was not for the problem we were asking. It is so great to have some family help on this.” It was so positive and authentic that it quickly got the student out of embarrassment.

In the advanced class, four students had Individual Education Plans (IEPs) to accommodate their learning disabilities. One of them, David, regularly stayed after class to get every problem correct, no matter how long it took. He was in special education all through elementary school and was told that he couldn’t even be at the general level. But through the program and his relentless effort, he was on the advanced track and doing well. One day after the class, he wrote, “I am always at my happiest when I get to think about math problems.” That made the day for Ms. N and me.

Ms. N and I often tried to invite, cajole, and lure students to participate, but to little effect. I asked my son Matt for advice and came up with a program called Participation Appreciation. Every week, I gave out two $10 gift cards for those participating the most in a class (more than $10 was deemed inappropriate). It resulted in better participation and more interactions with the students, who opened up to me about some of their life situations in follow-up conversations.

A few students saw through my trickery and refused to be bribed by the prize, but most took it with gratitude. Some students wanted the money for video game credits, but several students wanted gift cards for groceries to help their families rather than things for themselves. I was surprised, but Ms. N knew better. Her own children attended private middle school and came to help with her summer classes every year. They said these students were more mature than the privileged kids in their schools. Later, they both changed to public school.

One day in December, she appeared on the front page of our local newspaper with a feature story. Some of the details added to what I already knew about her:

She started as a lawyer at General Electric in New York City, in a department that she said was dominated by men. She remembered coming home after a long day, taking a shower at 2 a.m., and thinking: What am I doing with my life? Her father, a teacher, died, and she saw how former students honored him; it made her think about what her tombstone would say: ‘She was a really average lawyer,’ perhaps. Her office closed, and she moved back.

She noticed that only 5% of advanced math students were Black, even though they made up nearly half the school’s enrollment. In her quest to change that, she found an ally in W, who was then the principal. These kids, she thought, were all capable of the same academic achievement — they just hadn’t been given the opportunity. So she and W tried to accelerate all kids in math.

Ms. N was a little concerned about how the school would react to it. The administrators’ responses to the news story were generally positive, but some teachers were jealous. One asked her in a confrontational way: “Why were you the one who was interviewed and written about? We all worked hard.” Reader responses were enthusiastic: a donation was made to support a scholarship in her name; several lawyers messaged her saying that they wished they had the courage to do the same; online commentaries were full of admiration, and one person offered to pay the full cost of upgrading her garage where she conducted her online teaching. Ms. N’s mother was so proud that she hung the clip of the story prominently at home. As her protegee, I sent the story to all my friends in excitement.

As we became friends, Ms. N told me more about her decision to make the dramatic career change and move from Manhattan to here, the hometown of her then-husband (she was divorced when we met). The one thing that influenced her decision most was her father’s funeral. On that day, the whole town emptied and lined up around the block to show their respect. After teaching at the same school for forty years, he touched on the lives of so many people in the tightly knit community that practically everyone knew someone he had taught.

It reminded her of an earlier situation when she worked at GE Capital in the 1990s. General Electric was widely regarded as the world’s greatest company at the time, and GE Capital was its crown jewel, although later, it would become the problem child that brought GE down from its glory. Under the leadership of then-CEO Jack Welch, it went on a spree of acquisitions. On one occasion, Ms. N was tasked with restructuring a business after a merger, a process that often involves layoffs or closures to boost earnings and satisfy Wall Street.

The business’s main factory was in a small town in the Rust Belt, similar to her hometown in Ohio. When it was shut down, many locals lost their jobs, and the town was devastated, much like what was described in the book Hillbilly Elegy, the best-selling memoir that launched the political career of Vice President J.D. Vance. The decision was not hers to make and would have been executed regardless, perhaps more ruthlessly than she had. But seeing the misery of so many families while her colleagues celebrated with $1,000-a-bottle champaign, she carried the guilt all these years and worked hard to redeem herself.

Her journey as a teacher was not all smooth. She always knew she wanted to teach at a low-income school. Other people warned her that a school like this would doom her career, but she was determined to prove them wrong. In the first few years, she worked at a school that didn’t value her and provided little support. She didn’t even have a dedicated classroom like all the other teachers. She kept learning and improving the craft of teaching during those years. After she moved to her current school, she found a supportive environment where her resourcefulness, determination, and result orientation, combined with deep empathy for underprivileged students, enabled her to spearhead the math empowerment program.

The past several years had been stressful for her, and she was no longer the Wall Street lawyer who could routinely work until 2 a.m. At the end of my six-month student teaching under Ms. N, she wanted to hire me as her eventual replacement, and I was very interested. However, even with her help, I couldn’t overcome the bureaucracy to get the position. The position required years of teaching experience and an additional certificate, which I didn’t have. Nevertheless, she gave me the highest rating for my student teaching and ability to connect with students, and she continued to be my mentor after our formal relationship had ended.

During my six months of teaching with Ms. N, I never stepped inside her classroom. I walked around the school once during that period. It was located south of downtown, in a neighborhood largely populated by recent immigrants. The school facility was new, modern, and well-equipped. It seemed that the school district was throwing a lot of money at improving education. Yet, it didn’t work, as fewer than 35% of students were on track for college-level learning, and regular attendance declined from 85% before the pandemic to less than 70% afterward and never recovered.

Perhaps it was the pandemic, but I didn’t feel unsafe and wondered why several police cars were parked nearby. Soon, I found out the reason. After the curriculum night, right before the school was scheduled to reopen for in-person learning, there was a shooting outside the main building. A fourteen-year-old student from the school was arrested with a gun, and the school reopening ended up being postponed for a week. When I heard about it in the news, I was concerned for Ms. N and texted her immediately.

She was inside a classroom when she heard the shooting. She was only a little shaken because, over the years, she had dealt with similar situations, sometimes going to court to make depositions for her students facing criminal charges. She took a look outside to figure out the situation, then calmly returned to her work preparing for the upcoming semester.

Notice anything unusual?

No Chinese mathematician has won the award before and this year two did (it is awarded every four years, thus rarer than the Nobel Prize). Both Wang and Deng are Chinese citizens who attended K-12 and undergraduate in China (Deng transferred to MIT later). Tsimerman is a Canadian educated in Canada and U.K. A side story: the only American winner, Pardon, speaks fluent Chinese and loves Chinese culture.

This is not a “China is dominating” scare—three of them are currently working at American universities showing their attraction for top talents—but it does serve as a warning shot. The fundamentals of K-12 math education in the U.S. are slipping while the rest of the world is moving up (as I provided some details in this introduction). A large number of STEM majors in a major university (UCSD) couldn’t even do middle school math, and national assessments are pointing to multi decade low levels, as the system lacks any serious accountability.

To a large degree, this is masked by the large inflow of skilled immigrants from the rest of the world. Yet, they are often denigrated as outsiders who take away American jobs. With the current administration pushing to bring manufacturing jobs onshore, Bloomberg reported this month that a growing nationwide shortage of high-skilled workers threatens to delay construction of new semiconductor plants across the US.

Do we really want to wait for the “Sputnik moment” to wake up to the fact that our nation is losing its competitiveness because of its STEM education?

(Thanks to people like , , Huang, , , who tirelessly advocate for better STEM education, we are not completely losing hope, but it’s long and hard road).

(I am immigrant and a math teacher who tried to improve math education in the U.S., and here is my own story:

)

Note: I am continuing to post stories of the teachers I have encountered on my own teaching journey. The second teacher is Ms. Mel, who was my older son’s favorite teacher in middle school, and probably of all times. She inspired my son to be his best, but in the end, she was burned out at a young age.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you enjoy what you are reading, you can purchase the eBook for $2.99; or you can email me for a copy at yellowheight@hotmail.com)

CHAPTER 9 TO LIVE ON AN ISLAND

Ms. Mel was my son Matt’s science teacher in sixth and seventh grade. She was in her early thirties, good-looking, with short hair, dimples, and bright eyes. She had just moved from California to teach at Matt’s school, a high-achieving middle school where over 60% of the students came from relatively affluent Asian immigrant families. Parent expectations were high, but Ms. Mel always exceeded them.

On curriculum night, I sensed in her an earnest, capable, and kind—if somewhat nervous—young teacher. Her brilliant course introduction made many parents wish they could take her class, including my wife and me.

In her teaching, she incorporated many state-of-the-art teaching practices and content from Stanford, where she received her teacher’s training. When I told Matt about the groundbreaking genome-editing technology I had read about in The Code Breaker by Walter Isaacson, he replied that Ms. Mel had already taught it and had even allowed the students to experiment with this cutting-edge technology in her class.

Matt couldn’t stop talking about Ms. Mel. On her birthday, he gave her a gift card and organized a class surprise. Ms. Mel sent us an email that warmed our hearts. She said whatever we did to raise Matt, we should just keep doing it.

While vacationing in Costa Rica, Ms. Mel was impressed by the natural wonder and educational potential of the region. She organized a science field trip to Costa Rica for the 2020 spring break. Many students signed up enthusiastically, including Matt. Single-handedly organizing such a complex international trip was a daunting task. So many forms and procedures, so many questions from parents, so many meetings to figure out the preferences and trip details. I couldn’t imagine taking on such a task as a future teacher myself.

Then COVID hit. Ms. Mel postponed the trip several times and finally gave up. When the semester ended, I heard that Ms. Mel took a year of leave from school to take care of her mother, who suffered from Alzheimer’s. I also heard that Ms. Mel got divorced.

During my program for master’s degree in teaching at ULW, I remembered that Ms. Mel had graduated from a similar program at Stanford and decided to call her up to ask for advice. I ended up knowing more about her from the hour-long conversation than the two years when she was my son’s teacher.

Stanford’s program had a similar size and structure to ours but more prestige, with a roster of star professors who had invented many leading teaching theories used in our program. Like our program, half of the time was spent on field experience as an assistant teacher in schools, and half was dedicated to teaching theories.

Ms. Mel loved her program and thought everything was just wonderful. Through the program and later in work, she saw how inequities play out in schools and society. In her teaching, she tried to use what she learned to make sure every student had an equal opportunity to learn. She believed in the mission of creating a more just world through teaching and pursued it wholeheartedly.

Growing up, she always wanted to make her mother proud and exceed her expectations. Going to Stanford and becoming a teacher accomplished that, but the path she had chosen was hard. During her teacher’s training, she often had to work a full day as a teacher, then went to Stanford to continue theoretical learning until late evening. Some evenings, she didn’t even have the energy to drive home. She taught in California public schools for six years before moving to our state. Her second school was in Sunnyvale, a city known for its high home prices and big high-tech companies, in stark contrast to the fact that half the students in her school came from low-income families, and most were Latino.

Despite her dedicated efforts, many students simply didn’t listen to her. She tried her very best to accommodate and motivate them to learn, but every day, when she was driving home, she was crying. Though she did not feel it, the students felt the differences, and some told her she was the best teacher they ever had. She won the Teacher of the Year award in the district. Yet she still left. She wondered aloud to me if some of those students were still dealing drugs today: “Did I really change their lives?”

Every time I saw Ms. Mel, she looked tired, preoccupied, and a bit nervous. She looked like she was always trying hard but was unsure if the world would be satisfied. Teaching 6^th and 7^th graders was tough for her because students that age were very energetic, curious yet uncontrolled, constantly trying to test the boundaries. She treated every student patiently and gave them learning opportunities, but she thought a few students including Matt really gave her a lot of support and comfort and thanked me again.

She wanted to get away from the crowd and the expectations weighing on her shoulders, expectations she had been trying so hard to exceed. She had just left her cramped commuter condo and bought a house on an island a few hours away. Inadvertently, she said, “Costa Rica was the last straw.” Throughout our call, she had been earnest and positive; this was the only moment when a trace of sadness slipped into her voice. Her expression, too, looked a little lost.

Suddenly, I understood: the pressure of all those years had never left her, and she was burned out.

As we were finishing the conversation, I looked out the window onto the lake. The sun was just about to set behind the clouds, a light mist began to gather on the surface, and the opposite shore was barely visible. Far beyond the lake, across another body of water, was Ms. Mel. Would we ever see her again? Would she ever return to teaching? How had this happened to one of our best teachers?

Ms. Mel stayed on the island and never returned to the classroom. After that, I only heard bits about her life from Matt: she was tutoring online and had just had a baby, though there were no mention of her getting married. A few years later, when Matt traveled to the island for a track and field event, I wanted to give him a nice surprise. I texted Ms. Mel to let her know, she replied but didn’t say whether she would come.

During the event, I was busy working as a parent volunteer and didn’t watch Matt run. When I got a moment to see him afterward, he was jubilant. He saw Ms. Mel right before starting his run and chatted with her briefly after finishing. Then she disappeared into the crowd, just as mysteriously as she had appeared for him, like a guardian angel. Matt broke his 400-meter personal record under her gaze. To this day, Ms. Mel still inspires him to be his best.

The first teacher was Mr. Ase, a young African American teacher teaching middle school math in a working class neighborhood. I did my observation in his classroom as a prerequisite for the “Master’s in Teaching” program in a leading university. He impressed me and influenced me in multiple ways, and I have a lot to thank him for.

This is an excerpt from my recent book, “Unbalanced: Memoir of an Immigrant Math Teacher”. If you enjoy what you are reading, you can purchase the eBook for $2.99; or you can email me for a copy at yellowheight@hotmail.com)

CHAPTER 8 “THIS IS NOT ABOUT ME”

The ULW teaching program requires the applicants to complete forty hours of classroom observation. As the head of the math department at a public middle school in a poor, working-class neighborhood, Mr. Ase offered me his classes for observation. When we first met in person, I hadn’t expected to see this young, self-assured, and muscular African American man, just as he hadn’t expected a middle-aged, bespectacled, nerdy Asian male: he addressed me as ‘Ms. X’ in an earlier email.

Mr. Ase was in his mid-thirties, with a well-trimmed beard covering his face. He smiled often, but even in his relaxed state, he radiated strength and confidence. His classroom was minimalist, with few decorations aside from a large, framed photo of him and his wife on the wall. In the photo, his wife, a beautiful, kind-looking blonde, was sitting on a swing in a backyard with half-finished landscaping, while he stood beside her.

American classrooms shocked my senses. The last time I experienced a middle school classroom was thirty years earlier, in a different education system. The classrooms I attended encouraged concentration and focus, where students were engaged in their studies, only making necessary noises and interactions. Students had a deep respect, even fear, for the teacher.

U.S. classrooms had an energy level that was simultaneously exciting and distracting. Students were seldom quiet, and there were often multiple threads of conversation going on, with several tasks and procedures involved. Teachers spent significant time and energy directing but not demanding student attention. Students explored concepts through various tools and activities, while teachers acted as conductors, guiding the flow of ideas and discussions.

Despite Mr. Ase’s ability to command more respect than most of his peers, trouble would inevitably emerge in the classroom. Some stemmed from typical juvenile behavior, but a few students simply refused to do whatever they were asked to do and talked back in defiance. At times, I could sense the tension rising in Mr. Ase, but he always kept his cool. He could be stern and appear serious, but he never lost his temper. Despite his outward sternness, Mr. Ase never ceased to respect or care for any of them.

Mr. Ase was a natural storyteller. He could talk about his life and family naturally at the right moment and make you feel close to him. He could draw real-life examples to illustrate math. One day, some students were playing with chairs somewhat dangerously during class. Instead of chiding them, he began discussing the mechanisms of insurance for broken chairs as an example to illustrate the concept of proportions. The students stopped and listened.

I often stayed after classes and got to know him more. His early jobs were in daycare and pre-schools. For several years, he worked with children with special needs. Even though he loved them, he was sometimes hit by those children and returned from work with his arms all scratched raw by the kids. After that, he worked as an assistant teacher for seven years before earning his teaching certificate from the same community college where he had completed his undergraduate studies.

When he started formal teaching, he had already accumulated over ten years of experience working with children. Within five years, he became the head of the math department, part of the union’s negotiating team with the school district, and the school leadership team. In his spare time, he ran for a city council seat on a platform on local issues including education. He lost but planned to re-run.

As part of our ULW teacher’s program, I needed to interview stakeholders in education for a project and contacted Mr. Ase in the summer of 2020. He gladly agreed to an interview.

I asked him how his students were doing during COVID. He said they were just coping. Over 70% of the students were students of color, and more than half came from low-income families. 5% of them were classified as “homeless” because they lacked a regular place to sleep and often lived in places like cars or emergency shelters. COVID exacerbated the inequities that already existed. When it hit, the school district ensured that every student had access to a computer and the Internet. Mr. Ase posted his recorded lessons along with assignments, but by the end of the semester, only 20% of students had accessed them.

He didn’t blame the students. School was a place where the presence of adults motivated students to learn. For students from lower-income families, both parents often had to work—sometimes two shifts—leaving them without supervision or help while also caring for younger siblings. His priority was to make sure students were healthy and being themselves. In his class session, he allowed all students to share their screens with the whole class at the same time, and it was a fantastic experience when the students competed to share their goofiest moments.

Talking about missing math content due to COVID-19, Mr. Ase was understanding of the obstacles faced by his students. He emphasized that sixth-grade math was universal and important for being a good citizen, while seventh and eighth-grade math was only important to keep some doors to opportunities open. If students missed it, they could pick it up later by learning how to apply old math to new concepts. Student well-being was more important than learning just a little more math during this unique time.

To him, the purpose of education is to provide opportunities for all children. For many of his students, especially those from low-income families, the doors to life’s opportunities start to shut as they progress through the education system. Teachers can show them all these available opportunities. By staying on track and learning in school, students can keep as many doors open for as long as possible.

During our call, he stepped away briefly to tend to his wife and one‑year‑old son. It was July 2020, a time of turmoil and uncertainty in the outside world, yet he sounded completely calm and composed. When I asked how he remained so serene, he chuckled and said he felt accomplished. With everyone confined to their homes by the pandemic, he had used the time to finish landscaping the backyard, which gave him a deep sense of personal fulfillment. We soon transitioned to the topic of social justice. He said:

"We need to prepare students for what they will face in society. Make them aware of the current situation, the history, the progress made, and things still unfulfilled. We need to hold our peers accountable to high standards. Don’t let little things slip.

Let me give you an example: a student called another student ‘gay’ in a derogatory way. This is not acceptable. However, every student will have a shot with me. The students had a chance to explain why they had done it and commit to change. It is important to hear them out first.

Follow the path of why they did it, what points the students were trying to convey, why it was inappropriate, and what actions should be taken. If repeated, there will be a big group meeting where the issue will be discussed during class.

Even if a student calls another the N-word, it is still not straightforward. You will need to know the context and their relationship in order to give the right message.”

Mr. Ase observed that a lot of African American students were treated either too softly (because teachers were too gentle and afraid to hold them to high standards), or as total write-offs. Many of them were suspended primarily for disruption or defiance toward teachers. He never wrote any of his students up for these infractions but instead tried to find the underlying reason and showed great understanding and tact in dealing with the trouble-making students. I learned from Mr. Ase not to use discipline as a teacher in my classroom later.

On being a new teacher, he offered:

“First-year teachers have it tough. Building meaningful connections with students is a skill that takes time to develop. On top of that, the ability to teach math effectively, especially in a school setting, must also be learned. Managing both at once can be overwhelming, particularly at our school, where many students came from challenging family backgrounds and required extra effort to engage. Three out of the four new teachers in my department left after their first year. One of them, a good person and a friend, transferred to a less demanding school to focus on content teaching first.

I almost quit several times during my first year as a certified teacher. Many days, I returned from school and cried to my wife because it was so hard. There were three main reasons: Number one was due to stress from parents. They could be very unreasonable and do real damage to a teacher. The second one was due to my own standards not being met. This one time, 70% of my class failed the state standards, but I was congratulated because this was better than any of the previous years. I felt really bad.

The third is that there will be a lot of unexpected things. To be a good teacher, you have to be internally driven but also learn to let go. For example, this twelve-year-old girl in my class once did an elaborate drawing especially for me, then threatened to stab me with a knife a month later. You have to be able to say to yourself: ‘This is not about me.’”

As Mr. Ase described the incident, his voice and expression never changed, as if it hadn’t disturbed him at all. Perhaps it was because he understood it wasn’t about him and didn’t take it personally. He explained that the girl hadn’t acted out because of anything he had done, but because something in her life had gone wrong, and those problems had spilled into school. A teacher must be able to recognize situations like this, separate their own emotions, and resist the urge to force a quick turnaround in the student’s behavior. Reacting hastily can backfire and even harm the student, whereas patient communication is often far more effective.

To fend off burnout, he became the volunteer coach for the school’s wrestling and track team, allowing him to connect with students outside the classroom in a different role without the pressures of teaching. Those moments helped him rediscover the joy of working with kids and rejuvenated him.

When I became a new teacher, I immediately signed up for tennis coaching, which helped with my student connections. Later, when faced with challenging student situations as a teacher, I often recalled Mr. Ase’s strong presence in the classroom and heard him gently say: “This is not about me.” It always helped me stay calm.

First, the balance between accountability through quantitative metrics and wholistic education that emphasizes qualitative assessments.

Second, the balance between supporting students in need (especially those with diagnosed learning disabilities) and challenging students to develop grit.

Third, the balance between education’s social goals such as equity and emphasizing rigorous learning goals.

Idealists would say that we could do all, but in practice, the boundaries between those opposing forces are often drawn in less visible ways yet have significant ramifications. This chapter from the book discussed the third imbalance, how the quest for equity in education has compromised excellence.

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CHAPTER 45: THE EDUCATION SYSTEM

· The Political Influence in Education

The U.S. education system is highly decentralized, fragmented, and susceptible to political influence. While the federal government plays a role, public schools are regulated and funded primarily at the state and local level, with voter-elected school boards making operational decisions. The existence of over ten thousand school districts in the U.S. underscores the system’s fragmentation, lack of standardization, and resistance to top-down reform.

This complex structure has given rise to massive bureaucracies and systemic inefficiencies. As day-to-day administrators, school principals have limited power to make long-lasting decisions, and even public schools with long histories and academic traditions find it hard to control their own fate.

Lowell High School in San Francisco is one of the oldest public high schools in the U.S. and accepted students based purely on merits before the COVID pandemic. Its academic tradition was responsible for the extraordinary achievements of its graduates, including Nobel laureates and Supreme Court justices. During COVID, the school board changed its admission policy to a full lottery system and later voted to make it permanent.

A New Yorker article, “What Happens When an Elite High School Becomes Open for All,” depicted teachers’ heroic efforts to educate a drastically different student body amid limited resources and abrupt policy shifts. Reading it as a new teacher in a similar situation, I strongly resonated with the teachers’ profound sense of powerlessness.

This decision triggered widespread protests from parents and the alumni association, who sued the school district. Meanwhile, they organized the largest special election in the history of the San Francisco school district and successfully recalled three board members, ultimately restoring the admissions policy to what it had been before the pandemic.

Throughout this multi-year process, students were the biggest victims. The continuity and stability in learning that they most needed were severely disrupted by the political turmoil. In the first graduating class admitted by lottery, over 10% of students dropped out before graduation—more than twice the rate of previous years. Among those who remained, the average GPA fell from 3.69 to 3.45, and SAT scores declined by 78 points compared to prior years.

• The Goals of Education

According to education scholar David Labaree, U.S. education has three social purposes: democratic citizenship, social efficiency, and social mobility.

Throughout much of the U.S. history, public schools have aimed to create relatively equal conditions for citizens through education access and a strong culture of civic commitment. The first purpose, democratic citizenship, remains especially important today through civics education and has the potential to unite the nation and combat increasing political polarization.

The second purpose of education, social efficiency, is to provide the economy with skilled workers. Economic needs for labor evolve over time, and education needs to keep pace. In today’s information age, when access to learning resources is widely available, education should evolve from imparting knowledge to teaching students how to think and learn for themselves. The looming AI age challenges education to fulfill the need to provide students with new skills, such as asking the right questions or judging answers from outside sources logically.

The third purpose of education, social mobility, is to prepare students to compete for social positions and realize individual dreams. Americans have generally equated fairness with social mobility, which is the very fabric of the American dream. This is related to meritocracy, which assumes that if opportunities can be awarded based on merit, a fair society can be achieved.

However, as opportunities became scarce, competition intensified, and education started to resemble a consumer product in the marketplace. Parents wanted the best for their children’s education and were willing to use their resources to get it for them. This resulted in good educational opportunities going disproportionately to students with more family resources and served to diminish social mobility achieved through education.

Through the power of education, which was increasingly influenced by students’ family backgrounds, economic stratification has become more extreme and rigid in recent decades. In 2023, the top 1% held 31.5% of total wealth in the U.S., while the bottom 50% held only 2.5%. The Federal Reserve Bank of Minneapolis (2021) reported that only 50% of individuals born in the 1980s earned more than their parents, compared to 90% of those born in the 1940s, indicating that the social mobility goal of education has regressed significantly.

• The Quest for Social Justice

Throughout most of America’s history, the struggle for justice has centered on equal rights for women, people of color, and other disenfranchised groups, with significant progress achieved on this front. However, the quest to improve the condition of the worst-off members of society has faltered, resulting in economic and social inequalities unparalleled in U.S. history.

The term “equity” remains confusing to many, including certain Supreme Court justices. The dictionary defines equity as “the quality of being fair and impartial” or “justice according to natural law or right.” Here, “just” and “fair” generally align with John Rawls’s book A Theory of Justice, which asserts that members of society have equal rights, and inequality should only exist if it improves the situation of the worst-off members while offering equal opportunities to all.

In practice, equity is related to, but not the same as, equality. It can mean three things: equal treatment, equal outcomes, or equal opportunities.

Equal treatment is an important part of procedural justice but is often insufficient. For example, disabled students need special accommodations, and simply applying common sense judgment is inadequate.

Equal outcomes are rarely cited as an explicit objective of equity, as they are often associated with communism. However, as Thomas Sowell’s writings point out, the pursuit of vaguely defined “equity” often leads to policies aimed at equal outcomes, such as a quota system.

The concept of equal opportunity has widespread support, but its interpretations can vary. Before the 1954 Supreme Court decision on Brown v. Board of Education, educating students in segregated schools was considered “separate but equal.” Societal attitudes have since evolved, but the boundaries of equal opportunity are still being tested and reshaped by current events. For example, the 2023 Supreme Court decision on college admissions found the race-based affirmative action policies at Harvard and the University of North Carolina to be unlawful.

In response to increasing wealth disparity and its failure to achieve social mobility, education has embraced a fourth goal: equity and social justice. From this perspective, education is less about learning and more about using its power to allocate social access and opportunities to achieve distributive justice—correcting the inequalities created by social and market forces.

· Equity In Education

The focus on equity, written into the mission statements of many public (and some private) schools, deeply influences who gets hired, what programs are adopted, what learning standards are used, what and how to teach, and what behavior is rewarded and punished in schools.

In teacher hiring, subject competence is considered far less important than the full embrace of equity and social justice. Teachers are encouraged, sometimes required, to incorporate social justice into teaching materials and grading practices. Those who express views inconsistent with equity are criticized or even fired, while those espousing such views are publicly praised.

Such prominent focus on equity in education can have serious consequences. Education has a clear duty to contribute to a fair society. Even considering the differences in the interpretation of equity and social justice, the current distribution of wealth and privilege is an urgent issue that threatens to destabilize our society. However, education cannot address the larger forces behind inequity, as distributive justice must be achieved primarily through the economic and social systems that distribute rewards, not the education system.

In doing so, we are asking education to fix social and political problems that we are unwilling or unable to address through political action. Elevating equity to such a high-profile goal gives education a moral calling—one that can transform rational issues into moral ones, often at the expense of logical reasoning. While this creates urgency to change the status quo, it risks backfiring, if not accompanied by broader political and economic reforms.

First, it detracts from the goal of excellence. When equity overtakes as the top priority in education, individual excellence becomes de-emphasized, even stigmatized. This explains why some schools are removing learning differentiation and limiting access to advanced academic programs. It also contributes to the lack of accountability, or the wrong kind of accountability that focuses purely on reducing the achievement gap within the student body and sacrifices excellence.

Public school districts that have most actively pursued equity goals have experienced relatively steep declines in high school student achievement, including New York City, Los Angeles and Chicago. Seattle Public Schools—one of the nation’s most equity-focused districts—spends 30% more per student, yet its students’ proficiency levels remain 17 to 25 points lower across all subjects, compared to those in its neighboring, less progressive school district.

Second, some equity policies in education can produce more inequitable or less inclusive outcomes. The decline of education quality caused parents to flee public schools. Even Mr. Felix and Ms. Mira—both ULW (the college I went for an education degree) instructors and prominent advocates of social justice in education—have withdrawn their children from public schools for private school or homeschooling. Equity-focused school districts such as Los Angeles, Chicago and Seattle have all seen over 10% decline in the past decade, forcing school closures—a painful and disruptive process disproportionately harming disadvantaged students.

The decline in educational quality not only undermines excellence but also fails to advance equity as it deprives disadvantaged students of opportunities through excellence. Pathways for these students to pursue academic excellence through schools once existed, but when schools lower standards in the name of equity, these opportunities start to close off.

The lowering of learning standards in school forces students to seek alternatives outside the school system, such as afterschool tutoring and enrichment programs. According to the Afterschool Alliance, annual spending on these programs has ballooned to an estimated $23–25 billion in 2025, up from $5–7 billion two decades ago. This trend undermines equity because, unlike privileged students, those from disadvantaged backgrounds often cannot afford these programs.

Well-intentioned but ill-conceived education policies favoring underprivileged students often leave them unprepared for workplaces that offer no such accommodations. This occurs when educators, in the name of equity, lower expectations, give inflated grades, and provide excessive assistance—practices that do not reflect how the outside world operates.

Several friends at Microsoft shared similar stories: a member of their team had ADHD and received more than normal time for assignments while in school. However, such accommodations couldn’t be sustained in a workplace environment, eventually leading to dismissal or resignation after years of underperformance despite significant support from the company.

Diversity should include differences in life experiences and opinions, not just those related to identity. Ironically, the emphasis on narrowly defined diversity has resulted in a severe lack of diversity in social and political views within education. The ULW teacher candidates and the teachers I encountered at work represented a narrow band of our society’s spectrum of political and social perspectives, but as educators, they disproportionately influence the next generation.

Third, equity policies often contribute to student fragility by overemphasizing the needs of the vulnerable. Vulnerability and fragility have subtle differences: vulnerability is the state of being exposed to harm, while fragility is the quality of being easily broken. One is an inner attribute of a person, and the other is often externally caused.

While society has a responsibility to protect its vulnerable members, education also has a duty to make students less fragile. Given appropriate exposure, students can often grow out of their fragility, even vulnerability sometimes, by developing natural resistance and immunity. Shielding them from challenges deprives them of the opportunity to build that resilience.

Excessive fear of harming the vulnerable often leads to practices that increase the fragility of students. Under the culture of fragility, teachers are discouraged and afraid to challenge students with learning issues. They feel obligated to provide accommodation, often beyond what is officially required, to demonstrate kindness. The resulting excessive kindness often led to coddling and reinforced a culture of fragility, which became even harder to change.

Fragility can manifest in various ways. Statements such as “these words are ideologically violent,” which I often encountered at the ULW education school, can reinforce fragility and provoke disproportionate reactions to discomforting perspectives. Simplistic and strong views on sensitive issues can turn students into activists for perceived injustices, leading them to take uncompromising stances. They often appear belligerent and tough on the outside while remaining intellectually and emotionally fragile inside.

Fourth, the focus on equity leads to accountability issues. It not only contributes to a lack of accountability for student learning but also promotes the wrong kind of accountability. In 2023, the principal of Princeton High School was suddenly dismissed, prompting widespread protests from students and parents. This principal was not only highly dedicated and exceptionally skilled at community building, but he also placed strong emphasis on students’ academic achievement. The official reason given by the school district was unconvincing, and many parents believed the real reason was his failure to meet the district’s “equity” targets. The school board had expected him to narrow the achievement gap within the school, but he was unwilling to do so at the expense of high-achieving students’ performance.

Lastly, it threatens to erode the future competitiveness of our country. The nation’s quest for AI dominance critically depends on talents from STEM field and the current system is failing especially in this respect. In a March 2025 interview, investor Ray Dalio stated that while 1% of the U.S. workforce is world-leading and 10% is highly capable, 60% lack the basic reading and math skills required for modern manufacturing jobs—a view largely consistent with national assessment data.

For now, the U.S. talent base remains strong, primarily due to its ability to attract top talent from around the world. For example, more than 60% of the top math talents in the U.S. got their undergraduate degrees from abroad, and it is reasonable to believe that a large portion of the remainder finished their K-12 studies outside of the U.S. as well. Without a more robust K–12 education system, serious challenges are likely to emerge for us to compete with other nations—particularly in light of current trends in de-globalization and shifts in immigration policy.
—-

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“In 2024, one out of nine of Huang’s students finished her advanced math class with proficiency – or grade-level understanding. But at the end of last school year, 11 of 17 of her students were proficient. Huang credits student growth to Xu and the district.”

A few weeks back I visited the middle school mentioned in this article and got to see the environment in which Ms. Huang was teaching in. It is indeed a Herculean effort to get the student so excited about math. I wish more school districts would be interested and open to try Mr. Xu’s method.

Congratulations to both teachers!

To better understand this phenomenon, I searched Google Scholar to see how often “School Accountability” and “Educational Equity” appeared in educational publications from 1995 to 2024. A clear pattern emerged: 2010-2015 was the peak for “School Accountability”, after which it gradually declined, while “Educational Equity” rose to be the dominant policy over the past decade.

This fits with the historical context. After the “A Nation at Risk” report in 1983 raised the alarm, the accountability movement in education began. The No Child Left Behind Act of 2001 (NCLB) mandated that schools be held accountable for measurable learning outcomes, ushering in what I call “the accountability era.”

Meanwhile, resistance began to build against NCLB, with the Every Student Succeeds Act of 2015 (ESSA) marking the beginning of “the equity era.” ESSA was hailed for its advancements in educational equity, its promotion of holistic education frameworks, and its emphasis on equal access and student well-being. Accountability measures were watered down.

***

A decade has passed since then; let’s look at the scorecard. The National Assessment of Educational Progress (NAEP) is the gold standard for evaluating K-12 academic outcomes, and the latest report came out at the end of 2024. The graph below shows the NAEP mathematics scores for 12th-grade students from 2005-2024. We can identify three patterns in this graph, which also exist in other subjects such as reading and science.

First, the average score declined from 150 to 147 during this period. Second, the decline was not linear. It increased by 2 points from 2005 to 2015, before decreasing by 5 points from 2015 to 2024. Third, the top 10% of students’ average score increased by 2 points, while the bottom 25%’s declined by 7 points.

The fact that K-12 education has declined in the past decade is undisputed. Less appreciated is the contrast between “the accountability era” (2005-2015) and “the equity era” (2015-2024): the former achieved across-the-board improvement in learning, while the latter more than squandered the hard-won gains.

What’s more, “the equity era” has worsened actual “equity” by widening the learning gap, as measured by the dispersion between top and bottom performers. This is especially disconcerting given that public education spending per student increased by 52% from 2015 to 2024, more than double the 19% increase from 2005 to 2015. More money was spent, yet worse outcomes were achieved across all dimensions.

***

The COVID pandemic played a large role, though the overall decline and widening gap were already evident in the 2019 data. So, it cannot be the only reason for the learning loss.

In order to remove the impact of COVID, a study of the relationship between equity focus and learning loss from 2015 to 2024 was conducted at the state level. Since there is no reliable data to measure “equity focus”, a widely respected index— the “State Policy Liberalism” score from the Caughey & Warshaw Dynamic Democracy project — was used.

The pattern is clear: states with more equity-focused policies experienced higher learning loss from 2015 to 2024. This correlation is highly significant: the probability of it being due to random chance is less than 0.1%.

While correlation does not equal causality, such a strong correlation deserves a deeper look. It is unlikely that learning loss caused equity-oriented policies during this period. It is possible that some common factors are driving both equity-focused policies and learning loss, although some obvious candidates, such as ‘economic inequality’ (as measured by state-level Gini coefficients), or ‘urbanization’ (as measured by population density), turn out to have much weaker correlations and are unlikely causes.

Combined with the strong evidence that learning gains occurred in “the accountability era” and learning loss happened in “the equity era”, we cannot escape the suggestion that the learning loss is to a large extent the result of replacing “accountability” with “equity” as the main focus of education. Indeed, there are clear mechanisms that translate the equity focus into learning loss through the erosion of accountability.

***

First, the goals of equity are often ambiguous and hard to measure. So, the system focuses on inputs instead, such as money spent, programs initiated, and representation of different groups, rather than on outcomes. When inputs become the primary focus, accountability for learning outcomes is invariably weakened.

Second, equity frameworks treat unequal outcomes as evidence of systemic failure. That creates pressure to inflate grades, reduce rigor, and discourage standardized tests, leading to practices such as unlimited retakes, minimum grading floors, and reduced penalties for missing work.

Third, equity-driven curricula deemphasize content, prioritize “identity,” “belonging,” or “student voice”, and avoid challenging coursework. As a result, standards become harder to evaluate, and assessments lose their meaning as objective measures.

Fourth, equity-focused reforms often remove student achievement from teacher evaluations and emphasize “culturally responsive practices” or “equity competencies” instead. When teacher accountability is decoupled from student learning, instructional quality becomes harder to monitor or improve.

***

My experience as an immigrant and a late-career teacher offers a unique window into the inner workings of schools as an outsider. At education school, when I tried to defend standardized testing as an objective measure that is important for accountability, I was attacked by the entire class. Almost all my job interview questions were centered on how to implement equity in teaching; few were about content mastery or pedagogy.

Having worked for two decades outside of education, I could not comprehend the lack of accountability at the school where I was teaching. There were no performance reviews, and all feedback was politely qualitative. When another late-career teacher asked for performance feedback, he was told by the head of school: “You must be doing a great job, since I have not heard any parent complaints.”

Teachers had great discretion in giving out grades, student grades did not reflect mastery, and student achievements mattered little to a teacher’s career. There was constant pressure on teachers to give good grades to students, especially those with vulnerabilities, while taking into account the life challenges those students faced.

My experience was not isolated. In November 2025, a faculty report from the University of California, San Diego (UCSD) found that nearly 900 freshmen (1 in 12) failed to meet middle school math standards. Yet 94% of these students had taken advanced high school math courses with an average GPA of 3.65. The grade inflation, often carried out for the cause of equity, forced professors to “reteach middle-school mathematics while simultaneously teaching the material students need for sciences, engineering, economics, and other quantitatively demanding fields.”

***

Is the current administration’s war on DEI (Diversity, Equity, and Inclusion) properly addressing the accountability issue? Hardly. While the federal government exerts significant leverage over colleges, its contribution to K–12 funding is relatively small, at 10%, limiting its influence. The equity-focused mindset is deeply entrenched in the education system and resistant to external pressures to change.

K–12 education is primarily funded and controlled at the state and district levels. Recent headlines have highlighted anti-DEI legislation at the state level, but in reality, more pro-DEI legislation has passed at the state level in 2024 and 2025. In May 2025, the San Francisco Unified School District introduced a new “Grading for Equity” policy, proposing that overall scores of 80–100 would receive an A, 60–80 a B, 40–60 a C, 20–40 a D, and 0–20 an E. Other recommendations included allowing multiple retakes and assigning a minimum grade of 50% for every task. While it was withdrawn quickly due to strong opposition, it reflected the still deep-rooted influence of equity ideology in education policy.

Diversity, equity, and inclusion are ideas that can be both productive and morally desirable. However, these words have been hijacked by advocates to mean different things and have often been carried to extremes. Setting them as the goals of education politicizes it and, as a result, weakens accountability. The past decade has shown that once accountability is lost, everything is lost, including equity.

There needs to be a balance between accountability and student experience, both of which are part of education. To correct the current imbalance, we must learn from the lessons of NCLB, which was a federal program that suffered from implementation issues. The recent Mississippi Miracle provides a good roadmap for state-level programs. Through grade retention, pedagogical change, and teacher support, Mississippi raised its 4th-grade reading ranking from 49th in 2013 to 9th in 2024. While 8th-grade improvements were less significant, it still ranked 4th nationally after demographic adjustments.

PS: Following is the link my recent memoir, which described my experience as a late-career teaching in “the equity era”.

I have written my recent book “Unbalanced: Memoir of an Immigrant Math Teacher” to tell a story that is similar to his, in the post-pandemic era. It’s a good comparison. You can either buy the book, or read free chapters HERE.

BTW, did everyone read the news of 1400 University of California professors petitioning to restore SAT/ACT math as requirements? “We now observe preparation gaps so severe that instructors must reteach middle-school mathematics while simultaneously teaching the material students need for sciences, engineering, economics, and other quantitatively demanding fields,” the professors wrote.

本章的中文版在后面部分，英文版之后)

Chapter 10 Spreading My Teaching Method

I want to dispel a common misunderstanding: my students excel in mathematics primarily because of my method, not because of my personality.

How many teachers truly recognize and want to learn my method? To be honest, many teachers come to seek advice from me, but most of them treat me as some kind of “sage from the East.” They come to me with math problems because I can not only provide correct answers, but also explain clearly why something works. Yet when it comes to my teaching method, they often attribute it to “Chinese wisdom,” assuming it is my personal talent rather than a system that can be taught and replicated. Principals praise my results, and even principals from other schools say I have achieved “astonishing success,” but no one has ever asked: Could you systematically train our teachers in your method?

I tried to promote my method within my school district, but there was no support from the administration. One reason is that teachers instinctively resist the IDK (“I don’t know”) method. They worry students will use it as an excuse to slack off and write IDK on every assignment. Because they have not yet built effective relationships with students, they rely on a “push-resist” model to maintain classroom order. My method requires the first two weeks of the semester to fundamentally reshape students’ thinking habits—something the current system finds difficult to accept and lacks the support to implement. So I received almost no formal recognition in the district. People knew I was a good teacher, but the culture of the education system tends to avoid highlighting individuals and instead emphasizes collective effort. Combined with the stereotype of “Of course, Chinese people are good at math,” my method was dismissed as “Eastern mysticism.” People praised me and respected me, but they did not truly want to learn my method.

I thought to myself: principals in K–12 schools often have limited mathematical and pedagogical background, so it’s understandable if they don’t understand me. But what if I approached the experts and scholars who study pedagogy and theory? Surely we could communicate. I had years of student achievement data, and my philosophy aligned with many of their theories closely. I could even point out why their theories were correct in principle but useless in practice, while I used the same principles to produce excellent results. They should understand immediately and adopt my suggestions, right?

***

So in 2014, I took early retirement, preparing to bring my teaching method to the outside world. I thought that all I needed to do was knock on the door of any school, show them my student data, explain where their theories were right or wrong, especially why their implementation failed. I could explain all of this clearly. Naturally, I assumed schools would invite me to speak; even places like Stanford should certainly want me to lecture. But the reality was: no one paid attention.

I gave a few talks at Arizona State University. People did come, but I was a nobody, so they put me in a very small classroom. The room was packed—people were even sitting on the windowsills—but that was it. Giving two or three talks wouldn’t lead to anything substantial. Teachers found it “interesting,” but once they went back to their schools, everything stayed the same.

I submitted proposals to speak at the annual conference of the National Council of Teachers of Mathematics (NCTM), but they never accepted me, not even once. On TED, a teacher could give a talk on something as simple as “students will not learn from teachers they don’t like.” I had at least achieved that, and my students’ results were right there, unmistakable, yet no one wanted to hear me. Not only did no one invite me, even my first-round applications were rejected. They politely asked, “Would you like to see the feedback?” I said yes, but I never received a single concrete comment. I truly couldn’t understand. Maybe they didn’t know how to reject me, or felt embarrassed to say it directly, so they simply gave no explanation.

I also applied for a position at Yale University related to K–12 mathematics education. I filled out many materials, but was rejected outright, again with no feedback.

Later, I recalled a profound insight from Schopenhauer: telling someone they are poor or foolish may not hurt them the most; the deepest wound comes from implying, “Your ideas are wrong, and your intelligence or wisdom is inferior to mine.” That is what people find hardest to accept. Only then did I truly understand why, when I brought my data to any university or institution and said, “You’ve been talking about closing the education gap for decades. Since the 1960s, the gap hasn’t narrowed; it actually has widened with every reform. I’ve taught for over twenty years, and my students have almost no gap. Do you want to know why?” no one wanted to listen.

Later, I learned about the experience of the famous math teacher Jamie Escalante. At Garfield High School, a low‑income school in Los Angeles, he dramatically raised students’ math achievement and helped many gain access to college. He became nationally famous, was featured in a movie, and even received a special award from President Reagan. But the attention he received made other teachers look incompetent, and that was something people could not tolerate. He faced numerous obstacles from the school system: some administrators disapproved of his strict classroom expectations, such as requiring students to answer a homework question before entering class. One assistant principal even threatened to fire him for arriving too early, staying too late, and raising funds for AP exams without permission. These tensions eventually led to his departure in 1991. After he left, student performance plummeted. Critics then claimed his success depended heavily on his personal charisma, long hours, and unique style, therefore his model could not be replicated.

***

Despite many setbacks, I continued trying in every way I could. Along the way, I met friends who wanted to improve American education, including a group called “SaveMath.” Through their introduction, I received support from the Palo Alto Rotary Club and began working with a school in East Palo Alto to help teachers adopt my method. At first, three teachers participated. Later, one left the school, and another had a baby and no longer had time, though both felt they had benefited greatly.

Helena has persisted for two years. At the beginning, I spent two weeks training her mathematical thinking. Although she had worked as an engineer for half her life and used plenty of math, her understanding was mostly at the level of tools. The training helped her grasp mathematical ideas more deeply. When she returned to school, she told me teaching felt easier, students were more willing to listen, and classroom disruptions decreased. I told her this was because once she understood math more thoroughly, her confidence grew and students could feel it, making them more willing to respect her. In addition, I visited Helena’s school to observe her teaching, give specific guidance and feedback, and even taught several demonstration lessons myself.

I hope more schools can work with me in this kind of deep collaboration. This is one of the reasons we are writing this book.

***

Feedback from 1^st adopter Teacher Ms. Helana (1)

I teach 6th grade mathematics at a public junior high school in a suburban town in Silicon Valley. Most of my students are from low-income immigrant families. The average of my students is at less than the 3rd grade math level, and several of my students are at the first-grade or kindergarten level. During my first four years as a teacher, the classroom was always noisy and fights broke out at any time. Most of the time, I couldn’t say a whole sentence and was constantly interrupted by students. This summer I met and learned from Mr. Xu, a retired transformative teacher from Phoenix. My classroom has changed day and night. I decided to record these truthfully and share them with more people who care about the growth of children!

I found that children growing up in the United States are very fond of sports, especially our students who do not have the confidence to become doctors, lawyers, and engineers. Most of the children dream of pursuing a career in sports. I had always thought about asking students to do some sports-related mathematics, but I didn’t expect that Teacher Xu directly integrated sports fun into mathematics teaching, making the children excited as soon as they entered the classroom.

It’s so common for us to fail to make a basket in basketball. We just wave our hands and keep shooting. People next to us often shout, “Nice try! Try again! You will get it!” But in math class, when we get the question wrong, the teacher gives you a red cross. The red cross could make the children’s mood hit rock bottom. Many were completely unable to muster the energy to continue studying. The core of Teacher Xu’s classroom spirit is to normalize “I don’t know”. There is no need to feel ashamed at all, but it is a great thing to discover your own flaws and find a direction to work towards!

Many teachers like the growth mindset very much, and I believe many parents have also heard of the growth mindset. I have taught students this growth mindset for many years and have not seen any impact. Unexpectedly, following Teacher Xu’s clever trick, I captivated the whole class’s attention within a few minutes! After the class bell rang, I gave the children a paper and asked them to read it through and write down I know (IK) or I don’t know (IDK). Then I gave them the first A and told them that you participated! Then let them write down how they solve the IK problems, and I will give them a second A because they worked hard. Next, the last A requires them to make progress. I asked them to exchange papers with the classmates next to them and talk about your problem-solving process. Either what you learned or what you taught others, this is your progress today. Because by teaching others you turn your knowledge into an ability, an ability that enables you to become a tutor and make money. Many children were so excited when they saw the three A’s on their first math paper in junior high school that they strongly requested to take it home to show their parents! In just 10 minutes, the children have already experienced the growth mindset intensely!

From then on, the children entered the classroom with one paper every day. Other classes were still making noise in the corridor and could not enter the classroom. My students were almost finished with the first paper. They compete for the rank in the class in answering math questions. Even if they write a lot of “I don’t know”, they can still compete with the kids who are the strongest in math, so everyone is participating.

Once you get involved, there will be growth! I’ve started to hear the kids whispering, “This lesson is so fun!” It was the first time in five years as a teacher that I heard such a heavenly sound!

***

Feedback from 1^st adopter Teacher Ms. Helana (2)

Wow! I can teach too? “ Diego, who is very smart but afraid of math, jumped up! He couldn’t finish the 3 Times table. He always felt that he couldn’t learn math and it was difficult to sit still in class. This time he was so happy that he found a student to teach. Later I found that he was so happy that he figured out the next question.

Last week I tested everyone’s ability to calculate the area of ​​a rectangle, and the situation was not ideal. I intentionally gave all the side lengths, and many of the children added them up mistaken perimeter as area. I also had students multiply all the numbers together and really show off their multiplication skills. It can be seen that they completely failed to understand the concept of length times width of a rectangle. Only about one-third of the children have mastered it.

I organized the whole class to discuss the test papers, and then asked the children to take a retest. This time there was some progress. About half to two-thirds of the students figured something out. I then decided to let students teach students as recommended by Teacher Xu. For students who got every question correct, I put a big red check mark on their retest. I told the class that if they get a red mark, they can become teachers. Then let the little teachers raise their hands high and let other students go to them to learn. I controlled the time for each major question and asked everyone to finish it in two minutes. When the time is up, I will add a couple minutes to wrap up. When they returned to their seats, they wrote down what they had learned and who they had taught and who had learned it. I walked around the classroom and didn’t find any students chatting and goofing off.

Many students are full of joy, especially the students with more active personalities who have a good relationship with each other and can almost achieve successful teacher-student matching. Many students who lacked motivation and energy in learning were very happy this time, likely because this is fun for them. Some introverts and a few students need my help.

Thanks to Teacher Xu’s guidance this year, every time I gave a quiz, there were questions of different difficulty. Among them, the easy questions are often particularly helpful in helping students see concepts intuitively. In previous years, I was very distressed that students learned concepts at the beginning, but after learning abstract math operations, they seemed to throw away the concepts and made weird mistakes. As a result, in this test, students like Diego, who did not understand abstract math operations at all, were able to grasp concept-based methods to correct those children who learned by imitation. With more such discussions, children will naturally become better and better at solving problems based on concepts instead of memorizing the math operations.

In the next step, I hope to make this more routine and have group discussions after every day’s practice. Once more students learn it, they can teach each other!

第十章 尝试推广教学方法

我想打破一个误解：我的学生数学学得好，主要是因为我的方法，而不是因为我的个人。 有多少老师真正认同并学习我的方法？老实说，向我取经的老师很多，但大多数人只是把我当成“东方的智者”，他们有数学问题会来问我，因为我会讲清楚“为什么这样做”。对于我的教学法，他们往往归结为“中国智慧”，认为这是我个人的天赋，而不是一套可以普及的方法。校长会称赞我的成果，甚至别的学校的校长也会说我“取得惊人的成绩”，但从来没有人提出：你能不能系统地给我们的老师普及推广一下你的教学方法？

我曾尝试在我所在的学区推动普及我的方法，但几乎没有得到任何行政层面的接纳。原因之一是老师们本能地反对 IDK（“我不知道”）：他们担心学生会借机偷懒，作业全写 IDK。因为他们本来就还没建立起有效的学生关系，于是只能靠“督促/对抗”这种模式来维持课堂秩序。我的方法需要在学期一开始的前两周就把学生的思维习惯进行“整体改造”，这在现有系统里很难被接受，也缺少支持。所以，我在学区里几乎没有得到正式认可。大家知道我教得好，但教育体系的文化倾向于“不要突出个人”，要强调集体力量，再加上“中国人数学好有什么奇怪的”这样的固化成见，于是别人都把我的教学方法当成“东方玄学”，夸奖我、尊敬我，却不真正想学我的方法。

我想，中小学的校长们数学基础和教学方法有限，听不懂我很正常，那我去找那些研究教学法、懂理论的理论大师，总能沟通吧？我拿着多年的学生成绩，再加上我的理念和他们很多理论其实“几乎一致”，我还能指出他们为什么“理论正确但落地成废品”，而我用同样的理念能做出“优秀产品”，他们应该一听就懂，会采纳我的意见吧？

***

所以，2014 年，我提前退休，准备到外面的世界推广我的教学方法。我原以为，只要敲任何一所学校的门，把学生数据拿出来跟他们讲，告诉他们你们的理论哪里对、哪里不对，尤其是执行层面上为什么达不到效果。这些内容我都可以讲得清清楚楚，那学校自然会请我去讲，像斯坦福这种地方也应该请我去讲课。结果呢，根本没人理我。

我在亚利桑那州立大学做过几次讲座，确实有人来听，但我就是个 nobody，所以给我安排的是一个很小的教室。虽然教室塞得满满的，连窗台上都坐了人，但也就那样，讲两三次不会有什么实质效果，因为听的老师只觉得“新鲜”，回去也就过去了。

我给美国教师协会（NCTM）年会写了 proposal，但他们从来没有一次让我去讲。TED 上一个老师连 “students will not learn from teachers they don’t like” 这种内容都能讲；我至少做到这一点，而且学生成绩摆在那儿非常清楚，却仍然没人愿意听。不仅没人请我讲，甚至我的第一轮申请都被拒绝了。他们还礼貌地问我：“你要不要看反馈？” 我回答说要，结果从来没人给我任何一条具体反馈。这让我完全搞不懂。可能他们也不知道怎么否定我，或者觉得不好意思说出来，所以最后干脆就不给解释了。

我还申请了耶鲁大学一个研究中小学数学教学的职位，填写了不少材料，结果也是直接被拒，没有任何反馈。

后来，我想到叔本华讲过一句很深刻的话，大致意思是：你说别人穷、说别人笨，对他的伤害都未必最大；最伤人的，是你暗示“你的东西不对，你的智力或智慧不如我”。这是人最难接受的。那一刻我才真正理解：为什么我拿着数据去任何大学、任何机构讲，“你们几十年来一直在说 closing the gap，从六十年代开始到现在，gap 不但没缩小，反而越来越大，每一次改革都让 gap 更宽。我教了二十多年，我的学生几乎没 gap。你想知道为什么吗？”，没有人愿意听。

后来，我了解到著名数学教师 Jamie Escalante 的遭遇。他在洛杉矶贫困学校加菲尔德高中，把学生数学成绩提高很多，让他们获得上大学机会，因此全国闻名，上了电影，还得了里根总统的特殊荣誉勋章。但他受到的关注太多，让其他老师就都显得更笨了，这是人们不能接受的。他受到学校系统的种种阻碍：一些行政人员不赞成他严格的课堂要求，例如要求学生在进入教室前先回答一道作业题。一位副校长曾威胁要开除他，理由是他来得太早、走得太晚，而且未经许可为 AP 考试筹集资金。这些紧张关系最终导致他在 1991 年离开了。他走了以后，学生表现急剧下滑。于是，批评者又说：他的成功高度依赖于他的个人魅力、长时间的投入以及独特的教学风格，因此他的模式难以推广。

***

虽然遇到不少挫折，但我依然尝试各种努力。在这个过程中，我结识了一些想要提高美国教育的朋友，包括一个叫做“拯救数学”（“Save Math”）的群。通过这个群的介绍，我得到 Palo Alto Rotary Club 的支持，在 East Palo Alto 的一所学校开始尝试让教师学习采用我的方法。最开始，有三个教师参加，后来有一个教师离开了学校，另外一个因为生了孩子，没有时间继续参与，虽然她们都觉得很有收获。

Helena 老师已经坚持了两年。最开始，我用了两周时间来训练 Helena 的数学思维。虽然她做了半辈子的工程师，用了不少数学，但其实对数学的理解只是在工具层面，而这个训练让她把数学思想掌握得更深了。回到学校以后，她告诉我感觉教得更轻松了，学生也更愿意听她讲，捣乱也变少了。我说，这是因为你数学明白得更透彻以后，你的自信心增加，学生都能感受到，对你也更加服气了。除此之外，我也到 Helena 的学校来观摩，给她具体的指导和反馈，也直接给学生上过几堂演示课。

我希望更多的学校能跟我有这样的深入合作，这是我写这本书的目的之一。我今年七十多岁，但头脑还很清楚，对数学教育充满热爱。教数学和培养数学教师，对我来说不是工作，而是一种爱好。

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首位实践者Helena老师的反馈（一）

我在硅谷郊区一所公立初中教授六年级数学。我的大多数学生来自低收入的移民家庭。学生的平均数学水平低于三年级，有几位甚至只有一年级或幼儿园的程度。在我当老师的前四年里，教室里总是吵闹不休，随时可能爆发冲突。我常常一句话都说不完整，就被学生打断。今年夏天，我遇到了来自凤凰城的退休教师徐老师，并向他学习。他是一位真正能改变学生的老师。从那以后，我的课堂发生了天翻地覆的变化。我决定如实记录下来，与更多关心孩子成长的人分享！

我发现，在美国长大的孩子非常喜欢运动，尤其是那些对成为医生、律师、工程师没有信心的学生。他们大多数都梦想从事体育相关的职业。我一直想让学生做一些与体育相关的数学，但没想到徐老师直接把体育的乐趣融入数学教学，让孩子们一走进教室就兴奋起来。

在篮球中投不进是再正常不过的事，我们挥挥手继续投，旁边的人还会喊：“Nice try! 再来一次！你一定能投进！”但在数学课上，一旦做错题，老师就给你一个红叉。这个红叉足以让孩子的情绪跌到谷底，很多人完全提不起劲继续学习。徐老师课堂精神的核心，就是把“我不会”正常化。孩子完全不需要感到羞耻，发现自己的不足、找到努力方向，本身就是一件了不起的事！

许多老师非常喜欢“成长型思维”，我相信很多家长也听说过。我教了学生多年成长型思维，却从未看到明显效果。没想到按照徐老师的巧妙方法，我在几分钟内就抓住了全班的注意力！下课铃响后，我给孩子们一张纸，让他们读一遍并写下“I know (IK)”或“I don’t know (IDK)”。然后我给他们第一个A，告诉他们：你参与了！接着让他们写下自己如何解决IK的问题，我再给他们第二个A，因为他们努力了。最后一个A要求他们取得进步。我让他们和旁边的同学交换试卷，讲讲你的解题过程。无论是你学到了什么，还是你教给别人什么，这就是你今天的进步。因为当你能教别人时，你就把知识变成了能力，这种能力甚至能让你成为家教、赚到钱。许多孩子看到自己初中第一张数学纸上出现三个A时激动不已，强烈要求带回家给父母看！短短十分钟，孩子们已经深刻体验了成长型思维！

从那以后，孩子们每天带着一张纸走进教室。其他班级还在走廊里吵闹、无法进教室时，我的学生已经快写完第一张纸了。他们在班里竞争解题排名。即使写了很多“I don’t know”，也能和数学最强的孩子竞争，所以每个人都参与。

只要参与，就会成长！我开始听到孩子们小声说：“这节课太好玩了！”这是我五年来第一次听到如此动听的声音！

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首位实践者Helena的反馈（二）

“哇！我也能教别人？” 聪明但害怕数学的 Diego 跳了起来！他以前连 3 的乘法表都背不下来，总觉得自己学不会数学，上课也坐不住。这次他因为找到一个学生可以教而兴奋不已。后来我发现，他高兴得连下一题都自己想出来了。

上周我测试了大家计算矩形面积的能力，结果不理想。我特意给了所有边长，但很多孩子把边长加起来，把周长当成面积。也有学生把所有数字相乘，兴致勃勃地展示自己的乘法能力。可见他们完全没有理解“长乘宽”的概念。只有大约三分之一的孩子掌握了。

我组织全班讨论试卷，然后让孩子们重考。这次有些进步，大约一半到三分之二的学生弄明白了一些。我于是决定按照徐老师的建议，让学生教学生。对每一题都做对的学生，我在重考试卷上画了一个大大的红勾。我告诉全班：如果你拿到红勾，你就可以当老师。然后让这些“小老师”举高手，让其他学生去找他们学习。我控制每道大题的时间，让大家在两分钟内完成。时间到了，我再加几分钟收尾。回到座位后，他们写下自己学到了什么、教给了谁、谁又向他们学习。我在教室里走来走去，没有发现任何学生在聊天或偷懒。

许多学生满脸喜悦，尤其是那些性格活跃、关系好的孩子，他们几乎都能成功配对。许多原本缺乏动力、精力不足的学生这次也非常开心，可能是因为这种方式对他们来说很有趣。一些内向的学生和少数孩子需要我的帮助。

多亏了徐老师的指导，今年我每次出小测都会设计不同难度的问题，其中简单题往往特别有助于让学生直观理解概念。往年我很苦恼：学生一开始理解了概念，但学了抽象运算后似乎就把概念丢掉了，开始犯奇怪的错误。结果在这次测试中，像 Diego 这样完全不懂抽象运算的学生，反而能用基于概念的方法纠正那些靠模仿学习的孩子。随着更多这样的讨论，孩子们自然会越来越擅长基于概念解决问题，而不是死记硬背运算步骤。

下一步，我希望把这种方式常规化，让每天的练习后都有小组讨论。一旦更多学生学会了，他们就能互相教学！

(This is Chapter 9 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, in the post-pandemic era. I want to explore what we can learn from a unique and successful teacher, especially on how to teach math effectively. 中文版在本文后面部分，英文版之后)

Chapter 9 A Comparison of Math Education in China and the U.S.

In June 1979, China sent a delegation to the United States to study its primary and secondary education. According to the 30,000‑word internal report they produced, the delegation observed the following features of American elementary education:

• Students were generally confident, even arrogant, and regardless of academic performance, they all displayed a strong sense of self‑expression.

• Foundational subjects (math, physics, chemistry) were weak; second‑grade students still used their fingers for addition and subtraction yet were enthusiastic about talking about “inventions and creativity.”

• Classroom discipline was loose; students walked around and chatted freely during lessons, with little effective unified management.

• Music, physical education, and art were emphasized, while academic training was relatively neglected.

Based on these observations, the Chinese delegation concluded:
“American basic education is in deep trouble. Give it another thirty years, and China’s science, technology, and culture will catch up with and surpass this superpower.”

That same year, as a reciprocal visit, the United States sent its own delegation to China. After visiting several schools in Beijing, Shanghai, and Xi’an, they also produced a report. Their conclusion was that Chinese students were among the most hardworking in the world—getting up the earliest, going to bed the latest. China’s education system had a solid foundation, but its heavy emphasis on uniformity, discipline, and examinations might suppress creativity.

More than forty years have passed. American basic education has deteriorated further; Chinese students are even more diligent than forty years ago: getting up earlier, going to bed later, and tackling more difficult problems. Yet China’s overall scientific, technological, and cultural strength still lags significantly behind that of the United States. Judging from top achievements in fundamental science (such as Fields Medalists and Nobel laureates) and from technological innovation, the United States remains ahead of every other country in the world.

In hindsight, there are at least three reasons why American science and technology have not collapsed:

First, American education is not uniform or centrally imposed. It includes homeschooling, a wide variety of private schools, and publicly funded charter schools. Public schools themselves vary widely from state to state. Even within the same state, school districts, individual schools, and even individual teachers enjoy considerable autonomy—although this autonomy has been gradually eroded by federal intervention over the past half‑century.

Second, there is a peculiar phenomenon, or paradox, in education: sometimes the worst teachers produce the best students. This reflects the remarkable individuality of learners. When teachers are poor, students complain; in that dissatisfaction, some students realize they cannot rely on the teacher and must rely on themselves. The best students are precisely those who push themselves. In today’s information‑rich society, motivated students can often learn very well on their own. This is somewhat reminiscent of Laozi’s idea of “governing by doing nothing,” which paradoxically leads to great achievement.

Third, the United States still retains a strong ability to attract global talent. This is the greatest pillar of its continued strength. The day the United States loses this attraction will be the day its decline begins. The scale and speed of social change in the U.S. over the past two decades have been astonishing. Today, American politicians seem to be working hard to hasten that day—thanks in no small part to a group of elites, educated under cultural‑Marxist doctrines, who are both naïve and self‑important.

China has not fulfilled the prediction made in 1979, and the reason is simple. China’s education system exhausts students physically and mentally. The government is trying to reform the system and reduce student workload, but these reforms treat symptoms rather than causes: addressing problems superficially and mechanically, without grasping the essence. Moreover, while reducing workload in one area, the system simultaneously tightens control in two others: enforcing uniformity and suppressing diversity, and maintaining top‑down authority that leaves little room for autonomy. As a result, students’ independence, creativity, and imagination are severely impaired. Chinese students are diligent, obedient, and hardworking, but their self‑motivation, autonomy, and innovation lag behind. A 2009 international assessment across 21 countries showed that Chinese children ranked first in computational ability, but last in imagination and fifth from the bottom in creativity.

So much for primary and secondary students—now let us look at students at the higher end. In 2000, China produced more than 9,000 PhDs in science and engineering—nearly half the U.S. number. Seven years later, China surpassed the U.S., and the growth has been explosive. By 2019, China produced more than 49,000 such PhDs, while the U.S. produced only about 33,000. Yet China’s academic output and technological innovation remain far behind the United States. The slogan “Chinese education produces high scores but not talent” has been repeated for decades. China has attempted various educational reforms, but the problem has not been solved—indeed, it has worsened.

Why? Is it a matter of Chinese genetics? Of course not. Ironically, the very reason Chinese students score so well is the same reason they often fail to become innovators. Students are obedient; curricula and textbooks are authoritative; teachers lecture endlessly; exams reward memorization plus test‑taking skills and drill‑based training. Obedient students who work hard and master the authoritative textbooks can score high or even perfect marks. Even weaker students, through repeated drilling, can grasp a portion of the material, so there are few students who perform disastrously or understand nothing. In contrast, American public schools have plenty of such students.

Under the heavy pressure of schools, families, and society, Chinese students must study whether they like it or not. Years of forced learning erode intrinsic motivation. Their academic goal is clear and concrete: get into a prestigious university. Once that goal is achieved, they can relax or even “lie flat.” As a result, aside from a few elite universities, the academic atmosphere in most Chinese universities contrasts sharply with the intense, minute‑by‑minute effort of Chinese middle‑school students. Many students simply stop working hard once they reach university.

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Yet in the past decade, there are signs that the Chinese delegation’s prediction may not have been entirely wrong; perhaps it was only premature. If current trends continue for another thirty years, China may well approach or even reach the highest levels of the United States in mathematics and technology.

The anti‑elite culture bred by American democracy has, to a large extent, evolved into an anti‑intellectual culture. American K–12 mathematics and science education has been made not only simple but sometimes almost absurd. Marina Ratner, a renowned mathematician and former professor at UC Berkeley, was baffled by her grandson’s math textbook:

“This requirement of visual models and creating stories is all over the Common Core. The students were constantly told to draw models to answer trivial questions, such as finding 20% of 80 or finding the time for a car to drive 10 miles if it drives 4 miles in 10 minutes, or finding the number of benches one can make from 48 feet of wood if each bench requires 6 feet. Here are some more examples of the Common Core’s convoluted and meaningless manipulations of simple concepts: ‘draw a series of tape diagrams to represent (12 divided by 3) × 3 = 12,’ or ‘rewrite (30 divided by 5) = 6 as a subtraction expression.’

This model-drawing mania went on in my grandson’s class for the entire year, leaving no time to cover geometry and other important topics. While model drawing might occasionally be useful, mathematics is not about visual models and ‘real world’ stories. It became clear to me that the Common Core’s ‘deeper’ and ‘more rigorous’ standards mean replacing math with some kind of illustrative counting saturated with pictures, diagrams and elaborate word problems. Simple concepts are made artificially intricate and complex with the pretense of being deeper — while the actual content taught was primitive.”

Like me, Marina Ratner was also an immigrant, originally from Russia. The strength of American science and mathematics has been supported to a very large extent by immigrants and their children. Mathematician Xin‑Yi Yuan points out that although the United States has the largest number of top mathematicians in the world, at least 60% of them are immigrants. The U.S. education system does not value math and science Olympiads; most participants are children of immigrants mainly from China, India and European countries.

For students participating in math and physics Olympiads, most schools neither supported them nor cared about the results. Most of the time, they didn’t even provide a testing site and students had to find one on their own. The organizing bodies (such as the Mathematical Association of America and the American Association of Physics Teachers), constrained by limited resources, are unable to effectively prevent cheating. In 2025, The Wall Street Journal published a special report on cheating in U.S. math competition.

Most Americans not only avoid Olympiads themselves but often dismiss top math and science enthusiasts as “nerds.” Yet in the age of AI, talent from math, physics, and computer Olympiads plays a decisive role. Winners of these Olympiads are 4,000 times more likely than the average person to become founders of unicorn companies. About twenty unicorn founders—including those behind OpenAI, Cursor, Stripe, Databricks, Perplexity, Ethereum, Cognition, and Hyperliquid—come from this group. Anthropic founder Dario Amodei was once a member of the U.S. Physics Olympiad national training team, and Alexander Wang, whom Facebook hired for billions to lead its AI efforts, was also a member of the U.S. Math Olympiad national training team.

Nearly half of Silicon Valley’s high‑tech workforce consists of immigrants, yet U.S. policy in recent years has become increasingly unfriendly to them. Immigrants are blamed for “taking American jobs,” and numerous obstacles have been added to work visas and immigration status. As a result, many top math and science talents no longer want to study in the United States, or cannot remain after graduation.

Meanwhile, federal funding for university research—traditionally a cornerstone of American scientific strength—has been sharply cut. For example, in 2025 the National Science Foundation’s budget was reduced by 8%. Some departments at top universities no longer have funding to admit new graduate students, especially international applicants.

In contrast to America’s self‑sabotaging approach to science education, China’s emphasis on foundational education, combined with economic and cultural development and the return of large numbers of overseas‑trained scholars in the past decade, has led to steady improvement in research capacity and talent development. The DeepSeek research team that shocked the world is composed almost entirely of math and science talent trained in China. The number of Chinese mathematicians invited to speak at the International Congress of Mathematicians (held every four years) has risen steadily from 5 in 2006 to at least 22 this year, second only to the United States. Among the top five candidates for this year’s Fields Medal, two are Chinese: Wang Hong, who has almost a lock-in the medal, and Deng Yu, who also has a strong chance.

The rise of Chinese manufacturing is one of the main sources of friction in U.S.–China relations. The U.S. government is eager to bring manufacturing jobs back to America—but are there enough Americans capable of performing world‑class industrial work? In a March 2025 interview, investor Ray Dalio noted that among the entire U.S. labor force, about 1% have world‑leading capabilities and can drive major breakthroughs; about 10% are highly capable and can meet demanding professional standards; but the bottom 60% have very weak skills and lack the basic literacy and numeracy required for modern manufacturing. These are the long‑term consequences of the U.S. education system.

Elon Musk has repeatedly said that China has “a lot of very smart, hardworking people,” that the engineering teams he met there were “world‑class,” and that China’s manufacturing is “the best in the world.” According to research by economist Shan Jianwei, Tesla’s Shanghai Gigafactory and the Fremont factory in California are essentially “twin” factories: they use the same technology, the same production lines, and produce the same Model 3 and Model Y vehicles.

In terms of output, the Shanghai factory’s roughly 20,000 workers produced nearly one million cars in 2024—about 50 cars per worker per year. Fremont, with a similar number of workers, produced about 560,000 cars—28 per worker. Under nearly identical hardware conditions, the physical productivity of Shanghai workers is almost twice that of their American counterparts.

Considering wages, a Shanghai Tesla worker earns about $14,000–$15,000 per year, while a U.S. Tesla worker earns about $82,500. Overall, China’s labor “cost‑performance ratio” is 8 to 14 times higher. This pattern is widespread across manufacturing: China’s advantage is no longer merely “cheap labor,” but a systemic advantage built on highly efficient workers, a world‑class supply‑chain ecosystem, excellent infrastructure, and a massive engineering talent pool (China produces four times as many STEM graduates as the U.S.).

Regarding AI—the most important arena of future U.S.–China competition—NVIDIA CEO Jensen Huang has said that China is extremely strong in AI talent, infrastructure, and chip development, because it has a vast number of highly trained engineers, strong computer science graduates, and world‑class AI researchers. Huang attributes this to China’s education system: its emphasis on mathematics, engineering, and science; its ability to produce large cohorts of technically skilled graduates; and its support for rapid industrial and technological development. China’s STEM pipeline is one of the largest and strongest in the world.

***

Rational mathematical thinking is the soul of science and rationality. If too many people fail to develop this capacity, the rational components of a society’s governance will be squeezed out by emotion and impulsiveness. Such a society may descend into disorder due to the lack of rational, effective management; or it may split into a class of rational elites who govern and a class of people who are governed. Before the emergence of rational social rules—from the Magna Carta to the U.S. Constitution—human societies largely existed in these two forms: the former chaotic and backward, the latter relatively well‑governed but rarely free or equal.

The lack of sufficient rational thinking is a serious problem in contemporary Western politics. The explosive growth of psychology, combined with the influence of film, television, and social media, has elevated emotion and sentiment to an extreme. Society, especially the academic elite, often dismisses or even scorns rationality. Repeated failures in math‑education reform have caused the quality of American K–12 mathematics to decline steadily. Under the banner of politically correct “equitable education,” some have even publicly called for eliminating high‑school mathematics altogether. If that were to happen, the United States would inevitably revert to one of the two societal forms described above.

***

Appendix: A Comparative Illustration of Math Tests in the U.S. and China

From two examinations, one from the United States and one from China, I selected representative problems in four areas: algebra, geometry, functions, and statistics. Although not comprehensive, these examples provide math-minded readers with a direct and intuitive comparison of the actual level of math education in the two countries.

The U.S. exam I used is the National Assessment of Educational Progress (NAEP), the most authoritative test for measuring academic achievement among American K–12 students. It is administered roughly once every two to four years. From the official website, I selected items categorized as “hard questions” on the 12th‑grade mathematics assessment. The four sample problems I obtained correspond to algebra, geometry, functions, and statistics.

The statistics question merely tests whether students remember the definition of a box‑and‑whisker plot—pure memorization. The geometry question is also essentially a memory check. The algebra and function questions are extremely simple.

For China, I selected problems from the 2025 Beijing High School Entrance Examination (“Zhongkao”) in mathematics. This is the exam taken by students transitioning from middle school to high school, representing the level of students in developed regions of China, roughly equivalent to 9th grade in the United States.

The algebra question is relatively simple but still more complex than the U.S. example. The geometry question is far more difficult, requiring both computation and proof. The function question has two parts: the first requires calculation, and the second requires a deeper understanding of the definition of a function. The statistics question involves a large amount of reading and requires understanding and computing standard deviation and related concepts; even I find it challenging, and I was a high school math champion in China who taught math competition recently!

This Chinese middle‑school exam is several levels more complex and difficult than the U.S. high‑school NAEP assessment. In Beijing, only 10–15% of students fail the exam (scoring below 60 out of 100). In contrast, on the 2024 NAEP national 12th‑grade math test, 62% of American high‑school seniors failed to meet the basic standard (scoring below 140 out of 300).

第九章 中美数学教育比较

1979年6月，中国派代表团去美国考察中小学教育。根据当年的三万字内部报告摘要，访问团观察到美国初级教育中的以下现象：

· 学生普遍自信，甚至有些“趾高气扬”，无论成绩好坏都很有自我表达意识。

· 基础学科（数理化）薄弱，小学二年级学生加减法仍靠掰手指，却热衷于谈论“发明创造”。

· 课堂纪律松散，学生上课随意走动、聊天，缺乏有效的统一管理。

· 重视音乐、体育和美术，相对轻视学科训练。

基于这些观察，中国访问团得出结论：“美国的基础教育已经问题重重，再用30年，中国的科技和文化必将赶上并超过美国这个超级大国。”

同一年，作为回访，美国也派考察团来中国。他们看了北京、上海、西安的几所学校，也写了一份报告。他们得出的结论是：中国的学生是世界上最勤奋的，起得最早，睡得最晚。中国教育基础扎实，但过于强调统一、纪律和考试，可能会抑制创造力。

而今四十多年过去了，美国的基础教育比原来更烂了，中国学生比四十年前更勤奋，起得更早，睡得更晚，做的题目更加难了。但中国的科技和文化综合实力，离美国还有相当的距离。从基础科学的顶尖成就（如菲尔兹奖、诺贝尔奖得主来看），以及科技领域的创新来看，美国还是领先于世界所有其他国家。

作为事后诸葛亮，美国的科技还没有垮掉，至少有三个原因：

第一是美国教育不会一刀切。它有家庭教育，有形形色色的私立学校，和私立公助的特许学校。公立学校也是各州自行其是。即便在同一个州内，各学区、各学校乃至各个老师，都有相当大的自主权，尽管这种自主权在过去半个多世纪里被联邦政府一步步削弱。

第二是教育中的一个奇特现象，或者说是一个悖论，就是常常会有最差的老师培养出最好的学生。这体现了个体差异的奇特性。老师太差，就会有学生不满、抱怨；在这种不满和抱怨中，有些学生会发现老师靠不住，只好靠自己。最好的学生，就是那些自己奋发图强的学生；而在当今信息社会，学习资源充裕，有动力的学生往往可以自学得很好。这有点类似老子所说的“无为而治”，反而可能取得大成。

第三是美国仍然保持着对世界人才的吸引力。这是美国保持先进或强大的最大支柱。哪一天美国失去这种吸引力，那天就是美国衰退之日。美国社会在最近一二十年的变化之大、速度之快都十分惊人。目前，美国的政治家们正在十分努力地追求这一天的早日到来。这在相当程度上要“感谢”美国那批由文化马克思主义教育培养出来、既天真又自负的精英。

中国没有能实现当年的预言，原因也很简单。中国的教育把学生压得身心疲惫。政府正在努力改革，要为学生减负，但这种改革只治标不治本，头痛医头、脚痛医脚，所以改得生硬，不得要领。而且，在改这一条的时候，却在另外两条上加强：一条是一刀切，不允许多样性；另一条是权威高压，不允许有自由放任的空间。这样，学生的自主性、创造力、想象力就会严重受损。所以中国学生认真、听话、努力都不错，但自觉、自主、创新就跟不上。2009年，教育进展国际评估组织对全球21个国家进行的调查显示，中国孩子的计算能力排名第一，想象力却排名倒数第一，创造力排名倒数第五。

这是中小学生，我们再来看看高端的学生。2000年，中国培养的理工数类博士生9千多，接近美国的一半。7年后超过了美国，而且增速迅猛，2019年更达到4万9千多人，而当年美国仅3万3千多人。然而，中国人才的学术成果和科技创新却离美国很远。“中国教育出高分不出人才”这个口号喊了几十年，耳朵都听腻了。为此中国进行了各种教育改革，可问题不但没解决，还有越演越烈的趋势。

为什么呢？中国人种的问题吗？当然不是。说起来几乎有点搞笑，中国学生成绩好和不出人才，实际上是同一个原因造成的。那就是学生听话，教学大纲和教材有权威，教师满堂灌，考试就是凭知识记忆加上考试技巧和刷题训练。学生听话，只要用功，把权威教材掌握了、记住了，就可以考高分甚至满分。再差的学生，通过不断的重复训练也能掌握一部分，所以很少有极差、甚至完全不懂的学生；而美国很多公立学校里，这样一问三不知的学生可是有一大堆。

中国学生在学校、家庭和社会的高压下，你爱不爱都得学习，使劲学习。长久的被迫学习，让很多学生的主观能动性被磨没了。他们学习的目标也非常明确和具体，就是考取名牌大学。目标一旦达到，他就可以松口气，甚至躺平不干了。结果是，除了几所名牌大学，大多数国内大学的学习风气，和中学生起早贪黑、分秒必争的努力形成鲜明对比，学生在大学阶段反而不再用功了。

***

然而，最近十年有迹象表明，也许当年中国代表团的预言并非完全没有道理，只是时间的问题。再过三十年，如果这种趋势继续下去，很可能中国能在数学和科技领域接近或达到美国的最高水平。

美国的民主滋生出的反精英文化，已经在很大程度上演变成了反智文化。他们把中小学数学和科学教育弄得不仅简单，有时甚至到了近乎荒谬的地步。Marina Ratner 是美国著名数学家、加州大学伯克利分校数学教授。她对孙子的数学教科书充满了不解：

“对视觉模型和编故事的要求随处可见。学生被不断要求画模型来回答一些极其简单的问题，比如求 80 的 20%，或者一辆车 10 分钟行驶 4 英里，那么行驶 10 英里需要多少时间。还有这样一些例子：‘画一系列条形图来表示 (12 ÷ 3) × 3 = 12’，或者‘把 (30 ÷ 5) = 6 改写成一个减法表达式。’

这种画模型的狂热在我孙子的班级里持续了一整年，以至于他根本没有时间学习几何和其他重要内容。虽然画模型偶尔可能有用，但数学并不是关于视觉模型和‘现实世界故事’的学科。《共同核心标准》所谓的‘更深刻’和‘更严谨’，实际上意味着用充满图片、图示和冗长文字题的某种‘展示性计数’来取代数学。简单的概念被人为地弄得复杂繁琐，假装更深刻——而真正教授的内容却非常原始。”

和我一样，Marina Ratner 也是移民，来自俄罗斯。美国的科技和数学之所以发达，极大程度上是移民及其后代支撑的。数学家袁新意指出，美国虽然拥有世界最多的顶级数学人才，但其中至少 60% 以上是移民。美国教育系统对奥赛不重视，参与者绝大部分是受父母影响的移民后代。学生参加数学和物理奥赛，大部分学校既不支持他，也不关心结果，甚至都不提供考点，需要到外面自己找。主办机构（数学学会、物理学会等）由于资源有限，无法有效制止作弊现象（2025 年《华尔街日报》对美国数学奥赛作弊做了专门报道）。

大部分美国人不仅自己不搞奥赛，还经常不屑地把数理尖子叫做书呆子（Nerd）。但在 AI 时代，数学、物理和计算机奥赛人才起着决定性的作用。数学/物理/计算机奥赛得主成为独角兽创始人的概率是普通人的 4000 倍，其中包括约 20 家独角兽企业的创始人，如 OpenAI、Cursor、Stripe、Databricks、Perplexity、Ethereum、Cognition、Hyperliquid 等。Anthropic 创始人 Dario Amodei 曾入选美国物理奥赛国家集训队，而被 Facebook 以百亿美金挖来管理 AI 的 Alexander Wang，也曾是美国数学奥赛国家集训队员。

硅谷的高科技从业人员几乎一半是移民，但美国近年的政策对移民越来越不友好，不仅在舆论上责怪移民抢走了美国人的工作，在工作签证和移民身份问题上也设置了很多障碍。这导致很多顶级数理人才不想来美国留学，或者毕业后不能留在美国。美国传统上非常重视的大学科研，近年来联邦支持经费被大砍（比如，2025 年国家科学基金的经费支出减少了 8%），以致于著名大学的一些系里都没有经费招新的研究生，尤其是从海外申请的研究生。

对比美国对科学教育的自我削弱政策，中国对基础教育的重视，以及经济文化的发展，加上最近十几年大量海归人才带来的理念，导致科研和人才水平总体稳健上升。震惊世界的 DeepSeek 研究团队，基本上都是由中国本土培养的数理人才组成的。在每四年一次的国际数学联合大会上做报告的中国数学家人数，从 2006 年的 5 位稳步上升到今年至少 22 位，仅次于美国。今年令人瞩目的菲尔兹奖候选人前五名里，有两位是中国数学家：王虹几乎已经锁定了这届的菲尔兹奖，而邓煜也有比较大的机会同时获得。

中国制造业的崛起是中美关系摩擦的主要领域之一。美国政府千方百计想要把制造业工作带回美国，但有足够多的美国人可以胜任世界工厂的工作吗？在 2025 年 3 月的一次采访中，著名投资人雷·达里奥表示，在所有美国劳动力中，有 1% 的人能力处于世界领军水平，可以做出伟大的变革，有 10% 的人能力非常强，可以胜任标准很高的专业工作，而底部 60% 的人能力则很差，完全缺乏现代制造业所需要具备的基本读写和数学技能。这些正是美国教育系统长期积累的后果。

埃隆·马斯克曾经多次表示，中国有“许多非常聪明、勤奋的人”，他在那里的工程团队是“世界一流”的，中国的制造业是“世界上最好的”。据经济学家单建伟的研究：特斯拉的上海工厂和加州弗里蒙特工厂，可以说是“兄弟”工厂。它们使用相同的技术、相同的生产线，制造相同的 Model 3 和 Model Y。论产出，上海工厂大约 2 万名工人，在 2024 年生产了接近 100 万辆车，人均年产约 50 辆车。弗里蒙特工厂同样约 2 万名工人，年产量约 56 万辆，人均年产约 28 辆车。在几乎相同的硬件条件下，上海工厂工人的实物生产效率是美国同事的近两倍。

再考虑到一名上海特斯拉工人的年薪大约是 1.4 万至 1.5 万美元，而一名美国特斯拉工人的年薪大约是 8.25 万美元，综合算下来，在劳动力“性价比”上，中国工厂的优势是美国的 8 到 14 倍。这个现象在制造业领域普遍存在，中国的竞争优势早已不仅仅是“人工便宜”，而是一个由高效率的劳动力、顶级的供应链生态、完善的基础设施和庞大的工程师红利（中国的 STEM 毕业生数量是美国的四倍）共同构成的系统性优势。

关于中美两国未来最大的竞争领域 AI，英伟达总裁黄仁勋说，中国在 AI 人才、基础设施和芯片研发方面都极其强大，因为中国拥有大量受过高水平训练的工程师、优秀的计算机科学毕业生，以及世界顶尖的人工智能研究员。黄仁勋指出，这归功于中国的教育体系：强调数学、工程和科学，培养大量具备技术能力的毕业生，支持快速的工业化和技术发展。中国的 STEM 人才培养体系是全球规模最大、实力最强之一。

数学的理性思维是科学和理性的灵魂。如果太多民众没有开发出这种智力，那么整个社会管理系统的理性成分就会受到感性化和情绪化的挤压。或者，这个社会因为缺少理性支持的有效管理而乱象丛生，最终导致整个社会的混乱；或者，有一部分具备理性思维能力的人成为企业和政府的管理者，另一部分人成为被管理者的阶级社会。在从英国《大宪章》到美国宪法这些理性的社会规则诞生以前，人类社会基本就是这两种形态：前者基本上是混乱和落后的国家，后者基本上治理得相对较好，但很难是一个自由平等的国家。

缺乏足够的理性思维是当今欧美国家政治遇到的一个严重问题。由于心理学的超常发展和影视及多种媒体的推波助澜，人们的感性和情绪被推崇到极致。整个社会，特别是精英学术界，对理性几乎不屑一顾，甚至鄙夷。而多次失败的数学教育改革，使得整个美国中小学系统的数学水平越来越差。在追求政治正确的“公平教育”旗帜下，甚至有人公开喊出要取消高中的数学教育。真要那样，美国就必然会回到上面讨论的两种国家之一。

附录：美国和中国数学教育水平的实例比较

我从中美两国的两份考试中，代数，几何，函数，统计四个领域各选了一个实例代表题目，来展示中美数学教育的实际水平差距。虽然不够全面，但对关心数学的读者，这可以直观体现两国数学教育的区别。

美国的考试所用的National Assessment of Educational Progress (NAEP)，是美国最标准的用来衡量中小学生学业水平的考试，大致每四年一次。我从官方网站上选择12年级数学“难题”类别，以下是我得到的四个样题，分别是代数，几何，函数和统计。统计题目考的仅仅是学生是否记得Box-and-Whisker这种绘图的定义，纯属记忆。几何基本上考的也是记忆。代数和函数问题都极其简单。

中国的考试我选的是2025年北京市中考数学题目。这是初中到高中的升学考试，代表的是中国发达地区初中毕业生的水平（相当于美国九年级）。代数题比较简单，但也比美国题目复杂。几何题目更是比美国题目复杂得多了，要求计算和证明。函数题目的第一部分需要计算，第二部分需要对函数定义的深入理解。统计题目阅读量极大，而且涉及标准方差等概念理解和计算，要我做都比较困难。

这份中国初中生的考试比美国高中生的统考题目，要复杂和困难好几个档次。北京市中学生里，只有10-15%不及格 (就是100分得60分以下)。而美国高中生，在2024年全国统考里，有62%达不到基本标准（就是300分里得140分以下）。

(This is Chapter 8 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, in the post-pandemic era. I want to explore what we can learn from a unique and successful teacher, especially on how to teach math effectively.

I may not fully agree with all Mr. Xu says, but I found it a great learning journey to show the world his thoughts.

中文版在本文后面部分，英文版之后)

(home, sweet home)

Chapter 8: Some Harmful Practices in K–12 Mathematics Education

Learning mathematics requires an emphasis on understanding. This is a point on which education experts generally agree, and one that I also believe is correct. However, many education experts have a serious misunderstanding of what “understanding” actually means. They completely discard and even deny the role of memory in learning mathematics, going so far as to oppose students memorizing the multiplication table. This is a typical example of experts’ “arrogance of rationality”—an overconfident belief in their own theories. Some of the practices they promote are extremely damaging to students.

The first is allowing elementary students to use calculators for addition and subtraction. On the surface, this seems to save students a lot of trouble: just press a few buttons and the answer appears effortlessly. But this does nothing to help them build number sense, develop an understanding of mathematics, or cultivate any affection for the subject. Instead, it fosters dependence on calculators and creates unfamiliarity and even fear toward mathematics. The joy of learning math comes from the sense of accomplishment and satisfaction gained by discovering solutions through one’s own effort, and from the pride and confidence of realizing one is becoming smarter. In this regard, calculators are harmful to students.

The second problem is that, because calculators exist, students no longer need to memorize the multiplication table. Addition and subtraction are intuitive and do not cause conceptual confusion. But multiplication and division embody the most fundamental principles of mathematics; without them, it is difficult to truly understand the subject. I once conducted a survey among students, and the results showed that those who could not remember the multiplication table had a much more negative impression of mathematics than those who could. This is not surprising: I have seen students compute 12×8 by adding 12 repeatedly.

Students who cannot recite the multiplication table cannot do multiplication, let alone division; square roots and exponents are even further out of reach. The feeling of “the calculator can do it” is completely different from “I can do it myself.” It is like a basketball game in PE class: the feeling of scoring a basket yourself is not the same as watching someone else score. At the very least, such students lack a sense of security and confidence. When they see others answer effortlessly, they feel incompetent and foolish. Without a calculator, they become helpless, as if their soul has been taken away.

Many progressive educators explicitly oppose having students memorize the multiplication table, claiming that scientific studies show rote memorization is unhelpful or even harmful to children’s brain development. Stanford professor Jo Boaler, who has enormous influence on K–12 math education, holds this view. Because of the popularity of this idea, many American students cannot recite the multiplication table. Yet I have never met an Asian student in fourth grade who still could not recite it, nor have I met an Asian parent who would tolerate such a situation. Asian students generally outperform other groups in mathematics. There are many explanations for this, but one obvious fact is that nearly all Asian students can fluently recite the multiplication table.

I would like to ask these experts who oppose memorizing multiplication facts: do you also oppose requiring students to remember which pairs of numbers make ten? Without remembering 1 and 9, 2 and 8, 3 and 7, and so on, students cannot perform addition with regrouping. Of course, these progressive experts would say, “It doesn’t matter; let them use calculators.” Indeed, many states are now promoting calculator use in elementary school, claiming it frees students’ minds for more advanced abstract reasoning.

The idea that students need not memorize multiplication because calculators can do it rests on the assumption that the purpose of learning mathematics is simply to get the correct answer. If a calculator can produce the answer, then why bother finding it yourself? The irony is that these same professors do not claim that the purpose of learning mathematics is to find answers. They also advocate for conceptual understanding.

As we discussed earlier, the purpose of learning mathematics is to train the mind and develop strong reasoning abilities. Gaining a skill has two effects: first, the skill can be used to solve problems; second, realizing that one has acquired a new ability brings a sense of achievement, joy, and confidence. Saying “you don’t need to memorize multiplication because you have a calculator” is like telling someone learning the violin, “Just buy two CDs—you can listen to violin music, so why bother playing it yourself?” It is absurd.

In fact, the multiplication table is not difficult to memorize, even though it appears to have 81 entries. Ones and twos require no memorization; threes are mostly learned through addition. Five is half of ten: since 8×10=80, 8×5 is half of that, 40; 7×5 is half of 70, which is 35. For students who attend math class every day, memorizing these is not a problem.

Nine also has a clever trick: it is one less than ten. Four nines are not forty; they are thirty‑something. Five nines are not fifty; they are forty‑something. Among four nines, three of them “become” tens by adding one to each; the fourth nine “gives” each of the three a “hat,” leaving 9−3=6. The answer is 36, and 3+6=9. This works for every multiple of nine. For example, seven nines cannot be seventy; they must be sixty‑something. The seventh nine gives each of the six nines a hat, so 9−6=3, giving 63, and 6+3=9.

Thus, the only products that truly require memorization are those involving 4, 6, 7, and 8, which is sixteen entries in total. Four of these are squares: 4×4=16, 6×6=36, 7×7=49, 8×8=64; these are facts that should absolutely be memorized because they are useful later. The remaining twelve are duplicates (e.g., 4×6 and 6×4). So strictly speaking, there are only six unique rows, each with four numbers, twenty‑four numbers total. A phone number has ten digits; memorizing these is like memorizing two and a half phone numbers. Add the four squares, and you get forty numbers; it’s just four phone numbers. Are we really to believe that memorizing four phone numbers will damage a child’s brain?

Much of mathematics consists of tools like the multiplication table, tools that help us rise to higher levels of reasoning. Learning mathematics is not only about developing abstract logical reasoning; it is also about discovering more efficient, simpler, and more accurate ways to solve problems. Skipping these tools has a profoundly negative impact on learning mathematics.

The third harm that progressive American education inflicts on students is providing all the mathematical formulas needed for every exam question during the test itself.

The stated rationale for this practice is opposition to rote memorization in learning math, on the grounds that memorizing formulas is a form of rote learning. In fashionable mathematics pedagogy, rote learning, or “drill and kill”, is a label for backward and misguided teaching, something that must be firmly opposed and even banned outright. This led to no longer requiring students to memorize the multiplication table, and the next logical step was no longer requiring students to memorize mathematical formulas.

Education theorists with even a modicum of common sense can still recognize that memory has a necessary role in learning mathematics. What they oppose is learning math through memorization alone, without understanding. In many elementary school math classrooms today, a mode of learning does exist in which students gain mechanical memory not through understanding but through repetitive drilling for correct answers, and this is indeed something that ought to be stopped. But the grand experts have overcorrected: by labeling every exercise that involves building memory as “rote learning” and banning it wholesale, they have gone too far.

Memory is indispensable in learning. It is both the starting point and the outcome of learning. When we become curious about information received through our senses, we form momentary memory. If we find it interesting and continue to dwell on and explore it, we form short-term memory. The content of that short-term memory may strike us with great force, or it may connect in multiple ways with existing structures in our understanding, or it may simply have recurred in short-term memory often enough to generate long-term memory. That is what we have acquired — what we have learned. Of course, in this process we also gain two other things: the mental exercise that comes from inquiry, and the emotions that arise in the course of seeking and gaining new knowledge — a sense of achievement, confidence, love or aversion, and so on.

Teachers resort to the method of rote memorization without understanding for various reasons: some are pressed by curriculum pacing requirements and have no time for deep exploration and careful explanation. But more often it is because the teachers themselves do not understand mathematics and therefore do not know how to explore it deeply or explain it carefully. Without conceptual understanding of their own, they naturally cannot explain the logical substance of mathematics. It may be alarming to hear, but more than half of the elementary school teachers I have personally encountered could fairly be described as mathematically illiterate. I have many times had such teachers earnestly ask me why a negative number plus a negative number is negative, while a negative number times a negative number becomes positive. I sometimes ask them in return: why does dividing fractions require flipping the divisor and then multiplying? I have almost never met an elementary school teacher who could fully explain this to me.

The reason so many elementary school teachers have become mathematically illiterate is the “New Math” curriculum developed under the guidance of those lofty theories, and the “new” pedagogical methods designed according to those theories and that curriculum. Students who never learned mathematics properly under those methods often became math teachers themselves, and so it has been passed down, each generation weaker than the last. Letting elementary students use calculators for addition and subtraction, not requiring students to memorize the multiplication table, providing all formulas on exams — these are the very instruments that manufacture mathematical illiteracy.

Relying solely on plugging numbers into memorized formulas without pursuing deep understanding of the problem is bad. But relying on plugging numbers into provided formulas is equally devoid of deep understanding. In fact, how a formula is derived and how it works is the best path to a deep understanding of the problem. Memorizing a formula is at least one step closer to understanding it. Not memorizing formulas does nothing positive for conceptual understanding. For the simple formulas found in K-12 mathematics, the most reliable way to remember them is to understand the principles behind them. Without that understanding and not knowing the why, things memorized through rote learning alone are easily forgotten over time, and newly learned, similar material easily causes confusion. A formula derived from deep understanding has logical support behind it; it is alive, and even if forgotten, can easily be re-derived from that understanding.

The area of a triangle is an excellent example. In our current exams, students are typically given a triangle with the lengths of its sides and asked to find the area using a provided formula. All they need to do is look up the formula for the area of a triangle, “1/2 × b × h” on the formula sheet they have been given. No understanding or logical reasoning is required. What they must do is recall what they were told: which side is the base and which line is the height, then substitute into the formula: base times height divided by 2. This requires nothing more than the clerical ability of a data-entry worker: identify which side is the base, which line is the height, substitute them into the formula, multiply the two numbers, divide by 2, and the answer is done.

The common mistake students make is to take the bottom-most line as the base and the shortest line pointing most directly upward as the height. Teachers and textbooks have certainly explained which side serves as the base, that it has nothing to do with physical position in space, but rather that the base and height are the two sides that form a right angle. Yet calling the lowest line or surface “the bottom” is common sense formed by countless real-life experiences, and the triangles drawn in textbooks and by teachers in class are most often drawn with the bottom side labeled as the base. If the triangle is actually tilted at an angle, making it hard to say which side is lowest, students will often take the longer side as the base and the shortest line as the height — a judgment rooted in long-accumulated practical experience, not logical reasoning.

What should actually be assessed regarding the area of a triangle is the ability to explain the formula. The conceptual understanding needed to find area requires first being able to clearly explain several concepts: What is a surface? What is area? What is a square unit? Why is it called a “square”? Then comes how to transform a triangle into a parallelogram, and how to transform a parallelogram into a rectangle. Only then does the rectangle have its “squareness.” And with that squareness, one can precisely measure the number of square units. This is a remarkably interesting and ingeniously clever method of solving a problem. The process is a model of mathematical thinking in action, and an example that can lead students to appreciate the beauty of mathematics and come to love it.

Given all the concepts and the derivation process discussed above, it is truly unbelievable that any expert would say students must not be required to memorize this formula. Having students take the formula given to them, find two numbers in a diagram, multiply them together, and divide by 2 — and calling that proof of conceptual understanding? This single example vividly illustrates why, even as our educators’ theories grow ever more sophisticated and their curricula ever more refined, the methods they have designed continue to produce American students whose mathematical ability grows ever weaker.

These practices have produced a strange and troubling phenomenon in recent years: despite policies of preferential treatment and economic support aimed at achieving equality, schools in impoverished areas of the United States have not only failed to improve and close the gap; in fact, they have fallen even further behind. In 1989, the National Council of Teachers of Mathematics (NCTM) released new teaching standards, reigniting the Math Wars. I arrived in the United States during this very period, and so witnessed firsthand the new teaching standards, the theoretical arguments supporting them, and how they actually played out in schools. More importantly, I saw the results: students’ mathematical performance became worse than before.

The goal of the new teaching standards was to shift from “rote memorization and procedural learning” toward a more comprehensive understanding, with mathematics viewed as a tool for reasoning and solving complex problems. I fully agree with this in principle, but the standards left out one vital function of mathematics, and in doing so committed a grave error. Beyond being a “tool for reasoning and solving complex problems,” mathematics, in my view, must also be recognized as a discipline for training and developing rational thinking ability. Just as with music, philosophy, and physical education, some students study these subjects as practical tools, even as means of livelihood. But others study them to cultivate character, strengthen the body, sharpen the mind, or simply because they find them genuinely enjoyable and fascinating. The grave error lies in the conclusion drawn from the shift away from “rote memorization and procedural learning” toward comprehensive understanding: namely, that since we now have tools like the calculator, students can be entirely liberated from the need to memorize basic mathematical facts.

Although education experts advocate against rote memorization, in actual practice teachers frequently teach students procedural rules to memorize — without adequate understanding — to help students quickly find the “correct” answer. In many cases, these rules do nothing to promote genuine understanding and produce long-term negative effects.

PEMDAS, discussed earlier, is a mnemonic created to help students remember the order of operations. In fact, understanding the order of operations is far more useful than memorizing this acronym. While “Please Excuse My Dear Aunt Sally” has a pleasing ring to it, it carries a serious flaw: it leads many students to fail to grasp that addition and subtraction, and multiplication and division, are actually of equal priority, resulting in absurd calculation errors. PEMDAS has now been banned in many school districts, yet many similar mnemonics, which appear helpful on the surface, are still widely used by teachers who do not themselves fully understand mathematics, as a substitute for genuine mathematical understanding.

Most middle school teachers I know use a simple mnemonic ( “Keep! Change! Change!”) to help students remember the rule for subtracting a negative number: keep the minuend unchanged, change the subtraction sign to addition, and change the negative subtrahend to a positive. This rule ensures that students can get the right answer even without understanding. But the rule has both a time-decay problem and an interference problem. Time decay means that after a while, students forget whether “Keep! Change! Change!” applies to subtraction or addition, whether it is used when the subtrahend is negative or when the minuend is negative. With time, the details fade. The interference problem arises when students learn other rules for addition, subtraction, multiplication, and division, memorize new mnemonics, and find them blurring together. For instance, the “Keep! Change! Flip!” rule, taught for dividing fractions, can easily get confused with this rule.

In fact, there is a better way to help students understand: negative numbers can function as a verb, meaning to subtract, use up, or take away; or as an adjective, meaning deficient, missing, below, or in debt. With this in mind, one can design quick-response games to help students understand the difference between reducing income and reducing expenses — and that reducing expenses has the same effect as increasing income. Language also offers many examples from which students can discover and understand the logical principle that a double negative equals a positive: “right,” “wrong,” “not wrong”; “good,” “bad,” “not bad.” In English: “Make sense,” “Nonsense,” “That’s not nonsense”; “Not bad.” These can make “negative times negative equals positive” a matter of common sense in students’ minds, a natural habit of thought, so that eventually, seeing a negative sign in front of a negative number immediately registers as positive.

Another example: why do so many students conclude that 7 ÷ 21 = 3? Our textbooks explain that division is sharing or grouping, then describe how to convert a horizontal expression into a vertical one: write a division bracket (shaped like a little house, or the Chinese character “厂”), place the dividend inside and the divisor in front; then use multiplication to work digit by digit from the leading figure to find the quotient; once found and multiplied, subtract the product and move to the next digit. The whole process is a tedious, mechanical procedural routine, where the teacher follows the textbook demonstration, students dutifully follow along, repeating day after day.

This repetition sometimes continues for years, since elementary math is taught in a spiral, gradually deepening. The examples in the textbook, or those used by teachers, almost always place the larger number inside the bracket and the smaller number in front, and are always division-without-remainder problems. In theory this is to avoid confusion; in practice it is to get through the process as quickly as possible. The result is that students develop the habit of placing the larger number inside the “little house” and the smaller one in front, so that 7 ÷ 21 becomes the operation 21 ÷ 7, yielding the answer 3.

I once gave students a division problem to solve. One high-achieving student told me: “Your problem has a mistake.” I asked why. He said it was because it didn’t divide evenly — there was a remainder of 3. I asked him to consider: division is sharing or grouping; if we genuinely have this many things to divide into this many groups, what do we do? He said it couldn’t be done, as you simply cannot divide those numbers that way. High-achieving students speak with confidence, because they are usually right, and indeed, according to the textbook, every example at that grade level divides evenly. When only such clean, tidy examples are given to make things feel easy and smooth for teachers and students alike, this is exactly the kind of student it produces.

Chapter 9: A Comparison of Math Education in China and the U.S.

第八章 中小学数学教育的几个误区

学数学需要强调理解，这是教育理论家们一致同意的，我也认为是对的。但教育专家们对“理解”的认识存在严重误区，把数学学习中的记忆部分完全摈弃、彻底否定，甚至反对学生记乘法口诀。这是专家们典型的“理性的自负”，是一种过头的自以为是。他们鼓励的一些做法，对学生非常不利。

第一个是允许低年级学生用计算器做加减法。这个做法表面上看让学生省去很多麻烦，按按键钮，轻而易举地就能找到答案。但这对他们建立数感、形成对数学的理解乃至喜爱，都不会有任何帮助，反而会使他们对计算器产生依赖，对数学产生陌生和恐惧。学数学的乐趣来源于通过自己的努力找到解决问题的方法所带来的成就感和满足感，以及发现自己变得越来越聪明的自豪和自信，而计算器在这方面对学生是有害的。

第二个问题是，因为有了计算器，学生就用不着背诵乘法口诀。加减法是直观的东西，比原来多或者比原来少，不会在概念上造成混乱。但乘除法包含着数学最根本的原理；没有它们，很难对数学有真正的理解。我曾经在学生中做过一个问卷，结果显示，记不住乘法口诀的学生比记住的学生，对数学的印象更加负面。这一点不难想象：我就见过学生算 12×8，是把8个12逐次加起来的方法得出答案的。

不能背乘法口诀的学生做不来乘法，更无法做除法，开方平方的问题当然就更别想了。计算器能做和自己能做，心里的感受完全不一样。就像体育课上篮球比赛中，是你自己投进一个球，还是看见他人投进一个球，感受肯定不一样。至少，他没有安全感，没有自信心；看着别人轻松愉快地就能说出来，他能感受到自己显得无能、显得笨。当他没有计算器的时候，他就像丢了魂似的无助、抓瞎。

不少进步派教育家明确反对让学生背诵乘法表，并声称这种反对是有科学实验依据的：死记硬背对学生的学习无助而有害，甚至伤害儿童的大脑功能。目前对中小学数学教育有着极大影响的斯坦福大学教授玖-薄勒（Jo Boaler）就持这种观点。由于这种观点的流行，美国学生中记不住乘法口诀的人不在少数，但我从来没有遇到过一个四年级还记不住乘法口诀的亚裔学生，也从来没有遇到过一个能容忍学生记不住乘法口诀的亚裔家长。亚裔的数学普遍优于其他族裔，造成这种局势的原因有各种说法，但有一个明显的事实是，几乎所有的亚裔都能熟练背诵乘法表。

我想问一下这些主张学生不记乘法口诀的大师们：难道不要求学生记住“十是由哪两个数字构成”吗？如果不记住 1 和 9、2 和 8、3 和 7 等，学生就不会做需要进位的加法。当然这些进步派大师们会说，不会做不要紧，让他们用计算器。目前确实有好多州在推行让小学生用计算器，理由是可以解放他们的脑力，去做更高深抽象的推理。

因为计算器能够做乘法，学生就用不着自己记住乘法口诀，这种想法的前提是认为学数学就是为了找到正确的答案。既然按计算器能够找到答案，那么你就没必要自己再去找了。让人啼笑皆非的是，这些教授自己也不承认学数学是为了找答案。他们也追求让学生对数学有概念性的理解（Conceptual Understanding）。

我们前面讨论过，学数学的目的是为了训练思维能力，让学习者有一个好脑袋。获得一种能力会带来两个效果：第一，这种能力可以用来做事、解决问题；第二，发现自己增加了一种能力会带来成就感、愉悦感和自信。所以，“有了计算器就用不着记乘法口诀”，就像你去拉小提琴，有人说：“你买两张 CD 就可以听小提琴了，干嘛要自己去拉呢？”一样荒唐。

其实，乘法表并不难背，虽然看起来有 81 行。但一不用记，二和三在学加法时基本都记住了。五是十的一半，即八乘十是 80，八乘五就是 80 的一半，40。七乘五就是 70 的一半，35。对于每天都要上数学课的学生，记住这些不会有问题。

九也有个很好的诀窍，就是它比十少了一。即 4 个九不会是 40，只会是 30 几；5 个九不会是 50，只会是 40 几。4 个九中，那 3 个九要变成十，每个都加一个，或者说每个都得给它戴一顶帽子，把它变成 10。这第 4 个九给了那 3 个九每个一顶帽子，自然剩下的就是九减三，成了六。答案是 36，三和六加起来是九。这种情况适用于每一个乘九的数字。比如，7 个九不会等 70，只能等 60 几。那第 7 个九得给那 6 个九每个一顶帽子。所以九减六是 63，因为 6+3=9。

这样，真要记住的，就是 4、6、7、8 四个数字的乘法。总共就是 16 行，每行 4 个数。其中 4 个是平方数，即 4×4=16，6×6=36，7×7=49，8×8=64，这些无论如何都应该记住，因为它们在以后的学习中会有很多功能。

余下 12 行是重复的，即 4×6 和 6×4 都等于 24，是一回事。所以严格地说只有 6 行，每行 4 个数（比如 4×6=24 就记住 4624），一共 24 个数字。我们一个电话号码是 10 个数字，所以要记得加起来，就是 2 个半电话号码。把四个平方数加进来，总共 40 个数字，就是 4 个电话号码。我们的这些大教授说，记住这 4 个电话号码会把人的脑子记坏了，真的是这样吗？

数学里有很多内容都如乘法口诀一样，是帮助我们上升到更高阶思维推理的工具。学习数学本身，不只是发展抽象的数理逻辑推理能力，也是去发现更便捷、更简明、更准确的解决问题的工具。直接跳过这些工具，对学习数学有很大的负面影响。

目前美国进步派教育给学生带来的第三个伤害，是考试时给学生提供所有试题需要用到的数学公式。

这样做的理论依据，是反对学生学数学靠死记硬背，而记住公式就属于一种死记硬背。在时兴的数学教学法中，死记硬背（”rote learning” or “drill and kill”）是一个落后的、错误的教学法的标签，一个必须坚决反对，甚至必须完全禁止的方法。于是出现了不要求学生记住乘法口诀的做法。下一步，也就理所当然地不要求学生记数学公式了。

稍微有点常识的教育理论家，还是分得清记忆在数学学习中是需要的。他们要反对的，是只靠记忆，不靠理解来学数学的方法。在当今很多的小学数学课堂中，不靠理解，只靠重复刷题对答案，让学生练习多了，产生机械记忆的学习方法是存在的，也是应该阻止的。但大专家们矫枉过正，把所有需要产生记忆的练习，都挂上死记硬背的标签，全面禁止，这就过犹不及了。

记忆在学习中是不可或缺的。它既是学习的起点，也是学习的结果。当我们对感官接收到的信息产生好奇，就生成瞬间记忆。觉得很有意思，继续玩味和追寻探究，就生成短期记忆。这短期记忆的内容，可能会给你带来重大的冲击，或者可能和你认知结构中原有的内容有多重的关联，或者仅是这个内容在短期记忆中重复出现过多次，让你生成了长期记忆。这就是你获得的，或说学到的东西。当然这个过程中你还获得另外两个东西，一个是在探究过程中锻炼了你的脑力，另外一个是在探寻和获得新东西的过程中产生的情绪，如成就感、自信心、爱憎等。

教师使用只靠死记硬背、不求理解这种方法的原因，有的是为了完成教学大纲的进度要求，没有时间来深入探讨和仔细讲解。但更多的是因为老师自己不懂数学，当然也就不懂怎么深入探讨和仔细讲解。老师自己没有概念性理解，自然没有能力去解释数学的逻辑内涵。说起来有点吓人，我个人接触过的小学教师，超过一半可以说是数学盲。我碰到过很多次这种老师很认真地问我，为什么负数加负数是负数，而负数乘负数就变成正数了。我有时也会问他们，为什么分数除法要把除数颠倒过来，然后用乘法去做？我几乎没有碰到过一个小学老师可以把它解释给我听。

让这么多的小学老师成为数学盲的原因，就是那些高深的理论指导下开发出来的”新数学”（New Math）教育大纲，以及根据这些理论和这个大纲设计出来的”新”教学法。这些方法教出来的**、没学好数学的学生去做了数学老师，然后一代不如一代地**传承下来。让小学生学做加减法可以用计算器，学乘法不要求学生记住乘法表，考试提供所有公式，这些都是造成数学盲的利器。

单靠代入死记住的公式来解题，而不追求对问题深刻的理解，是不好的。但是靠代入被提供的公式来解题，同样不需要对问题的深刻理解。其实，公式的形成和它的工作原理，是深刻理解问题的最佳道路。记住公式，至少是向理解公式进了一步。不记公式，对概念的深刻理解没有任何正面的帮助。中小学数学中的那些简单公式，要记住它们最牢靠的方法，就是搞懂它们的原理。没有搞懂它的原理，不知其所以然，仅靠死记硬背记住的东西，一是时间久了容易忘记，二是新的、类似的东西进来很容易产生互相混淆。由深刻理解推导出来的公式，有逻辑支撑，是活的，就是忘了也很容易根据理解再推导出来。

三角形的面积就是一个很好的例子。我们目前的考试题目，一般是给出一个三角形和它各条边的长度，让学生用提供的公式去求面积。学生所要做的，就是到发给他们的公式表里找出求三角形面积的公式：1/2×b×h。学生不需要动任何理解或是逻辑推理的脑筋。他们所要做的**，就是回忆老师告诉他们的，什么叫底，什么叫高，然后代入公式，底乘高除以2。这只需要学生有资料查询员的一般事务处理能力，找出哪条边是底，哪一条线是高，把它们代入公式，两个数相乘，再除以2，答案就算出来了。

学生通常的错误，是找最底下的线做底，找一条最往上伸的、最短的线为高。把哪条边定为底边，老师和教科书肯定都解释过，它和具体空间位置没有联系，而是和形成直角的另外一条边互为底和高。但把最底下那条线或者那个面叫做底，是现实生活中无数次实际经历形成的常识，且教科书上和老师在课堂上画的三角形，也多是把最下面那一条叫底边。如果这个三角形确实是斜放在那儿，很难说清哪一条在最下面，学生常会把较长的那条线定为底，把最短的那条线作为高。这是他们长时间积累的实际经验所形成的判断，而不是逻辑的推导。

三角形的面积，真正应该考核的，是对这个公式的解释。求面积需要的概念性理解，是先得讲得清楚这几个概念：什么是面，什么是面积，什么是平方？为什么叫”方”？然后是怎么把三角形变成平行四边形，又怎么把平行四边形变成长方形。然后长方形才有了”方”。有了”方”，就可以精确量出平方数来。这是一个非常有意思的、非常巧妙的解决问题的方法。这个过程是启发人思考的一个典范，也是让人欣赏数学美妙而喜爱上数学的例子。

有了上面讨论的这几个概念和这些概念的推导过程的理解，难以置信的是，竟然有专家说不许要求学生记住这个公式。让学生用给他们的公式，在图形里找两个数字乘一下，再把乘得的积除以2，就证明他们有概念性的理解了？就这么一个例子，就充分展示出为什么我们的教育家们的理论越来越高深，设计出来的教学大纲越来越完善，而他们设计出来的这一套方法，却把美国学生的数学能力培养得越来越差。

这些做法造就了近些年来，为了追求平等，通过种种的政策倾斜和经济支持，美国贫困地区的学校不但没有好转以接近平等，反而差距越来越大的怪现象。1989年，美国数学教师协会（NCTM）发布了新的教学标准，重新点燃了数学战争。我就是在这个时期来到了美国，所以身临其境地看到了新教学标准，它的理论支持，它在学校里的实际操作状况。更重要的是，我看到了它的效果：学生的数学成绩比原来更差了。

新教学标准的目标是要从”死记硬背和程序化”转向一种更全面的理解，数学被视为推理和解决复杂问题的工具。这一点我也完全赞成，但是它少讲了一个数学的功能，犯了一个天大的错误。数学是”推理和解决复杂问题的工具”之外，我认为还得加上一条，它是一门”锻炼和发展理性思维能力的学科”。就像音乐、哲学、体育等课程，有的学生是在学一门”工具”，甚至是谋生的工具。但有的人学它们是为了陶冶性情，锻炼身体，锻炼脑力，就是为了好玩，觉得它很有意思。那个天大的错误，就是由”从’死记硬背和程序化’转向一种更全面的理解”所引申出的结论：既然有了计算器这样的”工具”，学生就可以完全解放，不需要记住基本的数学事实。

虽然教育专家主张不要死记硬背，但在实际操作中，教师经常会在学生不充分理解的情况下，教给学生一些记忆性的程序性规则，来帮助学生快速找到”正确”答案。很多时候，这些规则并不有助于真正的理解，会产生长期的负面效果。

比如，前面讲到的PEMDAS，是为了帮助学生记住四则运算的顺序而创造的一个口诀。其实理解这个顺序，比记住这个口诀要管用得多，而PEMDAS（”Please Excuse My Dear Aunt Sally”，即”请原谅我亲爱的姑妈莎丽”），虽然朗朗上口，但有很大的缺陷。它导致很多学生无法明白加法和减法、乘法和除法其实是同等优先级的，导致出现很荒唐的计算错误。目前，PEMDAS在很多学区被禁用了，但还有不少类似的、看似对学生有帮助的口诀，被那些没有充分理解数学的教师广泛使用，以代替对数学真正的理解。

我知道的大部分中学老师会用一个简单的助记口诀：”保持！更改！更改！”（Keep! Change! Change!）来帮助孩子记住负减数的减法规则：即”保持”被减数不变，将减法”更改”成加法，将负数的减数”更改”成正数。这条规则可以保证学生不需要理解，也可以把含负数减数的减法做对。但这条规则有时效性和易受干扰性。时效性就是时间稍微久一点后，学生就忘了这个”保持！更改！更改！”是只用在减法上，还是加法上；是减数是负数才可以用，还是被减数是负数时才用。时间一久，就会淡忘了这些细节。易受干扰性，就是学了其他加减法或者乘除法，记住了一些新的助记口诀，容易互相混淆。譬如学分数除法时老师教过的”保持！更改！倒过来！”（Keep! Change! Flip!），就容易跟这条规则相混。

其实可以这样让学生理解：负数可以是一个动词，意思是减去、用掉，或者是少去；也可以是形容词，意思是欠缺的、少掉的、低于，或者是债务，等。这样，就可以设计一些抢答游戏，让学生理解减少收入和减少支出的区别，而减少支出和增加收入是同样的效果。另外在语言上有很多例子，可以让学生发现并理解否定之否定就是肯定的逻辑关系。如”对”、”错”、”不错”，”好”、”坏”、”不坏”。英文中有”Make sense、Nonsense和That’s not nonsense”、”Not bad”等。这些可以让”负负得正”成为学生意识中的一个常识，一个正常的思维习惯。最后一眼看见一个负数前面有一个负号，就知道它是个正数。

再举一个例子：为什么那么多学生会得出 7 ÷ 21 = 3 这样的结论？我们的教材会解释说除法就是分享（sharing）或者是分组（grouping），然后就是描述如何把横式变成竖式：先写一个像小房子（像”厂”字）的除法格式，被除数放在”厂”字里面，除数放在”厂”字的前面；然后用乘法依次从首位数或首几位数找可能的商；找出来、乘出来之后，把乘积减掉，然后移入下一位。整个过程就是个无聊乏味的技术操作流程，老师照着教科书上的示范教，学生老老实实地跟着，日复一日地重复练习。

这个重复有时会持续几年时间，因为小学数学是螺旋式逐渐加深的。教材里的例子（或者老师用的例子）一般都是大数字放在”厂”字里，小数字放在”厂”字的前面，而且是没有余数的——理论上是为了避免太复杂、怕把孩子搞混了，实际上是为了让这个过程赶快结束。结果就是，学生养成了习惯，看见大的数字就放到”小房子”里面，小的数字放在前面，这样 7 ÷ 21 就变成了 21 ÷ 7 的实际操作，得到 3 这个答案。

我有一次出了一道除法题让学生做，一个平时成绩很好的学生对我说：”你的题目出错了”。我问为什么，他说因为除不尽，剩下个 3 是余数。我跟他讨论说，除法是分享或者分组，如果我们确实有这么多东西要分成这么多组，该怎么办？他说分不了，这个数字就不能这么分。成绩好的学生说话比较自信，因为他们通常是正确的，而且根据教材，这个年级的教科书上确实所有的例子都是除得尽的。如果只给出这样简单又干净的例子，让老师（和学生）觉得轻松容易，学生就会学成这个样子。

(link to buy my book, “Memoir of an Immigrant Math Teacher”

(I am tempted to post a picture of Mr. Xu’s lecture, but alas, student privacy prevails)

Appendix to Chapter 7: The Socratic Method

In 2025, a decade after his retirement, Mr. Xu was invited to give a demonstration lesson at a low‑income middle school near Stanford University. Several people observed one of the classes, including Patricia Devaney, former Associate Dean of Research at Stanford. Before the observation, Patricia was skeptical of Mr. Xu’s methods; afterward, she became a complete believer. She said, “I really appreciate how Mr. Xu treats every student equally, regardless of whether their answers are right or wrong. Several times I almost couldn’t hold back and wanted to tell the students the answer myself, but he always maintained a poker-face, patiently guiding students to explain their thinking to the whole class until they eventually reached a consensus. In this process, all the students were fully engaged in learning through thinking and expressing. Thinking and explaining are the core goals of every one of his lessons.”

Through a friend, I recently watched the second half of the video of this lesson. The class was about fraction operations, with examples of addition, subtraction, multiplication, and division, as well as discussions of the concept of fractions. At the end, the class launched into a whole‑group discussion based on one problem: 7 – ¼ = ? A teacher I deeply admire (Mr. Ase, whom I wrote an entire chapter about in my new book “Unbalanced: Memoir of an Immigrant Math Teacher”) once told me, “Teaching is like giving a performance.” What Mr. Arthur meant was not that teaching is like acting and requires no sincerity, but that treating the classroom like a compelling performance can maximize student engagement and interest, thereby achieving better results. Mr. Xu’s lesson fully demonstrated his gift for “performance.”

In the twenty‑plus minutes of video, Mr. Xu’s expressions, language, and body movements indeed had a touch of theatricality, and the students were his audience. For example, when he spoke to students, his eyes were always on them, and he kept his body at their eye level as much as possible, sometimes naturally bending his knees. He almost never turned fully toward the board. His arm movements were wide—sometimes stretching, sometimes waving, sometimes tapping his head. Interestingly, some students even unconsciously mimicked his gestures during the lesson.

He gave students a great deal of attention and never ignored their questions or comments; the students, in turn, stepped into their roles and interacted with him. He tried to let every student speak, and when a student had nothing to say, he would naturally move on, keeping the class progressing.

When a student not only gave an answer but also clearly explained the reasoning, he would summarize: “See, she not only knows the answer, but she also knows the reason. She used reverse mathematical reasoning. What we are learning here is mathematical reasoning.”

When a student gave a wrong answer and later understood through discussion, he would say: “Mistakes are precious; they are opportunities for us to grow. If we all got everything right, there would be nothing left to learn. So, we should cherish every mistake. We come to school to learn, and we all have a license to make mistakes.”

He used questions that seemed very natural to draw out students’ insights, guiding them until they explained the problem clearly. At that point, he would say: “Look, I didn’t teach you anything; you figured out the math yourself. You taught yourself math.”

These behaviors encouraged and motivated students effortlessly. Keep in mind, this was Tier 1 school where students normally lacked the motivation to learn. Mr. Xu is neither tall and imposing, nor stern and forceful, yet he was able to create such a classroom atmosphere on the very first meeting. It is no small feat.

He had a well‑practiced method to ensure every student participated rather than simply watched. When a student gave an answer, he would ask each student: “Do you agree? Disagree? Or need more time to think?” If a student disagreed, he would ask: “What is your answer? How did you get it?” If a student agreed, he would ask: “You agree? What exactly do you agree with?”, naturally prompting students to express their own understanding. After the bell rang, he kept the students for thirty more seconds and said, “When you get home, think about what you learned today. I’ll ask you tomorrow.”

What Mr. Xu used most was Socratic questioning: Why do you think so? Does anyone have a different answer? Can you give an example? I noticed he almost never stated conclusions himself; instead, he asked questions at the right moments so students could figure things out themselves.

Socratic teaching method uses a sequence of logical questions to guide students to discover truth on their own. It cultivates “people who can think,” not “people who can do problems.” Its key points include:

1. Centering on questions, not answers. The teacher does not give conclusions but uses questions to help students continually refine their thinking.

2. Letting students “say it themselves.” Students must articulate, explain, and defend their conclusions.

3. Using dialogue to drive thinking. The teacher is not the source of knowledge but a guide; students reach conclusions through reasoning.

4. Treating mistakes not as trouble or shame but as learning opportunities.

Although the Socratic method has existed for thousands of years, it has never been widely adopted in mainstream education. Beyond institutional reasons, could it be that widespread adoption faces practical difficulties? Mr. Xu’s interactions with students looked natural on the surface, but behind them lay extraordinary control and the ability to respond to different situations. How high is the bar for teachers to achieve this? As a high school math teacher myself, while watching the video, there were several moments when I could barely resist the urge to explain the answer to students, yet Mr. Xu avoided doing so with complete ease, letting students reach the conclusion themselves.

In my own teaching, I have also used this method, but never as thoroughly as Mr. Xu, nor with such good results. I either worry we may be progressing too slowly for the topics needed to cover, or that student discussions may drift off topic, or that the chosen topic doesn’t match students’ level. I consider myself a fairly good teacher (to be fair, what teacher doesn’t think that about themselves?). If I can’t do it well, how many other teachers can? If this method is widely but improperly used, what would be the consequences?

These topics we will discuss in detail in the later part of this book.

第七章附录：苏格拉底式教学方法

2025年，徐老师退休多年后，受邀到斯坦福大学附近的一所贫困初中课堂做示范讲课。有一堂课有好几个人观摩，其中有斯坦福大学已退休的研究副院长（Associate Dean of Research）Patricia Devaney。观摩前，Patricia对徐老师的方法持怀疑态度；观摩后，她成了彻底的信徒。她说：“我很欣赏徐老师对每位学生一视同仁，无论学生的答案是对是错。他的好几次停顿都让我几乎忍不住想直接告诉学生答案，但他始终保持镇定的表情，耐心引导学生向全班解释自己的思考过程，直到他们最终达成一致的答案。在这个过程中，所有学生都通过思考与表达而全身心投入学习。思考与解释，是他每一堂课的核心目标。”

通过一个朋友，我最近看了徐老师这堂课后半部分的录像。这堂课讲的是分数运算，举了几个分数加减乘除的例子，也涉及分数概念的讨论。最后，通过一个题目：7 – ¼ =？展开了全班的课堂讨论。我敬佩的一名好教师（在我的新书《失衡》里有一章专门写到的阿瑟先生）曾告诉我：“教每堂课就像是做一场表演。”我想，阿瑟先生所说的，并不是讲课像演戏、不需要真诚付出，而是说，把课堂当作一场精彩演出，可以最大程度地调动学生的参与和兴趣，从而达到好的效果。徐老师的这堂课，充分体现了他的“表演”天赋。

在二十多分钟的录像里，徐老师的神态、语言、身体动作，看起来确实有一些戏剧表演的成分，而学生就是他的观众。比如，他对着学生说话的时候，眼睛一直看着学生，身体活动也尽量保持在学生眼睛的高度，有时会很自然地弯下膝盖，几乎没有完全转身去看黑板的时候。他的手臂活动范围比较大，有时伸长，有时挥舞，有时拍拍脑袋。有意思的是，有的学生在课程中竟然不自觉地模仿他的动作。

他给学生的注意力很多，从不忽视学生的问题和评论；学生也进入角色，跟随他一起互动。他尽量让每个学生都发言，当学生没有意见时，他有时也会自然跳过，让课堂一直保持推进。

当学生不仅给出答案，而且清楚地给出原因时，他会总结说：“你看，她不仅知道答案，她还知道原因。她用到了反向数学推理。我们要学的，就是数学推理。”

当学生给出错误答案，通过讨论弄懂以后，他会说：“错误是很宝贵的东西，它是让我们成长的机会。如果我们都做对了，就没有什么可以学习的了，所以我们要珍惜每一个错误。我们到学校来，就是来学习，我们都有犯错误的许可证。”

他用一些看起来很自然的问题，引出学生的见解，直到学生把问题解释得很清楚。这时候，他会说：“大家看，我并没有教你们什么，是你们自己搞懂了数学，你们教会了自己。”

这些行为让学生很自然地得到鼓励和激励。要知道，这是一所超贫困的学校，学生平时根本不愿意学习，徐老师既不高大威武，也不严厉逼人，第一次见面就能达到这样的课堂气氛，实属不易。

他有一套很熟练的方法让每个学生都参与，而不是简单观望。当有学生给出答案后，他会问每个学生：“你同意？不同意？还是需要时间考虑？”如果学生说不同意，他会问：“那你的答案是多少？你怎么得到这个答案的？”如果学生说同意，他会问：“你同意啊？你同意什么？”这样自然让学生表达自己的理解。下课铃声响了之后，他留学生三十秒，还会说：“你们到了家，也要想想今天学了些什么。明天我会问你们的。”

徐老师用得最多的是启发式提问：你为什么这样想？谁还有别的答案吗？你能举个例子吗？我注意到，他几乎从不说结论，而是在适当时候提出问题，让学生自己搞明白。我意识到，这是苏格拉底式教学。

苏格拉底式教学是用连续、有逻辑的问题，引导学生自己发现真理。苏格拉底式教学培养“会思考的人”，不是“会做题的人”，有以下几个要点：

1. 以问题为中心，而不是答案。老师不讲结论，通过提问，让学生在回答中不断修正自己的思考。

2. 让学生“自己说出来”。学生必须说、解释、辩护自己的结论。

3. 通过对话推动思考。老师不是知识的来源，只是引导；学生通过推理，自己得出结论。

4. 错误不是麻烦，也不是耻辱，而是学习的机会。

然而，苏格拉底式方法已经流传了几千年，一直没有被主流采用。除了制度原因之外，是否也因为在广泛使用中会碰到实际操作上的困难？徐老师在课堂上与学生的互动，表面上看起来很自然，但其实背后展示了超强的课堂控制能力，以及对不同情景的反应能力。做到这样，对教师的要求到底有多高？作为高中数学老师的我，在观看徐老师录像时，有几处都很难控制想要给学生详细解释的冲动，虽然徐老师每次都很自然地避开了，让学生自己得出结论。

我在实际教学中，也曾运用过这种方法，但从没做到像徐老师这么彻底，也没有达到这么好的效果。我要么担心进度太慢，要么担心学生讨论偏离主题，要么担心选择的主题与学生水平不匹配，无法及时调整。我自认为是个还不错的老师（当然，哪个老师不是这样觉得自己的？）如果我做不好，又有多少数学教师能做得好呢？如果这样的方法被推广，但被广泛地错误使用，会产生什么样的结果？

这些问题，书的后面部分会继续探讨。

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Chapter 7 What’s Required of a Teacher

There is a peculiar phenomenon in education: no matter what method you use, whether it is a “good method” praised to the skies by experts, or a “bad method” harshly criticized by scholars, there will always be students who adapt to it, and students who do not. Therefore, my conclusion is: there are no bad methods in education, only bad results. As long as the results are good, the method is good.

I often warn the parents, students, and teachers who come to me for advice: the shelves full of books written by experts on educational theories or teaching methods are often nothing more than “correct nonsense,” because they may not apply to a particular student. Take Amy Chua, the “Tiger Mom,” for example. She raised two outstanding daughters, and their mother‑daughter relationship is very harmonious. If you follow her method, you might get the same results, hallelujah! But it is also possible that you will raise a child who rebels against everything you do, not only disappointing you deeply, but even becoming your enemy.

In truth, education has only one method: look at the results. But there are two issues to pay attention to. First, results can be superficial or deep. Second, results can be immediate and short‑term, or long‑term and lasting.

By levels of results, I mean: when the outcome you expect appears, is it merely superficial and behavioral, or is it conceptual and internalized, something that truly enters the heart? For example, a mother—or, more commonly a grandmother—chases a child around trying to feed him. The child is happily playing, gets tricked into opening his mouth, and a big spoonful of food is shoved in. The grandmother feels satisfied, thinking she finally fed him something delicious. She completely ignores the fact that stuffing so much food into his mouth at once may make him uncomfortable, ruin his appetite, cause aversion to eating, and even make him dislike you.

When teaching young children math, too many parents feel happy and satisfied the moment they see the child answer 3 + 5 = 8 correctly. If the child answers 7, they become unhappy and immediately correct him. But in the learning process, whether he gets a question right or wrong is not important; what matters is whether he understands it in his mind. Even more important is whether he experiences joy or a sense of accomplishment after being right or wrong. And most important of all: after doing these problems, does he think, “Wow, math is so interesting!” Does he fall in love with the subject, or does he find it boring, meaningless, or even hateful? This effect is deep and long‑lasting. A teacher who can make students love learning math and become good at learning math is a truly good math teacher.

Therefore, I divide teaching into several levels.

The first is “teaching to the book,” meaning being familiar with the curriculum, mastering the textbook, preparing lessons carefully, and lecturing according to the script. Traditional teachers teach this way and are evaluated this way.

The second level I call “making students act.” In the Chinese school I founded, one rule for evaluating teachers was to have parents observe classes and write comments. All the parents’ comments focused on lesson preparation, lecturing, and board writing; everything was about the teacher’s performance. None wrote about the students’ performance. In fact, this is the more important aspect of evaluating K–12 teachers: what the teacher arranged, guided, attracted, or instructed students to do.

The third level is “training the brain,” meaning what happens inside the student’s mind: what thinking abilities are being trained and improved, what skills are being mastered, and what knowledge is being gained.

The fourth level is “shaping the heart,” meaning what emotions the student develops toward the subject through this class, love or dislike; what aspirations the teacher inspires; what mindset is formed. A teacher cannot accompany a student for a lifetime, but a good teacher influences a student for a lifetime. A love for mathematics can also accompany a student for a lifetime. Therefore, helping students clearly learn new things must have a definite positive impact on their emotions and psychology.

***

The great Tang‑dynasty writer Han Yu wrote the famous essay On Teachers, in which one sentence defines the teacher’s work: “A teacher is one who transmits The Way, imparts knowledge, and resolves doubts.” And so, teachers trained in psychology, pedagogy, and subject‑matter expertise walk into classrooms with carefully designed curricula and textbooks, face a group of naïve and ignorant young children, and begin “transmitting the Way, imparting knowledge, and resolving doubts”. This is how a typical process of “pulling up seedlings to help them grow” begins. Human nature already has a tendency to “love being a teacher.” Now, with official authority, their job is literally to be a teacher! Facing a group of ignorant children, and teaching them an extremely important but abstract formal science that they desperately need to “have their doubts resolved” about, responsible teachers exert all their strength, pulling and pushing, determined to teach them this lofty, abstract, formal discipline.

In our current class‑based teaching model, one teacher often teaches a large class. One person facing dozens of students with different backgrounds and levels naturally leads to an obvious solution: the teacher takes responsibility for the teaching content, but cannot possibly take responsibility for each individual student’s learning process and learning outcome. So teachers prepare lessons carefully, explain in detail, and hope all students listen attentively and study hard. As for how students learn, how much they learn, whether they merely know how to do the problems without understanding the principles, or whether they truly understand, teachers simply cannot manage that. Our exam system fully endorses this teaching model: as long as students can solve the problems and get the right answers, they are good students, regardless of whether they truly understand or merely imitate the steps from memory.

The problem with this method is that self‑disciplined, confident, and focused students tend to learn well, while others tend to learn poorly. Those who understand find math easy and enjoyable. Those who rely only on memorization and imitation find it exhausting, joyless, and increasingly difficult. The traditional solution was to emphasize more practice. After constant practices and rote memorization, you can “do” the problems. Although few students truly understand, most can score well on exams, so they are considered to have “learned.” Among those who rely on imitation and memorization, some will gradually understand the knowledge they previously held only mechanically, as their cognitive abilities develop. This is the miracle of the brain and human plasticity. But this learning process is absolutely like force‑feeding a duck: the throat feels the discomfort and even pain of being stuffed. Students dragged into this lofty “mathematics” in such a way will certainly not have a pleasant learning experience. This is also why many people dislike math.

***

In the lower grades of American elementary schools, most schools assign one homeroom teacher to a class, who teaches all major subjects. Other subjects such as art and physical education are taught by specialists. In the upper elementary grades, one teacher is responsible for language arts and social studies, and another teacher handles math and science. At the middle‑school level, subjects are generally separated, each taught by a different teacher.

In the U.S., teaching credentials are divided into two types: an elementary certificate for kindergarten through sixth grade, and a secondary certificate for grades seven through twelve. Being qualified to teach math in middle school already means the teacher is considered reasonably strong in math. These teachers often ask me questions they encounter, and through these questions, I realized how serious some of their conceptual gaps are.

It is normal for teachers to discuss difficult word problems among themselves. When they come to ask me, I usually not only tell them how to solve the problem but also explain the underlying principles, hoping they can understand more deeply and teach more effectively. This helps them see that understanding the principles behind a problem is indeed much clearer and deeper than merely finding the answer. As a result, some begin asking me questions they can solve procedurally but do not truly understand conceptually. For example: Why does a negative plus a negative give a negative, but a negative times a negative gives a positive? Why do fractions require a common denominator for addition and subtraction, but not for multiplication?

This made me see why so many people dislike math: they don’t understand it.

Because they “know the result but not the reason,” no matter how fluent they become, what they experience is a mechanical task, not a process of reasoning and inquiry. Learning becomes a meaningless cycle of repetition and memorization. Solving problems becomes nothing more than performing a technical procedure, without the joy of using the mind to explore, understand, and solve.

It is like climbing a mountain blindfolded: even if you memorize every step or receive verbal cues about when to climb or when to step up, you may eventually reach the top. But throughout the journey, you see none of the scenery. The climb becomes nothing but exhausting and dull labor. Without joy, you may still work hard and be disciplined, and you may even achieve good results; but you are merely a laborer of mathematics, not a player of mathematics. Human nature prefers ease over toil. If you are not “playing,” but “laboring,” you will inevitably feel tired and want to stop.

To some extent, Chinese students’ strong math performance is the achievement of excellent laborers. Parental, teacher, and societal pressure—combined with their obedience, discipline, and diligence—drive their effort. But because their success comes from hard labor rather than genuine love, their likelihood of achieving true long‑term excellence is not very high.

***

The principle I follow is closer to Socrates’ famous line: “I cannot teach anything. I can only make them think.”

With AI, humans will never outperform machines in calculation. The true purpose of mathematics is to make students think and to push their thinking to higher levels. Humans are born “knowing nothing,” but our capacity for learning is unmatched by any other species. In the age of AI, the focus of math education is not training students to compute faster than machines, but training them to think, and to push their thinking to the most abstract, rigorous, and reliable levels.

Mathematics is not becoming useless; it is becoming more useful. It helps human thinking become more precise, clearer, simpler, and deeper. Many people say math is too hard and too abstract. I believe the opposite: math has become so abstract precisely because it is so easy to get started, which allows it to advance very far.

For example, someone may ask whether violin or piano is easier to learn. To play “Happy Birthday,” the piano is certainly easier: you can play it with one finger. Because the piano is easy to start, people keep pushing further, to harder and more complex pieces, until they reach their limits. Math is the same: because it can express the most complex relationships with the simplest structures, it can advance extremely deeply.

I believe math trains thinking in at least three ways:

1. Deeper: the ability to think more deeply;

2. More logical / more accurate: more rigorous and precise reasoning;

3. Simplest way: the ability to capture the essence with the simplest method.

Because “the simplest” allows us to handle more complex problems; otherwise, thinking quickly reaches its limit. Mathematical thinking is micro‑level; mathematical ideas are macro‑level structures and systems.

Back in the classroom, content and standard methods are tools, not goals. Therefore, teachers must focus on two things:

First, teachers must truly understand mathematics, not just how to solve problems, but the logic behind them. Second, classroom content and “standard methods” are tools for training students’ thinking and mathematical reasoning, not the endpoint of teaching. Many classrooms today treat “being able to solve problems” and “raising test scores” as goals. But they should be pathways to show how mathematics analyzes relationships and pursues clarity, simplicity, and accuracy.

***

Teachers need sound judgments to let children use the knowledge they already have, take on learning tasks that are challenging, yet achievable, through discussion and collaboration. This is one of the most important abilities of a good teacher. And this ability is actually not difficult to cultivate. The problem is that our current teaching model requires teachers to follow the curriculum and textbook to stuff knowledge into children, rather than guiding children to “play” according to their developmental level. Over time, teachers lose this ability. Teachers become accustomed to teaching by looking at the textbook and curriculum, rather than looking at the child’s current level, their interests, and their feelings during the learning process. In fact, when people talk to others, they naturally try to anticipate how the listener might receive the message and adjust their wording and tone. This is what “teacher’s autonomy” really means.

Every time the government organizes experts and professors to design a curriculum to tell teachers what to teach and how to teach, it is actually weakening, or even stripping away, the teacher’s autonomy. This is one major reason why every new reform the government introduces ends up lowering, rather than improving, teaching quality. You could say that these officials and experts with a “God complex” not only harm students but also destroy teachers. As long as teachers regain autonomy in the classroom, and receive a bit of necessary training, they can recover this ability.

Modern fashionable educational theories often claim to shift from “teacher‑centered” to “student‑centered” instruction. But in reality, they have not shifted to student‑centered teaching. Instead, they have shifted to curriculum‑centered teaching, or expert‑designed content and pacing‑centered teaching. Conceptual understanding requires teachers to know the student’s existing knowledge structure, anticipate what methods the student can accept, and then step back to let the student explore.

The students we face every day are living individuals. Students in one district differ from those in another; students in one class differ from those in another; one student differs from another. Even the same student may be different today than yesterday. Without autonomy, when experts try to impose uniform content, pacing, and even teaching methods on diverse districts and students with vastly different backgrounds, teachers inevitably become robots who merely follow scripts. And students will struggle to develop conceptual understanding.

The discussion above may give readers two impressions: (1) Teachers must learn a completely new method to teach this way; (2) Most teachers probably won’t be able to learn it. The first point is true: this is a new method. The second point, however, is unnecessary worry. Teachers absolutely can learn it and learn it well.

What teachers need to learn is to let go and allow children to do things themselves, satisfied to be a “guide on the side”. Like a PE teacher on the playground, let children play on their own while you stand on the side offering occasional guidance, like a coach or referee who blows the whistle only when someone commits a foul. The rest of the time, they are the ones doing the work. I often join them in play. Of course, you must restrain your advantage of “knowing more and being more capable,” and instead become someone who needs their help the most. Give the joy of teaching others, even correcting others, to the smallest, weakest student.

What teachers need is simply to be a normal human being who can communicate with others, with basic empathy. When talking, they should be able to sense the other person’s thoughts. This is well within the scope of what every teacher learns in education and psychology courses. Of course, to help students develop conceptual understanding, teachers themselves must have it. After decades of “new education” reforms, many American K–12 teachers no longer possess this ability. Because young children’s cognitive development is limited, abstract mathematical concepts must be nurtured slowly; but for adults, understanding these basic concepts is actually not difficult.

***

Lesson plans should be written, but writing too much becomes a problem: the more detailed they are, the more they bury the essential ideas. What matters most is that before entering the classroom, the teacher must have clarity. At the macro level, there are three components:

First is the teaching objective, or what you want students to gain. My goal is not for students to score high on tests. My students’ good test scores are merely a side effect of effective teaching. The current school system, and most teachers, treat high test scores or meeting standards as the primary goal.

Second is the teaching method you have mastered, or are practicing, that best achieves your teaching goals.

Third is your deep understanding of the material needed to achieve the teaching goal; for K–12 math, this means the interconnected structure of the four types of numbers and the four operations.

At the micro level, the most important thing is understanding your students; then understanding parents, district expectations, the specific textbook used, and coordination with other teachers and administrators. Understanding your students means: (1) knowing the misconceptions they already hold about math; (2) knowing the bad habits they have developed over years of learning math; (3) knowing how much useful knowledge they currently have.

My lesson plan begins by outlining the components above, listing a few concrete items under each. From the first day I meet the students, I start talking. After each class, I record the day’s results—overall, specific details, class‑wide, individual—and then continue the “story” the next day or continue the conversation.

Most people are born with the ability to chat with others. More deeply, it means being able to adjust your way of speaking, tone, and content depending on the person. Strictly speaking, someone who cannot converse with others simply cannot be a teacher—though such people are rare. The traits of a good teacher are the same traits that allow someone to chat with a stranger at a party, in a bar, or with the person sitting next to them on a plane. Our current teacher training, however, almost forces teachers to abandon this natural ability, and instead follow expert theories, curriculum requirements, and textbook structures to write detailed lesson plans and then deliver them mechanically.

Teachers need two things: One is emotional connection: children must feel that talking with you is enjoyable. They must find you interesting, and also find what you talk about interesting, something that increases their knowledge and ability. Liking you as a person and liking what you talk about are two sides of the same coin.

The other is the “hardware” of professional knowledge: (1) a deep understanding of the foundational mathematical structure you are teaching; (2) an understanding of cognitive development, i.e., basic knowledge of education, psychology, and brain science.

This is the teacher’s true confidence, or what people often call “swagger.” This “swagger” strengthens the teacher’s presence in front of students and makes students more willing to follow you.

Chapter 8：Some Harmful Practices in K-12 Math Education

第七章 教师的职责

教育里有个奇特的现象，就是无论哪种方法，不管是被专家们吹得天花乱坠的好方法，还是被学者们痛批的坏方法，总会有学生适应，也总会有学生不适应。所以，我的论断是：教育里没有坏的方法，只有坏的结果。反过来说，只要有好的结果，就是好的方法。

我告诫找我咨询的家长、学生和老师：书架上成排的专家写的教育理论或者教学方法的书，往往只是些“正确的废话”，因为它们不一定适合这个特定的学生。比如说虎妈蔡美儿，她培养了两个优秀的女儿，而且母女关系非常融洽。你要是照她的方法来，有可能会有和她一样的结果，Hallelujah！但也有可能，你会培养出一个完全跟你作对的反叛者，不只是让你大失所望，甚至还可能与你反目成仇。

教育其实只有一种方法，那就是看结果。但这个结果有两个要注意的问题。第一个是：结果有表面的和深层次的。第二个是：结果有当下的、短期的，也有长期的、恒定的。

所谓结果的层次，就是你期望的那个结果的出现，是表面上的、行为上的，还是思想上的，是否真正深入到心里。一个例子是，一个母亲——更常见的是祖母——追着小孩子喂饭。孩子玩得正高兴，被用各种幌子哄得张开嘴，然后乘机塞进一大勺食物。祖母心里美滋滋的，觉得终于把这么好的美味喂给他了，完全不去考虑一下子塞那么多食物在他嘴里，会让他不舒服，倒了胃口，甚至产生厌食，甚至连带讨厌你。

教小孩子数学，太多家长只要看到他把 3 加 5 等于了 8，结果对了，就感到很高兴、很满足。他的结果等于 7，错了，你就不高兴，立马纠正他。但在学习过程中，做对一道题还是做错一道题并不重要；他脑子里是否真正搞懂才重要。更重要的是，在他的对错之后，有没有得到乐趣，有没有享受到成就感。而最重要的是，他做了这些题之后，有没有发现：“哇，数学太有意思了！” 是爱上这门课了，还是觉得这门课很无聊、没什么意思，甚至讨厌这门课。这个效果就是深层次的、长久的。能让学生热爱学习数学，并善于学习数学的老师，才是好的数学老师。

所以，我把教学分成几个层次。

第一个叫“教书”，就是指熟悉大纲，吃透教材，认真备课，在课堂上照本宣科。传统上的老师就是这么做，也是这么被评判的。

第二层我称为“使学生”。在我创建的中文学校里，有一项考察老师的规则，就是让家长听课，然后写评语。所有家长的评语都只写了备课、讲课、板书等老师课堂上的表现，从没有家长写学生的表现。其实，这才是中小学老师课堂更重要的考核方面：教师安排、指导、吸引或者命令学生做了些什么。

第三层是“练大脑”，就是指学生的脑袋里发生了什么，他们的思维能力受到了什么训练和提升，掌握了什么技能，增加了什么知识。

第四层是“使生心”，就是通过这门课，学生心里对这门课产生了什么样的情绪，是爱还是厌；教师引导学生立了什么志，生了什么心。老师不能陪伴学生一辈子，但好的老师会影响学生一辈子，对数学的爱好也会陪伴学生一辈子。所以，让学生清楚地学会新东西，对学生的情绪和心理有明确的正面影响。

唐朝大文豪韩愈写过一篇著名的《师说》，一句话把教师的工作指明了，即：师者，传道、授业、解惑也。于是，接受过各种心理学、教育学和所教学科专业训练的老师，抱着专家们精心设计的教学大纲和教材，走进教室，对着那些幼稚无知的懵懂幼儿，在讲台上侃侃而谈，开始传道、授业、解惑。一个典型的拔苗助长的过程就这么开始了。人的天性中本来就有“好为人师”的趋向。这下好了，他名正言顺的工作，就是为人之师。面对的一群无知幼稚的儿童，而且要授业给他们的，又是一门极其重要、但他们非常欠缺、亟待“解惑”的抽象形式科学。于是，负责任的老师们就使出浑身解数，生拉硬拽，一定要教会他们这门高深的、抽象的、形式科学。

我们现行的班级制教学模式中，是一个老师面对着一群，常常是几十个学生。一个人要面对几十个背景各异、水平不一的学生，一个显而易见的解决方法，就是老师认真地对教学内容负责，而不大可能对每一个不同的个体学生的学习过程和学习效果负责。于是，老师们都会认真备课、细致讲解，希望全体学生认真听、努力学。至于学生怎么学、学了多少，是仅仅会做但没懂原理，还是真正理解了，教师就顾不过来了。我们对学生的考试方式，也完全认可了这种教学方式的合理和效率：只要学生把题做出来，答案对了就是好学生，不管你是懂了，还是仅仅依葫芦画瓢、凭记忆复现解题步骤找到正确答案而已。

这种方法的问题是：自觉努力、自信心强、坐得住的学生容易学得好，反之则会学得差。懂了的，会觉得学数学很容易，而且学着有乐趣。仅靠记忆和模仿会做的，则学得很辛苦，累而无趣，而且结果会越来越差。从前的一个解决方法，就是强调多做练习。大量刷题、死记硬背之后，你也就算是会了。尽管真正懂得的学生不多，但大多数学生都能考试考得不错，算是“会了”。那些靠依样画葫芦、死记硬背的学生，在他们的认知能力发展和探索过程中，也会有一部分人逐渐理解他们先前仅靠模仿和记忆获得的知识。这也是大脑的神奇之处，或者说人的可塑性。但这个学习过程绝对像填鸭子，喉咙里会有被强塞食物所产生的不快，甚至痛苦的感觉。学生被这样硬拽进这高大上的“数学”里面，肯定不会是一个愉快的学习经历。这也就是很多人不喜欢数学的原因。

***

美国小学低年级，大部分学校是一个班配一个主科老师，教所有主要科目。其他如美术、体育之类由专科老师教。小学高年级阶段，通常是一个老师负责语文和社会科学，另一个老师负责数学和科学。初中阶段基本就各科分开，各由一个老师负责教。美国的教师资格执照分为幼儿园到六年级的小学教师证，以及七到十二年级的中学教师证两种。能够在初中专门教数学的，已经算是数学基础不错的老师了。这些老师常常会问我一些他们碰到的问题，通过这些问题，我意识到他们概念性问题的严重性。

在教学过程中遇到一些不好理解的应用题，老师们互相讨论，这是正常的。他们来问我，我通常不只告诉他们怎么解这道题，还会讲解一些背后的原理，希望他们能够理解透彻，更容易教会学生。这让他们看到，理解原理确实比仅仅找到答案要清楚得多、深刻得多。于是，有些人就会来问我一些他们都会做、但不甚明白其中原理的问题。比如：为什么负数加负数得负数，而负数乘负数却得正数？分数相乘不用通分，为什么分数加减却要通分？这使我看到了为什么那么多人不喜欢数学的原因：他们并不真正理解数学。

因为知其然不知其所以然，所以不管做得多熟练，他们所经历的只是一个事务性的操作过程，而不是一个思辨推理的过程。这样，学习过程就只是一个无意义的重复记忆过程。用它来解题，也就只是一件重复技术动作的劳作，而没有运用大脑的智慧去探索、理解、认知、解决问题的乐趣。就像一个人被蒙了眼睛去登山，虽然你记得很熟，或者有声音提示你什么时候上坎、什么时候爬坡，最后也能走到山顶。但是整个过程里，你看不到任何沿途的景象，登山就只是一个辛苦而乏味的劳作。

没有了乐趣，你可能很努力、很自律，也能学得不错；但你只是一个数学的劳动者，而不是一个数学的“玩家”。人都好逸恶劳，你不是在玩，而是在劳动，就随时有可能感到累了想歇歇。在某种意义上，中国学生数学好，是一种“优秀劳动者”的成就。由于家长、老师和社会的压力，由于他们的懂事、听话、自律和勤奋，促成了他们的努力。也正因为他们是靠勤奋劳作取得成绩，而不是出于热爱，所以他们最终取得真正成就的可能性就不会很高。

我遵循的原则更接近苏格拉底那句话：“我不能教给人们什么，我只能让他们思考”（“I cannot teach anything. I can only make them think.”）。

有了 AI 以后，数学计算你永远打不过机器；数学真正的功能是让学生开始思考，而且把思考推向更高层次。人类一出生其实“什么都不会”（born an idiot），但学习的潜力是任何动物都无法接近的。AI 时代更是如此：我们教数学的重点，不是训练学生算得比机器快，而是训练他们会思维，并且把思维推到最抽象、最严密、最可靠的层次。

数学不是没用了，而是更有用了：它帮助人的思考变得更精确、更清晰、更简化，并且能推得更深。很多人说数学太难、太抽象，我反而觉得：数学之所以能发展到那么抽象，恰恰是因为它“太容易上手”——所以才可以一路推进到很深很远的地方。打个比方：有人问小提琴好学还是钢琴好学？要弹《Happy Birthday》，肯定是钢琴，只需要一个指头就能弹出来。正因为钢琴容易上手，人们才会不断推进到更难、更复杂，直到能力极限。同理，数学也是：因为它能用最简洁的结构表达最复杂的关系，所以它才能一路走得极深，最后让外行看不懂。

我认为数学训练人的思维，至少有三个方向：

1. Deeper：能想得更深；

2. More logical / more accurate：逻辑更严密、更准确；

3. Simplest way：能用最简洁的方法抓住本质。

正因为“最简”，才能处理更复杂的问题；否则思维很快就到极限。数学思维偏微观，数学思想是宏观的体系与结构。回到课堂，内容与标准方法只是工具，不是目的。因此在课堂上，老师真正要重视的是两件事：

第一，老师必须真正懂数学——不是表面会做题，而是懂背后的逻辑结构（the logic behind it）。 第二，课堂内容与所谓“标准解法”只是用来训练学生思考、训练数学思维的工具，而不是教学的终点。

现在很多课堂把“会解题”和“分数提高”当作目的；但它其实应该是路径，是用来展示数学如何分析关系、如何追求更清晰、更简洁、更准确的工具。

***

教师需要正确做出判断，让小孩子用他们已有的认知，去做具有挑战性、但能够在相互讨论和协作中完成的学习任务。这是成为好教师最重要的能力之一。这个能力其实并不难培养。只是我们现行的授课方式，是让老师根据教学大纲和教材去填塞知识给孩子，而不是根据孩子的程度来引导他们“玩”。久而久之，教师就失去了这种能力。我们的老师习惯于看着教材和教学内容教，而不是看着孩子的现有水平、他们感兴趣的方式，以及他们在学习过程中的感受去教。

其实，每个人在与别人讲话时，自然而然都会揣摩听众可能的接受方式，来调整交流的词语和语气。这就是教师的自由支配度（teacher autonomy）。每次由政府组织、专家教授设计出来的教学大纲，用来指导老师教什么、怎么教，实际上是在削弱，甚至剥夺老师的自由支配度。这一点是每次政府努力制定的新教学法，最后导致教学质量没有提高、反而下降的一大原因。也可以说，这些患着“上帝病”的官员和专家，不但把学生伤害了，还把老师也毁了。只要提高教师的课堂自主权，再加上一点必要的培训，所有老师都应该能重新找回这种能力。

现代流行的教育学常常声称要把传统的“教师为中心”改成“学生为中心”。但实际上，他们并没有改成学生为中心，而是改成了教学大纲为中心、专家制定的教学内容和课程进度为中心。概念性的理解，需要教师了解学生现有的知识结构，揣摩学生能够接受的方法，并放手让学生自己探索才能产生。

我们老师每天面对的学生，是一个个鲜活的个体。这个学区和那个学区的学生不一样，这个班级和那个班级的学生不一样，这个学生和那个学生不一样。同一个班的学生，今天和昨天都可能不一样。失去了自主权，让形形色色的学区和背景极不相同的学生，都由那群专家教授来统一决定内容、进度、甚至教学方法，这样只会把教师训练成只会照本宣科的机器人，也很难让学生产生概念性的理解。

上面的讨论可能让读者产生两种感觉： 一种是老师要这样教，必须学习一种全新的方法； 另一种是：大部分老师估计很难学会。第一点是对的，这确实是一种全新的方法。第二点则是大部分现有教师完全不必担心的：你一定学得会，而且会学得很好。

教师需要学的是：放手让孩子去做，在旁边只做一个向导。就像体育老师一样，在运动场上让孩子自己去玩，你在旁边稍加指导，像个教练；或者像个裁判，有人犯规时出来吹一声哨子。其它时候都是他们自己在拼搏。我常常做的，是和他们一起玩。当然，你得克制住“你比他们懂、比他们能”的这个优势，反过来成为一个最需要他们帮助的人。把当老师、教别人、甚至教训别人的乐趣，让给那个最弱小的学生。

教师所需要的，就是做一个能和他人正常交流的人，具备一般人都会有的共情心。在谈话时，能够揣摩对方的心理活动。这些几乎没有超出所有教师都必须学习的教育学和心理学范畴。当然，要让学生有概念性的理解，教师自己也必须有。美国的中小学教师，经过几十年各种“新教育”的培养，很多人自己已经不具备这种能力了。由于中小学孩子的认知发展水平有限，抽象数学的概念性理解必须让他们慢慢成长和养成；但已是成年人的教师，要厘清、掌握这些基本概念，其实并不难。

教案当然应该写，但写得太多反而是个问题：越写越细，越容易把最重要的思想淹没。最重要的是，教师在走进课堂之前必须心中有数。从宏观上，有三个方面：

第一是教学目标，即你要让学生在这里得到什么。我的目标不是让学生考多高的分数。我的学生考试成绩好，只是我教学效果的副作用。现行学校的要求，也是绝大部分老师的教学目标，是让学生考试成绩优秀，或者过关、达标。

第二是教学法，即你已经练就的，或者正在操练的，最能达到你教学目的的方法。

第三是你对实现教学目标所需材料的透彻了解；对中小学数学而言，就是四种数字的四种运算相互关联的整体结构。

从微观上，最主要的是了解你面对的学生，其次是了解家长、学区要求、具体使用的教材，以及与其他老师和领导的配合等细节。了解你的学生，第一是了解他们心里已经形成的对数学的误解；第二是了解他们多年养成的学习数学的错误习惯；第三是了解他们现在脑子里知道多少有用的部分。

我的教案，首先是把握上面说的几个部分，每个部分列出几个具体项目。第一天跟学生见面就开始讲，每天下课后，把当天的结果记下来，包括整体性的、具体细节的、班级的、个体的，然后接着把这个故事讲下去，或者说，把这一天接着聊下去。

绝大部分人天生都具有与人聊天的能力。深入一点讲，就是能根据不同的人，调整聊天的方式、语气和内容。严格来说，不能和人聊天的人，就是做不了教师的人——这种人应该很少见。能否成为好教师的特质，是在派对、酒吧、飞机上遇到邻座，都能聊两句的能力。我们现在的教师，被培训、被要求的，几乎就是要丢掉这种天生具备的能力，照专家的理论、大纲的要求、教材的设定，去写出详细的教案，然后到课堂上照本宣科。

教师要具备的就是两个方面： 一个是情感方面，让孩子觉得跟你聊天很有趣。既是你这个人很有趣，他喜欢与你继续聊；也是你聊的内容很有趣，让他增长知识、提升能力。喜欢你这个人，和喜欢你聊的内容，也可以说是同一件事的两个方面。

另一个是教师专业知识方面的“硬件”。一是你对要传授的数学基础知识结构的通透理解；二是对大脑认知发展规律的了解，也就是教育学、心理学、大脑科学的基础知识。这是教师最根本的底气，也就是日常所说的“牛气”。这个“牛气”会增强教师在学生面前的气场，让学生更服你。

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The grand title, “How Learning Happens”, is not based on classic learning theories, but more of an empirical summary from I observed and experienced as a teacher and student.

It resonates with Mr. Xu’s approach in some respects yet differs in other. My book is primarily focused on stories and experiences, which differs from the book about Mr. Xu. Enjoy the reading!)

CHAPTER 28 HOW LEARNING HAPPENS

In my view, learning consists of knowledge, methods, and frameworks.

Take geometry, for example. Knowledge consists of definitions such as plane, parallel lines, and circles, and it includes notations and theorems. Methods include logical deduction, constructing additional lines, or multiple representations of the same object. The framework of building a system based on axioms (assumptions that seem obviously true) and logical proofs has been so powerful that Euclid’s book The Elements (of Geometry) has become one of the most influential books ever written, instrumental in the development of logic and modern science.

Learning can happen at different levels: conscious level, mental muscle level, and intuition level. Mastering a subject requires engagement at all three levels.

Conscious learning involves understanding the learning context, getting basic terminology, and memorizing important facts. Math is the gymnastics of the mind; its mastery is developed through sufficient practice to build muscle memory—similar to swimming or riding a bicycle—that, once learned, is not easily forgotten. When learning is mastered and integrated into a person’s existing system of knowledge, it becomes intuitions that can be retrieved quickly and accurately without a full thinking process. Mathematical intuition is important for generating new ideas and must be obtained through rigorous logic rather than memorization.

Effective learning takes thinking, practicing, and connecting.

In U.S. classrooms, students are encouraged to ask questions and explore knowledge but often not required to do sufficient practice. While I sometimes noticed creative ideas from students and encouraged them, most lacked depth because the lack of practice prevented them from thinking deeply. For most students in the U.S., more rigorous and well-designed practice is needed to develop proficiency; otherwise, new ideas risk being built on unstable foundations.

In China, on the contrary, asking questions is seldom encouraged, and exploration opportunities are not given in typical math classes. With mounting pressures to achieve high test scores, students are not often allowed the joy of discovering math, along with opportunities to develop originality, creativity, and intellectual risk-taking. They are required to practice a lot to master the specific methods that were taught to achieve mastery of those narrow skills.

Thinking and practicing are both essential. Practicing without thinking turns learning into a boring and mindless burden, while thinking without practicing can make learning empty and futile, like trying to learn tennis without wielding a racket. In the long term, preserving curiosity and creativity may be preferable to stifling these qualities with excessive, mind-numbing practice. The Chinese approach to teaching, prevalent in many East Asian countries, often produces students with stronger foundational skills, but fewer breakthrough ideas and more stressed mindsets.

When I read about the “Latticework of Mental Models” from renowned investor Charlie Munger, it immediately struck a chord. According to Munger, a comprehensive understanding of the world requires a widely connected mental model that includes knowledge from diverse disciplines such as psychology, economics, or history. The latticework metaphor implies an interconnected network of insights, each adding to the full understanding of complex issues.

The key to learning is integrating new insights and skills into the latticework of existing mental models in a robust, permanent, and relevant way. For that to happen, knowledge must be fully absorbed at a level that makes intuitive sense without laborious thinking. I can recall the math I learned in high school with confidence and wouldn’t forget those before I forgot my home address, because they were firmly integrated into my latticework of knowledge.

For teaching, providing multiple perspectives and connecting to students’ life situations, existing knowledge, or issues they personally care about, can be especially effective.

Figure: Three different ways to prove the centroid of a triangle

When teaching about the weight center of a triangle (called centroid), I demonstrated multiple approaches. I first showed the geometric way of proof, then mapped the issue to a problem on the coordinate plane that can be solved by algebra, exercising the mathematical brain on different branches of math. Additionally, I told the students that in physics, the centroid would be the unique point where the weight of the triangle can stay balanced. Sensing the students’ curiosity, I used math to explain the physics behind it.

I prepared and handed out triangles of different sizes and shapes, cut from thick cupboard paper, for each student to find the point where the triangle could stay flat on top without tipping. The students tried the act of balancing the triangles on all kinds of pointy objects (pins, pens, compass legs, fingertips, to name a few), engaging their sense of touch, moving around the classroom, and exchanging ideas with each other.

After finding the point of balance, the students drew the lines and identified the centroid according to the geometric definition. They verified that it was the same point they had just discovered. Although this exercise took more time, the students enjoyed the process, integrated knowledge from various sources, and retained the lesson well throughout the semester. It became one of the topics they continued to discuss even after the semester had ended.

第28章：学习是如何发生的（中文翻译）

在我看来，学习由三部分构成：知识、方法和框架。

以几何为例，知识包括平面、平行线、圆等定义，也包括符号体系和定理。方法包括逻辑推理、作辅助线、或用多种方式表示同一个对象。以公理（看似显然为真的假设）和逻辑证明为基础构建体系的框架极其强大，以至于欧几里得的《几何原本》成为史上最具影响力的书籍之一，对逻辑和现代科学的发展起到了关键作用。

学习可以发生在不同层次：意识层、心理肌肉层、直觉层。要真正掌握一门学科，必须在这三个层次都投入。

意识层学习包括理解学习背景、掌握基本术语、记住重要事实。数学是心智的体操；它的掌握依赖足够的练习来建立“肌肉记忆”——就像游泳或骑自行车一样，一旦学会就不容易忘记。当学习被完全吸收并融入一个人已有的知识体系后，它就变成了可以快速、准确提取的直觉，不需要完整的思考过程。数学直觉对于产生新想法至关重要，而它必须通过严密的逻辑获得，而不是靠死记硬背。

有效的学习需要思考、练习和连接。

在美国课堂中，学生被鼓励提问和探索知识，但往往不被要求做足够的练习。虽然我有时会看到学生提出有创意的想法并给予鼓励，但大多数想法缺乏深度，因为缺乏练习使他们无法深入思考。对大多数美国学生来说，需要更严格、设计更精良的练习来发展熟练度；否则，新想法可能建立在不稳定的基础上。

在中国，情况恰恰相反。提问很少被鼓励，探索机会在典型的数学课堂中几乎不存在。随着取得高分的压力不断增加，学生往往无法体验发现数学的乐趣，也缺乏发展原创性、创造力和智识冒险精神的机会。他们被要求大量练习，以掌握课堂上教授的具体方法，从而在这些狭窄的技能上达到熟练。

思考和练习都是必不可少的。只有练习而不思考，会让学习变成枯燥无味的负担；只有思考而不练习，会让学习变得空洞无力，就像不拿球拍却想学网球。从长远来看，保持好奇心和创造力可能比用过量、麻木的练习压制这些品质更为重要。中国式教学（在许多东亚国家普遍存在）往往培养出基础更扎实的学生，但突破性想法较少，心理压力更大。

当我读到著名投资人查理·芒格提出的“思维模型的格栅结构（Latticework of Mental Models）”时，我立刻产生了共鸣。芒格认为，要全面理解世界，需要一个广泛连接的思维模型网络，其中包含心理学、经济学、历史等不同学科的知识。格栅结构的隐喻意味着一个相互连接的洞见网络，每个洞见都为理解复杂问题增添一块拼图。

学习的关键在于：将新的洞见和技能以稳固、持久、相关的方式整合进已有的思维模型格栅中。要做到这一点，知识必须被充分吸收，达到无需费力思考就能直觉理解的程度。我仍能自信地回忆起高中学过的数学内容，甚至比记住家庭住址更牢固，因为它们已经牢牢融入了我的知识格栅。

在教学中，提供多种视角，并与学生的生活情境、已有知识或他们关心的问题建立联系，会特别有效。

图示：证明三角形重心的三种方法

在教授三角形的重心（centroid）时，我展示了多种方法。我先展示几何证明，再将问题映射到坐标平面，用代数方法求解，让学生在数学的不同分支之间切换。此外，我告诉学生，在物理中，重心是三角形能够保持平衡的唯一点。看到学生的好奇心，我进一步用数学解释背后的物理原理。

我准备了不同大小和形状的三角形，用厚纸板剪好，发给每位学生，让他们找出三角形能平稳放置而不倾倒的点。学生们尝试用各种尖锐物体（图钉、笔尖、圆规脚、手指尖等）来平衡三角形，调动触觉，在教室里走动，并互相交流想法。

找到平衡点后，学生画出中线，并根据几何定义标出重心。他们验证了这个点与他们刚刚通过实验找到的平衡点一致。虽然这个活动花费了更多时间，但学生们享受这个过程，整合了来自不同来源的知识，并在整个学期中牢牢记住了这节课。甚至在学期结束后，他们仍继续讨论这个主题。

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(This is Chapter 6 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, in the post-pandemic era. I want to explore what we can learn from a unique and successful teacher, especially on how to teach math effectively.

I may not fully understand or fully agree with all he says, but I found it a great learning journey to show the world his thoughts.

中文版在本文后面部分，英文版之后)

(sunset photo taken earlier this month)

Chapter 6 Playing with Math

Math teachers must remember this: they need to let students play with math.

In PE class, students play basketball or soccer to train and strengthen their bodies; playing math trains and strengthens the mind. So math class is actually Brain Education (BE). Just like in PE, the most important things in math class are participation, progress, and improvement—not merely whether answers are right or wrong.

The key is to let students participate, to think, to guess, and to try on their own. Whether something is right or wrong, clever or clumsy, good or bad should be judged by the students themselves. The teacher simply observes, asks guiding questions, pretends not to know, and always stands on the side of the students who haven’t figured things out yet, waiting to be convinced or taught by their classmates.

Taught this way, what students gain first is enjoyment, and only then learning. Nothing is more enjoyable than discovering something new or mastering a new method through one’s own effort—especially methods that are elegant, clever, efficient, and capable of solving problems. Over time, their willingness to participate actively in learning will increase. This is the biggest problem in American K–12 math education: students are not too stupid; they simply do not want to engage in real learning.

Curiosity is one of the three fundamental human desires, and children’s curiosity is universally strong. It is hard to find a child who does not like learning new things. If a child is not motivated to learn, the problem must lie in the content we give them or the way we give it. To solve this, we must first abandon the practice of following expert‑written curricula and textbooks word for word. These materials can only provide a broad vision of the professional level students are expected to reach eventually.

Experts are often too self‑assured and frequently invent methods that lack common sense—for example, teaching language by starting with the alphabet. In reality, letters come after learning many words and sentences, after comparing them and noticing which ones sound similar, identical, or different. Only then do people, through collective effort, distill hundreds or thousands of seemingly chaotic words into a small set of phonemes and then into a few letters. This process is actually very interesting and fun. The key is to avoid preaching, lecturing, or spoon‑feeding. Instead, teachers should watch with appreciation (while subtly guiding) as students discover, organize, and construct meaningful rules or systems.

There is far more to “play” with in math than in language. Elementary math is about comparing the differences and relationships among the messy things in the real world, then quantifying, refining, and simplifying those relationships, and finally combining them into structures and systems. Mathematical enlightenment should begin with understanding the quantities of things around us. Just like learning words and sentences in a language, only after enough experience with quantities can students, through play, distinguish and compare patterns and eventually “play out” the concept of sequences. Sequences introduce the idea of using a very limited set of digits to record infinitely many quantities—an elegant, clever, and efficient invention known as place value.

The focus of learning the four operations should be on the reasoning behind each operation and the logical relationships among addition, subtraction, multiplication, and division. Here is a concrete example: once students can all do 78 + 22 and know the answer is 100, you can play a quick‑response game to see who answers fastest. This is also part of “playing” math. With 78 + 22 = 100 written on the board, the teacher can ask: “If the first addend, 78, increases by 5, what will the final result be?”

Some students will compute 78 + 5 = 83, then add 22 to get the answer. Some will do it mentally, some on paper, and it may take seconds or tens of seconds. Others may answer immediately. Those doing mental or written calculations might make mistakes, but the ones who answer instantly are almost never wrong. If the addend increases by five, the answer should naturally increase by five. There is almost no room for error.

In this game, students write their answers on their own paper and raise their hands, saying “I got it!” instead of shouting out the answer. Once everyone is done, the teacher checks their work and ideally picks one or two incorrect answers to write on the board or say aloud. Then the whole class votes: thumbs up if they agree, thumbs down if they disagree, thumbs sideways if they’re unsure. This ensures everyone participates—everyone thinks, judges, and takes a stance.

Whether the chosen answer is right or wrong, the teacher can ask a student who disagrees to explain their reasoning and give their own answer. The class then discusses who is correct. During this process, the teacher must remain neutral, even side with the wrong answer, standing with the students who are still confused and waiting to be convinced or taught.

After the class reaches a consensus, they can discuss the different methods used. My standard for evaluating methods is: having any method is good; having more than one method is better; having a simpler, faster method that improves efficiency or reduces errors is best. Finally, we discuss why some students can answer instantly. I often joke with them: “Why does one person earn $10 an hour while another earns $100? Because one person can finish in one hour what takes someone else ten hours.” Learning math is about finding patterns so you can solve problems more quickly and accurately.

Once everyone understands the quick method, you can increase the complexity: the first addend increases by 5 while the second decreases by 5; or the first increases by 6 while the second decreases by 4, and so on. When to write answers, when to shout them out, when to increase difficulty, and by how much—all should depend on whether every student can participate and find it fun.

The same approach is even more meaningful in subtraction: when the minuend decreases, the result increases; when the subtrahend increases, the result decreases. This helps students later when dealing with expressions where the minuend is in parentheses or is negative. Of course, these relationships should not be told directly; students must discover them through enough examples, and then the teacher asks why.

After playing with addition and subtraction, you can explore their relationships. Using the same example (78 + 22 = 100), you can ask: What is 100 − 22? 100 − 78? Then 100 − 79? 100 − 77? Multiplication and division can be explored similarly, showing the relationships among divisor, dividend, and quotient, and the connections between division and subtraction, and between division and multiplication.

During teaching, several points must be remembered: First, students must truly understand the logical development of each operation and the relationships among them. Second, the entire understanding must come from students’ own discovery and summarization, not from being told. Third, “they” refers to every student in the class. Some learn fast, some slow, some early, some later, but no one should be merely told, memorizing mechanically, or left confused. Fourth, the entire process is “play”, a joyful, game‑like state. The joy comes from thinking: some enjoy making progress, some enjoy teaching others, some enjoy showing their ability. The goal is for every student to feel good—whether they enjoy the process, the math, their classmates, the class, the teacher, or themselves. This is what teachers should strive for.

These activities get students thinking, searching for clues, and building models. They feel engaged and accomplished. The teacher guides them in this direction but does not interfere or take over, even if they fail to find the answer. Instead, the teacher saves it for another time. It’s like PE: a coach wouldn’t take the ball from a student who keeps missing shots and score for them, then count it as the student’s point, right?

Math class encourages students to think, explore, and enjoy the satisfaction of success or progress. This not only makes learning enjoyable but also builds the belief “I can learn math.” It gradually develops the habit of approaching problems proactively, trying to understand, analyze, and explore. Unfortunately, most classrooms train students to be passive receivers. That’s why most students love PE but many dislike math.

How deeply students can “play,” and how long they want to play, depends primarily on their interest, and secondarily on their cognitive development and existing math foundation. The teacher’s role is to observe, guide, and support, while judging the situation: How many students are interested? Can everyone join and keep up? As long as they can keep up, even with lots of help, they will still feel accomplishment and joy. If the teacher senses things are going too far or some students can’t keep up, they can simply say: “This is a great question and we’ll learn it later.”

Some teachers may ask: Is it necessary to make a simple operation so complicated or spend so much time on it? When I taught in China, I often heard the phrase “teach concisely, practice extensively.” Most American textbooks are similar: each chapter begins with two or three examples followed by dozens of exercises. Those examples seem clear, but they mainly teach procedures, not deep understanding or playful exploration. The “play” in “play with math” is exactly what deepens understanding. Through layer‑by‑layer exploration and playful thinking, students’ understanding becomes deeper and more solid.

Throughout the process, students are constantly exploring, searching, and understanding new things. This is far more interesting—and far more effective—than spending large amounts of time doing repetitive exercises without true comprehension.

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To let students truly play with math, they must understand what they are doing and feel interested in it. That means using the real world they already know, and connecting math as much as possible to the other subjects they are learning. For example, you can use money as an example. Some students already know how to use money; others are interested in money, want to know more about it, or want to get some. Even actually handing out a few coins in class can greatly increase interest, aid understanding, and strengthen memory.

Another example: in geometry, the concept of a point is the most basic form—or rather, it has no form at all. A point is zero‑dimensional, a line is one‑dimensional, a plane is two‑dimensional, and a solid is three‑dimensional. So introducing the point first seems natural. But in students’ real‑world experience, the “point” in their everyday language is not the same as the “point” in geometry. Therefore, the geometric point must be rebuilt from scratch. And the way to build it should start from something they already know: the “plane.” We can begin by saying that a table has a tabletop, a classroom has a floor and then ask students to find other examples of “planes” they know.

The first rule of playing is that everyone must participate. In the “plane” activity, each student must think and identify at least one plane in the things they know. If no one mentions a non‑flat plane, the teacher can ask, “We all have a face. Does a face count as a plane?” Then let the students discuss. As long as their reasoning makes sense, all viewpoints are encouraged. The final conclusion is that both flat and non‑flat surfaces are planes. But we start with flat planes, and at this point we can introduce the concept of “Euclidean geometry.” We can also mention that non‑Euclidean geometry can be learned later.

Once the idea of a plane is clear, we can have each student place a sheet of paper on their desk and make a dot on it with a pen. Then ask: “What is this?” Many students will likely answer, “It’s a point.” Then ask, “Is this sheet of paper a point?” They will probably say, “Of course not.” Then continue: “Why not? Because it’s big? Because it’s a plane? Or for some other reason?”

We can explain that the dot has a diameter of about one millimeter—very small—so it is a point, not a plane. Then you can joke: a soccer field is bigger than a sheet of paper, so it should be a plane, not a point, right? But for a tiny insect or a bacterium, that one‑millimeter dot is so large that it could play soccer on it. So from its perspective, that dot is a plane—a very large plane.

The class can then have a lively discussion about what counts as a plane and what counts as a point. Someone might mention that in the photograph taken by a space probe as it was about to leave the solar system, Earth appears as a tiny, almost invisible dot. This leads naturally to the idea of the absoluteness of mathematics: it does not change with environment or perspective. This is one of the reasons people study mathematics.

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When I walk into an unfamiliar classroom as a substitute teacher, I look up at the ceiling, as if thinking deeply, and start by saying, “87.” Then I slowly say, “84,” still looking thoughtful, as if I’m reasoning something out, and then say, “81.” After thinking for a moment, I say, “78.” At this point, some students start watching me, wondering what I’m doing. I continue counting to myself, and students begin to get curious. Some may notice that I’m subtracting 3 each time, and one of them might guess, “Is it 75?”

Humans have innate curiosity. When a student follows that curiosity and discovers the pattern, he becomes excited and blurts out the next number: “72!” Other curious students who haven’t yet figured out the pattern will wonder, “How does he know the number the teacher hasn’t said yet?”

At that moment, I show some excitement and praise, but I don’t allow him to say the next number. My praise encourages him, but if he continues answering, the rest of the class becomes spectators or passive recipients, which are boring and unengaging roles. So I ask the students who have figured it out to raise their hands, giving others the chance to discover it on their own. Discovering something by oneself is enjoyable; being told the answer not only removes that joy but also forces a student to accept the discouraging feeling that “someone else is smarter than me.”

In this activity, I must estimate the students’ mathematical ability. Based on that, I design the difficulty so that someone can figure out the answer before their brief curiosity fades. If it’s too easy, students lose interest. For example, if I count 1, 2, 3, 4, 5, 6, 7 in front of middle schoolers, unless a student is extremely obedient or eager to please the teacher, most won’t care. If it’s too hard, like using the earlier example with first graders, their curiosity will drop to zero before they understand what I’m doing. They’ll simply conclude that I’m talking to myself, mumbling a string of meaningless numbers.

If I misjudge their level and make it too difficult, I can always start over and write the first number on the board. Math doesn’t have the vivid and instant magical appeal of an animated movie, so I must spark curiosity quickly enough for students to make a discovery. Many teachers, especially math teachers, often accuse students of “not thinking,” but in reality, it’s very hard to stop a student from thinking. If a teacher finds students not thinking, it means the classroom activity wasn’t designed well. Either the problem feels boring or meaningless, or it’s too difficult, and the student knows they won’t understand it. Since any effort will only lead to failure, frustration, or embarrassment, they choose to give up.

To get more students involved and help them gain confidence and enjoyment, I cannot let the first student who figures it out tell the others. In most math classrooms I’ve seen, that’s exactly what happens. On the surface, it seems to achieve the teaching goal: everyone “understands,” time is saved, and efficiency is improved. But if teachers keep in mind that the purpose of math class is not merely to teach skills, but to train the mind, bring enjoyment, and build confidence, then this approach becomes naturally justified.

In my classroom, every student must have paper on their desk and a pencil in hand, writing their answer on their own paper. Students who understand may indicate that they understand, but they cannot say the answer aloud. This gives those who haven’t figured it out time to keep trying and the chance to discover it themselves. If a student who thinks they understand actually gets it wrong, that becomes the best learning opportunity for the whole class—a true “teachable moment.” I treat the wrong answer as if it were correct and have the entire class vote. I often ask everyone to raise a hand: a fist or one finger means “agree,” two fingers means “disagree,” and three fingers means “not sure.” Students love voting this way.

Next, students begin discussing or explaining why an answer is right or wrong, and the final conclusion is entirely theirs to decide. During this process, I almost never directly answer their questions. I use many “psychological tricks.” For example, when a student is clearly wrong, I might deliberately say, “Very good, you’re right,” because I need that one student who refuses to accept it, someone who will jump up to challenge and question it. Only then does real thinking begin.

If the whole class arrives at the wrong answer, I still don’t deny it. Instead, I design a new problem for them to discuss. The conclusion of the new problem will conflict with the previous one, leading to an even livelier discussion. Throughout the process, everyone must participate. I try to ensure that each student discovers the answer step by step on their own, and that each one develops the ability to persuade others. I often side with the wrong answer until the very last student who finally understands can explain to me why their answer is correct and why mine is wrong. They must convince me; only then does the discussion end.

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Sometimes, when students get deeply engaged in “playing” with mathematics, they will even take the initiative to ask about topics that aren’t in the textbook. I have a class rule: whatever students want to learn—even if it’s not already in the textbook—they can bring it up. The first thing I write on the board or projector every day is: What question do you have? I tell the students: you can ask about anything, even yesterday’s news, as long as you’re interested. What we do here is think; anything that gets you thinking is welcome. If the first questions aren’t about math, after a few minutes I’ll try to bring mathematics into the discussion and guide them back to the main topic.

For example, if a student asks, “Can we learn trigonometry?” I have them do two things:

1. Go look up: What prerequisites does trigonometry require?

2. Come back and discuss: What exactly is it? What do we need to know before we can learn it?

If more than five students are interested, we cover it in class. If fewer than five, they can form a club on their own, outside of class time. My principle is: class time can be used to “open their eyes,” to let more students see that mathematics can be learned and thought about in many different ways.

One year, something interesting happened: a few students felt that the regular high‑school placement track wasn’t challenging enough, so they wanted to take the SAT. They actually went and took it. Some scored over 600; for seventh graders, that’s already higher than the average score of many local high‑school graduates.

The Phoenix school district is a large urban district with nearly five million people, mostly Mexican immigrants. IB is the international college‑prep program. Most good high schools have an IB class, but at that time the entire Phoenix area had only one IB class that taught calculus. Every year, that class struggled to recruit enough students because most students simply couldn’t handle the material. So one year, the district asked each school to have its math teachers recommend the students most likely to succeed in calculus for a summer training program, about thirty students in total.

The instructor was a Chinese teacher, a physics PhD who truly understood mathematics. He spent about a month and a half during the summer giving these students intensive training, hoping they would have the confidence to register for calculus in the fall. Gradually, he noticed something strange: among the thirty‑plus students, most were confused and struggling, but six stood out—they understood everything he taught quickly and deeply. He later discovered that these six students all knew each other, were from the same school, and were all my students.

When he came to me in surprise and asked why, I told him that this is the characteristic of my students: they didn’t get there by drilling endless exercises. They became students who can think mathematically because they learned through playing with math.

NEXT CHAPTER

第六章 玩数学

数学教师首先要记住的是，要和学生一起 “玩” 数学，让他们参与到真正的数学学习中去。 学生在体育课（Physical Education, PE）上打篮球、踢足球，是在锻炼和增强身体功能；而玩数学是锻炼和增强脑力，所以数学课其实是 脑力锻炼课（Brain Education, BE）。像体育课一样，数学课上最主要的是参与、进步和提高，而不仅仅是结果的对错。关键是要让学生参与进来，自己去想、去猜、去尝试。对错、巧拙、好坏由学生自己分析评判，教师只是旁观，进行诱导性提问，假装不知，并永远站在还没搞懂的那些学生一边，等待被其他同学说服或教会。

这样上数学课，学生得到的首先是愉快，其次才是学到了什么：没有比通过自己的努力学到新东西、掌握新方法更愉快的事，特别是那些聪明、美妙、高效、能解决问题的方法。长久下来，他们愿意主动参与学习的积极性会得到提高。这正是美国中小学数学教育的最大问题：学生不是太笨，而是不愿意参与真正的学习。

求知欲是人的三大欲望之一，尤其小孩子的求知欲是毫无例外的旺盛。我们很难找到一个不爱学习新东西的小孩子。如果小孩子学习不积极，那一定是我们给他的内容或者方式不对路。要解决这个问题，我们首先得改掉照着专家编写的大纲和教材、在课堂上照本宣科的做法。教育专家们编写的大纲和教材，只能是一个宏观的远景，是希望学生最终达到的专业水平。

专家们大多太过自以为是，经常发明一些缺乏常识的方法，比如从学习字母表开始学语言。字母是在学了很多单词和句子之后，通过比较和发现哪些词发音相似、相同或相差，然后将成百上千看似杂乱无章的词汇，通过大家群策群力，整理出不多的几个音素，再书写成那么几个字母。这本来是一件很有意思、很好玩的事。要点是避免用宣讲、布道、灌输的方式，不要由教师告诉他们，而是要充满赞赏地看着（实际上是诱导）他们去发现、组织、构造出很有意思的规则或系统。

数学里可以玩的东西，比语言要多得多。小学生的数学，就是把杂乱无章的现实世界中各种事物的差别和关系比较出来，然后把这些比较出来的关系量化、精确化、简化，再把它们组合起来，形成结构和系统。数学启蒙应是认识身边各种事物的量的概念。正像学语言的单词或句子，只有到了一定经验积累之后，才会在玩中区分和比较，然后 “玩” 出数列这个概念。数列带来了如何用非常有限的数字去记录无穷无尽的数量，也就是 “进位” 这个精简、美妙、聪明的设计。

***

学习四则运算的重点，应该放在每个运算中的推理过程，以及加、减、乘、除四种运算之间的逻辑关联上。这里可以举个具体的例子：78+22=? 等学生都会了，都做完并且知道答案是100以后，可以跟他们玩一个抢答的游戏，看谁答得快，这也是 “玩” 数学的一部分。看着黑板上写着的 78+22=100，这时教师可以问：“如果前面那个被加数 78 增加了 5，那么最后的结果是多少？”

有的学生会开始做 78+5=83，再将 83 加上 22，然后凑成正确的答案。有的心算，有的笔算，可能要几秒到几十秒才能完成，而有的学生可能立刻就能回答出来。用心算或笔算的同学有可能会算错，但不用一秒钟就能抢答出来的，一般不会错。被加数增加了五，答案自然也应该增加五。这里几乎没有犯错的空间。

这里所谓的抢答，是让学生在自己的作业纸上写下来，然后举手，说 “I got it!”（我得到了答案），而不是马上说出答案。当所有人都做出来以后，教师来巡视学生的答案，最好挑一两个没做对的，让他把答案写在黑板上，或者口头报出来，然后让全班同学表态：同意他的举手（或拇指往上），不同意的举手（或拇指往下），不知道对不对的举手（或拇指平放）。这样就让每一个人都参与进来，没有例外；也就是说每个人都动了脑筋，做了判断，表了态。

不管老师挑的答案是对的还是错的，都可以请一个不同意的同学说出他的理由，并给出自己的答案，然后全班同学讨论到底谁对。在这个过程中，老师一定要保持中立，或者站在错误的一边，和其他同样做错的同学一道，等着被说服或者被教会。最后答案统一以后，再来讨论各人使用的不同方法。关于不同方法的评判，我的标准是：有方法就是好的，得满分；有不止一种方法更好；有更简便的方法能提高效率或减少错误概率最好。最后大家一起讨论为什么有的人这么快，不用一秒就能得出答案。我常跟学生开玩笑说，为什么有人工作一小时能挣 $10，而有的人一小时能挣 $100？就是因为有人一小时完成了别人要十小时才能完成的工作。学数学就是要让你找到规律，能够更快捷而又精确地解决问题。

等大家都明白了这个快速简便的方法之后，就可以真的抢答，并且增加复杂度：比如被加数增加 5，加数减少 5；或者被加数增加 6，加数减少 4 之类。何时写出答案、等待其他同学完成以后再讨论，何时直接喊出答案，何时要增加难度，以及难度增加多少，都要根据教室里每个同学都能参与、都能感到好玩为标准。

同样的方式在做减法时更有意义：被减数减少，答案会增加；但减数增加，答案会减少。这对以后做被减数是带括号的表达式运算，或者是一个负数，都会带来理解上的帮助。当然，这个“被减数和差的正相关性”以及“减数和差的负相关性”的道理，也不能由教师直接告诉他们，而是要提供足够的案例，让他们自己察觉出来，然后老师再问他们为什么会这样。

玩完加减法之后，就可以玩加法和减法之间的关系。同样用前面的例子：78+22=100；则可以抢答 100−22=?，100−78=?，然后是 100−79=?，100−77=? 等。后面的乘法除法，也以类似的方式，展示除数、被除数和商的关联性，以及除法与减法、除法与乘法的关联性。

教学过程中要记住几点：

第一，要让学生真正理解加减乘除各自的逻辑发展过程，以及它们之间的逻辑关系；

第二，整个理解过程必须是学生自己察觉和总结出来的，而不是被告知的；

第三，“他们自己察觉和总结出来”中的“他们”，指的是整个班级的每一个人。他们的理解过程，有的快，有的慢，有的先，有的后，但不能出现有些同学觉察、有些同学总结、有些同学被告知、有些同学囫囵吞枣死记硬背、甚至有些同学云里雾里不知所云的情况；

第四，整个教学过程是在“玩”，也就是大家在一种愉快的游戏状态下进行。这种愉快来源于每个人都动了脑筋：有的人因获得了长进、学会了新知识而愉快；有的人因教会别人而有成就感；有的人因为展示自我、出了风头而开心。总之，要让每个学生感觉良好——不管是喜欢学习过程、喜欢数学本身、喜欢同学、喜欢这个班级、喜欢老师，还是喜欢自己——这是教师在课堂教学中要尽力追求的效果。

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这些玩法是让学生动脑筋、找线索、建模型，让他们感到有趣，并且有成就感。教师尽可能地引导学生朝着这个方向发展，即使他们没有找到，也不去干涉，不去插手越俎代庖；而是留待以后合适的时候再让他们去尝试。打个比方，在体育课上，体育老师不会因为一个学生老投不进篮，就接过球来帮他投一个，然后算作那个学生得分，对不对？

数学课让学生动脑筋去尝试探索，然后享受到成功或者进步的快乐，这不仅是让他们学得愉快，更是让他们建立“我能学会数学”的自信心；并且逐渐养成遇到问题时会积极主动地迎面而上，尽力去理解、分析、探索的主动学习习惯。遗憾的是，我们绝大多数教室里的学习过程，都是让学生养成被动接受的习惯。这就是为什么大多数学生喜欢上体育课，而许多学生却不喜欢上数学课的一个重要原因。

学生们能玩多深、想玩多久，主要取决于他们的兴趣，其次是他们目前的认知发展水平，以及已有的数学基础。教师要做的是观察、引导、辅助；并在行进过程中做出判断：有多少学生怀着多大兴趣？其他所有人是否能够加入、能够跟上？只要能跟上，哪怕是靠了同学很多次的帮助，最后得到的也是成就感，也是“玩”会了的喜悦。如果教师觉得会出现扯得太远，或者判断一部分人无法跟上，那就可以非常明确地说：这个题目很好，我们以后会学到。

有的老师可能会问：一个简单的运算，有必要搞得那么复杂，或者花费那么多功夫、那么多时间吗？我在国内教书时常听到一句流行的话，叫做“精讲多练”。美国大部分教科书也一样，你打开看，基本上是每章开头有两三个例题，然后跟着几十道练习题。那几道例题好像讲得很清楚，不过主要讲的是解题步骤，并没有在学生真正的理解上深入探讨和充分玩味。这里“玩味”的“玩”字，也就是“玩数学”的意思。在层层深入的探讨和玩味之中，学生的理解也一步步深入和巩固。

整个过程中，学生是在不断探索、寻找和理解新的东西；这比他们花大量时间去做单调重复、但没有真正理解的练习要有趣，也有效得多。

***

要让学生能 “玩” 起数学来，得让他们理解自己在做的是什么，让他们感兴趣。这样就得用他们熟知的真实世界来说事，尽可能地和他们正在学的其他科目联系在一起。比如，可以用钱来作例子。这些学生要么已经有人会用钱了，要么也会对钱感兴趣，愿意知道，或者有意愿得到钱，甚至在课堂上真实地给大家分几毛钱，对提高兴趣、帮助理解和加强记忆都有很好的效果。

再举个例子，在几何学的概念中，点是最基本的形，或者可以说它没有形：点是零维度的，线是一维，面是二维，体是三维。所以先介绍点，好像顺理成章。但在学生的真实世界经验中，他们语言中的“点”和几何学的“点”不是一回事。所以，几何中的点，是要帮他们从头开始重新建立的。而建立的方式，应该首先从他们熟悉的“面”开始。我们可以先说，桌子有个桌面，教室有个地面，然后让学生自己去找，他们还知道些什么“面”的例子。

玩的第一规则，就应该是每个人都必须参与其中。“面”的玩法，首先是每个人都必须开动脑筋，在认识的事物中至少找出一个“面”来。如果没有人提到任何非平面的面，讲师就可以提出来问：“我们每个人都有个脸，脸面，算不算面？”然后，答案让同学自己去讨论。各抒己见，只要言之成理，一律赞赏。最后的结论是：平面和非平面，都是面。不过我们先讨论平面，这时候可以把“平面几何”这个概念提出来。同时，也可以介绍说，非平面的几何以后也可以学。

面搞清楚了，我们可以让每个同学在桌子上放一张纸，在纸上用笔点一个点；问：“这是什么？”很多学生很可能回答：“是个点。”这时候，可以问：“这张纸是不是个点？”回答很可能会是：“当然不是。”这时可以接着问：“为什么它不是？是因为它大？是因为它是个面？还是有别的什么原因？”

我们可以分析说，这个点直径大概只有一毫米，一毫米很小，所以这只是个点，不是面。这时，你可以开玩笑说，足球场比纸还大，应该是一个面，不是一个点吧？但对于一个很小的虫子，或者一个细菌来说，你画的这个一毫米直径的点，它在上面踢足球还嫌大。所以在它看来，这是一个面，一个很大的面。

课堂里可以七嘴八舌地讨论，到底什么算面，什么算点。也许有人会提到太空探测器在即将飞出太阳系时回头拍的照片上，地球就是一个几乎看不见的小点。这时就可以引出数学的绝对性：它不会因为环境不同、观点不同而产生不同解释。这也是人为什么要学数学的原因之一。

这样继续讨论下去，细菌小下去是病毒，然后是分子、原子、夸克，这里把科学也联系上了。一直无限小下去，点就得小到不能再小，最后剩下没有尺寸，只是空中的一个位置。这个讨论好像比直接告诉学生花费相较多的时间，但它把数学的绝对性和抽象都淋漓尽致地表现了出来。学生花的这个时间是在动脑筋，是在获得新事物、建立新概念的愉悦之中。这样“玩”过之后，数学对他们不会再含糊，也很难忘记。多年的教学实践经验告诉我，学生这样实际上学到的东西会更多、更快、更扎实。

***

当我作为一个代课老师走到一个陌生的教室，我会望着天空，若有所思，开始说：“87”，然后慢慢说出“84”，继续若有所思，好像在推论，然后说“81”，想一下之后再说：“78”。这时会有学生看我在干什么。我接着一边自顾自地数，学生就会开始好奇，可能发现我是在每次减 3往下数，然后他可能会猜：“是 75 吗？”

好奇心是所有人天生有的。跟随着好奇心，发现了其中的奥妙，他就会有些兴奋，并大声说出下面一个数字：“72！”其他那些有好奇心、但还没有发现规律的人，就会想：“他怎么知道老师还没说出的数字？”

这时候我会显出一些兴奋和赞赏，但不允许他再说出下面的数字。我的赞赏鼓励了他，但如果他接着说下去，其他人就成了旁观者或接受者——这些都是很乏味的角色。这时我会让懂了的学生举手，给其他人机会自己去发现。自己发现是一件有乐趣的事情，而被别人告知答案，不仅失去发现的乐趣，还要接受“你不如别人聪明”的挫折感。

在这个活动中，我必须首先估算学生的数学能力。根据学生的数学能力设计问题的难度，让学生的短暂好奇心消失之前，有人能看出答案。如果太简单，学生会没有兴趣。比如，对着初中生数 1、2、3、4、5、6、7，除非是特别听话或特别想讨好老师的学生会跟着我数，大概率是没人感兴趣。如果太难，比如把上面那个例子用在一年级的小学生身上，他们的好奇心已经衰减到零，还没搞懂我在干什么，就会断定我是在自言自语，说一串没意义的数字。

万一对学生水平估计错误，设计得太难，也可以重来一遍，回到起点，把第一个数字写出来。数学不像动画电影那样有声有色、具有吸引人的魔力，所以必须在很短时间内，让学生的好奇心引导出发现。很多老师，特别是数学老师，经常会指责学生不动脑筋，但实际上让学生不动脑筋是很难的。如果教师发现学生不动脑筋，那一定是课堂活动没有设计好。要么是学生觉得问题无趣、无意义，要么是题目太难，学生知道自己搞不懂，任何努力只会带来失败、挫折甚至丢脸，所以会选择放弃。

为了让更多学生参与、获得成就感和乐趣，不能让最早发现的学生告诉其他人。我见到的大多数数学课堂几乎都是这样的。从表面上看，达到了教学目的，大家好像都懂了，而且节约了时间，提高了效率。但如果教师能时时记在心上：数学课不是为了让学生学到什么技能，而是让他们锻炼脑筋、得到乐趣、增长自信，这样做就顺理成章了。

在我的课堂上，每个学生桌子上必须有纸，手上必须有笔，把答案写在自己的纸上。懂了的学生只表示自己懂了，但不能说出答案，让没懂的人有继续努力的时间和自己悟出来的机会。如果自认为懂的学生有错误，那是全班最好的学习机会（Teachable Moment）。我会把那个错的答案当作对的答案，然后让所有学生投票。我常常让每个人都举手：举起的手握拳或伸出一个指头的是赞同，伸出两个指头的是反对，伸出三个指头的是不确定。学生很乐意这样举手表决。

接下来，学生开始讨论或阐述为什么对、为什么错，最后的结论完全由他们自己决定。在这个过程中，我几乎从不直接回答学生的问题。我会用很多“心理战”的方法，比如学生明明错了，我偏说：“Very good，你是对的。”因为我需要那种“不信服”的学生，需要有人跳出来反驳、质疑，然后大家才会开始真正思考。

如果他们一致得出了错误的答案，我也不加否认，而是设计一个新的题目让他们讨论。这个题目的结论会和刚才那个题目的答案冲突，结果就会导致更热闹的讨论。整个过程是每个人都必须参与，尽可能让每个人都一步步自己悟出来，每个人都必须具有说服对方的能力。我常常支持错误的一方，直到最后一个搞懂的人跟我解释为什么他认为的那个答案是对的，为什么我的答案是错的。他得把我说服，这个讨论才会结束。

***

有时候学生玩得深入了，甚至会主动提出想学一些教材里没有的内容。我有个规定：学生想学什么，只要教材上没有，都可以提出来。我每天上课的第一个问题，就是在黑板或投影上写：What question do you have? 我告诉学生：你可以问任何事情，甚至是昨天的新闻，只要你感兴趣。我们在这里做的就是思考——只要能思考就行。如果刚开始问的不是关于数学，过几分钟我会试图把数学引入讨论，把学生带入正题。

比如有学生问：“我们可不可以学 trigonometry（三角函数）？”我就让他们做两件事： 第一，回去查：三角函数需要什么 prerequisites（前置知识）？ 第二，回来讨论：它到底是什么？我们要先会什么才能学？

如果有超过五个人感兴趣，我们就在课堂上讲。如果少于五个人，学生可以自己组织一个 club，不占课堂时间。我的原则是：课堂时间可以拿来“开眼界”，让更多同学看到数学还能这样学、还能这样想。

有一年出现过这种情况：有几个学生觉得高中 placement 已经不过瘾了，他们想去考 SAT，后来真的就去考了。有的考到 600 多分，七年级考到这个水平，已经比附近很多高中毕业生的平均分都高。

凤凰城学区是个大学区，差不多有五百万人口，主要是墨西哥移民。IB 是国际入大学的高中预备课，大部分好一点的高中本身都会有一个 IB 班，但整个凤凰城那时候只有一个 IB 班，高中会学微积分。每年这个班都很难招够，因为学生一般学不会。于是有一年，学区要求每个学校让数学老师推荐最有希望学微积分的学生暑假去培训，一共有三十多个学生。

负责培训的是一个华人老师，一个真正懂数学的物理学博士。他在假期里花了大概一个半月强化培训这些学生，希望下一年开学时，他们有勇气注册。慢慢地，他发现一个奇怪的现象：这三十多个学生，大部分都学得稀里糊涂，但有六个学生比较出众，教什么就懂什么。他后来发现这六个学生互相很熟，都在一个学校，而且都是我的学生。

当他惊讶地来询问我时，我告诉他，这就是我学生的特点：他们不是刷题刷出来的，而是通过玩数学，变成了会动脑筋做数学的人。

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Supplement to Chapter 5: Mr. Xu’s Thesis on Learning a Second Language for His Master’s Degree

Shortly after writing chapter five, I came across a rare treasure: Mr. Xu’s thesis for a master’s degree at Arizona State University, written in the spring of 1992. At that time, Mr. Xu was almost forty years old, just about to become a new teacher in the United States. What strikes me most is the length (at 15,000 words, referencing over 100 academic papers) and the usage of academic language, which contrasts with the usual directness of his communication style. He told me that it was the urgent need to get his degree on time that enabled him to overcome his serious dyslexia and finish such a long thesis in a language that was not native to him.

In the thesis, Mr. Xu examines Communicative Language Teaching (CLT) — a way of teaching a second language that focuses on real communication rather than memorizing grammar rules. His core argument is that the best way to learn to communicate is to actually communicate, rather than going through the “theoretical” foundation of grammar first. What makes language stick for learners is whether it feels genuinely meaningful to them, not just whether it’s grammatically understandable.

The paper dovetails well with both his earlier methods of teaching English in China and his approach to math teaching in the U.S. later, shedding light on the theoretical foundation of his seemingly unconventional teaching that was so powerful. There are too many educational theories around, and in my view, only those validated by reliable and significant data deserve wide attention. Mr. Xu’s teaching record of over twenty years provides one such data source.

***

According to Mr. Xu’s literature review, there are two approaches to learning a new language. “A grammatical approach is one that is organized on the basis of linguistic, or grammatical forms (i.e. phonological forms, morphological forms, syntactic patterns, lexical items) and emphasizes the ways in which these forms may be combined to form grammatical sentences. A communicative (or functional/notional) approach, on the other hand, is organized on the basis of communicative functions (e.g. apologizing, describing, inviting, promising) that a given learner or group of learners needs to know and emphasizes the ways in which particular grammatical forms may be used to express these functions appropriately.”

Three main methods fall under the grammatical approach: the Grammar-Translation method (memorize rules, translate texts — dominant from the 1840s–1940s), the Audio-Lingual method (drill patterns by repetition, developed from the WWII US Army’s language training), and the Cognitive-Code method (grammar with more attention to meaning, post-Chomsky). Today, most school systems in the world use some combination of these methods under the grammar-based approach. The communicative approach, which CLT falls under, is less mainstream and more often used in commercial schools.

My own learning of English fell squarely under the grammatical approach. Even after scoring higher than 96% of students in English grammar (including native speakers), I couldn’t order a burger at the airport without struggling. Having lived and worked in the U.S. for almost thirty years, my social communicative skills are still not optimal. Although some immigrants, like my wife, managed to do better by watching a lot of soap operas, I suspect that the way we learned English is part of our cultural barrier.

Remember Mr. Xu’s experiment on two groups of Chinese students learning English with different approaches? It was almost exactly grammatical vs. communicative, where the communicative approach produced far superior results. Mr. Xu figured it out all by himself, a decade before he learned about the theory in an American university. Coincidentally, it also happened at a commercial school outside the school system.

***

Mr. Xu argues that to master a language, knowing grammar isn’t enough; knowing “when, how and with whom to say what” is more important. Those are learned through experience, and “meaningful input” is the key ingredient. Input that actually matters to you, that compels your attention and participation, is what truly drives learning. Two-way communication (where both sides need information from each other) produces far more of this meaningful negotiation than one-way communication does.

He notes that, “Pattern drills such as:

Teacher: Coffee.

Students: There is some coffee in the cup.

Teacher: Water.

Students: There is some water in the cup.

are meaningless, for there is no coffee or water there. These utterances contain a message but the message is not meaningful and does not function in communication… This utterance is only meaningful to those who, for any reason, need the message or will be affected by it, for example someone who is thirsty and is looking for something to drink.”

I pause at his insight: “Meaningful input is what learners receive in the target language that is so effective for the receivers that they will be attracted or compelled to involve themselves in the communication.” One key distinction of Mr. Xu’s math teaching is the high level of student engagement in real learning; creating meaningful inputs is an important part of achieving it.

***

Mr. Xu’s teaching practice, based on the intuition that necessity is the best motivator to learn, finds a theoretical basis in his thesis too. Macnamara claimed that “the main thrust in language learning comes from the child’s need to understand and to express himself.” He also pointed out that the child in the street was successful in learning language because “the reward for success and punishment for failure is enormous.”

There are two versions of CLT implementations. The weak version just adds some communicative activities to a grammar-based course. The strong version says language is acquired through communication — you learn English by using English, not by studying it and then practicing it. In other words, the learner can figure out the rules as they go.

Something similar is happening in the Artificial Intelligence (AI) arena as well. While earlier implementations of AI attempted to crystallize centuries of human wisdom into predefined rules, their performance never reached an impressive level. Modern AI takes a different approach by building a self-reflective, deep learning network and letting it figure out the rules using data only. As we all know, it outperforms humans in all kinds of tasks including coding and math, sometimes exhibiting great creativity.

***

Although Mr. Xu did not write a thesis on math teaching theory, much of his novel approach is closely related to his thesis on language learning. On “meaningful input,” his attempts to make math meaningful to students are multifaceted: telling them the superpower of math in helping their thinking, showing them that they can succeed in math and that it is not a subject for “someone smarter than me,” and relating problems to things that matter to them such as money and pizza.

His IDK method centers around relinquishing classroom control and encouraging students to figure out the answers rather than teaching them too quickly. On the one hand, he creates a classroom where every student is actively engaged in discussing and figuring things out through communicative activities: arguing, persuading, learning, explaining to each other and to the teacher, who innocently claims “I don’t know; teach me.” This is no easy achievement; he is almost a great director in this sense.

On the other hand, he creates “information-gap” activities that simulate learning challenges and subsequent growth. According to his thesis, “in such activities, each participant possesses information which is unknown to fellow participants and is required by them to carry out the activity successfully. Thus, all participants are compelled to participate in the activity. They found that there were significantly greater amounts of negotiation than had been found in the decision-making activities of both group work and teacher-directed interactions.”

(For those really interested, I have Mr. Xu’s permission to send you a private copy of thesis. Just email yellowheight@hotmail.com, or leave a comment)

第五章补遗：徐老师关于外语学习的硕士论文

在写完第五章后不久，我偶然发现了一件罕见的珍宝：徐老师 1992 年春在亚利桑那州立大学完成的硕士论文。当时，他将近四十岁，正准备成为美国的一名新教师。最令我惊奇的是论文的篇幅（约一万五千字，引用了一百多篇学术论文）以及他使用的学术语言风格，这与他平时直接、简洁的沟通方式形成鲜明对比。徐老师有严重的阅读障碍（dyslexia）。他告诉我，是为了按时拿到学位的紧迫需求，使他克服了阅读障碍，用一门非母语的语言完成了这样一篇长篇论文。

在论文中，徐老师探讨了“交际语言教学法”（Communicative Language Teaching, CLT）——一种强调真实交流、而非死记语法规则的外语教学方式。他的核心观点是：学习沟通的最佳方式就是实际进行沟通，而不是先走一遍语法的“理论”基础。语言之所以能真正被学习者掌握，关键在于它对学习者是否具有真实意义，而不仅仅是语法上是否完全正确。

这篇论文与他早年在中国教授英语的方法，以及后来在美国教授数学的方式，都高度契合，为他看似“不按常规却极其有效”的教学提供了理论基础。教育理论汗牛充栋，而在我看来，只有那些经过可靠且显著数据验证的理论才值得广泛关注。徐老师二十多年的教学记录，正提供了这样一个数据来源。

***

根据徐老师的文献综述，学习一门新语言主要有两种路径。“语法式教学法是以语言学或语法形式（如语音形式、词法形式、句法结构、词汇项目）为组织基础，强调这些形式如何组合成语法正确的句子。交际式（或功能/意念式）教学法则以学习者需要掌握的交际功能（如道歉、描述、邀请、承诺）为组织基础，强调如何恰当地使用特定语法形式来表达这些功能。”

语法式教学法下有三种主要方法：语法翻译法（记规则、译文章——1840s–1940s 的主流）、听说法（通过重复操练句型，由二战时期美军外语训练发展而来）、认知—代码法（在语法基础上更关注意义，受Chomsky影响）。今天，世界上大多数学校体系仍采用这些方法的组合。交际式教学法（CLT 属于此类）则较少成为主流，更常见于商业语言学校。

我自己学习英语完全属于语法式路径。作为一个在GRE英语语法考试中（包括母语者）得分超过 96% 的学生，我第一次在机场点个汉堡仍然会手忙脚乱。在美国生活和工作近三十年后，我的社交沟通能力仍然不算理想。虽然有些移民（比如我妻子）通过大量观看肥皂剧而做得更好，但我怀疑，我们学习英语的方式本身也是我们文化障碍的一部分。

还记得徐老师当年对两组中国学生进行的英语学习实验吗？那几乎就是语法式和交际式的对比，交际式方法的效果远远更好。徐老师完全凭自己摸索出这一点，比他在美国大学接触到相关理论早了整整十年。巧合的是，那次实验也发生在体制外的一所商业学校。

***

徐老师认为，要掌握一门语言，仅仅懂语法是不够的，知道“在何时、如何、对谁说什么”更为重要。这些能力是通过经验习得的，而“有意义的输入”是关键因素。真正对你重要、能迫使你投入注意力并参与其中的输入，才是真正推动学习的动力。双向交流（双方都需要从对方那里获得信息）比单向交流能产生更多这种有意义的协商过程。

他指出：“如下的句型操练：

教师：Coffee.

学生：There is some coffee in the cup.

教师：Water.

学生：There is some water in the cup.

是没有意义的，因为那里根本没有咖啡或水。这些话语包含信息，但信息本身并无意义，也不在交流中发挥作用……只有那些出于某种原因需要这一信息或会受到其影响的人，才会觉得这些话语有意义，例如一个口渴、正在寻找饮料的人。”

我在读到他以下时停下来思考：“有意义的输入，是学习者在目标语言中接收到的、对他们产生强大效果，从而吸引或迫使他们投入交流的内容。”徐老师数学教学的一个关键特点，就是学生在真实学习中的高度参与；创造有意义的输入是实现这一点的重要部分。

徐老师基于“需要是最好的学习动力”这一直觉的教学实践，也在他的论文中找到了理论基础。Macnamara 指出，“语言学习的主要推动力来自孩子理解和表达自我的需要。”他还指出，街头的孩子之所以能成功习得语言，是因为“成功的奖励和失败的惩罚都极其巨大”。

CLT 有两种实施版本。弱版本只是把一些交际活动加入语法主导的课程中；强版本则认为语言是通过交流习得的——你通过使用英语来学英语，而不是先学规则再练习。换句话说，学习者可以在使用过程中自己摸索出规则。

人工智能（AI）领域也出现了类似的现象。早期的 AI 试图将人类数百年的智慧提炼成预设规则，但其表现始终不够理想。现代 AI 则采用完全不同的方法：构建一个自我反思的深度学习网络，让它仅凭数据自己归纳规则。众所周知，它在包括编程和数学在内的各种任务中都已超越人类，有时甚至展现出惊人的创造力。

***

虽然徐老师没有写过数学教学理论的论文，但他许多创新的教学方法与他关于语言学习的论文密切相关。

在“有意义的输入”方面，他通过多种方式让数学对学生变得有意义：告诉他们数学在思维中的“超能力”，让他们看到自己能学好数学、数学并不是“比我聪明的人”才能学的科目，并把问题与他们关心的事物联系起来，如金钱和披萨。

他的 IDK 方法的核心，是放手课堂控制，鼓励学生自己找出答案，而不是过快地教给他们。一方面，他营造了一个课堂环境，让每个学生都积极参与讨论和探索，通过交际活动进行争论、说服、学习、向同伴和老师解释，而老师则认真地说：“我不知道，你们教我吧。” 要把这一点做好并不容易；从某种意义上说，他像一位出色的导演。

另一方面，他设计了“信息不充分”的活动，模拟学习中的挑战与随后的成长。根据他的论文，“在这类活动中，每个参与者都掌握着其他参与者不知道、但他们完成任务所必需的信息。因此，所有参与者都被迫参与活动。他们发现，这类活动中的协商量显著高于小组合作或教师主导的决策活动。”

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(This is Chapter 5 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, in the post-pandemic era. I want to explore what we can learn from a unique and successful teacher, especially on how to teach math effectively. It’s such a joy to find a common link between his method and that of , whose books and online course on learning has been widely known and respected.

中文版在本文后面部分，英文版之后)

Chapter 5 From Learning Language to Learning Math

Most people learn to speak in a simple way: their mothers teach them. If a child were raised by wolves, that child would not learn to speak, but would only howl like a wolf. So how does a mother teach a child to speak? Or rather, how does a mother know how to teach a child to speak?

First, there is maternal love, which gives a mother deep empathy. A mother would not simply say to an infant, “Why are you crying? Do you want milk, or do you want to sleep?” A mother senses that an infant does not have the ability to understand adult language, so she has a special way of speaking to the baby—baby talk. Perhaps this is something mothers have worked out through practice; perhaps it is a system the Creator has already built into a mother’s heart. Thank goodness baby talk was not “scientifically” designed by experts and professors and then taught to mothers.

A large part of the vocabulary in baby talk is actually taught by the infant to the mother. An infant’s speech organs are not yet very flexible, so the complex coordination of the tongue, teeth, and lips is still immature. The easiest sounds to make are labial sounds, produced simply by opening the lips, such as “ba-ba” and “ma-ma.” When a child is with the mother, the child is often nursing, so the air has to come out through the nose; that is how the sound “ma” is formed.

Given the right learning environment, even a person with relatively low intelligence can learn a language. Learning K-12 math is the same. Everyone can learn to speak and become literate, even if not everyone can become a poet or a writer. From my teaching experience, students who are not severely impeded intellectually are born with the capacity to learn math. But their way of learning—or rather, the way they are taught to learn—leads many of them into a dead end.

***

Because infants are born with a need to communicate, mothers naturally teach them language. In a child’s early development, math does not present such an obvious need, so it is regarded as a skill to be acquired only later in life. And the teaching methods we know are those “scientifically” designed by experts and professors.

Let me cite my own experience learning English. I was fortunate enough to attend the first year of middle school—which is cut short by the Cultural Revolution—and that included English class. In the first lesson we learned the letters ABCD, and then spent about a week learning all 26 letters. After that, we began learning single-syllable words with closed syllables. Then came open syllables, two-syllable words, and then simple sentences like “This is a map.” Experts took language, this immensely complex system, apart and analyzed it from complex to simple—from texts to paragraphs, from sentences to words, from syllables to letters. So we began learning English from the alphabet, proceeding from simple to complex, step by step, and it all looked very “scientific.”

I began learning this foreign and novel thing with enormous curiosity, but the whole process was unbearably dull, utterly lacking in interest or meaning. In the end, we learned to read sentences like “This is a table.” Yes, “This is a table”—so what? Did you think I was a fool, that I didn’t know this was a table? That may sound prickish, but it shows that throughout this whole learning process there was no real function being expressed. This kind of learning lacks meaningful communication, while the main function of language is precisely communication. Another function of language is thinking, and “This is a table” does not give one much to think about either.

Later, when I was teaching at a Chinese university, I also ran a private educational training center teaching English. This was in the 1980s. Under the normal school sequence, English was not introduced until middle school, but many parents wanted their children to get an early start (even to this day, the phrase “don’t lose out at the starting line” is quite popular among Chinese parents), and at that time teachers who could teach English were scarce. While teaching a group of elementary school students English, I carried out an experiment. One teacher followed the textbook and started with the alphabet, while I began by teaching the children how to quarrel in English, like the kind of everyday exchanges two sisters might have in real life—for example: “My apple—did you eat it? Give it back!” “It’s seven o’clock! Get up! We’re going to be late! Go brush your teeth! Where is my comb? Did you steal it?” and so on.

After one month, the students taught from the textbook could haltingly recite the alphabet, and could pronounce “This is a table”, although listeners couldn’t distinguish if they were talking about an apple or a table. The children who learned English through quarrelling, working in groups of two or three, were already able to create and perform their own situational dialogues. They could all stand on stage and carry on conversations for five to fifteen minutes. When writing their own scripts, they could ask the teacher about words they did not know. Very often, once they asked about a word, they remembered it, and then used it fluently on stage. After two months, the students in the performing group could almost all reenact their scenes, whereas the students who had learned English by memorizing the alphabet had already forgotten nearly half of what they had learned.

The good news is that today English teaching methods have basically shifted to approaches like situational teaching and immersion. The bad news is that math teaching is still largely organized the way experts have prescribed: start with counting, then one-digit addition, then two-digit addition, then subtraction, and continue on in that way. It was only after I observed children’s confusion and listened to their questions in my own teaching that I began to realize the defects of this method, and how harmful they were. So I set about changing it. And just as when I improved the way English was taught, improvements appeared immediately.

***

Before children’s abstract thinking is fully developed, they find intuitive arithmetic easy to understand, but abstract concepts and reasoning operations feel difficult and confusing. What we call “intuitive” is really just “familiar.” For young children, even “1 + 1 = 2” is abstract at first. But once they become familiar with it, they can make the conceptual turn: “1 + 1” means one object plus another identical object, which makes two such objects. This process is the same as learning a language.

Language learning begins with an urgent need to communicate. That need drives active effort—trying things out, guessing, and forming concepts through use. The successes and failures in communication, the different effects produced, and the feedback received all help the learner quickly become familiar with and remember things.

Once they remember the pronunciation and spelling of the words that express a concept, they can use those concepts just like adults. What they internalize is not just single words or two-word expressions. They go on to produce three-word, four-word expressions, and gradually infer the logical structure of how concepts combine—grammar—so they can form sentences and speak like adults. Of course, to become a writer, poet, or linguist, learning from predecessors and receiving expert guidance is extremely important and beneficial.

Learning math is essentially the same process as learning a language: first internalize basic concepts—memorize them deeply—then combine these concepts into “sentences” according to logical relationships. The difference is that language originates in concrete expression and then develops into figurative, emotional, rational, abstract, and all-encompassing forms of expression. In the early stages, this entire learning process happens without expert guidance or a prescribed curriculum. It comes purely from the child’s interaction with loving parents and from their own attempts, guesses, and continuous inference and summarization.

But the mathematics we now force-feed students in school is the distilled, purely abstract logical system created by experts. Children do not yet have abstract thinking ability. Their mathematics—or arithmetic—should be entirely concrete: the quantitative relationships in the countless events of their world, and the clever, simple methods they invent, explore, or discover to display, extend, and connect those relationships. Yet we adults—especially professors and experts—after discovering the beauty and power of mathematics, turned it into a discipline focused entirely on abstract reasoning, and then used the mathematics familiar to adults to teach young children. This is a serious mistake.

***

Imagine taking newborn children away from their parents and having linguists teach them letters, phonemes, morphology, and syntax. Do you think they would grow into good writers? I suspect most of them wouldn’t even learn to speak, let alone become literary figures. Our schools teach children to count and then immediately jump to “1 + 1 = 2,” dragging them abruptly into the abstract world of mathematics.

Another analogy is throwing someone straight into the ocean to learn to swim, with no training or preparation. For most people, that would be a negative experience. Of course, because individuals differ and environments vary, some children who initially feel overwhelmed will gradually “wake up” as their abstract thinking develops and eventually grasp the ideas. Some children, thanks to supportive environments and their own effort, can still learn well. But students who lack confidence or grow up in environments where people around them mostly failed at math—parents, relatives, friends—are easily discouraged and give up after repeated setbacks.

Math teaches logical structures and reasoning. Although the reasoning is abstract, like language it uses very concrete concepts, with clear structures and explicit rules. These are like vocabulary and grammar in language: they must be memorized and internalized, or one cannot speak fluently. In mathematics, number sequences, addition tables, subtraction tables, multiplication tables, and the procedures for the four operations with rational numbers—these are what I call the “building blocks” of mathematics. They should be mastered to the point of intuition, just like vocabulary, grammar, and language sense. In other words, one should be able to produce answers without deliberately going through the full thinking process. In language, this fluency comes from subconscious procedural memory; without it, language would not flow smoothly, just as those learners who learned primarily through translated texts struggle to speak foreign languages fluently.

For example, fraction multiplication and division are topics that most need to be trained to an intuitive level. Fractions intimidate many students because they seem hard to understand. But that’s simply because a fraction is itself a division expression, and multiplying or dividing fractions means performing three operations within one expression. If a student thinks division is not difficult, then doing division again is not a qualitative change—it just requires one more mental step. Reaching this level is not hard, but it requires two things: first, truly understanding the concept; second, practicing—or “playing with math”—until it becomes familiar. For these building blocks, there is a distinction between “performing a task” and “understanding it,” and then a further step from “understanding” to “mastery.”

The idea is similar to the “chunking” concept proposed by Barbara Oakley. From elementary school through high school, Oakley struggled with math—she could barely pass. At the time, she felt that mathematics was like a plague that humanity should avoid at all costs. She majored in Russian in college, and after graduating, she became a captain in the military, leading a team in the Signal Corps. Her colleagues were all very strong in mathematics, which had a profound impact on her. At age 26, she decided to rebuild her brain, overcome unimaginable challenges, and learn mathematics and engineering from scratch. After earning her bachelor’s and PhD in engineering, she became a distinguished engineering professor, an IEEE Fellow, wrote more than a dozen bestselling books on learning and education, and created the most popular online courses to date (“Learning How to Learn”), gaining enormous influence in the United States.

Oakley drew lessons from her experience learning Russian and applied them to relearning mathematics. When learning a language, she needed to “chunk” common expressions (such as “Could you tell me”), grammatical structures (such as the fact that “I have been meaning to” must be followed by a verb), and common phrases. In mathematics, basic facts (like multiplication tables), basic operations (like division), basic tasks (like solving linear equations), and word‑problem frameworks can also be learned as “chunks.” Through deep understanding, repeated practice, linking ideas, and applying them in different contexts, learners can reach a level of fluent, intuitive understanding.

The theory of “chunking” holds that learning becomes easier when scattered bits of information are integrated into a meaningful whole. The brain can then recognize and reuse it as “one thing,” rather than many separate pieces. Once a chunk is formed, it reduces cognitive load because human short‑term memory is limited. Instead of handling five separate steps, we process them as one familiar structure.

***

After the assistant principal observed my class, he no longer gave me trouble about my lesson plans. But not long after, another conflict arose between us. This incident also put a lot of pressure on me, but the outcome was unexpected.

To improve math education, the district planned to adopt new textbooks. Each school sent one or two strong math teachers to attend the selection meetings, which were held several times a week—sometimes almost daily. At these meetings, publishers laid out dozens of textbook sets, and each teacher took one home to try before returning to discuss. A teacher from my school, someone I got along with well, also participated. Later we learned that he had been hired as a spokesperson by one of the textbook teams. That textbook, the Connected Math Project (CMP), was written by professors at Michigan State University. Its goal was to help students learn mathematics by connecting problem‑solving, reasoning, communication, and conceptual understanding.

The philosophy was actually quite valid: math should not be detached from life; it should connect to real‑world contexts. So the textbook read like a storybook—each lesson began with a narrative that led into the math. When the teacher representatives voted on whether to adopt the textbook, I was the only one who voted no. My reasoning was simple: 80% of our district’s students speak Spanish and have not yet mastered English. Asking them to begin each lesson by reading a long English story creates a huge, artificial barrier to learning math.

For example, one lesson described a group planning to bike around a lake. It spent several long paragraphs describing how beautiful the lake was, how they divided into groups, how they set off, and so on—only then did it get to the math: wheel diameter, how to calculate circumference, distance traveled, speed, time, etc. Just getting to the actual math took a long time. Students worked in groups to complete tasks. Only one or two students did the real thinking; the others were busy copying notes. After a week, students had learned very little math, but everyone had “busily engaged in activities,” so observers thought the teaching was highly successful: Look, every student is engaged! They’re even learning from each other’s strengths, because every student has unique insights worth learning from.

In the end, the district adopted the textbook. The superintendent came to inspect several times. I insisted on using no more than about 10% of the new textbook. The teacher who served as spokesperson used 80%. Other teachers used between 30% and 50%. The district kept pressuring the principal: every teacher must use the new textbook, and those who don’t will be disciplined. The principal privately asked me twice, “Could you pretend to use the new textbook? Before the superintendent visits, I’ll let you know, and you can just put on a show.” I said, “No. I can’t act in front of my students.”

Over the next year, the assistant principal repeatedly called me in to talk about this issue. Although he never directly threatened to fire me, he kept pressuring me and encouraged the principal behind the scenes to dismiss me. I told him at the time: I am responsible to my students, not to a bureaucratic system. Most of my students are English learners; this “storybook textbook” is useless to them and actually hinders their learning.

After one year of using the new textbook, the state standardized test results came out. The teacher who used 80% of the new curriculum had a pass rate of only 10%. My class had a pass rate above 90%. Other teachers also showed the same pattern: the more they used the new textbook, the worse their students performed. The teacher who had endorsed the textbook soon left our school. The assistant principal eventually resigned and left the school district altogether.

Yet that textbook continues to be used not only in our district but in many districts across the United States.

NEXT CHAPTER

第五章 从学语言到学数学

大多数人能学会讲话，其原因很简单，因为是“妈妈教的”。如果一个小孩被狼养大，这个小孩就不会讲话，只会像狼一样嚎叫。那么，妈妈是怎么教孩子说话，或者说，妈妈是怎么知道该怎么教孩子说话的呢？

首先是母爱，无与伦比的母爱使母亲具有深度的同理心。母亲不会对婴儿说：“你哭什么哭！你到底是想要吃奶，还是想睡觉？” 母亲能感知婴儿不具备听懂成人语言的能力，所以她有一套跟婴儿讲话的儿语。这也许是母亲从实践中摸索出来的办法，也许是造物主在母亲心中已经设置好了的系统。谢天谢地，儿语不是专家教授们“科学地”设计出来，教给母亲的！

儿语中的很大一部分词汇，是婴儿教给母亲的。婴儿的发音器官不是很灵活，所以舌头、牙齿、嘴唇的各种复杂运动配合还不熟练。最容易发的音是唇音，嘴唇张开就来，如“爸爸、妈妈”。孩子跟妈妈在一起的时候常常是要吃奶，所以气只好从鼻子里出，这就是“妈”的发音。吃饭的儿语中，婴儿发音为“曼曼”。妈妈会说“吃曼曼”，上厕所叫“窝吧吧”，都是唇音。孩子长大一些后，“曼曼”变成了“饭饭”。“饭”的发音需要让上牙出来，对收回来的下唇，口腔运动复杂度要求就高了；“吃曼曼”改成“吃饭饭”的过程完全与婴儿的认知和身体发展密切配合。

学语言只要有正确的学习环境，即使智商差的人也能学会。中小学数学也是一样的，就像人人都会讲话，能读写，虽然不是每个人都能成诗人或文学家。从我教学的经历看，非特殊智障的学生，天生是具备学习数学能力的。但他们的学习方法，或者说是他们接受的教育方法，让很多学生走进了死胡同。

因为婴儿天生就有交流需求，所以母亲天然就会教语言。数学在孩子的早期发展中，没有明确的需求，所以被认定是长大后才要获得的技能。而我们知道的那些教学方法，都是专家和教授“科学地”设计出来的。

这里引用我学英语的经历。我有幸上了不到一学年的初一，其中就有英语课。第一课学字母ABCD，然后用大概一个星期把26个字母学完。接着，我们开始学闭音节的单音节单词。然后是开音节、双音节，接着开始简单句，像“This is a map.” 专家们把语言从繁杂的语言系统拆开分析，从复杂到简单，从文章到段落，从句子到单词，从音节到字母。我们就从字母开始学英语，从简单到复杂，循序渐进，看起来很“科学”。

我带着极大的好奇，开始学这个洋玩意，可整个学习过程乏味无比，毫无趣味和意义。最后我们能学会读“This is a table”。是的，“这是一张书桌”，然后呢？你以为我是傻子，不知道这是一张书桌吗？这虽然有点抬杠，但至少说明这整个学习过程中，没有任何实际功能的体现。这种学习缺乏有意义的交流，而语言的主要功能正是交流。语言的另外一个功能是用来思考，而“This is a table”也实在没多少好思考的。

我在中国大学教书时，兼职开了一个教育培训机构，专门教英语。那是在八十年代，按正常进度初中才有英语课，但很多小朋友小学就想提前学，那时会教英语的老师又很稀缺。我在教一群小学生英语时，做了一个实验。一个老师照教科书从字母表开始教，而我则先教小孩子用英语吵架，就像两姐妹在生活中的日常交流，譬如“我的苹果，你吃啦？还给我！”“七点啦！起床啦！迟到啦！漱口！我的梳子呢？”之类。

一个月之后，按教科书教出来的学生，能结结巴巴地念出字母表，连“This is a table”里说的是苹果（apple）还是桌子（table），都不是很分得清。通过吵架学英语的小孩子，两个或三个一组，已经可以自编自导一个场景对话。他们全都能在舞台上讲五到十五分钟的对话。他们自己编剧本的时候，不懂的单词可以来问老师。常常他们问了一遍就记住了，然后就在舞台上流利地说了。两个月之后，演出的那一组同学几乎都能重新再演，而那些靠背字母表学英语的学生，已经忘了几乎一半。

好消息是，现在的英语教学法基本都改成情景教学、浸入式教学等方式了。坏消息是，现在的数学教学仍然是按照专家拟定的方式，从数数开始，然后一位数加法、两位数加法，然后减法，这样一路走下去。我也是在教学中观察到小孩子的不解和他们的反问，才意识到这种方法的毛病，以及其严重性。然后，我就着手把它改了。正像我改进英语教学方法一样，同样的效果一下就出来了。

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小孩子的抽象能力还没有发展完全之前，他们对直观的算术容易理解，对抽象的概念和推理运算会感到吃力，容易搞混。所谓直观（intuitive），实际上就是熟悉。对小孩子来说，一加一等于 2 就是抽象的，但熟悉了以后，他们就能转过这个弯来：一加一就是意味着一样东西，加上另一个一模一样的东西，就等于两个这样的东西。这个过程和学语言是一致的。

学语言首先是由于有急切需求，然后就会有主动的努力，在运用中尝试、揣摩、建立概念。运用的成败，交流中不同的效果，以及得到的反馈，很容易让实践者熟知并且记住。他们记住了表述这个概念的词汇的发音和拼写以后，就和成年人一样可以使用这些概念了。他们所熟知的，也叫内化了的，不只是他们尝试着用单个字、两个字的表述。他们会接着发展成三个字、四个字，然后揣摩出这些概念组合的逻辑结构方式，即语法，这样他们就能组织句子，和成年人一样说话了。当然，要成为文学家、诗人或语言学家，学习前人的积累，获得专家的指导是相当重要而有益的。

学数学和学语言实际上是一个相同的过程，都是先把一些基本概念内化，即熟记于心，然后把这些概念根据逻辑关系组成句子。它们的区别在于：语言起源于具象的表述，然后发展出形象的、感性的、理性的、抽象的、包罗万象的表达。整个学习过程在初期没有专家的指导，没有大纲的规定，纯粹是学生和无比关爱的父母的互动，以及自己通过努力尝试和不断揣摩推断，总结发展出来的。

我们现在学校里强行灌给学生的数学，是专家们提炼出来的、纯抽象的数理逻辑。小孩子没有抽象能力，他们的数学，或者说算术，应当是完全具象的，即他们所在的万千世界中的各种事件的数量关系，以及为展示、推衍、链接这些关系而设计出，或者说探索出、发现的各种精巧简便的方法。但是我们大人，特别是教授专家们，发现数学的精美和强大的功能后，把它独立成一门完全侧重于抽象推理的学问，然后用成年人熟知的数学去教小孩子。这是非常错误的。

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比如，如果小孩子一出生就把他们从父母身边分开，让语言学家教他们字母、音素、词法、句法，你看他们能否学成文学家。我估计基本都不会讲话，不要说成为文学家。我们的学校教小孩子数数，然后是 1+1=2，把小孩子们生硬地拽进了数学的抽象世界。

再打个比方，这样做就像不做任何培训和准备，一下就把人直接扔进海里学游泳。这对多数人来说，绝对是个负面的经历。当然，由于人有个体的差异，有环境不同的影响，有的孩子学晕了之后会慢慢醒过来，随着抽象思维能力的发展，能悟出其中的道理。有的小孩子由于周围环境的影响，通过周围人的帮助和自己的努力，还是可以学得很好。但自信心不足的、周围环境影响负面的学生，比如其父母、亲戚和朋友中，大多是没有能学会数学的，他们就容易因挫折而放弃。

数学教的是数理的逻辑结构和推理。虽然是抽象的推理，但它和语言一样，会使用非常具体的概念，有清晰的结构和明确的法则。这些就像语言中的词汇和语法，必须是熟记于心，否则就不能顺畅讲话。数学中的数列、加法表、减法表、乘法表和有理数的四则运算过程，我称之为数学的建构组件（building blocks），是应该像语言中的词汇、语法、语感一样，被熟悉掌握到“直观”程度。也就是说，不需要再经过主动思考过程就可以直接得到答案。这在母语中是通过下意识运作的程序性记忆（procedural memory）实现的，否则语言就不会流畅，正如很多用语法翻译法学会外语的人一样。

比如，分数乘除法就是一个最需要训练到“直观”（intuitive）程度的内容。分数让很多学生畏惧，觉得它不好理解，其实就是因为分数本身是一个除法式子，它们之间相互乘或者除，就等于一个式子里包含了三个运算。如果学生认为除法不难，那么除法再除一次并没有发生什么质的变化，无非是脑子里要多转一道弯。要达到这个程度其实并不难，但需要做两步。一是真正透彻地理解了，懂了；二是通过练习，或者“玩数学”，把它学熟。所以对于这些 building blocks（建构组件），就有“会”和“懂”、从“懂”到“熟”的要求。

这个道理很接近芭芭拉-奥克利（Barbara Oakley）提出的分块学习（chunking）概念。奥克利从小学到高中数学一路开挂，几乎都不能及格。她那时觉得，数学跟夺命瘟疫一样，是人类应该不惜一切代价躲避的。她大学主修俄语，毕业后在军队里做到上尉，在通信兵团领导一个团队，周围都是数学很强的同事，这对她触动很大。于是，26 岁的她决定重塑自己的大脑，克服不可想象的挑战，从头重新学习数学和理工科。获得工程学学士和博士之后，她成为一名杰出的工程学教授、IEEE 会士（Fellow），还写了十几本关于学习和教育的畅销书籍，开设了至今最受欢迎的在线教程（“学会怎么学习”），在美国有极大的影响力。

奥克利从自己学俄语的经历里总结出经验，并且应用到重新学数学的过程。学语言时，她需要把常用表达式（比如，“Could you tell me”）、语法结构（比如，“I have been meaning to” 后面要接动词）、常用语等进行“分块”学习。学数学时，基本事实（如乘法表）、基本运算（如除法）、基本任务（如解一元一次方程），以及应用题框架，这些也可以作为“分块”来学习，通过充分理解、反复练习和链接，以及在不同场景下反复使用，达到熟练的“直觉”理解。

“分块学习”理论认为：当零散的小信息被整合成一个有意义的整体单元时，学习会变得更容易。这样，大脑就能把它识别并重复使用为“一个东西”，而不是许多个彼此分散的部分。一个 chunk（分块）并不只是死记硬背下来的零碎片段，它是一个有意义的紧凑模式。一旦这种分块形成，它就能减轻认知负担，因为人脑的短期记忆有限。这样，我们不再需要分别处理五个独立的步骤，而是把它当作一个熟悉的整体结构来处理。

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自从副校长听了我的课以后，他没有再就教案问题为难我了，但不久后，我们之间又发生了矛盾。这件事也给了我很大压力，但结果却是意想不到的。

学区为了振兴数学教育，要换新的教材，于是每个学校都派了一两个数学不错的老师去参加选教材的会议，每周固定几次开会，有时候几乎天天都有会议。在会上，出版商摆出几十套教材，老师每人拿一套回去试用，然后开会讨论。我们学校有一个跟我关系不错的老师也参加了，后来我们才知道，他被其中一个教材团队雇做代言人，不知道有没有拿回扣。那套教材是密歇根州立大学的教授们编的，叫 Connected Math Project（CMP），目标是帮助学生通过解决问题、交流推理和建立概念之间的联系来学习数学。

它的理念其实很正确：数学不应该脱离生活，应该连接现实情境。所以它的教科书翻开像故事书——先讲一段故事，再从故事引出数学。最后教师代表投票决定要不要用这个教材时，全场只有我一个人反对。我的理由很简单：我们学区 80% 的学生只讲西班牙语，英语都还没学好。你让他们翻开教材先读一大段英文故事，他们根本读不进去，这对他们学习数学是很大的人为障碍。

比如，书里有一个例子：一群人决定环湖骑自行车，先讲湖有多漂亮、怎么分组、怎么出发，讲上几大段，然后才进入数学：车轮直径多少、周长怎么算、骑多少公里、速度多少、时间怎么算等等。光是进入正题就要花很久。在这个过程中，学生一起合作完成任务。真正动脑筋的学生就一两个，其他的只是忙着抄写记录。一个星期下来，数学没学到多少，但大家都在参加“做活动”，于是观摩者就觉得教得很成功：看，每个学生都参与学习了，他们还相互学习对方的长处，因为每个学生都有独特的、值得学习的见解。

最后，学区选定了用这套教材，学区总监（superintendent）也来检查过好几次。我就坚持：我最多用 10% 左右。那个代言的老师用了 80%。其他老师有用 30% 到 50% 的。学区一直给校长压力：每个教师必须用新教材，不用的老师就要进行处罚。校长私下两次求我：“你能不能假装用一下新教材？学监来检查前，我提前通知你，你就摆个样子出来。”我说，“不行，我不能在学生面前演戏。”

接下来的一年里，副校长就这件事情多次找我谈话，虽然没有直接威胁要解雇我，但还是不停对我施压，在背后鼓动校长解雇我。我当时对他说：我需要对学生负责，而不是对官僚系统负责。我的学生大部分是英语学习者，这种“故事书教材”对他们没用，反而是阻碍。

新教材使用了一年之后，学生的州统考成绩出来了。新教材使用 80% 的那位老师，班上学生的及格率只有 10%，而我班上学生的及格率超过 90%。其他教师也是新教材用得越多，学生的分数越差。那位给这份教材代言的老师，很快就离开了我们学校。那位副校长则最后直接辞职，离开了我们学区。

然而，那份教材不仅在我们学区，而且在全美国范围内很多学区都在继续使用。

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