I spent three days, on and off, finishing Mathematics Awakening: Learning to Think More Clearly, and throughout the reading my wife and I argued intensely. By historical grades and by the math ability I still retain, I was a top student and my wife was a poor one, but the two of us could discuss this book passionately, because it isn’t a math book in the ordinary sense. It tries to reveal the mental landscape of a mathematician doing mathematics; it’s closer to a book of psychology or philosophy.

When I finished the book, I put down my phone and said to my wife: I feel a little like crying. But I also saw quite a few negative reviews on WeChat Reading, and I thought: to understand a book, or to resonate strongly with its author, takes a certain alignment of circumstances. If my first forty years were to be described with a few keywords, “mathematics” would have to be one of them. It built my early, powerful self-confidence, and later it also plunged me into deep doubt and self-doubt.

My clear memories of learning — mathematics especially — start in the middle years of primary school. In that era, when the countryside was still very poor and short of everything, supplementary workbooks were not a common thing. But my parents spent money to buy me one in the Xinhua Bookstore in the city, called A Quarter Hour a Day. The math problems in it were especially hard; by today’s standards they’d probably put fairly easy Olympiad problems in there. My parents had no idea how to explain such problems in a way a primary schooler could understand, but they had at least studied a little algebra properly, and so that summer they spent a few afternoons hiding from the heat teaching me to solve those problems with equations. Primary school math back then didn’t teach equations at all (today’s textbooks introduce them earlier, and it’s been proven that primary schoolers don’t find simple equations all that hard).

Suddenly I felt I had a very handy weapon in my hands. Besides finishing that workbook, for the many folk math puzzles my grandfather set me, solving them with equations was like cutting melons and chopping vegetables — trivially easy. Get them right and there was a red envelope, plus praise. Math class was the same. People generally assume that someone good at math, with good math grades, is smart, and that smart people can learn anything — physics, chemistry, or Chinese and English. And so it became a self-fulfilling expectation: I always felt I was ahead of my classmates and even the teachers, so I basically didn’t listen in class — math and science classes especially, I spent them reading on my own, thinking, and doing exercises ahead of the schedule. The results felt like a cheat code all the way: from a national second prize in the primary school math Olympiad, to the first district-wide exam in middle school, where my total score beat the second in the grade by over a hundred points without much effort, to being specially recruited into the city’s No. 1 high school.

In high school math I began to feel the ceiling of my ability. Daily coursework and exams were no problem; I still often got full marks. But when the problems got harder, that effortless cutting-through-melons feeling from primary school equations was gone — high school Olympiad math especially was beyond me. At first this didn’t make me too anxious: in my third-year mock exams my math was still 145-plus, near perfect. My homeroom teacher was a math teacher, and he even told me that given the current gaokao situation, there was no need to spend extra time going deeper into math; better to put more time into Chinese and English, where the return would be higher.

Then came the 2003 gaokao, the one that is mildly famous even in the history of the exam: the math paper was far harder than in previous years. A male classmate of mine, whose math was usually decent, came out of the exam room crying. I wasn’t much better. From the multiple choice and fill-in-the-blank questions onward I felt strong discomfort, and by the first big solid geometry problem I hadn’t even fully solved it — traditionally that question is called a free-points question. I tried every means to steady my frame of mind, even spending time attacking the final bonus problem, but the final result was merely passable: a bit over 110. Fortunately in the later comprehensive science paper, though the questions were also hard, I held my nerve, and my final total was about 5 points short of the Tsinghua/Peking University cutoff. At the time I actually knew nothing about universities; I had simply absorbed from public opinion the claim that Tsinghua and Peking were good schools. Given how I usually performed, I ought to have gotten in. I didn’t, and I naturally interpreted that as a failure. I did end up in the computer science department at USTC, but the shadow of that failure hung over me for many years — so much so that my college life and my earliest career choices now look like a mess.

Only after thirty, after marrying and establishing a career, did I slowly come to terms with these things. The first thing that released me was realizing this was a question of breadth of vision. For me at that time, entering USTC was a resigned choice made after the correct option had been taken away; my limited horizons meant I didn’t know how to begin my college life. Though in a broader sense, getting into USTC was undoubtedly a success — I just couldn’t feel that success at the time, and no one could make me truly understand that university study is inherently far harder than high school. I remember stepping onto the USTC campus and seeing seniors at stalls outside the cafeteria selling books, many of them TOEFL and GRE vocabulary books. At the time I was baffled and repelled by the sight, not realizing that those people aiming to go abroad — whether or not their choice was right — at least had a plan, had a choice.

Then I thought: breadth of vision doesn’t seem to explain everything either. There are also kids from the countryside who, once in university, take to it like fish to water and quickly reach heights beyond mine. So is all of this, then, about intelligence? Was my intelligence insufficient to support my university studies, so that my college math was only half-understood — the exam scores were fine, but I never fully grasped it — so that years later, when I was studying artificial neural networks, I always felt I’d missed too many classes in mathematics? But setting aside how hollow the concept of IQ is, it can’t even explain this: if I got into USTC by underperforming, why did so many of my classmates, who entered USTC with similar scores, live fuller college lives than I did and develop better after graduation? You can’t very well say they were smarter than me all along and simply underperformed by a wider margin.

Finally I came to suspect confidence and courage — that the gaokao had damaged my confidence and courage, producing some subtle sense of defeat, and that this was what made my college life muddled along. But this is a hidden psychological experience not fit to be told to outsiders; I had only ever discussed it with my family. Having dissected myself this far, I can return to Mathematics Awakening: Learning to Think More Clearly and explain why it moved me so much. The book comes from a French mathematician, who analyzes the mental landscape of his own mathematical research and that of Einstein, Descartes, and a whole company of first-rate mathematicians and scientists. It let me finally let go and make peace with myself.

The author, an established mathematician, bravely admits that he finds it almost impossible to read any professional mathematics book written by a mathematician, and cannot follow the thesis defense of a math PhD student. Einstein told a middle schooler that the mathematical difficulties he faced were no fewer than the student’s — these are not modesty or politeness but honest statements. Mathematicians, like ordinary people, face enormous difficulty before mathematics. So why is the gap in mathematical ability between people so vast? Su Bingtian’s 100 meters is 9.83 seconds; an ordinary able-bodied person cannot beat Su Bingtian, but at worst they can finish in 20 or 30 seconds. In mathematics, though, the gap between people seems to be that some need an hour to cover 100 meters.

Mathematics is about imagining and manipulating abstract objects, and that is a capacity everyone is born with. But this imagining and manipulating of abstract objects in the mind, unlike language and movement, is hard to learn by imitation, and so many people give it up early and define themselves as having no talent for it. Logical reasoning is only one expressive language of mathematics, a finger pointing at the moon, not the whole truth of mathematics. Mathematics is first about curiosity and imagination, and ultimately about intuition; sooner or later all of these hit a “ceiling.” The people who break through that ceiling again and again are called mathematicians. It isn’t about IQ here; it’s about courage and confidence. In the face of abstract concepts, a vast sense of ignorance easily knocks a person down — but when you know that the ignorance faced by the greatest mathematicians is equally daunting, and then look at the heights they reached, at least some courage rises in you unbidden. Aim high and you will land in the middle; of course few can become mathematicians as great as Descartes, but it is certain that in mathematics the vast majority of people — myself included — vastly underestimate themselves.

Only by believing can something exist and be realized. Mathematics, in a sense, is a self-fulfilling expectation. Isn’t life the same?

This is a collection of excerpts and a short review of The Gardener and the Carpenter . I recommend the paper edition, or the e-book on WeChat Reading.

The author of The Gardener and the Carpenter is an outstanding researcher in child learning and development, a psychologist and a philosopher. The book’s scholarship is high. A parenting book that argues from the heights of evolutionary theory and philosophy is a rare thing.

Are you a gardener, or a carpenter?

The author flatly rejects the anxious, mainstream middle-class model of raising children — the so-called “parenting” model. “Parenting” is a word that only took shape in the 1950s and 60s. It was a trend that set in after the middle class rose and the traditional way of raising children could no longer hold. Perhaps this model played a positive role in industrial production, but facing a more complex future, it has to be rethought. Gardener and carpenter may not be the best metaphors in Chinese: a carpenter does sand things into shape, but the masterpiece of the Chinese-style gardener — the bonsai — is no better, if anything worse. The gardener here should be one who lets plants grow wild. Having no “ideal” child to aim at is unbearable for many Chinese parents, I imagine. Raising a child involves two paradoxes: dependence and independence, play and learning. Neither is a matter of simply walking in one direction.

“Parenting” is a bad modern invention

“Chaos is a ladder”: children’s chaos is the ladder of human development. Exploration is a child’s nature, you could even call it a child’s mission. Drawing a blueprint for a child and setting out the rules and boxes is a fairly arrogant act — true in the unstable living conditions of primitive society, and just as true in the fast-moving present and future. The traditional Chinese idea that “jade cannot become a vessel without carving” dovetails neatly with “parenting” under industrial civilization, and makes it worse. This “parenting” model designs the child according to school-like rules and boxes. Traditional education still had intuition grounded in the parent-child relationship; today even that intuition gives way to doctrine. Whether to pick up a crying child — parents have no instinctive judgment left, they follow what the parenting books say. Never mind the anxiety of comparing “super kids”.

Childhood, a key strategy in human evolution

The author raises the importance of childhood to the level of an evolutionary strategy for the species. In reality, many parents treat childhood as nothing but an unavoidable preparatory period for adulthood. What they want is for the child to grow sensible fast, get good grades, and stop being a worry. The author analyzes the meaning of human childhood from an evolutionary standpoint, and I think he has hold of the crux. This vindication of childhood is not merely humanistic concern; it is a crushing superiority of perspective. Once you start from “don’t lose at the starting line”, a child either competes for a lifetime without ever learning how to be happy in life, or collapses at some point and goes after pleasure out of revenge.

Love, the guarantee of continued evolution

The author discusses three layers of love for children: parents, grandparents, alloparents. From an evolutionary angle he explains how humans broke through the prisoner’s dilemma in the form of love, planting the code for caring for children deep in both genes and culture. It is not that we care because we love; we love because we care. My own sense is that our current social culture has to some degree alienated this tradition. There is blanket, allergic condemnation of “bratty kids”, a fear of the unpredictability of children — people who are not fully capable of acting for themselves. Blaming the mother of a crying two-year-old on a plane, for instance. The whole society’s myth of “parenting” runs like this: as long as the parents teach well, even a tiny child will obediently keep public order like an adult. That is precisely against human nature.

Learning by watching

This chapter is about how children learn by observation and imitation. The existing education system is only a few hundred years old. Middle-class parents today have adapted to schooling and want to apply the same thing to their own children as early as possible, while paying too little attention to the older, more natural learning by imitation. A common explanation is mirror neurons, but that is a lazy, oversimplified one. Children’s imitation is complex and fine-grained, beyond adult perception (you never know when they learned it). First, a child’s imitation is not just copying a process: they try to understand the adult’s intention, dig out cause and effect, fold in their own thinking, and end up imitating a result they find interesting. If the adult does not look like an expert, the child discards more of the chaff. When the adult looks more like an expert, the child imitates more of the details — and sometimes that is not a good thing. Imitation also has a social dimension. Children prefer to imitate people similar to themselves, and imitate ritual gestures with no practical effect, in order to fit into a group. The lesson for parents today: the old Chinese saying that example outweighs instruction never goes out of date. Rather than keeping your mouth on your child all day, do your own part well, build a healthy interaction with the child, and create ample conditions for them to take part in family and even social activities.

Learning by listening

This chapter is about how children learn through language, which is where humans far surpass other animals. Children listen to adults, but they do not swallow everything. They judge whether an adult is worth trusting, and that judgment rests on many factors: whether the speaker is confident, whether the speaker is an expert in the field, what the relationship with the speaker is. In short, trust has to be established. Children love to ask why, and most of the time they really want to get at how the world works. Often, asking directly is a more efficient way to learn than observing — to find out why it rains, say. We should not fob children off, and need not worry about the occasional question we cannot answer, because their appetite for inquiry is strong. Children extract bias from adult wording: “Japanese people are all bad” and “some Japanese people are bad” have completely different effects. What parents should do is not try to teach something, but keep a good interaction going, build a good relationship in conversation, answer more whys, and ask the child why too. Let the child talk with more people. This back-and-forth is real life, not something staged. Because children are sensitive, parents cannot keep pretending to be learned, pretending to be fair and objective; everything can only be what you actually feel.

Learning by playing

The effect of play is hard to measure, because once you measure it, it may no longer be play. It is not only humans: any animal with developed intelligence likes to play. One common form is rough-and-tumble play — puppies biting each other, children bumping into each other. This kind of play improves social skills. From what I see, most Chinese parents today forbid it; at the first hint of physical conflict between children, the adults are ready to start a fight themselves. Play is driven by curiosity. Children at play are somewhat like scientists running experiments: hypothesizing, verifying, exploring at random. Humans are better than other animals at pretend play. That is a capacity for counterfactual reasoning, for building a virtual world with some internal logic — literature has the same power. Children who are good at pretending understand other people’s minds better; many of them even have an imaginary friend, which often frightens adults. In a child’s play, parents should support rather than guide. Guidance tends to cut off the child’s own exploratory ideas and turn into dull, step-by-step imitation of the adult. What parents can actively do is create an environment with more things to play with — and that does not necessarily depend on money. A stick can be a fine toy. As for adults, once they drop their dignity, they may well enjoy joining the child’s game.

Learning by practicing

This chapter is mainly about school-age children and schooling. The very notion of school age is based on schooling, but children’s psychological development really does shift in this way: from learning mainly by exploration to learning mainly by mastery, laying a foundation for adult life. Mastery learning requires practice; it is a restructuring of the brain’s neurons, a process that turns a skill into something unconscious — and from another angle, a kind of regression. Today’s schooling came along with industrial production; it is not the human norm. Before it, earlier education was apprenticeship, and that model still dominates in some fields, such as sport and music. Schooling produces people who are good at school. The format of concentrated classroom teaching requires a high capacity for attention, and in reality, past and present, that capacity is not necessarily an advantage. Yet the modern American education system increasingly defines children who cannot concentrate for long as having ADHD, and China follows the same trend. On the spectrum of attention, every child sits somewhere different; this is just evolutionary diversity. The author argues that schools should adapt to all kinds of children, not medicate them. From an evolutionary view, adolescence is the human passage to adulthood; it concerns our motivational and control mechanisms. Adolescents are more affected by how peers judge them, and value rewards that calm adults find bizarre. Today’s children reach puberty earlier, but they are far behind children raised in the apprenticeship model in having real work experience and in being able to control their behavior and produce something genuinely intended. A child who wants to be a scientist cannot actually work alongside scientists until his twenties. Schooling, like family upbringing, should be the gardener model, but in reality it is more often the carpenter model.

Technology and your child’s future

Your child is a native of the information age, and you are only an immigrant. Watching my child, barely able to speak, operate an iPad with ease, I agree strongly with the author’s judgment. Today’s parents’ fear of technology is not a new fear. Socrates expressed deep worry about writing, the new technology of his day, and his worry came true: we rely more on written transmission than on oral, yet human culture was not destroyed — it flourished. Where technology will take the next generation, the only certain answer is that we do not know. Trying to control everything about your child and arrange his future life is absurd. Teenagers grow up in the new culture and face the same difficulties of human relations as before, or worse: they have to face the malice of strangers online. All of this sliding toward an unavoidable future is the fate of this generation. As parents, what we can do is neither control nor the so-called letting things be, but to serve as a bridge between past and future. The child grows up in tradition, draws nourishment from it, and will have his own innovations. We can keep reading with our children, but we can never cut them off from electronic devices, because that is what the world really looks like. Writing this, I feel a little sad. In the end the new waves of the Yangtze push the old ones along, and the old waves die on the beach. Every generation has its responsibility, its own grief and worry.

The meaning of raising children

Philosophically, the meaning of raising children cannot be defined, because the you before and the you after having a child are simply not the same person. A pluralistic set of values respects the choice to be child-free, but never pretend that what you have not had is not worth having. From the perspective of social development, the way society currently runs does harm to the parent-child relationship. Parents’ work has almost nothing to do with their children; society does not reward raising children; children become a burden in adult competition. In the close-knit society of the past, members of the community treated children well. In cold, independent city life none of that exists; there is only pressure on children to act mature too early. The merciless online attacks on “bratty kids” may start from the infantile behavior of some parents, but they also reflect a kind of discrimination against children across society. After the inevitable breakup of the close-knit society, government and society must take on the responsibility of raising children. Whether raising children or supporting the elderly, the whole society needs to rethink: beyond naked economic calculation, is there anything else to consider?

Being a parent is the art of finding balance among contradictions

I think no good parenting book can give a firm answer, and this book’s conclusion is no exception — because that is the truth. Every day I make demands on my child, soft or hard: maybe he disturbed someone, maybe I simply think some way is better for him. And every day I doubt whether my demands are reasonable, whether they are necessary, whether I am just imagining things. There are thousands, tens of thousands of parenting books that begin with “how”, and they usually hand you methods with certainty. But as long as you are capable of reflecting on yourself, you will never be that certain when facing your child. Even the most authoritarian parent will admit he wants his child to stand on his own one day, yet a small child really does need his parents’ help. This contradiction between dependence and growing independence is eternal and has no universal solution. My own sense: a child is not porcelain, he will not shatter because of one or two parental mistakes. What is frightening is the parents’ certainty — “this is definitely for your good”. Lack of reflection is the origin of every tragedy. A good parenting book like this one gives you a starting point for reflection, but when you finish it, everything has only just begun.