Introduction1

I once asked an auditorium of undergraduates to recall theirearliest memories of math. One of them shared a scene so peculiar-and yet souniversal-that it burrowed deep into my unconscious, to the point where it hasbegun to feel like a memory of my own.

我曾经让一个大学生听众回忆他们最早的数学记忆。其中一个分享了一个如此奇特但又如此普遍的场景，以至于它深深地钻入了我的潜意识，以至于它开始感觉像是我自己的记忆。

At age five, this student was assigned worksheets of additionproblems. Trouble was, she didn't know how to read the funny symbols on thepages, the 2's and +'s and such. No one had ever taught her. Too intimidated toask, she found a work-around, memorizing each sum not as a fact about numbers,but as an arbitrary rule about shapes. For example, 8 + 1 = 9 was not thestatement that 9 is one more than 8, but a coded set of instructions; if youare shown two stacked circles (8), followed by a cross (+), a vertical line(1), and a pair of horizontal lines (=), then you must fill in the blank spaceprovided by writing a circle with a downward-curling tail (9). With painstakingdiligence, she taught herself dozens of rules like this, each as baroque andpointless as the last. It was mathematics by way of Kafka.

五岁时，这个学生被分配了加法题的工作表。问题是，她不知道如何阅读页面上的有趣符号，2 和 + 之类的。从来没有人教过她。她不敢问，于是找到了一个解决方法，记住每个总和不是关于数字的事实，而是关于形状的武断规则。例如，8 + 1 = 9 不是 9 比 8 多 1 的陈述，而是一组编码的指令;如果显示两个堆叠的圆圈 （8），后跟一个叉号 （+）、一条垂直线 （1） 和一对水平线 （=），则必须通过书写一个带有向下卷曲尾巴的圆圈 （9） 来填充提供的空白区域。她煞费苦心地自学了几十条这样的规则，每一条都和上一条一样巴洛克式而毫无意义。这是卡夫卡的数学。

Few people learn 8 +1 = 9 this way. But sooner or later,almost every math student suffers a similar sense of confusion and resorts tosimilarly desperate work-arounds. Whether in preschool, middle school, or gradschool, befuddlement eventually descends, and mathematics becomes, in the wordsof mathematician David Hilbert, a game of "meaningless marks onpaper."

很少有人以这种方式学习 8 +1 = 9。但迟早，几乎每个数学学生都会遭受类似的困惑感，并求助于同样绝望的解决方法。无论是在学前班、初中还是研究生院，困惑最终都会降临，用数学家大卫·希尔伯特 （David Hilbert） 的话来说，数学变成了一场“纸上无意义的标记”游戏。