How to Read Mathematics

Table of Contents

link: http://www.people.vcu.edu/~dcranston/490/handouts/math-read.html

1 Reading Protocol

reading protocol就是指reader/writer之间建立好的协议/默契,这点很重要。对于不同的文体/内容/文章,protocol是不同的,因为侧重点是不一样的。小说并不强调用词的精简和准确,更加着重用词是否可以勾勒出画面,这点和科技文章不同。而科幻小说与非科幻小说也有很大的差别,protocol也不尽相同。protocol还强调作者已经有某些基础和背景知识,这样交流起来就可以更加关注于高层次和深层次的内容。

A reading protocol is a set of strategies that a reader must use in order to benefit fully from reading the text. Poetry calls for a different set of strategies than fiction, and fiction a different set than non-fiction. It would be ridiculous to read fiction and ask oneself what is the author's source for the assertion that the hero is blond and tanned; it would be wrong to read non-fiction and not ask such a question. This reading protocol extends to a viewing or listening protocol in art and music. Indeed, much of the introductory course material in literature, music and art is spent teaching these protocols.

小说的reading protocol以及阅读体验是下面这样的。小说家关注说如何选择一个个故事,然后将这些故事联系起来,不断地勾勒出每个人物和整个氛围。

When we read a novel we become absorbed in the plot and characters. We try to follow the various plot lines and how each affects the development of the characters. We make sure that the characters become real people to us, both those we admire and those we despise. We do not stop at every word, but imagine the words as brushstrokes in a painting. Even if we are not familiar with a particular word, we can still see the whole picture. We rarely stop to think about individual phrases and sentences. Instead, we let the novel sweep us along with its flow and carry us swiftly to the end. The experience is rewarding, relaxing and thought provoking.

Novelists frequently describe characters by involving them in well-chosen anecdotes, rather than by describing them by well-chosen adjectives. They portray one aspect, then another, then the first again in a new light and so on, as the whole picture grows and comes more and more into focus. This is the way to communicate complex thoughts that defy precise definition.

数学则强调用简洁的语言,将事物背后的复杂性,准确地表达出来。所以数学语言要求简洁准确,所以阅读的时候一定要全神贯注。阅读小说的时候可以分心并不影响,但是阅读数学文章则不行。

Mathematical ideas are by nature precise and well defined, so that a precise description is possible in a very short space. Both a mathematics article and a novel are telling a story and developing complex ideas, but a math article does the job with a tiny fraction of the words and symbols of those used in a novel. The beauty in a novel is in the aesthetic way it uses language to evoke emotions and present themes which defy precise definition. The beauty in a mathematics article is in the elegant efficient way it concisely describes precise ideas of great complexity.

2 Common Mistakes

2.1 Don’t Miss the Big Picture 深入浅出

“Reading Mathematics is not at all a linear experience …Understanding the text requires cross references, scanning, pausing and revisiting” 2

Don’t assume that understanding each phrase, will enable you to understand the whole idea. This is like trying to see a portrait painting by staring at each square inch of it from the distance of your nose. You will see the detail, texture and color but miss the portrait completely. A math article tells a story. Try to see what the story is before you delve into the details. You can go in for a closer look once you have built a framework of understanding. Do this just as you might reread a novel.

2.2 Don’t be a Passive Reader 参与其中

带入一些例子,或者是重新推导,又或者是故意给作者找茬

A math article usually tells only a small piece of a much larger and longer story. The author usually spends months discovering things, and going down blind alleys. At the end, he organizes it all into a story that covers up all the mistakes (and related motivation), and presents the completed idea in clean neat flow. The way to really understand the idea is to re-create what the author left out. Read between the lines.

Putting too little effort into this participation is like reading a novel without concentrating. After half an hour, you wake up to realize the pages have turned, but you have been daydreaming and don’t remember a thing you read.

2.3 Don’t Read Too Fast 放慢速度

阅读速度和个人对这方面的熟悉程度有关。另外如果刻意强调速度的话,可能会让你受挫最终让你没有动力。所以掌握好速度和深度是很有必要的。

Reading mathematics too quickly results in frustration. A half hour of concentration in a novel might net the average reader 20-60 pages with full comprehension, depending on the novel and the experience of the reader. The same half hour in a math article buys you 0-10 lines depending on the article and how experienced you are at reading mathematics. There is no substitute for work and time. You can speed up your math reading skill by practicing, but be careful. Like any skill, trying too much too fast can set you back and kill your motivation. Imagine trying to do an hour of high-energy aerobics if you have not worked out in two years. You may make it through the first class, but you are not likely to come back. The frustration from seeing the experienced class members effortlessly do twice as much as you, while you moan the whole next day from soreness, is too much to take.

2.4 Make the Idea your Own 消化吸收

用自己的语言描述出来,讲给其他人听,确认你完全吸收了这个idea. 费曼学习技巧

The best way to understand what you are reading is to make the idea your own. This means following the idea back to its origin, and rediscovering it for yourself. Mathematicians often say that to understand something you must first read it, then write it down in your own words, then teach it to someone else. Everyone has a different set of tools and a different level of “chunking up” complicated ideas. Make the idea fit in with your own perspective and experience.

2.5 Know Thyself

Know yourself. 了解自己是否有足够的背景知识。

Most math books are written with assumptions about the audience: that they know certain things, that they have a certain level of “mathematical maturity,” etc. Before you start to read, make sure you know what the author expects you to know.

文章里面作者以“T.S.Eliot"的一首诗为例,说明了解背景知识的必要性(readig protocol)

  • 诗里面的名词
  • 诗想要描绘出的意境
  • 诗在格式上的优美之处

For example, Eliot’s poem pretty much assumes that its readers are going to either know who Simeon was or be willing to find out. It also assumes that its reader will be somewhat experienced in reading poetry and/or is willing to work to gain such experience. He assumes that they will either know or investigate the allusions here. This goes beyond knowledge of things like who Simeon was. For example, why are the hyacinths “Roman?” Why is that important?

Elliot assumes that the reader will read slowly and pay attention to the images: he juxtaposes dust and memory, relates old age to winter, compares waiting for death with a feather on the back of the hand, etc. He assumes that the reader will recognize this as poetry; in a way, he's assuming that the reader is familiar with a whole poetic tradition. The reader is supposed to notice that alternate lines rhyme, but that the others do not, and so on.

Most of all, he assumes that the reader will read not only with the mind, but also with his/her emotions and imagination, allowing the images to summon up this old man, tired of life but hanging on, waiting expectantly for some crucial event, for something to happen.