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.
Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics. Students need to learn how to read mathematics, in the same way they learn how to read a novel or a poem, listen to music, or view a painting. Ed Rothstein’s book, Emblems of Mind, a fascinating book emphasizing the relationship between mathematics and music, touches implicitly on the reading protocols for mathematics.
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.
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.
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.
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.
Why is this idea true?
Do I really believe it?
Could I convince someone else that it is true?
Why didn't the author use a different argument?
Do I have a better argument or method of explaining the idea?
Why didn't the author explain it the way that I understand it?
Is my way wrong?
Do I really get the idea?
Am I missing some subtlety?
Did this author miss a subtlety?
If I can't understand the point, perhaps I can understand a similar
but simpler idea?
Which simpler idea?
Is it really necessary to understand this idea?
Can I accept this point without understanding the details of why it
Will my understanding of the whole story suffer from not
understanding why the point is true?
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.
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.
For example, consider the following theorem from Levi Ben Gershon’s manuscript Maaseh Hoshev (The Art of Calculation), written in 1321.
例如，考虑 Levi Ben Gershon 1321 年的 Maaseh Hoshev 手稿（计算的艺术）。
“When you add consecutive numbers starting with 1, and the number of numbers you add is odd, the result is equal to the product of the middle number among them times the last number.” It is natural for modern day mathematicians to write this as:
“当你从 1 开始连续的累加数值，累加奇数个，其结果等于中位数与最后一个数 的积。”现代数学写作:
A reader should take as much time to unravel the two-inch version as he would to unravel the two-sentence version. An example of Levi’s theorem is that 1 + 2 + 3 + 4 + 5 = 3×5.
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.
A well-written math text will be careful to use a word in one sense only, making a distinction, say, between combination and permutation (or arrangement). A strict mathematical definition might imply that “yellow rabid dog” and “rabid yellow dog” are different arrangements of words but the same combination of words. Most English speakers would disagree. This extreme precision is utterly foreign to most fiction and poetry writing, where using multiple words, synonyms, and varying descriptions is de rigueur.
In other words, the construct, when used correctly, is a signal to the reader that what’s involved here is perhaps tedious and even difficult, but involves no deep insights. The reader is then free to decide whether the level of understanding he/she desires requires going through the details or warrants saying “Okay, I’ll accept your word for it.”
Now, regardless of your opinion about whether that construct should be used in a particular situation, or whether authors always use it correctly, you should understand what it is supposed to mean. “It follows easily that” does not mean
but a person who doesn’t know the lingo might interpret the phrase in the wrong way, and feel frustrated. This is apart from the issue that one person’s tedious task is another person’s challenge, so the author must correctly judge the audience.
Lord, the Roman hyacinths are blooming in bowls and
The winter sun creeps by the snow hills;
The stubborn season has made stand.
My life is light, waiting for the death wind,
Like a feather on the back of my hand.
Dust in sunlight and memory in corners
Wait for the wind that chills towards the dead land.
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.
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.
To allow an opportunity to experiment with the guidelines presented here, I am including a small piece of mathematics often called the birthday paradox. The first part is a concise mathematical article explaining the problem and solving it. The second is an imaginary Reader’s attempt to understand the article by using the appropriate reading protocol. This article’s topic is probability and is accessible to a bright and motivated reader with no background at all.
A professor in a class of 30 random students offers to bet that there are at least two people in the class with the same birthday (month and day, but not necessarily year). Do you accept the bet? What if there were fewer people in the class? Would you bet then?
Assume that the birthdays of n people are uniformly distributed among 365 days of the year (assume no leap years for simplicity). We prove that, the probability that at least two of them have the same birthday (month and day) is equal to:
假设 n 个人的生日平均分布在一年的 365 天中（简单起见，不考虑闰年）。我 们证明，至少有两人生日相同的概率为：
What is the chance that among 30 random people in a room, there are at least two or more with the same birthday? For n = 30, the probability of at least one matching birthday is about 71%. This means that with 30 people in your class, the professor should win the bet 71 times out of 100 in the long run. It turns out that with 23 people, she should win about 50% of the time.
Here is the proof: Let P(n) be the probability in question. Let Q(n) = 1 – P(n) be the probability that no two people have a common birthday. Now calculate Q(n) by calculating the number of n birthdays without any duplicates and divide by the total number of n possible birthdays. Then solve for P(n).
The total number of n birthdays without duplicates is:
This is because there are 365 choices for the first birthday, 364 for the next and so on for n birthdays. The total number of n birthdays without any restriction is just 365nbecause there are 365 choices for each of n birthdays. Therefore, Q(n) equals
这因为第一个人有365种可能的选择，第二个人有 364 种可能，依次递推直至第 n 个生日。所有 n 种无限制的生日就是 365n。因此，Q(n) 等于
Solving for P(n) gives P(n) = 1 – Q(n) and hence our result.
根据 P(n) = 1 - Q(n) ，求得 P(n)，解之可得前述答案。
Our Reader Attempts to Understand the Birthday Paradox 阅读理解
In this section, a naive Reader tries to make sense out of the last few paragraphs. The Reader’s part is a metaphor for the Reader thinking out loud, and the Professional’s comments represent research on the Reader’s part. The appropriate protocols are centered and bold at various points in the narrative.
My Reader may seem to catch on to things relatively quickly. However, be assured that in reality a great deal of time passes between each of my Reader’s comments, and that I have left out many of the Reader’s remarks that explore dead-end ideas. To experience what the Reader experiences requires much more than just reading through his/her lines. Think of his/her part as an outline for your own efforts.
Reader (R): I don’t know anything about probability, can I still make it through?
Professional (P): Let’s give it a try. We may have to backtrack a lot at each step.
R: What does the phrase “30 random students” mean?
“任意的” 30 个学生是什么意思？
“When I use a word, it means just what I choose it to mean 当我使用一个词，它就是我为它选择的意思”
P: Good question. It doesn’t mean that we have 30 spacy or scatter-brained people. It means we should assume that the birthdays of these 30 people are independent of one another and that every birthday is equally likely for each person. The author writes this more technically a little further on: “Assume that the birthdays of n people are uniformly distributed among 365 days of the year.”
P: Yes the assumption is kind of obvious. The author is just setting the groundwork. The sentence guarantees that everything is normal and the solution does not involve some imaginitive fanciful science-fiction.
P: For example, the author is not looking for a solution like this: everyone lives in Independence Land and is born on the 4th of July, so the chance of two or more people with the same birthday is 100%. That is not the kind of solution mathematicians enjoy. Incidentally, the assumption also implies that we do not count leap years. In particular, nobody in this problem is born on February 29. Continue reading.
R: Oh, I see. I can pair up each top term with each bottom term, and do 365/365 as the first term, then multiply by 364/365, and so on for 30 terms. This way the product never gets too big for my calculator. (After a few minutes)... Okay, I got 0.29368, rounded to 5 places.
R: So that’s why I should read mathematics, to make a few extra bucks?
P: I see how that might give you some incentive, but I hope the mathematics also inspires you without the monetary prospects.
R: I wonder what the answer is for other values of n. I will try some more calculations.
我对 n 为其它值的答案很有兴趣，我想再多算几个。
P: That’s a good idea. We can even make a picture out of all your calculations. We could plot a graph of the number of people versus the chance that a duplicate birthday occurs, but maybe this can be left for another time.
R: Oh look, the author did some calculations for me. He says that for n = 30 the answer is about 71%; that’s what I calculated too. And, for n = 23 it’s about 50%. Does that make sense? I guess it does. The more people there are, the greater the chance of a common birthday. Hey, I am anticipating the author. Pretty good. Okay, let’s go on.
R: It seems that we are up to the proof. This must explain why that formula works. What’s this Q(n)? I guess that P stands for probability but what does Q stand for?
看起来我们做出了证明。我们一定要弄清公式的原理。什么是 Q(n) ？我猜 P 代表概率的意思，不过 Q 代表什么？
P: The author is defining something new. He is using Q just because it’s the next letter after P, but Q(n) is also a probability, and closely related to P(n). It’s time to take a minute to think. What is Q(n) and why is it equal to 1 – P(n)?
R: Q(n) is the probability that no two people have the same birthday. Why does the author care about that? Don’t we want the probability that at least two have the same birthday?
Q(n) 是没有生日相同的概率，为什么作者强调这个？我们不能去考虑至少两个 人在同一个生日的概率？
P: Good point. The author doesn’t tell you this explicitly, but between the lines, you can infer that he has no clue how to calculate P(n) directly. Instead, he introduces Q(n) which supposedly equals 1 – P(n). Presumably, the author will proceed next to tell us how to compute Q(n). By the way, when you finish this article, you may want to deal with the problem of calculating P(n) directly. That’s a perfect follow up to the ideas presented here.
P: Yes, he does, but what’s worse, he doesn’t even tell us that it is obvious. Here’s a rule of thumb: when an author says clearly this is true or this is obvious, then take 15 minutes to convince yourself it is true. If an author doesn’t even bother to say this, but just implies it, take a little longer.
R: It’s 2/3, because the chance of something not happening is the opposite of the chance of it happening.
是 2/3 。因为没发生某事的机率是发生此事件的机率的互斥数。
Make the Idea Your Own 建立你自己的思想
P: Well, you should be careful when you say things like opposite, but you are right. In fact, you have discovered one of the first rules taught in a course on probability. Namely, that the probability that something will not occur is 1 minus the probability that it will occur. Now go on to the next paragraph.
R: It seems to be explaining why Q(n) is equal to long complex-looking formula shown. I will never understand this.
看来这解释了为什么 Q(n) 等于看起来这么复杂的公式。我可能永远也理解不了。
P: The formula for Q(n) is tough to understand and the author is counting on your diligence, persistence, and/or background here to get you through.
R: He seems to be counting all possibilities of something and dividing by the total possibilities, whatever that means. I have no idea why.
P: Maybe I can fill you in here on some background before you try to check out any more details. The probability of the occurrence of a particular type of outcome is defined in mathematics to be: the total number of possible ways that type of outcome can occur divided by the total number of possible outcomes. For example, the probability that you throw a four when throwing a die is 1/6. Because there is one possible 4, and there are six possible outcomes. What’s the probability you throw a four or a three?
R: Well I guess 2/6 (or 1/3) because the total number of outcomes is still six but I have two possible outcomes that work.
我猜是 2/6 （即 1/3）。因为所有可能结果的总数是6，但是我有两种可能的结 果。
P: Good. Here’s a harder example. What about the chance of throwing a sum of four when you roll two dice? There are three ways to get a four (1-3, 2-2, 3-1) while the total number of possible outcomes is 36. That is 3/36 or 1/12. Look at the following 6 by 6 table and convince yourself.
R: That’s a relief. Well, I know what you mean now. To answer your question, I can get a seven in six ways via 1-6, 2-5, 3-4, 4-3, 5-2, or 6-1. The total number of outcomes is still 36, so I get 6/36 or 1/6. That’s weird, why isn’t the chance of rolling a 4 the same as for rolling a 7?
P: Because not every sum is equally likely. The situation would be very different if we were simply spinning a wheel with the sums 2 through 12 listed in equally spaced intervals. In that case, each one of the 11 sums would have probability 1/11.
P: There’s a lot of information implied in a small bit of mathematics. Yes, the author expected you to know all this, or to discover it yourself just as we have done. If I hadn’t been here, you would have had to ask yourself these questions and answer them by thinking, looking in a reference book, or consulting a friend.
The total number here is 365 × 364 since each row now has 364 pairs instead of 365.
总共有 365 x 364 种，因为现在每行有 364 对而不是 365。
P: Good. You are going a little quickly here, but you’re 100% right. Can you generalize now to 30? What is the total number of possible sets of 30 birthdays? Take a guess. You’re getting good at this.
很好。这次你做的很快，不过完全弄对了。30 人生日的总的样本数是多少？猜 猜看，你很擅长这个的。
R: Well if I had to guess, (it’s not really a guess, after all, I already know the formula), I would say that for 30 people you get 365 × 365 ×... × 365, 30 times, for the total number of possible sets of birthdays.
OK让我猜一下，（也不算真的猜，毕竟我知道公式），30 人生日的所有可能样 本是 365 x 365 x 365 ... x365，共 30 次。
P: Exactly. Mathematicians write 36530. And what is the number of possible sets of 30 birthdays without any duplicates?
很好，数学家会将其写做 36530。那么 30 个不重复的生日 组合会是多少？
R: I know the answer should be 365 × 364 × 363 × 362 × ... × 336, (that is, start at 365 and multiply by one less for 30 times), but I am not sure I really see why this is true. Perhaps I should do the case with three people first, and work my way up to 30?