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Angels on a Pin---Alexander Calandra

Some time ago, I received a call from Jim, a colleague of mine, who teaches physics. He asked me if I would do him a favor and be the referee on the grading of an examination question. I said sure, but I did not quite understand why he should need my help. He told me that he was about to give a student a zero for his answer to a physics question, but the student protested that it wasn't fair. He insisted that he deserved a perfect score if the system were not set up against the student. Finally, they agreed to take the matter to an impartial instructor. And I was selected.

I went to my colleague's office and read the examination question. It said: "Show how it is possible to determine the height of a tall building with the aid of a barometer." The student had answered: "Take the barometer to the top of the building, tie a long rope to it, lower the barometer to the street, and then bring it up and measure the length of the rope. The length of the rope will be the height of the building."

I laughed and pointed out to my colleague that we must admit the student really had a pretty strong case for full credit since he had indeed answered the question completely and correctly. On the other hand, I could also see the dilemma because if full credit were given to him it could mean a high grade for the student in his physics course. A high grade is supposed to prove competence in the course, but the answer he gave did not show his knowledge on the subject. "So, what would you do if you were me?" Jim asked.

I suggested that the student have another try at answering the question. I was not surprised that my colleague agreed, but I was surprised that the student did, too.

I told the student that I would give him six minutes to answer the question. But I warned him that this time his answer should show some knowledge of physics. He sat down and picked up his pen. He appeared to be thinking hard. At the end of five minutes, however, I noticed that he had not put down a single word. I asked him if he wished to give up, but he said no. He had not written anything down because he had too many possible answers to this problem. He was just trying to decide which would be the best one. I excused myself for interrupting him and asked him to go on. In the next minute, he dashed off his answer, which read: "Take the barometer to the top of the building and lean over the edge of the roof. Drop the barometer and time its fall with a stopwatch. Then, using the formula S = 1 /2 at2, calculate the height of the building."

At this point, I asked my colleague if he would give up. He nodded yes, and I gave the student almost full credit.
When I left my colleague's office, I recalled that the student had said that he had other answers to the problem. I was curious, so I asked him what they were. "Oh, yes," said the student. "There are many ways of getting the height of a tall building with the aid of a barometer. For example, you could take the barometer out in a sunny day and measure the height of the barometer, the length of its shadow, and the length of the shadow of the building, and by the use of a simple proportion, determine the height of the building. The beauty of this method is that you don't have to drop the barometer and break it."

"Fine," I said. "Any more?"

"Yes," said the student. "There is a very basic measurement method that people will like, because it is so simple and direct. In this method, you take the barometer and walk up the stairs. As you climb the stairs, you mark off the length of the barometer along the wall. You then count the number of marks, and this will give you the height of the building in barometer units. The only trouble with this method is that it doesn't require much knowledge of physics."
"Of course, if you prefer a more sophisticated method, a method that will really show some knowledge of physics, you can tie the barometer to the end of a rope, swing it as a pendulum and determine the value of'g' at the street level and at the top of the building. From the difference between the two values of'g' the height of the building can, in principle, be worked out."

Finally, he concluded that while there are many ways of solving the problem, "Probably the best and the most practical in a real-life situation is to take the barometer to the basement and knock on the superintendent's door. When the superintendent answers, you speak to him as follows: Mr. Superintendent, I have here a fine barometer. If you will tell me the height of this building, I will gladly give you this barometer!"

At this point, I asked the student if he really didn't know the expected answer to this question. He smiled and admitted that he did, but said he was fed up with standard answers to standard questions. He couldn't understand why there should be so much emphasis on fixed rules rather than creative thinking. So he could not resist the temptation to play a little joke with the educational system, which had been thrown into such a panic by the successful launching of the Russian Sputnik. At that moment I suddenly remembered the question: How many angels can dance on the head of a pin? We teachers are always blaming the students for giving wrong answers. Perhaps we should ask ourselves whether we are always asking the right questions.

参考译文

针尖上的天使--亚历山大·卡兰德

不久前，我接到一个教物理的同事的电话，问我是否愿给一考题的打分作一个仲裁。我说可以，但我不明白为什么需要我帮忙。他说他要给一个学生做的物理题答案判零分，而学生认为这不公平，声称如果该评分体制的建立不是与学生作对的话，他就该得满分。导师和学生同意将此争议交给一位无私的裁决者，于是选择了我。

我去了那位同事的办公室，看了那道考题：“说明怎样才能用气压表来测定一栋高层建筑物的高度。”那考生答道：“把气压表拿到此楼顶，在表上系一根长绳，将气压表降至街上，然后提起来，测量绳子的长度。绳子的长度就是该楼房的高度。”

我大笑，告诉同事必须承认这个学生确实有得满分的强有力的证据，因为他完全正确地回答了问题。另一方面，如果给他满分，就可能陷入两难的处境，因为如果给该生满分，这意味着该生在物理课程取得了高分。而高分就应证明该生在物理学习方面的能力，但答案却无法证实这点。“那么，如果您是我，您会怎么做呢？”吉姆问我。

我建议这个学生再次尝试回答这个问题。我的同事同意了我的建议，对此我并不感到意外，令我感到意外的是，那个学生也同意了。

我给了他六分钟的时间回答这个问题，并提醒他必须用一些物理知识。他坐下来，拿起笔。显然在努力思考，但是五分钟过去了，他还一个字没写。我问他是否放弃，他说不。什么都没写是因为：他对这个问题有很多答案，他只是在想最佳答案。我道歉说打扰了他，并请他继续。在最后一分钟里，他匆匆写出了他的答案：把气压表拿到建筑物最顶层，倚着楼扔下表，用跑表计算其降落时间，然后用公式计算建筑高度。

这时，我问我的同事他是否愿放弃。他让步了，我给了那学生几乎一个满分。

在离开同事办公室的时候，我想起那个学生说他还有其它解法，便感到好奇。于是我问他还有哪些解法。“啊，有，”他说，“有很多种方法，都可以利用这只表得出一栋高楼的高度。例如，你可以在一个大晴天把它拿到户外并测量表高和影长，还有该大楼的影长，使用一个简单的比率，就可以测定该楼的高度。这个方法的好处是不用把气压计扔下去而摔碎。”

“好，”我说道“还有其它方法吗？”

“有，”那学生说道，“有一种人们会喜欢的非常基本的测量方法，因为它很简单直接，该方法是，你拿着气压表开始爬楼梯。在你爬楼梯时，你在墙上标出气压表的长度，然后再数这些记号，你会以气压表为单位得到楼高，一种非常直接的方法。这个方法的唯一缺点是用不了太多物理知识。”

“当然，如果你想用一种更复杂的、真正用物理学知识的方法，可以把表系到绳子的一端，让他像个钟摆一样，然后确定街面上和楼顶的摆值。通过两个摆值的差别，原则上可以算出楼高。”

最后，他推断道：“还有很多其它办法来解题。也许，现实生活中最好最实用的办法是把表拿到地下室并敲主管人的房间。等他开门后，你对他说：‘我这儿有一个精致的气压表。如果你告诉我这栋高楼的高度，我就把它送给你！”

这时，我问那个学生是否真不知道这个问题的常规答案。他笑了笑，承认他知道，但是他说他已厌倦了标准问题和标准答案，他不知道为什么总是强调那些固定的规则而不是创造性的思考。因此忍不住对那些对苏联人造卫星惊慌失措的美国课堂开个小玩笑。那一刻，我突然想起了一个问题：一根大头针上可以放几个跳舞的天使？我们老师们总是责备学生回答错误。或许我们应该反问自己，我们是否一直在寻求正确的答案？

词汇释义

1.academic adj. 学院的,理论的,学术性的

例句：
She already has good academic qualifications under her belt.
她已获得良好学历.

2.barometer n. 气压计,晴雨表

例句：
The barometer began to fall.
气压计的读数开始下降。

3.basement n. 地下室

例句：
In the basement water streams down the walls.
在地下室里，水顺着墙不断地流下来。

4.calculate v. 计算,考虑,计划

例句：
The scientists calculated when the spacecraft would reach the moon.
科学家推算出宇宙飞船抵达月球的时间。

5.deserve vi. 应该得到 vt. 应受,值得

例句：
You've been working all morning---you deserve a rest.
你已经干了一个上午了，该休息一下了。

6.dilemma n. 困境,进退两难

例句：
Faced with the dilemma, he opted for soldiering on.
面临着进退两难的困境，他决定顽强地坚持下去。

7.impartial adj. 公平的,不偏不倚的

例句：
People in the city held the impartial judge in high regard.
这个城市的人们都很尊敬这位公正的法官。

8.inner adj. 内部的,里面的,内心的 n. 里面,内部

例句：
He has no inner resources and hates being alone.
他没有内在的精神寄托，因而害怕孤独。

9.pendulum n. 摆,钟摆

例句：
The pendulum of the clock in my room is a bear.
我房间的钟摆是小熊形状的。

10.superintendent n. 监督人,管理人,所长

例句：
His brother is appointed superintendent of the security force.
他哥哥被任命为安全力量的负责人。

11.temptation n. 诱惑,引诱 v. 引,诱

例句：
She yielded to temptation and had another chocolate.
她禁不住诱惑, 又吃了一块巧克力.

词组解释

1.dash off vt. 匆忙完成(匆匆写)

例句：
She dashed off two letters in thirty minutes.
她在30分钟内写好了两封信。

2.with the aid of 借助于, 通过...的帮助

例句：
We may travel with the aid of a good map.
我们可以靠一幅好地图旅游。

3.point out v.指出

例句：
No matter who point out our shortcoming, we will correct them.
不管谁指出我们的缺点，我们都改正。

4.at the end of 在...结尾,在....末端

例句：
The town lies at the end of the valley.
这个镇坐落于河谷的尽头。

5.put down v.放下, 拒绝, 镇压, 羞辱, 削减, 记下, 制止, 取缔

例句：
Put down your address here.
请在这里写下你的地址。

6.mark off v.划分出

例句：
The boundaries are clearly marked off on the map.
边界的划分在地图上标得很清楚。

7.be fed up with 饱受, 厌烦

例句：
I'm fed up with all those nonsense.
所有那些胡言乱语我真听腻了。

8.emphasis on 强调

例句：
The teacher laid emphasis on the precision of the translation from the outset.
老师从一开始就强调翻译准确性。