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并非搞笑的回答  

2012-02-08 22:36:14|  分类: 一葉知秋 |  标签: |举报 |字号 订阅

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孩子做数学测试卷遇到这样一道题:“从不同方向观察同一物体,最多能同时看到物体的()个面?”——答案:3

孩子有疑问,找我讨论,我也开始怀疑了。

如果这个物体是长方体,我们只能看到正面、侧面、上(或下)面。但是,如果这个物体不是长方体呢?比如说长方体的一个角被一刀削去,多出来一个面,那我们应该就能看四个面了。再比如说,正八面体(见下图),我们可以看到四个面;正十二面体可以看到六个面;正二十面体可以看到十个面;钻石那种更多面的就更不用说了。

 

并非搞笑的回答 - paths - 微覩著秋       

正四面体    正六面体  正八面体  正十二面体  正二十面体

 

我不解,“物体”既然没有指明是长方体,何以最多只能看到三个面呢?

这个问题有另一种问法:“从不同的位置观察同一个立方体,最多可以看到几个面?”网上有不同的回答。有网友说:五个面。大概这个立方体放置在桌面,而且桌面又不透明。既然你可以从“不同的位置”观察,除底面外,那个面都能看到,那“最多”自然就是五个面。桌面若透明,底面也看得到,那就是六个面。

网友说,十二个面。如果立方体是纸箱,除了外面六个面,里面还有六个面。

    我倾向于把这题的问法改变一下:把笼统的“物体”改成具体的几面体;把“从不同的位置(方向)”改成“从一个位置(方向)”,这样,或许能更精确而无歧义了。

 

附:Angels on a Pin 及汉译:

Angels On A Pin  by Alexander Callandra

Some time ago, I received a call from a colleague who asked if I would be the referee on the grading of an examination question. He was about to give a student a zero for his answer to a physics question, while the student claimed he should receive a perfect score and would if the system were not set up against the student. The instructor and the student agreed to submit this to an impartial arbiter, and I was selected.

I went to my colleague’s office and read the examination question:” 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, attach a long rope to it, lower the barometer to the street, and then bring it up, measuring the length of the rope. The length of the rope is the height of the building.”

I pointed out that the student really had a strange case for full credit, since he had answered the question completely and correctly. On the other hand, if full credit were given, it could well contribute to a high grade for the student in his physics course. A high grade is supposed to certify competence in physics, but the answer did not confirm this. 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.

I gave the student six minutes to answer the question, with the warning that his answer should show some knowledge of physics. At the end of five minutes, he had not written anything. I asked if he wished to give up, but he said no. He had many answers to this problem; he was just thinking of the best one. I excused myself for interrupting him, and asked him to please 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, timing its fall with a stopwatch. Then, using the formula S=1/2at?, calculate the height of the building.”

At this point, I asked my colleague if he would give up. He conceded, and I gave the student almost full credit.

In leaving my colleague’s office, I recalled that the student had said he had other answers to the problem, 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 on 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.”

“Fine,” I said. “And the others?”

“Yes,” said the student. “There is a very basic measurement method that you will like. In this method, you take the barometer and begin to 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. A very direct method.”

“Of course, if you want a more sophisticated method, you can tie the barometer to the end of a string, 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 calculated.”

Finally he concluded, there are many other ways of solving the problem. “Probably the best, “he said, “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, here I have a fine barometer. If you will tell me the height of this building, I will give you this barometer.”

As this point, I asked the student if he really did not know the conventional answer to his question. He admitted that he did, but said that he was fed up with high school and college instructors trying to teach him how to think, to use the “scientific method”, and to explore the deep inner logic of the subject in a pedantic way, as is often done in the new mathematics, rather tan teaching him the structure of the subject. With this in mind, he decided to revive scholasticism as an academic lark to challenge the Sputnik-panicked classrooms of America.


前不久,我接到一位同事的电话,问我是否愿意充当一名评判员,对有争议考题进行评判。他打算给一名学生的物理答卷打零分,而那位学生则声称如果考试制度不是用来对付学生的,他应该得高分。教员和学生都同意把这个问题交给一位公证的仲裁人,所以选中了我。 

我来到同事的办公室,查阅了考题:论证强何用气压表算出一座大楼的高度?” 

学生的答案是:把气压表拿到顶楼,拴上一条长绳,把气压表垂直落至街面,然后把它拉上来,测量绳子的长度。这个长度即为这幢高楼的高度。” 

我指出,既然这位学生完整正确地回答了这个问题,那么他完全有理由得满分。另一方面,如果给他打了满分,他的物理这门课就能得高分。高分应该能证明物理方面的能力,但他的回答却不能证明。我建议让这个学生再答一次。我的同事接受了这一建议,我并不感到惊讶,可让我惊讶的是那位学生也同意了。


我给这个学生6分钟时间来回答这个问题,并告诫他必须运用物理知识。5分钟过去了,这个学生一直未动笔。我问他是否想放弃。他说不。他说对于这个问题,他还有多种解法,他正在考虑用最好的答案。我请他原谅我打断了他的思考,请他继续。在最后一分钟里,他匆匆写出了如下答案:把气压表拿到大楼的顶楼,靠在楼顶边上,让气压表落下,用一块跑表测出其下落时间,然后用公式S12 at2,从而计算出大楼的高度。” 

我问同事是否愿意做出让步,他让步了。我给这个学生几乎打了满分。 我正准备离开办公室的时侯,突然回想起这个学生说他就这一考题还有其它的解法,所以我又回过头来问他。他说:是的,借助于气压表还有许多种方法可以得到这幢大楼的高 度。比如说,你可以在一个阳光灿烂的日子里,测量这个气压表的高度、气压表阴影的长度、以及大楼阴影的长度,然后用一个简单的比例就可以得出大楼的高度。” 

“很好”,我说, “还有其他方法吗? 

“当然。”这位学生说,“还有一种你会喜欢的最基本的方法:你拿着气压表上楼,一边爬楼,一边以气压表为长度单位在墙上标出,然后数一下记号个数,就可得出大楼的高度。这是一种非常直接的方法”


  “当然,如果你想用一种更复杂的方法的话,你可以把这个气压表系在一根细绳的一端,让它像钟摆一样摆动,然后用它测出其在地面上和楼顶的g值。根据这两值之差,按公式就可计算出大楼的高度。” 

最后他总结道,解决这个问题还有其他很多方法。他说:也许其中最好的一种,就是拿着气压表到地下室去敲大楼管理员的门。在他开门时,你这样说:管理员先生,我这儿有一个上好的气压表,如果你告诉我这幢大楼的高度,这个气压表 就给你了。” 

说到这里,我问这学生他是不是真不知道这道试题的常规答案。他说他知道,但他讨厌高中和大学的老师总是试图教导他们如何思考、如何运用科学的方法,用学究式的方式来探讨事物内在的逻辑性。出于这种想法,他决心再用学究主义作为学术恶作剧,来向因苏联人造卫星而惊慌失措的美国课堂教学挑战。 

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