Estimation and Fermi QuestionsDate: 04/10/2002 at 13:10:05 From: Kenneth S. Kwan Subject: Very general estimation question Dear Dr. Math, I'm currently learning about Estimation techniques similar to those used by the famous scientist Enrico Fermi, who proposed the question, "How many piano tuners are there in Chicago?" The question I invented was the following: If there are 12,000 students who attend a certain college, how many professors are employed by the college? I came up with the following estimation. Can you tell me if my reasoning is reasonable? Thanks very much! 1) The average professor teaches about 1 hour a day, so in one week (Monday - Friday) he teaches 5 hours. 2) Each class takes about an hour, so in one week he teaches 5 classes. 3) But he doesn't teach 5 different classes in one week; the same classes are held 2-3 times a week (either Monday-Wednesday-Friday classes or Tuesday-Thursday classes). To make it simpler, let's say a professor teaches each class 2 times a week (assume only Tuesday- Thursday classes exist). Therefore, he sees the same class 2 times a week, meaning every half week he sees the same students, but that also means every week he only sees the same students because the classes repeat. 4) If a professor teaches for 2.5 hours per half week (5 hours per week), where each class takes about an hour, and assuming that a typical class consists of 50 students, then he sees 2.5 classes x 50 students = 125 students per week. 5) Since there are 12,000 students for all professors to lecture, then there are probably 12,000 / 125 = close to 100 professors on campus. From a scale of 1-10, how would you rate this estimation? Any suggestions or comments are greatly appreciated. Cordially, Kenneth S. Kwan Date: 04/10/2002 at 17:03:38 From: Doctor Peterson Subject: Re: Very general estimation question Hi, Kenneth. It seems to me that you have left out one important factor: each student attends more than one class. This can be tricky to describe clearly: in your model each class has 50 students, and each student has, say, 5 classes. So is it 50 students per class, or 5 classes per student? One way to avoid this trouble is to think of a specific name for the relation of a student to a class. One I've thought of is "seat." Each class has 50 seats (students in the class); each student has 5 seats (in different classes). You can diagram this: Student ------> Seat <------ Class <------ Prof 1 5 50 1 2.5 1 Looking for more on Fermi questions, I ran across these pages which may be of interest: Classic Fermi Questions with annotated solutions - Sheila Talamo http://mathforum.org/workshops/sum96/interdisc/classicfermi.html How To Solve Fermi Questions - Norman Rothery http://aries.phys.yorku.ca/~rothery/fermi/fermi.faq.html But I didn't find your question among these. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/10/2002 at 19:56:01 From: Kenneth S. Kwan Subject: Very general estimation question Thanks very much Doctor Peterson for answering my "Fermi" question, but now I have a question about your answer: The diagram you gave me was: Student ------> Seat <------ Class <------ Prof 1 5 50 1 2.5 1 I'll understand that as: Each student has 5 seats. There are 50 seats in one class. But does that mean each class holds only 10 students? I'm not sure I understand this concept. And I thought that the total number of students (12,000) attending the college would matter in the estimation. Can you explain this further? Thanks very much! Cordially, Kenneth Date: 04/10/2002 at 22:28:44 From: Doctor Peterson Subject: Re: Very general estimation question Hi, Kenneth. Maybe you can get a better understanding if you try to tell me what role the number of classes each student takes should play in your estimate. It takes a bit of wrestling on your own before you can quite pin this idea down. Obviously I didn't say that each class has only 10 students; but if each student took only one class of ten students, you would need the same number of professors, so in a sense it is equivalent to that. Or, instead of 12,000 students taking 5 courses each, you could have 60,000 students taking one course each; that would require the same number of professors, which is five times as many as you estimated. Does that help? And I didn't say the number of students doesn't matter; my diagram (a variety of Entity Relationhip Diagram) only shows relative numbers, not absolute numbers. It says that for each student there will be 5 seats (in different classes), each of which is 1/50 of a class, each of which needs 1/2.5 of a professor, so 12,000 students will need 5 seats 1 class 1 professor 12,000 students * --------- * -------- * ----------- = 480 professors 1 student 50 seats 2.5 classes There's a lot of useful math and logic in this question! - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ Date: 04/11/2002 at 14:43:16 From: Kenneth S. Kwan Subject: Very general estimation question Dear Dr. Peterson, Thanks very much, I understand your solution. One last general question on how I can sort of think the way you do, so to speak. What you said about how I missed a link between the students and the class, where we're not sure if each student has 5 classes or if each class has 50 students, makes me ponder about how you realized and picked that up. Have you worked with problems such as these repeatedly, so you know all the nitty gritty details? Or is it because I'm illogical or have less ability to reason that I totally neglected that fact. If so, is there a particular math course I can take to help open up my mind to incorporate more reason and logic into my mathematical thinking? I feel incompetent when I see other students being able to answer such questions while I'm struggling. Can you give some advice? Thank you once again, Dr. Peterson! Cordially, Kenneth Date: 04/11/2002 at 16:17:42 From: Doctor Peterson Subject: Re: Very general estimation question An interesting question! I suspect a lot comes from experience - and that experience is probably what Fermi was trying to develop with his questions. Logical ability is not just something you are born with (though it may be that some of us naturally gravitate toward it, and therefore develop the skills without having to be forced into it); I think everyone has to develop it by practice. In this case, as with the piano tuner problem, a lot of the thinking needed depends on specific knowledge of the subject matter. You have to picture a university and know that there are classes and professors and so on, or picture a piano tuner's job and see that he will do more than one piano a day, and they will be in different homes, and so on. In this case, I just thought about what factors would play a role, saw that each student would be in several classes, and expected to see that somewhere in your analysis. When I didn't, a red flag went up. No logic, just visualization. But I probably would have noticed it anyway, by going through your presentation in order and falling off the end when you didn't mention what each student does. Maybe you can call that "follow-through" - you can't stop your logic just because you've made contact with the goal, but have to keep thinking until you can't think any further. You did fine until then; you just didn't take it all the way. Assuming that "problem-domain" knowledge, you have to be able to think through the problem, both in a straight line (each professor teaches N classes; in each class there are N students; ...), and also sometimes coming at it from all sides, just brainstorming to think of all the relevant factors. The former requires the ability to stay focused and think in an orderly way; the latter requires defocusing and letting wild ideas come in. Both have their place, and some of us are probably better at one than the other. In this case, to find my own answer, I just went through it sequentially, starting at the student (since I knew the hard part was at that end). Now I happen to have an advantage over you in doing a Fermi problem, one that I haven't seen mentioned in discussions of them. I am a computer programmer, and part of my work involves designing relational databases, where you might have one table listing all the students, another listing all the classes, and so on. That's where my Entity Relationship Diagram came from - it's a tool used to see how these tables relate to one another, and design additional tables that capture the information needed for these relations. While I was thinking about how to explain the "5 classes per student and 50 students per class" problem, that method popped into my mind - I suppose that is an example of the non-linear type of thinking, pulling a tool out of my toolbox because the kind of thinking I was doing reminded me of it, not because it was the next thing to think of. That's not essential for this kind of problem, but it can always be handy. Speaking of databases, if I had actually been trying to design one, I would have based my thinking on the paperwork involved in a university. Each of the 12,000 students has a course schedule, listing his or her 5 subjects. Each of those 60,000 subject lines corresponds to one seat in one class; 50 of them together form one of 1200 classes. Each professor has a schedule listing 2 or 3 courses, among which the 1200 classes will be found. In the database, you would add a table containing the information from all the students' schedules, which relates students to classes. (That's why it's called a "relational" database.) This image of actual pieces of paper can often make the abstract ideas of numerical ratios more concrete, and help in logical thinking. Where can you build these skills? Many math classes will incorporate them implicitly, in geometric proofs or word problems, for example. Other fields need it too, from Fermi's physics to law school. There are all sorts of puzzles you can find in books or elsewhere. I went back to the Web to look for articles on Fermi problems that go a little deeper in talking about the kind of thinking involved. This is a long discussion that seems very helpful: Fermi Problems - Discovery Learning Project http://www.ph.utexas.edu/~gleeson/httb/section1_3_3_5.html This looks good, too: 'Back-of-the-Envelope' Calculations - Paul Francis http://msowww.anu.edu.au/~pfrancis/astr1001/approx/ - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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