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This reminds me of a problem that was on the entrance examination to the University of Tokyo in 2003: “Prove that the ratio of a circle’s circumference to its d

This reminds me of a problem that was on the entrance examination to the University of Tokyo in 2003: “Prove that the ratio of a circle’s circumference to its diameter is greater than 3.05.” The problem was believed to have been posed as a protest against the dumbing down of the government-mandated curriculum, as elementary school children in Japan were being taught to use 3 as a substitute for pi in some calculations [1]. The elegance of the problem is that test-takers were unlikely ever to have encountered it before and that various creative solutions are, it seems, possible. [1] https://ja.wikipedia.org/wiki/円周率は3 https://ja.wikipedia.org/wiki/円周率は3

koolba 6y ago
What disgrace if a math program would use 3 as an estimate for pi? Not even 22/7?!
- wizzwizz4 6y ago
  3 is no more of a disgrace than 22/7, in my book.
  - angelgcuartero 6y ago
    Wait... --> https://en.wikipedia.org/wiki/Indiana_Pi_Bill https://en.wikipedia.org/wiki/Indiana_Pi_Bill
- TrackerFF 6y ago
  I get the feeling that using 3 for pi, is mostly beneficial when speed and volume/throughput is more important than precision. I.e - doing 100 problems with Pi = 3 probably doesn't even require a calculator, whereas doing the same problems with a decimal approximation will require students to pull out their calculators, and therefore students can do more problems in a shorter time, using pi = 3. This would certainly make sense in countries where rote learning is the king, and students are pretty much churning through problems. Or, it could be that the vast majority of people will never need the use of an accurate pi value, and it's more beneficial to make them remember some easy value like 3, rather than a confusing decimal or fraction. The people that are going to use pi in any serious manner, will obviously know its value and derivation.
deleted 6y ago
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908B64B197 6y ago
> as elementary school children in Japan were being taught to use 3 as a substitute for pi in some calculations That's robbing them of getting an (intuitive) understanding of a whole class of numbers. And limits too!
- ajsnigrutin 6y ago
  pi=3 is a totally usefull approximation for when the students are old enough to know, when pi=3 is good enough. Definitely not elementary school.
  - tkgally 6y ago
    According to the Japanese Wikipedia article I cited above, children were not taught that pi equals 3; rather, children who had not yet learned how to calculate with decimals were told that they could use 3 as a rough approximation for pi when calculating by hand. This minor change in the official curriculum came at a time when the Japanese government was trying to de-emphasize rote memorization and promote critical thinking in school education. Because that policy also reduced class hours and the amount of material covered, it was widely criticized for supposedly dumbing down school education. The notion that children were being taught “pi = 3” was an exaggeration, but it resonated with other objections to educational policy and apparently led the university to pose that problem on its entrance exam in protest. I still think it's an interesting problem. I wonder how far I could have gotten with it if, at the age of 18 or 19, I had encountered it on an exam and had just twenty or thirty minutes to try to come up with a proof.
    - 908B64B197 6y ago
      Ah! That's more nuanced. Sadly I do not speak Japanese so I didn't read the original source. > I wonder how far I could have gotten with it if, at the age of 18 or 19, I had encountered it on an exam and had just twenty or thirty minutes to try to come up with a proof. It's a nice one. Especially because since pi shows up in so many places there are multiple ways of doing it.
gizmo686 6y ago
I wonder why they chose 3.05 instead of just 3. The best I could come up with is related to a student using the inscribed n-gon method for computing a lower bound. The two obvious choices for an inscribed n-gon are a 6-gon and a 12-gon. Computing their perimeters involves computing cos(pi/3) or cos(pi/6) respectivly. Both of these are generally found the same way. By bisecting an equalateral triangle, you get a right angle triangle with angles pi/3 and pi/6, from which sin/cos/tan can easily be read. Once you have the relevent cos, it is straightforward trig and algebra to find With a 6-gon: pi>3 With a 12-gon: pi>6sqrt(2-sqrt(3)) Using a calculator, the 12-gon lower bound is pi>3.1058... Computing by hand (which I did), you end up having enough wiggle room to get away with some further approximations; but still need to go through some somewhat tedious computation. Maybe they were concerned that some students solution for pi>3 that I didn't, but as far as I have been able to see, the extra .05 just makes the problem more messy.

koolba 6y ago

What disgrace if a math program would use 3 as an estimate for pi? Not even 22/7?!

wizzwizz4 6y ago
3 is no more of a disgrace than 22/7, in my book.
- angelgcuartero 6y ago
  Wait... --> https://en.wikipedia.org/wiki/Indiana_Pi_Bill https://en.wikipedia.org/wiki/Indiana_Pi_Bill
TrackerFF 6y ago
I get the feeling that using 3 for pi, is mostly beneficial when speed and volume/throughput is more important than precision. I.e - doing 100 problems with Pi = 3 probably doesn't even require a calculator, whereas doing the same problems with a decimal approximation will require students to pull out their calculators, and therefore students can do more problems in a shorter time, using pi = 3. This would certainly make sense in countries where rote learning is the king, and students are pretty much churning through problems. Or, it could be that the vast majority of people will never need the use of an accurate pi value, and it's more beneficial to make them remember some easy value like 3, rather than a confusing decimal or fraction. The people that are going to use pi in any serious manner, will obviously know its value and derivation.

deleted 6y ago

[deleted]

908B64B197 6y ago

> as elementary school children in Japan were being taught to use 3 as a substitute for pi in some calculations That's robbing them of getting an (intuitive) understanding of a whole class of numbers. And limits too!

ajsnigrutin 6y ago
pi=3 is a totally usefull approximation for when the students are old enough to know, when pi=3 is good enough. Definitely not elementary school.
- tkgally 6y ago
  According to the Japanese Wikipedia article I cited above, children were not taught that pi equals 3; rather, children who had not yet learned how to calculate with decimals were told that they could use 3 as a rough approximation for pi when calculating by hand. This minor change in the official curriculum came at a time when the Japanese government was trying to de-emphasize rote memorization and promote critical thinking in school education. Because that policy also reduced class hours and the amount of material covered, it was widely criticized for supposedly dumbing down school education. The notion that children were being taught “pi = 3” was an exaggeration, but it resonated with other objections to educational policy and apparently led the university to pose that problem on its entrance exam in protest. I still think it's an interesting problem. I wonder how far I could have gotten with it if, at the age of 18 or 19, I had encountered it on an exam and had just twenty or thirty minutes to try to come up with a proof.
  - 908B64B197 6y ago
    Ah! That's more nuanced. Sadly I do not speak Japanese so I didn't read the original source. > I wonder how far I could have gotten with it if, at the age of 18 or 19, I had encountered it on an exam and had just twenty or thirty minutes to try to come up with a proof. It's a nice one. Especially because since pi shows up in so many places there are multiple ways of doing it.

gizmo686 6y ago

I wonder why they chose 3.05 instead of just 3. The best I could come up with is related to a student using the inscribed n-gon method for computing a lower bound. The two obvious choices for an inscribed n-gon are a 6-gon and a 12-gon. Computing their perimeters involves computing cos(pi/3) or cos(pi/6) respectivly. Both of these are generally found the same way. By bisecting an equalateral triangle, you get a right angle triangle with angles pi/3 and pi/6, from which sin/cos/tan can easily be read. Once you have the relevent cos, it is straightforward trig and algebra to find With a 6-gon: pi>3 With a 12-gon: pi>6sqrt(2-sqrt(3)) Using a calculator, the 12-gon lower bound is pi>3.1058... Computing by hand (which I did), you end up having enough wiggle room to get away with some further approximations; but still need to go through some somewhat tedious computation. Maybe they were concerned that some students solution for pi>3 that I didn't, but as far as I have been able to see, the extra .05 just makes the problem more messy.