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Its pointless in the context of understanding. The capital is the capital due to historical reasons/declared by the government of the country. On the other hand

Its pointless in the context of understanding. The capital is the capital due to historical reasons/declared by the government of the country. On the other hand the circle has the maximum area not because some one declared it but because its deducible from base principles. Teaching children to learn these by rote memory prevents them from learning them from first principles. Which is the point the author is trying to make. Her point is Maths != trivia and even if Maths can be learnt by memory like trivia, it serves no purpose because the same problem might comes back in a different form and then the students wont be in a position to solve for it. For example, Draw the shape that has maximum area when all fence links are joined at and angle that's between 40 and 80 degree (all these problems have practical relevance for example while maximizing the usable area for a odd shaped plot).

Tainnor 5y ago
Fields Medalist Terence Tao famously said: "There is more to mathematics than rigour and proofs".[1] Look, I love proofs. But realistically, you can't expect most people to ever learn multidimensional calculus before you teach them the basic fact that a circle maximises the area: something that can be intuited by drawing polygons of increasing degree. Discovering this fact through inductive reasoning, even if mathematically unsound, is still a good teaching method (and will stick well in memory) and is not just "memorising trivia". Incidentally, working inductively is also crucially important for working mathematicians, it's just that the insights gained by doing that need to be verified (and refined) by constructing a rigorous proof; but working towards a proof when you don't even know what you want to prove yet is often difficult, it might work in an optimisation case like this, but it won't in many other areas of mathematics. Also, if you compare how other mathematics is taught in high school (and in the US, even in undergraduate university classes), it's not like there is a heavy emphasis on proof either. You're not deriving calculus from the real number axioms, you're handwaving the existence of derivatives and integral and the formulas to compute them. I'll be the first to say that knowing how this actually works behind the scenes is fascinating, but realistically, that's not what most people will need. I'm not sure why kids discovering that circles maximise the area through inductive reasoning is worse somehow than them memorising calculus rules and applying them to find the result. [1]: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...
- option_greek 5y ago
  I feel we are stating the same thing but may be using different terms: > something that can be intuited by drawing polygons of increasing degree That's not same as just remembering that the circle has maximum area. Its a good starting point to prove it in a more rigorous way. I guess it depends on the approach to solving the problem (which as per the article is stated plainly as 'Circle' without going through the mental exercise above).
  - Tainnor 5y ago
    Yeah I don't know what the curriculum actually argues or how it would be put into practice (which are two quite different things anyway), but I just felt like the justification from the article - "you can't solve this problem in high school because it involves too many advanced concepts" - not particularly convincing.
harperlee 5y ago
Yo can show it physically by inflating a balloon or pushing things in a weak cardboard box (or marbles on a table and a cord, if you insist on circles). Intuition building is very useful to build concepts, even perhaps more than proofs. See https://betterexplained.com/articles/adept-method/ https://betterexplained.com/articles/adept-method/ for a discussion of this. You can also introduce the idea of problem decomposition: first identify target shape, then argue how to approach it with 36 fences, then try to derive the formula for the area. At the end of the day, a good teacher will try to do their best with the curriculum, and a bad one will find a way to proselitize wokeness if they chisse to do so; in my opinion the good direction is to empower teachers with resources and curriculum discretion, and in turn require them better results.
zepto 5y ago
> Teaching children to learn these by rote memory prevents them from learning them from first principles. Does it? I have never heard of evidence of this.

Tainnor 5y ago

Fields Medalist Terence Tao famously said: "There is more to mathematics than rigour and proofs".[1] Look, I love proofs. But realistically, you can't expect most people to ever learn multidimensional calculus before you teach them the basic fact that a circle maximises the area: something that can be intuited by drawing polygons of increasing degree. Discovering this fact through inductive reasoning, even if mathematically unsound, is still a good teaching method (and will stick well in memory) and is not just "memorising trivia". Incidentally, working inductively is also crucially important for working mathematicians, it's just that the insights gained by doing that need to be verified (and refined) by constructing a rigorous proof; but working towards a proof when you don't even know what you want to prove yet is often difficult, it might work in an optimisation case like this, but it won't in many other areas of mathematics. Also, if you compare how other mathematics is taught in high school (and in the US, even in undergraduate university classes), it's not like there is a heavy emphasis on proof either. You're not deriving calculus from the real number axioms, you're handwaving the existence of derivatives and integral and the formulas to compute them. I'll be the first to say that knowing how this actually works behind the scenes is fascinating, but realistically, that's not what most people will need. I'm not sure why kids discovering that circles maximise the area through inductive reasoning is worse somehow than them memorising calculus rules and applying them to find the result. [1]: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-...

option_greek 5y ago
I feel we are stating the same thing but may be using different terms: > something that can be intuited by drawing polygons of increasing degree That's not same as just remembering that the circle has maximum area. Its a good starting point to prove it in a more rigorous way. I guess it depends on the approach to solving the problem (which as per the article is stated plainly as 'Circle' without going through the mental exercise above).
- Tainnor 5y ago
  Yeah I don't know what the curriculum actually argues or how it would be put into practice (which are two quite different things anyway), but I just felt like the justification from the article - "you can't solve this problem in high school because it involves too many advanced concepts" - not particularly convincing.

harperlee 5y ago

Yo can show it physically by inflating a balloon or pushing things in a weak cardboard box (or marbles on a table and a cord, if you insist on circles). Intuition building is very useful to build concepts, even perhaps more than proofs. See https://betterexplained.com/articles/adept-method/ https://betterexplained.com/articles/adept-method/ for a discussion of this. You can also introduce the idea of problem decomposition: first identify target shape, then argue how to approach it with 36 fences, then try to derive the formula for the area. At the end of the day, a good teacher will try to do their best with the curriculum, and a bad one will find a way to proselitize wokeness if they chisse to do so; in my opinion the good direction is to empower teachers with resources and curriculum discretion, and in turn require them better results.

zepto 5y ago

> Teaching children to learn these by rote memory prevents them from learning them from first principles. Does it? I have never heard of evidence of this.