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You are a missing a proof of max area. Exactly what author complains about. According her (and not only) teaching math problems just to memorize is not very use

by betaby 5y ago

You are a missing a proof of max area. Exactly what author complains about. According her (and not only) teaching math problems just to memorize is not very useful.

option_greek 5y ago
Yes. Just remembering that circle has the maximum area for same perimeter is pointless like remembering capital of a country: https://math.stackexchange.com/questions/461853/given-a-fixed-perimeter-which-shape-will-have-the-maximum-area https://math.stackexchange.com/questions/461853/given-a-fixe... To truly understand it, students have to go beyond and those concepts aren't available for them yet.
- zepto 5y ago
  How is it ‘pointless’? It’s applicable and useful whether or not you understand the proof. Similarly, remembering the capital of a country is useful when you hear that capital mentioned in a discussion. Both of these can be scaffolding that enables you to learn the deeper concepts. A far greater problem is students faced with nobody willing or able to teach the deeper concepts when they want to know about them.
  - option_greek 5y ago
    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.
AlanYx 5y ago
I don't really understand the author's comment that "it cannot be solved without very sophisticated math, rarely studied even in college". That would be true if the lengths of the fences were arbitrary or the fences were potentially curved, but it's relatively straightforward to develop a proof of max area given that all the fences are the same length. Step 1 would be to demonstrate that the shape with max area has to be convex. This is pretty straightforward, and a pretty decent example for advanced middle school kids. Once you've got convexity proven, then the total area covered becomes the sum of triangular wedges. There are a variety of ways here to arrive at the maximum, either by a sequence of replacements (probably most appropriate for advanced middle schoolers) or a formulation in terms of Langrange multipliers (for advanced high school students). I think a lot of students would find the proof pretty cool.

by betaby 5y ago

You are a missing a proof of max area. Exactly what author complains about. According her (and not only) teaching math problems just to memorize is not very useful.

option_greek 5y ago

Yes. Just remembering that circle has the maximum area for same perimeter is pointless like remembering capital of a country: https://math.stackexchange.com/questions/461853/given-a-fixed-perimeter-which-shape-will-have-the-maximum-area https://math.stackexchange.com/questions/461853/given-a-fixe... To truly understand it, students have to go beyond and those concepts aren't available for them yet.

zepto 5y ago
How is it ‘pointless’? It’s applicable and useful whether or not you understand the proof. Similarly, remembering the capital of a country is useful when you hear that capital mentioned in a discussion. Both of these can be scaffolding that enables you to learn the deeper concepts. A far greater problem is students faced with nobody willing or able to teach the deeper concepts when they want to know about them.
- option_greek 5y ago
  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.

AlanYx 5y ago

I don't really understand the author's comment that "it cannot be solved without very sophisticated math, rarely studied even in college". That would be true if the lengths of the fences were arbitrary or the fences were potentially curved, but it's relatively straightforward to develop a proof of max area given that all the fences are the same length. Step 1 would be to demonstrate that the shape with max area has to be convex. This is pretty straightforward, and a pretty decent example for advanced middle school kids. Once you've got convexity proven, then the total area covered becomes the sum of triangular wedges. There are a variety of ways here to arrive at the maximum, either by a sequence of replacements (probably most appropriate for advanced middle schoolers) or a formulation in terms of Langrange multipliers (for advanced high school students). I think a lot of students would find the proof pretty cool.