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From the article: Take the example of a geometry problem in Chapter Two: “A farmer has 36 individual fences, each measuring one meter in length … the farmer w

From the article: Take the example of a geometry problem in Chapter Two: “A farmer has 36 individual fences, each measuring one meter in length … the farmer wants to put them together to make the biggest possible area.” This problem is completely inappropriate: It cannot be solved without very sophisticated math, rarely studied even in college. Really? It's been a while since I've had high school math, new or otherwise, but how hard can it be? Obviously you arrange the fences in a 36-sided polygon that's as close as possible to a circle. That can be represented as 36 pointy isosceles triangles, each with the pointy bit having 10 degrees angle and the base having one unit of length. High school trigonometry to work out the area of one of these, multiply by 36, and done. Easily within the skills of the high school curriculum I had. What am I missing?

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-...
- 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.
alexanderchr 5y ago
That it's not at all obvious that a circle maximizes the ratio area/boundary.
- sfink 5y ago
  It's fairly obvious: first, it's pretty clear that you need to connect all of the fences together to form a single continuous fence (no branches). Second, if you think of it physically, you can take any configuration and pour more "space" (or animals or whatever it's containing) into the enclosed area so that they apply pressure. It's pretty clear that they'll all push outward, and thus that they'll tend towards a circular configuration as you increase the area enclosed. But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof. Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking. There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.
  - Tainnor 5y ago
    You're getting downvoted because the same thing happens every time someone wants to fix mathematics education: People who learned it "the old way" insist that there can be no other way. Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).
- rasz 5y ago
  yes, with all those rectangle balloons etc
camgunz 5y ago
I think it's because the circle is intuitive to you, but other likely commonly guessed shapes are probably rectangles and stars. So like, from there you're faced with "well, OK given any polygon how do I calculate the area", and then "without measuring the area, how can I say that one polygon has a greater area than another" All of that is pretty tricky IMO. I think if you took your average SWE and asked them to work this out, they'd either google it or futz around in a buffer for a couple hours. To me that means "too hard for your average 15-16 year old", although maybe the US has super lowered my standards.
datr 5y ago
It's and old joke but: You can enclose a larger area by having the farmer take 4 pieces of fencing, constructing a square around himself, and defining him to be outside the fence.

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-...
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.

alexanderchr 5y ago

That it's not at all obvious that a circle maximizes the ratio area/boundary.

sfink 5y ago
It's fairly obvious: first, it's pretty clear that you need to connect all of the fences together to form a single continuous fence (no branches). Second, if you think of it physically, you can take any configuration and pour more "space" (or animals or whatever it's containing) into the enclosed area so that they apply pressure. It's pretty clear that they'll all push outward, and thus that they'll tend towards a circular configuration as you increase the area enclosed. But it's obvious, and only obvious. I have no clue how to prove it. I could prove that if you're close to a circular shape, that shifting to more circular will increase the area. But that's not a complete proof. Still, I have to say that this seems obvious enough to me that the difficulty in proving it doesn't really bother me too much. I would have to read the actual proposed standards to know whether these complaints are cherry-picking. There's long been a battle between conceptual understanding and rote learning, and I lean towards conceptual understand and yet don't want it to go too far -- you need practice in order to cement things and to provide an opportunity to cover all the little subtle gaps that you won't fill in with a top-down approach. A student has to work actual concrete problems no matter how well they grasp the general concepts.
- Tainnor 5y ago
  You're getting downvoted because the same thing happens every time someone wants to fix mathematics education: People who learned it "the old way" insist that there can be no other way. Somehow people think that applying calculus rules is "real maths" and "experimenting with shapes and intuiting a result" is not, but that's not true, and it's especially not true when those calculus rules were arbitrarily memorised at some point (because most people never derive calculus from the real number axioms).
rasz 5y ago
yes, with all those rectangle balloons etc

camgunz 5y ago

I think it's because the circle is intuitive to you, but other likely commonly guessed shapes are probably rectangles and stars. So like, from there you're faced with "well, OK given any polygon how do I calculate the area", and then "without measuring the area, how can I say that one polygon has a greater area than another" All of that is pretty tricky IMO. I think if you took your average SWE and asked them to work this out, they'd either google it or futz around in a buffer for a couple hours. To me that means "too hard for your average 15-16 year old", although maybe the US has super lowered my standards.

datr 5y ago

It's and old joke but: You can enclose a larger area by having the farmer take 4 pieces of fencing, constructing a square around himself, and defining him to be outside the fence.