1488: "Flowcharts"

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Re: 1488: "Flowcharts"

Postby Flumble » Thu Feb 19, 2015 12:07 pm UTC

Neil_Boekend wrote:Edit: I kinda forgot the point.

There is no point, there is only discussion.
Or the point is the discussion.

Anyway, you're doing a good job. I, for one, just learned a thing or two about lightning. :)

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Re: 1488: "Flowcharts"

Postby Kit. » Thu Feb 19, 2015 1:23 pm UTC

Neil_Boekend wrote:This ionized channel is formed over the path of least resistance.

Not really, because it would make stepped leaders impossible.

Neil_Boekend wrote:It is drawn to tall conductive structures.

That's because of this.

Electric field strength of air breakdown is about 3MV/m. Lightning's voltage is in the order of a hundred megavolts. That gives the order of 30 meters of distance at which the lightning's leader can directly "sense" tall conductive structures.

My understanding is that the leader just follows some local (in the range of its "sense" ability) disturbances, creating a path that will be "the path of least resistance" for the second phase of the lightning, because, well, an ionized path has much smaller resistance than a non-ionized one.

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Re: 1488: "Flowcharts"

Postby Neil_Boekend » Thu Feb 19, 2015 1:49 pm UTC

I was giving a quick and dirty version. I should have qualified as such.

Point is: the ionized path will be the (local) path of least resistance. Then the main current takes that ionized path, which is now of far far lower ground-cloud resistance then all other paths.
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Re: 1488: "Flowcharts"

Postby blademan9999 » Fri Feb 20, 2015 3:10 am UTC

Holy crap.

It' the HELIX FOSSIL!

TPP is doing their anniversary red run, this can't be a coincidence!
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Re: 1488: "Flowcharts"

Postby Pfhorrest » Fri Feb 20, 2015 4:01 am UTC

oh god no not that again
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Re: 1488: "Flowcharts"

Postby blademan9999 » Fri Feb 20, 2015 5:08 am UTC

Image
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Re: 1488: "Flowcharts"

Postby orthogon » Fri Feb 20, 2015 8:52 am UTC

Neil_Boekend wrote:I was giving a quick and dirty version. I should have qualified as such.

Point is: the ionized path will be the (local) path of least resistance. Then the main current takes that ionized path, which is now of far far lower ground-cloud resistance then all other paths.

Thanks both, this is all good stuff, but I can't believe that's what people are taking about when they say the "path of least resistance" - that the path it takes is the path it's already "decided" to take. It's more or less tautological. And it's a particularly special case of resistances in parallel in that one resistor is finite and the others are infinite. Surely people have in mind some concept of the charged cloud and all the possible ways it could discharge to the earth, in other words at best it's a misuse of the well defined technical term "resistance" to describe a very different and nonlinear process of leader formation.

rmsgrey wrote:
orthogon wrote:
DougDean wrote:Given that electricity takes the path of least resistance, if a circuit with 5,000V potential difference divides into two paths each with 50Ω of resistance, will Schrödinger's cat be electrocuted?

Yeah, I don't even know why people say that. Electricity takes all the paths, with currents weighed in inverse proportion to the resistances. There might be nonlinear cases where something like the path of least resistance applies, like with lightning where you're looking at dielectric breakdown. But in your example, even if they're 40 and 60 ohms the current will very definitely flow through both paths.


Because resistance usually varies by multiple orders of magnitude (at least), so the current that flows down other paths is negligible...

It's only when you have two or more paths at roughly equal resistance that the division is worth mentioning.


Agreed, but then again, this is more relevant to a current source. Of you have a reasonably stiff voltage source, like the mains, the current flowing in other legs has no effect on the current in a particular leg. Touching the live terminal of a light fitting, a 3kW electric heater or an iPhone charger will all do you equal damage. (Assuming you touch the heater connection at a point where you don't get physically burned into the bargain!) The electricity doesn't choose not to fry you because there's a parallel path with a thousandth of the resistance. I dunno, the whole thing feels like a meme that's not very helpful to the general understanding of science.
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Re: 1488: "Flowcharts"

Postby Pfhorrest » Fri Feb 20, 2015 9:33 am UTC

blademan9999 wrote:[img]redacted[/img]

I need to invent a way to stab you in the face over the internet.
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Re: 1488: "Flowcharts"

Postby PinkShinyRose » Fri Feb 20, 2015 12:10 pm UTC

orthogon wrote:
Neil_Boekend wrote:I was giving a quick and dirty version. I should have qualified as such.

Point is: the ionized path will be the (local) path of least resistance. Then the main current takes that ionized path, which is now of far far lower ground-cloud resistance then all other paths.

Thanks both, this is all good stuff, but I can't believe that's what people are taking about when they say the "path of least resistance" - that the path it takes is the path it's already "decided" to take. It's more or less tautological. And it's a particularly special case of resistances in parallel in that one resistor is finite and the others are infinite. Surely people have in mind some concept of the charged cloud and all the possible ways it could discharge to the earth, in other words at best it's a misuse of the well defined technical term "resistance" to describe a very different and nonlinear process of leader formation.

rmsgrey wrote:
orthogon wrote:
DougDean wrote:Given that electricity takes the path of least resistance, if a circuit with 5,000V potential difference divides into two paths each with 50Ω of resistance, will Schrödinger's cat be electrocuted?

Yeah, I don't even know why people say that. Electricity takes all the paths, with currents weighed in inverse proportion to the resistances. There might be nonlinear cases where something like the path of least resistance applies, like with lightning where you're looking at dielectric breakdown. But in your example, even if they're 40 and 60 ohms the current will very definitely flow through both paths.


Because resistance usually varies by multiple orders of magnitude (at least), so the current that flows down other paths is negligible...

It's only when you have two or more paths at roughly equal resistance that the division is worth mentioning.


Agreed, but then again, this is more relevant to a current source. Of you have a reasonably stiff voltage source, like the mains, the current flowing in other legs has no effect on the current in a particular leg. Touching the live terminal of a light fitting, a 3kW electric heater or an iPhone charger will all do you equal damage. (Assuming you touch the heater connection at a point where you don't get physically burned into the bargain!) The electricity doesn't choose not to fry you because there's a parallel path with a thousandth of the resistance. I dunno, the whole thing feels like a meme that's not very helpful to the general understanding of science.

Did the 'path of least resistance' thing come from an electricity context at all?

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Re: 1488: "Flowcharts"

Postby Neil_Boekend » Fri Feb 20, 2015 12:14 pm UTC

Good question.
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Re: 1488: "Flowcharts"

Postby mathmannix » Fri Feb 20, 2015 8:21 pm UTC

blademan9999 wrote:Image

Is that some sort of flying spaghetti monster thing?
Spoiler:
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I hear velociraptor tastes like chicken.

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Re: 1488: "Flowcharts"

Postby Eebster the Great » Fri Feb 20, 2015 11:10 pm UTC

PinkShinyRose wrote:Did the 'path of least resistance' thing come from an electricity context at all?

You mean the oldest usage of the phrase? I'm almost certain it referred to water flowing downhill.

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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sat Feb 21, 2015 1:52 am UTC

Copper Bezel wrote:I honestly never thought of it as really an electricity metaphor in the first place. I pictured objects rolling downhill and some semantic handwavery. So "take the path of least resistance" was two puns for me.

Oh, good, it would have been tragic if folks had been sidetracked by that very silly argument for several posts because no one had posed the question.
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Re: 1488: "Flowcharts"

Postby orthogon » Sat Feb 21, 2015 9:11 am UTC

Copper Bezel wrote:
Copper Bezel wrote:I honestly never thought of it as really an electricity metaphor in the first place. I pictured objects rolling downhill and some semantic handwavery. So "take the path of least resistance" was two puns for me.

Oh, good, it would have been tragic if folks had been sidetracked by that very silly argument for several posts because no one had posed the question.

Sorry, CB, but you were at the bottom of the [old]page, which probably makes you some kind of antipope.
Fwiw I had seen your post, but whether or not it's the original use, or the use Randall had in mind, people definitely use it that way. The wiki page on the subject may have some issues, but it makes some good points and describes the concept as "folk physics", which I guess is what it is. The principle doesn't even apply to objects rolling downhill, which will smash through a fence rather than go through the open gate 1m to the left. I guess that's the kind of thing you meant by " semantic handwavery"?

I think this kind of thing really is worth discussing, since public misunderstanding of science is a significant problem. It's a can of worms, but the 911 "Truth" Movement uses the "path of least resistance" meme to argue that the towers should not have collapsed straight down, for example.
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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sat Feb 21, 2015 9:48 am UTC

Why are they doing that? That's really quite stupid. People who are both conspiracy theorists and base their entire understanding of science and logic around figures of speech are quite probably beyond saving.

But yes, it's an inaccurate description of objects rolling downhill. It's also an inaccurate description of electrical currents. It's probably at its most literally accurate when I actually use it, when it means the most socially acceptable and expected action in a given situation or the one requiring least interaction with and imposition on people around me, and thus the one least likely to raise hackles or incur, well, resistance, which probably makes it very bad at being a metaphor.

If people are taking it literally and thinking it's a scientific law, then that is itself the misunderstanding to be corrected. You may also need to explain to them that neither pots nor kettles are universally black since the introduction of stainless steel alternatives (though teflon certainly adds some nuance) and that apples falling from trees on hills may indeed come to rest at some distance from the parent organism.
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Re: 1488: "Flowcharts"

Postby orthogon » Sat Feb 21, 2015 12:03 pm UTC

:-) Your third sentence is brilliant - I would quote it if I weren't doing this on my phone. I was going to cite an example but I think I may have sightly misrepresented their argument, to the extent that they actually have one.

Yes, it's the misconception that this is a physical law that needs correcting. That was the point I was trying to make. Apologies if I failed to make it clearly!
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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sat Feb 21, 2015 1:06 pm UTC

Ah, gotcha. Cool. = D
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Re: 1488: "Flowcharts"

Postby rmsgrey » Sat Feb 21, 2015 4:59 pm UTC

orthogon wrote:
Copper Bezel wrote:
Copper Bezel wrote:I honestly never thought of it as really an electricity metaphor in the first place. I pictured objects rolling downhill and some semantic handwavery. So "take the path of least resistance" was two puns for me.

Oh, good, it would have been tragic if folks had been sidetracked by that very silly argument for several posts because no one had posed the question.

Sorry, CB, but you were at the bottom of the [old]page, which probably makes you some kind of antipope.
Fwiw I had seen your post, but whether or not it's the original use, or the use Randall had in mind, people definitely use it that way. The wiki page on the subject may have some issues, but it makes some good points and describes the concept as "folk physics", which I guess is what it is. The principle doesn't even apply to objects rolling downhill, which will smash through a fence rather than go through the open gate 1m to the left. I guess that's the kind of thing you meant by " semantic handwavery"?


There's a distinction between least resistance locally and least resistance globally - an object rolling downhill will smash through a fence (or be stopped by it) rather than detour to go through a gate because that's where following the path of least immediate resistance has taken it. To find global minima rather than local minima, you need quantum phenomena...

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Re: 1488: "Flowcharts"

Postby sevenperforce » Sat Feb 21, 2015 5:29 pm UTC

I think "path of least resistance" is a colloquial extrapolation of the second law of thermodynamics -- that a physical system will follow the most immediate path to a net increase of entropy.

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Re: 1488: "Flowcharts"

Postby Eebster the Great » Sun Feb 22, 2015 2:30 am UTC

rmsgrey wrote:
orthogon wrote:
Copper Bezel wrote:
Copper Bezel wrote:I honestly never thought of it as really an electricity metaphor in the first place. I pictured objects rolling downhill and some semantic handwavery. So "take the path of least resistance" was two puns for me.

Oh, good, it would have been tragic if folks had been sidetracked by that very silly argument for several posts because no one had posed the question.

Sorry, CB, but you were at the bottom of the [old]page, which probably makes you some kind of antipope.
Fwiw I had seen your post, but whether or not it's the original use, or the use Randall had in mind, people definitely use it that way. The wiki page on the subject may have some issues, but it makes some good points and describes the concept as "folk physics", which I guess is what it is. The principle doesn't even apply to objects rolling downhill, which will smash through a fence rather than go through the open gate 1m to the left. I guess that's the kind of thing you meant by " semantic handwavery"?


There's a distinction between least resistance locally and least resistance globally - an object rolling downhill will smash through a fence (or be stopped by it) rather than detour to go through a gate because that's where following the path of least immediate resistance has taken it. To find global minima rather than local minima, you need quantum phenomena...

Basically this. It's always following a sort of resistance gradient. People who don't know the correct mathematical terms for the phenomenon presumably still understand that water (or electricity) doesn't "know" all possible paths beforehand, only the immediate surroundings, and flow accordingly.

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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sun Feb 22, 2015 6:07 am UTC

Yeah, it's taking the most direct immediately available path toward "down," which is ... sort of like seeking least resistance.

It's still true that nature abhors a vacuum. Mine is getting clogged all the time, and it's 99% cat hair.
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Re: 1488: "Flowcharts"

Postby orthogon » Sun Feb 22, 2015 10:33 am UTC

But the rolling ball could crash through the fence even if there were a local path available that was more downhill and less obstructed, because its momentum was already carrying it that way. So we are obliged to include inertia in our definition of "resistance". Thus the state of motion of the ball itself distorts the "resistance field". Even if classical mechanics can be expressed as a resistance field in some way (can it?), the way in which the moving mass modifies the field would be rather complex. And you'd have a system of differential equations for a field rather than just a single body.
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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sun Feb 22, 2015 11:58 am UTC

Right, well, it's not a very good turn of phrase, though, is the thing.
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Re: 1488: "Flowcharts"

Postby Eebster the Great » Sun Feb 22, 2015 2:15 pm UTC

I think you could compare the principle of least action to a sort of "path of least resistance." Well it's a stretch, but at least the syntax is similar.

Anyway, actual water does tend to flow along a path close to the elevation gradient when it flows for instance during a flood, and in most of the types of situations people see significant amounts flowing relatively slowly over land, or at any speed in rivers. Obviously the rebound at the bottom of a waterfall has nothing to do with the "least resistance," but that's not really what people are talking about.

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Re: 1488: "Flowcharts"

Postby Flumble » Sun Feb 22, 2015 5:05 pm UTC

orthogon wrote:But the rolling ball could crash through the fence even if there were a local path available that was more downhill and less obstructed, because its momentum was already carrying it that way. So we are obliged to include inertia in our definition of "resistance". Thus the state of motion of the ball itself distorts the "resistance field". Even if classical mechanics can be expressed as a resistance field in some way (can it?), the way in which the moving mass modifies the field would be rather complex. And you'd have a system of differential equations for a field rather than just a single body.

Yes, we must therefor restrict the "path of least resistance" to things with infinitesimal or no momentum. Like heat or, more general, entropy.

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Re: 1488: "Flowcharts"

Postby Copper Bezel » Sun Feb 22, 2015 5:42 pm UTC

I feel like "entropy happens" would be a metaphor for an entirely different thing....
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Re: 1488: "Flowcharts"

Postby PinkShinyRose » Mon Feb 23, 2015 3:17 am UTC

What if it describes behaviour during a gas/dust explosion? Where some lid/roof blows of because it is weaker (doesn't resist the pressure as much) as other walls. If you're lucky the pressure even falls enough when such an exit is provided to offset further expansion of the explosion medium causing the other paths to remain unused. Of course this is analogous to the lightning example since excess pressure causes an extremely low pressure path which in turn causes such an uneven distribution of 'resistance' that the path indeed becomes the only noticeable path.

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Re: 1488: "Flowcharts"

Postby orthogon » Mon Feb 23, 2015 12:21 pm UTC

PinkShinyRose wrote:What if it describes behaviour during a gas/dust explosion? Where some lid/roof blows of because it is weaker (doesn't resist the pressure as much) as other walls. If you're lucky the pressure even falls enough when such an exit is provided to offset further expansion of the explosion medium causing the other paths to remain unused. Of course this is analogous to the lightning example since excess pressure causes an extremely low pressure path which in turn causes such an uneven distribution of 'resistance' that the path indeed becomes the only noticeable path.

That's a good example, though there's also the "weakest link" concept that might apply there.

Having reflected on this a bit, I think it may relate to a problem in the way science is taught. I can only speak from my own experience in the English system, and that more than two decades ago, but what stands out in my recollection is a distinct lack of quantitativity in science education up to GCSE level (16+). I remember experiencing a kind of revelation when we were given the formula for the period of a pendulum and compared it with experimental measurements. I'd always been into science, but somehow I hadn't appreciated just how powerfully and accurately it can model real-world phenomena until that moment. But even that experiment was the exception rather than the rule. There were a few simple formulae, like F=ma and V=IR, but other than that the emphasis was very much on a qualitative description of the phenomena. Once you eschew the language of mathematics, you naturally end up with trite and oversimplified (if not wholly incorrect) statements of how things work, like "electricity takes the path of least resistance". (Another example I remember was about structures: we were taught that "structures are most likely to break at the joints", and the question in the exam was "where are structures most likely to break?" After studying some basic structural engineering at university, this statement strikes me as plainly incorrect: an efficiently designed structure ought to be as likely to fail in a member as in a joint.)

Since many people leave formal academic education at 16, and many of those who carry on to A-Level drop sciences at that point, this qualitative view of science is all the majority of people have to go on. At A-Level (16-18), things suddenly get very quantitative; to some extent this is because calculus is introduced at this stage, making an enormously wider range of derivations possible. Actually, this might be a clue to what happened: from the '60s onwards an important principle in education has been that the student should understand why things are as they are: this approach would frown on the idea of teaching a formula whose derivation was beyond the student's mathematical ability. The pendulum formula, requiring calculus, would have violated this principle. (The quadratic formula wasn't derived either, but it was taught as a kind of "necessary evil" alongside the use of factorisation).
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Re: 1488: "Flowcharts"

Postby Jorpho » Mon Feb 23, 2015 5:24 pm UTC

orthogon wrote:The pendulum formula, requiring calculus, would have violated this principle.
It does? I can only recall that it required the small-angle approximation (sin x = x). But then, it has been much too long.

We never really practiced calculating moments of inertia; I remember that much.

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Re: 1488: "Flowcharts"

Postby orthogon » Mon Feb 23, 2015 5:34 pm UTC

Jorpho wrote:
orthogon wrote:The pendulum formula, requiring calculus, would have violated this principle.
It does? I can only recall that it required the small-angle approximation (sin x = x). But then, it has been much too long.

We never really practiced calculating moments of inertia; I remember that much.


Well, you prove that the acceleration is proportional to the displacement and directed back towards the equilibrium position, i.e. =-ω2x. It's only proportional under the small-angle assumption (a mass on a spring doesn't need the approximation, provided the spring is perfectly Hookean). Then you either quote the standard result for Simple Harmonic Motion, or you solve the differential equation. Quoting the standard result would fall foul of the same criticism, and you still need differential calculus to express the equation of motion anyway.

[Edit: beautified the equation]
Last edited by orthogon on Tue Feb 24, 2015 5:11 pm UTC, edited 1 time in total.
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Re: 1488: "Flowcharts"

Postby Quercus » Mon Feb 23, 2015 7:00 pm UTC

orthogon wrote:At A-Level (16-18), things suddenly get very quantitative; to some extent this is because calculus is introduced at this stage, making an enormously wider range of derivations possible.


It always strikes me as extraordinary that calculus is introduced so late and made out to be some scary advanced thing. Geometry and algebra are both introduced several years before that, and as soon as you grasp basic geometry and algebra you can grasp differential calculus, and as soon as you grasp differential calculus you can grasp integral calculus. I guess there might be a little stumbling block over the concept of a limit, but personally I found that no less intuitive than the concept of zero in the first place, which is introduced at about age 5.

Edit: To clarify, I can easily see people having a problem with calculus because they have a problem with algebra and/or geometry, but it's far trickier in my mind to go from arithmetic to algebra or geometry than it is to go from algebra and geometry to calculus, yet for some reason you wait several years after people are meant to have done the former before introducing the latter.

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Re: 1488: "Flowcharts"

Postby Copper Bezel » Mon Feb 23, 2015 8:10 pm UTC

Thing is, I know of a lot of people who never "got" algebra and just worked their way up by rote. I somehow ended up doing very little calculus myself, and I regret that, but I sort of feel like we have a system where algebra is treated as something that people are expected to learn at x level, instead of as the underpinning for a bunch of useful tools. So long as that's the case and people can "learn" algebra without actually being able to connect it to any real-world applications, I think the other problems are all natural consequences.
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Re: 1488: "Flowcharts"

Postby orthogon » Tue Feb 24, 2015 5:19 pm UTC

Copper Bezel wrote:Thing is, I know of a lot of people who never "got" algebra and just worked their way up by rote. I somehow ended up doing very little calculus myself, and I regret that, but I sort of feel like we have a system where algebra is treated as something that people are expected to learn at x level, instead of as the underpinning for a bunch of useful tools. So long as that's the case and people can "learn" algebra without actually being able to connect it to any real-world applications, I think the other problems are all natural consequences.

This is where I think the relatively modern "understand what you're doing rather than get the right answer" approach is vindicated. I look at old O-level papers and can't help thinking that enormous amounts of material have been removed over the years, particularly in the move to GCSE. But I'm very doubtful that most people of my parents' generation really had a clue what they were doing, nor that they retained much of it in a way that was practically useful. I suspect they could integrate and differentiate functions in the same way that they could conjugate and decline Latin verbs and nouns. I.e., just as a rote-memorised incantation.
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Re: 1488: "Flowcharts"

Postby rmsgrey » Tue Feb 24, 2015 6:50 pm UTC

orthogon wrote:
Copper Bezel wrote:Thing is, I know of a lot of people who never "got" algebra and just worked their way up by rote. I somehow ended up doing very little calculus myself, and I regret that, but I sort of feel like we have a system where algebra is treated as something that people are expected to learn at x level, instead of as the underpinning for a bunch of useful tools. So long as that's the case and people can "learn" algebra without actually being able to connect it to any real-world applications, I think the other problems are all natural consequences.

This is where I think the relatively modern "understand what you're doing rather than get the right answer" approach is vindicated. I look at old O-level papers and can't help thinking that enormous amounts of material have been removed over the years, particularly in the move to GCSE. But I'm very doubtful that most people of my parents' generation really had a clue what they were doing, nor that they retained much of it in a way that was practically useful. I suspect they could integrate and differentiate functions in the same way that they could conjugate and decline Latin verbs and nouns. I.e., just as a rote-memorised incantation.

The trouble with going for understanding rather than rote memorisation is what happens to the people who don't understand and then are expected to figure out something that builds on it - if someone's uncertain in algebra, your explanations of calculus are going to be all Greek (rather than just having a scattering of Greek letters) to them. On the other hand, just repeating the same material at them is going to turn them off pretty quickly because they know it's incomprehensible mumbo-jumbo...

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Re: 1488: "Flowcharts"

Postby Pfhorrest » Tue Feb 24, 2015 7:03 pm UTC

The people who don't understand need remedial education to help them understand before they can move on to understanding things that hinge on things they already don't understand.
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Re: 1488: "Flowcharts"

Postby Quercus » Tue Feb 24, 2015 7:45 pm UTC

Pfhorrest wrote:The people who don't understand need remedial education to help them understand before they can move on to understanding things that hinge on things they already don't understand.


This is especially true for something like calculus, because there is absolutely no point in rote learning calculus - any career which requires calculus is going to require an understanding of calculus. Rote learned calculus is just useless knowledge. It's not even beautiful for it's own sake, because the beauty lies in the understanding of what it means.

There's an argument to be made for rote-learning things like the rules of grammar and how to calculate compound interest, because rote-learning those things will get you by even if you never manage to understand them.

I think it does a great disservice to children to have them spend their time rote-learning anything beyond the basics they need to know for everyday life. School time would be far better spent teaching people how to research things, how to learn new skills most effectively, how to think critically and assess evidence etc. Of course, you end up teaching kids a lot of specific stuff in the process of teaching them those things, but the emphasis is completely different - you're teaching the process by using an example, rather than just teaching a list of facts. For example teaching kids to code is great at teaching them to understand logic, and teaching them how to design a scientific experiment or assess the reliability of historical sources is great at developing critical thinking skills.

It's the old "give a man a fish vs. teach a man to fish argument". If you teach someone the skills of acquiring new skills, knowledge and informed viewpoints (and believe me, those are learned skills) then they can apply that to pretty much anything, whatever they end up doing in life. If you only teach them particular bits of knowledge, then they're stuck with just those bits of knowledge.

The problem, of course, is that it's harder to teach, and harder to assess, processes and skills compared to facts and figures.

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Re: 1488: "Flowcharts"

Postby Jorpho » Wed Feb 25, 2015 2:53 pm UTC

Quercus wrote:It always strikes me as extraordinary that calculus is introduced so late and made out to be some scary advanced thing. Geometry and algebra are both introduced several years before that, and as soon as you grasp basic geometry and algebra you can grasp differential calculus, and as soon as you grasp differential calculus you can grasp integral calculus.
Actually, geometry doesn't strike me as having much to do with calculus at all.

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Re: 1488: "Flowcharts"

Postby mathmannix » Wed Feb 25, 2015 3:01 pm UTC

Jorpho wrote:
Quercus wrote:It always strikes me as extraordinary that calculus is introduced so late and made out to be some scary advanced thing. Geometry and algebra are both introduced several years before that, and as soon as you grasp basic geometry and algebra you can grasp differential calculus, and as soon as you grasp differential calculus you can grasp integral calculus.
Actually, geometry doesn't strike me as having much to do with calculus at all.

Maybe not per se, but trig functions are pretty much ubiquitous in calculus, and you really need a good grasp on geometry before trigonometry. Also, for things like the area under a curve and so on.
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Re: 1488: "Flowcharts"

Postby Quercus » Wed Feb 25, 2015 3:02 pm UTC

Jorpho wrote:
Quercus wrote:It always strikes me as extraordinary that calculus is introduced so late and made out to be some scary advanced thing. Geometry and algebra are both introduced several years before that, and as soon as you grasp basic geometry and algebra you can grasp differential calculus, and as soon as you grasp differential calculus you can grasp integral calculus.
Actually, geometry doesn't strike me as having much to do with calculus at all.


I was using a rather broad definition of geometry - I was thinking about understanding graphs of functions, tangents, slopes, the area under a curve, that sort of thing.

Edit: Ninja'd by mathmannix.

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Re: 1488: "Flowcharts"

Postby orthogon » Wed Feb 25, 2015 4:01 pm UTC

Whilst we could debate just how hard or easy certain concepts are, I think we could probably agree that mathematics, and to a lesser extent natural sciences in general, are viciously cumulative. Properly understanding one thing requires a pretty thorough grokking of the principles on which it builds. This for me is a very strong argument for that other unfashionable idea in education: ability-based streaming. Time spent teaching concept B to a student who doesn't get the prerequisite concept A is almost completely wasted. (This might not always be quite the whole story. Many concepts have some kind of circular dependence. Normally you'd think of matrix arithmetic as a prerequisite to geometrical transformations, but it's possible that somebody might suddenly "get" matrices as a result of learning about transformations. But even there there's a kind of cumulative effect resulting from iteration: by going back and forth between two topics you can make a small progress in your understanding of each in each iteration, like climbing up the inside of an alcove by moving your feet alternately on facing walls - I forget the climbing term for this).
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