But preferring not to deal with esoteric arguments, there are concrete examples of how this multifaceted approach works in our everyday (collective) life.
A specific case in point would be how we think of the Moon and its orbit around the Sun. I found a good discussion about this at The Orbit of the Moon Around the Sun is Convex! , and it serves as a good example for this post about Word Problems.
Summarily, there are some who believe the Moon’s orbit around the Sun is loopy, having negative curvature at times, and others who think it traces some elliptical or circular path, when in fact (in a 3 or 4 dimensional context) it approximates a convex polygon with rounded corners.
Based on a sidereal month the polygon would have about 13 ‘corners’ since 365.25 days to orbit the Sun/27.32 days per month in orbit=13.37 months per orbit.
There are those who believe that God Creates With His Words, and others who profess His Applied Language is Geometry or Calculus; but some of us have discovered that God does all of the above because He likes working out Math Word Problems.
Someone recently hinted that everybody already knows what the problems are and that one’s efforts are best directed at finding more solutions. The short answer is ‘ewes guys shore expect a lot out of 1 brain cell clinging desperately to all the missing parts of its anatomy.’ But even in this situation, the old adage holds that ‘If you give a man a fish, he eats for a day; but if you Teach a man how to fish he eats for a Lifetime (and can feed others).’
So, learning how to get a solution is more important than just providing someone with an answer, because then they can find out answers to things they think important and not just settle for solutions to things someone else feels they should know.
This is where it gets personal; during my school years, I remember having to work out Math Word Problems from as early as grammar school (thanks to Mr. Comer, Principal, PS 229; and for all those Irish Blessings that at first glance looked like curses), then deliberately throughout middle school, and by the time high school and college rolled around it seemed every math problem was initiated with a word problem.
We were not given math problems to merely hone technique or just math’s sake but the majority of the problems were tied in to real world phenomena and required having to depict or translate the narratives into mathematical relationships.
This is where those of you following this site get to LOL because you see how abysmal my math skills are right now, but that won't keep me from trying; I know I have forgotten the bulk of all that I learned—I even forgot how to integrate, but somehow I have to make this work…
And having put two children through school I am surprised at the limited exposure they each have had to applying the math they learn by solving word problems. It was only until they each reached beyond Algebra II that their textbooks or homework assignments included a slim number of word problems that they needed to work, if any, and not until Calculus did working word problems become common.
The trick is to learn how language describes math, as simple as ‘and’ means ‘+’ and ‘a collection’ means ‘a set’; higher level concepts would be like ‘marry’ means ‘unify’ and if you are working out a problem in physics it may mean ‘equalize two strands before splicing (uniting, adding, linking, merging, etc)’ ; or the word ‘soul’ may mean ‘any one' or ‘nonphysical, i.e, imaginary' entity.
Why such an emphasis on solving word problems?
For the simple reason that 'a picture is worth a thousand words'; it is simply a fast way to get a detailed concept across to a large audience in a language everyone can understand.
This is best illustrated with an example:
(وَإِنَّا لَجَمِيعٌ حَاذِرُونَ (26:56
(وَجُمِعَ الشَّمْسُ وَالْقَمَرُ (75:9
Signs 26:56 and 75:9 are written in Arabic, a language that only about 4% of the present day world population can understand.
The first translation in English (a language spoken by about twice as many as those who can understand Arabic) works out to:
26:56 And, truly, We are Wholly Vigilant
75:9 And the gathering of the Sun and the Moon
And the interpretation of these same Signs in arithmetic syntax works out to:
26:56 And, True, We are Discretely Computing/Adding Up/Gathering
75:9 And combining/calculating the 'Center of the Orbit' and the 'Time in Months'
The words discrete and calculating and adding up do indicate some type of integration is taking place.
The Sun is a metonym for the point around which other bodies revolve or the 'center of an orbit' (aka focus); and the Moon is the only natural Earth satellite and is closely linked to timekeeping.
In fact, the words moon and month are cognates and a month was used by early civilisations to mark out the natural time period of the Moon (in fact, some languages use the same word for both Moon and month, like in Farci 'mah').
Based on the foregoing, it's not a major leap to think that Sign 75:9, in addition to its prima facie message, is also a math problem. If one were to follow the arithmetic syntax and use a handy-dandy computer program to do the integration, such as the ubiquitous but quasi-reliable Excel tool, it becomes evident the problem is basically calling to attention the geometry of the orbit of the Moon around the Sun and what happens if one were to integrate or sum up the discrete values (given by the Sign numbers 75:9) from 75 to 9.
What is not clear is whether these upper and lower bounds are to be taken in Degrees or Radians (
), or simply Time. To keep the math manageable, an approximate circular orbit is assumed but that shouldn't keep anyone with more time and better tools from working out the case for an ellipse. The radial plot and the edge view (line chart) along with the calculations appear below.
Both the Radians v R and the T v (cos ^2 + sin ^2) plots are generated based on the assumption that the Moon's orbit is given by (400 cos t+ cos 13t, 400sin t + sin 13t) with the Sun at the geometric center (the focus).
The line graph depicts a close-up of the edge view and shows that the curve undulates (has 'rounded corners') without any negative curvature...
...and I hear someone say, "So, what?!" and another call "Dibs!" and Old Faithful say, 'F#*$ me!'...
The Radar plot T v R^2 indicates that a sector about 1/3 of the whole is missing. Interestingly, there is a beltway that partially circles Lexington, Kentucky evocative of this undulating orbit (generalized ellipse unwrapped from a conic surface for more than one period with interval length 2pi/k) with a similar size bite taken out of it:
________________
Depeche Mode
Policy Of Truth song lyrics
Songwriters: Gore, Martine …
It's too late to change events
It's time to face the consequence
For delivering the proof
In the policy of Truth
Never again
Is what you swore
The time before…
You'll see your problems multiplied
If you continually decide
To faithfully pursue
The policy of Truth
Never again
Is what you swore
The time before …
Surely there must be saner ways to go fishing (get it--de peche mode?)!
I guess somebody out there doesn't like working out Math Word Problems or feels threatened by something as simple as my commuting home, and that's why I have included the image below that has nothing to do with what I am on about here. And it also occurred to me (with a great deal of sadness) that nobody gets what this is really all about since I am having to deal with this type of tomfoolery still.
Just because I am slow to anger, does not mean that I don't get angry; and just because I prefer to walk away does not mean that I am chicken. But I know now why I don't see them coming, I guess it would give me a rather sizable unfair advantage. But then, that's what Mikey is for and I take back what I said earlier about your artwork Michael, please feel free to submit it prolifically; and I don't need to hear any more about my bunker mentality since after this morning's commute it appears to be well founded (the angle of incidence was absolutely perpendicular to the direction of travel (vehicle speed about 70 mph)--did You see who did it?--And, it's no use trying to break the news to me slowly and gently when the kids blurt it out at the breakfast table).
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