Further my post dt. Jan 21, 2013 7:18 AM (which is copied for reference below my signature), I should perhaps clarify the current state of my very small understanding of this matter: > > The way one's mind is 'formally organized' (some > small part of which possibly may be written down) is > almost certainly NOT what "enables one to see". It > may help a bit, true, in 'triggering' other ideas, > which help one "see". > What one may possibly write down about the 'formal organization of the mind' is, as earlier stated, only a 'minimal representation' of that formal organization. Further, the 'formal organization of the mind' is, most likely, only a very minimal and cursory mapping of what really goes on in the mind.
We (i.e. , even the best of of our science and scientists) in any case today understand only the very barest outlines of even this 'formal organization'.
It was, perhaps, Shakespeare who gave the best utterance of my feelings (thoughts/ideas) on this matter (and that was several centuries before the 'cognitive sciences' appeared!):
- -- "And therefore as a stranger give it welcome. There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy".
GSC posted Jan 21, 2013 7:18 AM: > Robert Hansen (RH) posted Jan 21, 2013 1:01 AM (GSC's > remarks interspersed): > > > > On Jan 20, 2013, at 11:51 AM, Joe Niederberger > > <email@example.com> wrote: > > > > > - --------------------------------- > > > Because, if > > > - - -- DEAN - NED = A, then > > > - - -- DEAN - NED should not = D unless you > really > > wish to foncuse those pre-algebra kids! > > > and > > > - - -- ARIEAL - BLAIRE = B is also likely to > > foncuse. > > > - --------------------------------- > > > > > > That's why I said that I'd like to see their > > scratch paper. Kids have different talents at > > organizing their scratch paper work (and double > > checking thereof). Its very related to algorithm > > building, and that old fashioned view of math that > > insists that the correct answer shows up at the > end. > > > > > > Cheers > > > Joe N > > > > The original name was actually "ARIEL" not > "ARIEAL". > > > Yes. > > > > On a puzzle like this your interest is in checking > > the work? To me, this isn't a very good problem for > > "checking the work" because the only significant > > detail is "seeing the trick". I don't know how the > > teacher framed this problem when they presented it > to > > the class. If the teacher gave away the trick and > > after that only 2 students got it then I am indeed > > very saddened. > > > No need to be sad. This is NOT what happened! > > > > If the teacher didn't give away the > > trick and only 2 students got it then I would > > understand, although, I wish we could do better. > > > Yes. > > > > 99% of this puzzle (and all puzzles) is "seeing the > > trick". I don't think that has anything to do with > > organization and double checking. > > > Yes. > > The 'guesses' one might make do have plenty to do > with "seeing the trick" (or 'algorithm-building in > the mind' - see below). > > > > In fact, in order > > to "see" solutions to puzzles, organization is > > probably the last thing you want in your mind. It > > won't allow you to "see". > > > The way one's mind is 'formally organized' (some > small part of which possibly may be written down) is > almost certainly NOT what "enables one to see". It > may help a bit, true, in 'triggering' other ideas, > which help one "see". > > > > A puzzle can be defined as a problem that is > alleged > > to have a solution but that solution is not > obvious. > > However "not obvious" can mean... > > > > 1. There is no obvious solution. > > 2. There is an obvious solution, but obviously not > > the one intended, because it would be impractical. > > > > There isn't any algorithm I am aware of to find non > > obvious solutions. You must rely on "seeing in the > > dark" which involves instincts not generally used > > when "seeing in the light". Indeed, you must even > > suppress your "seeing in the light" instincts, as > you > > do when walking through a dark room, or else they > > will fool you with "not seeing" and you will run > into > > something. > > > > Bob Hansen > > > Joe N. was not talking about any specific algorithm > that could be formally written down, I believe, but > about 'algorithm building in the mind', which is a > horse of a different colour entirely, a rather > mysterious process that goes on in our minds (about > which science and scientists still know very little). > The child's scratch papers surely would have helped > d the teacher to understand what might have been > going on in that child's mind, which is fairly > important in the process of 'teaching' (or 'helping > to understand') - or at least so some of us believe. > > My own response to Joe N. > (http://mathforum.org/kb/thread.jspa?threadID=2430190) > was in reference to this mysterious process of > 'algorithm building in the mind'. The teacher I > referred to insisted that we do our 'scratch work' > not on different bits of paper, but in a broadish > 'rough work' column along the right side of each page > of all our math work. I believe this simple > discipline helped us all enormously, then and later, > in this mysterious process of 'algorithm building in > the mind' - though we never wrote (or very rarely > would have written) down any 'formal algorithms'. > Those 'formal algorithms' that we might write down > n are, I daresay, only a minor fraction of what goes > on in our minds. > > GSC