Date: Jan 20, 2013 8:48 PM
Author: GS Chandy
Subject: Re: To K-12 teachers here: Another enjoyable post from Dan Meyer

Robert Hansen (RH) posted Jan 21, 2013 1:01 AM (GSC's remarks interspersed):
>
> On Jan 20, 2013, at 11:51 AM, Joe Niederberger
> <niederberger@comcast.net> 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 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 are, I daresay, only a minor fraction of what goes on in our minds.

GSC