Date: Dec 31, 2012 1:11 AM
Author: GS Chandy
Subject: Re: A Point on Understanding
Robert Hansen (RH) posted Dec 15, 2012 10:34 PM (GSC's remarks interspersed):
> Earlier I posted a link to the observations of a
> calculus teacher regarding students understanding of
> infinite sequences and series...
> One of his observations was that students will accept
> that 0.333... is 1/3 but have trouble accepting that
> 0.999... is 1.
I have been unable (despite several attempts) to view (and hear) the youtube video, so my remarks are based only on what's been written here.
I believe that the teacher's observations to the effect that:
- -- "students will accept that 0.333... is 1/3 but have trouble accepting that 0.999... is 1"
is not correct at all.
To the best of my understanding, a student who has accepted (underrstood) how "0.333... 'would go to' 1/3" as a limit should have have no difficulty at all in accepting that "0.999... 'would go to' 1" (in the limit).
Of course, that student must arrive at a clear understanding just how those repeating decimals 'go to' the fractions 1/3 and 1/1 respectively. Given this, the rest of RH's argument makes no sense at all to me.
> Students "accept" that 0.333... is 1/3 because they
> are familiar with the algorithm of long division and
> that algorithm will readily produce the result of
> 0.333... when you divide 1 by 3. When I say "readily"
> I mean that it is easily seen by the student that
> every subsequent iteration of the algorithm will add
> another "3" to the decimal result and return you
> right back to the same state you started from. This
> doesn't mean that they necessarily understand why
> this occurs, but they certainly accept it.
> However, if we attempt to look at this from the other
> direction by determining the sum of 3/10 + 3/100 +
> 3/1000 + ... then this is another matter altogether.
> We could show that this series is the same as
> 0.333... and that since that is the result of
> dividing 1 by 3 then this series must equal 1/3. But
> that is not an exercise in understanding, that is an
> exercise in convincing and acceptance.
> I propose that an increase in understanding must
> involve an increase in formal thinking. In this
> particular case an increase in understanding must
> involve the understanding of what a limit is and how
> it applies equally to these two series.
Makes no sense to me.
> Even though the student accepted that 0.333... is
> 1/3, that acceptance was based on nothing more than a
> presentation (long division). Without being able to
> produce a similar presentation for 0.999... the
> student is stuck.
> I am not saying that we (as students) do not
> appreciate being convinced during all of this formal
> development but, being convinced is applicable only
> to the convincing presentation while understanding is
> applicable in general. Being convinced is obviously
> important (to learning) somehow but being convinced
> is certainly not the crux of understanding. Formal
> thinking is the crux of understanding and that
> requires the development of many things (habits of
> mind) but mainly the development of a theory
> involving more precise concepts and more precise
> Some will confuse "formal thinking" with "formal
> mathematics" and actually, I am talking about the
> same thing, but not at the same level. It is all
> formal, all the way from learning how to count, but
> it is also leveled and takes time to develop. A
> student playing simple Mozart pieces on the piano may
> not be playing Liszt etudes (yet) but they are still
> Bob Hansen
It is easy to become extremely foncused in regard to issues related to "thinking about thinking", as seen in RH's original post (http://mathforum.org/kb/thread.jspa?threadID=2420856). I agree that it is not easy to "think about thinking".
The tools described in the attachments to my post at http://mathforum.org/kb/thread.jspa?threadID=2419536 could be most helpful to remove most (if not all) such foncusions.
I am not able to play the piano at all (whether it be a Mozart piece or Liszt or 'Mary had a little lamb') - so the analogy (or metaphor?) is not meaningful to me.
But I believe I am not wrong in regard to the issue of "thinking about thinking" and how to get over such foncusions about it.