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.

See above.
> 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
> reasoning.

See above.
> 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
> playing.
> 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 ( I agree that it is not easy to "think about thinking".

The tools described in the attachments to my post at 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.