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...

>

> http://www.youtube.com/watch?v=bcWImSmBTzQ

>

> 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 (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.

GSC