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".