Date: Dec 15, 2012 12:04 PM
Author: Robert Hansen
Subject: A Point on Understanding
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.
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.
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 reasoning.
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.