[RUME] analysis and teachers
Ed Dubinsky
edd at mcs.kent.edu
Mon Mar 1 06:45:47 EST 2004
I agree with everything David says here except that, other than an "abus
de langue" I don't even think that from the standard mathematical point of
view, it is necessarily true that 0.999...=1. It is at best ambisuous
because the expression 0.999... refers either to a sequence or its limit
(I think this distinction is important in mathematics). Of course, in the
latter case, the equation is true (mathematically) but in the former case,
it is not.
What is interesting from the point of view of math ed research is that
there have been some studies of students views about this equality. In
all of them (except in a recently submitted paper of which I am a
co-author), any response other than that the equation is true has been
taken as an error to be corrected by instruction. I think that the
discussion on this list shows us that it is possible that students who
assert that the equation is false may be expressing some of the ideas we
are talking about and not necessarily having a difficulty with a
mathematical idea.
Ed
On Sun, 29 Feb 2004, David W. Henderson wrote:
> This thread that has been continuing for the past week has been
> interesting; but I find something missing -- the mathematics.
>
> It is not a FACT that 0.999...=1.0. There are many views of numbers
> currently being used successfully by mathematicians in which 0.999... is
> not one. To name a few:
>
> Conway numbers (see J.H. Conway, _Numbers and Games_, A.K. Peters 2001)
> which were developed to analyze problems in game theory. Interestingly,
> Conway numbers are too large be a set so they go beyond set theory. See
> also, J. Barwise and L. Moss. Hypersets. Mathematical Intelligencer,
> 13(4):31--41, 1991
>
> Non-standard Analysis (introduced by Robinson in 1961 and used as the basis
> for a calculus text by Keisler, see an informative discussion of its uses
> in "The Infidel is Innocent", by Adrian Simpson, Mathematical
> Intelligencer, pp 43---51, Vol 12, number 3, 1990). Non-constructive,
but
> with lots of properties loved by logicians.
>
> Laugwitz and Schmieden's constructive non-standard analysis which embeds
> the reals in a ring that include includes infinitesimals. See _Advances in
> Analysis, Probability and Mathematical Physics: Contributions of
> Nonstandard Analysis_ (edited by S.A. Albeverio, W.A.J. Luxemburg, M.P.H.
> Wolff), Kluwer, 1994.
>
> Of course, we do not want to teach students or teachers the details of
> these theories, BUT we should be honest to them by telling them that there
> are modern useful theories in which 0.999... is not equal to 1. I was in a
> middle school classroom where they were discussing 0.999... and one student
> (let call him Keith) answered the argument that there is no number
> between 0.999.... and 1 by inventing such a number 0.999...5
> (infinitely many 9's, then a 5). This is in the spirit of Conway numbers;
> but more importantly, Keith was excited and being mathematically
> creative. Then another student, Mary, sees 0.999... = {.9, .99, .999, ...}
> as a sequence of approximations and points out that {.99, .9999, .999999,
> ... } (each term has twice as many nines as the corresponding term in
> 0.999...) is between 0.999... and 1. This is close to the idea of
> Laugwitz and Schmieden. If encouraged Keith and Mary are excited and
> creative. If they are squashed by being told "it is a fact that 0.999... =
> 1", then likely they will look somewhere else for excitement and
> creativity. Both of these students when told about the Archemedean Axiom
> (if the difference between two numbers is less than 1/n, or 1/10^n, for
> every positive integer n, then the two number are assumed to be equal) had
> no trouble in accepting and using this assumption.
>
> In terms of Ed Dubinsky's notions of "process" and "object". Both Keith and
> Mary described their ideas in terms of processes, but they also had (at
> least partially) objectified them in order to assert inequalities.
>
> There was no need to talk about "limits", "epsilons/deltas", and other
> formalisms. But they could answer the question: "Why is 0.999... = 1?" by
> saying "they are equal because we assume the Archemedean Axiom." Not only
> are they satisfied but they are being mathematically accurate, and their
> intuition and creative thinking is not being squashed.
>
> I do not advocate putting the formalism of analysis into schools, BUT I do
> advocate being honest to the students and teachers and encouraging creative
> ideas/discussions and using an enlivened notion of proof ("a convincing
> communication that answers -- Why?"). It is, I believe, possible to use
> this notion of proof at all levels of mathematics classes.
>
> David
>
>
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