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precision/fuzziness
Posted:
Jun 5, 1995 5:44 PM
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msouth/habits of mind/1
Perhaps a compromise of sorts might be in order. I think that
a) one of the most important things I have gained from my
mathematics education is the ability to identify the fuzz in
a fuzzily worded problem, since this carries over, for example,
into identifying the fuzz in political rhetoric, legal contracts,
studies and surveys, etc b) Conway is right about students needing
to see precisely worded problems.
a)'s importance is obvious (I think). Anyone with even the slightest
amount of analytical skill can see how often logical fallacies are
employed by politicians, advertisers, etc. Learning to think about
the statement of a problem (instead of just assuming that it is
well stated because the teacher wrote it, for example) is a vital
skill not only in mathematics, but in almost every aspect of real
life. A mathematician would immediately recognize that the
problem "find the largest prime" contains the assumption that
there _is_ a largest prime (which is false). Likewise, when
a politician says he/she/it will "support education", a thinking
individual should wonder what the person means by "support" (speak
in favor of it often? throw money at it? hire someone to look
at it? increase teacher salaries? etc etc). These are simple
but fairly illustrative examples. Lots of things we hear and
read don't mean anything (like an (uneducated) artist I just heard about
who signed a contract with an art dealer in which the arist was
to receive "reasonable compensation" for the pieces that were sold).
One thing math should do is teach people precision in their thinking,
and one way to do this is to encourage them to question the questions.
Often when a student asks about a poorly worded question, the teacher
essentially denies that the question is poorly worded with a "you
know what they are asking for" response instead of "what do you
think they meant to ask?" or "how could the question be worded more
clearly?" One example is a question from [I think] MathBlaster that
said, in effect, "3 students play tennis, 4 play baseball, 10 run
track. How many play spring sports?" The answer, of course,
should be "not enough information given to answer." MathBlaster
wanted 17. Lots of kids are bright enough to realize that some
kids might play two or three sports, and some realize that there may
be other spring sports not listed. The joy of discovering that there
are other ways to think about this question is frequently destroyed
with a "you know what they meant."
In my opinion, stuents need both the experience of thinking about
precisely worded problems _and_ identifying ways a less precisely
worded problem could be made clearer. Much of what a mathematician
does is figure out what the correct question to ask about a particular
problem is.
mike
(entire old post included below for reference)
>A long time ago, John Conway posted a very thoughtful response
>to a posting of mine about mathematical habits of mind. Below
[Quoting John]
>`` However, there is something about Michelle's paper that
>worries me TREMENDOUSLY. I'm sure it's mostly accidental, but
>it doesn't seem to mention what I regard as the most important
>thing of all about the mathematical experience, from both the
>practical and theoretical standpoints.
> This is the habit of PRECISE thinking about PRECISELY worded
>problems. This is the most important thing to teach, and should
>precede ANY kind of thinking about FUZZILY worded problems, in my
>view (and I'm speaking here as a teacher, rather than as a
>professional mathematician).''
From usenet@forum.swarthmore.edu Mon Jun 5 11:10 CDT 1995
Date: Mon, 05 Jun 95 11:08:25 EST
From: Michelle@edc.org
Encoding: 115 Text
To: nctm-l@forum.swarthmore.edu
Subject: Re[6]: Habits of Mind: Paper Summary
A long time ago, John Conway posted a very thoughtful response
to a posting of mine about mathematical habits of mind. Below
is a belated response from my project. I would ver much like to
continue this discussion. Thoughtful criticism from intelligent
people is the best way to improve the work we're doing.
-michelle
_______________________________________________________________________________
John says,
`` However, there is something about Michelle's paper that
worries me TREMENDOUSLY. I'm sure it's mostly accidental, but
it doesn't seem to mention what I regard as the most important
thing of all about the mathematical experience, from both the
practical and theoretical standpoints.
This is the habit of PRECISE thinking about PRECISELY worded
problems. This is the most important thing to teach, and should
precede ANY kind of thinking about FUZZILY worded problems, in my
view (and I'm speaking here as a teacher, rather than as a
professional mathematician).''
He is right (of course); his worry is due to an accidental slip (or, more
precisely, to a lack of a precise description of what we mean) in the
paper.
Here's an annotated version of the paragraph in question:
``There is another way to think about it, and it involves turning
the priorities around. Much more important than
specific mathematical results are the habits of mind used by the
people who create those results, and we envision a curriculum
that elevates the methods by which mathematics is created, the
techniques used by researchers, to a status equal to that enjoyed
by the results of that research.
[This includes (1) precise thinking about precise problems and
(2) precise thinking abut not-so precise problems. It also includes using
(3) heuristics and intuition to come up with plausible conjectures.
Examples from some curriculum materials we are developing:
(1) Students are given the explicit task of cutting a rectangle up to form
another rectangle on a different base. (2) Students are asked to define
what they mean by ``best'' if the want to find the best spot for an
airport that will serve three cities. (3) Students are asked to imagine a
line segment, starting at one base of a trapezoid, moving up parallel to
that base (connecting points on the non-parallel sides), stopping at the
top base, and are then asked to use this thought experiment to
conjecture a general formula for the median of a trapezoid.]
The goal is not to train large
numbers of high school students to be university mathematicians,
but rather to allow high school students to become comfortable
with ill-posed and fuzzy problems, to see the benefit of
systematizing and abstraction, and to look for and develop new
ways of describing situations. ''
[ We should have emphasized precision when we wrote ``to see the
benefit of systematizing and abstraction, and to look for and develop
new ways of describing situations;'' it was certainly on our minds.]
Right after that,we say:
``While it *is* necessary to
infuse courses and curricula with modern content, what's even
more important is to give students the tools they'll need to
use, understand, and even make mathematics that doesn't yet
exist.''
A curriculum organized around habits of mind tries to close the
gap between what the users and makers of mathematics *do*
and what they *say*. Such a curriculum lets students in
on the process of creating, inventing, conjecturing, and
experimenting; it lets them experience what goes on behind the
study door *before* new results are polished and
presented. It is a curriculum that encourages false starts,
calculations, experiments, and special cases.
Students develop the habit of reducing things to lemmas for
which they have no proofs, suspending work on these lemmas and
on other details until they see if assuming the lemmas will
help. It helps students look for logical and heuristic
connections between new ideas and old ones. A habits of mind
curriculum is devoted to giving students a genuine research
experience.
[There are two main thrusts of the paper: (1) to call for a mathematics
experience based on the *interplay* between deduction and experiment
that is so crucial to doing mathematics (that's the integration of
the logical (precise) and heuristic (less-precise) ways of thinking) and
(2) to concentrate on the way mathematics is *developed* as opposed
to the way it is presented. For example, students are introduced
to proof as a method for communication *and* as a
technique for discovery. The communication of an argument is a
rather precise activity, and we show students a few ways to do this
(two-column, paragraph, as well as some presentation techniques from
Russia, China, and Israel). The *search* for an argument is a more
fuzzy activity, which makes it harder to teach, but we spend a great
deal of time developing some general principles for navigating through
the morass of detail that you face when trying to establish a
conjecture.]
So, the point is: we certainly agree that precision and precise thinking
are central to mathematics, and, if that didn't come across in the paper,
it's due more to shoddy writing than to shoddy intent. The paper will
be appearing soon in JMB. Maybe some letters to the editor published in
subsequent issues can clear this up.
--
Michelle Manes
Education Development Center
michelle@edc.org
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