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Re[6]: Habits of Mind: Paper Summary
Posted:
Jun 5, 1995 12:08 PM
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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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