Date: Jun 5, 1995 12:08 PM Author: Michelle Manes 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