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Topic: Do we need a full year of Geometry?
Replies: 29   Last Post: Nov 21, 1997 11:13 AM

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Michelle Manes

Posts: 64
Registered: 12/3/04
Habits of Mind: Paper Summary
Posted: Apr 14, 1995 4:29 PM
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Thinking about the future is risky business. Past experience
tells us that today's first graders will graduate high school
most likely facing problems that do not yet exist. Given the
uncertain neeeds of the next generation of high school graduates,
how do we decide what mathematics to teach? Should it be
graph theory or solid geometry? Analytic geometry or fractal
geometry? Modeling with algebra or modeling with spreadsheets?

These are the wrong questions, and designing the new curriculum
around answers to them is a bad idea.

For generations, high school students have studied something in
school that has been called mathematics, but which has very
little to do with the way mathematics is created or applied
outside of school. One reason for this has been a view of
curriculum in which mathematics courses are seen as
mechanisms for communicating established results and
methods---for preparing students for life after school
by giving them a bag of facts ... Given this view of
mathematics, curriculum reform simply means replacing
one set of established results by another one (perhgaps
newer or more fashionable) ...

There is another way to think about it, and it involves turning
the priorities around. Much more important than the 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. The goal is not to train large
numbers of high school studnets 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. 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.

Mathematical Habits of Mind:

Pattern sniffing

(Tinkering always needs more explanation. If you take a situation
apart and put it back together, what happens if the pieces are put
back slightly differently? For example, if you experiment with
rotating a figure, followed by translating the figure, a natural
next question is what happens if you translate and then rotate?
Since we know every integer is the product of primes, we should
wonder if every integer can be expressed as a sum of primes.
If so, is it a unique summation, as the factorization is unique?)


Mathematical Approaches to Things:

Mathematicians talk big and think small.
(Talk in generalizations, but relate these to familiar,
specific situations).

Mathematicians talk small and think big.
(Seeing simple problems as special cases of deeper mathematical
theories. Or developing deeper mathematical theories to work
on seemingly simple problems, e.g. Fermat's Last Theorem.)

Mathematicians use functions.
(Algorightms, dependences, and mappings.)

Mathematicians use multiple points of view.
(Multiple approaches to the complex number system through
algebra, analysis, arithmetic, and geometry.)

Mathematicians mix deduction and experiment.
(Mathematicians experiment with problems, but never stop
there. They are always working towards proof. The experiment,
if your lucky, may point in a good direction for a proof, and the
proof might bring up more questions that can be the bases of
future experiments.)

Mathematicians push the language.
(Example: Definitions for zero and negative exponents come
from wanting the rules for positive integral exponents to
hold in other cases.)

Mathematicians use intellectual chants.
(When engrossed in a problem, a mathematician may alternately
stare into space and scribble on paper. The staring has to do
with a mental search for logical connections.)

Geometric Approaches to Things:

Geometers use proportional reasoning.
(Self explanatory?)

Geometers use several languages at once.
(Complex numbers, analytic geometry, turtle geometry, ...)

Geometers use one language for everything.
(The language of points, lines, planes, angles, surfaces, areas,
and volumes, gets applied to seemingly non-geometric

Geometers love systems.
(Finite geometries, systematizing so that special cases are combined
into one large result, etc.)

Geometers worry about things that change.
(Reasoning by continuity, as shapes are deformed, what happens?)

Geometers worry about things that don't change.
(Looking for geometric and numeric invariants in all kinds
of situations.)

Geometers love shapes.
(Classification, analysis, representation, ... Check out Majorie
Senechal's chapter in _On the Shoulders of Giants_.)

Why Habits of Mind?

If we really want to empower our students for life after
school, we need to prepare them to be able ttouse, understand,
control, and modify a class of technology that doesn't yet exist.
That means we have to help them develop genuinely mathematical
ways of thinking. Our curriculum development efforts will attempt
to provide students with the kinds of experiences that will help
develop these habits and put them into practice.


There is also a section on algebraic approaches to things, but I'm
already running horribly late...

I'd be happy to answer questions and engage in discussion based on
this paper. Thanks for asking about it.

Michelle (All Typos are Mine) Manes
Education Development Center, Inc.

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