Contexto(s) para a Reforma do Cálculo

 Date: Wed, 5 Apr 1995 16:21:47 -0400
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From: John Pais 
To: Multiple recipients of list 
Subject: [CALC-REFORM:2363] Context(s) for Calculus Reform
X-Comment: From the CALC-REFORM discussion list.

Creating a Polya-Thurston Learning Context

All reasoning, teaching, and learning occurs in a context.
In a recent article (where??) Paul Halmos commented that
"there is no such (one) thing as calculus," and in this
sense there can be no such one thing as calculus reform.
There are as many different versions as there are contexts.
Even within the subject matter itself there are conceptual contexts. In 
[3] William Thurston points out that there are multiple ways people 
understand the derivative concept,
such as (though not limited to):

1. "Infinitesimal" change ratio
2. "Symbolic" derivation
3. "Logical" epsilon-delta argument     
4. "Geometric": slope of attached tangent line
5. "Rate": instantaneous speed
6. "Approximation": best linear approximation
7. "Microscopic" view of higher and higher power

There are rich pedagogical questions to deal with regarding
this list alone for the context of a specific course, in
a specific place, at a specific time, for a specific 
population of students:

Mathematically, which one first? 
In terms of ease of understanding, which one first?
In terms of ultimate applications, which must be addressed?
How many the first time through?
.....?
.....?

In the learning context that I deal with, a key observation
that leads me to choose 4. as the primary focus is that human
reasoners (my students) are accustomed to using all their
faculties to understand the world. In particular, they
often effortlessly absorb sophisticated visual information
from a diagram and automatically generate a linguistic 
representation of the information. This is nicely described
by Barwise and Etchemendy in [1].
 
"...even a relatively simple picture or diagram can
support countless facts, facts that can be read off
the diagram. Thus a diagram can represent in compact
form what would take countless sentences to express." 

This is a very powerful human mode of representing,
processing, and understanding information, which has yet
to be fully exploited in helping human learners acquire
knowledge. 

But what to do to exploit this insight, educationally? 
Barwise and Etchemendy have developed an approach to
teaching (and doing) logic that uses both linguistic
and visual representations, which has resulted in their
(excellent) courseware: "Tarski's World" and "Hyperproof". 

In response to my learning context, I am developing an
interactive Maple-based text stressing student-active
learning, visualization, writing to learn, guided
discovery, gradual abstraction, and mathematical
understanding.

In [3] William Thurston challenges all mathematicians to
recognize and focus on this human perspective. 
 
"How do mathematicians advance human understanding of
mathematics? We are not trying to meet some abstract
production quota of definitions, theorems, and proofs.
The measure of our success is whether what we do enables
people to understand and think more clearly and effectively
about mathematics."
 
In the learning context I am trying to create I also find 
George Polya's [2] perspective extremely helpful.

"...learning begins with action and perception,
 proceeds from thence to words and concepts, 
 and should end in desirable mental habits...
 For efficient learning an exploratory phase 
 should .. [be the first phase in which learners]
  discover by themselves as much as is feasible
 under the given circumstances."

In addition, I find George Polya's [2] Ten Commandments for
Teachers to be a very useful set of guiding principles that
help me keep pedagogically focused (can you guess the 11th?):   

1. Be interested in your subject.
2. Know your subject.
3. Know about the ways of learning: the best way 
   to learn anything is to discover it for yourself.
4. Try to read the faces of your students, try to
   see their expectations and difficulties, put 
   yourself in their place.
5. Give them not only information, but "know-how,"
   attitudes of mind, the habit of methodical work.
6. Let them learn guessing.
7. Let them learn proving.
8. Look out for such features of the problem at hand 
   as may be useful in solving the problems to come, 
   try to reveal the general pattern.
9. Do not give away your whole secret at once. Let 
   the students guess before you tell it. Let them 

   find out by themselves as much as is feasible.
10. Suggest it, do not force it down their throats.
 
Alright then, what is my definition of the activity
I am engaged in that I refer to as "calculus reform".
I guess I will plant my feet and say that I am trying
to create a Polya-Thurston Learning Context. But
more precisely what is this and how do I do it?
I don't know. Its complicated. I am still trying. 
How about technology? It is a tool and not a magic
bullet. It is very powerful, but using it effectively
and appropriately requires much more effort and creative
insight on my part. It requires me to reform the way I
think about how I can help human learners acquire
mathematical knowledge. When I focus on this question
and try to get it right, technology issues seem to take
care of themselves.

Best wishes,

John Pais
Mathematics Department
Saint Louis College of Pharmacy
4588 Parkview Place
Saint Louis, MO 63110, USA
paisj@medicine.wustl.edu


References

[1] J. Barwise and J. Etchemendy. Visual Information and
    Valid Reasoning. In W.Zimmermann and S. Cunningham
    (eds.), Visualization in Teaching and Learning   
    Mathematics, MAA Notes 19, Washington, DC: The
    Mathematical Association of America, 1991.
[2] H. Taylor and L. Taylor. George Polya: Master of
    Discovery. Palo Alto: Dale Seymour Publications, 1993.
[3] W. Thurston. On Proof and Progress in Mathematics.
    Bulletin of the American Mathematical Society 30 (2),
    161-177, 1994.

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