Descartes
once remarked that "common sense" is a quality with which all persons
are equally well endowed. No one seems to want any more of it than he already
has. Interactive Mathematical Texts
Descartes
once remarked that "common sense" is a quality with which all persons
are equally well endowed. No one seems to want any more of it than he already
has.
Galileo
did battle, it is well known, with the "common sense" of the Aristotelian
philosophers who would deny the evidence of their senses in order to preserve
a preconceived order of ideas. (But as Galileo pointed out, Aristotle would
have been the first to revise his view...if only he had a telescope!).
"Education"
is the analog for the "environment" in an evolutionary process. In
this context at least, ontogeny always recapitulates phylogeny. The learner
must retrace the steps of her cultural history. Consider, for example the remarkable
step that every learner must make as a child to reconcile the notion of "whole
number" with the new conception of "ratio." Later, this beleaguered
learner must extend this notion of ratio to include also the "irrational"
numbers of the Real Number line. Some never take this last step, which is so
necessary for an understanding of Calculus. And no wonder -- it took Western
Europe 200 years to assimilate it.
So
we must do daily battle with the "common sense" of our students which
strives (quite sensibly!) to fit new ideas into old and comfortable patterns
of thought. But there are critical moments of "invention" (or "discovery")
when the learner adjusts her own schema of thought, by questioning it...
It is a truism that a student
will be better prepared to understand the meaning of the answer to a question
that she herself asks, than she will be to understand the answer
to a question that a teacher or a textbook asks for her. It is because when
she finally asks the question, she is posing a challenge to her own conceptual
scheme, her own "common sense" view. These pregnant moments are rare,
and so we must take great pains, as teachers, to prepare them. We do it with
examples, with pictures, with exhortations and hijinks, and with a great deal
of verbiage.
Now,
one of the real promises of computer-based learning environments is that it
is possible for those environments actually to elicit (and answer) questions
from students. They should involve the learner as an active participant, and
place at her disposal the "virtual manipulatives" that will allow
her to experiment in a natural and intuitive way. In particular, while there
will be a "story line," the learner should be free to focus on that
part which is meaningful to her, and temporarily to ignore the rest. Computer
environments that can support "Piagetian learning" of this sort must
be versatile, responsive, and smart. For example, it must be "smart"
enough to be able to interpret (and compare) a wide variety of input, since
one does not know in advance what questions a student might ask. It should be
able to respond at least to the meaningful ones. And, what is most important,
it should present to the student an open-ended opportunity to pursue her thoughts,
rather than a limited tree of prepared alternatives.
Perhaps
the boldest and most successful examples of environments of this sort were the
Logo Turtle learning environments of the Eighties that offered
the learner essentially unlimited opportunities to explore her ideas. Logo environments
told only rudimentary stories because of the limitations of the technology then
currently available, but, in the hands of experienced teachers, they were powerfully
interactive, allowing the students to pursue "what-if" scenarios to
their hearts' content. The WorkBooks at the Mathwright Library attempt to be
interactive in this sense. In fact, the base language of Mathwright
is LISP, which is also the parent of Logo.
We
can, of course, now shape the environments in our WorkBooks to a much wider
range of topics than the elementary geometry and computational logic to which
Logo restricted itself. But the idea is the same. Place the student in a virtual
setting in which she can explore in a free-form way the questions that she has
about the story. This is quite different from reproducing a text on the computer
screen.
Well
designed Java applets can be interactive in this sense also. But because of
their size constraints, and the lack of a simple development platform for authors,
they tend to be difficult to write. Authors must, as they did in the Days of
DOS, frequently "reinvent the wheel." And the result can be the terse
and Spartan environments that lack range and expressivity. MathwrightWeb
demonstrates that this is not necessary.
Of
course, calculators are also interactive in this sense, but the questions that
calculators get asked are usually not very powerful pedagogically. Questions
are pedagogically powerful if their answers can "disequilibrate" the
student (to use a Piagetian term). In other words, we strive in our WorkBooks
to allow students to ask questions, the answers to which may surprise
them. Only these questions have value in the long term.