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THE MATHWRIGHT LIBRARY NEWSLETTER, Feb 2002, VOL 4,
#2
A publication of Bluejay Lispware
James E. White, Editor
The official publication of the New Mathwright Library and Café:
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In this issue:
Future Math: Good news for Teachers and Home Scholars
1) Future Math: Good news for Teachers and Home Scholars
We
cannot predict the future, but we may be able to affect it.
Of all
the subjects we study in school, mathematics enjoys a special place. In
"Reading, 'Riting and 'Rithmetic" we can be sure that ancestral
students learned pretty much the same arithmetic that we do today, long before
our language itself was born. Of course, some modern conventions, algorithms
and styles of calculation were added to the curriculum as recently as five
centuries ago, but all in all, arithmetic has not changed much since Babylonian
times. And even Babylonian children learned to do 'word problems' just as we do
(or ought to do) today.
But it would be wrong
to think that the teaching of mathematics has not changed over the centuries.
Our curriculum is greatly enriched in subtle ways by the deductive and
axiomatic methods introduced (say 25 centuries ago) in the pure synthetic
geometry of the Greeks, and extended in the very recent past (4 centuries ago)
to the beautiful and self-consistent notion of the continuum, as expressed in
the methods of algebra, analytic geometry, and calculus. These ideas certainly
would have appeared strange to an Athenian student of the Golden Age.
In the single span of
the second half of the last century, we have seen a number of very dramatic
changes in teaching styles and approaches in mathematics, beginning perhaps
with the optimistic and premature introduction of the 'New Math' in the '50s in
the primary grades, and continuing to the unfortunate gradual de-emphasis of
deductive reasoning and axiomatic geometry (a la Euclid) in the schools.
A guiding beacon in
the history of mathematics has always been that learners should 'recapitulate'
in their own minds the continuous and causal sequence of mathematical ideas on
which their present knowledge is based, in order to extend that knowledge.
Galileo owed much to Euclid for his insight into uniform acceleration. Kepler
might never have discovered the laws of motion of the planets around the Sun if
he were unaware of Apollonius' discoveries concerning conic sections, and in
particular, ellipses. Isaac Newton could not have discovered the Calculus if he
did not wed the pure Greek notion of Ratio with the 'modern' idea of the
continuum, and 'ultimate ratios'. And on, and on.
And while we are not
all Giants of the stature of these visionaries, we tend to follow this beacon
in our teaching also. Until recently, the course of studies in mathematics
followed closely the historical development of those ideas. In simplistic
terms: 'Ontogeny Recapitulated Phylogeny' in the mind of each student. One
learned Euclidean Geometry and Algebra before Cartesian Geometry, and had a
thorough understanding of these, and of measurement and number, before studying
Calculus. This usually required a great deal of active reflection,
experimentation, backtracking and integrative thinking on the part of the
learner.
Our text books in
recent years tend, by their modular reduction of topics into easily assimilable
'behavioral goals' to discourage this wholistic approach to learning
mathematics, presenting it instead as a series of 'recipes' to be memorized and
used at the appropriate time in the course. Unfortunately, taught in this way,
they are just as easily forgotten. One sees this by looking at almost any
algebra text written before the '40s, and counting the number of 'word
problems'. These word problems (when well chosen) require the learner to think
about the 'meaning' of the computational technique, and they often require the
student to look back to an earlier chapter, or even to a different book to
understand that meaning. The style of many (but not all) modern texts actively
discourages this integration of ideas in favor of a rapid delivery and
expeditious packaging of 'the facts'.
Unfortunately, until
very, very recently (the last decade, or so), technology has not contributed
much of a solution to this tendency. If anything, the 'programmed texts' of the
'70s and '80s only enforced the behaviorist paradigm. And even today, there are
multimedia electronic programmed texts built on the same design. The difficulty
is that computer-based texts could generally present only very simplistic
branching choice trees or multiple-choice questions. But that is changing. And
it points to an encouraging possible future for the role of computers in the
design of mathematics books.
Here at the Library,
we have attempted to offer in our WorkBooks and Microworlds an alternative
paradigm. While a few of our authors sometimes follow the behaviorist style of
presenting multiple-choice questions and telling you whether you got the right
answer, the vast majority of our authors attempt to write their books in such a
way that you can ask your own questions. And there are appearing on the market
a number of new books crafted in this new way. These books usually appear on
CD, but some are also becoming available on the web using Java applets, and
even JavaScript. We feel that the web offers a real opportunity to explore this
new style of mathematical pedagogy in which the reader becomes (once again) an
active participant in the construction of her own knowledge. Certainly, such
books cannot solve the structural problems outlined above, because those
problems belong more to the curriculum than to the particular books written to
teach it. But it may happen that with the appearance of more Interactive Web
Books, and with the easy hypertext navigation that can connect these books, one
to the other, that eventually the curriculum itself will be positively changed
by them.
We cannot predict the future, but we can affect it.
In that spirit, we present a glimpse of the possible shape of a mathematics
book of the future. The book is called Cardano,
and you may go directly to it. It is for anyone who ever wondered how you solve
Cubic Equations, and why the quadratic formula is not followed by a "cubic
formula" in the textbooks. This book will show you the "cubic
formula" and will show you why you do not have to memorize it. Cardano is
free, and does not even require a password to read it. It presents its story in
the familiar way that a static text might present it, but with the exception
that the pages of the story "come to life" and offer the reader the
opportunity to make and test hypotheses, to experiment and explore in a visual
and interactive way many of its main constructions and concepts. For many
readers, this active participation in the story can add a dynamic dimension
that will help them visualize certain of its ideas for the first time. Of
course, you may download and extract the Word
2000 version in order to print and read it in the traditional way as static
text offline, but these interactions are not ancillary; they are from the
beginning an essential part of the narrative.
Cardano is not
actually intended to teach the mathematical topics it develops to any
particular student audience. It covers a range of topics from high school to
graduate-level mathematics. And it moves at warp speed across several centuries
of mathematics, beginning perhaps in the 16th Century, and ending with some
results obtained by the authors, and published in the November 2001 issue of
The American Mathematical Monthly. But it was written so that almost anyone who
has studied a little algebra can "jump in" and use it as a starting
point for future studies. Some topics that it covers are accessible to high
school students, others to university students of Modern Algebra and Theory of
Equations, and others may be of interest to graduate students, teachers, and
professional mathematicians. So the aim is not only to teach the mathematics
but to demonstrate the range and the efficacy of a new style of web pedagogy
and of authorship.
James E. White, Ph.D.
Library Director