In response to my recent posting [Friday, January 5, 2007] on "Why
you should learn algebra," Prof. Walter Whiteley [Director of Applied
Mathematics, York University, Canada] <whiteley@mathstat.yorku.ca>
has the following thoughts. FYI.
Jerry
As a geometer and educator, I am struck, once again, by the confusion
represented in the piece, between mathematical thinking, as a rich
range of activities including the associated ways to learn effective
reasoning for mathematics, physics, etc., and the narrower slice
associated with algebra.
Perhaps the author and the readers would find the story of Michael
Faraday helpful for context. Faraday apparently showed both dyslexia
and dyscalculia. He did not reason with formulae (algebra).
Nevertheless, he could do engineering, and built the first electric
motor, using visual reasoning recorded in his note-books. (See, for
example, the studies of David Goodings.) This alternative approach,
too, is part of the range of what mathematics and science include.
James Clerk Maxwell (the physicist and geometer) at one point said
that some of his key early work was translating Faraday's pictural
reasoning into formulae. Both versions are what I would call
'mathematical reasoning'. Good science can be expressed in multiple
ways and we are more effective working with such multiple
representations.
It is a sad carryover of the curricular excesses of the last four
decades in mathematics (including excesses at the University level),
that people would speak as if algebra is central and, implicitly,
geometry and associated visual reasoning are marginal. As long as we
exclude people who excel at visual reasoning, and struggle with
algebraic reasoning, we exclude people of enormous potential to
contribute to science, engineering and mathematics. For some
reflections on these choices (nuanced by interviews with a number of
current mathematicians), I suggest some chapters in very recent book:
The King of Infinite Space by Siobhan Roberts, subtitled: Donald
Coxeter: the man who saved geometry.
In the web of connections within human cognition correctly evoked in
the piece, algebra, geometry, probability, and the connections of
these (and other) areas all play a role in developing mathematical
reasoning. One should take care not to narrow the entrances to
mathematics to the single door associated with algebra, nor to
restrict the wanderings of interested students within the landscape
of mathematics. The capacity of this richer landscape to
entice, motivate, and train students, including students interested
in the trades and applications, should be used to the full.
In fact, one area that supports the rise of geometry these days, are
the deep, unavoidable needs of so many applied areas to use geometry
in the solution of their problems. My own work in discrete applied
geometry has included collaborations with structural, mechanical and
electrical engineers, as well as computer scientists, biochemists and
biophysicists. Algebra is not the way this communication starts, or
ends, through it has a parallel role in suggesting and supporting
connections and insights.
Walter Whiteley
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