Report on the Future of the Teaching and Learning of Algebra
In early December 2001 the 12th ICMI Study Conference on The Future of the Teaching
and Learning of Algebra met at the University of Melbourne, Australia. One of
the working groups of the conference was devoted to the history of algebra.
Each of the members submitted a paper, the contents of which will eventually
be worked into the published study. I report here, however, on the papers as
submitted.
Stephen Campbell, of the University of California, Irvine, USA wrote on Number
Theory and the Transition from Arithmetic to Algebra: Connecting History and
Psychology. He notes that number theory is generally viewed in two ways, as
a generalization of whole number arithmetic and as a specialized part of algebra.
As such, it is now virtually missing in the teaching of secondary mathematics.
He argues, however, that because of its central position between arithmetic
and algebra, and because historically it played an important role in the transition
between the two subjects, it should play a more central role in education at
the secondary level. He suggests that research should be done to test the use
of number theoretic ideas as a psychological gateway for learners moving from
arithmetic to algebra (This paper is also mentioned on page 7 of the Newsletter).
Fulvia Furinghetti, of the University of Genoa, Italy, in collaboration with
Annamaria Somaglia of the Liceo Scientifico of Genoa, submitted a paper on The
Method of Analysis as a common thread in the History of Algebra: reflections
for teaching. They described the outline of a course for in-service teachers
on the teaching of algebra. Among the critical points stressed in this course,
especially by the use of original sources, were symbolism, the relation between
arithmetic and algebra, the relation between geometry and algebra, giving meaning
to manipulation, and, as a general theme, the obstacle of formalism. To pick
the original sources to discuss, they used as a thread the method of analysis,
following this through the algebraic work of Diophantus, al-Khwarizmi, and Viète.
Aurora Gallardo, of CINVESTAV-IPN, Mexico, dealt with George Peacock and a Historical
Approach to School Algebra. The author notes that in a recent research project
on the introduction of negative numbers, the conclusion was reached that all
the machinery necessary for calculations with negative numbers was present long
ago. The crucial step, however, was the acceptance of negative numbers as "real"
objects. The author then analyzes Peacock’s Treatise on Algebra (1845),
especially as he explains the non-existence of negative numbers in arithmetical
algebra and their necessity in symbolical algebra. The question is then raised
as to whether Peacock’s analysis may still be useful in today’s teaching.
Victor Katz, of the University of the District of Columbia, Washington, D.C.,
USA, discussed Using the History of Algebra in Teaching Algebra. He presented
an outline of the history of algebra from ancient times to the nineteenth century,
including Islamic algebra, Medieval and Renaissance algebra in Europe, and the
development of symbolism in the seventeenth century. At each stage, he emphasized
the importance of certain historical ideas in the teaching of algebra today.
In particular, he emphasized that there should be a clear focus in algebra teaching,
based on the solution of equations. And the fact that the solution of equations
from earliest times was based on geometrical ideas means that it may well be
useful to use these ideas in the teaching of algebra today.
Israel Kleiner, of York University, Toronto, Ontario, Canada, presented A Historically
Focused Course in Abstract Algebra. He outlined a course designed for an in-service
program of mathematics teachers. This course is based on the study of certain
problems – their basic solutions, the abstractions to which these solutions
led, and how the solutions to the original problem usually led to solutions
of other, often more important, problems.
The leaders of the History of Mathematics working group were Luis Puig, of the
University of Valencia, Spain, and Teresa Rojano, of the Research Center in
Advanced Studies, Mexico. A full report on the results of the working group
will appear in a later issue of this Newsletter.
Victor Katz
Washington, U.S.A.
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