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Vlll Preface
wish to broaden their mathematical experience and discover possible ma-
terial for use with their regular or enriched students. In particular, I am
concerned about two groups of students.
There are those who romp through the school curriculum in mathemat-
ics while they have yet to complete other subjects. A standard response to
this situation is to accelerate them, either into calculus or into college pre-
maturely. While this is undoubtedly appropriate for some, my experience
is that very often such acceleration is counterproductive and leads to an
unsettled academic experience.
Then there are those who get caught up in contest activity. It is now
possible to spend much of the spring semester preparing for and writing
contests, and this may have some value. However, there are some for whom
contests are not congenial and others who emphasize the short-term goal
of solving problems and winning contests at the expense of proper mathe-
matical growth.
What seems to be needed is a mathematical enrichment which starts
with school mathematics, broadens it and yet is sufficiently down-to-earth
that the student can explore it in an elementary way with pencil and paper
or calculator.
The theory of equations seems to fill the bill. There is a large algorith-
mic component, so that students can enjoy technical mastery. At the same
time, they are led through their experiences into an appreciation of struc-
ture and a sense of historical and mathematical context. Beginning with
topics of high school-factoring, theory of the quadratic, solving simple
equations-polynomial theory looks forward to central areas of the uni-
versity curriculum. Having seen the derivative and the Taylor expansion
in an algebraic setting, and having graphed polynomials and appreciated
the role of continuity of polynomials in root approximation, students will
then see in calculus how these ideas can be adapted to a wider class of
functions. The algorithms of evaluation, factoring and root approximation
will provide a base of experience upon which a college numerical analysis
course can be built. The ring of polynomials provides a concrete model
of an abstract structure encountered in a modern algebra course. Having
studied the role of the complex plane in the analysis of polynomials, stu-
dents will better be able to appreciate the richness of a complex variable
course and see many of the results there as extensions from polynomials
to a wider class of functions. Other areas, such as combinatorics, geometry
and number theory, also make a brief appearance.
I offered a course on polynomials for four successive years to high school
students in the Toronto area. They were given a set of notes, a monthly
problem set for which solutions were submitted for grading, a monthly lec-
ture at the university and a set of videotaped lectures. It was advertised
for those who had completed school mathematics, but were still in high
school. Many students enrolled in the course, some stuck with it and only
a few wrote the optional examination at the end of the year. However, the