New mathematics -- a controversy;

              w RITING ABOUT the new mathematics these days is a
safe and popular venture: everybody is talking about it,
and no one really knows exactly what it is. Such erudite
institutions as The New York Times, The Saturday Re­view,
The American Mathematical Society, and The
Mathematical Association of America have entered the
foray with the result, surprisingly enough, of adding new
dimensions to the confusion.1
The widespread self-consciousness about education in
the United States, as Sputnik I orbited the Earth in 1958,
focused national attention on the importance of mathe­matics.
Though mathematicians were pleased to find that
their traditionally "dull" subject had become the object
of public concern and support and even catapulted to
editorial pages, the sudden glare of the limelight took
them by surprise. A sense of urgency surrounded the ever-present
task of revising curricula and courses; and, for
the past five years, mathematics teachers have found
themselves beset by new proposals to do this and do that
and by a vast array of definitions, sometimes contradic­tory,
of just what new educational problems need to be
solved. The controversy and confusion which now sur­rounds
the new mathematics is due in part to the haste
in which this reappraisal was undertaken.
There are, of course, a number of other reasons for the
controversy; and it is hardly necessary to comment that
any attempt to bring order out of the present confusion
must trace the causes of it. As mathematics teachers are
painfully aware, the definition of a problem is a major
part of its solution.
As I see it, the sources of the arguments surrounding
the new mathematics are the following; the problem of
SEMANTICS, which, for example, finds different writers
attaching to the same word quite different meanings; an
honest difference of opinion as to what the CONTENT of
the new mathematics should be; and a failure to give the
new mathematics its proper HISTORICAL perspective (for
example, when did it begin, how "new" is it really, and
where is it going) with the result that its importance and
innovations are often exaggerated.
Any attempt to treat these three extensive problems in
an intensive manner is really quite difficult, since they
have been widely misunderstood, but I shall comment on
each of them briefly and in the order in which they are
stated. Since almost all of the present debate has con­cerned
itself with secondary school mathematics, my
remarks shall be confined, for the most part, to mathe­matics
at that level.
With regard to the problem of SEMANTICS, the first
error here is in the use of the word new in connection
with mathematics at the secondary school level. I presume
that this use of the word would imply to the layman that
the mathematics now being taught has just been discov­ered;
whereas nothing could be farther from the truth.
Professor W. W. Sawyer, a noted mathematics scholar
and author, in a recent denunciation of the use of the
word new in this connection, commented, after noting
that most of the mathematics now being introduced into
high school curricula was known by the nineteenth cen­tury
at the latest, "We do not serve the cause of education,
of mathematics, or of honesty by calling old things new,
by making simple ideas appear imposing."2 What is NEW
is the emphasis being given to topics which were previ­ously
taught and the introduction of topics which were
not previously treated; and I shall discuss these further
in connection with the content of the present secondary
school curriculum.
To press the semantic problem somewhat further, since
it is the basis of so much of the present debate, I would
like to cite several other instances of words in mathe­matics
which have come to have widely divergent conno­tations
:
—To high school students the word algebra denotes a
subject (probably epitomized by quadratic equations)
which is studied in the ninth grade and, more often than
not, over again in eleventh grade; whereas to professional
mathematicians algebra denotes an extensive area of
higher mathematics which is presently alive with research
and new results. Little wonder that freshmen become con­fused
when a professor tells them that he has written his
Ph.D. thesis in algebra.
—The Pittsburgh Public Schools refer to their brand-new
and sophisticated course for high school seniors as
analysis, but to students at Ohio State University, for
example, analysis means the first really high-powered
graduate course in functions of a real variable. And now,
just recently, our own Department of Mathematics at
Carnegie has chosen to relabel its freshman and sopho­more
analytical geometry and calculus sequence simply
analysis.
—For the first time in the public schools, children are
taught about sets, sometimes as early as in the fourth
grade, and they are told that the word sets refers to such
collections of things as the children in their classroom or
the states of some union. But to graduate students in
mathematics the word sets suggests such things as, perhaps,
a geometric manifold with strange topological properties.
—To mathematicians the word topology refers to a
14 THE QUARTERLY