**The Mathematics Teacher of March 1962**
**American Mathematical Monthly of March 1962**

The following memorandum was composed by several of the under-signed and sent to 75 mathematicians in the United States and Canada. No attempt was made to amass a large number of signatures by can-vassing the entire mathematical community. Rather, the objective was to obtain a modest number from men with mathematical competence, background, and experience and from various geographical locations. A few of the undersigned, whose support is indeed welcomed, volun-teered their names when they learned about the memorandum from a colleague.

The mathematicians of this country now have a more
favorable climate in which to develop and gain acceptance of improvements
in mathematics education. Indeed a number of groups have recognized the
opportunity and are working hard and with the best of intentions to utilize
it.

It would, however, be a tragedy if the curriculum reform should be
misdirected and the golden opportunity wasted. There are, unfor-tunately,
factors and forces in the current scene which may lead us astray. Mathematicians,
reacting to the dominance of education by professional educators who may
have stressed pedagogy at the expense of content, may now stress content
at the expense of pedagogy and be equally ineffective. Mathematicians may
unconsciously assume that all young people should like what present day
mathematicians like or that the only students worth cultivating are those
who might become pro-fessional mathematicians. The need to learn much more
mathematics today than in the past may cause us to seek shortcuts which,
however, could do more harm than good.

In view of the possible pitfalls it may be helpful to formulate what appear to us to be fundamental principles and practical guidelines.

**1. For whom**. The mathematics curriculum of
the high school should provide for the needs of all students: it should
contribute to the cultural background of the general student and offer
professional preparation to the future users of mathematics, that is, engineers
and scientists, taking into account both the physical sciences which are
the basis of our technological civilization, and the social sciences which
may need progressively more mathematics in the future. While pro-viding
for the other students, the curriculum can also offer the most essential
materials to the future mathematicians. Yet to offer such sub-jects to
all students as could interest only the small minority of pros-pective
mathematicians is wasteful and amounts to ignoring the needs of the scientific
community and of society as a whole.

**2. Knowing is doing**. In mathematics, knowledge
of any value is never possession of information, but "know-how." To know
mathe-matics means to be able to do mathematics: to use mathematical language
with some fluency, to do problems, to criticize arguments, to find proofs
and, what may be the most important activity, to recog-nize a mathematical
concept in, or to extract it from, a given concrete situation.

Therefore, to introduce new concepts without a sufficient background
of concrete facts, to introduce unifying concepts where there is no experience
to unify, or to harp on the introduced concepts without concrete applications
which would challenge the students, is worse than useless: premature formalization
may lead to sterility; premature introduction of abstractions meets resistance
especially from critical minds who, before accepting an abstraction, wish
to know why it is relevant and how it could be used.

**3. Mathematics and science**. In its cultural
significance as well as in its practical use, mathematics is linked to
the other sciences and the other sciences are linked to mathematics. which
is their language and their essential instrument. Mathematics separated
from the other sciences loses one of its most important sources of interest
and motivation.

**4. Inductive approach and formal proofs**. Mathematical
thinking is not just deductive reasoning; it does not consist merely in
formal proofs. The mental processes which suggest what to prove and how
to prove it are as much a part of mathematical thinking as the proof that
eventually results from them. Extracting the appropriate concept from a
concrete situation, generalizing from observed cases, inductive arguments,
arguments by analogy, and intuitive grounds for an emerging conjecture
are mathematical modes of thinking. Indeed, without some experience with
such "informal" thought processes the student cannot understand the true
role of formal, rigorous proof which was so well described by Hadamard:
"The object of mathematical rigor is to sanction and legitimize the conquests
of intuition, and there never was any other object for it."

There are several levels of rigor. The student should learn to appreciate,
to find and to criticize proofs on the level corresponding to his experience
and background. If pushed prematurely to a too formal level he may get
discouraged and disgusted. Moreover the feeling for rigor can be much better
learned from examples wherein the proof settles genuine difficulties than
from hair-splitting or endless harping on trivialities.

**5. Genetic method**. "It is of great advantage
to the student of any subject to read the original memoirs on that subject,
for science is always most completely assimilated when it is in then ascent
state." wrote James Clerk Maxwell. There were some inspired teachers, such
as Ernst Mach, who in order to explain an idea referred to its genesis
and retraced the historical formation of the idea. This may suggest a general
principle: The best way to guide the mental development of the individual
is to let him retrace the mental development of its great lines, of course,
and not the thousand errors of detail.

This genetic principle may safeguard us from a common confusion: If
A is logically prior to B in a certain system, B may still justifiably
precede A in teaching, especially if B has preceded A in history. On the
whole, we may expect greater success by following suggestions from the
genetic principle than from the purely formal approach to mathematics.

**6. "Traditional" mathematics**. The teaching
of mathematics in the elementary and secondary schools lags far behind
present day requirements and highly needs essential improvement: we emphatically
subscribe to this almost universally accepted opinion. Yet the often heard
assertion that the subject matter taught in the secondary schools is obsolete
should be closely scrutinized and should not be taken simply at face value.
Elementary algebra, plane and solid geometry, trigonometry, analytic geometry
and the calculus are still fundamental, as they were fifty or a hundred
years ago: future users of mathematics must learn all these subjects whether
they are preparing to become mathematicians, physical scientists, social
scientists or engineers, and all these subjects can offer cultural values
to the general students. The traditional high school curriculum comprises
all these subjects, except calculus, to some extent; to drop any one of
them would be disastrous.

What is bad in the present high school curriculum is not so much the
subject matter presented as the isolation of mathematics from other domains
of knowledge and inquiry, especially from the physical sciences, and the
isolation of the various subjects offered from each other; even the techniques
and theorems within the same subject appear as isolated, disconnected tricks
to the student, who is left in the dark about the origin and the purpose
of the manipulations and facts that he is supposed to learn by rote. And
so, unfortunately, it often happens that the material offered appears as
useless and boring, except, perhaps, to the few prospective mathematicians
who may persist despite the curriculum.

. **7. "Modern" mathematics**. In view of the lack of connection
between the various parts of the present curricula, the groups working
on new curricula may be well advised in seeking to introduce unifying general
concepts. We think, too, that judicious use of sets and of the language
and concepts -of abstract algebra may bring more coherence and unity into
the high school curriculum. Yet, the spirit of modern mathematics cannot
be taught by merely repeating its terminology. Consistently with our principles,
we wish that the introduction of new terms and concepts should be preceded
by sufficient concrete preparation and followed by genuine, challenging
application and not by thin and pointless material: one must motivate and
apply a new concept if one wishes to convince an intelligent youngster
that the concept warrants attention.

We cannot enter here into detailed analysis of the proposed new curricula,
but we cannot leave unsaid that, in judging them on the basis of the guidelines
stated above (Sections I-5), we find points with which we cannot agree..
Of course, not all mathematicians have the same taste. Mathematics has
many aspects. It can be regarded as an instrument to understand the world
around us: mathematics presumably possessed this value for Archimedes and
Newton. Mathematics can also be regarded as a game with arbitrary rules
where the principal consideration is to stick to the rules of the game:
some such view may be considered suitable for certain problems of foundations.
There are several other aspects of mathematics, and a professional mathematician
may favor any one. Yet when it comes to teaching, the choice is not a mere
matter of taste. We may expect that an intelligent youngster would want
to explore the world around him, but we cannot expect him to learn arbitrary
rules: why just these and not others?

At any rate, we fervently wish much success to the workers on the new
curricula. We wish especially that the new curricula should reflect more
the connection between mathematics and science and carefully heed the distinction
between matters logically prior and matters which should have priority
in teaching. Only in this way can we hope that the basic values of mathematics,
its real meaning, purpose, and usefulness will be made accessible to all
students, including of course, the prospective mathematicians. The recently
expressed "widespread concern about a trend to excessive emphasis on abstraction
in the teaching of mathematics to engineers" points in the same direction.

Lars V. Ahlfors, Harvard University

Harold M. Bacon, Stanford University

Clifford Bell, University of California, Los Angeles

Richard E. Bellman, Rand Corporation

Lipman Bers, New York University

Garrett Birkhoff, Harvard University

R.P. Boas, Northwestern University

Alfred T. Brauer, University of North Carolina

Jack R. Britton, University of Colorado

R.C. Buck,University of Wisconsin

George F. Carrier, Harvard University

Hirsh Cohen, IBM

Richard Courant, New York University

H. S. M. Coxeter, University of Toronto

Dan T. Dawson, Stanford University

* First Summer Study Group in Theoretical and Applied Mechanics Curricula,
Boulder, Colorada, June 1961.

The Testimony of Tests

Avron Douglis, University of Maryland

Arthur Erdelyi, California Inst. of Technology

Walter Freiberger, Brown University

K. O. Friedrichs, New York University

Paul R. Garabedian, New York University

David Gilbarg, Stanford University

Sydney Goldstein, Harvard University

Herman Goldstine, International Business Machines Corp.

Herbert Greenberg, International Business Machines Corp.

John D. Hancock, Alameda State College

Charles A. Hutchinson, University of Colorado

Mark Kac, Rockefeller Institute

Wilfred Kaplan, University of Michigan

Aubrey 1. Kempner, University of Colorado

Lucien B. Kinney, Stanford University

Morris Kline, New York University

Ignace I. Kolodner, University of New Mexico

Rudolph E. Langer, University of Wisconsin

C. M. Larsen, San Jose State College

Peter D. Lax, New York University

Walter Leighton, Western Reserve University

Norman Levison, Massachusetts Institute of Technology

Hans Lewy, University of California, Berkeley

W. Robert Mann, University of North Carolina

M. H. Martin, University of Maryland

Deane Montgomery, Institute for Advanced Study

Marston Morse, Institute for Advanced Study

Zeev Nehari, Carnegie Institute of Technology

Jerzy Neyman, University of California, Berkeley

Frederick V. Pohle, Adelphi College

H. O. Pollak, Bell Telephone Laboratories

George Pôlya, Stanford University

Hillel Poritsky, General Electric Co.

William Prager, Brown University

Murray H. Protter, University of California, Berkeley

Tibor Rado, Ohio State University

Warwick W. Sawyer, Wesleyan University

Max M. Schiffer, Stanford University

James B. Serrin, University of Minnesota

Lehi T. Smith, Arizona State University

I. S. Sokolnikoff, University of California, Los Angeles

Eli Sternberg, Brown University

J. J. Stoker, New York University

A. H. Taub, University of Illinois

Clifford E. Truesdell, Johns Hopkins University

R. J. Walker, Institute for Defense

Analyses and Cornell University

Wolfgang Wasow, University of Wisconsin

André Weil, Institute for Advanced Study

Alexander Wittenberg, Laval University