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