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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 selfconsciousness 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 everpresent 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 brandnew and sophisticated course for high school seniors as analysis, but to students at Ohio State University, for example, analysis means the first really highpowered 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
Object Description
Title 
New mathematics  a controversy 
Author 
Strehler, Allen F. 
Subject 
Mathematics  Study and teaching  United States 
Citation 
Quarterly, Vol. 09, no. 4 (1963, December), p. 1417 
DateIssued  1963 
Source  Originally published by: Touche, Ross, Bailey & Smart 
Rights  Article not under copyright 
Type  Text 
Format  PDF image with OCR under text, scanned at 400dpi 
Collection  Deloitte Digital Collection 
Digital Publisher  University of Mississippi. Digital Accounting Collection 
DateDigitally Created  2009 
Language  eng 
Identifier  Quarterly_1963_Decemberp1417 