The Role of Axiomatics and Problem Solving in Mathematics, CBMS 1966



Notes on the Symposium Report, The Role of Axiomatics and Problem Solving in Mathematics, published by the Conference Board of the Mathematical Sciences in 1966: 


This volume contains papers written by solicited American authors, in­tended for submission to the International Commission on Mathematical Education for its quadrennial meeting as a sub­section of the International Congress of Mathematicians, held that year in Moscow.  The coordinator of this volume was E.G. Begle, head of SMSG and a member of the United States Commission on Mathematical Instruction.  Other members of that Commission were R.C. Buck, Burton Jones, Phillip Jones, Henry Pollak, and R.J. Walker.


There were 19 papers, of which the first eleven were avowedly about axiomatics while the remainder were scheduled to concern “problem solving”.  I list them all below, but shall describe only the papers concerning axiomatics, and one more, the paper by Peter Lax (# 15 below), which is really about the overuse of axiomatics in the schools, and not really about problem solving as construed by the other speakers.  The list of papers follows:


0.  Preface, Edward G. Begle (Stanford)

1.  The Use of the axiomatic method in teaching high school mathematics, Frank B. Allen (Lyons Township high school, Western Springs, IL)

2.  The use and abuse of the axiomatic method in high school teaching, Albert A. Blank, NYU

3.  The role of a naive axiomatics, R. Creighton Buck, Wisconsin

4.  Mathematics:  Its Structure, Logic and Method, Irving Allen Dodes, Kingsborough Community College of CUNY

5.  Axioms, Postulates and the teaching of elementary mathematics, Andrew M. Gleason, Harvard

6.  The axiomatic method in mathematics courses at the secondary level, Leon Henkin, UC Berkeley

7.  Mathematics and Axiomatics, Morris Kline, NYU

8.  The axiomatic method and school mathematics, Merrill E. Shanks, Purdue

9.  The axiomatic method in high school mathematics, Patrick Suppes, Stanford U

10. A use of the axiomatic method in teaching algebra, Herbert E. Vaughan, U of Illinois

11.  The role of postulates in school mathematics, Gail S. Young, Tulane




12. Some thoughts on problem solving, Nathan J. Fine, U Penn

13. The role of problems in the development of mathematical activity, Florence D. Jacobson,  Albertus Magnus College

14. The role of problems in secondary school mathematics, Phillip S. Jones, Michigan

15. The role of problems in the high school mathematics curriculum, Peter D. Lax, NYU

16 On individual exploration in mathematics education, Henry O. Pollak, Bell Telephone Labs

17. On teaching problem solving, George Polya, Stanford

18. Problem making and problem solving, Paul C. Rosenbloom, Columbia

19. Problems in the teaching of mathematics, Frantisek Wolf, UC Berkeley



Most of the authors were research mathematicians, principally logicians or pure mathematicians working in univer­sities, though Henry Pollak, who appeared in many educational conferences of the time, was an applied mathematician at the Bell Telephone Labs. Peter Lax and Morris Kline were professors at NYU’s Courant Institute, where the mathematics department had a strong applied character.  Kline was also known for his wide interests relevant to mathematics and its place in society, and in mathematics education as well, where he was already famous for his vociferous opposition to the New Math[1].  Others were pri­marily tea­chers (Frank Allen in particular, a high school teacher and later President of the National Coun­cil of Teachers of Mathematics, 1962-1964), or professors of college mathematics or of mathematics educa­tion.


The level of education addressed in the papers was mainly the high school, with particular attention to students intending to attend university, but even so, much in these papers had im­mediate application to school mathematics as well; the New Math controversies of the time, concerned with school mathe­matics, were undoubtedly uppermost in the minds of the au­thors.  Even had all the papers been explicitly directed towards university instruction, that would not have changed the audience for this report, an audience of mathematics educators, professors in teachers colleges, members of curriculum projects for the schools, and public school officials; such lessons as could be drawn from these papers would affect students right down to the kindergarten level.  The "sides" that had been taken from the beginnings of the New Math, and that had emerged in public disputes since even before the publication of the CEEB Report of 1959, and that had publicly pitted Begle and Kline against one another with such vigor in the seven years since that report, are clearly visible in the present vol­ume.



Frank Allen


Perhaps the most enthusiastic proponent (in this symposium) of the rigorous use of axiomatics in school math was Frank Allen, himself the author of several high school textbooks of algebra and geometry.  His paper begins with a "merciless analy­sis" (Allen’s words), or "flow chart", proving that for each real num­ber x, |x| = |-x| and -|x| £ x £ |x|, each step in his argument explicitly appealing to one of the axioms for a field or some rule of logic (or the definitions, of course).  His purpose here is, he says, to illustrate "a relatively new phenomenon in school math­ematics -- proof in algebra."


Allen asserts that high school textbooks on the American market are showing this trend, "that publishers now believe that there is a substantial demand for texts that emphasize structure and proof in algebra", and that "this belief is amply vindicated by an examination of the courses of study which fifty high schools have submitted with their applications for membership in Mu Alpha Theta since September 1965."  He concedes that since Mu Alpha Theta is a national organization of high school and junior college mathematics clubs his sample of fifty is not random.  "Nevertheless the contrast between current programs in these schools and the programs found in most high schools ten years ago is striking and highly sig­nificant.  One recent application, typical of many received, lists 'logic' and 'a closer look at proof' in its tenth-grade program and 'Statements and Sets', 'Ordered Fields', 'Math­ematical Induction -- Sequence and Series,' 'The Algebra of Vectors,' 'Functions,' and the 'Field of Complex Numbers' in the 12th grade sequence...  The change is so profound and far reaching that it can only be described as a revolution."


Allen immediately follows this with, "The writer is among the many teachers of high school mathematics who welcome this revolution.  We believe the axiomatic method of exposition will help pupils acquire a deeper understanding of elementary mathematics.  While we are aware that a vocal minority of mathematicians have expressed dismay with this method by denouncing 'excessive formality', 'trivial proofs', 'logical gems', and 'long lists of properties', our confidence in the method is, we believe, endorsed by the majority of mathe­maticians who have contributed to the improvement of school mathematics during the last decade by participating in the various writing groups.  Most important of all, this confidence is sustained by our daily experiences in the classroom."


The paper continues with detailed examples of classroom activities, some familiar high school material, such as the derivation of the quadratic formula, some less familiar, such as the analysis of the occurrence of an “extraneous solution” of an equation involving radicals, and some quite new to the schools of the 1960s, such as a formal proof of the statement, “If a2 is an even integer, then a is an even integer,” via the contrapositive version, “If a is an odd integer, then a2 is an odd integer” and the preceding theorem, that “t is an odd integer if and only if  t = 2k + 1 where k is an integer.”  In this statement it can be seen that Allen was even old-fashioned for his time; a logician would have introduced quantifiers here, instead of using “k” before introducing it, writing “where k is an integer” as a sort of afterthought, as was common in the early part of the 20th Century.  The more formal New Math advocates would have written, “…if and only if there exists an integer k such that t = 2k + 1.” 


Allen’s formal proofs, however, are not casual and contain no genuine afterthoughts or ambiguities; what’s more, they bristle with logical notation:  symbols for disjunction, conjunction, negation and implication, though for some reason he avoids the symbols for quantification and writes out “for every”, “there exists”, and “such that” in English.  Instead of the familiar “two-column” proof format generally prescribed for Euclidean proofs in the schools of his time and earlier, he introduces a “flow-chart” schema, with arrows for implication, each arrow surmounted with a numeral referring to some earlier known truth needed at that point.  (In Allen’s examples these numerals are invitations to the student, to supply the reason for the truth of the implication.  He makes much of the transitivity of implication, among other logical comments.)  The flow-chart arrangement saves much space, and is particularly transparent when hypotheses contain several clauses, either conjoined of disjoined; for the traditional scheme would require separate two-column displays for each part of the theorem.  And as Allen himself mentions, such a chart does mirror much that is found in computer programming, and can be more transparent to a professional than even the more casual “paragraph style” that modern mathematical exposition uses in professional journals.


Allen’s paper includes proofs in Euclidean geometry, though here the obstacles to rigor are more formidable than when “proving the obvious” in elementary algebra.  Allen does not avoid the battle, however, and takes an SMSG system of axioms, a mixture of statements of Euclidean flavor with others invoking knowledge of the real number system, as his starting point.  Taking the bull by the horns, he writes:


“We know that long lists of carefully worded postulates are often criticized and derided.  The following ‘Protractor Postulate’ from the SMSG Geometry with Coordinates, ‘If M is any plane and if VA and VB are noncollinear rays in M, then there is a unique ray coordinate system in M relative to V such that VA corresponds to 0 and such that every ray VX with X and B on the same side of VA corresponds to a number less than 180,’ will seem pretty formidable to a teacher whose ‘protractor postulate’ is a statement to the effect that every student must have a protractor.  Nevertheless the SMSG ‘Protractor Postulate’ is an essential link in a very significant exposition which enables the student to use his knowledge of real numbers in the study of geometry.


“We know, too, that formal proofs are often denounced as a particularly pedantic and artificial way to belabor the obvious – though I have never heard this charge made by a teacher who has presented such proof in the classroom.  Experienced teachers know that it is the ‘obvious’ that often blocks student understanding.  Flow-diagram proofs do not create new difficulties.  They merely expose the difficulties that are already there.  Is it better to ignore these difficulties or to examine them forthrightly?  We know of course, that the student will not encounter flow-diagram proofs in advanced courses in mathematics.  For this reason we encourage the use of essay proofs during the last semester of the twelfth grade.  However, this writer is convinced that there is no substitute for the flow-diagram format when the student is learning the structure of proof in grades nine through eleven.  The construction of such a proof demands the same level of understanding that is required to program for a computer.”


“Those who believe that teachers should encourage the development of intuition and the construction of plausible arguments should have no quarrel with this axiomatic method.  Every formal proof is preceded by many introductory exercises, experiments, and conjectures.  Many plausible arguments are presented by both teacher and pupil.


“Those who carry the banner for ‘discovery’ and for ‘multiple attack’ on problems should be particularly enthusiastic about the axiomatic method.  As noted earlier the multicontrapositive concept 10 [a reference to earlier exposition of his format of proof] suggests as many as n+1 different attacks on the proof of a theorem the hypothesis of which is a conjunctive statement having n clauses.  Some of these may be very easy to prove while others are difficult or even impossible.  Students are intrigued by the problem of selecting the one that is easiest to prove and by the fact that one proof will suffice to establish n+1 mutually equivalent statements.  After a student has verbalized all of the n (partial) contrapositives of a theorem having n clauses in the hypothesis, he begins to understand what the theorem says.”


Other examples of classroom exercises are given, and I will cite one more to make quite clear what sort of thing Allen is expecting the high schools to accomplish.


“We are to prove the ‘Coordinate Systems Theorem’ which is stated as follows in the SMSG text, Geometry with Coordinates.    Let a line L and two coordinate systems, C and C’, on L be given.  There exist two numbers A and B with A ≠ 0 such that for any point on L, its coordinate x in C is related to its coordinate x’ in C’ by the equation x’ = Ax + B.’


“My students decided to try to construct a general proof because they were not satisfied with the ‘proof by example’ shown in the text.  The essential postulate here is Postulate 13: 


“Let A and A’ be any two distinct points and let B and B’ be any two distinct points.  Then, for every pair of distinct points P and Q in space, [PQ (relative to (A,A’))] / [PQ (relative to B,B’)] is a constant.”


I’ll cut short the exposition here, except to say that Postulate 13 would be regarded by any professional mathematician as already and evidently saying the “Coordinate Systems Theorem” in slightly different words – provided he understood the terminology of Postulate 13 and a good bit else about the relationship of real numbers and geometric congruence as elucidated in the other Postulates which underlie the SMSG “Geometry With Coordinates” system.  Frank Allen’s proof of the Coordinate Systems Theorem, or, rather, the proof his students allegedly “decided to try to construct . . . because they were not satisfied with the ‘proof by example’ shown in the text”, runs to a page of very tedious formulas in logical format, saying the obvious in far from obvious language.  Even so, the final line in Allen’s proof is, “This assertion is readily verified by means of an 8-line truth table.”


Allen does not say, by the way, that the students who had allegedly demanded a proof to satisfy their doubts had been the ones to write it.  As described by Allen, the proof sounds very much like his work.  To say, as Allen does earlier in his paper, “that it is the ‘obvious’ that often blocks student understanding” is surely correct if by ‘the obvious’ Allen is referring to statements of the sort found in this theorem, or in Postulate 13, statements which obscure the intuitive notions underlying all analytic geometry as students of his time and ours (and scientists who use it daily) understand it.


An American mathematician of forty years later, reading Allen’s exposition, wonders what he could have said to Frank Allen in 1964, to stop this runaway train of New Math.  Probably experience is the only answer.  Yes, the so-called obvious is indeed a common stumbling block in students’ understanding; yes, one cannot sensibly talk of anything, even the graph of a linear equation, without knowing the meaning (if not the notation) of the statement, “Let L be the line {(x,y) є RxR | Ax + By = C}.”  But the obvious is best left alone in most contexts, unless it gives rise to a misapprehension the experienced teacher knows will bring trouble.


As for the “set-builder” notation used in describing the line L just above, that is certainly a gain.  Not every notational or pedagogical novelty of the New Math era was a mistake.  During the 1960s, in fact, many books used it to good effect, for even when the notation is no longer needed it can be a valuable introduction to more casual language.  I remember from my own experience (1941) in freshman “analytic geometry” that I did not at first understand that a phrase of the form, “the curve f(x,y) = k” really meant “the set of all (x,y) in the plane such that …”, and only after many examples (never explained as such) did I catch on.  It would have been better to teach me the idea of “{(x,y) such that …}” for once and all, with examples, before springing “Let z = f(x, y, z(x,y)) be …” in physics courses.  “If the thing on the right is a function of x and y,” I would think, “Then x and y are the cause of z.  But then, how can the z appearing on the right be part of the cause of the z appearing on the left?”  Thermodynamics was full of this sort of thing, and the writers of New Math textbooks were determined to remove the ambiguities and sloppy phrasings.  But precise phrasing, too, can be carried beyond reasonable bounds.


Would I have learned it faster if the full logical notation had been taught me in high school?  Perhaps so, but I know many of my classmates would not have learned it at all, maybe not even with Frank Allen as a teacher.  We know this now, for thousands of teachers were forced, by the temper of the times and the textbooks they had been given, to try to imitate Allen’s lessons, or those of Beberman or other master teachers, men who understood what they were saying as well as how children behave in classrooms.  But many teachers did not have a coherent textbook (for all that the market was crowded with new textbooks called New Math by their salesmen), and did not have a sufficient education in mathematics itself, beyond what they could collect from an NSF summer institute, to make a success of it all.  Frank Allen, with his Masters degree in mathematics and much more, was not the product of a teachers college.  What Allen was urging was not laughable; it was merely – when attempted on a mass scale – impossible.  And probably not even wise for most students, compared with what other sorts of things mathematical they might have been taught in the same time by the teachers already in place, were the text materials improved in some other direction.


There is a difference between the unnecessary rigor of Frank Allen and the simulacrum of rigor found in so many commercial textbooks of his time.  The former did have some successes;  the latter not.  I have known people who did take an SMSG high school geometry or algebra course and found it thrilling; they have credited such early instruction with leading them to become mathematicians (or economists, or scientists).  On the other hand, what would they have made of the following lesson, written by a teacher in a book designed as a high school geometry?  It was published two years after Frank Allen’s paper and surely reflected the new emphasis on rigorous proof, but it entirely missed the point of Allen’s prescriptions:


From Geometry:  A Modern Approach by Marie Wilcox, Addison-Wesley 1968.


Page 98:


Theorem 5-6

Any two right angles are congruent.

Given Ang(A) is a rt .angle; Ang (B) is a rt. angle

Prove Ang(A) = Ang(B)



1.  Ang(A) is a rt. angle                                 1. Given

2.  Ang(B) is a rt. angle                                 2. Given

3.  m(Ang(A))=90m; m(Ang(B))=90             3. A rt. angle is an angle with measure 90

4.  Ang(A) cong Ang(B)                                 4. Congruent angles are angles that have the same measure


We may now use this theorem to prove that certain triangles are congruent...


    Frank Allen was annoyed when his opponents claimed that SMSG was overkill, that such rigor was tedious and unnecessary, and that lessons invoking formalities of logic were boring to students, but as an experienced teacher, having used some of the foolish textbooks of the 1930s and 1940s, he might have foreseen that the foolishness would not automatically change just by virtue of his “revolution”. 


Marie Wilcox, the author of the quoted theorem and proof, had been a high school teacher  (and perhaps more) in an Indianapolis high school, and had been in 1954-1956 President of NCTM.  (Frank Allen was President in 1962-1964.)  She had been appointed (as had Allen) a member of Begle’s original SMSG advisory committee in 1958 and was a member of the SMSG summer writing session in 1960.  (Allen was a member of the summer writing sessions in their earlier years as well.)  Marie Wilcox was also a member of the SMSG Committee on Gifted Students. 


It had been Begle’s avowed intention to make the SMSG an organization bridging the gulfs between the colleges of education, the university mathematics professors, the active schoolteachers, the professional organizations representing these parties, and the world of commercial textbook publishing.  SMSG itself did not itself print anything as banal as appear above in the Wilcox book; SMSG was a deliberate effort to provide a model for commercial textbook writers, drawing from the experience of all parties.  What the commercial publishers then did with it all depended initially on the writers they recruited, but ultimately depended on sales.  A name such as that of Marie Wilcox, SMSG writer and past President of NCTM, was commercially valuable on a textbook of the 1960s, no matter what imbecilities it contained.  Begle’s name would have been worth more, but he never participated in a commercial school textbook series.  (He would never in a million years have written Theorem 5-6 as Wilcox did.)  Everyone associated with SMSG, and many who were not, were eagerly sought by commercial publishers to accomplish the purpose of SMSG, which was to serve as a model for other writers out in the competitive world of new textbooks.  If ever there was a time and a market for good mathematics textbooks, the middle 1960s was such a time. 


But neither an impressive name nor a good text was sufficient to make a book profitable.  Frank Allen’s Modern Algebra – A Logical Approach (Books I and II, with co-author Pearson, and published by Ginn & Co. in 1964 and 1966) was reputedly a financial failure.  One might well quarrel with its text, which was tedious for its intended audience but was at least correct, which Van Engen’s[2] middle school algebra was not, nor were Allen’s books trivial, as that of Marie Wilcox was. I do not know anything about the sales of the Wilcox book.  The SMSG author who outdistanced them all was Mary Dolciani, whose work was conservative by Allen’s standards, but which included, in its early editions, much of the SMSG structure; then gradually toned down its excesses in succeeding editions.  Her name still heads the list of authors of the posthumous editions of her high school analysis textbooks, which by 1990 became called “traditional” as distinguished from the styles, perhaps temporary, of the following decades.  Dolciani’s own work did have a lasting effect, but it was only at the high school, even “college prep”, level, but the axiomatics that spurred their creation remain hardly visible in their 21st Century versions, while Frank Allen’s books, axiomatic to the core though not a bit silly, were long unknown by then.


It was a great disappointment to Allen that the influential authorities in the world of mathematics education, both in the teachers colleges and the NCTM, repudiated his sort of effort once the 1960s came to an end.  He persisted all his life in trying to get school mathematics to be rigorous and coherent, becoming especially voluble on the matter following the publication of the 1989 NCTM Standards.  In 1995 he published an Open Letter to Jack Price, the then President of NCTM, chastising the mathematics education establishment for its advocacy of what became known as “constructivist” teaching, and for its sponsoring of what it called “reform” textbook series, in which axiomatics played no part whatever, but which with the help of the National Science Foundation gradually worked their way into the schools.  His 1995 letter had no noticeable effect, at least in the following ten years.


R. Creighton Buck


Creighton Buck was Chairman of the mathematics department at the University of Wisconsin at Madison, Wisconsin, and his paper for this meeting was entitled The Role of a Naïve Axiomatics.  He begins by citing the work of the CEEB Commission (1958), approving its call for “more attention to the deductive structure of algebra and for a greater reliance upon general principles rather than upon special tricks.”  But he immediately sounds a warning:


“However, in the hands of some who perhaps do not understand the role of axiomatics in mathematics, these points have been exaggerated and carried to extremes that are certainly unwise and probably harmful.  Unfortunately, we as mathematicians are at fault in that we have not communicated our attitudes toward our subject to the general community.  Too often, we have allowed others to speak in our behalf, and in so doing have allowed a distorted picture of the nature of modern mathematics to be widespread.  A concern for axiomatics represents only a small portion of the activity of a professional mathematician, and even less for the professional scientists for whom mathematics is a tool.”         


In particular, Buck notes that in his time the axiomatic structure of mathematics was, in the schools, being elevated to a definition of what mathematical material ought to be taught.  For Buck it is the modeling aspect of mathematics that is most important.  Euclidean geometry as construed by Euclid was a science much as classical mechanics is:  one begins with axioms not only as if one believed them, let alone as arbitrary rules for a pointless game, but because they are a summary of experience or experiment.  They are formulated rigorously in order to be able to deduce from them with full confidence things that are not so easily experienced directly, or that are too many to memorize as facts.  Modern logicians, on the other hand, proceed from axioms as if they were no different in principle than the description of the legal moves of a bishop in chess.  There is no harm in this; indeed it is a necessary and valuable attitude for the study of matters like consistency, for the behavior of the real world is irrelevant to the study of the inner structure of the logical or mathematical system; but in a complex subject like Euclidean geometry the investigation of whether some of the axioms are deducible from others is, for children in school, an arid study. 


Time and effort spent “proving the obvious” are a hindrance to progress, says Buck; “The course in geometry should be a study of geometry, not abstract axiomatics for its own sake.  Go as quickly as possible to the theorems on concurrence.  Prove that the process for constructing a pentagon works...”  It is worth observing that Buck’s two examples here are not chosen at random; they represent two very fundamental aspects of Euclidean geometry, the projective and the metric.  Each example, or class of examples, opens out into its own wonderful vista of modern geometry, which (alas) only a few students of high school geometry will ever get a chance to see.


In the case of the number systems, Buck would begin the study of negative numbers by examples from temperatures, and yardage lost in football plays, to arrive at useful definitions for subtraction and multiplication, say, and only after some experience with what comes naturally should the teacher formulate the rules (“axioms”) for a field, from which the procedures of algebraic manipulation follow.  But even then, he cautions against proving such things as that for every x, y, z, and w, (x+y)+(z+w) = (x+w) + (y+z).  “These statements are dull, and the student learns very little about the number system from them.  A little of this goes a very long way, and even a very good student is to be forgiven if his interest flags.”  Buck suggests elementary number theory instead.  As for fields, he notes there is little of interest to prove for fields as such, and that polynomials on the one hand, and certain special fields such as the reals and complexes on the other, are the things that have both interest and importance for students; one should get to those as quickly as possible, except in advanced work for potential mathematicians, where the study of the logical systems themselves is in question.  Even “the very subtle problem of how one can make a logical analysis of the nature of logic” had  Buck’s approval, but for mathematicians and philosophers, not the schools.


Though naming no names, Buck argues against those who would begin the systematic study of the rational number system with Landau’s formulation, that begins with Peano’s axioms for N and creates from them via suitable definitions a model for the rational field.  To put the matter brutally, the rationals are shown by Landau’s construction to be isomorphic with certain equivalence classes of ordered pairs of certain other equivalence classes of ordered pairs of positive integers, when these things are fitted with certain quite technical definitions of equivalence, and then addition and multiplication.  A proper development, such as Landau himself gives in a famous little book[3], necessarily requires careful definitions followed by an ordered sequence of small theorems developing the arithmetic of the system via tedious proofs of the properties of addition and multiplication of the defined objects.  This doesn’t make these monsters the actual rational numbers, which we all know from an early age, and use freely when teaching children how to cut up pizzas and compare batting averages; all Landau does is to show that the properties of fractions, as we know them from elementary experience and certain traditional models, are logically consistent with what we take to be obvious about the counting numbers. 


Buck warns against any effort to import all this into the classroom (except for future mathematicians, who really do need to understand it): “Those curriculum designers who have attempted to follow this Landau pattern in developing the number system in elementary and junior high school may have felt that such a formalized treatment gave more meaning and concrete substance to the nonrigorous and intuitive concept of negative number and “fractions” which children brought to the classroom.  On the contrary, I believe that children have a strong intuitive feeling for the number line, and are quite willing to use it as a basis for a model of the number system ... If they must see a demonstration of relative consistency, let it be that of Euclidean geometry, done analytically in terms of the number system.”


Buck does not offer this particular caution without experience.  A year earlier he had been joined by several members of his mathematics department at the University of Wisconsin (located in Madison, Wisconsin) in a struggle with the Madison school authorities about the adoption of a series of textbooks[4] for the middle school grades, a series that did exactly what he was objecting to here.  The development of the rational number system from the positive integers (assumed already known) had been taken by Van Engen and Hartung, the leading authors of this middle school series, from Landau’s exposition, including (apparently) all the rigor that goes with ordered pairs and equivalence classes.  However, the actual wording of their text and examples often betrayed a failure to understand precisely the distinctions that the Landau development was designed to clarify, so that only a competent mathematician already knowing how should have been written (and had been, by Landau) could make sense of it.  It is really painful to read.  (Scott, Foresman printed and sold a Canadian edition, too.) 


For school children the series was a double disaster, since it consumed time and destroyed student interest by the attempted elucidation of something neither needed nor understandable for students that young, and then garbling it so that it was incomprehensible as written.  Buck and his colleagues testified before the Madison school system’s book selection committees, wrote commentaries and appeals, and lost.  The school board took its advice from (among others) Van Engen himself, and Van Engen was a professor of Education and Mathematics at the same University of Wisconsin as Buck.  The written response of the school board rejected the appeal on the grounds that the Van Engen books were more modern than those Buck favored, and had the approval “of mathematicians” – as if Creighton Buck, Walter Rudin and Richard Askey were not.  The Madison authorities knew nothing of mathematicians and mathematics, but as both Van Engen and Hartung held PhD degrees in mathematics, Van Engen even holding a joint appointment in both Education and Mathematics (surely a double credential!), at the University, it must have appeared to them a mere in-house dispute among mathematicians.  Professor Buck might have made a better case had he indulged in some sociological discussions here.  Instead, he confined himself to  the case itself, while Van Engen’s party added the claim that a “vendetta” had been launched against Van Engen (Van Engen actually believed this, in these words); and some unknown person around town even generated a rumor that Buck had a financial interest in the outcome.


Irving Allen Dodes


The paper of Dodes (p. 27-43) immediately follows that of Buck, and while its placement is the result of the alphabetical ordering of authors in the volume it could hardly have been better placed to illustrate what Buck had in mind when writing, as quoted above, “In the hands of some who perhaps do not understand the role of axiomatics in mathematics, these points have been exaggerated and carried to extremes that are certainly unwise and probably harmful.” 


Dodes taught at Kingsborough Community College in New York City at the time of this conference, but had earlier been Chairman of the Deprtment of the famed Bronx High School of Science.  He was also the author of Mathematics: A Liberal Arts Approach (1966) and several later books intended for early college-level students, with applied subject-matter in computer programming, finite math, and statistics; but in the subject matter of his paper, Mathematics:  Its Structure, Logic, and Method, prepared for this symposium, he was out of his depth.  Like Frank Allen, Dodes was an enthusiast for axiomatics for students in the schools, and shared none of the qualms of Buck, Pollack or Peter Lax, who also spoke at this conference; yet the latter three, along with many of their colleagues who also worked in the most rarefied domains of pure and applied mathematics, were certainly not unaware of anything Dodes had to say.  Among other things, they were competent to notice that Dodes himself didn’t get it quite right, and that whatever the psychology of students might be, no pedagogical skill or experience is sufficient to get children to understand what is not correct, even if it can command their agreement. (Mathematicians call this “proof by intimidation”, echoing the more familiar “direct proof”, “proof by induction,” and “proof by contradiction”.)


Dodes is addressing the question of axiomatics at the level of k-12, and structures his own paper carefully, with one section each for “structure”, “logic”, and “method”, intending to describe how much of each can or ought to be presented to students at varying stages of “mathematical maturity”.  Mathematics, he says, has a “liberal arts aspect”, a “propaedeutic aspect”, and a “service aspect”; and of course these distinctions are often better appreciated by a professor in a community college than by one who devotes the major part of his life to mathematical research.


Under “structure” Dodes outlines the axiomatic scheme in principle, and explains that a student should at all times know the “status” of the items of his present knowledge: whether a statement is a definition, an axiom, or a theorem, and whether a term is an undefined term or one defined in terms of other knowledge or conventions.  So much is correct and important, but then he gives some examples, rhetorically asking what is the status of each of the following:


                                                           i.         If equals are added to equals, the results are equal;

                                                         ii.         Radii of a circle are equal;

                                                       iii.         –1 X –1 = +1.


He says that “in the usual development, the first of these should be called a theorem, the second is a conventionalized part of a definition, and the third is part of the definition of multiplication in a field and is, therefore, a postulate.”  Here the reader, if he is a mathematician, begins to suffer, and if he is a student he should begin to tune out.  The part about the radii of a circle is correct; that is indeed part of the definition.  But the other two are quite problematic.  (i) is discussed at some length in the chapter concerning ignorance in school mathematics[5], and while it is usually called an axiom it is really not a statement having mathematical content at all.  It is a conventional way of reminding students that two symbols for the same thing (number, usually) can be substituted for one another when (among other operations) addition is in question; once addition known to have a definition at all, it becomes something like saying a rose is a rose is a rose.  Dodes does provide a proof just the same; it leans heavily on the statement that the system under discussion is “closed”, in this case meaning that if x and y are members, “addition” is presumed to have meaning for them. 


Well, yes.  There is some historical warrant for this caution, and it goes back to the 19th century development of the very idea of a group, when groups were preeminently groups of transformations and the ones under study at any moment were in general proper subsets of the group of all transformations of a certain kind (e.g. permutation groups, in the Galois theory), with composition the group operation.  A certain set of transformations, in this setting, would be called a group if, to begin with, it was closed under this pre-existing operation.  This mind set persisted into later expositions of group theory, and Dodes must have studied from some text or teacher who listed the abstract axioms of a group in such a way as to include the word “closure” in Axiom I, where a really abstract beginning would begin by merely positing the existence of a mapping from GXG to G with certain properties.  By 1966 it was really indefensible to speak of groups in the abstract, as Dodes was doing here, as if they were necessarily subgroups of something else, as if “closure” were something to be ascertained rather than known from the mere fact that “+” was taken to have a definition.  To caution high school students that it might not, in which case “x+y = x+y” might sometimes fail for want of a meaning for “+”, is plain silly.  If “+” is not defined, why are we arguing about it?  The other fact his proof requires is the “reflexivity of equality”.  One must wonder that anyone thought all this would do anything in a high school math class but infuriate the students.  Even Gertrude Stein didn’t posit so banal a proposition as that “a rose is a rose implies that a rose is a rose”, unless that was the intent of her more famous dictum.


(iii) is not one of the usual statements given in the axioms for a field, but can be proved from them.  “In the usual development” it is a theorem, actually.  Dodes here avoids an essential part of the explanation of an axiomatic system, if it is to be explained to students at all, which is that the distinction between theorems and the axioms need not be the same for all presentations.  If a student is to be aware of the status of his knowledge, which he should, he should not be told such things as these without more elaboration.  (iii) can be one of the postulates for a field, but some other of the usual postulates can be dropped or changed if that is the case.  The most usual postulates for a field F state that with the two operations, + and *, F is a commutative group under + and F \ {0} a group under *, with 0 and 1 the names of the identity elements, respectively,  -a and 1/a denoting the inverses of a under + and *, and the usual distributive laws holding.  For these axioms, (iii) becomes a theorem.


More important than this misapprehension of Dodes, such as it is, is the fact that students have known the number “– 1” before taking whatever education in axiomatics Dodes has in mind, and that these students should have been given good reason to want the product in (3) to be what it is long before ever hearing about a field.  As we will be reminded by Peter Lax in a later paper in this symposium, it is not that (3) is dictated by the field axioms so much as that the field axioms were selected to codify such properties as lead to (3); if they did not, the world would not have much use for them in school algebra.  Along with the status of his knowledge from an axiomatic point of view, then, the student should know the status of the axioms themselves in the framework of the understanding of mathematics and its uses.  The complaint of the 1960s was that students knew what a field was but could not calculate .47712 + .30103.  This complaint had some justice, and arose exactly from the shifting of emphasis exemplified by the Dodes commentary upon these examples.  Time spent on axiomatics might be well spent if the connection between axioms and mathematical necessities for modeling were made plain, and if the students had the maturity and time to appreciate both the axioms and the attendant mathematics.  When the first is garbled and the second scanted, all that time is doubly lost, for a bad lesson must be unlearned if the good one is to replace it successfully.


There are of course two more parts to the Dodes paper:  Logic, and Method.  Under the first of these headings he gives a list of nine devices (he could list more, he writes) that he calls the “media” of proof:


By arithmetic

By algebra (often called a “derivation”)

By truth table

By informal “direct proof”

By informal “indirect proof”

By use of symbolic logic, e.g., contrapositives

By formal “direct proof”

By formal “indirect proof”

By mathematical induction          


He gives examples -- at an International Congress of Mathematicians.  The naiveté of this list needs no explaining, and to reproduce any of his written-out illustrations would be supererogatory.  The flavor of his recommendations for the classroom is sufficiently conveyed by the opening of his next section, which is headed (p.35) Proof in the Algebras:


On the whole, the best place to start proving is in the first course in algebra.  The course should start with the elements of sets and symbolic logic.  One of the first proofs is the following:[6]


             (iv)    {x ε G | x + a = b} = {b – a}


This sentence has no period at the end, but more seriously, it has no quantification for “G” or “a” or “b”.  One supposes that we may assume G is a group and a and b members of G.  The announcement of (iv) is followed by a two-column proof in nine steps that (for a given x, a, and b in G) if x+a = b, then x = b - a; but Dodes doesn’t notice that the statement of his theorem (iv) requires proof of the converse.  Following the half-proof he writes,


In actual practice, some students insert more steps and some insert fewer; this is a matter of taste.  (The more perceptive tend to use in-between steps to show that  –a exists in the group, too.  However, this is not so important.  The important fact is that in this proof there is a chance for a class to find out what is needed to convince a mathematician....


I don’t quite know what Dodes means by saying the existence of –a is problematic.  It is clear in any case that there is something garbled in his understanding of “group” and the “minus” symbol, along with what every mathematics teacher should know even at the expense of never studying a truth table: that solving an equation requires more than the uniqueness part of the exposition.  This algebraic “proof” is then followed by similar examples taken from geometry and calculus.


Under the section labeled Method the reader is even more surprised.  Nothing philosophical turns out to be meant here; to Dodes, “method” is not what Poincare was alluding to in his well-known Science and Method.  Instead, Simpson’s Rule and MacLauren’s Series are “methods”, where the paper continues with allied suggestions for high school lessons, or calculus problems, solved in the language of proof, again always one-way, i.e., without “checking the answer”, something of a comedown after the nine “media of proof”.  When all is said and done, Dodes turns out to be a teacher after all, concerned with what should and should not be included in “Algebra I” and “Algebra II”. 


With historical hindsight one can see Dodes returning to the classroom and, after a chapter of New Math, continue to teach as he has always done, sometimes “by mathematical induction”, sometimes “by arithmetic”. Then, when the 1970s arrive he will, with the rest of the country, stop trying to teach the chapter of New Math axiomatics found at the front of his textbooks, because after all he didn’t find it helped much in the trig course.  The publishers discovered the same thing as their field representatives reported back, and almost all of them dropped that chapter from their “revised, improved, modernized” 1980 editions of the same old books.  But this is hindsight; in 1966 that development was not yet visible to all.  Those who saw this pretense of modernity -- and its misunderstanding -- and who were offended by it as Peter Lax and Creighton Buck were, were struggling hard to counter it with reasonable arguments, but for the moment, for the decade of the 1960s, and certainly in Madison, Wisconsin,  it was without success.


The Dodes paper has a final section called “A Blueprint for the Future”, and it is not at all the future I conjectured for Dodes above, nor is it the future that actually took place after 1966.  In this section Dodes becomes particular, and under the headings Algebra I, Geometry, Algebra II (including trigonometry), and Twelfth Year Mathematics, he presents explicit list of topics.  The whole presentation takes a printed page, but the flavor of the suggested curriculum may be had by reading the first and third of the four segments:


Algebra I

    It should now be possible to carry students through: an introduction to symbolic logic and the nature of proof; an introduction to sets, functions, relations and number systems leading to the concepts of groups and fields; methods for finding solution sets of equations and inequalities, including systems of these; graphs; and some right triangle trigonometry.  The present anomalies leading to the lack of definition for numbers like √(-4), hence to a lack of understanding of the number or roots of a quadratic with complex roots, would be avoided.  Inclusion of a discussion of the complex field will clarify the entire course.  If possible, enrichment topics should include simple determinants and simple problems in probability.


Algebra II (including trigonometry)

    The greatest changes should probably take place in the second algebra course, including trigonometry.  At the present time, this course proceeds as though there were almost no content in Algebra I. With the first course firm, the symbolic logic can be continued to quantifiers, abstract algebra can be taken through groups, rings and fields in a somewhat more rigorous fashion; set theory can be expanded to include the Boolean algebras and more advanced  probability (including the Kemeny tree); the exponential, logarithmic and trigonometric functions can be discussed as isomorphisms over the reals; there is ample time for an introduction to vector and matrix spaces; and a considerable time can be spent on a substantial unit in plane and solid analytics, for which the students would now be ready.


What is notable about this part of his outline, and true of the rest, is the entire absence of mention of any reason for wanting to know these things, that is, the absence of mention of either the human experience of the world that makes it convenient to reduce empirical beliefs to lists of axioms, or the human experience of the world that is then clarified by using these axioms, via logical deduction, to discover other truths, perhaps not immediately evident to observation.  There are some mathematical solecisms in the text (trigonometric functions as isomorphisms?) which, coupled with the occasional old-fashioned language (“analytics”), signal a rather recent conversion of Dodes to the abstract approach to school mathematics, but these are only part of what make his presentation an important indicator, that part being this:  If Dodes sometimes failed to understand the import of his abstractions even in the construction of logical arguments, how much could the world expect of the more ordinary high school teacher?  (Dodes himself was teaching in a junior college, actually, and beyond the level of even the more ambitious high schools). 


It is a mark of the times, argued by many teachers in the educational journals as well as by mathematicians in conferences like this one, that the so-called revolution in mathematics was omitting any applications of mathematics as something to be studied under the heading of mathematics.  In this Dodes presentations, probability is mentioned, but only by name.  Many more words are devoted in his essay to the meaning of the √(-4), something that his own classroom experience teaching about quadratic equations told him was crucial – but crucial to what?  To Dodes himself, educated under an earlier dispensation, these applications were natural – everyone knew arithmetic and how to model a problem about a rock falling under the influence of gravity – while it was the structure of fields, especially the complex field, that illuminated so much of what had been obscure in his own earlier experience.  Neither Dodes nor other partisans of truth tables and quantifiers imagined their students could end up learning logic or algebra without the substratum that had nourished themselves in pre-New Math years; but in the event many did. 



Andrew Gleason


Gleason was a professor of mathematics at Harvard and a distinguished pure mathematician.  His paper is short and makes a single point, though he was capable of making many more, and in other venues did.  Granting that all mathematics is necessarily deductive, he still distinguishes between what he calls the “axiomatic” method and the “postulational” method, the words harking back to Euclid.  Axioms, for this discussion, are things we begin with because we believe them to be true, while postulates are merely formal starting points, the rules of the game.  Mathematics begins, he says, with axioms:  We believe that 2+2=4, and by much practice with elementary arithmetic (of positive integers) we come to believe in the ring axioms for these numbers.  So much of mathematics is, like the very beginnings of Euclidean geometry, something by which children can link mathematics with the world about them, and learn how to command that world in some degree.  Later, when negative numbers are introduced, it is neither convincing or psychologically attractive to students to show them that the ring axioms imply that (-1)X(-1) = 1.  In their minds, as in the minds of physical scientists who use mathematics, it is not the ring axioms that govern the world, but experience.  One can only take the ring axioms as the beginning of the study of all the integers (or all the reals) if we and they first have some experience telling us that (-1)X(-1) = 1 is consonant with what we believe to be true for some informal observations and interpretations; only then does the discovery that this rule is consistent with the ring axioms no longer seem arbitrary.


“It is confirmation that we have set down the facts correctly.  Concomitant with axiomatic mathematics is the realization that we may not have set down the facts correctly.  A rethinking of the fundamentals is, therefore, never totally excluded.

“A great merit of the axiomatic approach is that it admits the possibility of a mixture of deductive and empirical thinking.  Euclidean geometry is notoriously deficient in dealing with the order of points on lines.  The remedy, as far as elementary geometry instruction is concerned, is not to introduce half a dozen highly technical axioms for order, but to admit candidly that these questions will be handled by inspecting a carefully drawn figure.  This mixed approach is characteristic of all branches of science.  Axiomatic mathematics is simply the application of the scientific method to problems classically recognized as purely mathematical”


Gleason remarks here that modern pure mathematics is always presented from a purely postulational point of view, but it doesn’t follow that mathematicians are really that indifferent to the actual truth of what they are saying, not many of them, anyhow.  “Yet there is a movement which would direct elementary education toward postulationalism.  The proponents of this movement do not suggest that we teach irrelevant mathematics.  But they fail to appreciate the importance to a beginning student of a justification of the postulate system.  In their hurry toward the more sophisticated ways of viewing mathematics, they forget the importance in their own education of the hundreds of examples which, boring as they may have been, did show that mathematics has something to say about the real world.”


He concludes with recommending that longer chains of reasoning be introduced only after suitable preparation, so that students will never think the development “irrelevant’ to their world.  Having seen the utility of a chain of reasoning eventuating in a usable theorem, in simplifying the problems they must solve, they will have more patience with a longer chain of such reasoning when it comes up in later work.


“Organized deductive methods in mathematics should be introduced as soon as they can legitimately contribute to the student’s ability to understand and simplify the problems before him.  The approach must be axiomatic in the sense I have described above because the overriding consideration is relevance.  There is no necessity for completely organized axiom systems as long as we are honest about where we are using deduction and when we are being empirical.”


Leon Henkin


Leon Henkin was a logician, a student of the famous Alonzo Church at Princeton and himself a longtime professor at Berkeley.  He was, however, no stranger to applied mathematics, having spent four youthful years on the mathematical analysis of military problems during World War II; but his academic research was always in abstract logic.


His paper for the conference on axiomatics (and problem solving) begins with the assumption that he is speaking about secondary students only, and at that he is limiting himself to those who will be able to understand the material he is recommending.  He does say he believes this will include “most” secondary students, which is probably mistaken, but this does not affect the validity of his remarks.  It does affect the way in which those remarks would get used by persons impressed with his ideas but who do not understand them completely, and who do not remember the early warning he quietly offers concerning the audience for such work.


Henkin would like the axiomatic method to be learned, used and understood even by people who have no use for formal logic in daily life, but who “should have some appreciation of the basic means by which scientific knowledge is gained and applied.”  Of course, future scientists and the like are included in his potential audience for logic a fortiori.  Henkin begins by listing what he considers the most important features of an axiomatic presentation:  (a) its organizing power, (b) the possibility of more than one model for the axiom system under study (e.g. there are many fields in algebra), and (c) the idea of isomorphism. 


He notes, as did many other participants in school mathematics debates of the time, that Euclidean geometry is a rather clumsy way to introduce axiomatics, even though it had been traditional to do so for two thousand years.  Euclidean axioms suffer from being very numerous (if given fully, as Hilbert was finally able to do only sixty years before), and seemingly rather artificial, so that school geometry tends to avoid mention of enough of them to make the resulting geometry really illustrate the axiomatic method in full rigor.  The student must use intuitive methods, as Euclid did, alongside the few axioms that had been stated, however rigorously.  The second deficiency of Euclidean geometry, said Henkin, was that when given in full the Euclidean axioms permit only one model, and when given without (say) the parallel axiom, the alternative models are not already within the students’ experience, and so become a difficult study of their own, diverting attention from the purpose of the exercise, which here was the appreciation of the idea of axiomatics itself.


Henkin therefore advocates – for this purpose – the study of number systems.  His primitive system is the structure P of positive rational numbers.  He explains that these numbers are already known to us from their application to arithmetic and measurement, so that this is a good start.  He lists a number of properties they enjoy, properties we all have taken as obvious from childhood, and calls them axioms.  He explains that he could have started with other axioms, but for his purpose these seem convenient.  The first axioms he chooses (and numbers (i) through (iv))  are these properties of addition and multiplication: associativity, right and left distributivity, and solvability of the equations ax=b and xa=b for all a and b in P; also that P is not empty.  He deliberately fails to introduce commutativity at this point, but shows that the four axioms he has are sufficient for the theorem stating the existence of a unique right identity for multiplication, and the same on the left, and that they are the same.  He therefore introduces a symbol for the identity and so on to the usual properties of multiplicative inverses. 


At this point one might expect some more field axioms but instead Henkin shows his origins as a logician, by giving as his Axiom (v) the statement that any subset of P closed under addition and multiplication must be equal to P.  This is a second order axiom, involving as it does the class of subsets of P.  (Henkin remarks here that it can be proved that it is not possible to generate P with an axiom system entirely of first order statements.  Since the statement is not as intuitive as the others, he shows, using a form of the principle of mathematical induction in his admittedly intuitive proof, that the system of positive rational numbers we are all used to does have this property.) 


Next he deduces the commutativity of multiplication and the basic rules for computation with fractions, and is ready to produce the two axioms needed to finish the system.  They are (vi) that for all x, y in P, if x+1 = y+1 then x=y (from which he is able to prove the commutative law for addition), and the final axiom (vii) that P contains more than one element, from which he is able to conclude the story.  His final remark, that any two realizations of P must be isomorphic, however, would rather spoil one of his criticisms of the Euclidean system as a learning device, were it not for the pedagogical discussion which follows.


Returning to his ideas for a high school presentation of these ideas, Henkin writes,


“Our experience suggests that the first explanation of the purpose of axiomatization should be limited to the desire to study the logical relations among mathematical propositions holding in a given structure, with the question of self-evidence handled in an incidental way (as in our discussion of Axiom (v) above).  Only later should the applicability of given axioms to a variety of models be brought out.  The latter concept is more meaningful to the student after he has actually worked a while with logical derivations; the delay also provides an element of dramatic impact which is valuable in maintaining and enlarging the student’s interest in the work.


“It is to be emphasized that a thorough understanding of the axiomatic method cannot be achieved if a student’s study is limited to working within a given axiom system.  He should also be asked to find definitions.  Furthermore, after working through the development of given axioms he should be asked to construct an axiom system for some other theory.  (For example, an axiomatic theory of the integers is a natural counterpart to the system given above.)  Discussion of the considerations affecting the construction of a set of axioms gives opportunity for considering a host of fruitful concepts – including questions of mathematical esthetics.


“The program discussed above can be profitably carried through in average high school classes today during a second year of algebra.  As with other truly fundamental concepts, however, we should look toward a future development of the curriculum in which at least rudimentary forms of the axiomatic method appear early in the elementary curriculum, reappearing in increasingly significant forms through the grades.  When such an elementary curriculum has been achieved the secondary student will require a more sophisticated treatment of the axiomatic method than the one we have outlined.


“One possible direction would consist in allocating to the secondary schools responsibility for a complete and thorough treatment of the real number system.  This would involve axiomatic treatment of the natural numbers, the rationals, and the reals, as well as definitional constructions of each of the latter systems on the base of its predecessor system.  Such dual treatments have several advantages:  the construction provides a relative consistency proof of the axiomatic version, and the axiom system shows what are the essential properties of the construction.


“Another direction for deepening an understanding of the axiomatic method after a suitable groundwork has been laid in elementary school would consist of an axiomatization of the algebra of sets, leading through Boolean algebras to a general introduction to representation theorems and their significance.”


This is the end of Henkin’s paper.  His suggestion that the program he outlines, concerning P and its variants, is suitable for the average high school Algebra II class is quite breathtaking, yet he doesn’t hesitate to say ex cathedra that it can be done, and more, once the elementary school curriculum is beefed up with a sufficiency of axiomatics in preparation. The peroration is begun with the phrase, “Our experience suggests...”  One wonders about that experience.  I have never heard it referred to.  That some such things, though with a bit less reference to formal logic, were believed by others, however, is undeniable.  John Kelley said much the same thing a few years later, at least as to the construction of the reals beginning with the Peano axioms for N, averring that the better high school textbooks were already doing this[7].  In this he was somewhat correct, if, with a suitably inappropriate interpretation of the word “better”, the Van Engen series is taken as an example.  But however ambitious he was for American school mathematics programs he couldn’t have read Van Engen’s middle school textbooks with approval, for Kelley knew how to write modern mathematics correctly and even with grace and style.  It is curious that a generation of graduate students in mathematics was brought up on Kelley’s General Topology, and yet Kelley imagined that axioms of the same degree of abstraction were known to him as featured in “the best currently available commercial textbooks”. 


      Morris Kline


      After all this it is surprising to read the paper of Morris Kline, the gadfly of the New Math, who had been a severe critic of the new directions as far back as the 1950s, while the CEEB Commission was preparing its report, and who had been the leader in the preparation of the “Letter of the 75 Mathematicians” published in 1962[8] in formal attack on – essentially – the program being put forward by SMSG, a letter that brought into the open the split in the community of mathematicians concerning the advisability of the abstract approach to school mathematics that had been developing since Beberman’s early work at Illinois. Kline was called intemperate, inaccurate, “an alley fighter” and worse, by those he offended, and was much criticized for his sweeping characterization of all the New Math programs as if they were all the same, and as if they all included every inexcusable) feature he rather grandly attributed to them.


    In this symposium, which was for an international audience, Kline could not assume familiarity with the exact dimensions of the current American efforts at reform, nor did he have any interest in attacking anyone in particular, for the politics of American pedagogy was not something to debate in Moscow.  His short paper was bland and moderate in tone, but got his message across nonetheless. 


He began with reference to Euclid, regretting that the  traditional study of the iconic Euclid had given modern students the impression that axiomatics was of the essence, as if the work of mathematicians began with axioms and proceeded deductively.  “This superficial and largely erroneous view of mathematics has become especially popular in our generation….  Had other works of the classical Greek period survived, the Western world would surly have gotten a better understanding of the true nature of mathematics.  The Greeks did possess it.  They knew that conjectures must precede proofs and that analysis precedes synthesis…”  And, “We have only to look at the development of mathematics from 1500 on.  The greatest creations since that time have been in the fields of algebra and analysis and these subjects had no axiomatic basis until about 1900.”


    Recognizing that he is speaking of children now, students and not professional mathematicians, he continues to insist on the point:  “Even if instruction in mathematics is intended merely to transmit knowledge, the totally deductive presentation gives no insight into what is accomplished.  It is the finished structure with all the scaffolding removed, as Gauss once remarked, … in other words, the ideas underlying a proof are difficult to discern in the stilted deductive formulation.”  Even philosophically, he says, the 20th Century studies of the foundations of mathematics have shown – and he cites Godel’s theorem – that “no significant branch of mathematics can be encompassed in an axiomatic approach.”  Thus, citing Hermann Weyl, he says that “mathematics rests on intuitions and the proof can be no more than the hygiene.” 


    Kline gets a bit more vigorous as he goes along.  Speaking of creativity, he lauds those who seek their ideas from far afield:  “guessing, conjecturing, blundering, trial and error, induction from concrete evidence and all the other diverse and haphazard processes which enter into thinking.   “A particular axiomatic structure … acts as a straitjacket on the mind.”  And there is also the matter of the applications of mathematics.  We wish “to understand and master the physical world.  The more immediate goals in a branch [of mathematics] whose structure is already established may be purely mathematical such as generalization and abstraction, but one must never lose sight of the larger objectives.  To understand and pursue these intelligently is also part of mathematics.”


    Now, Kline allows a place for axioms, as “a check on thinking and a systematic organization of the knowledge gained” -- but for school children?  “Here pedagogy rather than a priori consideration dictates what we must do.  Proofs must be of what is not obvious, on the basis of what is obvious, or it will mean nothing to a student.”  Here Kline is attacking the kind of textbook that was becoming popular in his time, where exactly the obvious, such as that a line segment has a midpoint, was being proved in geometry books on the basis of dozens of axioms and lemmas, but being defended partly on the grounds that such simple proofs were needed to exhibit the structure of mathematical thinking.  “The capacity to appreciate rigor is a function of the mathematical age of the student and not of the age of mathematics.  This appreciation must be acquired gradually and the students must have the same freedom to make mistakes that the mathematicians had.”


    Thus Kline advocates going easy on axiomatics, especially the mathematicians’ desire to make do with a minimal independent set of them for each structure under study.  “These conditions on, or properties of, axiom systems must be sacrificed in behalf of pedagogy.”  He notes that the teacher can work himself into a trap with his love of a minimal set of axioms, finding that it is then terribly difficult and time consuming to prove anything significant.  The textbook writer then cheats a bit by adopting as an axiom something that could be a theorem, for example, that a circle has area πr2.  It is true, after all.  And thus, he observes, by working their way out of one trap (the minimal set of axioms) into another, current school textbooks in geometry now had come to contain as many as 70 or 80 “axioms”.


    Even with a reasonable set of axioms, understandable and familiar to the student because he has been using them all his life (the field axioms for the rational numbers, for example), Kline cautions against making them explicit and of undue importance as part of the curriculum.  They would then have to be used with explicit mention in common practice (else why mention them so fervently?), and “to show, for example, that 3ab(ab + 2ac) = 3a2b2  + 9a2bc [sic], he will require an interminable amount of time to do even the simplest algebra.” 


    That much is certainly correct, though downright concealing the commutative law from the students is not necessarily the answer.  One can know the reasons for computations such as he illustrates here and still, after some practice,  perform them without thought as rapidly as one who has been taught the procedures without having been given the reasons.  Here Kline rather goes off the deep end, extending his argument beyond what it can withstand:


    “We should be grateful that the students accept such facts without question, and, in fact, do all we can to make the elementary operations so habitual that one does not have to think about them at all, any more than one thinks when he ties his shoelaces.  If students do not see readily that 3 times x = x times 3, it is not because they lack familiarity with the commutative principle, but rather because they fail to understand that it [“x”] is just a number (or, more properly these days, a placeholder for the name of a number).  Those who are concerned that commutativity may become so inbred that students will refuse to multiply matrices are not only needlessly fearful but are trying to rush the educational process.”


    But then Kline returns to more reasonable considerations.  He does note that historically algebra and geometry have been taught quite differently, with students under the impression that geometry proceeds from axioms and that algebra is a bag of disconnected tricks, amounting to a game.  “The proper reform should certainly uniformize the treatment of the two subjects.    Some deductive structure should be introduced in algebra but by no means a rigorous one.  Axioms such as equals added to equals and the like can certainly be use to accustom students to the idea of proof.  On the other hand, we must add especially to the geometry, but also to the algebra, the motivated, constructive approach which will permit students to see why we investigate topics such as congruence and take part in fashioning theorems and proofs.”


    “… We do not seem to recognize or admit that students have the right to question why we establish particular theorems or techniques.  The student who complains, ‘Who cares whether the altitudes of a triangle meet in a point?’, is wiser than the one who passively memorizes the proof and hands it back on an examination…”  Here emerges Kline the gutter fighter:  The student “who questions why”, and the student who “memorizes the proof for regurgitation”, are not the only kinds of students; posing this as a dichotomy is unfair.  It was this kind of argument with a straw man that so infuriated Begle that he once wrote a letter[9] to Max Beberman asking his advice on the question of how to handle “the Morris Kline problem”.  Begle does not seem to have succeeded in his effort to create a united front with Beberman against Kline’s attacks, and it may be noted that Beberman maintained good relations with Kline throughout his life, despite all.  Though Kline attacked Beberman’s programs as vigorously as he did Begle’s, Kline and Beberman did partly share a pedagogical philosophy.)


    Kline concludes with a further attack.  Given the baneful influence of a foolish axiomatics in the new programs in the schools, he asks rhetorically, “Why is the axiomatic approach so popular?” and answers his own question in a way unlikely to earn him any friends in the New Math movement: “The answer seems to be that it is the easy way out.  Hardly any work is required to present axioms and their consequences in the approved fashion.  One merely repeats the steps and the reasons.  On the other hand, to know why a theorem should be proved at all, to penetrate to the basic idea of the proof, and to know why one proof is preferable to another or why one order of theorems rather than another is necessary calls first of all for a depth of understanding.  Moreover, having acquired the understanding, it is far more difficult to present the mathematics so that the students take part in the divination of theorems and proofs, and then in the casting of these discoveries into the axiomatic mold.  To state it otherwise, to motivate and to teach discovery are difficult tasks and require much thought and preparation.  It is far easier to take the position which Samuel Johnson once adopted when he refused to explain further something he had said, ‘I have found you an argument, but I am not obliged to find you an understanding.’”


    Unkind as these remarks may be, Kline could have gone further.  He could, also with justice, have said that many of those who wrote materials called New Math were not necessarily doing so from laziness only, making things easier for themselves, but were merely too ignorant to do anything else.  We shall always have the ignorant with us.  In the pre-1950 generation of math educators we had those who didn’t understand the “standard curriculum” of the time, as can be seen in, e.g., the case of Dean Minnick[10] who imagined he had found an easy way to explain “limits” and the irrational numbers.  No amount of teaching skill and experience could have helped him write a coherent “methods” book for future teachers.  The easy way out, though Minnick wasn’t looking to save energy and was merely honestly unaware of what he was doing, was to take the standardized school math of his time and repeat it.  It is not only in mathematics that such a phenomenon is seen, for in every domain of human life even the honest follower of a current orthodoxy, when questioned, has to take refuge in the formulas of the time, for he knows no other response than to repeat them.


    Van Engen did this, though he didn’t quite get his formulas right.  Another formula of the time can be seen in the following careful, two-column proof by Marie Wilcox[11], that any two right angles are congruent, because by her own definition of “right angle” they both have measure ninety degrees, and because the definition of congruence of angles is equality of angle measure in degrees.  Her argument much resembles that of Molière’s imagined physician, who explains that morphine puts one to sleep because it has a dormitive virtue.  As written out in the Wilcox book the two-column display is:


Theorem 5-6 Any two right angles are congruent.



Given Ang(A) is a rt.angle; Ang(B) is a rt.angle

Prove Ang(A) = Ang(B)



1.  Ang(A) is a rt.angle .........…………1. Given

2.  Ang(B) is a rt.angle .........…………2. Given

3.  m(Ang(A))=90; m(ang(B))=90 …....3. A rt.angle is an angle with measure 90

4.  Ang(A) cong Ang(B) .........…….…4. Congruent angles are angles that have the same measure


 One might object here that Ms. Wilcox has forgotten here to include, between (3) and (4), the steps



3a. (for all  numbers x), x = x ……….3a.  Reflexive property of equality;

3b. 90 = 90 ………………………….3b. Substitution rule for universally quantified statements.


As will be seen below, Suppes would have had high school students learn at least enough logic to see why the extra steps are needed for a complete logical proof.  (“90=90” is not one of the axioms for the real number system, after all, and hadn’t yet been proved as a theorem or lemma in the Wilcox book before its use here.)  But Suppes also cautioned that the student, while entirely and necessarily mindful of such details, should also be taught the conventional ways in which mathematicians abbreviate proofs.  Such a student, of course, wouldn’t bother to prove the Wilcox “theorem” at all, it being trivial in view of the axiomatic scheme Wilcox thought she had in mind.  (It is curious that Euclid took the statement that all right angles are congruent as a postulate, a prescient choice; but Euclid’s system was such as made the statement non-trivial.  Perhaps it is from this historical fact that Wilcox imagined the statement to need proof in her book.)


    Beberman, Begle and Kelley of course knew what the mathematics behind their axiomatics was, and what the real world behind the mathematics looked like as well, and were equipped to answer the hypothetical student questioner (‘Who cares whether the altitudes of a triangle meet in a point?’) posited by Kline.  But not all mathematicians are good at this, and the majority of teachers in the field, often stifled further by ignorantly written textbooks created for a sudden 1960s market for books featuring “sets”, and names of numerals, were not.




      With the Kline paper and its predecessors, the reader of this 1966 symposium would probably have found all the points of view of the time already expressed, the later papers essentially repeating the same arguments for and against the use of axiomatics in school mathematics, but even the following, less extensive review of the remaining papers on axiomatics will still be of some value, if only to indicate the relative volume of the voices for and against full logical rigor at the level of the public schools.


    Merrill Shanks


    Shanks was a professor of mathematics at Purdue University. His  life-long interest in school mathematics, and in the education of future teachers of school mathematics, is evidenced in the fact that Purdue offers an annual Memorial Prize in his name, to an undergraduate student in mathematics education. His papers included reports on the progress of the Ball State school mathematics program, in which he participated.  The Ball State program, led by Charles Brumfiel, was probably the best known curriculum programs of the time, after Begle’s SMSG and Beberman’s UICSM.  Shanks was, therefore, an educator and a research level mathematician both, for he  also did significant work in pure mathematics,  and directed several PhD theses.


    His paper for this symposium is entitled The Axiomatic Method and School Mathematics, and he appears not to be at all troubled by the math wars going on about him. He takes as normal that the current “revolution” should finally be taking place, pointing out the traditional place of Euclid in the school curriculum, though deploring its ever diminishing rigor in the schools during the first half of the 20th century, and deploring even more the pretension of rigor in the most recent trextbooks.  He explains that the axiomatics he describes is not to be a straitjacket, but amounts only to reasoned discourse in the English language.  He emphasizes that the success of the axiomatic method in teaching depends on the insights of the teacher, who “can either reinforce or distort the aims of the best designed syllabus.”


    “My major claim for good axiomatics in the classroom is that students actually thrive on it…. They enjoy seeing underlying reasons…  Axiomatics need not be deadly.  I recall seeing a selected class discussing extemporaneously and with keen interest a proof that a ray with endpoint in the interior of a triangle must meet the triangle.”  He states that student “discovery” is as present with the axiomatic method as it is in free exploration or in problem solving, evidently approving without discussion that ‘discovery’ is good pedagogy.  He then offers the usual cautions, in answer evidently to the popular notion that theoretical learning in the schools was displacing “basics”: 


    “First, there is the matter of technical skill, of algorithmic competence.  No program of instruction is viable that allows the pupil to push on without this competence…  Second, there is the matter of applications…”  At this point he acknowledges that many of the new programs are deficient in their integration with the study of science, but notes that one difficulty is that the sciences are not presented, in the schools of his time, in a sufficiently axiomatic manner to allow of mathematical intervention.  But then he goes on to the more serious problems of teacher competence and the writing of textbooks.


    “It is instructive, and disheartening, to read through many school texts and fail to find any march of ideas.”  His amplification of this observation for the next few paragraphs is not, he says, a plea for explicit axiomatics in the textbooks (or not until an advanced level), but for the textbooks to be written by authors who themselves have a clear underlying axiomatic structure in mind.  Without it, students cannot know “the ground on which they walk”; with it students will find it easier to learn.  And he lays out the order of ideas, briefly but clearly.  One can recognize all the ills of the textbooks of his time in his simple enumeration of desiderata from kindergarten to high school pre-college mathematics courses, and those ills are not much different from those of forty years later.


    Patrick Suppes


    Patrick Suppes, a professor of philosophy at Stanford, had been a student of Ernest Nagel at Columbia and was at the time of this conference already known for his book, Axiomatic Set Theory (Princeton, 1960).  During the decade 1960-1970 he was quite active in research in logic, and directed the PhD theses of six or more students at Stanford.  During the 1960s he directed a project, for which he wrote text and workbook materials, in the teaching of geometry to very young (first and second grade) children, and as of February, 2005, he continues to be the director of the computer-based Education Project for Gifted Youth (EPGY), based at Stanford since 1985.


    After an introduction stressing the importance of “the axiomatic method” in mathematics itself since the time of Eudoxus, and an acknowledgement that his advocacy of axiomatics in the schools was being challenged (probably a reference to critics such as Morris Kline), he begins, “The point of view that I want to present here is to make a [sic] vigorous defense as I can of the importance of teaching the axiomatic method in high school mathematics.”  Suppes means exactly what he says here, as it turns out, that is, he is not merely advocating, as Shanks had done, that an axiomatic structure underlie each part of a school mathematics program, as a structure that must be present in the teacher’s mind, to govern the order of ideas as presented to the student; he advocates a formal study of the major axiom systems for school algebra and geometry and indeed a formal study of the very idea of an axiom system.  Naturally this requires some attention to the rules of logic, and his paper outlines all this as well.


    He gives three reasons:  Firstly, that it makes the subject easier to learn, in that the basis is definite and seen by the student to be limited; thus that the student is made confident that he has in his possession all he needs to proceed.  Relying on intuition, he says, makes a student uncertain, especially in geometry based on the real number system. 


    Secondly, the study of axioms develops the intuition needed to prove theorems in mathematics.  The writing of mathematical proofs in college mathematics, he says, is notoriously difficult for students, and the study of axioms and the writing of simple elementary proofs, using simple logical devices, is the best preparation for the more complex work ahead. 


    Thirdly, axiomatics teaches students to “think mathematically”. 


    He also comments that there is a close relationship between the working out of the implications of an abstract axiom system and the working out of a problem in applied mathematics.  The marshalling of data in an applied problem is very like the construction of an axiom system; thus the teaching of students to invent a system of axioms for some purpose is in effect a sort of applied mathematics.


    Then the paper goes into some technical detail concerning axioms used in logic, algebra, geometry and calculus, not to defend his thesis but to illustrate some of the material.  In the case of logic, he shows how the axioms of mathematics at the high school level can be arranged (using “iff” in his definitions) to avoid the existential quantifier, which so bedevils students of algebra and analysis.  Then the axioms for an ordered field  (“Euclidean field”) containing square roots of positive elements are given as the basis for his algebra. “The beauty of these axioms is that their intuitive content should be familiar to the students.”  At a later stage, he says, one can introduce  “the concept of a real closed field, that is, a field that is Euclidean and is such that every polynomial of an odd degree with coefficients in the field has a zero in the field.”  This sort of thing goes on when he comes to geometry and calculus, and will not be further described here.  The paper concludes with an explosion of formulas in a footnote containing an axiom system due to Joseph Schoenfield.


    It is hard to believe that Suppes was imagining a real high school mathematics class, with blackboards and examinations, in which his quantifier-free logic system was being used to help students understand Euclidean geometry via vector spaces over Euclidean fields.  On the other hand, no real high school classes actually tried this, though there were a few, encouraged by people of Suppes’s leanings, who did try to smuggle the Landau construction of the real number system into middle school textbooks,[12] albeit with existential quantifiers in the old style.


    Herbert E. Vaughan


    Vaughan was a young assistant professor of mathematics at the University of Illinois in 1952, when Max Beberman arrived there to inaugurate the UICSM and in effect the entire New Math phenomenon in the schools of the following twenty years, and since Vaughan worked with the Beberman UICSM project and co-authored some of the texts used in the high school program, his influence was considerable there.  Vaughan’s principal mathematical interest was logic, and it showed up in the joint Beberman-Vaughan textbooks[13] and program, especially in Beberman’s insistence on precision in mathematical statements, the distinguishing of “number” from “numeral”, and the use of set-theoretic language and notation.


    Vaughan’s paper for this Moscow conference is entitled, A Use of the Axiomatic Method in Teaching Algebra.  He refers to the common opinion among high school teachers that axioms are for geometry and not for algebra, as indeed was implicitly suggested by the textbooks of the first half of the century, which presented the rules of algebra as something quite different from axioms, though their origins were seldom questioned by either student or teacher.  Even the axioms of geometry, he points out, are not usually construed (by the high school student) in the way modern logic treats them; the student takes them as descriptive of the physical space he lives in, and in a scientific spirit sees that a limited number of observations he can verify with the ruler and compasses in his hands can lead, via logical reasoning, to complicated statements he might never have imagined true or sought to test empirically.  The objects of algebra, Vaughan points out, seldom appear in this light to such students.


    “During all his school life,” Vaughan writes, “[the 9th grade student] has been learning about various kinds of numbers and operations on them.  If he could be brought to understand the question, “What is the cardinal number 2?” he would be likely to consider it a dull one…. One way in which students can be made acquainted with the real number is by suggesting to them situations in which the ‘unsigned” – or nonpolar – numbers with which they are already acquainted prove inadequate.”  Here Vaughan imagines they might, on learning about negative numbers and, seeing that they appear to obey the commutative and distributive rules (etc.), become interested in formalizing what they know about these abstract objects known to them in a manner reminiscent of the way the axioms of geometry formalize what they know about the concrete objects of their world.  Though he denies that ordered fields and such should be imposed on students at this level, he is clearly looking for a way to justify that final formality when the time is ripe.


    On meeting a new sort of number (the negatives, here), the students should be encouraged to invent names for them, and to take possession of them in other obvious ways.  Vaughan argues that there is a great difference in attitude between being given axioms for invented entities, and playing with those entities as if they were there all the time (negatives are immediately found and interpreted, after all) and trying, a posteriori, to find statements characterizing them.  The same is possible, he says, for the reals, though earlier he denies the advisability of introducing Cauchy sequences, and doesn’t elaborate on how a “prior” acquaintance with real numbers is to be obtained, as might lead students to a desire for axiomatization.  Nonetheless (Vaughan imagines), now that the students have “come to have a sense of personal involvement in the development of the subject [they now appreciate] the subject’s value to the point where they are not inclined to ask, “What good is it?”  Anyone who has done research can testify that something in which he has had a hand is, for that reason if for no other, important!”


    “A more pertinent point is that, as it is used in such a course, the axiomatic method shows up as a powerful way of organizing knowledge.  Students learn that one’s choice of axioms is somewhat arbitrary – one man’s axiom is his neighbor’s theorem.  This emphasis on organization furnishes motivation for studying the process of deduction. …”


    Back and forth it goes, explaining how to motivate high school students to axiomatize, to learn how to prove theorems (but not trivial theorems, which would lose their interest) by their use, with a view to an ultimate understanding of the whole process, which has the valuable side-effect of developing one’s “power to think clearly and to organize his knowledge”, which is, after all, widely agreed to be one of the goals of the study of mathematics in general.  Vaughan makes out the case that axiomatization in algebra is probably better suited to these purposes than is Euclid’s geometry.


    And he adds, “Finally – in answer to an objection which is sometimes voiced – just as a deductive treatment of geometry does not hinder a student in learning to solve geometric problems, neither does a deductive approach to algebra make it more difficult for him to develop manipulative skills and to use algebra for solving problems.”


    Here it is apparent that Vaughan has not addressed the point of these objections.  In solving a geometric problem, say, to prove that the medians of a triangle are concurrent, one has before him all the ingredients of the problem:  a picture of the triangle and of the medians, with all the data visible there and a clear question stated in terms of the picture. 


But the “algebraic” problems that really trouble the high school student, in a real high school class, are not so easily related to axiomatic algebra, since they are often not mathematical problems as such.  They have to do with the age of the sister five years ago, the speed needed for the second mile, or the amount of water delivered to a cistern by a hosepipe.  Later on, the high school level “problems” in “algebra” might be based on geometric descriptions, such as asking the distance of (a,b) from a line passing through (c,d) and perpendicular to L.  Solving a given equation is the easy part, as every teacher knows, while distributivity has virtually nothing to do with translating these things into equations. 


It simply does not occur to Vaughan that the “objection which is sometimes voiced” against the New Math of his time, in which he was a pioneer, was that “using algebra to solve problems” usually meant “to solve problems which are not algebraic in origin.”  These latter problems are not about quadratic equations or the quotients of polynomials; they are engineering problems, finance problems, geometric problems, problems in the solution of which these equations and quotients might be used.  Though this matter does not concern axiomatics as such, it does concern “the role of axiomatics in school mathematics”, which was the subject of this symposium.


    At the end, Vaughan remains concerned about another logical question, a distinction that never occurs to the practicing mathematician, and so he adds an APPENDIX which begins,


    “I realize there is still a possibility that some readers will not be ready to grant that there is a difference between studying the theory of complete ordered fields and studying the real number system axiomatically.” 


He goes on to explain that while all realizations of complete ordered fields are isomorphic, this doesn’t mean that “the real numbers” refers to any old complete ordered field, but only to the one we call “the reals”. Well, then. Philosophy. With the elucidation of this point Vaughan lays to rest the last possible niggling objection a high school teacher might have to his description of the proper use of axiomatics in the case of the reals.

    Gail S. Young


    Gail Young was topologist and had been a student of R. L. Moore in Texas.  Moore had been famous as an exponent of what might be called “discovery” in the teaching of mathematics, though at the collegiate and graduate student level.  Though this method had demonstrably led to a certain narrowness of outlook among his students, there was no denying that he had been singularly successful as a teacher, and discovery learning was one thread of the New Math period often ignored by later commentators.


(It does appear in the latter part of this symposium, the part on “problem solving”, however; and Max Beberman had taken to discovery learning – as a pedagogical device -- with enthusiasm, side by side with the formality of his written textbooks.  Begle’s SMSG showed no signs of it, though the SMSG textbooks did not command any particular pedagogical approach.) 


Young was a professor of mathematics in several universities, the author (With J.G. Hocking, who had been a doctoral student of his at the University of Michigan) of a topology textbook for college or beginning graduate level mathematics students, and as time went on he was increasingly involved with pedagogical matters


    As with most of his commentary on mathematics education, Young appears here as a measured observer, pointing out the pitfalls in the extreme positions and weighing the balancing involved in any choice of program.  To begin with, he gives as a minimal requirement for introduction of rigorous reasoning in the schools the necessity for the teachers to have mastered the idea themselves, and points out that this condition was hardly or not at all met in his time, even by teachers fortunate enough to have spent a summer or two in one of the NSF Institutes which provided some of the best mathematical experiences such teachers were likely to have had.  Yet, Young notes,


    “It is natural that many teachers found in their retraining the idea of rigorous mathematics to be very difficult and did not master it in the couple of summers they spent in study.  Not surprisingly, some of these overemphasize rigor in their own teaching.  Exactly the same thing can be said of some college teachers or textbook writers.  This has been responsible, I believe, for many distortions in school mathematics.  When I see a seventh-grade text (age 12-13) that treats rational numbers as equivalence classes of ordered pairs of integers, and another that defines them as formal solutions of equivalence classes of equations mx = n, m and n integers, I can only account for the books by such an explanation.”


    The textbooks Young had in mind were well known to his American audience, and certainly included the Van Engen[14] series and such things as the Marie Wilcox geometry[15], even though Van Engen and Hartung could have known better.  Even John Kelley imagined the construction of the reals to be a desideratum for high school students, albeit those of superior ability.  On the other hand, even though they were the leaders among reformers harshly criticized by opponents of excess rigor in the school classrooms, Beberman and Begle never imagined to include such things as Van Engen’s equivalence classes and Kelley’s limits in the actual curricula written for UICSM or SMSG.


    Young argues, quite reasonably, that there is a useful distinction between a convincing argument and a rigorous proof, and that for school purposes the former is to be sought. 


    “The real spirit of contemporary mathematics is that of creative understanding, of abstraction for greater clarity of thought and ease of proof, or experimental study of the relatively concrete.  It is not rigorous deduction of theorems from fixed postulate sets.  That is a tool, not the goal.”  How much of this tool can we expect students to learn at this or that stage of their mathematical education?  Young assumes preparation for first year college is the goal, and doesn’t expect much change there in the immediate future:  Calculus is likely to stay where it is.  Beyond that, however, he sees things that were not traditional successors, e.g. linear algebra, game theory, group theory (for physical chemistry, he says) and computer science. 


How should the school program prepare for this?  One suggestion is “to begin the study of algebra with some simple set of axioms, say, those for a group.”  Young disapproves because such a beginning places all the emphasis on proof, leaving the student with no idea of the purpose of all this.  To begin with a rich enough experience of groups themselves before getting abstract, which is the way mankind did it historically, would take a year.  One reason the necessary prior experience takes so long is that the group axioms are not categorical; there are too many different structures that are groups.  Then, what about N?  (The Peano axioms are categorical.)  Again, Young disapproves.  The positive integers are at the other extreme, too familiar.  Proving things about N would seem like proving the obvious, while the truly important things here, the principle of mathematical induction and the prime factorization theorem, don’t need all that formality. 


Young therefore settles on the reals as the proper substratum for the use of the axiomatic method, with the axioms themselves presented as a handful of properties, many of them already known and considered natural by the students, chosen to be referred to in any subsequent theorems and proofs.  The properties of a field (distributive law, inverses) have mostly already been isolated in pre-high school grades in modern programs, he notes, and further discussion of these in some other connection, modular arithmetic, say, can serve as a good introduction to algebraic ideas, while other properties of R can be isolated as needed, over a period of two or three years, without axiomatic formalities.


    The most traditional axiomatic structure in the schools has been geometry, of course, and Young rather lightly approves the Euclidean system and rather lightly outlines the difficulties it offers; yet it is clear that he really recommends a year of some sort of geometry, maybe with the Birkhoff-Beatley axioms  (as SMSG did) or maybe using “the European axioms in terms of rigid motions,” which he doesn’t explain further, though probably referring to the Dieudonné suggestion of Rn with Pythagorean metric, from which rigid motions are then defined by matrices. They are all good, he says, but he hastens onward:  “There is a movement to cut down sharply on axiomatic geometry in the schools, to perhaps 10 or 12 propositions, in favor of an increase in analytic geometry.  The case is utilitarian, I understand, for better preparation for calculus.  However, it seems to me to be a loss of intellectual content, and of mathematics.  The pressure for more mathematics in the same time is certainly great.  But “more mathematics” should mean more real mathematical power, more familiarity with logic and structure, more hard problems solved, rather than more facts learned and techniques mastered.”


    At this point Young observes that two sorts of critic will disagree with what he has said.  The first group would like more postulational work, much more “pure mathematics”.  Of them he asks (again) where they will find time for enough examples to make all this meaningful, and what they would do with students who really need enough down-to-earth analysis to prepare them for calculus.  The other group is the users of mathematics, “the engineer, physicist, etc.”  To them he says first that, yes, most of their concerns are not denied by what he outlined, since he was concentrating on only a portion, the postulational portion, of the school curriculum; and second, that they must accept a partial displacement of the traditional “pre-calculus” program because modern mathematics is a plain necessity even in applied fields, there having been a revolution in mathematics in the preceding thirty years.  “We are training for the future,” he writes, “One must try to get something of the spirit of modern mathematics over to our 15-year-olds, because when they are 45, that is part of what they will have to understand as physicists, engineers, etc.”


    One may read Young’s presentation without knowing exactly which part of the disputed territory he wishes to favor, or even how much of it is disputed.  He himself disputes almost nothing, his only forceful recommendation being that the algebraic axioms most of us never saw before graduate school should not be given and developed rigorously in the schools.  Less forcefully, he recommends Euclidean geometry as a good place to learn logical reasoning, albeit with some vagueness about the axioms themselves.  And finally, he recommends everything else.  It is impossible to look back on this paper, after forty years, and see just where he was wrong.


    The role of problem-solving: Peter Lax


    The remaining papers are avowedly about problem-solving in school (or college) mathematics, and with one exception I shall not discuss them, even though this means missing out on the presentations of Henry Pollak, George Polya and Paul Rosenbloom.  The exception is Peter Lax, then as forty years later a professor at the New York University Courant Institute of Mathematical Sciences, a renowned center of research in applied mathematics and in much of pure mathematics as well.  Lax was almost unique among the panelists in having had virtually no professional concern with the problems of mathematics education in the schools during his career up to that point, and a similar lack of connection thereafter.  (Creighton Buck is probably the only other example among the participants in this symposium.)


    The title of Lax’s paper is The Role of Problems in the High School Mathematics Curriculum, and his expertise as to problem-solving being beyond dispute he was placed in that part of the symposium; but what he said was “about problem-solving” in the same sense that a nature of summer is illuminated by a talk dealing with winter alone.


"I will take my illustrations from the lower end of the high school curriculum:  the multiplication of fractions and the multiplication of negative numbers, two somewhat dry subjects...."  He multiplies a/b by c/d by partitioning a rectangle into suitable subrectangles, counting them, and seeing how many are needed to represent the product of a/b and c/d; he thus "proves" the definition ac/bd to be the right one.  As for the arithmetic of possibly negative numbers, he imagines a train on a track, on some point of which a controller takes measurements of distances and times from the control station at a certain moment, using ∆x/∆t as a measure of average velocity; to avoid using four different formulas, as might be required to avoid the use of negative numbers, he measures time from 0 and distance from some point along the route (+ or -) and gets a single formula that serves the purpose no matter where he stands relative to the two positions and the two times.


To Lax these two problems are what dictate the algebraic rules concerning fractions and negatives, and not the fortuitous fact that the definitions preserve some of the algebraic laws already known to be valid for whole numbers, and positive numbers.  “Had we defined multiplication or division otherwise than we did these operations would be useless for these particular applications; on the other hand if these new operations did not share the usual properties of multiplication and division, then we would be unable to manipulate them [conveniently].”


p115:  "In contrast to this approach through problems, the current trend in new texts in the United States is to introduce operations with fractions and negative numbers solely as algebraic processes.  The motto is: Preserve the Structure of the Number System.  I find this a very poor educational device: how can one expect students to look upon the structure of the number system as an ultimate good of society?"


In another place (p. 115) Lax writes, "Of course I agree that many of the traditional problems of high school algebra are just as artificial [as the logical exercises leading from given hypotheses to predictable conclusions, usually of trivial interest to students], such as the ones involving perverse children who, instead of disclosing their age as asked, relate it in obscure ways to that of their brothers, sisters, parents, etc.  The remedy is to stick to problems which arise naturally; to find a sufficient supply of these, covering a wide range, on the appropriate level is one of the most challenging problems for curriculum reformers.  My view of structure is this:  it is far better to relegate the structure of the number system to the humbler but more appropriate role of a device for economizing on the number of facts which have to be remembered."


    And in conclusion (pages 115, 116):


    "What motivates textbook writers not to motivate?  Some, those with narrow mathematical experiences, no doubt believe those who, in their exuberance and justified pride in recent beautiful achievements in very abstract parts of mathematics, declare that in the future most problems of mathematics will be generated internally.  Taking such a program seriously would be disastrous for mathematics itself, as Von Neumann points out in an article[16] on the nature of mathematics – the most perceptive ever written on the subject -- it would eventually lead to rococo mathematics.  As philosophy it is repulsive, since it degrades mathematics to a mere game.  And as guiding principle to education it will produce pedantics [sic], pompous texts, dry as dust, exasperating to those involved in teaching the sciences.  If pushed to the extreme it may even cause the disappearance of mathematics from the high school curriculum along with Latin and the buffalo."




Lax was and is not alone in his view that presenting mathematics in strictly axiomatic form is poor pedagogy at grade school levels, for all that it is philosophically necessary at some point, and enables some part of our profession, maybe even all of us, to pay close attention to the details of reasoning behind all putatively correct mathematical conclusions.  Lax had already seen some of the “dry as dust” textbooks produced by some of the folks “with narrow mathematical experiences.”


Me, too.  I once inadvertently conducted a small experiment by asking a number of acquaintances the question, “How do you tell students why the product of two negative numbers is positive?”  I wasn’t looking to verify some hypothesis; I was just interested.


From Charles Rickart, a Yale mathematician, I got this answer (this was in about 1956):  “I tell them that if you have two dollars deposited in each of three banks, you are ahead by six dollars; if you owe two dollars to each of three banks, you are behind by six dollars; but if there are not three banks to each of which you owe two dollars, why, then you are ahead by six dollars, aren’t you?”


    From Henry Pollak, a Bell Laboratories mathematician I questioned on this and related matters many years later, in a telephone conversation of about 1996, I got two different such stories before I cut him short, since our conversation was really about something else, that I was more anxious to get to in the time we had.  I no longer remember his illustrations in detail but both of them involved such things as the distance advanced by a train unwittingly headed in the wrong direction, but forced to back up for a while.


    Neither Pollack nor Rickart mentioned the field axioms!


    But when I asked a math teacher of less mathematical experience or insight than Rickart or Pollak, his almost instinctive answer (this was also in 1996) was that the proposition was an easy consequence of the distributive law and the definitions concerning negatives; and he was certainly able to produce a lucid proof on that basis.  Curiously, however, he was unable to understand my next question at all: I asked him how he knew the distributive law applied to negatives, or mixtures of negative and positive numbers.  He became angry, thinking I was mocking him. 


I can think of no better illustration than this, of the danger Peter Lax was calling to our attention in the final sentences of his article.  The field axioms are not the cause of ab = (-a)(-b); they are a summary – a miraculous fact – of properties we have other reasons to observe and want to codify for computational purposes.  It is wonderful that so many things fall under the field axioms, but they aren’t a law of the universe, after all.  There are rings that don’t happen to be fields.  Josiah Willard Gibbs had to give up several properties that would have been handy for cross products, could they have been made consistent with the rest of his system of vector analysis.  He was driven by the geometry, not a preconceived axiom system, to stick with the ungainly product he seemed to have, whether he liked it or not.


It was a bit of a fad in the 1990s, among NCTM devotees, to point out that children who used the mistaken formula a/b + c/d = (a+c)/(b+d) got a result that had meaning in baseball, for example:  If a batter scores a “hits” out of b “at bats” on one day his batting average is a/b, and if he scores c out of d on the following day, his average for the following day is c/d.  To get his average for the two days taken together, the computation is (a+c)/(b+d), and not the average of the two earlier averages (or their sum).  Wonderful, but why call this combination by the name “a/b + c/d”?  In adding fractions, “+” already has a meaning, a different meaning, and one of many uses.  It is as idle to call the new operation “addition” as to say “one plus one is eleven” – an old schoolyard joke. 


Even if this new operation (of “averaging the averages in baseball”) were of such importance as to warrant a new symbol, and some systematic study of its arithmetic properties, one must recognize that while it says something about fractions, it says nothing sensible about rational numbers.  There is more value in daily life in calling equivalent fractions equal, and making sure that our comments on such numbers are independent of the representative fraction used, than in trying to use every fraction as a separate entity in computation.  I don’t believe Lax would argue with that.  In the daily measurement of lengths, adding ¾ of an inch to ¾ of an inch will not give a total length of 6/8 inches.  Our rules for addition of fractions can be derived from suitable axioms, to be sure, but the source of the rules is in the measurement, not the axioms.  Just as dictionaries are compiled after a language has been in use, and not the other way round, so it is with mathematics and its axioms. Their main value in mathematics, as elsewhere, is to codify what you have already taken to be true, i.e. taken as a sufficient, though preferably simple, description of what is at issue.  Axioms are a way to keep the basics in mind, and later help to understand truths one might not have thought of without the clarity brought on by our codification; but to teach the grammar before looking at any of the literature is a mistake of philosophy and pedagogy alike.


    Ralph A. Raimi

    Revised April 25, 2005


[1]  I shall use the phrase "New Math”, with capitalization but not quotation marks, to refer to the phenomenon of the period 1955-1975  generally known as “The New Math” but variously phrased.  The New Math phenomenon was the effort in many textbooks and schoolrooms of the time, albeit not always consistent or successful, to introduce the language of logic and set theory into school mathematics, and to make use of them and of axiomatic bases for algebra and geometry to teach these subjects more rigorously than before.  Sometimes the introduction of topics such as statistics, probability and finite combinatorics, which had not previously been common in the schools, was considered part of the same movement at the time, but by 1990 or so it became generally forgotten that this had been part of the attempted reforms of that era.


[2] Van Engen, Hartung, Trimble, Berger, Cleveland, Seeing Through Mathematics, I, II, III, Scott Foresman 1964-1966.


[3] Landau, Edmund, Foundations of Analysis, NY, Chelsea 1960 (translation of Grundlagen der Analysis, Leipzig, Akad.  Verlag 1930)


[4] Van Engen, Hartung, Trimble, Berger, Cleveland, Seeing through Mathematics, I, II, III, Scott Foresman 1964-1966.


[5] See


[6] Here Dodes has a footnote of his own, a reference: “A. Dodes and S.L. Greitzer, Algebra 1: Its Structure, Logic and Method.  New York, Hayden Book Company, 1966.”


[7] See Mathematics Education: The Sixty-ninth Yearbook of the National Society for the Study of Education, Part 1, edited by Edward G. Begle and published by the Society in 1970 (U Chicago press).  Section 1 is headed Historical Background and Psychological Bases, with chapters by R.L.Wilder celebrating axiomatics and Lee Shulman celebrating developmental psychology.  Section 2, Curriculum Content and Pedagogy, leads off with a 56 page chapter by John Kelley, called Number Systems of Arithmetic:

      "This is a brief description of the development of the number system.  This presentation requires very little mathematical background; it is an attempt to provide, for nonspecialists, an overall view of the successive enlargements of the concept of number which take place in the early school years. The approach we use is very near that of the best currently available commercial textbooks.  These comprise the so-called second generation new math programs..."

      He then begins with some axioms for sets, including one that assures the finiteness of all the sets he calls sets:  "Trichotomy Axiom, If A and B are sets, then precisely one of the following occurs:  A matches a proper subset of B, A matches B, or B matches a proper subset of A."  As an example he shows why N cannot be a set in this sense.  He defines a counting number as the class of all sets matching a given set.  He defines "less than" among counting numbers to mean proper inclusion for some choice of  representatives, and shows it is independent of the choice.  After some discussion he produces "AXIOM ON COUNTING:  Suppose that n is a counting number and that d is a counting number other than zero.  Then each set of n members can be split into a collection of sets, each having d members, and a remainder set which has fewer than d members..."  and so on.  Kelley, a well-known mathematician and author of some excellent textbooks for graduate students in mathematics, was long a member of the Advisory Board of SMSG.


[8] Kline, Morris, et al, On the Mathematical Curriculum of the High School, American Math Monthly 69 (1962), 189-193.


[9] November 24, 1961.  “Dear Max:  I keep telling myself that we ought to consider the possibility of consolidating our efforts on the Morris Kline problem…  Do you have an comments or suggestions?  Best regards, Ed.”  This letter is in the UICSM archives (Max Beberman  papers) at the University of Illinois, Urbana, Illinois).


[10] Minnick, John Harrison, Teaching Mathematics in the Secondary Schools, New York, Prentice-Hall, 1939


[11] Wilcox, Marie, Geometry: A Modern Approach, (Teachers Edition), Addison-Wesley, 1968, p.98


[12]  Van Engen et al, Op. cit.


[13]  E.g., Beberman and Vaughan, High School Mathematics, Courses I and II. Boston, D.C. Heath & Co 1964, 1965.


[14]  Van Engen et al, Op. cit.


[15]  Wilcox, Marie, Op. cit.


[16] Von Neumann, The Mathematician, reprinted in vol. 4, p.2053ff of The World of Mathematics, an anthology edited by James Newman and published by Simon and

Schuster, 1956