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)

 

Proof:     

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 trad