__ __

The College Entrance Examination Board (CEEB) was established in 1900 by a consortium of private high schools and colleges called the Association of Colleges and Preparatory Schools of the Middle States and Maryland[1], one of several such consortia which in the preceding fifteen years had been trying to establish uniform entrance requirements for universities which had earlier had no reliable criteria for judgment except their own entrance examinations. The consortium also wished to influence the public high schools’ curricula, that those of their graduates wishing to enter college might not find they had wasted too much of their high school time. CEEB soon became an independent body, and its famous Scholastic Aptitude Examinations (SAT) became and remain a standard of academic value for much of the country. Naturally, the CEEB has had to study the changing currents of school education, as necessitated by the changing demands of society itself, as time went on, and to adjust its examinations accordingly.

The Commission on Mathematics was appointed in 1955. According to the Forward, written by Frank H. Bowles, President of CEEB, “Prime movers in giving life to the Commission [and] strongly recommending its creation were the chairman of the [CEEB] Committee on Examinations. Dean Albert E. Meder, Jr., of Rutgers University, and the two distinguished educators with whom he consulted, Professor Howard F. Fehr of Teachers College, Columbia University, and Professor Albert W. Tucker of Princeton University.”[2]

Tucker was Chairman of the Commission throughout, but in 1957 Dean Meder took an 18 month leave of absence from Rutgers to serve as Executor Director of the project, and was succeeded by Robert Rourke for the final year. (Meder later became a prominent spokesman for the Commission when the Report turned out to be the subject of substantial controversy.)[3]

The
Commission published its Report, __Program for college preparatory
mathematics,__ in 1959 in two paperbound volumes. The Report proper was the shorter (63 pages), while the
Appendices (231 pages) detailed what the Commission considered a model
mathematics program for “college-capable” high school (grades 9-12) students,
though with comments on some possible substitutions for its sample chapters.
Early versions of the Report had been circulating in the mathematics community
for criticism for a long time, so that the debate that attended its formal
publication had been well prepared for, and some of the attacks and defenses of
the CEEB stance were written or delivered at conferences of mathematics
educators, if not yet published, even before the Sputnik shock of October,
1957, let alone the date of publication of the Report.

The members of the Commission were:

Albert W. Tucker (Princeton) as Chairman,

Carl B. Allendoerfer (U of Washington),

Edwin C. Douglas (The Taft School, Watertown, CT),

Howard F. Fehr (Teachers College, Columbia University),

Martha Hildebrandt (Proviso Township High School, Maywood, IL),

Albert E. Meder, Jr. (Ex Officio, CEEB and Rutgers),

Morris Meister (Bronx High School of Science),

Frederick Mosteller (Harvard),

Eugene P. Northrop (U of Chicago),

Ernest R. Ranucci (Weequahic High School, Newark, NJ),

Robert E. K. Rourke (Kent School, Kent, CT),

George B. Thomas, Jr. (MIT),

Henry Van Engen (Iowa State Teachers College), and

Samuel S. Wilks (Princeton).

The fourteen
members were a carefully balanced group.
Four of them, Tucker, Allendoerfer, Mosteller and Wilks were practicing
mathematicians, that is, persons working in current __mathematical__
research, but they were also university professors with known interest in
undergraduate mathematics education, which many mathematicians are not. Five others, while also university
professors (Meder was a Dean of Administration, though he had earlier taught college
mathematics for about twenty years), were either professors of education itself
or were concerned primarily with education of undergraduates in mathematics. Of
these, Northrop, Van Engen and Thomas did hold PhD degrees in pure mathematics
(from Yale, Michigan and Cornell, respectively) but had devoted their careers
to teaching of undergraduates or future teachers: Northorp at the famous
undergraduate college of the University of Chicago, Van Engen (though he had
published one research paper in pure mathematics, based on his thesis) as a professor
in teachers colleges, and as a writer of textbooks for school mathematics and
for the education of teachers, and Thomas,
though a professor at MIT and the author of the most popular calculus
book of the era, not active in __mathematics__ research as such. The fourth in this group, Howard Fehr, held
a PhD (from Columbia) based on research in school mathematics education, a
subject to which he devoted all his subsequent work as well. The other five, Douglas, Hildebrandt,
Meister, Ranucci and Rourke, were or had been actual teachers at the level (college-preparatory high school programs)
of concern to the CEEB, some at private schools and some at public schools.

Thus, while the results of the
Commission’s work were widely interpreted by its opponents as a brief for pure
mathematics for the elite alone, written by “mathematicians” intending only to
create more mathematicians or mathematical scientists, and disregardful of the
needs of more ordinary students, the original make-up of the Commission was designed
to avoid any such prejudice. It was
true that only “college-capable” students were the objects of their attention,
but with the college-bound fraction of the American high school children
growing at so enormous a rate, this so-called elite was in fact a substantial
part of the population and was not concentrated within any geographical or
ethnic enclave. Mr. Bowles’s __Forward
__goes on to say:

“In addition, the following persons generously participated in the Commission’s deliberations by attending one or more of its meetings: Frank B. Allen, chairman, Secondary School Curriculum Committee, NCTM; Max Beberman, director, UICSM, Edward G. Begle, director, SMSG; William Betz, supervisor of mathematics (retired), Rochester, NY; R.S. Burington, chief mathematician, Bureau of Ordnance, Department of the Navy; S.S. Cairns, professor of mathematics, University of Illinois; Robert L. Davis, member, Committee on the Undergraduate Program, MAA; George E. Hay, professor of mathematics, University of Michigan; Saunders MacLane [sic], professor of mathematics, University of Chicago; John R. Mayor, director of education, AAAS; Eugene Nichols, associate professor of education, Florida State University; David A. Page, assistant professor of education, University of Illinois; G. Baley Price, president, MAA; Thomas D. Reynolds, Assistant Program Director for Summer Institutes, NSF; and Herbert E. Vaughan, associate professor of mathematics, University of Illinois.”

Furthermore, while the “new math” that emerged in part from the recommendations of the Commission was adversely criticized as leaning too heavily on the logical and set-theoretic underpinnings of modern mathematics, none of the mathematicians on the Commission was a logician, though logic had once been a particular interest of Dean Meder. Wilks and Mosteller were statisticians, in fact, and Tucker and Allendoerfer worked in topology and geometry. Of course, logic and the language of sets are important to every sort of 20th century mathematician, where they had not been of such moment in earlier times, but this is not to say that the mathematicians on the Commission were unrepresentative of mathematics as a whole, or that the other members, the teachers and the professors of education, were unrepresentative of their own categories.

It would be fair to say, in view of the subsequent history of the Commission’s recommendations, that a notable absence from both the Commission and its list of invited part-time participants was that of Morris Kline, an applied mathematician from NYU, who was from beginning to end a vociferous opponent of much that the Commission stood for or recommended. His reaction to the Commission’s work[4], publicized even before the final version of the Report became public, will be described after the following account.

__The Report__

** Program
for college preparatory mathematics** begins with some justification for
its own existence, a discussion of the “urgent need for curricular revision”,
based on several differences between the world of 1955 and the world of
1900: new mathematics which had emerged
during that interval, reorganization of older mathematics, new uses for
mathematics both new and old, demand for improved instruction, and so on. The summary chapter ends with this quotation
from another recent study[5]
(the “Rockefeller Report”) of American educational problems:

“The immediate implications for education may be briefly stated. We need an ample supply of high caliber scientists, mathematicians, and engineers… We need quality and we need it in considerable quantity.”

Such an appeal, plus the avowed purpose of
the Commission to produce a high school program for the “college-capable”
students, has led later observers to dismiss the resulting program, and its
impetus in particular, as unsuited to general educational needs in an
egalitarian and democratic society.
True, the Commission does explicitly state that its detailed program is
for the “college-capable”, but it
recognizes there is another audience for mathematics in the secondary schools,
and after giving a brief summary of what that other mathematics might be,
leaves “to other hands” the details of what modifications of its own
recommendations would suit that other audience. This did not mean that the Commission felt its recommendations
were without __any__ implications for the wider population. Indeed, for the entire literate population
of secondary school students, whether college-capable or not, it lists
“objectives of mathematics in general education” as in fact not much different
from what they would be for the special audience contemplated by the
Report. These objectives are[6]:

** 1.
An understanding of, and competence in, the processes of arithmetic and
the use of formulas in elementary algebra.
A basic knowledge of graphical methods and simple statistics is also
important.**

** 2.
An understanding of the general properties of geometrical figures and
the relationships among them.**

** 3.
An understanding of the deductive method as a method of thought. This includes the ideas of axioms, rules of
inference, and methods of proof.**

** 4.
An understanding of mathematics as a continuing creative endeavor with
aesthetic values similar to those found in art and music. In particular, it should be made clear that mathematics
is a living subject, not one that has long since been embalmed in textbooks.**

Except for the implied emphasis on the axiomatic method, these objectives continue to command general agreement; and even “deductive method” is given lip-service in general (i.e., non-elite) education today. The Commission, however, also addresses the common, egalitarian-minded objection to a rigorous program of high school math by insisting on its inclusivity, not exclusivity, as follows:

** It is a mistake, in the American
educational system, to think in terms of the “college-bound”. Financial or other external considerations
too often affect the question of who is bound for college. As a policy in education, the nation cannot
afford to do less than attempt to recognize and prepare appropriately every
student who is capable of college work.
The Commission prefers to call such students the “college-capable.” **

This statement is immediately followed by
the admonition (to the schools) to demand that college-capable students study
mathematics in high school for __at least__ three years, preferably all
four. The Commission here feels it
necessary to defend this proposition, which by the 21st Century has become
regarded as a truism, and not only for the “college-capable”, because, though
it is hard today to believe this, the educational orthodoxy of the mid-20th
Century “progressive” educators tended to consider “too much mathematics” a
distraction from a well-rounded education.
The Commission thought otherwise: “Young people in high school do not
always know what careers they will follow. What they should know beyond all
doubt is that lack of mathematical preparations closes many doors – not only
doors that open the way to engineering and the natural sciences, but also newer
doors….”

One thing the Commission much too briefly
attended to, was that study of their program for the college-capable should
inform the __primary school __educational program in mathematics
accordingly. In the opening of Chapter
3, where the program as a whole is outlined, a set of prerequisites in
arithmetic, geometry and algebra is briefly listed, with a comment that
students who are college capable can accomplish this during Grades 7 and
8. __How__** **it can be done
during these years, and how even that can be built on what came before, the
Commission leaves to other curriculum projects, and it mentions SMSG as a
model. (SMSG had only come into
existence in the year of completion of the Commission report.) But the college-capable are not necessarily
genetically determined, as if waiting until high school for the authorities to
discover their properties; they are in large part __created__ by proper
instruction and discipline from the beginning.
Looking forward to what the high schools will be demanding of the
college-capable __elementary school__ children should therefore be of
advantage to those in charge of preparing their “college-capability”. Should not the elementary schools see to it
that the door to college-capable status is not shut early, in the same way the
Commission was warning the high schools not to shut the door to college
mathematics by an insufficiency of high school instruction? Was not the Commission remiss in not more
forcibly warning of the obvious gap between what the pre-high school math program
was accomplishing and what the Commission was recommending for its products,
such as they then were?

This omission by the Commission was not unnoticed, however; and there were in the years immediately following the Report many hurried efforts on the part of “new math” K-8 curricular projects to do exactly this, to jump the gun, as it were, by importing some of the more ambitious ideas of the high school programs envisioned by the Commission directly into the early grades. (David Page, an early factotum to Max Beberman and by 1958 the director of one such elementary school projects, “The Arithmetic Project”, at the University of Illinois, had been one of those consulted by the Commission, and Robert B. Davis, another invitee, devoted most of his career to the study of elementary school mathematics teaching.) The importation of such ideas as set intersections, base-k arithmetic and number-numeral distinctions into the early grades often (some would say invariably) had unfortunate results, as Max Beberman himself was to point out in his Montreal lecture of 1964[7].

That the
recommendations of the Commission would lead so much of the nation in an
unprofitable direction was not anticipated by many but it was anticipated by
some; and the debate over the curriculum projects which followed actually had
more to do with the new programs later undertaken by the elementary schools
than with what was in the purview of the Commission report. In retrospect one can say the Commission was
remiss in not warning the educational authorities against such an excess of
democratic idealism. If __everyone__
is potentially college-capable, it becomes hard to resist an elementary program
filled with vocabulary and drills that cause everyone to__ sound __college-capable,
to school boards and parents, anyhow, and yet be hollow at the core.

Probably the Commission considered any
such warning to be obviously superfluous.
Its detailed program was __explicitly__ for a “college-preparatory”
group; surely nobody would decide to import its most daring recommendations ** verbatim**
into the lower schools? Concerning its
intended audience, the Report said (p.13),

Students studying college preparatory mathematics should, in our opinion, be taught in groups with similar interests and similar intellectual abilities… The injustice done to capable students by failure to use ability grouping is well pointed out in the Rockefeller Report: “Because many educators reject the idea of grouping by ability, the ablest students are often exposed to educational programs whose content is too thin and whose pace is too slow to challenge their abilities.”

In addition, this limitation of intended
audience is followed by another implicit warning, that the recommended program __designedly__
omits many practical applications of everyday mathematics:

**College-capable students who follow the
recommendations of this report will not be exposed to certain so-called
“practical” courses such as consumer mathematics, installment buying,
principles of insurance, and so forth.
The Commission does not regret this fact. For it believes that these students will ordinarily develop
sufficient mathematical power to acquire such information independently if they
need it. One must not assume that an
individual’s knowledge consists solely of what he has been specifically taught
in school.**

The actual recommendations of the Report
begin with Chapter 3, headed __Recommendation: the Commission’s Program__.
The Commission immediately recognizes the existence of two schools of educational
theory in America, the “traditional” and the “progressive”, and wishes to make
its work available to both, and perhaps to encourage the latter in particular
while not offending the traditional, or losing it as an audience.

**While the
direction of the proposed changes is oriented to the future, the changes
themselves are not radical. No attempt
has been made to uproot the traditional curriculum… The result is a revision
that the Commission believes can begin at once gradually and move forward rapidly… **

**[The]
flexibility of the recommended program includes its manner of
presentation. Members of the Commission
would decry an authoritarian approach to method and practice, but a teacher who
believes that such an approach is most effective may present this material in
the same way that he has, presumably, taught the traditional content. Most if not all of the Commission members
would prefer to see a developmental approach, which would encourage the student
to discover as much of the mathematical subject matter for himself as his
ability and the time available (for this is a time-consuming method) will
permit.**

It is noteworthy that the program outlined in the remainder of the report does not include any calculus. Early in the Report, before the recommended content of Grades 9-12 is made explicit, there is a section devoted to explaining why. Acknowledging that some students are well enough prepared for an “Advanced Placement” calculus course, and can complete it with profit, it warns that such a course must be preceded by a sufficient preparation in other parts of mathematics, indeed something much like the full program being described in the present Report. A program of such depth was quite rare in the high schools of 1958, and all the more rare as a program that could be completed by the end of Grade 11, opening the twelfth year to calculus. In addition (p.15),

The Commission cannot recommend the practice of exposing college preparatory students to formal calculus for a short time at the end of the 12th grade. Such anticipation tends to breed overconfidence and blunt the exciting impact of a thorough presentation…

Chapter 3 begins by outlining the
“Prerequisite mathematics”, subject matter regarded as essential, and which
should be accomplished or completed in Grades 7 and 8. “The Commission is convinced that it can be
mastered by all college-capable students during grades 7 and 8… [indeed,] “The
better students can cover this work in one to one-and-a-half years and move on
to the Commission’s program during the eighth school year.” Then follow detailed lists of topics under
the headings, __Arithmetic__, __Geometry__, and __Algebra and Statistics__,
a fairly modest list of competencies the nation as a whole today, fifty years
later, professes – on paper, at least -- to take as a matter of course, with
even more asked of “gifted students”.
The CEEB prerequisite list says nothing of quadratic functions, or any
other functions as such, and as to geometry it only asks for common mensuration
procedures and formulas, and for the vocabulary of triangles, circles,
parallels, etc., some symmetries, and a knowledge of the Pythagorean
relation. In 1958 this list would have
been considered ambitious, but not unreasonable.

The level of detail in the Report here can be deduced from some quotations (still concerning Grades 7 and 8):

Under __Arithmetic__:
“Mastery of the four fundamental operations with whole numbers and fractions,
written in decimal notation and in the common notation used for fractions. This includes skill in the operations at
adult level (i.e. adequate for ordinary life situations) and an understanding
of the rationale of the computational processes … The meaning and use of an
arithmetic mean. In addition, a
knowledge of square root and the ability to find approximate values of square roots
of whole numbers is desirable.”
[Newton’s method is suggested.]

Under __Algebra and Statistics__, the
following is the entire entry:

“Graphs and formulas: Use of line segments and areas to represent numbers. Reading and construction of bar graphs, line graphs, pictograms, circle graphs, and continuous line graphs. Meaning of scale. Formulas for perimeters, areas, volumes, and per cents – introduced as generalizations as these concepts are studied. Use of symbols in formulas as placeholders for numerals arising in measurement. Simple expressions and sentences involving “variables”.

The only novelty, at the middle school level, was the suggestion that binary notation “and perhaps other bases” might be useful “to reinforce decimal notation” in teaching the rationale for place-value arithmetic. Apparently the Commission thought such training by middle school years would clarify, and render easier to understand, our conventional decimal system, which the students had been learning – presumably – in earlier years. As things turned out in it did no such thing, and base-k arithmetic in the elementary and middle schools became a target for attack by opponents of “the newmath”.

The program for grades 9, 10 and 11, the
heart of the Report, takes pages 20-30 and is given in outline, and with
philosophical commentary, rather than by listing topics, with references to the
much longer __Appendices__ for curricular detail. To appreciate the controversy with which the Report was greeted
it is necessary to read a good many of the actual words of the report; it would
not suffice to cite (e.g.) the algebra topics as “the nature of a function – in
particular, the linear, quadratic, exponential and logarithmic functions…” and
“the meanings of conditional equations and identities and inequalities”, which
everyone took as reasonable. (Curiously, the exponential and logarithmic
functions are not mentioned in the Appendices, though a substantial chapter on
the circular functions is included.) Such general phrasing alone would have
been insufficient both to carry the Commission’s message and to raise the red
flags seen by Morris Kline even before a word of SMSG text had been
published. Nor would mention of the
distributive law of the real number system, or some generalities about
deductive reasoning, have generated anxiety among traditional teachers. But consider this, from Page 21:

** **

** In making its recommendations for
increased emphasis upon algebra and for instruction oriented toward a more
contemporary point of view, the Commission is influenced by the fact that
algebra is a subject that has been largely transformed by the mathematical
research of the past quarter-century.
This transformation has been brought about through the systematic
axiomatic development of algebra or, more accurately, of algebras, and the
light thus thrown on the study of mathematical structures. Such structures, or patterns, have been
thrown into sharp relief. The Commission
is influenced too by the fact that many recent applications of mathematics to
areas hitherto regarded essentially as non-mathematical are algebraic in
character.**

** In the proposed program, the mechanics or
formal manipulations in algebra are the same as hitherto taught, and the
subject matter is largely the same. The
difference is principally in concept, in terminology, in some symbolism, and in
the introduction of a rather large segment of new work dealing with
inequalities treated both algebraically and graphically. Solution sets of inequalities involving two
variables are also studied.**

** The new emphasis in the study of algebra
is upon the understanding of the fundamental ideas and concepts of the subject,
such as the nature of number systems and the basic laws for addition and
multiplication (commutative, associative, distributive). The application of these laws in various
number systems, with emphasis on the generality of the laws, the meanings of
conditional equations and identities and inequalities, is stressed…**

** One way to foster an emphasis upon
understanding and meaning in the teaching of algebra is through the
introduction of instruction in deductive reasoning. The Commission is firmly of the opinion that deductive reasoning
should be taught in all courses in school mathematics and not in geometry
courses alone.**

Deductive reasoning alone, however, as a subject in its own right (usually called “formal logic”) is not recommended, and the Commission takes care to say so. Though by 1958 the phrase “the new math” was already common in the newspapers, Max Beberman in particular having attracted the attention, often unfavorable, of editorialists in the cities he visited[8], the Commission Report cautioned (p. 21):

**It is not,
however, an alternative of either skill or understanding that
confronts teachers. Both skills and
concepts are essential... [**emphases in original

On the other hand, there is no question about which of these comes first:

** **

**Strong
skills are surely needed; but they must be based on understanding and not
merely on rote memorization. Once
meaning has been achieved, then drill should be provided to establish
skills – skills that can be performed, as Whitehead says, “without
thinking.” In this way the mind is
liberated to grapple with new ideas.**
[emphasis added]

The priorities could not have been made
clearer. American textbooks from the
mid-Nineteenth Century onward invariably began with an introduction explaining
that the text was written to make use of “the inductive method”, that poor
teaching (and previous textbooks) had proceeded by first presenting a “rule”
and then following with illustrations and drill in its use, a method (so these
authors said) sure to fail. Here the
CEEB Commission was reversing that accepted wisdom, though in considerable
disguise: a set of axioms and some abstract “understanding”, rather than a
“rule”, were to precede the drill. The
Commission did not consider the possibility that the axioms and understanding
it expected could be turned into “rules” taught as rigidly and mindlessly as
any that had been condemned, fruitlessly for the most part, in the preceding
century. (Probably no author of a
program or textbook can make allowance for bad teaching, after all.) In both dispensations, however, whether algebra and geometry were learned
via axiomatics (newmath) or by “the rules” (19^{th} Century math), the
drill came second.

This theoretical foundation for its proposed college preparatory curriculum occurs not only by way of preface, but (as in the quotations above, from pages 20 and 21) within the first of the major subjects, in this case, Algebra, that the Report takes up in its outline for Grades 9, 10, and 11. The case of algebra was particularly important because algebra was not, in 1958, ordinarily considered an axiomatic, deductive science in the schools, as the Euclidean system for geometry obviously was, as it had been for centuries. The Commission wished by calling attention to reasoning in algebra to make sure that the schools understood the place of deduction in all of mathematics.

On page 22 the Commission gives a telling example of the kind of mindlessness current in its time, which could not occur in an algebra program having the careful logical underpinning it recommends:

**For example, one sometimes reads: “If the discriminant of a quadratic equation
is a perfect square, then the roots of the equation are rational.” Yet the equation**

** **

** 2x ^{2} + (2√6)x +
1 = 0**

** **

**has a
discriminant equal to 16, and irrational roots. Wherein lies the discrepancy?
A student who approaches algebra from the contemporary point of view is
accustomed to thinking in terms of the set of numbers admissible in a
particular problem. The quoted
statement holds only for quadratic equations with rational coefficients: it fails for the given example because
(2√6)x is not a member of the set of rationals. **

** **

(An amusing
sidelight: The Commission itself is
here guilty of carelessness, for the popular quoted rule about the discriminant
holds only for equations with **integral** coefficients, unless fractions
such as 9/16 are construed as “perfect squares”, an uncommon usage. Such an objection here is really trivial,
since equations with rational coefficients can always be converted to
equivalent equations with integers as coefficients; but still --- to be
complete, and true to its own announced principles in exactly this place, the
Commission should have made clear the answer to the question “perfect square of
** what**?” Every positive
real, and every complex, number also has a (“perfect”?) square root in its own
system, after all. Critics of the
Commission’s report had many occasions to observe that the precision in
language recommended in the report was not always convenient, and that strict
adherence to fully defensible statements would require an ungainly prolixity,
or generate philosophical subtleties not suitable for high school study. Many a
textbook of the following decade, anxious to follow the Commission’s advice,
exhibited both of these negative qualities.)

But none of this is to say that the Commission neglected the traditional substance of the algebra it recommended:

**In the proposed program, the mechanics
or formal manipulations in algebra are the same as hitherto taught, and the
subject matter is largely the same ... [In addition, we introduce] a rather
large segment of new work dealing with inequalities treated both algebraically
and graphically. Solution sets of
inequalities involving two variables are also studied.**

And in truth these uses of equations and
graphs were a substantial innovation, and by no means of purely linguistic or
philosophical import. The phrase
“solution set” was an innovation of the newmath era, one that clarified much
that had been obscure to students during the preceding epoch. (E.g., are there really two solutions to a
quadratic equation? How can there be
two answers to a mathematical question?
Is the answer “r_{1} __and__ r_{2}”, or is it “r_{1}
__or__ r_{2}”?) Yet this
phrase is, like almost all the terminology of school mathematics, definable as ordinary English; it use in
clarifying the language of equations didn’t really need or use the technical language of the algebra of
sets. Yet that language came to be seen
as one of the defining characteristics of the new era.

Next, the Report takes up Geometry, advocating a mixture of analytic (coordinate) methods and the traditional synthetic, and a reduction in the number of theorems to be learned. The Commission had in mind here that Euclidean geometry had itself become a home for rote learning in recent years, with students memorizing theorems and proofs for such ordeals as the New York Regents Examinations, thereby losing entirely the theoretical lessons supposed to be inculcated by the study of Euclid. The Commission also suggested that plane and solid geometry not be as separated as was traditional, and that other geometries (including topology!) deserved mention, if not systematic study.

The question of rigor in geometry was harder to attack, since the Euclidean system is incomplete in its system of postulates, which can only be completed in synthetic terms by adding a thicket of statements difficult to understand and use. There are two ways to avoid this difficulty. One way is by merely assuming the first few classical theorems (about congruence and similarity of triangles, for example), taking as obvious what the eye tells us is obvious about these situations as well as what we see concerning intersections of lines and ordering of points on a line. From there one can begin deductions in rigorous form, albeit with a non-minimal set of assumptions. The other way is to take the real numbers for granted and house Euclidean space in a Cartesian frame with Pythagorean metric. The second method, which was advocated by Jean Dieudonné in a famous speech in the following year, makes some of the most charming theorems about circles, chords, angles and so on rather difficult, whereas the Euclidean method makes them visible and interesting. On the other hand, the Report recognizes that the first method, purely synthetic, suffers from logical difficulties, or at the very least from a superabundance of “postulates”.

The Commission recommended as a compromise the use of both methods, but regretted that no suitable textbooks as yet existed. It warned against asking students to memorize more than a few theorems because to ask more had in the past shown itself to be an invitation to the rote learning of proofs. The Commission wanted most theorems, after the fundamental ones, to be construed as “originals”, not something to be memorized; thus the ordering of the theorems became somewhat inessential. “Of course, no teacher would accept such circular reasoning as basing the proof on two propositions each upon the other,” it wrote (p.25), but “each upon the other” is rather hard to define, and can be concealed via intermediate theorems. The Commission of course knew this, but since it regarded geometry as something needed for the information it supplied, and as exercise in reasoning, though not as a complete and airtight axiomatic system (in the schools, at any rate), it ended by relying on the judgment of teachers to maintain a reasonably logical structure, especially if decent textbooks could be found or created. Finally, combining Euclid with coordinates would reinforce the students’ algebraic skill and memory, an important consideration, for experience showed all teachers that a year of Euclid often meant that a very time-consuming review of algebra would be needed at the beginning of the following high school course (“Algebra II”) in the traditional sequence. (Much of the detail in the Commission’s Geometry is referred to the Appendices, which amount to a virtual model textbook.)

The geometry segment in the Report continues with a section on solid geometry, recommending a limited exposure, one-third of a semester to be exact, and where possible interleaved with plane geometry rather than being regarded as a separate study. “The objective should be the development of concepts of spatial relations, and of spatial perception. Mensuration should have been covered intuitively earlier; however, lengths, areas, and volumes of standard figures should be reviewed. Deductive proofs of mensuration formulas are not in order at this time…” Since the traditional high school sequence had devoted a whole semester to solid geometry, and another whole semester to the trigonometry of triangles, with calculations via logarithms, compressing each of these to a part of a semester constituted a semester’s saving at least.

Finally, to conclude this segment there is a section headed “Other geometries.” Here nothing specific is recommended, nothing on which the College Board would feel entitled to ask examination questions, but apparently for cultural value only, it is suggested that students be made to realize that the Euclidean geometry is not the only one, and that, e.g., spherical geometry might be invoked to show that the sum of the angles of a triangle need not be 180 degrees in a system that still carried meaning. Projective geometry and even topology receive mention, too. “Simple pictorial material from these geometries provides intriguing foils to stimulate the imagination,” and the Report here pictures a torus and a plane, each cut by a circle but with a different effect on the connectivity of the complement.

The summary of the section on geometry recommends that students be made explicitly aware of the nature of a system of axioms and deductions, with geometry as one example, but without any extended series of Euclidean theorems. This omission, it admits, will be regretted by some teachers, but has compensating advantages. The Commission is persuaded that short bursts of theorems, with a mixture of analytic and synthetic reasoning, will suffice to exhibit the nature of both geometry and of axiomatic systems. Some beauty is missing? “Books in which [students] can encounter this particular form of mathematical beauty are easily available.” And there are plenty of occasions for original thinking even within the Commission’s recommendations.

The report then continues with an outline
of trigonometry, with the study of the circular functions as such reserved for
the 12^{th} grade, which was to be devoted to the elementary functions,
vectors, complex numbers, and other topics needed before calculus (in
college). There are then sections on
probability and statistics, as optional topics, and on the education of
teachers, and at the end of the Report a rather detailed list of topics in the
form known in the schools as “scope and sequence” outlines. While most of this is traditional, at least
in the titles of subjects to be learned, the spirit of logic and axiomatic
structure is found throughout the prose.

In the section on the training of secondary school teachers, especially, occur certain emphases that led to serious controversy when the Report became public. On page 56, for example,

**In graduate courses offered for
teachers of secondary mathematics, there is need for colleges and universities
to give careful consideration to the point of view from which instruction
proceeds. The chief contribution that
some courses (analysis, for example) have to offer at the graduate level is the
development of a feeling for the spirit of modern mathematics. The teacher should understand the importance
of making certain fine distinctions and the need for precise terminology and
adequate symbolism, and he should develop an appreciation for mathematical
rigor. All of these are more important
than the precise content of the course, since the content in itself will be of
less professional significance to the teacher.**

This recommendation should be compared with an earlier one for the education of teachers of elementary mathematics, something given several pages of detail but then succinctly summarized (p.58) as follows:

**Unless a teacher of arithmetic has a
mathematical background at least equivalent to the first three years of the
secondary school content outlined in this report, he cannot be said to have an
understanding of mathematics adequate for successful, meaningful, and assured
teaching of arithmetic. Additional
college work in mathematics as well as in methodology is highly desirable, if
not essential.**

** **

Now the Commission’s report was overwhelmingly devoted to secondary school mathematics for the college-capable, and necessarily also to the education of those who were to teach this population, but mention of elementary school arithmetic involves the entire school population, with problems much different from what the Commission was in a position to handle, even in summary. To say an arithmetic teacher should know what comes next might be correct, but what the Commission prescribed here was plain impossible, from the demographics of the teaching profession, indeed the demographics of their charges as well. The majority of elementary school teachers in 1955, like those of fifty years later, simply did not know the material of “the first three years of the secondary school content outlined in this report”, and the Commission had no recommended mechanism by which to change that rather hard fact. The recommendation as to content was surely correct, even minimal, but to prescribe an impossibility can – and did – give rise to meretricious substitutes: programs devised during the era that followed, to make elementary education “teacher-proof” by writing curricula that would permit elementary students to obtain what CEEB and everyone else thought those students could learn if only their teachers and programs were wise enough.

Teachers
already on the job (“in-service”) can and do learn things they need and haven’t
learned earlier, but the third grade teacher who understands and can somehow
use the functional properties of quadratics, logarithms and exponentials
(recommended by the Commission for Grade 11) is almost non-existent. If such a teacher is suddenly made to try to
learn such things, he will most probably have wasted time that should instead
have been devoted to the study of the inverse nature of multiplication and
division of common fractions, and the use of these operations in meaningful
applications.[9] At the same
time, college professors faced with future elementary school teachers, and
taking seriously what the Commission was in fact recommending for** **“the
first three years of the secondary school content outlined in this report**”**,
abstract and rigorous as it all was (though intended for another audience),
often ended by giving potential elementary school teachers a very mistaken
notion of what should be __taught__ in the early grades, and the publishers
of elementary school textbooks then followed the apparent demand.

In this, the Commission exceeded its
mandate. Even though the education of
elementary school teachers in mathematics is of extreme importance, the
Commission did not give the matter sufficient consideration; and the
mathematics profession itself was not slow to initiate a debate about the
wisdom of asking, in effect, for “modern mathematics” to be forced into the
primary grades. To be sure, this is not
quite what the Commission was asking here, since it was recommending for the
education of elementary **teachers**, not their students, but there were
those who could foresee the connection that ultimately did take place and
warned of its dangers, and there were many more who later decried that
connection when it became visible.[10]

The Report as a whole makes a complete and
quite comprehensible prescription, and
apart from its spirit and its insistence on the axiomatics for school
algebra it was really quite traditional as to subject matter. The purpose of the Report, after all, was to
tell the schools what the __colleges__** **wanted, and what the College
Board examiners should be testing in the coming generation – in all, a very
practical purpose. The CEEB had no
intention of describing a program for __everyone__, or a program to be
considered “practical” for business or the mechanical trades, and certainly not
intended for students who voluntarily abandoned the study of mathematics past
Grade 10, as was common at the time, especially for the college-intending
future elementary school teachers. That there would be students uninterested in
this subject matter, or unwilling or unable, teachers too, went without
saying. The CEEB wanted a program
making sure nobody would be deprived of access to the material needed for a good
college preparation; it could not at the same time prescribe a program everyone
was guaranteed to be able to master when the time came. That it overprescribed what was feasible for
future elementary school teachers was an unfortunate oversight.

The second
volume of the CEEB Commission Report, the __Appendices__, went even further
in detail, becoming almost a textbook for major parts of 11^{th} and 12^{th}
grade mathematics, or for the teachers of those grades, even including
exercises at the end of many of its sections and chapters. Yet it does not outline a full curriculum,
nor does it intend to. For example, the
logarithm and exponential functions are not given even an outline of a
treatment such as would be expected in a high school, though the trigonometric
functions are. Both subjects are,
however, outlined in the Report itself as expected subject matter for the 11^{th}
and 12th grade levels. Furthermore, the logarithm as inverse to the exponential
is presumed understood by the reader in a section (in the __Appendices__,
page 31 ) on irrational numbers, where an indirect proof is given for the fact
that for any positive integer k which is not a multiple of ten, log_{10}k
is irrational. As the introduction to
the __Appendices__ states,

**The
Commission suggests that teachers might use these appendices for independent
study, or as a basis for cooperative, in-service group study (in which teachers
with special training might take special responsibility for the presentation of
certain topics)… Finally, the Commission does not claim to have created in
these appendices anything basically new…**

Just the same, the appendices do treat of some topics that were not common in even the best high school programs of the time. It is divided into three sections: Algebra, Geometry, and Trigonometry, but manages to include sets and set notation, complex numbers, the definition of “function” as a set of ordered pairs, “limits”, the binomial theorem, proofs by mathematical induction and indirect proofs, even the algebra of sets (under inclusion, intersection and union) --- all this under “Algebra”. The geometry section includes three-dimensional figures, axioms for order in synthetic geometry, geometric transformations, “theorems having easy analytic proofs”, and deductive geometry with vectors, all this seguéing into the final chapter on trigonometry and the trigonometric functions, with the usual formulas and identities carefully proved. An “intuitive” derivation of the infinite series formula for the exponential function, given as “an optional topic” in outline, completes the book.

I will in
a separate chapter tell the story of how I once attempted to use the __Appendices__
as a textbook for an in-service course for high school teachers.

Ralph A. Raimi

August 24, 2005

[1] See Willoughby, Stephen S., __Contemporary teaching of
secondary school mathematics__,__ __NY John Wiley 1967, p.5

[2] Quotations will all be, unless otherwise
indicated, taken from the text of the __Report of the Commission on
Mathematics__, CEEB, 1959. Only the
Forward is signed, since Frank Bowles was not a member of the Commission; the
rest of the Report is taken to be unanimously endorsed by the members of the
Commission.

3 See
Meder, Albert E., __The ancients versus the moderns – a reply__, The
Mathematics Teacher 51 (1958), p.428-433, following the attack by Morris Kline, __The Ancients versus the moderns, a new battle of the books__. The Mathematics Teacher, 51 (1958), 418-427.

[4] Kline, __op.
cit__.

[5] __The
Pursuit of Excellence: Education and
the Future of America__ (Special Studies Project Report V, Rockefeller
Brothers Fund), Doubleday & Co., Inc., 1958, pp 27-28.

[6] __Report__, p. 11.

[7] Schwartz,
Harry, __Peril to
Doing Sums Seen in 'New Math'__, NY Times 31 December 1964, p.1

[8] E.g., __Math
in Abstract__, an opinion piece in the Rochester Democrat & Chronicle,
Jan 15, 1958, followed by a rejoinder a few days later from a mathematics
professor at the University of Rochester, explaining that abstraction was what
made mathematics comprehensible, contrary to the opinion of the editorialist.

[9] The much
more recent book, Ma, Liping, __Knowing and Teaching Elementary Mathematics__, Lawrence
Erlbaum Associates (“LEA”), 1999, presents a persuasive argument for educating
elementary school teachers in quite a different fashion from that envisioned by
CEEB, even under the assumption that, unlike the American practice, these
elementary teachers will be specialists in __mathematics__ teaching.

[10] Max Beberman himself, for example, in Schwartz,
Harry, __Peril to
Doing Sums Seen in 'New Math'__, NY Times 31 December 1964, p.1; and in any
of the relevant polemics of Morris Kline..