**Chapter 1: Max **

Max Beberman is generally regarded as the father of the New
Math, his teaching and his curriculum project having achieved nationwide fame
long before the 1957 Sputnik raised mathematics education to the level of a
national priority. He was born in 1925,
a Jew from Brooklyn: Stuyvesant High
School and a BA in mathematics from CCNY in 1944, Army Signal Corps until 1946,
upon discharge staying in Alaska, where he had served as a meteorologist in the
Army, to teach math and science in the schools of Nome until 1948. He then returned (married, now) to Rutgers
for an MA and, finally, Columbia University Teachers College for a PhD in
mathematics education in 1953. His thesis director at Columbia was Howard Fehr,
famous then and later as an authority on the teaching of school mathematics,
and a man who directed the PhD theses of many future professors of mathematics
education.

But as was common in the days when PhD degrees were fewer
than they became later, Beberman was already an Instructor in education at the
University of Illinois well before his degree work at Columbia was done. He arrived at Illinois in 1950 and with few
interruptions spent his entire career there, dying at the age of 45 in early
1971. His early death was predictable,
for in 1966, already seriously ill, he had gone to the Mayo clinic for a heart
valve replacement; it was the failure of this valve that caused his death five
years later, years during which he did not slacken in either his work or bon
vivant style of life.

Max Beberman was an enormous worker, and the light in his
campus office in the middle of the night was a beacon. People returning from late parties would
sometimes drive past just for the pleasant reassurance that Max was still up
there. As early as 1955 his friend and
frequent correspondent on mathematics education, Bruce Meserve, wrote him a
serious letter[1]
begging him to let up on his terrific activity lest he work his way into an early
grave. (Meserve was on the mathematics faculty at Illinois
when Beberman got there, and was an early collaborator, but left Illinois in
1954.) The fact that Max's girth increased continuously during his
lifetime, approaching sphericity, was also of no help
to his health.

Beberman's principal work at Illinois was split: He was a professor of education at the
University and a teacher in the University High School. His reputation was immediate; it was said
that he could teach mathematics to a stone wall. In his classroom there were no back seats,
for he engaged the entire class with what he was getting across. It was more than a Socratic dialogue, and he
called it "discovery learning".
The idea -- or at least the phrase -- has been institutionalized since
Beberman's time, and is often turned into something of a fraud, and a cover for
the failure to impart any real information or make any real intellectual
demands on students; but for Beberman it was real and it was seen to work,
though within limits.

Not all his teaching was in the discovery style; he would
lecture like other professors when that seemed called for,
and even hector students into saying things his way. To Beberman, no matter how the process of
"doing mathematics" was begun -- or discovered -- by a student, it
had to eventuate in mathematically accurate speech. By the end of his career thousands of people
had seen him in action, for his curriculum project (UICSM) produced a number of
16 millimeter films of his classroom performance. In addition, his own
personal appearances in schoolrooms around the country were frequent and
legendary.

Yet being a fine teacher alone will not make anyone famous,
or cause his work to be remembered beyond his time. There have doubtless been other teachers as
good as Beberman, but he happened along at a propitious time, and in a
propitious environment. 1950 was near
the end of a dismal era in American public education in mathematics and
science. The flood of war veterans that
had engulfed the universities on "GI Bill" dollars from 1945 to 1950
was now subsiding, and these unusually mature students were being replaced by
high school graduates of a more customary age, school-educated no worse in
mathematics than their predecessors, but without the discipline and
understanding that a few years of military service affords, or the seriousness
about knowledge that a few more years of anything tends to bring. The new, postwar wave of
college students without these advantages were not the same thing.

The professors of mathematics in the universities, by 1950,
were therefore suddenly shocked by the wave of ignorance that seemed to have
swept over their new freshmen. At
Illinois it was William Everitt, Dean of the College of Engineering, who took
an initiative to see if anything could be done to improve high school
mathematics education, at least in Illinois, and at least for students
intending to study engineering.[2]

At his initiative a faculty committee, including two
professors in each of mathematics, engineering and education, was appointed to
study what was lacking
and issue a report to be disseminated among the high schools of
the State of Illinois. Thus, __Mathematical Needs of Prospective Students in
the College of Engineering of the University of Illinois__ came to be
published as __University of Illinois
Bulletin, vol 49, No 18__ in October 1951, and given wide circulation.[3]

The committee writing this report not only brought three
points of view to the problem, but conducted surveys of students and professors
in engineering at the University, with some attention to the statistical
validity of the results there, as well as to their own predispositions. Their 18 page report was printed and sent to
high schools all over the State of Illinois Three pages were devoted to a
simple listing of topics, 97 of them labeled "indispensable". Some of these, marked with an asterisk, were
considered more advanced than the others, and while it was recognized that not
all high schools might offer them, or students study them, and could be excused
for not including them, all were warned that these topics would then have to be
studied remedially once the student entered the engineering program, under the
traditional rubric, "College algebra and trigonometry." Another 13
items called "not so fundamental", were
recommended for superior students or those who had time for them in high
school.

Typical of the "indispensable" items were

8. Scale drawing

9. Concept of an approximate number,
precision of a measurement, significant digits, and rounding

17. Common special products, i.e., a(a+b), (a+b)(a-b),

18. Factoring such expressions as a^{2}
– b^{2},

56. Mensuration of plane figures

60. Concept of locus

68. Cylinders, cones, spheres; Concept of a definition,
a postulate, and a theorem

69. Deductive proof

74. Solution of right triangles

while some
of the indispensable items marked for "College algebra and
trigonometry" were

38. Change of base of logarithms

39. Solution of exponential and logarithmic
equations

44. Geometric progressions, both finite and
infinite.

77. Definition of the trigonometric
functions of any angle

More interesting today, perhaps, is the following
(complete) list of topics classified as desirable but not indispensable, to be
studied "if there is
time available or by high-ability students whose rate of learning
warrants supplementary work."

1. Extraction of square roots by means of
the algorithm

2. Slide rule

3. Binomial theorem with fractional and
negative exponents

4. Permutations

5. Combinations

6. Probability

7. The inverse, converse, and
contra-positive of a statement

8. Polyhedral angles

9. Line values of trigonometric functions

10. Formulas for tangents of the half angle

11. Multiplication and division of complex
numbers in polar form

12. De Moivre's theorem

13. Exponential form of a complex number

Even for the year 1951 one might wonder why the listing is
so detailed. Items 11-13 are part of a package
that can hardly be partitioned, after all.
It is amusing that the archaic algorithm for taking a square root, a
device that ceased to have practical value centuries earlier and whose
intellectual underpinnings by 1950 were no longer understood by anyone but
antiquarians, should have even received mention, but in fact the high school
textbooks of 1950, many of them later editions of books thirty years old, did
mention such things, and "college preparatory" students were still
being put through it as "good for the mind." And whatever a "line value" of a
trigonometric function might be, the committee didn't want the schools to waste
time on it at the expense of (say) the "concept of locus" or any
other item mentioned earlier as indispensable. The purpose of the document was
not to make efficient mathematical sense, but to make it unmistakably clear to
high school teachers across the state, teachers familiar with the kind of
textbook available at the time, which sections of these books it would be safe
to omit and which it was necessary to include. The committee's list had to use the jargon
of the time to serve such a practical purpose.
For example, the more sensible mathematical language that could have
combined the three "topics" concerned with DeMoivre's theorem under
one heading might well have failed to carry the proper message to a
schoolteacher uncertain about complex numbers altogether.

Despite the appearance in this listing of such logical
ideas as "contrapositive" and "axiom" (and the mysterious
"inverse" of a proposition, which turns out to be the same as the
converse, but expressed in contrapositive form), the context makes it plain
that these notions were to be part of geometry, not algebra, and that nobody
then intended the sort of axiomatic algebra that later became a hallmark of
"the new math". The nearest thing
to mathematical reasoning contemplated by the algebra entries was perhaps
suggested by the item

47. Concept of
equality including the symbol, and the postulates of equality

But this appears between the mention of
geometric sequences and, after a similar item involving the word
"inequality",

48. Use of the
protractor

49. Use of the
compass and straight edge

betraying the
then popular confusion between equality as Euclid construed it, and posed
axioms for, and equality meaning identity, as the word is used in algebra in discussing the solution of equations,
in which context there simply are no postulates, after all. (cf. the later
discussion of Minnick's methods book[4])

The opening of the report to Dean Everitt deplored the
inadequacies of mathematical preparation of engineering freshmen, vigorously
recommending more mathematics. It quoted
from a paper by W.C. Krathwohl in the Journal of Engineering Education, vol
27 (1938), "Nobody can say what
mathematics an engineer does not need",
and "In particular, [the engineer's training] should emphasize
general mathematical principles and methods of analysis rather than dexterity
in few specified fields." And the remainder of the report was devoted
to administrative comments on the University's admissions policies and so
on. Despite Krathwohl's words, and the
presence of the two young mathematicians, Assistant Professors Bruce Meserve
and William A. Ferguson, on the committee writing this report, there was no
great originality in this document. More
mathematics for engineers, and earlier, but much the same sort of mathematics
the schools were used to, and no mention at all of what a non-engineer might
want with mathematics. Yet this report
of limited purpose was destined to segué into a revolution.

Soon after the appearance of the report, the Principal of
the University High School took steps to implement its recommendations locally:

** 102 University High School**

** December 18, 1951**

**Mr. Daniel S. Babb**

**Dr. Bruce Meserve**

**Mr. Max Beberman**

**Dr. R.E. Pingry**

Gentlemen:

I wish to ask your help in planning for revision of our
University High School mathematics offering.

The recent publication, "Mathematical Needs of Prospective
Students of the College of Engineering of the University of Illinois,"
faces high schools with a number of problems.

1. The topics listed
as essential (pages 12 to 14) are not accompanied by descriptions of the degree
of competence required. While this was
not within the scope of the committee who developed the publication, it seems
essential to the organization of appropriate high school courses. Your work on this problem will undoubtedly be
related to that of another interdepartmental committee to be concerned with
evaluation of the mathematical competence of prospective University students.

2. The publication
suggests that the competences to be required may be learned in three years of
mathematics if the learning experiences are properly organized and taught,
rather than in the four years of conventional mathematics courses mentioned in
the bulletin.

3. All of you, I am
sure, believe that mathematics should be a part of the general education of
high school pupils as well as preparation for such specialized fields as
engineering. This belief raises, for all
but the largest high schools, the question of how to organize the mathematics
offering so that both sets of needs are met.

4. The largest
problem of all may be that of selecting and arranging the mathematical
experiences of high school pupils to achieve the most effective learning.

I request that you serve as a committee (1) to study the
problems involved and propose revisions in the mathematical program of our High
School and (2) that after a new program has been approved by our staff, you
continue your study of it as it goes into effect. I am asking Mr. Beberman to serve as
chairman. Your committee can be of great help, not only to our school, but to
others in Illinois as well.

<signed>

Beberman, though
only a year at Illinois, and as yet without the dignity of a PhD, was a member
of a faculty committee that was formed to systematize the effort in 1951;
indeed he thereupon immediately became the first director of what was called,
modestly, the University of Illinois Committee on School Mathematics, and whose
acronym, UICSM (initially UICSSM, for "Secondary School
Mathematics"), is today remembered more as the name of a curriculum, and a
system of teaching, than as the name of a group of people.

UICSM early, but not immediately, received financing from
the Carnegie Foundation, and in later years, when it had grown much larger,
from the U.S. Office of Education (OE), under its Cooperative Research Program,
and even the National Science Foundation (NSF) for particular studies
associated with its work (and of its junior offshoot, the University of
Illinois Arithmetic Project, headed by David Page, also of the school of
education). Beberman, already an
experienced teacher of children, was now also a professor teaching future
teachers to follow his methods if possible.

The quality of teaching in the schools was not the only
problem UICSM had to face, for it was immediately apparent that the curriculum
was even more fundamental than classroom procedure. Any Dean of engineering could see at a glance
that what was being printed commercially for the high schools in 1950 was not
what was needed, and any mathematician could also see that it was usually ignorant
and often downright wrong, mischievous in its misrepresentation of the nature
of mathematics even at the most elementary levels. UICSM therefore began with
two threads of activity: training
teachers and writing textbooks.

Now, Beberman was not a mathematician. Almost nobody
associated with school mathematics in 1950 was a mathematician, and the
textbooks showed it. Beberman's first friend at Illinois had been Bruce Meserve,
a mathematician and later an author of some fine textbooks; but Meserve left
Illinois in 1953 and Beberman's principal associate in UICSM for purposes of
writing texts was Herbert Vaughan, a professor of mathematics whose specialty
was logic. Some commentators on the
failures of the New Math, in later years, attributed that failure to this
single fact, that Vaughan persuaded Beberman that the proper study of
mathematics at the high school level begins with some set theory and formal
logic. Without a clear idea of how a list of axioms gives rise to the
particular structures of mathematics -- algebra and trigonometry as well as
geometry -- high school students intending to go to college would be limited,
so the argument went, to a repertoire of handy rules rather than a mathematical
education one could build on. This emphasis was later to be taken up by the
most prominent of the other 'new math' projects, most notably SMSG.

That Beberman took Vaughan's advice can be seen in the
textbooks they wrote together for the UICSM courses, initially simply produced
by mimeograph for trial and ultimately published commercially [e.g., High
School Mathematics, Course 1, 1964, and Course II, 1965, Boston, D.C. Heath
Co.] Beberman's personal correspondence with both friends and critics often
amounted to discussions of the language to be used, to make mathematics both
logically correct and understandable to high school students.

A less than kindly account of the UICSM devotion to logical
niceties has it that Vaughan, even before the establishment of UICSM, already
had one or more book manuscripts in his files, rejected by commercial
publishers, which he dusted off for UICSM when the opportunity arose. He might have had them, since certainly
before the arrival of Beberman there would have been no publisher willing to
take a chance on such a project; but there is also no doubt, as can be seen
from Beberman's correspondence with mathematicians all over the country, that
Max himself was a genuine co-author of what ultimately appeared, and not just
the spectacular classroom performer of another man's scripts.

Furthermore, to say that Vaughan was "at fault"
for the logical excesses of the Beberman program implies
two questionable assumptions: that there
was such fault, and that Vaughn was responsible. As to fault: Had Beberman asked almost any
other relatively young mathematician of the time to serve as Vaughan did, as
co-author and mathematical guide, he would likely have got much the same
advice, for any mathematician looking at the usual school textbooks of the time
would have noticed first off that they were lacking in logic and structure, and
that indeed they were often downright foolish.
While it does not follow that this mathematician would have prescribed
much theory, or first-order functional calculus, with the technical notations
and language used by professional logicians, something needed to be prescribed,
and it is hard to see how the prescribers could avoid the tension between
logical structure and pedagogy that characterized the programs known generically
"the New Math", except by abandoning the call to reason -- as indeed
happened later on in the reaction of the 1970s.

In point of fact, Vaughan was not initially a member of
UICSSM, which consisted of a professor of education (Beberman), a mathematician
(Bruce Meserve, a
geometer, not a logician), a professor of
electrical engineering (Daniel S.
Babb), and a practicing high school teacher (R. E. Pingry). The initial committee was formally appointed
on December 18, 1951 by Charles M. Allen, Principal of the University
High School (affectionately called "Uni" by the students and their
parents) of the University of Illinois at Urbana, the high school which was to
serve as the initial laboratory for what
became UICSM. Though Meserve left the
University in 1953, the initial work on the syllabus was done by Meserve and
Beberman:

For example, A Grade Placement Chart (tentative) was drawn
up by Beberman on February 26, 1953[5]
. Meserve had been granted a half-time
"release" the preceding December by Professor Cairns, his chairman in
the mathematics department, to work with Beberman, and this document was one of
the early results of that collaboration, and not the work of Vaughn at all; it
is a typescript signed by Beberman alone, and Vaughn was not to become part of
the committee or its works until 1955, after Meserve left the University and
the project.

(Actually, in consequence of some administrative dispute
the details of which have probably died with the participants, both Beberman
and Meserve left the University of Illinois in 1953, but Beberman, to the
dismay of wife and children, returned to Illinois the following year from
Florida, where he had already bought a house which then had to be sold
again. Of course, his return was
attended by a promotion and a raise.
Meserve never returned, but remained a close friend of Beberman's to the
end. This sort of thing is common in academe.)

Beberman's own draft placement chart (1953) names all the
topics that should occur in Grades 9-12, as follows: Algebra and Arithmetic, Geometry,
Trigonometry, Logic, and Statistics; and then, devoting a large double-size page to each of these
topics, specifies what subheadings should occur in each of the four high school
years. The page devoted to logic reads as follows:

NINTH YEAR

Generalizations

Definitions, conventions

Postulates

Theorems, corollaries

Undefined terms

Hypotheses

Inductive reasoning

Deductive proof

If-then statements

Converses

TENTH YEAR

As in ninth year

Indirect proof: reductio ad absurdum

Contrapositive

method
of elimination

Inverses

Circular reasoning

Analysis and synthesis

Fallacies

Multiconverses

Necessary and sufficient
conditions

Functional dependence

Class inclusion

ELEVENTH YEAR

Functional dependence

Implication

TWELFTH YEAR

Functional dependence

Implication

Mathematical induction

Postulational thinking

This document is clearly
tentative, and clearly Beberman's, containing some ideas that a mathematician
might not have put into those words
(especially 'functional dependence', which by 1950 was an old-fashioned word,
that Beberman might have been concerned with in his own study for a doctorate
at Columbia Teachers College. It is the
phrase used to designate "function theory" in the MAA 1923 Report[6],
that is, it refers to the idea of a function being a relation of dependence
between independent and dependent variable, and does not, in Beberman's
listing, mean the rather specialized notion of the that name that occurs in
multidimensional calculus.

Bruce Meserve, at about the
same time, also published an article[7] outlining the topics a high school college
preparatory program should contain, 67 of them, with reference to the report,
among others, of the NCTM Commission on Post War Plans[8]
, another document that would be deemed
archaic by the "new math " standards of a very few years later.

The main problem, that of the
tension between mathematical imperatives generated by the advance of
mathematics since the 19th Century, and pedagogical imperatives, or what seemed
to be imperatives in the "progressive era" that followed, could not
help emerging during the 1950s, since all mathematicians in the first fifty
years of the 20th century had learned to write in a certain style, in which
axioms and close logical reasoning were predominant, and where no statement
goes unproved, while the school teachers were still being taught a la
Minnick. It seemed natural to most
mathematicians that one should organize school texts tightly,
making no statement that was less than complete and less than fully supported
by logical reasoning from previous assumptions.
As is visible in the talks given in some of the math education
conferences of the 1960s, many mathematicians were even more fervent than Beberman
and Vaughan in demanding a full-scale Bourbaki approach to school
mathematics.

While Beberman (and Meserve,
for that matter) had been conservative in the initial formulations of what
freshmen in engineering needed to know, the 1955 team of Vaughan and Beberman
were riding the spirit of the times, not yet realizing how difficult it would
be in practice to assure high school students the benefits, such as they might
be, of the rigorous approach. Nor could
Beberman initially realize that this problem would arise at all in mathematics
for the earlier grades, right down to Kindergartens. UICSM began, after all, as
a University High School program, with an audience of college-bound students
who even had to compete for entrance to the program. But while Beberman's own UICSM remained a
college-preparatory high school math program until the end, "new
math" reformers in the lower grades were constructing imitations,
textbooks and examinations in which novel words such as "set" and
"Venn diagram" appeared as newly necessary preludes to arithmetic, to
bedevil the elementary school teachers and the parents of the children they
were teaching.

Teaching modern mathematical discourse to children is not easy, and perhaps impossible on a mass scale even for the
best teachers with the most clearly written textbooks. The fact that Beberman himself was so great a
teacher probably obscured the necessary lesson for a few years, but those few
years were crucial, since with the 1957 launching of Sputnik and the NSF
creation of SMSG in 1958 the same taste for logical excesses was implemented on
a truly enormous, and irrevocable, scale, before the lessons of the Beberman
experiment were visible in contexts more ordinary than the University of
Illinois laboratory school.

To begin with, Vaughan and Beberman didn't begin by writing
"textbooks" at all. They wrote
chapters and sections, mimeographed and given immediate trials by Beberman and
other teachers in the University High School, and then subjected to immediate
revision in light of classroom experience.
This became the pattern in the later curriculum projects of the
"new math" era as well, especially the School Mathematics Study Group
as led by Edward Begle at Yale and Stanford, beginning in 1958 -- well after
UICSM was established. As time went on, UICSM enlisted other high schools to do
the same as at the Illinois base, but only after their teachers had come to
Illinois for special summer instruction in the use of these materials. In later years "Summer Institutes"
were held outside of Illinois especially for teachers of UICSM materials, and
schools all over the country applied for permission to participate.

Beberman personally conducted correspondence with all these
schools and the participating teachers, many of whom were known to him from his
incessant travels around the country.
His travels were initially intended to recruit schools and teachers for
UICSM, in fact, but Max the showman never passed up a chance to secure public
acceptance of his ideas via local newspaper reports -- with photographers -- of
his visits. From beginning to end,
Beberman did not permit use of UICSM materials (in classes affiliated with his
project) whose teachers did not receive such training, and from beginning to
end these teachers submitted systematic reports to UICSM on their success or
failure, as they saw it, along with suggestions for improvement. To the newspapers it was a travelling show,
worth at least a column and a photograph, and sometimes an editorial commending
or condemning his approach, but to UICSM Beberman's travels were more than
propaganda; they were for the education of the educators who watched him work,
and who would later be training to imitate him.

By 1957 the UICSM materials were ready for a tryout on a
broad basis for the first time, when, according to Willoughby
[9],
12 pilot schools cooperated in a program involving forty mathematics teachers
and about 1700 pupils. Thereafter, the
number of experimental schools and classes increased rapidly. By 1958 the UICSM curricular materials, which
now included lengthy teachers' guides as well as school texts, were made
available to a wider public, that did not formally participate in the program
or have the advantage of the kind of training Beberman insisted on for his
experimental use. The results cannot be known, of course, but one can infer
from the growing failure of the uses of these and allied (generally
bowdlerized) materials over the next ten years that the initial intention, to
keep the program strictly experimental and controlled, was wiser than could be
maintained during the post-Sputnik clamor for "new math". Among other things, the experimental attitude
had generated much of the enthusiasm UICSM teachers felt for the project, while
the sudden wide introduction of "new math" materials into unprepared
classrooms inspired fear more than enthusiasm at the grassroots level.

In due course, then, since much of the commercial material
published during the 1950s were poor imitations or bowdlerizations of what
Beberman and Vaughan intended, there were also produced overtly commercial
versions of most of the tested and seasoned UICSM material: Beberman, M. and Vaughan, H.E. High School
Mathematics, Course 1 (Boston: D.C.
Heath, 1964) and Course 2 (Plane Geometry with appendices on Logic and Solid
Geometry 1965). As these were commercial
texts anyone could now use; but while the UICSM materials were produced only
after long experience in class use by thousands of students, other textbooks of
the 1960s were less carefully done and less correct. By that time the whole country was
enthusiastic about "the new math", and every publisher had to have
something that could be described this way.

The books by Beberman and Vaughan had their faults, but
they were not the worst of the genre.
The commercial publishing houses had a wealth of models from which
authors able to write more attractively for ignorant audiences could take their
material. SMSG, the post-Sputnik NSF-financed
School Mathematics Study Group headed by Edward Begle, a much larger effort
(which will be described below), including all grades K-12, was also such a
source; but while UICSM and SMSG were at least mathematically correct many
other authors simply didn't understand the purpose of this sort of writing, and
offered embarrassing simulacra of logical discourse, advertised, to be sure, as
"New Math." Furthermore, most
teachers of "new math" were untrained for the task, let alone coached
by Beberman.

Beberman himself never stopped learning mathematics as he
went along, or at least elementary mathematics from a modern point of
view. His trial drafts were sent to many
mathematicians around the country, even to Morris Kline, the New York
University mathematician who was the New Math's most articulate and frequent
critic, for comment and criticism. In
innumerable letters Beberman discussed details of nomenclature, the best axiom
system to use for geometry and the best sequence of theorems and exercises,
whether this or that topic should precede or follow that one or this, and so
on. These epistolary debates were
learned and serious, and Beberman never showed a closed mind to criticism, so
much so that his colleagues sometimes wondered if he were straying from what
they took (from his earlier attitudes) as the True Path.

******************************************************

The hallmark of the New Math, Beberman's to begin with, but
all the others to follow, was logical language.
A typical conundrum in high school mathematics is the question of how
one proves a trigonometric identity, that is, why the procedures by which one
solves equations are suddenly prohibited in "the proving of an
identity." Somewhere early in his
career a student learns some rules of the form, "You can do the same thing
to both sides of an equation." Most
often this kind of statement is referred back to Euclid's "If equals be
added to equals, the results are equal."
This is true enough if the things in question are known to be equal, but
if algebraic equations are to be solved, or identities proved, the objects
being manipulated in this way are not yet known to be equalities, despite the
presence of an "equals" sign in the middle of each line, and the
manipulations are problematic in a way hardly mentioned in the traditional
textbooks.

The experience of generations has shown that children
taught "to solve equations" this way, taught without attention to the
logical standing of their materials, do indeed develop reflexes rather than
understanding. The more recent invention
of the hand calculator, even the graphing calculator, has done nothing to
change things. Give the dutiful child
"x²-y²" as an exercise and he will write "(x-y)(x+y)
ans." Show him "(x-y)(x+y)" and he will (unless he makes a mistake!) write
"x²-y² ans." -- all without having been asked a question at all. Before 1950 in American schools, and to a
distressing degree today as well, school mathematics was simply not construed
as a collection of statements written in English, whether questions, answers,
or theorems, and the feeble attempts to put order in the cook-books of high
school algebra even since the ill-fated experiments of the 1960s are still an
embarrassment.

In 1951 Beberman was one of the pioneers in deciding it
could be otherwise. Why do freshmen come
to college apparently believing that (A+B)²=A²+B²? Obviously because they have
learned the wrong reflex. How can
they be persuaded of the correct formula, that the answer is A²+2AB+B²? There are several ways to go about it.

A skilled traditional teacher of 1940 would first try
numerical examples: If A is 3 and B is
4, then A²+B² is 25, while (A+B)² is 49. Clearly the
badly remembered formula is wrong. Then
he goes on, this traditional teacher, perhaps after other numerical examples
illustrating the correct and incorrect formulas, to show the well-known diagram
of a square of side length A+B, partitioned to show four smaller pieces, one a
square of side A, one a square of side B, and two rectangles of sides A and B.

This
diagram was obviously known in Babylonia four thousand years ago, and still has
the power of conviction, but it teaches only that one lesson. Furthermore, from the point of view of a
logician this diagram is irrelevant, since the correct formula concerns
numbers, while the areas of the diagram are an application dependent on the
properties of Euclidean space and a number of conventions by which we apply
numbers to the study of that space. The
real reason, goes the New Math argument, is that by the right-distributive law
of multiplication, (A+B)(A+B) = A(A+B)+B(A+B),
following which, by applying the law on the other side, we obtain
(AA+AB)+(BA+BB). Further dickering with
the associative law of addition, and application of the commutative law
(AB=BA), produces the desired answer.

A complete rendition of this argument in the two-column
format used for Euclidean proofs of the same era would occupy a full page of a
textbook, yet one should not be too hasty in laughing at this example of
logical overkill, since this particular formula is not the only point of the
exercise. In fact, the distributive and
associative laws must be understood by the 9th grade student of elementary algebra
for many other problems of importance.
Some of them are quite important even in daily life, some for future
developments in mathematics that at least some of the University of Illinois
High School students would be studying in a few years' time, whether for
physics or medicine or finance, and some for the mere filling-in of the
cultural background that all education, whether mathematical or literary,
should provide.

But should one belabor the
"obvious"? The
distributive law is in fact understood by the average adult citizen, as may be
illustrated by the laughter that invariably follows anyone's telling this
allegedly true story:

A certain drug store was advertising "15% OFF -- ALL
ITEMS!", and the narrator, having chosen her purchases, took her place in
line at the check-out desk while the clerk was serving those in front. The clerk entered each price into the cash
register, then calculated 15% of that price (as a
negative entry in the cash register, i.e. as a subtraction from the total) before
entering the next item. Each customer
waited while this tedious string of entries and 15% subtractions went on, and
it took a long time indeed, before each customer's grand (discounted) total
emerged. When the narrator came to the
front she suggested to the clerk that it would save a lot of time if she
entered all the items as marked, summed them, and only at the end took 15% from
the total. "Oh, I thought of
that," said the clerk, "But it doesn't always work."

Now if failure to grasp so simple a principle evokes
laughter, perhaps it is unnecessary to teach it in the schools? Apparently not, as Dean
Everett of Illinois observed of his 1950 freshmen, who were unable to expand a
power of a binomial or graph an elementary function. How abstract and how extensive the necessary
instruction should be, on the other hand, is a question that was settled one
way by Vaughan and Beberman, and in other ways by other reformers of the time;
and the details of the matter are a center of serious disagreement in the
community of mathematics educators to this very day. The last time there was general agreement, in
the *de facto* national high school curriculum of (say) 1940, the policy
was merely to ignore logical structure altogether, and to introduce rules as
necessary, offering no pretence of justification. Indeed, it was not even understood even by
teachers that there was a question.
Geometry had axioms and proofs, and algebra was simply a different ball
game. Everybody knows xy=yx, so why go on about it? This perception clearly had to change, and in
the 1950s it changed most drastically indeed.

One letter written by Max Beberman in 1957 can illustrate
the seriousness with which he took the necessity of precise language, in this
case nomenclature itself rather than logic.
In answer to what appeared to be a request that he criticize a chapter
named Basic Mathematics in some proposed encyclopedia called __Our Wonderful
World__, Beberman wrote a long set of comments on things he counted
misapprehended or wrong in the draft, citing them by page number. It is clear from what he wrote what the
original had said. Here are some of his
comments:

(4) This is quite
bad. Try this: The value of π can never be expressed exactly
by a decimal numeral.

p.53 (1)
You can write a numeral but you can't write a number, just as you
can write a person's name but you can't write a person...

p.56-57
This section should be thrown out... One set of symbols can never
be equal to another set of symbols, unless the symbols are identical (in
appearance). There is no such thing as
two equal numbers.

The first of these comments is clearly in answer to a
statement such is commonly seen in school math books, saying something like
"The number π can
never be exactly known." This is of
course untrue, since π indeed
is exactly a certain ratio of two geometric lengths, and the misstatement
Beberman is criticizing is not merely a trivial matter of English usage. School children learning such false
statements do come to think there is something inexact about the idea of π, just as later they come to
think the derivative of a function at a point is also some sort of
approximation, making an impenetrable mystery of mathematics altogether. Add to this the mystique of "variables"
(denoted x, y, etc.) being numbers that "actually vary", unlike those
denoted a, b, etc., which do not, though they might change from problem to
problem, and you have the beginning of a catalogue of horrors that every
college calculus teacher is well acquainted with.

What can be a number that "actually varies"? Beberman and Vaughan sought to lay such
nonsense to rest. In summer institutes
they taught teachers to talk sense, in preparation for demanding sense of the
high school children they would in their own turn be coaching. One exam
question in one UICSM summer institute for teachers asks for the insertion of
single quotes that make the following sentence meaningful and correct:

It is impossible to add 8 to 5 but it is easy to add 5 to
8.

The intended answer was

It is impossible to add “8” to “5” but easy to add 5 to 8

since “8”
and “5” (printed in quotation marks) denote numerals, not numbers, and addition
is something one does to numbers.

The distinction between number and numeral is far from the
only subtle point insisted on by Beberman's project and the Beberman-Vaughan
textbooks, though it is probably the one that attracted the most derision as
the "new math" worked its way towards its own extinction. The language of sets was also new at this
level, and while very little of what mathematicians call the "theory of
sets" -- a profound study not at all suited to children -- is needed by
daily mathematics, some of its more obvious nomenclature is important for
understanding what is meant, for example, by "solving an
equation".

That is, "Solve 3x-5=7" means "Describe the
set of all numbers such that thrice that number diminished by five will produce
7." Without the idea of a
"solution set", of which x is the name of a typical member, students in
the past had regarded "Solve 3x-5=7" as a mere prescription for
symbol juggling according to pointless rules.
Such a student will be quite stymied if the problem is changed to
"Solve 3x-5 = 1 - (6-3x)", which when treated in the same way
produces the curious result "0 = 0", rather than what the
traditionally drilled student would recognize as an answer at all. The correct
answer here is that *every* number replacing x in the second equation
yields a true statement of equality; but one would not think to investigate
that possibility until “Solve 3x-5 = 1 - (6-3x)" is translated into a real
question demanding an answer. In the
high school classrooms of 1950 that sentence was thought to be understandable
as it stood, but in truth “solve” is a word needing a good bit of
definition. Beberman saw that without a
language in which to ask mathematical questions there are no questions, only
exam rituals.

These distinctions, second nature to logicians, were new to
school mathematics and most of their teachers in the 1950s, but they seemed
essential to Max Beberman if the imprecision of the actual algebra and geometry
of the typical high school course were to be avoided, and it was the
imprecision in the previous way of doing things that he saw as the great stumbling
block to later learning of even the practical applications of mathematics, in
particular the calculus the Dean of engineering wanted his freshmen to learn at
Illinois.

Just the same, not everyone shared Beberman's point of
view, not even every mathematician.

© Ralph A. Raimi

Unfinished, May 6, 2004

[1] UICSM Archive at Urbana, IL, correspondence of Beberman with Brucr Meserve

[2] NCTM yearbook #32, 1970, p251ff

[3] Meserve,
Bruce E., __The____ University of Illinois list of
mathematical competencies.__ The School Review 61 (1953) 85-92

[4]John Harrison Minnick,
"Teaching Mathematics in the Secondary Schools" (New

York, Prentice-Hall, 1939).)

[5]Beberman Archive 10/13/1, Box 8

[6] MAA, National Committee on Math Requirements,
__Report: The Reorganization of Mathematics in Secondary Education__. (Published by "MAA, Inc." 1923)

[7] __The
School Review__ LXI #2, Feb, 1953, p85-92

[8]__ NCTM
Commission on Post-War Plans, Frist Report__,
Mathematics Teacher 38 (1945), 195-221, entitled __Improvement of Mathematics
in Grades K-14__); also its associated __ Guidance____ Report__, Mathematics
Teacher 40 (1947), 315ff.

[9]
Willoughby, Stephen S., __Contemporary teaching of secondary school
mathematics__, NY, John Wiley 1967