My Experience With the CEEB Commission Report

On July 1, 1959 I became Acting
Chairman of the department of mathematics at the University of Rochester, for
one year. My major task that year was
to find and hire a real Chairman, to be brought into our department from
outside, and I succeeded: Leonard Gillman was appointed in 1960 and in the fall
of that year I reverted to the ranks.

In about
January of 1960, while I was still the Chairman, I was visited by John Montean, a professor of science education
in our College of Education, asking me to find someone in my department,
preferably myself, who would accept NSF money for a “Math Institute”, something
I had only dimly heard of through reading the Report of the CEEB
Commission. I had recently read the
1959 Report of the CEEB Commission, which, along with its larger, companion,
volume called ** Appendices** had been mailed to all members of the
MAA. It had impressed me greatly, and
favorably. In particular, it includes a
long recommendation on the education of secondary school mathematics teacher
beginning on page 50, describing fairly well how such Institutes as Montean had
in mind should operate, and also urging courses, given by mathematicians, to provide
such teachers with actual teaching-material that they had not received in their
earlier training, mathematics that could be found in the Report’s

The resulting
NSF grant was to Montean, not me; he did all the administrative labor –
almost. The grant paid the tuition for
30 working high school teachers to come to a classroom each Saturday morning
during the school year 1960-1961 for a course in “modern mathematics”. The participants received a small stipend,
too, and I received some payment ----I
don’t remember how much, but it was useful.

There was one thing more: Montean (a professor of Education) insisted
that the course be given six hours of __graduate-level__ credit; that’s what
the teachers would want. In New York,
as in many other states, a Master’s degree is required for permanent
certification as a high school teacher, and young teachers, temporarily
certified, often take courses part-time towards that end, even if they finish
up (or begin) with a semester in residence somewhere. The law intends that high
school teachers be required to be academically more competent than one can expect
of elementary school teachers, but in the case of mathematics and the sciences
it is really impossible to ask a Master’s degree in the subject matter. Persons of that competence in math or
physics generally continue for PhDs or go into technical jobs paying much more
than teaching in school. Future
teachers therefore mostly get their Masters degrees in colleges of education,
where the mathematical content of their “methods” courses is not demanding. At least, this was the case in 1960, and as
for teachers already in service in Rochester-area high schools, virtually none
of them could qualify for any graduate courses at all in our own program. (If
they could, the entire Institutes Program would have been unnecessary.) On the other hand, an in-service course in
what is essentially high-school mathematics, however necessary and valuable,
could not honestly be labeled a graduate course in our math department.

Yet the course I had in mind was
what the __Appendices__ was written for. Except for its clipped exposition
and its lack of enough exercises for the student, it is an admirable text, at
least for the professor, who naturally finds it easy to expand on the
exposition and supply more examples.
Much of the __Appendices__ markedly resembles, in fact, a textbook that became exemplary in the
following decade or two: __Fundamentals
of Freshman Mathematics__ , by Carl Allendoerfer and Cletus Oakley
(McGraw-Hill, 1959). This was no
accident, since Allendoerfer was one of the more prominent members of the CEEB
Commission. But __Freshman__
mathematics – for graduate students?

Well, I had to solicit an
exception. Technically speaking, the
decision to award graduate credit in mathematics is a function of the Graduate
Committee of our college (not the college of education). They understood the delicacy of the problem
as well as I and Montean; but this was a special case. Since the course was of limited duration (it
would never be given again, in fact), there would be no danger of Arts College
graduate students discovering it as an easy way to pile up “elective” graduate
credits. Furthermore, I told the
Graduate Committee, the NSF, and myself, I would be teaching a rare and
necessary skill to these teachers, something the country would not have benefit
of without this Grant. Nobody in the
College of Education was as qualified as me, I said modestly, and besides, the
NSF requirement for the grant needed a __mathematician__. Being Acting
Chairman helped, too, and so my
recommendation to the College’s Graduate Committee for a six-hour donation of
credit for completion of my course was granted. Two courses, actually, Math 212 and 213, fall term and spring
term, three hours attendance each Saturday (9 to 12 a.m.) throughout the school
year, award of three hours graduate credit for each course.

The Arts College Graduate
Committee did understand the scholarly hypocrisy invariably attached to having
a College of Education graduate degree awarded equal status with one in math,
history or engineering. That hypocrisy, which seldom has to be confronted
as directly as in the time of the NSF Institutes of the newmath era, is
something every university has to endure.
Its history is a long one whose telling is beyond the scope of this
story.[1]

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Looking back
on this episode I see that I made one serious mistake. I suffered from a democratic virtue in those
days, the days of my youth, the 1950s, long before such democratic thinking
became obligatory in the university turmoils of the 1960s and 70s. Thinking
that the best educated high school math teachers had less need of my course
than the average ones, and that the least educated would probably not be able
to follow the course at all if its content was to be non-trivial, I selected an
intellectually “middle-level” thirty persons from my very large group of
applicants.

Montean had seen to it that this
Institute was advertised in all the high schools of the region, and I must have
received sixty or more applications.
Each one came with a ** Curriculum Vitae** and all the college
transcripts and course credit listings the applicant could amass for me. Strange transcripts are of course hard to
read, and I had to take account of the quality of the colleges my applicants had
attended, the grades received, their SAT scores, if any; but all in all I
probably did get the middle group. What
I didn’t get was what I had innocently expected to get: a group that would
learn what I thought I was teaching.

I should have
selected only the very best. I began
the first Saturday session with exactly the thirty students I had accepted; 28
finished the first semester with a passing grade (quite minimal, and indeed
fraudulently passing for some), and of these, only 20 finished the second
course with a passing grade. That was
democracy at work, I supposed; in my usual university courses I had never had
to face the problem of grading a group almost all of whom hadn’t learned the
subject. Fundamentally, the only __listed__
failures in the present case were those who were too often absent, or hadn’t
handed in at least half the written exercises (no matter how well done). Half the rest got the lowest grade, “p” for
“pass”, and the rest got A’s and B’s, sometimes decorated with a plus or a
minus.

I would
estimate that about five of my students really gained something from this
course, but I cannot be sure it was not more.
Spending a year hearing the words and looking at the diagrams, and
attempting to solve problems – even without success -- might itself have been
helpful to some of my near-failing students in later years, when called on to
teach “trigonometry”. For example, almost
none of these high school teachers had previously known that sine and cosine
could be defined for “angles” larger than 180 degrees; and, needless to say,
they had earlier known nothing of the graphs of the trigonometric functions
(radians were also a mystery), or of the exponential and logarithms, either
(base e was another mystery).

Those
teachers were my age, after all, some of them older. Probably all of them had learned, as I had, that “a logarithm is
an exponent”, an abbreviation of the ungainly, though complete, definition,
“The logarithm is the exponent to which the base must be raised to produce the,
er…, the number you began with.” This definition, which I remember we stood by
our desks and recited when I was in high school, suffered from the start by not
saying just what the logarithm was a logarithm ** of**, since the idea
of “function” did not appear at all in high school math of the 1940s. And since there was no sense, in such a
recited definition, of the idea of

I tried to teach such things. My goal was simplification, not filling the
air with newmath jargon. But I did not
succeed. The main trouble was that they were overworked. During the school year they taught a full
load, and then on Saturday had to drive to my class besides. They had little time for homework, the
exercises for which I wrote out for them every week, duplicating them on a
“Ditto” machine, something that no longer exists these days but serves well to
make copies of handwritten text. As the
year went on I received fewer and fewer responses. I expected the work I assigned to take three or four hours, plus
what time was needed to review the class notes. They didn’t have it, or didn’t have the strength for that much.

My class was not entirely
lecture and homework; I included student problem-solving at desk or blackboard,
but the results of in-class problem-solving were so dismal that I did that less
and less as the year went on, hoping that more leisurely work at home,
consulting familiar books of their own (They each had a copy of the CEEB __Appendices__,
too, courtesy of NSF) or talking things over with each other where possible,
would give them some ideas and some confidence. By the end of the year I was receiving next to no handed-in
solutions to my exercises.

The typical class began with a
discussion of the preceding week’s homework.
I would solve crucial problems at the blackboard, asking questions of
the class as I went along, sometimes digressing, sometimes a Socratic procedure
tending to the solution of the problem at hand. Then I would explain the next bit of text, exhibit some problems
with solutions, and have the students do something simple at their desks with
the new definitions, in preparation for doing their homework.

I’m afraid
most of them were lost. One day after
class, two of my students, women older than me, came up to me with a question,
something they had not understood the preceding week. I was appalled; what they didn’t understand was at the heart of
the matter. They were in no condition
even to begin that homework they were supposed to (but didn’t) hand in to me
that day. So I took a few minutes –
this was Saturday and no school bells were ringing to push us anywhere – to begin
at what I thought was the beginning, to gather up what they needed,
illustrating it on the blackboard, too.
They stood there dumbfounded – and I was being ever so plain! One of them said, “My, you’re so young, to
know so much.” It made me so sad, to
know that there was no way to reach them with the basis of this course, at
least in the time available. They knew
that, too. There was no animosity in
that exchange.

There were, however, some simple exams and a final exam. I needed __something__ on which to base a
grade, and I entered many little grades in my classbook (which I still have),
some for exercises and some for short, mainly factual, quizzes. By accident I have found a copy of what was
probably my last effort, a ditto sheet of particularly ambitious exercises I handed
out on April 15,1961. It has been
folded into my copy of the __Appendices__, which like some other of my old
textbooks I have kept all these years. (I reproduce it below.) Looking in my classbook today, I see that I
have listed no grades for that last assignment, which should have been handed
back to me the following week. Many of
the better students, who ordinarily handed something in, were absent altogether
that following week. Embarrassed,
perhaps, at having nothing to hand in. I
think practically the whole class took a bye on this one.

I have no later “homework”
grades, either. By April I had given
up. Except that I recall my discouragement
with the course, I no longer remember exactly what happened in response to this
or other assignments concerning elementary functions, the capstone of the
course. Nor do I remember how much of
the __Appendices__ I actually covered during the course, except that it was
far from all of it. Whether or not, or
how much, they understood was something I no longer dared investigate by
Socratic dialogue at the end of the course, as I was wont to do earlier. My Socratic dialogue grew more and more like
a monologue, my questions becoming more rhetorical than real as time went
on. But while there were no more than
four who were handing in the homework
by the end of the term, I continued to assign it.

While the CEEB Commission Report
lists exponential and logarithmic functions as part of their recommended 12^{th}
grade syllabus, the Appendices don’t have them. Evidently I tried to include them, somehow, for here is that last
assignment, verbatim, copied here from my ditto page:

__Math 213____Homework____April 15, 1961__

1. Use a table of
powers and roots to obtain a table of values for 3^{x }when x = -2, -1,
0, 1/3, 2/3, 1, 4/3, 5/3, 2; plot the points on a good piece of graph paper and
connect them smoothly to get a graph for **y = 3 ^{x}**

–2 ≤ x ≤ 2.

2. Draw a tangent
line at (0,1) and estimate the slope of the graph there.

3. Calling the
answer to Question 2 by the name c, show that the tangent line at (1, 3) has
slope 3c, and that the tangent line at (-1, 1/3) has slope (1/3)c.

4. Use the graph
to find an x such that **3 ^{x} = 2**, and to find a number

5. Take another
piece of graph paper and plot the log** _{3}** function, i.e., the
graph

Raimi

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Yet the
prescriptions of the CEEB Report remain today a good model for a pre-calculus
program, though to cover it all, whether in a part-time Institute or as a full
scale MWF college course, seems to me to be more than one year’s worth. On the other hand, the sections on complex
numbers are unnecessary for both the geometry and the analysis involved in a
calculus course, and could well be replaced by a chapter on the exponential and
logarithm functions; and I would certainly also include some graphing of
rational functions with factorable or factored numerators and denominators.

My Institute course didn’t try to include all the topics listed in
the Contents of the Appendices, some of which isn’t exactly “pre-calculus”; but
I did add the logarithms and exponentials.
I was anxious to give them some idea of the notion of function, with as
many accessible examples as possible, so I omitted most of the suggested
geometry

Here is the table of contents of
the Appendices:

An introduction to algebra

A classroom approach to irrational numbers

The linear function and the quadratic function

Introduction to complex numbers

Limits

Permutations, selections, and the Binomial Theorem

Mathematical induction

Some reasons for modifying the traditional treatment of
geometry

A note on deductive reasoning

The first sequence of theorems

Indirect proofs

Introduction to coordinate geometry

Theorems having easy analytic proofs

Outline of a unit in solid and spherical geometry

Geometrical transformations

Order relations in plane geometry

Introduction to vectors

Coordinate trigonometry and vectors

Trigonometric formulas

Circular functions

I have italicized all those sections – there are only two of them – that sound at all like what was, in the 1960s, being objected to by opponents of the newmath. Neither the CEEB Report nor its Appendices appear to me today to contain an overemphasis on structure à la Bourbaki, as the outcry that greeted its publication, especially in view of what came several years later, would make one think. Morris Kline would make one think it was pure Russell and Whitehead.

Yes, the notion
of “set” was essential to almost everything in the book; but those sections had
not been included for abstraction’s sake at all, or in ignorance of the fact
that mathematics has something to do with the real world. In writing of a solution of an equation, for
example – and this I saw myself while teaching my teachers – a student can be
confused if he is not told what the sentence “Solve x^{2} + 5x + 6 = 0” means. It does **not** mean, “Take 5 and divide it by two, …”; it
simply means, “Find all the x such that x^{2} + 5x + 6 = 0.

Yet, to my
audience questions remained. “All the x”?
Is x a number, is it two numbers?
(Try telling them “x is a literal number” as was common in 1950 and see
where that gets you!) The phrasing,
“Solve the equation …” is unclear unless there is the understanding of the
intent of the symbol “x” in the problem.
With a minimum of explanation, the matter is clarified by the
set-theoretic translation, “Find {x | x^{2} + 5x + 6 = 0}.” Corresponding symbolism was becoming common
in calculus books during that period, and if the teachers teaching it hadn’t
made such a long story of the notion of “set” itself they would have made a
better story out of such things as comparing {(x.y) | x^{2} + y^{2}
= 4} with {(x.y,z) | x^{2} + y^{2} = 4}. When I went to college I found it quite
unpleasant that “ x^{2} + y^{2} = 4” was entitled sometimes to
be a circle and sometimes a cylinder. Here in 1960 it became clear. Should not the high school teachers know
this?

Even
more elementary things sometimes are cleared up with set notation. A common “definition” given for the idea of
“circle” is “A circle is made up of all the points equidistant from the
center.” At least, you will sometimes
hear this from students. What is wrong
with this phrasing is that it doesn’t define a circle. A circle is a set of points, and to know a
circle one must know a rule by which a tested point is known to be on or not on
the circle. A __defined__ circle
must have a __defined__ center and radius, and a proper definition, in the
plane, is meaningless if it does not invoke both data and give us the
criterion, viz., “A circle with radius r and center C is {p in the plane | dist
[p, C] = R}.” From this the Cartesian
equation may follow, of course.

The
judicious use of the word “set” can be a great help in stating almost
everything in analysis; but it doesn’t need a chapter, or Venn Diagrams, for
such a purpose. The CEEB Commission
could have gone a bit easier on the subject, and probably would have done so if
it could have foreseen that its work would soon become something the authors of
textbooks for middle schools and even elementary schools would be trying to
imitate; but __as written__ – ** and it must be remembered that the
Commission recommendations were intended for the teachers of high school college-preparatory
students only** – the Report and its Appendices were not the call for
abstraction for abstraction’s sake that their critics imagined.

Even so, when I tried to get these ideas across in my teaching about algebra and the elementary functions to the high school teachers in my Math 212, 213 of the year 1960, I failed. I believe I would have done better if I had repeated the course with a group taken from my most skilled or best-prepared candidates. I see now that the seeding of the high schools with teachers who do understand these things would have been of benefit to their college-preparatory students, and maybe to other teachers. Too late to be wise; I never taught in another Institute, summer or winter, and in 1975 the whole Institute program was closed down. Federal money for Education, albeit under other headings, never was diminished, even in the Nixon administrations which closed the Institutes, but the invitation to mathematicians to participate in projects for the improvement of school teaching, and the improvement of textbooks, was replaced by 1975 by a corresponding invitation to professors of Education instead. And there it has remained.

Ralph A. Raimi

Revised 21 December 2006

[1]
See, for example, Cremin, Lawrence A.,
__The Transformation of the School:__ __Progressivism in American
Education 1876-1957__. NY, Knopf
1961; and Duren, Wm.L., the
chapter, __Mathematics in American__ __Society 1888-1988__, in Duren, Peter, et al (Editors), __A
century of Mathematics in America__ (Amer. Math. Soc. 1988), Part 2, pp
399-447, for some orientation concerning the intellectual decline of the
profession of teaching in America.
Other, more recent works to similar purpose are by Diane Ravitch: __The Troubled Crusade__, NY, Basic
Books 1983, and __Left Back: A Century of Failed School Reforms__, Simon
& Schuster, 2000. Finally, a polemic which damns the education schools from
top to bottom is Albert Lynd’s __Quackery in the Public Schools__, Little,
Brown, 1953.