A Prescient Letter to Frank Quigley Concerning The New Math

	I reproduce below, complete, verbatim and without amendment of any
kind, except that I have here divided the text into more paragraphs than
indicated in the original, a letter I sent to Frank Quigley, then
Assistant Professor of mathematics at Yale University.  The date of the
letter is a few weeks after the inauguration of the School Mathematics
Study Group (SMSG), headed by E. G. Begle, then a professor at Yale. 

	I had spent a postdoctoral year at Yale in 1955-56, and was back
at Rochester as Assistant Professor of mathematics.  But I still took an
interest in events at Yale and I had been reading in the newspapers about
the imminent formation of SMSG.  I thought Quigley was going to be
associated with the project, whose exact coutours and (needless to say)
ultimate fate were not yet known. 

	The footnotes of course are not in the original, and were added 
today, December 15, 1996.

                                          Sunday, June 29, 1958
Dear Frank,

      Today is the eleventh anniversary of my marriage, which is the
reason I did not attend the educationists' [note 1] meeting.  That seems
to make my wedding as important as an Orange Bowl game [note 2].  But it
is education I wish to write about, not weddings. 
      I was truly very sorry not to have been at the meeting, and even
sorrier not to be at Yale, where, I read in the newspapers, something is
being done about mathematical education.  I take it you are a member of
Begle's project, however, and I am pleased that a competent authority on
division facts [note 3] is in on it. It is important to keep the MAT boys
at Yale out [note 4], especially L__, at this stage of the game, but Begle
is capable of handling such necessary exclusions with his own stony grace,
and he knows L__ [note 5].
	If you do not know L__ (and if he is still at Yale), you must meet
him, perhaps under the pretext of asking his advice on something connected
with your project.  The full magnitude of the problem you face can only be
appreciated if you get to know those who teach the teachers.  L__ is
clearly a superior intellect among these, and has read enough
'anti-educationist' literature to impress a scholar, at first, as a
remarkably reasonable man.  Later, when you see the true convictions, and
lack of understanding, which underlie this veneer of sense, you will see
that hope does not lie in that direction. 
      The remodeling of elementary education cannot be revolutionary.  All
the violent revolutions of the past have generated their own reactions; I
have 1789 and 1917 particularly in mind.  One should contrast these
failures with the success of the evolution of English democracy and
justice.  There is a cybernetic problem involved:  not to build a perfect
machine, but to build a stable one, capable of correcting its own mistakes
in a stable way, and capable of growing towards perfection.  The first
stage of a revolution should be so constructed that while it may fall far
short of the goal, it renders inevitable that the next step will be taken
in the right direction, whoever may be around to take that step. 
      In the evolution of English law, this was done by insisting on
proper procedure, long before the particular statutory provisions, which
the procedure, after all, served, could have been called just.  It is sad
that witchcraft should have been called a crime, but it would have been
much sadder if witches had not been tried with all due process.  It was
inevitable that mankind should discover the non-existence of witchcraft,
but by no means inevitable that it should rediscover the value of due
process.  Preservation of due process, were a choice necessary, was better
than a fiat of the non-existence of witchcraft, given by a wise autocrat,
for then the next step in the evolution of official justice was
necessarily a good one. 
      Well, I don't wish to push this figure too far; I'd rather speak
specifically of our present concern.  We are faced not merely with an
outmoded, too slow, often false school curriculum, but with a staff of
teachers unable to teach a better one, headed by a bureaucracy which
doesn't want to, backed by a public which, having come from such schools,
doesn't much care.  If this were the precise picture, as it was ten years
ago, no action by a bunch of Yale professors could do any good.  But there
has been a break:  the public now thinks it cares.  There is a great deal
to be said for lip-service to a fine ideal; it permits underground labor
in behalf of that ideal some chance of freedom. 
      The public now demands better education.  It doesn't know what good
education is, and it fears every particular manifestation of it, but it is
so far committed that it will accept, if only for a while, a certain
number of genuine reforms.  When enough of these reforms have been
adopted, the public will feel free to call it a day.  The job, they will
say, is done. 
	Precisely this has happened once before:  1900-1950.  A transition
from public education for a few to public education for all was demanded,
and the noble ideal of equal opportunity was in the air.  The reformers
had their chance, and made some startling innovations, often on the basis
of good theory.  But their reforms did not carry any built-in guarantees
of further progress.  By 1930 an entrenched officialdom had taken over,
declared the revolution complete, and proceeded to deform the tenets of
that revolution to suit their own convenience. 
      If we are not careful, we will find a similar rigidity of doctrine
arising fifteen years from now [note 6].  The only difference will be that
a certain amount of mindless Boolean algebra will have replaced a certain
few pages of mindless trigonometric tables.  Our first task must be to
establish the procedure of reform beyond the possibility of repeal, even
at the cost of perpetuating a few barbarisms in the actual curriculum.  A
constant, continuous examination of the curriculum by the scholarly
community must be accompanied by a tradition of due honor to the
examiners; otherwise we shall have failed.  
	Of course this cannot be done in a vacuum; besides, it is not what
the public is currently demanding.  They pretend they want a better
curriculum, now; we must pretend we are giving it to them, now.  And we
must, in fact, change it, and do what we can towards changing the training
of teachers, and all the rest of what is needed.  But we must not forget
that our first purpose is to change the climate in the hierarchy of
education, so that our descendants will also be able to advance the true
      Thus we must, in all our public utterances, and in all our textbook
prefaces, and in all our conferences with educational leaders and
politicos, stupid or wise, emphasize the length and continuing nature of
our present effort.  
	There is great danger in this possibility:  On a given occasion we
may propose a certain kind of textbook, or a certain set of rules for
teacher certification, and encounter strong opposition.  In overcoming
this opposition we may be tempted to argue so persuasively the merits of
this particular proposal, and win in such a blaze of glory, that the cowed
opposition will mutter to itself, "OK, buddy, you won this time, but this
book, or set of rules, had better do the trick." 
	"Oh, it will, it must," we will have argued (otherwise, why so
fierce a struggle?), and for good or bad, our day will be ended.  We
ourselves will have been committed to rigidity, and to the vindication of
our momentary beliefs. 
      The public asks what to take for its educational malaise.  We must
answer, as the medicos do, 'Take advice, and come back every two weeks
thereafter.' This is the first advice they must take. 

      Sonya, Jessica, Diana, and I are well.  I hope to see you at the
Summer Meeting.  Would you like to take a weekend trip to Rochester this
summer?  We have a room, a piano, a flute, a shortwave radio, and lots of
good food and drink. 
	1.  I'm not sure what meeting this was.  The initial grant of
$100,000 from the NSF was dated May 7, and Begle's Advisory Committee was
already in place.  The meeting I missed was a more public affair than a
simple initial gathering of SMSG personnel, though I believe it included

	2.  This is a personal reference of no importance. 

	3.  When I was at Yale (1955-56), one of the younger professors,
perhaps Quigley himself, had found in a used book store a book about
teaching arithmetic in the schools.  In it were gravely listed the "number
facts" a child ought to know by this or that age.  Some of them were
called "division facts," e.g. that fifteen divided by five is three.  Of
course there were also addition facts, subtraction facts, etc.  At the end
of each chapter the author summed up the total number of facts that
constituted, apparently, a satisfactory accomplishment.  So it went.  We
passed the book around among ourselves at the time, making merry with it
and its whole philosophy of school mathematics, challenging each other
(for example) to "number fact" duels.  That book was a superb example of
the kind of idiocy that SMSG was later to attempt to remedy. 

	4.  "MAT" stands for "Master of Arts in Teaching," the name of a
graduate program in the school of education at Yale.  A certain 1956
neighbor of mine, an assistant professor of education whose name I am
suppressing here by calling him L__, was active in this program.  L__ was
particularly interested in science education, as I recall, and maybe
school mathematics too.  I never kept track of his later career, and 
cannot even remember his first name.

	5.  Begle's 'stony grace' was well-known, but I was quite wrong
about the exclusion.  SMSG welcomed the folk I characterized in my letter
as "the educationists", and I never understood why.  In later years Begle
complained bitterly, though never in public, that math education had been
taken over by the "psychologists", by which he meant mainly behaviorists. 
He did once say (publicly, I think) that on every occasion that he
included some person for politic reasons it turned out a mistake. It might
just have been as a diplomatic necessity that Begle's teams had to be "3
from Group A, 2 from Group B,..."  My own present view is that he thereby
got the intersection of all the talents in each group, and it appears I
believed so in 1958, since I was *crediting* Begle with the same view. 

	6.  Fifteen years!  A startling (though conditional) prediction. 
SMSG was disbanded 14 years after its founding, and the official
death-knell, Morris Kline's *Why Johnny Can't Add", was published the
following year.