__New Developments in
Secondary School Mathematics__

This paperbound
report, a small volume of 189 pages, is a collection of articles from the __Bulletin
of the National Association of Secondary-School Principals__, # 247, May 1959,
specially reprinted for the __National Council of Teachers of Mathematics__. According to the page just preceding “Page 1”
it was “Planned and prepared under the guidance of a committee of the __National
Council of Teachers of Mathematics__, a Department of the __National
Education Association.__” The chairman
of this committee of five was Myron F. Rosskopf, Professor of mathematics,
Teachers College, Columbia University.
However, while the committee that prepared this report was apparently a
committee of the NCTM, the latter is not a “Department of NEA” as stated.

There are 49 papers in this collection, presenting a good picture of the way the mathematics education profession saw itself at the dawn of the era called “The New Math” The title “New Developments”, however, is important to bear in mind. The year was 1959 and the papers were prepared in 1958, one year following Sputnik and about the time of the very beginning of SMSG. The developments described here as novel will not have had much time for assessment of their results, and in any case should not be taken as descriptive of most of what was going on in the average schools. And many of the papers deal with old problems in old ways, and are altogether unaffected by the dawn of the newmath.

I will give at least the names of all the Chapters and of all the papers and authors, but will review only those papers that would surely not have been possible five years earlier. The opening group, comprising four papers, is clearly the most important, as Mayor’s overview is magisterial and the other three represent what were clearly the most powerful intellectual influences upon the entire newmath era (1955-1972, essentially). Other influences, such as the failure of teacher education to keep pace, are not as strongly represented, but were perhaps of greater importance in the long run.

**Chapter I: New Programs in Secondary School
Mathematics. (pp. 5-31) **

1. Efforts to Improve Programs and Teaching in Mathematics (John R. Mayor)

2. A Secondary School Mathematics Program (Eleanor McCoy)

3. Proposals of the Commission on Mathematics of the CEEB (A. E. Meder, Jr.)

4. The School Mathematics Study Group (E. G. Begle)

**John R.
Mayor,** then Director of Education
for the AAAS (and a former President of NCTM ) writes generally about the
promise of the new programs, naming the “widely publicized” UICSM and the
nascent SMSG. He expects important
results from the MAA Committee on the Undergraduate Program in Mathematics (Wm.
L. Duren, Jr., not named here, was Chairman of CUPM) and from Albert Tucker’s CEEB Commission. Both the latter committees’ recommendations
concerned collegiate expectations (which was the at the origin of UICSM as well),
which indicates how much the demand for scientists was driving the world of
mathematics education in the early days of newmath. Mayor’s review of the progress of SMSG does
(in an aside later on) foresee that its program will in time penetrate to the
earliest grades, and though UICSM was never to do that, as it turned out, there
were already several elementary math projects under way (those of David Page
and Robert B. Davis, to mention two very
different kinds of example) that Mayor’s review fails to mention.

Yet Mayor does not slight all the curriculum projects for the schools that were already beginning or under way. He describes the early stages of what was to be the Maryland (UMMaP) project, of which he was already the director, though this was to be for the junior high schools. He also takes notice of new special committees of NCTM, and of a joint committee of MAA and the American Society for Engineering Education. The problem of teacher training, already addressed by the NSF program of Teachers Institutes, he says, shows promise of alleviating the problem of an insufficiency of teachers skilled in the new things coming up. Mayor admits the magnitude of the problem, however, and notes that in general the undergraduate teacher education programs are not being modernized at a sufficient speed; still, he envisions the Institute programs as the beginning of a “call to colleges to see that needed revisions are made immediately.”

Quite rightly, Mayor calls attention to the relative lack of attention to elementary school programs in mathematics. He does note the activities of NCTM, which had appointed a committee on the matter, and observes that the Carnegie Corporation was providing grants to individuals for exploratory studies, and that the AAAS had sponsored two interdisciplinary Conferences on Mathematical Instruction, which had pointedly recommended further studies in elementary school curricula and methods of teaching.

Altogether, however, Mayor’s review indicates that as of about 1958 it was high school (and collegiate) mathematics that mainly was on the national mind.

The last section of Mayor’s talk is
headed __Some Answers to Apprehensions__, and the opening lines (p. 10)
deserve full quotation:

Various groups have expressed varying degrees of apprehension about “what is going on in mathematics.” These concerns may be categorized as follows:

1. That the new programs are too abstract and will not provide the most efficient preparation for the other sciences and engineering.

2. That the studies should have started at the elementary level, or tat least be concurrent with elementary-level studies.

3. That teacher education programs even in 1958-59 are not preparing teachers to teach the new materials.

All of these have some basis for support, but, combined, these bases do not appear to be sufficient to detract from the apparent progress toward sound programs that is being made. Planning for the new School Mathematics Study Group already gives promise of directly alleviating any shortcomings due to these causes.

The various programs have of course differed in degree in abstraction, but the concerns of all of the studies for providing materials which pupils can learn by discovery and which in their emphasis on broad and underlying concepts should make understanding more within the grasp of the pupil is an important safeguard. There is no reliable evidence that the learning of abstract concepts, which is interesting and often exciting, will not enable the learner better to understand basic concepts and in turn be better able to apply the needed mathematics in appropriate situations.

Mayor doesn’t name names, but clearly he has in mind the Morris Kline attack on abstraction in school mathematics, which was published at about the time Mayor’s paper was prepared. To this (Point (1) above) Mayor replies in words that seem to contemplate college-preparatory mathematics only, and with studious care, in the last sentence quoted above: “There is no reliable evidence that the learning of abstract concepts . . . will not enable the learner better to understand … “ Mayor knew the difference between a double negative and an affirmative in ordinary rhetoric, and made full use of it there.

The third “apprehension” was that the programs should begin at the elementary level, not the secondary. Here, Mayor takes a combination position (p.11):

While it is to be hoped that work
at the elementary-school level is soon to be greatly increased, there is no
doubt that the current emphasis on the secondary-school level and resulting
improvement will have a salutary effect on the teaching in the elementary
grades even though no attention is given directly to this level. Parents of pupils in an experimental program
at the seventh-grade level raised questions about the desirability of a “new
program” in grade 7 when that which precedes in earlier grades remains
traditional. It is pleasing to find the
parents asking how one can be sure that the new topics should not have been
taught at an even earlier level.

Mayor’s upbeat conclusion celebrates the participation of scientists and mathematicians in school education matters: “Nevertheless, all school people should rejoice that top-level scientists and pre-college level teachers are joining forces in a major way . . . The new programs in mathematics which we will be teaching in the next ten years also promises [sic] to be exciting for teacher and pupil.”

As to “exciting” he was quite right; as to “salutary” he was not. The amateurish effort on the part of both publishers and teachers to imitate, for the primary level, the logical language that was so visible a feature of the new programs at the secondary level were a disaster that helped bring down the whole enterprise.

The
second paper is by **M. Eleanor McCoy**, who at Illinois was “Teaching Coordinator” for Max Beberman’s
UICSM. It mainly describes the project’s emphasis on rigorous language, e.g.,
the distinction between number and numeral, and indeed the “pronumeral” (where
most people would say “variable”), as a
sort of pronoun, the numeral being the name and the number the thing
itself. Thus, for example, five has the
name “5” (though it could have the name “4+1”, among others), but when we don’t
yet know what number we are talking about, or simply don’t want to be definite about
its name, we refer to it by assigning it a letter, e.g., x. “x” is therefore a pro-numeral.

“Discovery learning” is also given some attention by McCoy, as it was of some importance to the Beberman program. (Other programs, such as SMSG, did not suggest any particular manner of teaching.) The subject-matter outline of the four-course UICSM high school program completes her discussion.

In view of the importance of the idea of “discovery learning” in the years following the period under discussion, it is worth quoting a paragraph (on p.17) from McCoy’s paper:

Another
topic in which UICSM courses insist on discovery is that of solving simple
equations and inequations. When a
student understands what an equation is, and what it __means__ to solve one,
he can invent his own methods for solving many of them. Equations and inequations such as:

r+5 = 11, 7a = 21, x-3 < 10, (3n+2)/4 = 5, and |x+4| > 6,

are solved
with considerable speed by ninth-grade pupils in UICSM classes, even though no
formal rules for finding the solutions have been given. When some of the teachers in UICSM pilot
schools first read about this technique for teaching equation solving, they
were quite dubious as to its probable effectiveness. However, a first trial with the class
convinced them of its worth. The
students became intrigued by the problem of converting sentences like those
given above into true ones, and their interest is sustained by making them rely
completely on their own ingenuity. That
they can discover workable techniques for solving problems of this nature is
all the evidence needed to convince them that mathematics is a human endeavor
which requires creative energy and carries its own reward.

The
third paper, by **Albert E. Meder, Jr.**, reviews the proposals of the
Commission on Mathematics of the CEEB, which had been circulating in draft form
for two years or so before its 1959 publication, and which is regarded as a
formative document of the “new math” era.
These proposals are much discussed in other parts of my files, and so I
won’t summarize what Meder says here.

The
fourth paper is by **Ed Begle**, head of the new S__chool Mathematics Study
Group__ (SMSG). He notably credits the
University of Maryland project, the UICSM, and the CEEB as having pretty well
set the path he envisions for SMSG, the
first writing group (in the summer of 1968) having certified as much. He praises the cooperation of mathematicians
and teachers, not to mention psychologists and social scientists, whom he sees
as probably being of increasing importance to his project as it develops into
the middle school level. Begle cautions
that the high school materials already prepared by SMSG are intended for the
‘top half’ of the high school population, the population from which the future
scientists and technicians will be drawn.

He doesn’t intend to neglect the other half, however, and also mentions the gifted, the use of films, the special teacher education projects, and the preparation of ancillary materials (what became a commissioned library of special topics, mainly for teachers and enrichment of superior students). The plan as it developed is here entirely described, except for the elementary level, about which he writes (p.31), “Finally, the SMSG is about to come to grips with the fact that each child who enters the seventh grade has gone through the first six grades. What, if anything, should this Group do about the mathematics of these grades?” . [SMSG did soon in fact prepare a primary school program to answer this question.]

Following
these comments about Chapter I are chapters whose titles will be given with a
list of the titles of the papers within them, and in some cases a brief
indication of what they are about, but nothing more than the titles in the
cases where “the newmath” is neither particularly influential nor elucidated.
The names of the papers I shall __not__ describe in any detail might serve
as a reminder that the “newmath” era was not entirely devoted to what some saw
as a total revolution.

**Chapter II: Mathematics for a Liberal Education
(pp. 32-62)**

5. Mathematics in General Education (Herald P. Fawcett)

6. Problems of Mathematics in the Small High School (Milton W. Beckmann)

7. Experiences with Some Different Topics for Slow Learners (Stephen Krulick)

8. A General Mathematics Program for a Large High School (Ida A. Ostrander)

9. The Place of Mathematics in a Liberal Education (William H. Glenn)

10. A Look at a Multiple Track Mathematics Program (Howard L. Gallant)

11. A Junior High School Principal Views Mathematics in a Liberal Education (Donald W. Lentz)

Not much in this chapter has particular relevance to the newmath, though the flavor of the new era is in the air, and surprisingly appears here quite explicitly where one would not expect it. In the paper on experiences with slow learners Stephen Kurlick, described as “with the New York City High School Department”, gives as examples of things slow learners have learned very well in his classes, “the intersection of two sets” and “one-to-one correspondence”, including the symbols associated with these ideas. From page 45:

We discovered that the “union of the set of boy pupils and the set of girl pupils,” gave us the “set of all pupils”; the “union of the even integers with the odd integers (gave us) the set of all integers.” [sic] Similarly, we were able to discuss many problems with the intersection of sets. We reached the stage where pupils could find the answer to problems such as this one:

{1, 2, 3} ∩ {3, 4, 5} = {3}.”

… Aside from the academic results attained, the class attitude was radically altered. Cutting dropped off from two or three a day, to, at most, once a week. The class attitude towards homework was changed also. Each pupil did his homework, and usually it was quite correctly done, too. The atmosphere in class also changed. Where the pupils had been quiet, bored, and occasionally troublesome, they now became awake, attentive, and even volunteered information. ...

1. Here was a class in which the students were doing some real mathematics. This was new material. They were no longer just rehashing areas of previous failure.

2. The mathematics was meaningful to them. It was not just a series of symbols and a chore to accomplish. Much of their work was out of class, and interesting to them.

3. The prestige factor was enormous. Here were these slow pupils, always at the bottom of the heap, never showing any progress. Now they had some mathematics to “show off” to their less-fortunate brighter friends.

4. Finally, they came to understand and appreciate real mathematics. This was mathematics they could learn, they could understand, and they could enjoy!

If
we keep in mind that a slow learner ** can** learn, and if we try to
teach him some good mathematics, not

Here before the publication of the CEEB Report and before the work of SMSG had more than begun, Mr. Kurlick was giving lessons in what would later be taken as a defining mark of newmath, and presaging its failure with his own failure to recognize what was really being learned. Apart from the logical error he makes in calling the exhibited equation a “problem” – it isn’t, for it asks no questions – he is celebrating a trivial item of nomenclature, designed to make easier some later learning of mathematical substance, as if it were itself of mathematical substance. It is not surprising that his slow learners liked it, for it was totally disconnected with their own past schooling, which, as Kurlick recognized, had been a failure. Since with this group there was no hurry about producing any particular result, both teacher and pupils had nothing to fear, and the school system as well.

This confusion of lessons in set nomenclature and lessons in (say) computation or deductive reasoning in geometry was characteristic of what would follow in the 1960s, especially in the elementary schools, where teachers ignorant of mathematics would welcome being instructed in teaching such simple and pointless things to all students, not just the slow ones, happy in the knowledge that they were doing well (for such lessons are really quite easy) where before they had been struggling to teach such “old-fashioned” things as long division.

Gresham’s Law, that bad money drives out good, applies in education. Most teachers in 1959 were unschooled in mathematics, even those teaching at the middle school level, and tragically many even at the high school level. They did what they could with the textbooks and teachers’ guides presented to them, and so were easily persuaded that set intersections, being more modern than arithmetic, can be substituted for arithmetic. The lessons are easy, the children learning them are happy in their work, and the principals enjoy showing high marks on math exams taken by their students. Everything is in favor of the new math, except that the children were not learning mathematics. But who was there in the schools to recognize this?

The early critics did in fact recognize it, and among them were a large number of research mathematicians, led by Morris Kline. Later, when commercial bowdlerizations of the work of such people as Begle and Beberman reached the textbook market in quantity, there were more. Beberman in particular had made emphasis on logical language and symbolism a hallmark of his UICSM program, but only as preface to real mathematics to follow. By 1959 it should have been evident how counter-productively this lesson was being taken, but the SMSG did not give explicit warnings on the matter, and it was not until 1965 that Beberman himself, at the Montreal meeting of NCTM, gave voice to the problem, predicting a disaster in elementary mathematics education to result from such misapprehension of the place of “modern mathematics” in the early grades in the schools.

**Chapter III: Students
Talented in Mathematics (pp. 65-106)**

12. Math and the Gifted
Student (A.H. Passow, D.J. Brooks, Jr.)

13. Gifted students in Senior High School Mathematics (Frances
Freese)

14. The Rice Institute Summer Program for Talented High School
Students (L.K. Durst)

15. The JvN Memorial Math Summer Seminar for Talented High School
Students (D.E. Edmondson)

16. The Seattle Project for Talented Students (Elizabeth
Rondebush)

17. Mathematics in the Bronx High School of Science (Irving Allen
Dodes)

18. The University of Maryland Project and Talented Students (M.L.
Keedy)

19. Ohio State University Works with Talented High School Seniors
(P.V. Reichelderfer)

20. Meeting the Needs of Cincinnati’s Gifted Pupils in Mathematics
(Mildred Keefer)

21. The Advanced Placement Program of the CEED (Edwin C. Douglas)

22. A Junior High School Seminar for Talented Students (F.L.
Elder)

23. A Demonstration Class in a NSF Summer Institute (H.F.
Montague)

24. A Summer Mathematics Camp for Talented High School Students
(E.D. Nichols)

25. Mathematics Field Day (M.R. Kenner)

Since
all these papers describe programs for students called “talented”, it is to be
expected that advanced material would appear in them, even if they had taken
place in 1947 rather than 1957, and what material is considered worthy does not
always change with the times, the number theory of Gauss, for example, is
forever as living as it was in his time.
But one difference is certain:
The funding for some of them would not have been as available in 1947 as
in the years following Sputnik, and almost none of it would have been available
prior to the Second World War. That
there are more papers in this group than in any other exhibits a profound
difference between the way the profession saw itself in 1958, or saw the demands
up on it at that time, and the way it saw things not only ten years before, but
thirty years later, too, when school programs for the talented were seen less
as a virtue than as signaling neglect of the needy.

The Rice Institute Summer Program (p. 74)
was financed by the __Fund for the Advancement of Education__, which was not
otherwise described here but most probably was the division of the Ford
Foundation having that title, and which then had as one of its activities the
improvement of articulation between high school and college, including a
program for the early admission of talented high school students to college,
something that had been pioneered by the University of Chicago years
before. The program was one of five
parallel efforts, two of them in
mathematics and the others in science, taking place simultaneously at five
different Texas Universities. It had no
connection with “early admission”, or with what later became the CEEB program
of Advanced Placement. These summer math
or science programs were advertised to all the high schools in Texas, and
“nominations” in mathematics, apparently from the school authorities, were
entertained for students in the top one percent of Texas high school students
who had completed the geometry and algebra courses but still had one or more
years of high school ahead of them before graduation. Of 190 nominations, a class of 25 was
admitted, to a residential five-week session of the “college” style, eating and
living together with three of the faculty members. Durst writes, of the 1957 summer:

Two weeks were
devoted to symbolic logic … With this foundation, it was possible to discuss
several postulational systems, including finite and non-euclidean geometries,
and to give an introduction to the theory of ordered fields with special
emphasis on the field of rational numbers.
The step from the rational field to the real field was made by using
Dedekind sections, following Hardy’s __A Course of Pure Mathematics__. The final three days of the course were
devoted to the properties of functions continuous on closed intervals
(intermediate and extreme value theorems) and to calculus.

In addition to the
course work and exercises, with oral presentation of answers, each student
worked up a report on a special topic,
and some of these were in number theory, graph theory and geometries, as well
as analysis. Even with so ambitious a
program, Durst says, the students did not consider themselves overworked, and
they all enjoyed it:

Accordingly, when
the course was repeated in the summer of 1958, Dr. Allen Wilson presented a
series of lectures on finite mathematics in parallel with the lectures on logic
and analysis; the effect was to double the content of the course. The students in the 1958 course were kept
busy; they even appeared to enjoy their course more than the 1957 group.

The effect, and
hence the value, of a project such as this cannot be determined for several
years…

As with so many projects of
this sort, there seemed to be no systematic effort to evaluate the
results. Such summer schools were considered
an obvious good, and not an experiment; indeed this was the attitude also taken
by the traditional schools, winter and summer alike, at that time. Even Max Beberman took no systematic
evaluation results, to see if his methods were indeed an improvement. SMSG was almost unique in its “longitudinal
study” of the 1960s. Examinations are
not mentioned, in the case of the Rice Institute summers. It might be that the local professors did
evaluate their work, but one would have to dig deep into the archives of the
Ford Foundation to find more than what appeared in the Durst report.

The program was very ambitious indeed, and
it seems unbelievable that the listed topics were actually learned by the
students in the way graduate students in mathematics were learning the very
same things at about the same time. At
all but the most selective universities, the Dedekind cut construction of the
reals, for example, was the stuff of graduate, not undergraduate, study; here
it was being done by high school students not yet acquainted with
calculus. This is not to say these
teachings were fruitless during those summers, but one would like to know what
the students retained of it all.

**Irving Allen
Dodes**, Chairman of the mathematics department of the Bronx High School of
Science, describes (p. 81ff) the very ambitious program being developed there
(all Bronx High students are “talented” by American standards), in which
elements of newmath are prominent in the 12^{th} grade and schedules,
for the best students, to move down to the 11^{th} and even 10^{th}
as time goes by. Dodes was in the
process of developing an analytic replacement for the traditional Euclidean
geometry course at the time of this symposium, he wrote, and his ideas here
should be compared with what he presented seven years later, in 1966, at the
International Congress of Mathematicians in Moscow, under the heading of __The
Role of Axiomatics and Problem Solving in Mathematics__. (He was by then teaching in a junior high
school, with undiminished enthusiasm for teaching abstract mathematics.)

His course descriptions for the Bronx High
School of Science are worth quoting in full.
Beneath a chart with arrows indicating the partial ordering of courses
are the course descriptions:

M 3 Tenth year mathematics:
a NY State Regents course (enriched)

M 39 Tenth year mathematics (special): includes all of M 3, but
approaches the subject through a study of symbolic logic and the theory of
sets.

M 5 Eleventh year mathematics:
A NY State Regents course (enriched), which integrates intermediate
algebra, trigonometry, and the elements of plane analytics.

M59 Eleventh year mathematics (special): includes all of M 5 but approaches the
subject through a study of symbolic logic, theory of sets, groups and fields,
functions and relations.

M 87 Twelfth year mathematics:
The first half is an enriched version of the NYS Regents course in
Advanced Algebra. The second half is a
course in solid geometry with some vector analysis and solid analytics. Volumes are taken up through integration.

M 89 Twelfth year mathematics:
This is the NYS Experimental course which covers most of “Principles of
Mathematics” by Allendoerfer and Oakley and includes a sizable unit on vector
analysis and solid analytics.

M 99 Twelfth year mathematics (special): (prerequisite M 59) This
includes all the material in M 87 and M 89, and, in addition, advanced material
on abstract algebra, statistical inference, and programming.

There are also two
alternate courses, one in analytic geometry and calculus for Advanced
Placement, and another called Mathematical Techniques in Science, the latter an
add-on for students with the time, and the former a substitute for any of the
other “12^{th} year” courses and intended for very superior students. Any student who goes through the courses
simply marked “special” for each of grades 10, 11, and 12, however, will have
plenty to do, though without a course called “calculus” it is hard to
understand the line ‘Volumes are taken up through integration”, which appears
in both the special and non-special versions of Eleventh year mathematics. In fact, I don’t believe it can mean what it
apparently says, even for Bronx High School of Science students, who are an extremely
select group, since nothing resembling integration (or differentiation) is even
mentioned in the rest of the sequence, apart from the Advanced Placement
course.

Dodes goes on with
some comments on the reception of these new courses, by both students and
teachers. All the reports are good, and
students were so charmed with some of the 12^{th} grade material that
it has been, Dodes says, moved back into the 11^{th} grade – even into
the non-special sections. Then,

Last year, we experimented with symbolic
logic and set theory in four special tenth-year (geometry) classes. These classes were composed of entering
students who appeared to be very superior in intellectual ability. It is too early to make as clear-cut a
decision as we did in the eleventh- and twelfth-year classes, but there is no
doubt that the classes understood and appreciated the new material. They definitely liked it better than the
regular Euclidean geometry. As the news
filters to the other tenth-grade classes, requests pour into my office for
transfer to these classes. It is clear
that the new approach will soon pervade all the geometry classes.

Here is an omen, then: The students liked “symbolic logic and set
theory” better than “the regular Euclidean geometry”; yet, the course was still
called “geometry”, though with this “new approach”. Where’s the geometry? The Regents Examinations, which even Bronx
High students had to take, contained some question on standard Euclidean
theorems and procedures, but so few and so predictable that the average Bronx
High School of Science student could learn to answer them in a few weeks, and
then could afford to spend most of the time on the “new material” Dodes
considered modern. In the following
paragraph, indeed, Dodes says his department is considering replacing Euclid
altogether, with “a general switch … to
Descartes” in Grade 10. Apparently
Euclid was not a genuine necessity.

The other paper of note in this chapter,__
The Advanced Placement Program of the CEEB__, pp 92-95, is by **Edwin C.
Douglas**, who is listed as being from the Taft School (a private school near
Yale in Connecticut). On page 95 he
writes,

There have been many interesting
by-products of this program. It has unquestionably led to a move to re-examine
our secondary-school mathematics curriculum.
Tangible evidence of this is the NCTM Committee on Secondary-School
Curriculum, the CEEB Commission on Mathematics, and the popular interest in the
University of Illinois Committee on secondary-school Mathematics. But even more far-reaching in the writer’s
opinion has been the splendid way in which college professors and capable
secondary school teachers have come to a position of mutual understanding and
appreciation for each other’s problems.
Out of the Advanced Placement conference held annually in June has come
a warm respect for the abilities of those at both levels in our educational
structure . . .

That this comment
should have been made at all indicates a problem, and one which has not,
despite the efforts of Mr. Douglas, been solved to this day. That is, even if there did develop a mutual
respect between the “college professors” and “capable” secondary school
teachers involved in the AP program in 1959, there were other players in the
game, in particular, those college professors who are not professors of
mathematics, but of education. In 1959
this important class of educators, who do most of the teaching of future school
teachers, were by no means in a position of mutual respect with “college
professors” if by the latter term is meant mathematicians, or professors of
mathematics at four-year colleges giving degrees in mathematics, but who happen
to be taking an interest in school education, including high school
education.

If by “college professors” was meant
professors of education, which in a 1959 document contrasting them with
“capable” school teachers is unlikely, the “mutual understanding” that was
problematic would have had to have been between them and the professors of
mathematics. Between these two groups
the rift has, if anything, widened since then.
Certainly the mathematics research community is hardly aware even of the
existence of NCTM, the publisher of this symposium. Another important
constituency in the education world is formed by persons, most of them former
teachers or professors of education, who are the administrative officers of
public school systems: Principals,
curriculum specialists, supervisors, Chancellors, and many others. Mr. Douglas doesn’t mention them as a class,
for before the full flowering of the newmath under NSF financing they did not,
in 1958, have the authority in curricular matters they developed later.

For
those aware of the nature of the AP program only as it looked forty years
later, however, a few lines from the opening of Mr. Douglas’s paper are illuminating:

The present Advanced
Placement Program of the College Entrance Examination Board actually had its
origin in the School and College Study of Admissions with advanced
Standing. Educators and school
administrators will undoubtedly recognize this program under the more familiar
title of “The Kenyon Plan.” In its early
stages the program was supported financially by the Fund for the Advancement
of Education. In 1955 the College Board was asked to
administer this program on a national basis, and agreed to do so. Consequently, there now exists a nation-wide
program for education our gifted students as the secondary-school level, with a
set of examinations in college freshmen level work, suitable both to the
secondary schools and the colleges . . .
. . The tests are directed toward the
gifted pupil . . . In mathematics this means setting up a
curriculum whereby the gifted student can accelerate through the full four-year
secondary-school program in three years, thus leaving the twelfth year free for
a ** bona fide** college freshman course in calculus and analytic
geometry. Extreme care should be taken
to se that these students are in no way short-changed in the normal high-school
mathematics program. Acceleration which
prompted neglecting important topics in mathematics would be causing a distinct
dis-service [sic] to the student.

In the Dodes
program in the Bronx it is possible that this caution was being observed, but
for ninety percent of the AP calculus students of 2005 it is not, unless Euclid
and conic sections are not “important topics in mathematics” any more.

**Chapter IV: Articulation Between Elementary and
Secondary Schools (pp. 107-139)**

26. The importance of articulation (Phillip S. Jones)

27. Mathematics as a Man-Made Invention in the Elementary Classroom (M. V. DeVault)

28. The Supervisor, the Teacher, and Articulation between Elementary and Junior High Schools (R. R. Smith)

29. Concepts Pervading Elementary and Secondary Mathematics (H. Van Engen)

30. Providing for Individual Differences by Grouping in Depth (R. L. Morton)

31. Wise Use of Materials and Devices (F. E. Grossnickle)

32. Evaluation of Student Learning in Secondary School (J. J. Kinsella)

33. Evaluation of Learning in Arithmetic (L. K. Eads)

34. Articulating Geometry Between the Elementary and Secondary Schools (Ann C. Peters)

35. Evaluation in Arithmetic and Talented Students (V. J. Glennon)

36. Are We Sure Learning Has Taken Place? (E. Glenadine Gibb)

The first paper (p. 108-111) is by **Phillip
S. Jones**, a professor
of mathematics and education at the University of Michigan in Ann Arbor. Jones was soon to be. President (1960-1962) of the National Council
of Teachers of Mathematics, and later co-editor with Arthur F. Coxford, Jr. of
the NCTM Yearbook #37 (1970) __A
History of Mathematics Education in the United States and Canada__. In this paper Jones anticipates the Bruner
doctrine of the “spiral curriculum” but from an even wider point of view. His thesis is not that the partial lesson he
advocates here, for earlier stages of a student’s progress, should be
calculated to lead to a deeper treatment later on, with information withheld in
planned decrements, but that __anything__ added to the daily lesson, even ** en
passant** and incomplete, possibly quite incomprehensible, might be
valuable to that student if remembered at some later date.

As I like to put it, I have never in my
life learned something I was later sorry to have learned. Why should we not accord our students such
opportunities when they present themselves?
For example, I have met many a college student who has no idea of the
difference between “i.e.” and “e.g.”, let alone the meaning of ** mutatis
mutandis**. Since these phrases
represent ideas present (if not always signaled in the text) in all parts of
mathematics, it argues poorly for the humanistic values of school mathematics
that such ignorance can result. Jones,
though writing about certain basic mathematical ideas and not Latin tags,
expresses a similar thought:

For teachers to be able and willing to point out and use
these central continuing themes, they need first be aware of their existence
and scope, and secondly they must recognize that not all of these concepts can
or should be taught at their first appearance for the mastery and understanding
which ultimately we hope the student achieves.
By this we do not mean that mastery and understanding are unimportant,
that progress towards appropriate levels of achievement should not be measured,
nor that we should be content with less achievement than that of which the
individuals in our class are capable. We
do mean that the possibility of complete mastery and understanding is not the
sole criterion for the inclusion of a topic in a curriculum or the pointing out
of a relationship in a daily lesson plan.
Seeds which are not planted cannot be expected to grow, and teachers who
are delighted by the progress of a class should not lose their joy in their own
achievement, but should realize that just as they are reaping harvests sown by
their predecessors, so they too must at times sow where they may not reap.

As seems inevitable in all
papers dealing with “fundamental concepts”, whether in 1959 or forty years
later, the distributive law of arithmetic is presented as the example:

… if understanding is to precede the
formulation of algorithms and drill, he must first understand and use that
which years later he may learn to call the distributive law. This law is also the basis for many other
processes from elementary school through advanced mathematics such as
factoring, special products and the explanation for the definition of the sign
laws in junior high-school algebra. It
is also to be found in relationships between areas which may be noted both in
algebra and in the intuitive geometry of the earlier years.

Here Jones rather thoughtlessly has in
mind the formula (-a)(-b)=ab, the “sign law”, whose usual proof uses the
distributive law is an essential element. But exactly here is a conceptual
difficulty which many textbooks of the following era did less than nothing to
resolve. For positive numbers, construed
as lengths or points on the “number line”, the commutative and distributive
laws are not axioms, but are observed truths, proved (in the child’s mind) by
suitable diagrams (rectangular areas for multiplication, and so on). Of course, the distributive laws for positive
numbers are used as if they were axioms once they have been established as
describing the world correctly, just as Euclid’s axioms were for him a
distillate of observation, though he used them in building his system __as if__
they were a mere invention of rules for a certain game. There was just enough doubt about the truth
of those axioms, especially the fifth Postulate concerning parallels, to engage
centuries of European attempts to demonstrate that all Euclid’s postulates were
really a necessary system, at least as to consistency, and it was a shock to
the 19^{th} Century to discover they were not. Alternate systems were possible, if perhaps
not descriptive of nature.

When the negative numbers are introduced,
not taken as an arbitrary invention but as points on the left side of the
number line, additively inverse to the positives by definition, and applied to
credit *vs** *debit, past ** vs** future and so on, all
the usual rules of algebra are still “provable” in terms of these
interpretations and applications – except the product of two negatives. With the distributive law, however, the
proper formula is obtained. But how did
the distributive law, which had been derived empirically from diagrams
involving line segments and rectangle areas when positive numbers only were
being studied and used, suddenly and automatically become axiomatic for
negatives, which have no such interpretation?
What kind of proof of ab = (-a)(-b) can one have, using the distributive
law, when we do not yet even have a meaning for (-3)(4 –7), let alone that it
is necessarily equal to (-3)(4) + (-3)(-7)?
Not that the “algebraic” proof of ab = (-a)(-b) is a foolish exercise in
algebra; the proof does show an important fact, that the sign formula and the
distributive law are consistent, and indeed that if one wishes to have the
distributive law as part of the system one is driven to the sign rule as given
and no other rule will do.

Yes, the
algebraists of the 19^{th} Century did show that postulating the
distributive law, whatever the multiplication of negative numbers might mean,
resulted in a marvelously coherent system (a “field”, as it is now called),
which could extend to the complex numbers too, and so mathematicians simply
used the system before looking for interpretations that corresponded to the
operations. We can find them, yes, but
if there were no physical or other practical interpretation of distributive law
when negative numbers were placed in some of the terms, an interpretation which
extends the earlier one we had proved empirically for the positive numbers, the
idea that we could __prove__ the law (-a)(-b) = ab by invoking that
distributive law as an axiom would be as __ad hoc__ as to take the product
law itself as an axiom, and would have no trustworthy application to problems
of daily life.

If the
negative numbers can apply to debts and credits, surely the product of two
negatives should extend that interpretation, else our new number system would
have limited applicability, and the manipulation of formulas involving negative
quantities would become a dangerous practice, perhaps converting (via the
‘usual rules of algebra’) a meaningful formula into one that means something else. Without a constant appeal to physical
experience the spinning out of mathematical theories might turn into a vacuous
exercise. (This is a warning often
repeated by the most eminent mathematicians.
Peter Lax, in discussing this point in his paper for the 1966 symposium
volume, __The Role of Axiomatics and Problem Solving in the High School
Mathematics curriculum [1]__,
refers the reader to John Von Neumann’s essay

This
symposium volume was composed at the beginning of a decade of increasing
optimism about what could be taught to students in the way of abstract
mathematics. The purveyors of the newmath often didn’t think much about what
was axiom and what was experience, and were unreasonably emboldened to declare
axioms wherever convenient. (This was as
true in geometry as in algebra.) So the
real number system, painfully developed for children in grades K-9 perhaps by
beginning with counting, then defining (positive) fractions and developing
their arithmetic, then declaring the holes in the rational line “filled up”
with the irrationals and observing that most of the field axioms held so far
(including the distributive laws, though without additive inverses), did two
things simultaneously: introduce the
negatives as additive inverses, and declare the field axioms true for the
resulting system. This can be done
consistently, of course, though nobody suggested that the consistency should be
proved in the schools.

But while most school programs made a case
for additive inverses in commerce and mensuration, nobody, or almost nobody,
tried to make a corresponding case for the validity of the distributive law,
and most people simply took it as an axiom.
The result must have impressed many a school child to wonder what all
that algebraic manipulation was __about__.
At any rate, Phillip Jones, in the last quoted passage, merely considers
the distributive law a fundamental idea, while he later cautions against
excessive abstraction. He is willing to
take the distributive law for the reals as an axiom, and yet take it as a
warrant for numerical calculations which will ultimately be used (among other
things) to prove geometric theorems analytically. To take an algebraic __axiom__ as warrant
for believing that the medians of a triangle meet in a point with certain
properties is unfortunately to credit algebra with magical properties; Jones
does not notice this.

Jones’s paper includes the usual
pedagogical advice concerning the place of discovery and drill and all that,
and urges the teacher to take the best from various styles of teaching,
maintaining __continuity__ of instruction at the sacrifice, if necessary, of
some of the logical structure and abstractness such as is taught in graduate
schools. When it comes to specifics,
however, he has little to offer.

**Henry Van
Engen**, in the following paper (#29), takes only two or three pages to
outline some of the worst features of the newmath as his “pervading”
concepts. First, he writes of
enumeration systems: “… The child is
soon taught it is best to adopt a base (ten) for his enumerations scheme to
simplify the system. As he progresses …
he finds out that a base other than ten can be used … This opens the way for an
extensive study of enumeration systems in the elementary and secondary
school. Such a study would include the
binary system, which has assumed ever greater importance in the past decade due
to the extensive development of digital computers.”

An extensive
study of enumeration systems? In place
of arithmetic? Then, under __The Number
System__, Van Engen describes the “whole numbers” as something children first
learn by abstracting from sets, then used in counting, and then recognized as
having the properties of closure and associativity, under addition and
multiplication (but not subtraction or division), and the distributive
property. (These properties are described explicitly, with numerical examples
besides.) “Having learned these fundamental
principles in the elementary school, the pupil learns to develop a general way
by which to state these principles in the secondary school and he learns that
any collection of mathematical entities which possess the properties listed
above will be called numbers. Thus he
finds that fractions, positive numbers, negative numbers, and complex numbers
all possess these properties. Of course,
they possess others as well, but such things as 3+I and 2-5i are numbers
because they can be manipulated in the same way the whole numbers were
manipulated in the above examples.”

This sort of
thing goes on, with __Number Pairs and Sets of Number Pairs__ as the next
pervading concept.

He
learns to use the form a/b as a pair of whole numbers to describe situations
involving fraction ideas, and he learns to use this same form to describe ratio
situations. Late in the elementary
school, the pupil works with sets of number pairs…” The sets he has in mind are first of all the
sort that become __functions__, as when tickets cost 25 cents each and we
match number of tickets sold to total amount paid. “This is a number-pairing activity… In an
arithmetic class, the pairs may be graphed … Such activities serve as an
introduction to the idea of variable, formula, set of numbers and a set of
number pairs, and generally to the subject of algebra itself. Advanced work will introduce such topics as
relations, functions, and set-functions, all of which are based on the pairing
of numbers, writing rules to enable one to pair numbers, and similar
activities.

Geometry comes next, and after giving due mention of the necessity of learning the names of things Van Engen goes on to his favorite theme:

In the elementary school, pupils learn that it is possible to assign numbers to collections of points. For example, to each collection of points in space that make up a square they can assign the number 4 (for 4 sides and 4 vertices). This doesn’t prove to be of much interest. However, to each square a number may be assigned by use of other rules…” but he doesn’t say why area or length of diagonal is clarified or made more interesting by this functional approach. “In the secondary school, we continue this search for rules whereby numbers may be assigned to collections of points, which are usually given a name such as square or rectangle. In fact, we prove that certain rules must hold if we agree to accept sentences (postulates) as fundamental for our science.

His concluding sentences are, “Children can grasp abstract ideas; in fact, they enjoy them. The forward movement in mathematics can only come if we learn how to teach abstractions and how far pupils of varying abilities can go in making mathematical abstractions.”

An interesting feature of this paper is Van Engen’s effort to include the dogmas of “discovery learning” with the formation of abstract structures:

“As he progresses … he finds out that a base other than ten can be used …”

“the pupil learns to develop a general way by which to state these principles”

“Thus he finds that fractions, positive numbers, negative numbers, and complex numbers all possess these properties.”

“In the secondary school, we continue this search for rules whereby numbers may be assigned to collections of points…”

Van Engen’s
correspondence with Max Beberman exhibits his increasing interest in Beberman’s
ideas, though Van Engen goes farther with his abstractions, the business about
ordered pairs not yet, however, reaching the stage where he intends to
introduce negative numbers via ordered pairs of positive numbers. Nor does he, in 1959, yet speak of “numeral” **vs**
“number” at all, as Beberman did from the beginning. Van Engen’s 1965 elementary school textbook[3]
does include this for openers, but is mostly quite traditional, unlike his
three-volume middle school series[4].
The major influence on Van Engen up to 1959 must have been the CEEB Commission,
of which he was probably the least distinguished – as mathematician – member. By the time of his 1966 middle school algebra
series he will have absorbed the entire Landau construction of the real
numbers, though imperfectly.[5]

Some of the
other authors in this chapter, **Kinsella** and **Grossnickle** in particular,
were or would be well-known authors of “methods” books or textbooks. Grossnickle’s__ General Mathematics__, a
textbook published in 1949 and apparently intended for use at the 9th Grade
level, contains this remarkable problem, whose like is being repeated to the
present day (p.231ff):

Problem 9.
The area of a rectangle is 8a²‑ 6a. Since the width is one factor of the expression and the
length is the other factor, what are the dimensions of the rectangle?

An answer is of course “a by 8a-6,” though it is
not clear which is length and which is width.
Another answer, though, might be “a/7 by 56a-42”, a possibility not
mentioned. Fortunately, Grossnickle’s
paper for this symposium concerns “manipulatives” for use in the primary
schools, and not the use of polynomials in geometry and measure theory.
Kinsella writes about evaluation in high school classes.

Neither Kinsella nor Grossnickle show any sign of newmath influence,
nor do any of the others.

**Chapter V: Mathematics
and Educational Television (pp. 140-149)**

37. The production and Presentation of a Televised Mathematics Program (Joseph R. Hooten, Jr.

38. Teaching Algebra Via Television (David W. Wells)

39. Explaining a Mathematics Program to Parents Via Television (Louis Scholl)

Television came into common use in 1950, and by 1959 was still new enough to persuade some people that it could have some classroom use, perhaps replacing teachers, or partly so. There had been some experimental television courses given by that time, with registration and examinations, and the financing of some newmath programs included funds for experimenting in this direction; but these projects did not exhibit any particular intellectual novelties of the era. Certainly these papers do not, for they stay close to the problems of movie-making, skills of speech and demeanor and so on.

**Chapter VI: Mathematics
and Evaluation (pp. 150-168)**

40. A New Approach to the Evaluation of Competence (Sheldon S. Myers)

41. What Can the Classroom Teacher Do About Evaluation? (Donovan A. Johnson)

42. Mathematics Evaluation in a Large City (Marian C. Cliffe)

43. Evaluation in a Large State: New York State Regents Examinations (Frank S. Hawthorne)

Of these
papers, only the first, by **Myers**, includes a gesture towards the
newmath. Myers, who headed the
mathematics section of the Educational Testing Service, which develops and
administers the SAT college entrance examinations, noted that under the newly
developing definition of mathematics as the science of patterns, many of the
SAT “aptitude measuring” questions
involving analogies and logical deduction can now (1959) be construed as
mathematical in nature. In particular,
he expressed a conviction that the proper questions, not apparently questions
in traditional mathematics, can still predict future competence in learning
mathematics.

I found it amusing, in this connection, that he presents as an example of an SAT question that tests mathematical ability the following:

Which of the following numbers could not possibly be the square of an integer?

(A) 123454321

(B) 345676543

(C) 3086358025

(D) 4443555556

(E) 1111088889

and he gives the answer (B) with the explanation, “In solving this problem, one needs only to observe that the squares of numbers must end with 0, 1, 4, 5, 6, and 9, and not 3, as in the case of choice B above. The other choices happen to be perfect squares.”

Of course (B) is not a perfect square, and so answers the question, but the author of the problem really had no right to suggest that there are two classes of non-square numbers, those that fail to pass the test he used, and those that, like 14, do pass the test but are non-square nonetheless. Were it a strictly mathematical question, Mr. Myers’s question should have been worded, “Which of the following numbers is not the square of an integer?” But since this would not have given the hint that makes answering the question easy (checking a long number for squareness from the definitions took some tedious arithmetic in 1958), it would have spoiled his purpose. He ended up with a question that was less mathematics than mind-reading.

**Chapter VII: The
Mathematics Teacher, Present and Future (pp. 169-189)**

44. The Education of Teachers of Mathematics (Howard F. Fehr)

45. The Commission on Mathematics of the CEEB and Teacher Education (R.E.K. Bourke)

46. Academic Year Institutes (Burton K. Jones)

47. Industry Lends a Hand (G.A. Rietz)

**Howard
Fehr** was Head of the Department of the Teaching of Mathematics, Teachers
College, Columbia University, and past President of NCTM (1956-1958). His paper begins with the usual platitudes,
e.g., “It is evident that no teacher can
teach what he does not know,”, but it goes on to what were platitudes of the
newmath era even though they were not even considered true in the earlier days
of E.L. Thorndike and W.H. Kilpatrick at
Teachers College, for example (p.170), “Only if the teachers find recommendations
of mathematicians and educators worthy of favorable consideration, only if they
wholeheartedly accept them as having a large probable validity, and only if
they undertake to implement them in their classroom teaching can a modern
program of mathematics education be changed from a paper formulation to a
practical reality.” (And while this is
considered a platitude today, trotted out for political purposes, it is not
really believed by the present-day NCTM, at least as to the part about
“mathematicians”.)

Fehr’s paper recommends an ambitious teacher-education program, one he describes by course-names and credits needed for degrees and certification, and one which was hopeless in his day and has never come close to describing the average, let alone the least favored, teacher of his time or ours. He observes, “Frequently the high-school teacher of mathematics has never gone beyond the study of calculus and at present more than 25 per cent of the high-school teachers in the U.S.A. have never completed a course in the Calculus.” The dilemma is familiar: If one asks high school teachers to present a Master’s Degree, it is impossible for teachers this ignorant of mathematics, let alone modern mathematics, for the result to be a degree in mathematics, or even in a program featuring anything non-elementary in mathematics. “Since there is no obstacle in [a person with a B.A.’s getting] this degree, even though he does not get what he needs,” Fehr writes, “The dilemma of the prospective teacher in most cases is resolved in favor of straight professional educational courses.”

Therefore, Fehr recommends, the bachelor’s program must be strengthened. “The study of college mathematics should begin with a course in analytic geometry and the calculus running fro one to one and one-half years. A teaching major can then be rounded off by the study of eighteen semester hours beyond the Calculus. This work should include the study of modern algebra both of the polynomial type, and the abstract approach through the study of matrices, groups, rings, fields and a study of geometry from the real number approach, including projective, affine, Euclidean and non-Euclidean aspects….”

So it goes, on and on, with alternative programs and then the graduate year for future high-school teachers:

If
a teacher continues his study toward a Master’s degree, he should pursue at
least 15 more semester hours of mathematics.
Among the courses should be a year of professionalized high-school
mathematics, illustrated by the work of Felix Klein, **Elementary Mathematics
from an Advanced viewpoint**, but treating the concepts and spirit of
contemporary mathematics…. Real Functions, Calculus of Finite Differences,
Numerical Analysis, and a course in Applications of Mathematics to the Physical
and Social Sciences.

And, naturally, courses in pedagogy besides.

Well, there are high school teachers with such a background, to be sure, but to expect any large fraction of them to become so proficient? Later, Fehr considers the problem of in-service teachers:

These in-service teachers, for the most part, know how to teach. Giving these persons courses in topology, measure theory, theory of functions, theory of games, and so on is not the answer. These teachers have been away from the classroom study of advanced mathematics for years … most of their college mathematics is rusty. … There should be abbreviated courses in each of the following areas, in which the teacher should make himself a master of the basic elements and their relation to the high-school mathematics program:

1. Modern analysis including concepts of variables, function, relation, mapping, sentences, propositions, and point-set theory. Especially the setting up of functions and their inverses.

2. Modern algebra, including matrices, groups, rings, fields, number systems, Boolian [sic] algebra, and vectors, and also polynomial algebra.

3. Modern geometry and its foundations and the study of several types of geometries including a finite geometry.

4. Symbolic logic, axiomatic foundation, and nature of proof.

5. Mathematical statistics, stressing statistical inference, tests of hypotheses, and correlation theory.

This ends Fehr’s paper. The awkward statement of his curriculum topics will immediately strike a mathematician as evidence of Fehr’s own slender understanding of some of what he thought he was prescribing for the in-service teachers. Why mention “rings” and then “also polynomial algebra”? Why “Especially the setting up of functions and their inverses” – a detail --- after the grand though already redundant “function, relation, mapping”? If Fehr himself, at the top of his profession, had to strain so to herald the newmath, the prospects of its being understood by the teachers and future teachers he was contemplating were not good.

What Fehr’s 1959 paper mainly suggests is that the mathematics education establishment was trying hard to climb aboard the bandwagon it saw gathering steam. That bandwagon, unprecedentedly, was being led by mathematicians, and while the math educators (unlike Kilpatrick who had had contempt for mere mathematicians) were respectful, even intimidated, they didn’t quite understand.

**R.E.K.
Rourke**, of the Kent School in Kent, Connecticut, was the Executive Director
of the Commission on Mathematics of the CEEB, whose Report, published in 1958,
had established the principles of the newmath as exemplified by the Beberman
teaching and the newly established SMSG, as the major orthodoxy of math
education reform in the decade of the 1960s.
(Howard Fehr was also a member.)
Albert E. Meder, then a mathematician at Rutgers, another member of the
Commission, was in the same year of this paper (1958), publishing his side of a
debate in __The Mathematics Teacher [6]__,
replying to a paper by Morris Kline[7]
attacking the Commission report along with all other manifestations of newmath
philosophy of mathematics teaching.

Nothing in this Rourke account of the work of the Commission, however, reflects the bitterness of that debate, and despite the title of his paper there is no reference whatever to the Commission. Rourke gives a straightforward personal description for the proper education of an elementary school teacher, a middle school math teacher, and a high school math teacher, with a table summarizing the minimal requirements in terms of hours of credit. The titles he gives these courses or programs are quite conventional. His text does go into more detail than the table but the language of the table is worth noting. For example, the “Junior High School” teacher should have had all four years of high school math, 18 hours in college (“Calculus and Analytical Geometry, Algebra and Number Theory, Geometry”), and a three-hour or four-hour course in “Teaching of Junior High School Mathematics”. General education courses are not included in his chart but mentioned in a footnote there.

For elementary school teachers and high school teachers there is correspondingly less and more, respectively, except for one novelty: The six hours of college mathematics every elementary school teacher should have (following at least three years of high school math) should have for subject matter “Numbers and Number Systems”. That last item, “Number Systems”, is a specifically newmath importation, which in textbooks being produced in the 1960s usually meant one of the algebraic systems arrived at in the course of construction of the rational and, intuitively at least, real number systems from the positive integers, with negative numbers being introduced somewhere along the way. A second meaning for “Number Systems” was, in the Van Engen textbooks for middle schools for example[8], “a set of objects called numbers, along with two operations. Each operation must be closed, commutative and associative. One of the operations must distribute over the other.” Other newmath authors used the word somewhat differently, and badly educated teachers often used the term to mean systems of numeration, such as Roman numerals or any base-k place-value system.

But the CEEB
Commission had confined itself to prescribing for college-preparatory high
school programs only, and for a mathematical elite at that; its work did not
cover all Rourke’s concerns. Almost all
of what Rourke wrote here came out of his own teaching and learning experience,
and was not visibly affected by his experience as a Commission member. Like
every other thoughtful critic, newmath pioneer or not, he could see that the
traditional education of American teachers was inadequate. Rourke took special
care to address the difficulties faced by future schoolteachers, most of whom,
even those preparing for Junior High level, know very little mathematics. They
would rather not enter a typical college math course, however elementary
it appeared to the professor, since they were not prepared for **any**
college-level course in mathematics. So
Rourke urges the colleges to do something for this audience as well as the
usual “freshman calculus” audience of future scientists and engineers. He also notes that a large part of the
existing corps of teachers has as background education nothing like what he
recommends as minimal, and that in the future one cannot expect all teachers to
have it, either. He does what he can to
advise colleges and school districts accordingly. That the newmath of the following fifteen
years would actually founder on the lack of teacher preparation, however, was
apparently far from his thoughts.

**Burton W.
Jones**, the next author, was a distinguished topologist at the University of
Colorado. His paper is a brief summary of
the progress of the full-time NSF Academic Year Institutes program, begun in
1956 following the establishment of the Summer Institutes program in 1953. He celebrates the participation of so many
good universities and mathematicians, but does not describe the mathematical
content of any of the programs. Nothing
in what he writes here bears on newmath attitudes.

** G.A.
Rietz**, “Consultant, Educational Program Development, Educational Relations
and Corporate Support Service, New York, New York”, reviews industry support
for various programs of teacher training and career training, with no
indication that there is anything problematic about the contributions of the
schools and colleges to these purposes.
As he writes at the outset, “Business organizations are not educational
institutions. Therefore …” And the details, once industry has “lent a
hand”, are up to the educational establishment in place, in 1958 as in 1945,
when General Electric initiated a systematic program of generating enthusiasm
for science study with its “House of Magic” traveling science shows, followed
by its cooperative fellowship program
with Union College in Schenectady, setting a model followed by other
corporations and consortia.

**Robert B.
Davis** concludes the volume with a description of his __Syracuse Plan__,
a “released time” scheme by which in-service teachers in the Syracuse area
could take courses at Syracuse University, via appropriate scheduling and
funding. As Davis explains, this is a
program for the upgrading of the best teachers, not the marginal. “The concern here is with excellence.” In writing of the benefits of the plan, Davis
refers to a “new curriculum” being introduced into the American secondary
schools, following the lead and inspiration of Beberman’s UICSM and the
recommendations of the CEEB Commission.
He envisions a course in statistics as a standard high school terminal
course, except for AP Calculus for participating schools. Further, he says, “modern axiomatic algebra,
formal logic, digital computers, and newly developed social science
applications of mathematics are all leaving their mark on the modernized
curriculum. Considerable additional study by teachers – and especially by good
teachers, who can provide local leadership – is urgently needed. This need is ideally answered by
participation in the Plan.”

Davis became well known in later years as a leader in the newmath, though his research mainly took another direction, away from formulating abstractions for elementary textbooks, and towards a study of how children, small children especially, come to understand mathematical ideas at all. Following Beberman’s death in 1970, Davis became his successor as head of UICSM, which changed direction entirely as the newmath came to a gradual death between then and 1975 or so. As a teacher of future teachers he was said to have had no peer, and it was unfortunate there were not a thousand Davises to staff a thousand Syracuse Plans, or ten thousand, for the training of elementary school teachers was a more serious problem than the “knowledge gap” among high school teachers that had been highlighted in the time of Sputnik.

The opening guns of the newmath, however, were trained on the high school program and its teachers, and while elementary school programs soon were included, the attitude engendered by its early concentration on an elite body of students didn’t change. The result was a body of teachers and students quite mystified at almost all levels by what had been intended first (in Beberman’s UICSM, 1952) as a preparation for the University of Illinois college of engineering, and later taken up nationally as a response to the 1957 Russian challenge of Sputnik.

Ralph A. Raimi

7 August 2005

**[1]**** ****Published by the Conference Board of the
Mathematical sciences. Lax’s paper
dicusses this point on pages 115 and 116 of that report, and cites the Von
Neumann paper as the finest to have made this point.**

**[2]**** John Von Neumann, The**

**[3]**** Henry Van Engen, Maurice L.
Hartung, & James E. Stochl, Foundations of elementary school arithmetic,
Chicago, Scott-Foresman, 1965**

**[4]**** ****Van
Engen, Hartung, Trimble, Berger, Cleveland, Seeing through Mathematics, I,
II, III, Scott Foresman 1964-1966**

**[5]**** Here there will be a
reference to the chapter on Van Engen’s career.**

**[6]****
** **Meder, Albert, The ancients versus the moderns –
a reply, The Mathematics Teacher 51 (1958) 428-432.**

**[7]**** ****Kline, Morris,
The ancients versus the moderns, a new battle of the books, The
Mathematics Teacher 51 (1958), 418-427.**