The New Math, by Phillips, is subtitled as a “Political
History”. The New Math movement was an attempt to rejuvenate the teaching of
mathematics in secondary schools and ultimately in the Primary schools. To the
proponents of this movement there was staleness in the teaching of mathematics
which was reflected in their belief that it was merely an exercise in memorization
and lacked any true understanding of the elements of mathematical thinking.
For example, most elementary school students had memorized
multiplication tables, learned fractions and division as a mechanical process,
and dealt with “word problems” with abject terror. Mathematics was taught as if
it were some mechanical set of processes that one set to memory and the
students failed to have an understanding of what they were doing.
On pp 13-15 the author provides some baseline information on
the creation of the New Math. The intent was to imbue understanding of what
they student was doing not just a set of rote manipulations to produce an
answer. There was an abject fear that memorization was futile and that the
expression of what the processes were became the goal. Students must learn why
they could multiple 3 X 2 and then 3 X 2+4 and find an answer. They must
understand the processes of manipulations, at the risk of never memorizing that
3 X 4 are 12.
On p 27 the authors also discusses some of the political
movements pressing for improvement. It mentions Rickover and Doolittle as
applying their influence to promote improved education due to the need seen in
WW II to “educate” many enlistees to be able to perform what were technical
tasks. For example, to train an enlisted sailor in Fire Control Systems or in
Radio or Radar, there was a prerequisite in understanding Geometry, Algebra and
Trigonometry. Many schools never taught the skills to students, and thus the
need to re-educate. Thus on one hand the need was to have a better baseline
education and on the other hand to attempt to emphasize fundamentals as the
core of that education.
What then were the principles? It all depended on whom one
spoke with. What happened was that a group of “mathematicians” saw that they
needed understanding of set theory, complex rules of algebra, base n systems of
numbers and the like. This then changed the core of many of the courses.
The attack then also went to the teachers themselves.
Teachers were all too often the product of teachers colleges, often state run
institutions, to produce individuals to manage the utilization of the state
mandated texts and managed by state mandated exams. The Regents of the State of
New York was in many ways a classic example. Geometry was defined by them and
each instructor taught that material.
The author on p 31 refers to Hofstadter and his book on “anti-intellectualism”
and the argument that teachers had become “estranged” from academia. In reality
the Hofstadter book is a polemic of a Columbia Professor against what he
perceives is the “anti-intellectuals”, namely the Republicans, Catholics, and
anyone opposed to his politics. In Hofstadter’s book on pp 138-141 is one of
his many rants against Catholics and the Church, ironically because the
intellectual were also strong supporters of Kennedy.
Thus the intellectualism at Columbia and of Hofstadter was
at best problematic and use of the author of Hofstadter as a baseline is also
problematic. Likewise, for example, on p 394 of the anti-intellectual treatise
of Hofstadter, Hofstadter calls the Partisan Review the “house organ” of the “intellectuals”.
It is in William Barrett’s writings of his time at the Partisan Review that he
noted is strong Communist bent. Thus, using Hofstadter by the author is an
attempt to set up the New Math as the “intellectual’s movement” and then
subsequently to use this as the basis for arguing that its demise was the
result of some right wing attempt to defeat it appears as a bit of a straw man
strategy.
One of the problems I have is that the author fails to
clearly identify what he means by New Math and what the New Math was. In a
classic paper by Feynman in 1965 criticizing the New Math he states:
Many of the books go into considerable detail on subjects
that are only of interest to pure mathematicians. Furthermore, the attitude
toward many subjects is that of a pure mathematician. But we must not plan only
to prepare pure mathematicians. In the first place, there are very few pure
mathematicians and, in the second place, pure mathematicians have a point of
view about the subject which is quite different from that of the users of
mathematics. A pure mathematician is very impractical; he is not interested -
in fact, he is purposely disinterested - in the meaning of the mathematical symbols
and letters and ideas; he is only interested in logical interconnection of the
axioms
This was the problem of the New Math. A radar technician
does not need to understand set theory to understand the probability of a false
alarm and the signal to noise ratio. Specifically Feynman states:
What is the best method to obtain the solution to a
problem? The answer is, any way that works. So, what we want in arithmetic
textbooks is not to teach a particular way of doing every problem but, rather, to
teach what the original problem is, and to leave a much greater freedom in
obtaining the answer - but, of course, no freedom as to what the right answer should
be.
Feynman, a product of the New York City School System, and
then MIT and Princeton, is correct. His own technique was to intuit the answer
and then find the framework to support it. I doubt he ever used a single
element of set theory. The conclusion even in 1965 was the core of the New Math
was flawed as a pedagogical approach. It in fact was intellectualism gone
astray.
On p 103 the author describes some of the texts which
resulted from this effort. Take the Moise and Downs text on Geometry, and
compare it to text by Wells in 1908. Wells was brief and to the point and one
walked away understanding enough geometry to measure angles, understands
triangles and the like. The Moise and Downs book makes the development of Proofs
impossible. The simple example on pp 190-191 (of the 1982 edition) is a classic
obfuscation of the obvious, a proof of the existence of a perpendicular line.
Kline also discussed the shortcomings in his superb book “Why
Jonny Can’t Add” Mathematics is a tool for almost all of its users. It is “learned”
by application. No user of an Excel spreadsheet would benefit from the New
Math.
The author then proceeds to discuss the political opposition
from the right to New Math and the Back to Basics movement. On p 145 he opens
the Epilogue with the statement “Opponents of the new math won.” In reality the
weaknesses of the New Math caused its own demise. With the like of Kline and
Feynman against it than what chance would it have? It just did not work.
The last sentence is also worthy of comment:
“Yet math classroom will remain a political venue as long
as learning math counts as learning to think. Debates about American math
curriculum are debates about the nature of the American subject.”
It is not clear to me what he means by the phrase, “nature
of the American subject”. Is subject the material taught, the individual,
some broader idea not explained?
Overall this book has two tales. One is the intended one of
the birth and death of the New Math. It has not totally died but still is found
floating around a bit. It is also a tale of pedagogy in the state run schools
and who decides what students must know and why they must know it. For most of
us, mathematics is a tool, it is a way to express facts and explore reality. My
day is often driven by mathematical realities, albeit those of an engineer,
pedantic, utilitarian, and lacking in questioning principles. I assume
solutions exist; I do no really pay attention to uniqueness theorems, and use
them as a tool kit to gain knowledge. Almost all who rely on mathematics do so.
The pure mathematician asks fundamental questions, questions about fields,
convergence, existence, measurability and the like. They do affect reality from
time to time. But rarely, yet when the do the impact is significant, just look
at the analysis of the Wiener Process in dynamic systems, and the Ito integral.
However, for the most part we want students to understand
technique, to a point. Out of the mix will come the engineers, physicists, and
yes the mathematicians, the very few mathematicians who have the unique
capabilities to abstract thinking.
Overall the book is a reflection of the political processes
surrounding education. This has been all too common especially since the advent
of Dewey and the education movement he was so prominent in. This book is a
useful exercise in grasping with the tendencies to make material relevant on
the one hand and a facilitator to understand society for good citizens on the
other. The book has certain weaknesses but it also has certain positive points.
It allows one to see how the arguments can be made. One may then ask in a
similar fashion; if these same arguments and this same process will follow
through with Common Core?
The book also shows the break between the academic
practitioners and the practitioners who teach the subjects. At the University
level we still see a great deal of freedom. At MIT for example courses change
on almost an annual basis as the technology and science progress. The need for “standards”
is non-existent. At the secondary level this is hardly the case, due to the
size and complexity. That perhaps is worthy of a similar study.
But one of the important observations here is the movement of "protected" groups like the "mathematicians" who may very well have been used by political operatives to gain deeper control in schools. As indicated colleges and universities are somewhat protected. But if the Government extends its control to Community Colleges we can easily see the movement of Washington control to move there. That may very well be the unintended less of this book. Namely, beware the Politician, they ultimately want to control everything,
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