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Citation:
Kenneth R. Conklin, "Why Prefer The 'New Math'?," THE EDUCATIONAL FORUM, XXXV, 4 (May, 1971), pp. 439-446.
Reprinted from
The Educational Forum, May, 1971
Why Prefer the "New Math"?
Kenneth R. Conklin
During the last ten years there has
been a dramatic change in the
teaching of mathematics in the elementary
and secondary schools. Topics formerly
reserved for masters and doctors
degree candidates in pure mathematics
have now been simplified and introduced
into the elementary school curriculum,
sometimes even at the first or second
grade level. In addition to the introduction
of advanced topics at the elementary
level, there has been a change
in the methods employed in teaching
mathematics. Less emphasis is now
placed on memorizing formulas or standard
problem-solving techniques, while
heavy emphasis is placed on intuitive understanding
of basic patterns and abstract,
deductive reasoning from definitions
or axioms to theorems.
New math has largely replaced traditional
math in our public schools, although
this change has met considerable
resistance from parents and teachers
who sometimes wonder what was wrong
with doing math the way they learned it
when they were in school. Why should
new math be preferred to traditional
math? This question deserves to be answered,
for several reasons. Even
though we no longer need a rallying cry
to spark the now well-entrenched revolution
in school mathematics, we do
need to reassure parents and teachers
that the revolution has been worthwhile.
We also have an obligation to keep the
revolutionary spirit alive, so that the
ideals of the new math do not become
lost in the implementation of it. The
most brilliant and creative developments
in education can quickly become
stagnant routines, which are just as dogmatic
and destructive of creativity as the
old-fashioned system.
Traditionally, mathematics instruction
in the public schools has been designed
to meet the computational needs
of the average citizen. It was thought
that abstract mathematical theory would
be neither interesting nor useful for the
average person, and would be far too
dificult for children to understand.
Thus, abstract mathematics was confined
to the universities, and the only people
who encountered it were undergraduate
students majoring in pure mathematics
or graduate students in mathematics and
the physical sciences.
The first defense for traditional math
was the claim that students could readily
transfer what they learned in school to
practical situations outside the school.
Mathematics in elementary school consisted
mainly of practical computation
-----------
Although the "new" math is pretty well accepted, very few parents and perhaps not too many teachers can make out a case for preferring it to the traditional arithmetic. This article provides a rationale for such a preference. KENNETH CONKLIN is Assistant Professor of Educational Studies at Emory University in Atlanta, Georgia. He took his doctoral work at the University of Illinois.
[end page 439 / start page 440]
which would be directly useful in adding
up grocery bills, balancing a household
budget, figuring out the income
tax, or understanding the way insurance
premiums are computed. At the secondary
level, the character of mathematics
became quite different although still
computational: students would learn
how to solve algebraic equations or how
to prove simple measurement theorems
in Euclidean geometry. Sometimes these
algebra and geometry courses were defended
on the grounds that the ordinary
citizen would actually need to know
how to solve equations or measure areas
and spaces. More frequently, though,
such courses were defended by claiming
that students who mastered the logic of
geometry would be more logical about
solving the problems of everyday life.
It was also claimed that studying something
difficult and unpleasant would exercise
and toughen the mental faculties,
thereby improving the power of the
mind to deal with difficult or unpleasant
problems in life.
Since both computational skill and
mental discipline would transfer to a
wide range of situations outside of
school, it was argued that every student
should study mathematics. General education
is education which provides
knowledge and skill that is necessary or
useful for everyone, regardless of his occupation.
Since computational skills and
mental discipline are both necessary and
desirable for the ordinary citizen, and
since mathematical training transfers
such skills and discipline, it was claimed
that mathematics belongs in the general
education curriculum for all students.
On the other hand, abstract mathematical
theory (set theory, topology,
group theory, vector geometry) clearly
had no direct practical applications for
the ordinary citizen. Topology would
not be of much assistance in deciding
how to mow the lawn, and group theory
or modular arithmetic would only lead
to confusion in adding up the grocery
bill or figuring out the budget. Abstract
mathematics, therefore, clearly did not
belong in the general education curriculum.
Indeed, the only people who could
use abstract mathematics were mathematicians
or scientists. Vocational training,
broadly defined, is any kind of educational
program which is undertaken
for the primary purpose of producing,
maintaining, or improving one's income
or one's status in an income-producing
occupation. Abstract mathematics, therefore,
was thought to belong to the
vocational training program of mathematicians
and scientists, much as short-hand,
speed-typing, and the use of office
machines belonged to the vocational
training of secretaries.
The major arguments against the new
math can be seen quite clearly in this context.
Opponents of the new math claim
that it represents an unjustifiable shift of
topics from a specialized form of vocational
training into the general education
curriculum. Training in traditional
math easily transferred to practical applications
for the ordinary person, but
instruction in new math only prepares
the student for more math or for theoretical
science. While it may be true that
we need more mathematicians and scientists,
it seems unreasonable to demand
[end page 440 / start page 441]
that every public school student must
become one. Furthermore, students
trained in new math devote less time to
computational work, and often seem
slow or ineffective in solving practical,
everyday arithmetic problems. Parents
are unable to understand their children's
homework assignments, children cannot
count apples in the supermarket, and
teachers have to go back to school to
struggle with some wild, newfangled
fad in education.
If new math is to be defended as belonging
in the general education curriculum,
we must show that it is at least as
successful as traditional math in providing
the computational skills and the
mental discipline needed by the ordinary
citizen. In addition, we must show
that new math does something which
old math could not do, so that it is
worth a great deal of effort (not to mention
parent-teacher frustration) to substitute
new math in place of the traditional
approach.
There can be no doubt that a student
trained for 12 years in practical computational
techniques would do better at
practical computations than a student
who spends part of that time studying
abstract theory. But we may question
the extent to which computational techniques
are actually used by ordinary
people. Certainly everyone should be
able to add, subtract, multiply, and divide,
and everyone should be able to
perform these four operations on positive
whole numbers, negative whole
numbers, fractions, and decimals. Beyond
these four operations on these four
types of numbers, there is almost nothing else the ordinary person is called
upon to compute. Powers and roots
could be safely eliminated, along with
all of traditional algebra and all of geometry
except the simplest measurement
rules for line segments and polygonal
areas.
If the new math is deficient in training
students to do these simple computations,
the deficiency must be remedied.
Perhaps the mathematicians and educators
who planned the new math programs
got "carried away" with the possibilities
for doing theory and neglected
computational drill. Charges that new
math produces children who cannot
count apples in a supermarket seem a bit
extreme, but it certainly has been a common
complaint that children raised on
new math just haven't had enough basic
computational drill. This defect must be
corrected, but the correction will not require
great effort or inconvenience.
Much time in traditional math programs
was wasted learning computational
techniques which have no practical
applications in ordinary life, e.g.,
finding square roots, solving quadratic
equations, or determining the volume of
a truncated pyramid. The gap in computational
usefulness between traditional
math and new math will therefore not
be very hard to overcome.
The claim that traditional math exercised
the mind and increased the student's
power of logical thinking has
come under severe attack by psychologists.
The mental discipline or faculty
psychology theory has long been unacceptable.
Nonetheless, recent theories
of stages in cognitive development (e.g.,
[end page 441 / start page 442]
Piaget) have suggested that there are
certain basic kinds of reasoning that occur
only after the child has had an appropriate
background of experience together
with biological growth. Studying
mathematics may indeed facilitate the
development of these general reasoning
powers, but further research will be
needed. It seems safe to say, however,
that the new math does at least as well
as the old math in promoting the development
of general reasoning powers.
New math is characterized by abstract
thinking, formation of generalizations
based on observation and intuition, and
deductive reasoning from definitions to
proofs -- and these are precisely the
kinds of mental processes which the latest
psychological theories indicate must
be practiced by children and early adolescents.
Most teachers have noticed that students
are far more enthusiastic about
new math than they were about traditional
math. New math is interesting,
perhaps because it meets the cognitive
developmental needs of the students as
discussed above. Bright children do better
with new math, average children do
at least as well, and dull children seem
to suffer no more now than formerly.
Because almost all students find new
math more interesting than traditional
math, the job of motivation is made easier
and disciplinary problems are reduced.
Thus far we have seen that new math
requires only moderate improvement in
computational drill in order to be as useful
to the ordinary citizen as traditional
math, and new math seems to do better
at interesting the students and promoting the development of general reasoning
powers. New math, improved by
computational drill, therefore, is at least
as deserving as traditional math to be included
in the general education curriculum.
But new math offers something
very important which traditional math
could never begin to approach, and it is
this very important something that
makes new math worth all the frustrations
encountered by parents and teachers.
Not everything in the general education
curriculum is intended to be directly
applicable to solving the practical
problems of coping with the environment.
Darwin's law of the survival of
the fittest applies to animals whose sole
purpose in life is to get along with the
environment. Machines can perform
many tasks faster and more accurately
than man. Yet man is more than an animal
or a machine. Man can do more
than survive, grow fat for the slaughter,
or manipulate the world around him.
Man can appreciate his world. Man has
an aesthetic sensitivity to the beauty
around him enabling him to love and to
feel reverence.
Some parts of the general education
curriculum are designed to cultivate the
distinctively human ability to appreciate.
Art, music, literature, and drama have
traditionally fulfilled this function.
These subjects are not primarily defended
on the grounds that everyone
needs to know how to draw, play an instrument,
write novels, and act. Neither
are these subjects defended on the
grounds that they help students develop
their faculties of mental discipline. Art,
music, literature, and drama are appreciation
[end page 442 / start page 443]
subjects -- they help students appreciate
the beauty in visual forms,
sound, written language, and interpersonal
interactions. History also tends to
be an appreciation subject if properly
taught, giving students a respect for and
understanding of civilization and human
tradition. The "new" math, "new" science,
and "new" grammar are attempts
to enhance the appreciation of subjects
which are usually defended as being
practical or providing mental discipline.
The role of the appreciation dimension
of a subject in general education can
be better understood if we make a distinction
between two ways of using
knowledge./1 Sometimes knowledge is
used to reorganize the environment so
that a problem is solved. This applicative
use of knowledge occurs whenever
an engineer uses his knowledge of physics
to design a bridge, or a student uses
his knowledge of the elementary facts of
addition to decide whether he can afford
to buy all the things he wants. However,
knowledge can be used interpretively,
to understand a situation or to appreciate
a meaning. This interpretive
use of knowledge occurs when an ordinary
citizen uses his knowledge of physics
to understand and appreciate the
achievements of the space program,
even though the citizen could not actually
apply his knowledge to design the
space ship or compute its fuel requirements.
It is thus possible to have interpretive
use of a subject without having applicative
use of that subject. Nevertheless,
applicative use of a subject requires the
previous acquisition of the interpretive
use of it. A problem must be understood
and interpreted before it can be solved.
Vocational training in a subject there-fore
requires both the interpretive and
applicative uses of that subject, while
general education in a subject requires
only the interpretive use of it. The whole
range of general education may be seen
as the understanding of a broad spectrum
of subjects to be used interpretively,
coupled with the development of the
ability to cope with generally common
environmental problems.
Topics in set theory, topology, group
theory, and vector geometry were traditionally
limited to university-level vocational
training programs for mathematicians
and scientists. Knowledge of
these topics was meant to be used applicatively,
so the topics were studied in
depth and with considerable mathematical
rigor. In the "new" math these
same topics are studied in elementary
school as part of the general education
program required of all students.
Knowledge of these topics is meant to be
used interpretively, so the topics are
studied with greater fexibility and less
emphasis on mathematical rigor.
Interpretive knowiedge of mathematics
is important for the average citizen
in modern America because of the
importance of mathematics and science
in our civilization. A democracy cannot
succeed unless its citizens are sufficiently
well informed so that they can intelligently
share in the formation of public
------
1. Four uses of knowledge, including the applicative
and interpretive uses being discussed here,
were developed in Harry S. Broudy, B. Othanel
Smith, and Joe R. Burnett, Democracy and Excellence
in American Secondary Education (Chicago:
Rand McNally and Co., 1964), Chapters III and
IV.
[end page 443 / start page 444]
policy. Such a large portion of public
policy today is concerned with math and
science that being a good citizen requires
considerable interpretive use of knowledge
in these subjects.
Aside from the requirements of good
citizenship, the interpretive knowledge
of mathematics is important if life is to
be enjoyed to the fullest extent. A person
cannot enjoy life fully unless he understands
what is going on around him
and where he fits into the on-going
world. So much of what is taking place
today grows out of developments in
mathematics and science that awareness
and understanding of current events requires
the interpretive knowledge of
mathematics. Man's greatest achievements
today draw heavily upon math
and science, so that the proper appreciation
of those achievements is possible
only through an appreciation of mathematical
theory.
Perhaps the most important function
of the interpretive knowledge of mathematics
is that it makes possible the direct
appreciation of mathematics itself. The
masterpieces of mathematical theory,
like the masterpieces of art, have an inner
harmony and an outward magnificence
which are awe-inspiring. But the
eye must first be trained before it can appreciate
beauty. Mathematical beauty is
very much like beauty in art, music, or
poetry, and those who lack the interpretive
knowledge to appreciate it are missing
a profoundly meaningful experience.
The "newt" math attempts to make
the nature of mathematical knowledge
highly visible, so that mathematics can
be appreciated in the same way as an artistic masterpiece. All the "new" approaches
to traditional subjects have in
common this effort to lay bare the structure
of knowledge and the methods for
obtaining knowledge. Curricula in the "newt" physics, for example, emphasize
the use of scientific method and the relationship
between observations and theories.
The "newt" grammar is concerned
with the nature of language as a form of
communication and the methods whereby
linguistic patterns can be discovered.
The "new" history attempts to teach
students how historians do research.
Studying the structure of knowledge
in a subject is like studying the patterns
of color, balance, form, and texture in
an artistic masterpiece. Studying the
methods whereby knowledge is obtained
is like studying the artistic techniques
whereby masterpieces are created. Learning
about the structure of a product and
the method of its production helps one
understand and appreciate both the finished
product and the process which made
it. Interpretive knowledge of product
and process is thereby obtained.
When the ordinary person has interpretive
knowledge of a product, his enjoyment
of that product is enhanced. Interpretive
knowledge of a process enables
the ordinary person to share in and
perhaps control the work of the expert
who makes the product. Math, science,
language, and the social sciences are so
important that general education must
include interpretive knowledge for both
the humanistic enjoyment of these subjects
and enlightened cittzen participation
in democratic control of the creation
and use of subiect matter.
In every aiademic subject, the experts
[end page 444 / start page 445]
are divided on a basic philosophical
question: Is the subject a study of
real things which exist whether or not
anyone knows about them, or is the subject
something which has been invented
by the mind of man and says nothing
about outside reality? Are numbers and
lines real entities, or have we merely invented
them? (No number or line has
ever been seen) Do the theorems in
mathematics tell truths about the nature
of the universe, or are theorems merely
true by definition in a fairy-tale system
of definitions invented by man?/2
Philosophers and subject-experts ask
the same questions about the status of
scientific laws, grammatical patterns,
and historical trends. Do scientific laws
describe the "personality" of the universe,
or are they merely summaries of
man's responses and feelings? Is grammatical
structure an inborn shaper of
thought and perception, or is it merely a
set of rules or arbitrary conventions for
playing the language game? Are historical
trends real and inexorable, or are they
man-made myths which summarize
nothing real? The complexities of these
debates go far beyond the scope of the
present discussion, but we may note with
pride that new math, new science, new
grammar, and new social studies all provide
interpretive knowledge which
makes the debates much more open to
public understanding than formerly.
Although new math has largely replaced
traditional math in our public schools, we must continually re-examine
the reasons which justify this change.
Parents and teachers who understand
why new math has been adopted will be
more sympathetic toward its purposes
and better able to carry out those purposes.
Traditional math was defended as
part of general education on the
grounds that the computational skill and
mental discipline it produced were almost
automatically transferred to important
applications in ordinary life.
Topics in abstract mathematical theory
were thought to belong solely to the vocational
training of mathematicians and
scientists.
When compared with traditional
math, new math is found to have at least
as great a transfer value for mental discipline.
Overemphasis on theory has adversely
affected the computational skill
of new math students, but this defect
can be easily remedied by introducing
more basic drill. Furthermore, new
math provides a very important benefit
which was almost totally lacking in traditional
math: the interpretive knowledge
of mathematical theory. Interpretive
knowledge of a subject helps the ordinary
person understand and appreciate
the role played by that subject in modern
life. The well-rounded person
needs interpretive knowledge of important
subjects in order to enjoy life fully
and also to be able to exercise intelligent
judgment as a well-informed participant
in democratic policy formation.
New math, new science, new grammar,
and new social studies all share in a
common effort to teach students the
structure of knowledge in these subjects
and the methods wherebv subiect-experts
-------
2. For a collection of important articles on the
existence of mathematical objects and the nature
of mathematical truth, see Paul Benacerraf and
Hilary Putnam (eds.), Philosophy of Mathematics
(Englewood Cliffs, New Jersey: Prentice-Hall,
Inc., 1964).
[end page 445 / start page 446]
discover this knowledge. Students
are thus enabled to appreciate the humanistic
and aesthetic dimensions of the
subjects they study, and to feel a sense
of vicarious participation in the quest for
truth. The fundamental philosophical
dispute between the realist and conventionalist
views on the status of abstract
entities is also more open to public scrutiny
in the "new" subjects.
For all these reasons, new math
should be preferred to traditional math.
New math holds great promise as a part
of the general education curriculum, but
in order to fulfill its promise the new
math must be properly taught by teachers
who keep in mind its revolutionary
spirit. New math is characterized by abstract
thinking, formation of generalizations
based on observation and intuition,
and deductive reasoning from definitions
to proofs. All of these things
should be done in an open-minded spirit
of free inquiry and creative spontaneity.
New math must not be allowed to degenerate
into the rote memorization of
standard rules and procedures which
was so typical of traditional math.
Teachers can be flexible, creative, and
spontaneous only if they are thoroughly
trained in abstract mathematics and only
if they understand and are inclined to
apply the philosophical and psychological
bases of the new math.
[end of article]
==================
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