The “New Math” Curriculum
Reform Movements in 1960s and
1970s US Mathematics Education
Ben Robinson
The “Old Math”
• Until the late 1950s, elementary and
secondary math instruction had focused on
memorizing and applying procedures and
rules
• There was little effort to have students
understand how these procedures worked,
and why the rules were what they were; this
kind of reasoning was deemed more
appropriate for college studies
(Miller, 1990)
The Desire for Reform
• University professors, including William
Everitt, dean of the University of Illinois
College of Engineering, were frustrated with
the mathematical backgrounds of entering
students (Miller, 1990)
• Even students who were proficient at
computation were perceived by university
professors as having only a shallow
understanding of the subject, and as being
dependent on memorized algorithms and
methods whose meanings they did not
understand (Hill et. al., 1979)
In Particular
• 1957: Researchers at the University of Illinois
found that students had extreme difficulty
answering the following questions:
• What is a number?
• What is a variable?
• What is a function?
• What is an equation?
• What is geometry?
(Report of UICSM, 1957)
How the Researchers at
Illinois Interpreted these
Findings
• Without a curriculum that considers this
kind of question, “high school
mathematics would remain disjointed
and mysterious” and would suffer from a
lack of “consistent exposition”
(UICSM 1957)
The Proposed Solution(s)
• “The New Math” is in fact a term that refers to a large
number of reform curricula:
• Madison Project (Syracuse University)
• University of Illinois Committee on School
Mathematics (UICSM)
• Ball State Project
• University of Maryland Mathematics Project
• Minnesota School Science and Mathematics Center
• Greater Cleveland Mathematics Program
• School Mathematics Study Group (SMSG) (Yale
before 1961, Stanford after 1961)
Two Figureheads
• Max Beberman: Chair of UICSM
• Edward Begle: Chair of the SMSG
While there were other reformers whose
ideas had more success (in particular,
Robert Davis of the Madison Project),
Beberman and Begle were by far the
most prominent public and
congressional relations figures
What the Solutions Entailed
• Accelerate the introduction of abstract ideas
in the classroom and simultaneously increase
the level of abstraction
• Abstract notions that were introduced
included “axiomatic algebra, some matrix
algebra, […] a study of bounded monotonic
sequences, a treatment of complex numbers
via matrices, and a treatment of irrational
numbers based upon bounded monotonic
sequences”
(Davis 1965)
Exploration
• Emphasis on learning through “guided
independent exploration”
• “By plotting linear functions by trial and error,
[the student] discovers how to identify and
make use of the slope coefficient. […] The
course includes extremely little exposition-the workbook itself is mainly a sequence of
questions, such as: (1) Graph y=2x+1; (2)
Graph y=3x+1; (3) Graph y=5x+4; etc”
• (Davis 1960)
In Practice
• Prior to the introduction of “new math,”
algebra would have been introduced in 7th or
8th grade, and would have begun with
problems like “5+x=7. Find x”
• Proponents of “new math” began introducing
algebra to 4th grade classes, in the form of
questions like: “Find the set of ordered pairs
(x, y) such that 5+x=y,” and expected them to
give answers in the form of a list: “x=2, y=7,
x=3, y=8, x=4, y=9…”
(Ferster)
Reasons for Initial Successes
• As Tyack and Cuban argue, proposed
education reforms tend to be slow to gain
acceptance
• In its early years (early to mid 1950s) “New
Math” reforms were no different
• This all changed in 1957, when the Soviets
launched Sputnik, the first artificial satellite,
into space
(Miller 1990)
Impact of Sputnik
• “When the Soviets launched Sputnik, in 1957,
the small new-math experiment, previously
confined to a few schools, became a national
obsession. Parents went to night school to
learn the new approach. The press hailed the
reformers as the guiding geniuses of the most
important curriculum change since
Pythagoras”
(Miller 1990)
What Went Wrong
• University researchers were ultimately unable to affect teacher
education on a sufficiently large scale
• Go back to the earlier list of topics covered: “axiomatic algebra,
some matrix algebra, […] a study of bounded monotonic
sequences, a treatment of complex numbers via matrices, and a
treatment of irrational numbers based upon bounded monotonic
sequences” (Davis 1965)
• In many cases, reformers found themselves needing to spend a
significant amount of time teaching math teachers the content
described above; teaching the appropriate pedagogical methods
often fell victim to finite amounts of time (Miller 1990)
Nevertheless…
• Professors like Beberman and Begle were not oblivious to
the need to work closely with teachers (UICSM 1957)
• In 1958, Begle persuaded the National Science
Foundation (now desperately seeking an educational
answer to Sputnik) to fund summer teaching seminars for
high school teachers (Miller 1990)
• However, the number of mathematicians and professors
of mathematics education was small, and the number of
high school math teachers was huge; researchers were
unsuccessful in scaling their efforts with groups of
teachers in seminars to the nation as a whole
The Death of a Reform
• School districts rushed to adopt the “new
math,” often spending a great deal of money
on textbooks, but failing to invest
appropriately in teacher education
• Teachers often found themselves being
asked to teach material they had never
learned, using pedagogical methods they had
never experienced (either as teachers or as
students)
(Miller, 1990)
Other Problems (1)
• School districts overzealously embraced
the reforms: in many cases, the role of
drills (to practice computation) was
eliminated entirely, as opposed to
simply being reduced, as Beberman
and many other reformers had
advocated (Beberman 1966)
Other Problems (2)
• Parents were understandably unable to help
their children with homework problems about
set theory, axiomatic algebra, bounded
monotonic sequences, and other elements of
the new curriculum, leading to frustration with
school officials, and eventually, an insistence
on returning to old curricula
(Miller 1990)
Why Johnny Can’t Add (1973)
• Written by Morris Kline, who was chair of the
math department at NYU
• Agreed with Beberman and Begle that the
pre-1950 approach to mathematics education
was flawed
• Felt, however, that Beberman and Begle’s
approach was misguided; saw the new math
as placing excessive emphasis on formalism
and abstraction at the expense of practicality
and conformity to everyday experience
Moris’ Proposal
• “What we should be fashioning and teaching,
then, beyond mathematics proper, are the
relationships of mathematics to other human
interests […] Some of these relationships can
serve as motivation; others would be
applications; and still others would supply
interesting reading and discussion material
that would vary and enliven the content of our
mathematics courses.” (From Why Johnny
Can’t Add)
Effects of Moris’ work
• The actual content of his work, including his
opinion on how to move forward, was ignored
almost entirely
• Critics of the “New Math” used the book,
however, along with the “expert” authority of
its author, to malign the reforms of the past
decade and a half and advocate for a “Back
to the Basics” approach very much in line with
reforms of other academic subjects at the
time
• (Miller 1990)
New Math’s Answer
• Recall the earlier questions about numbers,
variables, functions, and equations
• Prior to the New Math Reforms, teachers and
textbooks often emphasized parts of the
definitions of these and other ideas that bore
particular relevance to the application being
considered. While such selective emphasis
might not be harmful in and of itself, it often
gives students the perception that
mathematics is a patchwork discipline riddled
with confusion and internal inconsistencies.
(Davis)
New Math’s Answer
Continued
• Illinois Researchers: “Students quickly detect
inconsistencies in the explanations supplied by
textbooks and teachers and decide that ‘math is
silly.’” (UICSM)
• Upon finding a way to solve a problem without relying
on an intimidating formula or procedure presented in
class, students tended to dismiss mathematics
altogether, deciding that “there is the math teacher’s
way of solving a problem and there is the commonsense way of solving it.” Thus, teaching the “basics”
does no good if students dismiss them as irrelevant
(UICSM)
New Math’s Answer
Continued
• “Rules,” “for removing parentheses,” “for
adding signed numbers,” “for combining like
terms,” and “for solving equations” have the
effect of making mathematics seem to be a
byzantine system of apparently arbitrary
“rules,” rather than as a unified body of
knowledge, that, while containing many
branches, stems from a common core of
ideas (Davis)
(Non) Effects of New Math’s
Answer
• The tide had turned; the academic arguments
made by Begle and Beberman fell on deaf
ears
• As public opinion turned against the new
math, NSF funds dried up
• Beberman died in 1971, Begle in 1978; while
these two men were by no means the only
reformers, their deaths left the movement
without a clear leader to advocate for the
necessary reforms in curriculum and teacher
education
Preliminary Conclusions and
Connections to the Quarter (1)
• There’s a clear parallel between the
groups of university math professors
that formed and instructed high school
teachers about what to teach and how
to teach it, and the Committee of Ten,
which served a similar, albeit more
general purpose
Preliminary Conclusions and
Connections to the Quarter (2)
• Political circumstances, which are partly
fueled by a generally impatient public, often
have a strong impact on education.
• Public opinion often derails the very reforms
that the same public demanded just years
before; many times, these calls for change
occur before the reforms have been in
practice for long enough to be fairly evaluated
Preliminary Conclusions and
Connections to the Quarter (3)
• The importance of giving teachers a role at
the forefront of any school reform efforts,
something that we’ve seen in both Tyack and
Cuban, as well as Rousmaniere, is
underscored once again; when reformers
make demands (be they asking teachers to
serve as social workers, or asking them to
direct a large class in independent inquiry in
order to learn an unfamiliar subject) without
providing the time and resources necessary
to meet these demands, efforts at reform are
likely to fail.