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 SMSGWhile 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 common-sense 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.