There is no shortage of analyses or critiques on the teaching of mathematics, especially since the 1950s. This paper, however, is not intended to be a survey of ideas or research, but will attempt to draw from textbooks, studies, and reviews in their historical context, a picture of life inside the mathematics classroom throughout the history of the United States. Admittedly, the accuracy of such inferences are severely limited by the glaring absence of data from within the classroom. Nevertheless, the historical perspective of how mathematics has actually been taught, versus the history of theoretical debates, is important to discover if possible.
Formal education in colonial America was primarily limited to teaching literacy and training the elite college-bound in the classics. One common form of schooling in northeast and middle colonies was the town school, an English institution for training clerks. Thus, the curriculum originally included arithmetic until puritan influence replaced this "non-academic," "skill" subject with religion and a greater emphasis on reading. In cities with business interests, certain mechanical mathematics skills were still needed and taught in a few schools(Willoughby, 1967, p.2). Later in the eighteenth century, Ben Franklin's crusade for utilitarian education and the rise of academies saw arithmetic and mechanical arts introduced as subjects for their intrinsic value in the real world.
This form of inclusion was quite distinct from the mathematics that migrated into the Latin Grammar Schools. College preparatory schools espoused the faculty psychology approach to education, believing that the mind trained on the most difficult subjects would be prepared for any task. Until 1726 this meant learning the ancient languages. In 1726, Harvard hired its first professor of mathematics and soon after began requiring proficiency in arithmetic as a requisite for entrance to the college (Willoughby, p.4). In response, arithmetic began to be taught in most secondary schools. It is very interesting to note that the order in which various topics in mathematics are taught in today's secondary schools is the same order in which Harvard began requiring such disciplines for entrance: arithmetic, algebra (1820), geometry (1844), and later advanced topics (p.4).
The rise of universal, free, compulsory education together with new college entrance requirements and the continuing need for basic commercial computational skills meant a significant increase in the number of students who were being taught arithmetic and a similar increase in the few privileged boys learning algebra and geometry. Unfortunately, there was no previous generation of citizens trained in mathematics to be available to teach these students. The common school movement in general suffered a shortage of teachers, and while the new normal schools helped to train this new workhorse and introduced a pedagogical aspect to the profession, they did little to develop the kind of mathematical understanding that effective teaching requires. Consequently, most teachers relied upon the "rule" method (Bidwell & Ciason, 1970, p.1-10) in which a particular rule for a particular problem was presented, memorized, and then drilled. Arithmetic during this time was considered an extremely difficult subject; and boys, if they even attempted to learn it, did not begin until age twelve or thirteen. Girls were never taught the formal rules and, like most citizens, any practical knowledge of numbers they attained came from life experience alone.
In 1821 the first edition of Warren Colburn's First Lessons in Arithmetic became available in the United States. Based on ideas derived from Pestalozzi's schools, it has been one of the most popular and influential arithmetic texts ever published (Bidwell & Ciason, p.13 ). As the first "new math," this program of study was designed to lead even very young children (five or six years old) through the discovery of the concepts of numbers and operations. The process is the converse of the old rule method in which abstract characters and patterns were presented first and exercised until one was proficient enough to try a practical problem. What most students learned under such instruction was how to follow examples. Very rarely did they understand an operation.
Colburn's system begins with practical problems, counting beans, making combinations with buttons, etc. and practices these until the child has grasped an understanding of the operation. Only then are the abstract numbers and signs introduced to help the child develop a general principle. The primary emphasis is on understanding. Colburn's personal ideas about the purposes of studying mathematics were first for its practical use, and second for the mental discipline value (Bidwell & Ciason, p.24).
There is evidence, if only in numbers of sales, that Colburn's book was widely used (Willoughby, p.3). Somewhere between the ideals of instruction and the actual classroom, however, something always seems to get lost. The remaining history of mathematics education is, in part, an ongoing struggle for the realization of Pestalozzi's ideals of learning through understanding first. One explanation for pedagogical disparities in mathematics education lies in the attitudes of teachers toward their subject. How mathematics is taught is often dependent on the teacher's or society's beliefs about what mathematics is. When mathematics is seen exclusively as a tool or set of skills, it is most often taught by drill. When mathematis is seen as a body of knowledge important to understanding one's surroundings, the teacher may also present a structure which helps the student to grasp the connection between various skills. But only when a teacher believes that the real value of mathematics is in the ongoing process of discovering new relationships will he naturally guide pupils to learn through analytical induction (Grouws, 1992, p.131)
Mathematics in school throughout the nineteenth century was believed to be a tool for exercising the reasoning faculties. Thus its teaching was characterized by such extremes of drill and discipline that up to one-half of every school day could be spent on arithmetic, without much learning occurring. In fact, arithmetic was the primary cause for non-promotion in the late 1800s (Grouws, p.13). At the same time, the secondary school population was rapidly increasing (Willoughby, p.20), which meant a growing number of students without plans for higher education who felt little need to develop such mental disciplines. Understandably, there was a growing public dissatisfaction with the formal methods and a revolt against the idea of mathematics as a subject worthy of study for its intellectual value (Grouws, p.10). From the 1840s to the 1950s, American society predominantly viewed the role of mathematics as solely for social utility. It wasn't until after Sputnik I that the public acknowledged the intrinsic values of mathematics for the common good (Barlage, p. 28).
In 1845, a societal hope in progress through the scientific method was being applied to the classroom through a streamlining of the mathematics curriculum. Two surveys (Stitt, 1845 and Wilson, 1919) by educators stand out for assessing what kinds of math were considered by the business community and the average citizen to be important in daily life. They found that only the smaller numbers and the most basic operations were used regularly and thus it was suggested that the more complex, perplexing, and tedious practices be eliminated from the curriculum (Grouws, p. 17)
Another foundation in science for the limited teaching of only immediately useful mathematics was in the budding field of psychological research. G. Stanley Hall's child studies in the 1880s were valuable for promoting the use of manipulatives and experience in teaching and for motivating research in cognitive development. His suggestions to postpone much of mathematics education to later years, however, was incorporated into a post-WW1 anti-intellectual movement which went so far as to threaten the role of mathematics as a standard school subject (Grouws, p.13). It was this devaluation of mathematics which led to the founding of the National Council of Teachers of Mathematics (NCTM) in 1920 (Willoughby, p.11).
Another blow in this struggle came from E.L. Thorndike in the 1920s. His research, though not extremely rigorous or conclusive, argued against theories of "transfer of learning." Transfer refers to the idea that reasoning skills learned by studying math can be generalized by students and thus be useful in all aspects of life. To a certain extent, this debunking of transfer theory was positive because it ended the reign of faculty psychology and the veneration of drill among educational theorists (Willoughby, p.16). Unfortunately, any student today will tell you that it didn't end drill in the classroom. And strangely, the form of mathematics that Thorndike suggested wasn't all that different. He joined with those who wanted much of the laborious, abstract, and unrealistic mathematics dropped from the curriculum, but instead of advocating understanding and structure, he championed a new rule and a simplified, problem-specific approach. This form of arithmetic without reasoning strengthened the anti-intellectual movement to teach only that mathematics that was immediately useful, if any at all (p.17).
At the turn of the century a new phenomenon appeared that, even today, continues to characterize educational reform - the committee of experts. Groups have been at times privately or publicly funded, professionally homogenous or diverse, and variously reported, researched, or recommended changes in organization, curriculum, and pedagogy. The abundance of commissions, boards, and studies would comprise a bibliography longer than this composition, but a few of the most influential reports from these groups will be presented here in their historical context. It is debatable to what extent any of these commissions affected life within the mathematics classroom, but some ideas of contemporary practices can be drawn from their criticisms and evaluations.
In 1892 the Committee of Ten on Secondary School subjects sponsored a Subcommittee on Mathematics. The final report sided with the social utilitarians for leaving the most perplexing and exhausting topics out of arithmetic, and for including courses such as bookkeeping for high school students without college aspirations or trigonometry for boys in the natural and technological sciences. On the side of meaningful mathematics, the committee recommended a general trend toward the decompartmentalization of subjects in mathematics and proposed that secondary schools teach parallel courses in algebra and geometry designed for integrating the subjects. While the parallel courses were attempted, they generally failed either because the teachers were more interested in one subject than the other or because they were unable to draw connections between the two (Willoughby, p.6).
In 1900, the College Entrance Examination Board (CEEB) was founded for the purpose of standardizing college entrance requirements. The official policy of this board was to never dictate public secondary school curriculums, but the influence of such an organization is inevitable. Their significant impact on education in the 1950s will be discussed later.
In 1908, the International Commission on the Teaching of Mathematics produced among its reports a survey of American education which gives us a picture of the status of secondary mathematics at the time. Recall, first, that algebra was not even required by colleges until 1820 and geometry was not taught until 1844. The 1908 study found that almost all secondary schools in the U.S. provided at least one year of algebra and geometry, that 50% of schools had one more semester of algebra, and that less than 20% of schools offered any higher mathematics (Willoughby, p.7).
The culmination of the progressive era for mathematics education was presented in a report by a branch of the Mathematical Association of America. The 1923 Report of the National Committee on Mathematical Requirements was responsible for formulating a plan of curriculum for the newly redesigned 6-3-3 school organization while incorporating the findings of educational research from psychology and various experimental school programs. The report included practical, cultural, and disciplinary justifications for the subject and outlined a variety of plans for junior and senior high curriculums which could be easily adapted to particular circumstances. Some topics in algebra and geometry were recommended to be introduced in junior high and it was suggested that all students complete the program through eighth grade, with only those who mastered continuing on. Courses in statistics, shop mathematics, surveying, navigation, or descriptive geometry were suggested for others who chose not to follow the college preparatory progression (Bidwell & Ciason, p.382-460).
Most of the history discussed thus far took place among a limited circle of professionals and interested parties; but if there was ever a time when the issues of mathematics education really attained national prominence and captured the attention of the average citizen, it was during the "New Math" movement of the late 1950s and 1960s. The common misperception then, however, was that the reform of mathematics instruction was a new idea. Obviously, such discussions had already been occurring even in the 1800s. Other factors in the continuum of ideas which have not yet been discussed are important to consider here before looking at the phenomenon of "new math."
In the nineteenth century, individuals like Dewey, Piaget and other cognitive psychologists often used arithmetic tasks for their research on learning (Grouws, p.8). One characteristic of their research was the laboratory school. In the twentieth century, experimental school programs continued to abound, increasingly within the context of the university. These programs encompassed such a diversity of purposes and methods that it is difficult to even choose one which is representative. Some of the more well-known university projects came from Illinois (UICSM, 1951), Maryland (UMMaP, 1957-58), Minnesota (Minnemath), and Syracuse (Madison Project, 1957) (Barlage, 1982). Most had only a regional influence, but the point here is that there was a great deal of research and effort aimed at reforming mathematics education long before "new math." One important unifying project was the Carnegie Corporation's Eight-Year Study (1932-1940) to assess the long-term effects of experimental curriculum changes.
The role that Thorndike's psychology played for progressivism in the early 1900s was eventually replaced by a new generation of psychological researchers. Myron Rosskopf overcame Thorndike's criticism of transfer of learning by developing a process by which generalization could be taught (Willoughby, p.21). This brought back the idea of mathematics as a subject useful for teaching reasoning skills adaptable to any task. B.F. Skinner's theories of programmed instruction significantly influenced the development of new programs into smaller units adaptable to individually-paced instruction. The impact of Wertheimer's gestalt psychology was to emphasize the importance of data organization and the ability to see patterns. This focus on insight was opposed to the behavioristic ideas of the previous century. Such intuitive ideas had been popular among mathematicians and educators since the 1920s, but were overshadowed by the emergence of standardized intelligence and proficiency tests and the demands of utilitarian mathematics for the new secondary school population. After World War II, Gestalt psychology became one of the foundations of the mathematics reform movement.
World War II also marked the beginning of the U.S. government's interest in mathematics education as a matter of national defense. Several committees during the war expressed concern over the inadequate mathematics skills of incoming officers. The NCTM Commission on Post-War Plans reported in 1944 and 1945 a series of recommendations aimed at achieving "functional competence" in mathematics for all who were able (Willoughby, p.11).
What made the projects and committees of the 1950s unique from their predecessors and those to follow was the increased involvement of mathematicians and their dominant influence over the ideas of educators. The twentieth century saw advances and discoveries in pure mathematics as significant as those in technology; and after the war, mathematicians became interested in education, especially with the hope that more mathematics could be taught before students began their undergraduate studies. At the same time, there was a general rising awareness that the job market was requiring increased technical competence. Research at the time showed that children were capable of learning quite advanced topics at much younger ages. What was not discussed at this time was whether or not such subjects as set theory, linear algebra, and formal deductive reasoning should be taught to most students (Barlage). Reform programs were designed for very capable students as if a whole generation of mathematicians were being trained.
In 1955 this university interest in secondary education found its voice in the CEEB Commission on Mathematics. The 1959 report of this group was the first national proposal for substantial reorganization of secondary mathematics curriculum to include what was termed as "modern mathematics." Modern mathematics was not an educational terminology, but literally referred to topics in mathematis, like linear programming and probability, which were discoveries of the twentieth century still being built upon by mathematicians.
This report might not have received any attention, except that in October of 1957 the Soviet Union launched the first satellite in space, Sputnik 1. The impact of this event can not be overstated. Suddenly, every American was intensely concerned with the quality of mathematics and science education (Barlage, p. 28). Not only was it a matter of pride, but one of national security. The consequence of that concern was money. In 1958 the National Defense Education Act was passed and for the first time, funding was available for programs concerned with developing new programs for mathematics education. The NSF channeled some of this money and in 1958 created the School Mathematics Study Group. This was the most influential of all projects at the time (p.29). Its primary accomplishment was the development a number of textbooks for all grade levels with emphasis on mathematical structure, the real number system, careful use of language and deductive proofs, discovery, experimentation, and scientific applications. These were meant as a model for commercial publishers who soon followed with appropriately revised textbooks (Willoughby, p.46).
The first phase of the reform movement as mentioned above was aimed at college bound students. The Cambridge Conference of 1963 marked a second phase in which the redesign of instruction for all grades and all levels of ability became important. The "Cooperative Research Act" of 1963 and the "Elementary and Secondary Education Act" of 1965 continued to provide funding for new program developments. This second round of programs, however, was still characterized by the same goals of the first ones -- more advanced, modern, and abstract mathematics at a younger age, and instruction through the discovery of structure versus memorization -- ideals that had originated among college mathematics professors for college-bound students. The devlopment of curriculum and tests were not guided by learning theories or educational research (McIntosh, 1971, p.22).
Quite a lot of controversy surrounded "new math" from the beginning (McIntosh, 1971, p.3-12). Despite all the talk of radical reform, the changes amounted to shifts in emphasis. Categorically, they consisted of:1. the rearranging of topics for better logical sequencing;
2. the earlier discussion of advanced ideas;
3. the removal of a few extraneous topics for the inclusion of new subject matter;
4. the introduction of set theory as a unifying theme;
and 5. a greater emphasis on formal logic, applications, and manipulatives for analytical induction.
From a historical perspective, it is easy to see that most of these changes are merely continuations of long-running themes in educational reform. Pushing subjects to earlier grades has been a slow but consistent evolution and the emphasis on discovery learning and application has been advocated since Colburn's First Lessons. So the only really "new" aspects of "new math" were the modern topics, a new demand for specialized mathematics teachers in the elementary schools, and a significant rise of in-service teacher training through conferences, workshops, and academic courses (Barlage).
In discussing the "new math" movement, it is easy to think that the curriculum and pedagogical changes were felt in every classroom in the nation. That wasn't quite the case. The greatest effects were felt in urban high schools of over 1500 students. But a limited 1963 survey of the College Board among 181 urban schools with a significant percentage of college-bound students still found that 30% of the schools did not teach a number of topics, such as set theory, probability, real numbers, and calculus, considered central to the new mathematics programs (McIntosh, p.21). Among rural schools, new math was not much more than a rumor, perhaps seen in the presence of set vocabulary in new textbooks.
The whole phenomenon of this attempt at such radical reform is quite a story all its own with volumes of recommendations and ideas about the essence of the problem. Most reflect Pestalozzi's ideas of discovery learning or specific schools of learning theory, like Skinner's programmed instruction. But despite all the great ideas and good intentions, standardized test scores in the 1960s and 70s actually decreased slightly and disillusionment abounded. Funding sources began requiring "Back-to-Basics" programs and established tougher standards of accountability. Such demands to show evidence of learning forced teaching back into rule and drill, leaving no time to foster interest in mathematics or time for students to achieve understanding and mastery through practical experiences. Application was reduced to a few word problems after practicing a method, a design used in the early nineteenth century.
The 1980s heard another call for "excellence" in the schools and another round of some experimental programs in some schools for some students. Today there is an enormous amount of research on mathematics education being conducted (Grouws, p.27-29), but relatively little widespread intervention in the schools by the universities. The fact that most urban high schools today provide the opportunity for top students to take calculus is evidence that in terms of content, mathematics education is continuing its historical trend of teaching more math to more students at younger ages. But the continued failure of average and low-ability students in the mathematics classroom points to the fact that instructors have not yet mastered how to daily teach mathematics for understanding to those who don't naturally grasp the concepts.References
Barlage, E. (1982). The New Mathematics: An Historical Account of the Reform of Mathematics Instruction in the United States of America. (ERIC Document Reproduction Service No. ED 224 703)
Bidwell, J.K., & Ciason, R.G. (Eds.). (1970). Readings in the History of Mathematics Education. Washington, DC: National Council of Teachers of Mathematics.
Broudy, H.S. (1985, March). Past and Future in Education. Paper presented at the annual meeting of the Association for Supervision and Curriculum Development, Chicago. (ERIC Document Reproduction Service No. ED 253 969)
Grouws, D.A. (Ed.). (1992). Handbook of Research on Mathematics Teaching and Learning. National Council of Teachers of Mathematics. New York: Macmillan Publishing Co.
Hayden, R.W. (1983, April). A Historical View of the "New Mathematics." American Educational Research Symposium, Montreal. (ERIC Document Reproduction Service No. ED 228 046)
McIntosh, Jerry A. (Ed.). (1971). Perspectives on Secondary Mathematics Education. New Jersey: Prentice-Hall, Inc.
Willoughby, S.S. (1967). Contemporary Teaching of Secondary School Mathematics. New York: John Wiley & Sons, Inc.