Nineteenth-Century Efforts to Promote Mathematics Education from Grade School to the University Level

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Nineteenth-Century Efforts to Promote Mathematics Education from Grade School to the University Level

Overview

Economic and political factors combined to foster an increased emphasis on mathematics education over the course of the nineteenth century. The demand for technical workers and for military officers who could understand the complexities of more powerful weapons resulted in new polytechnic schools and military academies throughout northern Europe and the United States. The increased need for teachers of mathematics provided employment for a vastly increased number of mathematicians. In addition, newly formed mathematical societies took an interest in the mathematics curriculum and teaching methods at the elementary level.

Background

Ever since Plato refused some 2,400 years ago to admit to his academy students who were ignorant of mathematics, there has been a connection between mathematics and education. The inclusion of three fields of mathematics—arithmetic, geometry, and logic—among the seven traditional liberal arts assured some emphasis on mathematical topics in the schools of the classical period and the middle ages. Printing with movable type became possible in 1454, and some of the first books printed were manuals of mathematics. The earliest arithmetic text was published in Trevisi, Italy, in 1478. The first printed edition of famed Greek mathematician Euclid (c. 330-260 b.c.) appeared in Venice in 1482. Prior to the nineteenth century, however, most mathematical research was conducted outside the universities. Mathematicians were supported by patrons and then by the scientific academies, which also produced the first scientific journals.

Expanded trade and new technologies called for increased emphasis on the mathematics of the marketplace: counting, weights and measures, currency exchange, and the calculation of integers—topics not normally included in the training of young gentlemen or future clergy. The industrial revolution and a demand for faster water and land transport called for the solutions of the mathematical problems of fluid motion, bridge building, and heat transfer—the bases of modern civil and naval engineering.

At the university level, revolutionary France led the way. The government in 1794 authorized the establishment of the schools that were to become the École Polytechnique, later converted into a military school by Napoleon, and the École Normal Superiore, to train future teachers. The École Normal would soon boast Joseph Lagrange (1736-1813), Adrien-Marie Legendre (1752-1833), and Pierre Simon Laplace (1749-1827) on its faculty. The École Polytechnique published the first journals devoted to mathematical research, when in 1810 Joseph-Diaz Gergonne (1771-1859) inaugurated the Annals of Pure and Applied Mathematics in French. German and English journals would soon follow. In 1826 August Leopold Crelle (1780-1855) inaugurated the Journal for Pure and Applied Mathematics, published in German. In 1841 the Berlin Mathematical Society began publishing The Archives of Mathematics and Physics with Special Consideration of the Needs of Teachers as a supplement to its Proceedings. Under the editorship of G. Grinert, this was the first journal to deal with mathematics teaching, as opposed to new research. In 1865 the London Mathematical Society was formed and began to publish its Proceedings.

The situation of geometry in English universities and preparatory schools was a special case. An English edition of Euclid's Elements had been published in 1758 by Robert Simson, a professor of mathematics at the University of Glasgow, and was to provide the basis for geometry instruction until late in the nineteenth Century, despite the fact that the inadequacies of many of Euclid's demonstrations were well known to mathematicians. In 1870 the Association for the Improvement of Geometrical Thinking was formed in England. In 1875 the Association published a Syllabus of Plane Geometry, which was based on Euclid's first six books but which was intended to remedy many minor deficiencies. The Syllabus gained the approval of the British Association for the Advancement of Science in 1877. In 1897 the Association changed its name to the "The Mathematical Association."

The development of university education in the United States proceeded in a very uneven manner. None of the colleges founded in the colonial period required extensive preparation or study of mathematics. A text, "A New and Complete System of Arithmetic Composed for the Use of Citizens of the United States," published in 1872 was adopted by Harvard, Yale, and Dartmouth colleges. It provided an overview of commercial mathematics, including currency conversions, weights and measures, the calendar, and the calculation of interest. Geometry was not required until the Civil War (1861-65). The first technical school to be established was the United States Military Academy at West Point, New York, in 1802. By the end of the century, the establishment of state universities and technical schools where the new engineering disciplines could be studied provided a basis for a growing American mathematical community.

Perhaps the most important influence on the teaching of mathematics in the modern elementary school was the work of Swiss educational reformer Johann Heinrich Pestalozzi (1746-1827), a disciple of the philosopher Jean Jacques Rousseau. Pestalozzi wrote that instruction should proceed from the familiar to the new, from the concrete to the abstract, and that the student of mathematics should retrace the path to mathematical or scientific knowledge that the original investigators trod in the first place. Pestalozzi was allowed to test his theories when the revolutionary French military forces gained control of the Swiss government for a time. Pestalozzi's influence was transplanted to the United states when William Maclure (1763-1840) published his First Lessons in Arithmetic on the Plan of Pestalozzi, with Some Improvements in Boston in 1921. In use for nearly a century, it has been described as the most popular arithmetic text ever.

Impact

It is perhaps ironic that the development of mathematics, perhaps the most austere and purely academic discipline of all, has been influenced in its development so strongly by political, military, and social factors. Throughout the twentieth century mathematics and mathematics education have been understood as components of national and economic power. Mathematicians played a prominent part in the Second World War in the development of atomic weapons and of electronic computers and computer languages. When the former Soviet Union appeared to have gained a temporary advantage in a "space race" with the United States by launching Sputnik, the first artificial Earth satellite, part of the American response was to develop new curricula for science and mathematics instruction. One component of the new curriculum was the "new math," an emphasis on abstract mathematical principles in place of rote learning of mathematical facts. The "new math" was at best a partial success. Ignoring the tradition of Pestalozzi and his followers, it expected students to deal with abstract concepts without having had the opportunity to become familiar with the facts and techniques of ordinary addition and multiplication at the concrete and operational level.

In hindsight, Pestalozzi appears to have been part of an educational tradition that includes Maria Montessori, John Dewey, and Jean Piaget. The research of these educators and psychologists has focused on the mental development of children and their growth through gradual progression through stages of concrete thinking, operationalism, and later abstract thinking, with the abstract conceptual stage considered the highest or most mature stage of development. This position is not without controversy. Any alternative school of thought holds that individuals differ in "learning style," and that individuals not comfortable with abstract relationships may function as well or better in all important human activities than "abstract thinkers" do. This group includes teachers who consider themselves "ethnomathematicians" and feel that traditional mathematics instruction depends too heavily on the assumptions of white European culture, and that more concrete ways of thinking also provide a valid approach to mathematical truth.

A number of important associations of mathematicians and mathematics teachers are now involved in decisions affecting mathematics education in the United States. The American Mathematical Society was originally founded as the New York Mathematical Society in 1888 and, adopting its present name in 1894, is devoted primarily to original research in mathematics. The Mathematical Association of America was organized in 1914 for individuals primarily interested in the teaching of mathematics on the college level. Concern with the teaching of mathematics at the primary and secondary levels would rest with the National Education Association , originally organized as the National Teacher's Association in 1875. This organization appointed a number of committees over the period 1892-1912 to make recommendations about the content of secondary school mathematics and the mathematics to be required for college admission. In 1920 the National Council of Teachers of Mathematics was formed by the National Education Association in order to bridge the gap between the more general concerns of the National Education Association and the Mathematical Association of America.

It is likely that debate about the proper way or ways to teach mathematics will continue for some time. In recent years the relatively poor showing of American students compared to students from other nations has served to renew debate on the best ways to tech mathematics in a culturally diverse society. With the number of employment opportunities for the mathematically literate increasing rapidly, means will need to be found, either to involve more professional mathematicians in the educational process or to provide the future mathematics teacher with a better understanding of mathematics and the methods of teaching it to contemporary students.

DONALD R. FRANCESCHETTI