The History Of The Function Concept

The History Of The Function Concept

In The Intended High School Curriculum Over The Past Century:

What Has Changed And What Has Remained The Same

In The Roles That Functions Are To Play?

Lisa Sheehy

July 1996

The National Council of Teachers of Mathematics (NCTM) proposed in the Curriculum and Evaluation Standards for School Mathematics that "one of the central themes of mathematics is the study of patterns and functions" (NCTM, 1989, p.98). In 1991, Froelich, Bartkovich, & Foerester made the more poignant statements "the concept of function is probably the single most important idea in mathematics" and "the idea of function is inherent in many parts of today's algebra and geometry programs" (p.1). It is clear that these proponents of the function concept view function as both central and essential in today's mathematics curriculum. The acceptance of the function concept as a crucial topic of study for the mathematics students of the nineties brings to mind questions of past views of the importance of the function concept and its inclusion in the high school curriculum. Are the recommendations for mathematics curriculum as we approach the end of the 20th century so radically different than those throughout the century? As I "trace the history of the function concept in the intended high school curriculum over the past century," I will show that not only are the recommendations for today not radically different from those of the past, but they are, in fact, strikingly similar to those made at the onset of the twentieth century.

So, where has function concept been in the intended curriculum over the past century? First the question "how does one define the intended high school curriculum in the United States?" needs to be addressed. Romberg (1992) reminds us

As a consequence of shared state and local control and shared state and local taxes to support schools, there are vast differences in the quality of programs, facilities, staff, and teachers both across and within states. There is no national curriculum, no national set of standards for the licensing or retention of teachers, no common policies for student assessment of progress or admission to higher education, and so forth. (p. 763)

In the absence of a national curriculum or imposed curriculum standards, the problem of defining an intended curriculum begins with the question "intended by whom?" Over the past century, college professors, pure mathematicians, politicians, teachers, psychologists, sociologists, and a variety of educational organizations have attempted to influence school mathematics curriculum. I will discuss how the function concept was presented in some of the most defining and influential curriculum recommendations throughout the century. This brief history of curriculum recommendations will be coupled with a discussion of the development of the function concept in the field of mathematics.

The 1890's began with a general dissatisfaction of secondary education and a call for a unified mathematics curriculum. As reported by Osborne & Crosswhite (1970), the National Education Association (NEA) appointed the Committee of Ten in 1892 to study secondary school problems and to provide a national force for standardizing the secondary school curriculum. The subcommittee of the Committee of Ten appointed to examine mathematics made no reference to the concept of function as a unifying theme. They did, however, promote the concept of equation as a unifying theme. Similarly, recommendations from the NEA College Entrance Requirements Committee and the American Mathematical Society (AMS) made little mention of the concept of function and emphasized equation as the important concept in Algebra.

In contrast to the above recommendations, Felix Klein ( a professor of mathematics in the University of Gottingen in Germany) gave an address at the International Congress of Mathematicians in Chicago in 1893. It was in this address that Klein first emphasized the vital importance of the function concept in school mathematics to teachers. In the years following this address, Klein began to develop and expand upon this idea of functional mathematics. In 1908, at the International Conference of Mathematicians in Rome, Klein claimed that the function concept "was, not simply a mathematical method, but the heart and soul of mathematical thinking "(Hamley, 1934, p. 53). Hamley (1934) claimed "the idea that the function concept should be made the central theme of school mathematics may be said to have originated with Klein" (p. 49). In E. H. Moore's presidential address to the American Mathematical Society in 1902 and in later writings about graphical representations, he echoed Klein's idea of function as a dependency relation.

Influenced by Moore, D. E. Smith and E. R. Hendricks argued for the elaboration of the function concept in the American school curriculum (see Hamley, 1934). According to the Mathematical Association of America (1923), in 1916 Hendricks appointed the National Committee on Mathematical Requirements(NCMR), of which D. E. Smith was an original member. The purpose of this committee was to give "national expression to the movement for reform in the teaching of mathematics" (NCMR, 1923). In 1923 the NCMR published a landmark report. Chapter seven of the report, entitled "The Function Concept in Secondary School Mathematics," was "recognized as the first authoritative statement of the case for functional thinking to be found in American mathematical literature" (Hamley, 1934, p. 78). The 1923 Report proposed that "methods for organization are being experimentally perfected whereby teachers will be enable to present much of this material more effectively in combined courses unified by one or more such central ideas as functionality and graphic representations" (NCMR, 1923, p. 38). From the above review of recommendations, it is evident that function was a topic in school mathematics that was receiving increased attention during the early part of the twentieth century.

But why was the study of function at the high school level becoming increasingly important? Why was there an apparent need to reform school mathematics curriculum with function as a unifying theme? Perhaps we should examine what was happening in the field of mathematics in relation to the function concept. From 1720-1820 a new subject, Analysis, began to take from in the field of mathematics in which the concept of function was central. Prior to this, Calculus seemed to be the topic that most affected how function was defined and applied. According to Kleiner (1989), the problem was that the concept was in a "state of flux." Was function to be represented geometrically (in the form of a curve)? algebraically (in the form of a formula)? or logically (in the form of a definition)?

Was there any agreement? It was Dirichlet's 1829 definition of function that was most widely accepted at the turn of the this century (Kleiner, 1989 and Malik, 1980). Dirichlet defined function as follows:

y is a function of a variable x defined on the interval a