From 1900-1920, concepts such as metric space, topological space, Hilbert space, and Banach space were introduced. These developments led to new definitions of function based on arbitrary sets, not just real numbers. In 1917, Caratherdory defined a function as a rule of correspondence from a set A to real numbers (Malik, 1980, p. 491).
Back in American school mathematics there was still tension. There were still those supporting the concept of function as a unifying theme in mathematics education and a unified approach to mathematics curriculum. For example, David Smith (1926) wrote in the first National Council of Teachers of Mathematics (NCTM) Yearbook "much has been written of the advance in appreciation of the function concept in recent years...It has of late come to be looked on as a kind a unifying principle running through all parts of algebra" (p.26). There were also those who are sharply criticized current trends and objectives in mathematics education. Some were even questioning the place of mathematics in general education. During the 1930's two studies of secondary mathematics curriculum were commissioned to address this very issue. One was a report of the Progressive Education Association (PEA) Committee on the Functionality of Mathematics in General Education and the second was a report from the Joint Commission of the MAA and NCTM. The PEA committee selected nine topics they felt were particularly applicable to life, one of which was functions. They suggested "the student should acquire understandings of the concept of variables, dependency, and the generality and power of the function concept" (Osborne & Crosswhite, 1970, p. 226). The Joint Commission formulated its recommendations around seven fields of mathematics which included graphical representation, elementary analysis and relational thinking (Osborne & Crosswhite, 1970). Both committees appeared to be upholding the idea that function was an important concept in secondary mathematics.
As the study of higher level mathematics became more and more abstract, so did the definition of function. The developing field of abstract algebra and topology gave way to more set-theoretic definitions of function. In response to the more modern definitions and applications of the function concept, Schaaf stated
The keynote of Western culture is the function concept, a notion not even remotely hinted at by any earlier culture. And the function concept is anything but an extension or elaboration of previous number concepts - it is rather a complete emancipation from such notions. (cited in Tall, 1992, p.497).
This so called emancipation from the old ideas was evident as the field of mathematics rapidly became more abstract. Bourbaki, a well known proponent of abstract algebra introduced a set definition of function that would eventually affect school mathematics curriculum for many years. In 1939, Bourbaki offered the following definition of function:
Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E and a variable element y of F is called a functional relation in y if, for all x in E, there exists a unique y in F which is in the given relation with x.
We give the name of function to the operation which in this way associates with every element x in E the element y in F which is in the given relation x; y is said to be the value of the function at the element x, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function. (cited in Kleiner, 1989, p.299)
Bourbaki also gave the well-known and textbook published definition of a function as a set of ordered pairs (the product of E x F). So as the study of mathematics and the definition of function at the college level were becoming increasingly more abstract, what was happening at the secondary level?
Markovits, Eylon, & Bruckheimer (1986) felt that the new formal definition of function was far too abstract for high school students, and yet they noted its influence was in fact felt at the school level. What happened over time was that a gap was widening between university mathematics and school mathematics (see Howson, Keitel, & Kilpatrick, 1981). There was a "new math" at the university level and high school graduates were not prepared to study it. College professors felt the need to take action. In the early 1950's, the Committee on School Mathematics (the UICSM) set up a project at the University of Illinois. As stated by Howson, et al. (1981), "it aimed to improve the teaching of mathematics to pre-college students, for the benefit of universities, so as to help overcome the gap between school mathematics and that at the university, and to secure a better qualified new generation of mathematicians"(p.133). In describing the courses that had been developed by the UICSM, the administrative head of the project, Max Beberman (1958), echoed the project's emphasis on precise language as he made the following observations about the treatment of function:
The semantics notion that a noun ought to have a referent has led us to give precise descriptions of relations and functions. The customary vagueness that surrounds the word 'function' in conventional courses vanishes when a student realizes that a function is an entity, a set of ordered pairs in which no two elements have the same first component. ( p. 22)
Other projects, textbooks, and recommendations in the late 50's and 60's echoed the emphasis of precise definitions which were set theoretic in nature. For example, the largest and most well known project of the new math era, the School Mathematics Study Group (see Howson, et al., 1981), was responsible for the authorship of a series of widely circulated and adopted textbooks written in the spirit of the new math movement. SMSG (1960) defined function as follows:
Let A and B be sets and let there be a given rule which assigns exactly one member to B to each member of A. The rule, together with the set A, is said to be a function and the set A is said to be its domain. The set of all members of B actually assigned to members of A by the rule is said to be the range of the function.
The above definition strongly resembles that of Bourbaki. It had taken twenty years, but the "abstractness" of the college mathematicians' definition of function was making its way into high school classrooms.
Because the definitions had become more abstract and precise, one might assume that no one continued to support the argument that function be a unifying theme. Not true. In fact, a priority of the new math movement was to provide unifying themes for mathematics in terms of an overall structure. The goal was for students to understand how different skills and definitions were connected in the overall structure of mathematics. In the 1959 the Commission on Mathematics of the College Entrance Examination Board published a report in which they described a nine point program for school mathematics reform in light of the new math movement. The fourth point was a call for the judicious use of unifying ideas, one of which was function. Kleiner, Moore, and peers had not been contradicted. As a matter of fact, May & Van Engen (1959) described the new definition of function as unifying all previous ideas of function. They sated
We have presented several different points of view from which functions (and relations) may be considered. We may describe them as sets of pairs, sets of points, tables, correspondences, or as mappings. We may emphasize the rule or we may concentrate attention on the set...The modern point of view (of function) is not contradictory to any of them, but unifies them all. (p. 87)
Function was taking on new meaning in the field of mathematics and the field of mathematics continued to take on new places in society, thus there was a perceived need to define and present it differently to students.
So, was this the answer to the definition of function in a state of flux problem...make it precise and theoretic. Well, it appears that the precise function definition stuck. Textbooks published even today are using similar definitions (see pp. 9-10 for a summary of texts). The New Math movement did not fair as well. With a decline in scores on tests that measure basic skills, a call for "Back to Basics" ushered in the 70's. Programs like the Individually Prescribed Instruction Project (see Howson, et al., 1981) were popular and the behaviorist approach to learning was evident in the skill and drill nature of the mathematics classroom. In this era of skill mastery, ideas of unifying themes took a backseat to the achievement of lists of objectives, one of which was the function concept.
In 1980, NCTM published their Agenda For Action which called for problem solving as the unifying theme for curriculum in school mathematics. As a result, problem solving became the new emphasis in mathematics education. Textbooks were quick to claim that their books carried problem solving as a unifying theme. So where was the function concept? According to Cooney (in press), textbooks in the early eighties continued to emphasize "functions as ordered pairs, basing such an approach on the structure and rigor of mathematics" (p. 1). Function had remained an abstract definition while the ideas of the connective structure of the modern mathematics movement had been lost.
There were educators who expressed concern. In 1980, Malik stated "the necessity of teaching the modern definition of function at the school level is not at all obvious and most of the instructors feel that pedagogical considerations were ignored while designing the course content and the mode of presentation. Markovits, et al. (1986), echoed these concerns when they stated
...the situation is today that the set definition has been taught in schools for about 25 years, and no one disputes the central importance of the concept, whatever the arguments about its definition. The natural question, therefore, arises: do the students 'understand' the new definition? (p.19)
Concerns stemming from mathematics educators about understanding and pedagogy were key issues in the 1989 NCTM Curriculum and Evaluation Standards for School Mathematics and in the 1991 Professional Standards for Teaching Mathematics. Since the publication of the Standards, there has been a renewed commitment to the teaching of mathematics so that students gain an appreciation for its applications and connections. With the emphasis taken off the formal definition of function, the focus of the study of function becomes more conceptual in nature. Froelich (1991) stated "the basic idea of function is that two quantities are related in some way" (p.1). This is the how the concept of function was first developed by Galileo when he studied physical problems of motion. Current trends in the study of function in school mathematics such as modeling, data analysis, real world and interdisciplinary applications are not new ideas. These ideas are seen at the onset of the development of the function concept. Neither are new pedagogical goals such as developing connections within mathematics through the use of function, using function as a unifying themes, and increasing an appreciation for mathematics in students a new idea. The goals today are so similar to the calls of the mathematicians and educators of the early 1900's that, just as it did for Klein, the question comes to mind "have we come full circle in the study of functions?" It appears that in terms of recommendations for the intended curriculum for school mathematics, we have come full circle.
The recommendations and intentions for mathematics curriculum have been varied as well as the impact they have had on the actual mathematics taught in schools. There has been a clear pattern throughout the history of curriculum reform efforts of misinterpretation or partial implementation of curriculum recommendations. Stanic and Kilpatrick (1992) concluded that intended outcomes of reform movements have been limited. Keeping in mind that none of the previously discussed curriculum intentions were implemented in direct accordance with the authors' visions and that textbooks tend to influence classroom teaching dramatically, it is important to examine the presentation of the function concept in a small sample of textbooks over the past hundred years. The following chart reviews eight Algebra textbook presentations of the function concept in terms of the definition and the position of introduction with respect to other topics of study.
While it is easy to see that the definition of function published in textbooks has changed minimally over the past century, it is not so easy to see the effects of the preceding curriculum recommendations. In a rather pessimistic view of textbook publishers, perhaps this is because publishers strive to sell a great number of textbooks. Because of this, they deem it necessary to include a certain amount of rhetoric as to claim alignment with professional recommendations while simultaneously reprinting traditional curriculum as to not "rock the boat" too much.
I believe, however, it is not the responsibility of textbook publishers to promote function as a unifying theme in mathematics education. No longer can teachers be dependent on textbooks to guide curriculum decisions. Teachers must take responsibility for what happens in their classrooms. Function is a powerful and unifying topic in secondary mathematics, but no textbook alone will help teachers come to this understanding. Teachers must pay attention to the recommendations being made by researchers. Mathematics and, more specifically, the study of function are rapidly changing due to new technologies available to students. As Cooney (in press) observes,
Today we have the advantage of technology and the many ways we can explore the behavior of functions using graphing calculators and various types of computer software. If not today, then very shortly, the teaching of functions will involve extensive use of various types of technologies. (p. 1)
Teachers can no longer wait for textbooks to guide the study of functions in their classrooms...classrooms are changing too fast for texts to keep up. Through a review of literature, discussion with professional colleagues, analysis of available technology, and self-reflection, teachers must reach a decision as to what they believe about the place of function in today's curriculum and then find the resources to implement their ideas.
So what can be learned from this trace of the history of the function concept in the intended high school curriculum over the past century? Certainly the observation can be made that we have gone full circle in our study of function: Galileo's dependence relationships to modern set theory and now back to an emphasis on relationships. But, this does not mean that we are right back where we started. To say that would be to infer that mathematics is a static field of study and that a circular path covers no distance. Mathematics is a dynamic field and the development of the function concept has traveled a great distance as it evolved. The meaning and use of the function concept has changed as society, technologies, and interests of mathematicians changed. The definition of function has been determined by its use in topics of study. Malik (1980) sums this point up well:
We note that the definition of function as an expression or formula representing a relation between variables is for calculus or pre-calculus; is a rule of correspondence between reals for analysis, and a set theoretic definition with domain and range is required to study topology. (p. 492)
The concept of function was created by mathematicians for mathematicians. Function does not exist outside of a mathematician's use or interpretation of it. The definition has only changed because mathematics has changed.
Since function is defined by what is needed for application or development of new fields of study, then in choosing the definition and context with which high school students should approach the function concept, teachers should constantly be examining the purpose of studying function in their own classroom. I do not see the fact that the function concept is still in a "state of flux" to be a problem at all. In fact, the question of "is 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)?" is a powerful one for teachers to consider as they make decision about their students' experiences with and studies of the function concept. As we continue along the path in search of "the definition" of function, let us always keep in mind, who we are defining function for and why.
Beberman, M. (1958). An emerging program of secondary mathematics. Inglis Lecture. Cambridge, Mass.: Harvard University Press.
College Entrance Examination Board, Commission on Mathematics. (1959). Program for college preparatory mathematics. New York: CEEB
Cooney, T. J. (In press). Developing a topic across the curriculum: Functions. In T. J. Cooney, et al. (Eds.), Mathematics, Pedagogy, and Secondary Teacher Education. Portsmouth, NH: Heinemann.
Froelich, G. W., Bartkovich, K. G., & Foerester, P. A. (1991). Connecting mathematics. In C. R. Hirsch (Ed.) Curriculum and evaluation standards for school mathematics addenda series, grades 9 - 12. Reston, VA: National Council of Teachers of Mathematics.
Hamley, H. R. (1934). The history of the function concept. In National Council of Teachers of Mathematics (Ed.), The ninth yearbook: Relational and functional thinking in mathematics (pp. 48-84). New York: Bureau of Publications, Teachers College, Columbia University.
Kilpatrick, J. & Stanic, G. (1995). Paths to the Present. In I. M. Carl (Ed.) Prospects for school mathematics (pp. 62-77). Reston, VA: National Council of Teachers of Mathematics.
Kleiner, I. (1989). Evolution of the function concept: A brief survey. College Mathematics Journal 20, 282-300.
Malik, M. A. (1980). Historical and pedagogical aspects of the history of function. International Journal of Mathematics Education in Science and Technology, 11(4), 489-492.
Markovits, Z., Eylon, B., & Bruckheimer, M. (1986). Functions today and yesterday. For the Learning of Mathematics, 6 (2), 18-28.
May, K. O. & Van Engen, H. (1959). Relations and functions. In National Council of Teachers of Mathematics (Ed.), The 24th yearbook: The growth of mathematical ideas grades K-12 (pp. 65-110). Washington, DC: National Council of Teachers of Mathematics.
National Committee on Mathematical Requirements. (1923). The reorganization of mathematics in secondary education. Oberlin, OH: Mathematical Association of America
National Council of Teachers of Mathematics. (1980). Agenda for action. Reston VA: Author.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston VA: Author.
Osborne, A. R. & Crosswhite, F. J. (1970). Forces and issues related to curriculum and instruction, 7-12. In Jones, P. S. (Ed.) The 32nd yearbook of the National Council of Teachers of Mathematics: A history of mathematics education in the United States and Canada. Washington, DC: National Council of Teachers of Mathematics.
Romberg, T. A. (1992). Problematic features of the school mathematics curriculum. In P. W. Jackson (Ed.) Handbook of research on curriculum (pp. 749-788). New York: Macmillan.
Smith, D. E. (1926). A general survey of the progress of mathematics in our high schools in the last twenty five years. The first yearbook of the National Council of Teachers of Mathematics (pp. 1-31). New York: Bureau of Publications, Teachers College, Columbia University.
Stanic, G. A. & Kilpatrick, J. (1992). Mathematics curriculum reform in the United States: A historical perspective. International Journal of Educational Research, 17, 407-417.
Tall, D. (1992). The transition to advanced mathematical thinking: Functions, limits, infinity, and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 495-498). New York: Macmillan Publishing Company.