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
1
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
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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
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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<x<b, if to every value of
the variable x in this interval there corresponds a definite value of the variable y.
Also, it is irrelevant in what way this correspondence is established. (cited in
Kleiner, 1989, p. 291)
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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.
5
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
6
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
7
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
8
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
9
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
10
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.
11
Textbook,
year, and
author
Definition of Function
Chapter Title
(In Chapter/
Total Chapters)
24. Properties of series
25. Binomial Theorem
26. Logarithms
New School
Algebra
(1898)
Wentworth
A function of a variable is an expression that
changes in value when the variable changes in
value. In general, any expression that involves a
variable is a function of that variable. For brevity
a function of x is represented by f(x) or F(x).
Graphs
(27 of 27)
Third Year
Mathematics
(1917)
Breslich
If two variables x and y are so related that to
every value of x there corresponds a definite value
of y, then y is said to be a function of x. The symbol
f(x) is used to represent a function of x.
Functions:
Equations in
one unknown
(1 of 15)
College
Algebra
(1939)
Rosenbach &
Whitman
If two variables are so related that to each
definite value assigned to one of the variables
there correspond one or more definite values of the
other variable, then the second (dependent)
variable is said to be a function of the first
(independent).
Functions
and Their
Graphs
(4 of 20)
Algebra and
Trigonometry
(1967)
Keedy, et al.
Intermediate
Algebra
(1972)
Lial & Miller
Algebra and
Trigonometry
(1989)
Rice &
Strange
Advanced
Mathematics:
A Precalculus
Approach
(1993)
Ryan, et al.
Secondary
Math: An
integrated
approach
(1996)
Charles, et al.
A function is a special kind of relation. A function
is a relation in which no two ordered pairs have
the same first coordinate (the domain) and
different second coordinates (the range).
A function is a special type of relation in which,
for each value of the first component, there is only
one value of the second component.
A correspondence f that assigns to each x in X
exactly one element y in Y is called a function. The
set X is called the domain and the set of y's is
called the range. We say that "y is the image of
x" or that "y is a function of x."
A function f is a relation in which each element in
the domain is mapped to exactly one element in
the range.
If two quantities, x and y, are related so that there
is only one value of the dependent variable (y)
associated with any value of the independent
variable (x), then y is a function of x.
12
Preceding Chapters
Following Chapters
Relations,
Graphs and
Functions
(2 of 15)
Relations and
Functions
(7 of 12)
The Idea of
Function
(3 of 14)
Functions
(1 of 16)
Functional
Relationships
(3 of 10)
2.Trigonometric Functions
3. Linear Equations
4. Quadratic equations in one
unknown
1. Fundamental Operations
2. Factoring and Fractions
3. Exponents and Radicals
5. Equations and Their
Solutions
6. Systems of Linear
Equations
7. Quadratic Equations
1. System of Real Numbers
3. Circular Functions
4. Trig Functions, Relations
and Angles
5. Linear Equations,
Inequalities, and Functions
4. Rational Expressions
5. Exponentials and Roots
6. Quadratic Equations
8. Systems of Equations
9. Logarithmic and
Exponential Functions
10. Sequences and Series
1. Basic Algebra
2. Equations and Inequalities
4. Elementary Functions
5. Exponential and
Logarithmic Functions
6. Trigonometry
2. Graphing Functions
3. Trigonometric Functions
4. Graphs and Inverses of
Trigonometric Functions
1. Data and Relationships
2. Patterns, change, and
Expressions
4. Solving Linear Equations
5. Analyzing Linear
Functions and Their Graphs
6. Connecting Slope and
Linear Functions
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 selfreflection, 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.
13
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
14
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.
References
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
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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
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Kleiner, I. (1989). Evolution of the function concept: A brief survey. College
Mathematics Journal 20, 282-300.
15
Malik, M. A. (1980). Historical and pedagogical aspects of the history of function.
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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
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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
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Washington, DC: National Council of Teachers of Mathematics.
Romberg, T. A. (1992). Problematic features of the school mathematics curriculum. In
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schools in the last twenty five years. The first yearbook of the National Council of Teachers of
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Mathematics (pp. 1-31). New York: Bureau of Publications, Teachers College, Columbia
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Stanic, G. A. & Kilpatrick, J. (1992). Mathematics curriculum reform in the United
States: A historical perspective. International Journal of Educational Research, 17, 407-417.
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