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THE CONCEPT OF FUNCTION
Epistemological remarks, didactical
questions, Students’ conceptions and
difficulties, new teaching trends
Nicolina A. Malara,
Mathematics Department,
Modena & Reggio E. University,
Italy
Points in the discussion
 Some epistemological remarks
 some consequent didactical questions
 Students’ conceptions and difficulties
 new teaching trends
Epistemological
remarks
The concept of function has a long history,
its objectivation has required many
centuries.
Its first definitions appear between the end of
XVII century and the beginning of the XVIII
century
The roots of these definitions are linked with
the exploration of curves
Curves were initallly described by proportions
between some auxliary segments (diameter,
axis,…) in the realm of specific problems
(Fermat, Descartes, Newton, Lebnitz…)
Curves were not regarded as graphs of
relationships between these ausiliary segments.
They were taken for what they appeared to our
eyes:
• geometrical objects
• trajectories of moving points.
In the course of the solution of the problems, the
proportions used lost their meaning and became
mere algebraic expressions on which formal
operations were performed
In the first definitions, the functions are
conceived as
analytic expressions
J. Bernoulli (1718) ‘Remarques sur ce qu’on a
donné jusqu’ici de solutions des problemes sur
les isoperimetres’
Function of a variable quantity is a
quantity composed in watever manner
of this variable and constant quantities
Eulero (1748) ‘Introductio in analysis infinitorm’
Functio quantitatis variabilis est expressio analytica
quomodocumque composita ex illa quantitate
variabili et numeris seu quantitatibus constantibus
A function of variable quantity is an
analytic expression composed in a
whatever manner by this quantity and by
numbers or constants
J. Bernoulli
Function of a variable quantity is a quantity
composed in watever manner of this varable
and constant quantities
The Eulero’ s concept of variable
Quantitas variabilis est quantitas indeterminata seu
universalis, quae omnes omnino valores
determinata in se complectitur
A variable quantity is an indeterminate quantity,
or an universal quantity, which includes all the
determinate values
Eulero conceives that the value of a variable
can range from Naturals to Complex Numbers
The XVIII century and the first half of the XIX century have
seen several studies and discussions which brought to
overcome the procedural-operative conception of function
We simply quote:
• The famous polemics among Euler, d’Alembert,
Daniel Bernoulli concerning the problem of
vibranting string;
• The development of the theory of trigonometric
series by Fourier;
• The notion of continue function (Chauchy,
Dirichlet, Abel, Bolzano, Weierstrass,… )
These questions have brought Dirichlet to
formulate a more general definition of function
Dirichlet (1837)
If a variable y is so related to a variable x that
whenever a numerical value is assigned to x
there is a rule according to which a unique
value of y is determined, then y is said to be a
function of the indipendent variable x
The Dirichlet definition encompasses very strange
functions:
- some of them are continuous and yet
nowhere differenciable;
- some of them cannot be represented by a
curve drawn by a free hand.
The Dirichlet definition was widely accepted and
used up to the middle XXth century.
However it started to provoke dicussions in fondationists’
circles already at the turn of the XIX century
Both constructivists and intuitionists as well as formalists
were against it, albeit for very different reasons:
• The former wanted to have a rule allowing
to find a y corresponding to a given x in
finite time or finite numer of steps
• The latter considered the definition not
sufficiently rigorous
The idea to refer the function definition to a
‘new’ (as to arithmetic) primitive notion
become inevitable
The Peano definition (1911) ‘Sulla definizione di funzione’
In tuning with the theory of the relations by Russell &
Whitehead (1910, Principia Mathematica)
Peano reduces the concept of function to the one
of the relation and introduces the notion of
univocal relation
The question is transferred to the concept of
ordered pair
assumed by Peano as a primitive concept
Sierpinska (1988) summarizes these first stages of the
development of the concept of function in the following
scheme
I
An implicit idea of
transformation (T) of
points or relationships
between magnitudes
IV
T described by
equations
II
T described by
numerical
tables
V
T described by
graphs and
equations
III
T described by
proportions
VI
An elaborate explicit
idea of relationships
between variables
She states that at school, as first steps, the students have
to do experiences and arrive to conceptualize the
Dirichlet definition.
In agreement of the development of the axiomatic
set theories, definitions of the ordered pair have been
given by:
Hausdorff (1914): (a,b) = {{a , 1}, {b , 2}}
Wiener (1914):
(a,b) = {a,, {a , b}}
Kuratowski (1921) (a,b) = {{a}, {a, b}}
The modern definition of map
- which generalizes the concept of function -
was born in the frame of the structuralism
(Bourbaki, 1939)
A map is a triad (X,Y, F) of sets where F is a subset
of XxY satisfying the following conditions:
1) x  X  y Y : (x, y)  F
2) [ (x, y)  F , (x, y’)  F ]  y = y’
The elements of the sets can be objects of any
type, they are not necessarily numbers
In this definition ‘time and action’
as well as the intuitive concept of
‘corrispondence rule’ disappear
The modern concept of function involves new
mathematical concepts such as:
• the domain, the codomain
• the injectivity and surjectivity of a function
• The image of a function
• The composition of functions
• The conditions of invertibility of a function
• the algebraic structure for the bijective functions
on a set
• The algebraic structure of the set of the functions
on a field
• …
Influences
on the
teaching
The modern definition of function has been
introduced into teaching at secondary level in the
sixties, during the period of the New Math reforms.
This definition overlaps on the previous ones, and
generates several delicate didactical questions
about
the coordination between
 the old (procedural-dynamic) concept
 the modern (relational-static) concept
The need
 to distinguish between the concept (in its
different acceptations) and its
representations
 to coordinate different types of
representations (tables, verbal sentences,
algebraic formulas, sets-arrows, cartesian
graphs, parallel lines. graphs, sets of
ordered pairs ) and the related notations
poses other important
didactical questions and amplifies the
difficulties of the students
For an expert is not difficult to consider the
concept of function in all its aspects
(definitions, representations, conceptions) and
(s)he can shift from one to another
this is not true for the student
The student has not the necessary ability to
master all the different aspects
Often the prevalence in the mastering
of a specific aspect inhibits the
development of the other aspects.
Concept definition and Concept image:
the case of the function (Tall & Vinner 1981, Vinner,
1983, 1992)
Concepts and notions
The term “concept” refers to an idea or a thought
in our mind.
Usually, a concept has a name, which denotes it.
It is called the concept name, or the notion.
Thus
• the concept is the meaning of a notion;
• The notion is a lingual entity.
(It is a word or a word combination.)
(Vinner, 1983, 1992)
In Mathematics definitions are verbal and based
on primitive concepts or previous notions.
They never are circular.
When a notion is introduced to a certain person
(intuitively or by definition), his mind reacts to it.
Various associations are evoked.
They might be verbal, visual or even vocal
(additional senses may be involved). They can
be emotional as well.
All these associations which are not the formal
definition of the concept are called the
concept image
(Tall & Vinner 1991; Vinner, 1983, 1992)
To acquire a concept means to form for it a concept image.
This means:
 to have the ability to identify examples of the concept
(ex.: to identify rectangles in a set of various polygons)
 to have the ability to construct examples of the concept
(ex.: to write a specific polynomial of degree 3)
 to be aware of the typical properties of the concept
(ex. an altitude in a triangle can lie outside the triangle)
 to know the common ways to denote or to represent
the concept
(for instance, a function can be denoted : A  B)
(Vinner, 1983, 1992)
The formal definition of a concept forms its image.
It is a tool:
- to construct examples in our mind
- to identify examples of the concept.
When a task related to the concept is given to us,
the concept definition is evoked in our mind and
we use it in order to perform the task.
This does not exclude the possibility that the
concept image are evoked in our mind as well.
However, the ultimate source by means of which
we come to our conclusions is the concept
definition and not the concept image.
(Vinner, 1983, 1992)
The concept immage is shaped by the common
experience, the typical examples, class
prototypes etc.
When the common experience is limited
there is a the
fixation of the concept
If a person’s concept image of a function contains
only straight lines, parabolas, graphs of exponential
functions, then this person may say that the graph of
a function cannot present any jump
The only knowledge of the definition of a concept
does not imply the knowledge of the concept
concept definition and concept immage can be
incompatible with each other and coexist in the
student mind
compartimentalization phenomenon
In the case of the function it is very easy to
happen for:
• the different conceptions generated
in the time,
• the different representations involved,
• the type of teaching usually made at
school
Italian syllabuses
Junior secondary school (6th-8th grade I.e. 11-14 years old)
the functions are presented as modelization tools of
simple phenomena in the realm of the relationships
This implies the prevalence of a vision of function as
a rule of corrispondence between quantities that
can be represented algebraically
But usually at the school this topic is not
faced (mainly in a constructive way)
Upper Secondary School
It is organized in:
Biennium
(grades 9-10)
Triennium
(grades 11 - 13)
The Biennium Sillabuses
privilege the structuralist
concept of function
Elementary Algebra and
hints of modern algebra
The Triennium Sillabuses
go back to the EulerDirchlet concept of
function
(calculus)
In the teaching the notions of relationship and
function are simply added to the old Algebra
track without any care of the students’
experience and of the inner choerence as to
the global educative plan
(Malik, 1980)
…. “A survey of problems and a
pedagogically accettable theory for a first
course of calculus shows that Euler’s
definition covers all the functions used or
required in the course”
Misconceptions in students’ mental
prototipes for functions and graphs
Bakar & Tall,
1991
Both secondary and university students have wrong
mental images of functions
They do not consider graphs of functions the following
On the contrary they consider functions: a circle, a
parabola with its symmetry axis parallel to the axis ‘x’
The cartesian connection
Knuth 2000
(A point is on the cartesian graph of a line L if and only
if its coordinates satisfy the equation of L’)
Several students have not the ability to connect
algebraic and graphical representations in the double
direction.
This connection is limited to translations into the
equation-to-graph direction.
It has been shown that 1st year university students in
front a simple straight line in the cartesian plane do
not recognize that the coordinates of a point of the
line constitute a solution for the equation of the line.
(Batshelet 1971)
Linguistic constructs used in the past become
ambiguous expressions
For instance the phrases such as
“y is function of x”
“ the function varies between 0 and 1”
are not formally correct.
A function is a relation and it cannot have numerical
values. Moreover, as an eshablished relation, a
function cannot vary.
But these phrases are often used at the school, and
they can hardly be eradicated.
An important question
The interpretation of the writing y=f(x)
as a predicative (not necessarily calculative)
expression
‘the rule’ is a ‘ two-places open sentence’
its characterization as function depends on the
cartesian product set where we interpret it
When we write y = 2x+1 we have to think about the
true-set of this predicate in RxR. This true-set is
represented in the cartesian plane by the graph.
But the same formula characterizes another true-sets
when we interpret it in another cartesian product set,
for instance ZxZ.
(Grugnetti 1994)
The aspect connected with notations used in
school mathematics must not be underestimated.
In a didactical prespective, the notation y=f(x)
reflects the classical tendency to consider “f” as
an operative symbol of a procedure which
applied to x produces y.
In fact, when this notation is used, it risks unfairly
connecting formulae and functions: pupils do not
realize that not all functions are represented by a
formula.
The case of the ‘Real functions of real variable’
(the question of the domain)
Hershowitz, Arcavi & Eisemberg 1987
In school mathematics the majority of the functions are
only numerical ones: that is the functions of which
elements of domain and codomain are numbers and the
rule is espressed by a formula.
In solving exercises the students’ attention focuses on
the formula and so there is the risk that the pupil
identifies the function with this formula and she/he
does not realize the importance of the assignement
of domain and codomain
Usually in our teaching it is often neglected
the passage from the old concept of function to the
new one
Generally the didactical interventions focus on
assigned simple predicative formulas, without any
care to domain and codomain, and the properties of
the associate functions ( injectivity, invertibility etc)
x
Classical students’ misconceptions depends on this
lack of care.

For istance, the students conceive:
- 1/x as the derivative of lnx
- arcosin x is the inverse of sinx
- equal functions
y x
and
y  (x)
1
2
About the different symbolic representations of a
same object - An example
the identity function in R can be represented
 by the aritmetical operators: ‘+ 0 ‘ and ‘1’
 algebrically, by the equation: y = x
 geometrically: by the bisector of the I and III
quadrants in the cartesian plane
 by the modern notation
i : R R
i(x) = x
by the set  =
(x, x) x  R
It is not easy for a student to identify all these
representations overcoming their specific sense

Notations, logical and syntactical aspects - Marchini 1998
The symbolic language requires a care control of
the meanings of the writings
Two representations of the same function
(1)
f  N N
f   n, 2 n n  N 
(2)
f : N N
n  2n
Static notation
Dynamic notation
function actually given
function potentially given

The writing
f: n  2n
Embodies and hides
the universal quantifier
Marchini 1998

(1)
f  N N
f   n, 2 n n  N 
(2)
f  N Z
f   n, 2 n n  N 
Are the 
functions in the cases (1) and (2) equal?
 consider N  Z and we consider the
 YES, whether we
function as a set of ordered pairs
NO, whether we consider the function as a triad

(the codomains are different)
Marchini 1988
From the polinomials to the functions
In R(x), field of the quotients of the polinomials of
R[x], we have the equality
x 1
x 1 
x 1
2
But the functions usually
represented by
y  x 1
The two terms are
equivalent from
the syntactical
point of view
x 1
y
x 1
2
are different because their domains are different
The equality sign
Marchini 1998
In the writing y = f(x) on the left side there is a
variable, on the right side an (usually algebraic)
espression
The meaning of the sign’ =‘ is not the equality
Given the formulas y = f(x) ; y = g(x)
nobody will consider the formula f(x) = g(x)
The sign ‘=‘ idicates an assignation
y is ‘given by…’
Its use comes from history, it highlights that the
functions are originated by the equations
The ‘empirical’ functions
Phenomena are studied collecting tables of finite
sets of data. They can involve as variables :
 discrete quantities (measured by natural
numbers)
 continuous quantities (measured by rational
numbers)
In both the cases, usually the phenomena are
modelled by continuous functions, with an implicit
jump to the numerical ambit of the real numbers.
Very often this jump is not clarified to the students.
Very often the students do not accept a table of pairs
of data in term of function (the function has be an infinite set
of pairs and has to have a continuous graph)
In summary
The different stages of the development of the concept
of function have to be underlined in the teaching, with
all the variety of associate notations and meanings.
Task of the teacher
She/he has to bring the students through opportune
reflections and comparisons:
• to distinguish in which stage a certain activity
is posed and which concept of function it
involves;
• to recognize ambiguity and to interpret the
possible meanings
• to identify different representations of the
same object
• to shift among different representations with
flexibility
It is an hard task
and requires
a metacognitive
teaching
Students
conceptions
and
difficulties
Sierpinska (1988)
Inquiry among Polish students (15-17 years
age old) about their conceptions of function
Study realized in small group sessions of work
through discussions on specific didactical
situations.
Sierpinska states that in Poland the notion of
function is introduced to pupils 13 years age
old in its abstract form, through different
symbolic and iconic representations
Sierpinska underlines that
• the definition and the examples given say nothing to
pupils who know little maths and even less physics.
• the meaning of the term ‘function’
constructed by the pupils has nothing or very
little to do with the most primitive but
fundamental conception of function as
relationship between variable magnitudes
She stresses the fact that for the students:
- a function (as corrispondence) has to be ‘regular’;
- a function cannot be defined though different
formulas;
- the domain of a function cannot be constituted
by disjoint sets.
She classifies the students’ conceptions of function in
‘concrete’ and ‘abstract’ conceptions according to the
hystorical stage to which they appear to fit
‘Concrete’ conceptions
Mechanical conception: A function is a desplacement
of points fruit of a mechanical transformation
Synthetic geometrical conception. A function is a
‘concrete’ curve, i.e. a geometrical object, idealization
of a line on paper or a trajectory of a moving point
Algebraic conception. A function is a formula with
‘x’ , ‘y’ and equality sign; it is a string of simbols,
letters and numbers
Abstract conceptions
Algebraic conception
A function is an equation or an
algebraic expression containing variables; by putting numbers
in place of variables one gets other numbers. The idea that the
equation describes a relationship between variables is absent .
Analytic geometrical conception A function is an ‘abstract’
curve in a system of coordinates, i.e. the curve is a
representation of some relation; this relation may be given by
an equation and curves are classified according to the type
of this relation (first degree, algebraic, transcendental,… ). It is
not the relation that is called function, it is the curve itself
Physical conception A function is a kind of relationship
between variable magnitudes, some variables are
distinguished as independent, other are assumed to be
dependent of these, such relationships may sometimes be
represented by graphs
Sierpinska states the need of cooperation between
mathematics and physics teachers, because the
most important conception of function is that of a
relationship between variable magnitudes.
If this conception is not developed, a deviation
from the genetic line is made.
She states that introducing functions to young
students by their elaborate modern definition is a
didactical error . Borrowing the Freudenthal’ s
expression (1983) , she says that this is
an antididactical inversion
Vinner’ s study about the students’conceptions of the
function (1992)
Enquiry on 146 Israelian Students of 10 and 11 grades
frequenting selected and high qualified secondary schools
Questions
1. Is there a function which assigns to each number different
from 0 its square and to 0 it assigns –1?
2. Is there a function which assigns 1 to each positive number ,
assigns –1 to each negatiive number and assigns 0 to 0?
3. Is there a function the graph
of which is the following?
4. What is a function in your opinion?
Please, explain your answers to the questions.
The main concept images
(1) A function should be given by one rule
This is expressed in the following answers to question 1
Is there a function which assigns to each number different
from 0 its square and to 0 it assigns –1?
• No, because such a function should give also the
square of 0
• No, because if you take the square then the result
is positive
• No, because it contradicts the concept of
function
• No, 02 = 0 and not -1
(2)
If two rules are given for two disjoint domains we
are concerned with two functions.
This is expressed in the following answers To question 2:
Is there a function which assigns 1 to each positive
number , assigns –1 to each negatiive number and
assigns 0 to 0?
• No, there are three different functions. One of them
assigns + 1 to all positive numbers, the second assigns
- 1 to all negative numbers and the third one assigns 0
• No, because such a function is a constant function
but the constant should be the same all the way
through
(3) A function can be given by several rules relating
to disjoint domains provided these domains are
half lines or intervals. But a correspondence as
in question 1 (a rule with one exception) is still
not considered as a function.
(4) A graph of
“reasonably”.
a
function
should
behave
Many students denied the graph in question 3 to
be a graph of a function because it is not regular.
They claimed that a graph of a function should be
symmetrical, persistent, always increasing or
always decreasing, reasonably increasing, etc..
This is expressed in the following answers to question 3
• No, I do not think so. I always believed that a
function is something persistent
• No. Since a function is constructed by means of a
fixed equation it is impossible for it to increase in an
unproportional manner
• No. A function either increases or decreases but
here it is neither nor
• No. There is no constant relation between x and y
No. There is no regularity in the graph and therefore
there is no function that can describe this graph. It
might be an arbitrary correspondence from x to y
without any regularity
Categories and percentage of the Answers
Category 1 The textbook definition sometimes mixed with
elements of the concept image.
Category 2 The function is a rule of correspondence.
Category 3 The function is an algebraic term, a formula, an
equation, or an arithmetical operation
Category 4 Some elements in the concept image are quoted
in a meaningless way.
Distribution of Students’ Concept Definitions
(N = 146)
Category 1
Category 2
Category 3
Category
4
No
answer
57%
14%
14%
7%
8%
Examples of the answers
Category 1 The textbook definition
• It is a correspondence of a number belonging to one set of
numbers (the domain) to a number in another set (the
range). To each number in the domain corresponds only
one number in the range, but numbers in the range can
have several numbers in the domain. The function does not
have to have numbers.
Anything can do (concrete
objects, animals, etc.)
• Every point in the domain has a point in the range
• Function in my opinion is that every x has one number
or one object in y but not vice versa
Category 2: The function is a rule of
correspondence.
(This eliminates the possibility of an arbitrary correspondence.
A rule and an arbitrary correspondence are contradictory. In
addition to the word “rule” the students also used the words “
law”, “ relation”, “ dependence between variables “, etc.)
• It is a relation between two sets of numbers based on a
certain law
• It is taking something and changing it by means of a
constant process determined by the specific function in
consideration
• A function is a method to obtain from one set of
numbers another set by means of a certain rule. I
denied 3 (in the questionnaire) from being a
function because generally in everyday life
applications, a function has a definite rule. It is not
defined for each point separately
Category 3: The function is an algebraic term, a
formula, an equation, or an arithmetical operation.
• A function is a set of numbers such that when
performing on them a certain arithmetical
operation we obtain another set of numbers, which
is functional to the first, set
• A function is an equation that has a range at one
side and a domain at the other side. To each
number (or factor) in the domain corresponds a
factor in the range
• A function is something like an equation. When
you put numbers instead of the unknown you get a
solution
Category 4: Some elements in the concept image are
quoted in a meaningless way.
The student says some words or picture associated with the
notion of function but he does not show understanding of
the concept.
• A function is a curved line in a coordinate system such
that to every point it corresponds exactly one point
• Values of one graph y depending on another graph
x according to y=f (x)
• A function is an expression corresponding elements
from one set to another under the condition that there
will not be more than one arrow for each number
Recent trends
The approach to the functions has to be faced
in a pervasive and gradual way
since the early stages of school
through realistic activity
In compulsory school high attention has to be
given to the esploration of situations and the
observation of the co-variance of two
magnitudes chosen among various ones
All the main stages has to be acrossed:
Eulerian approach; Dirichelet approach;
Modern approach
the students have to be brought
- to express an observed correspondence law,
possibly before in words, after using different
registers of representation:
arrows, tables, formulas, graphs, set of
ordered pairs, …
In this way they can:
• build a flexible and articulated concept image of
function
 gradually arrive at:
- conceiveing the functions as special sets of
couples in the frame of the binary relationships;
- understanding its formal definition.
In particular the young studens have to be
brought to manage the coordination of different
representations of a same relationship
 dwelling upon the interpretative aspects
of the representations
 highlighting in the algebraic sentences or in
the Cartesian graphs the representative
elements of
subject, predicate, object
of the verbal sentences that they translate
This facilitates the interpretation of
the writing y = f(x) in term of
a two-places predicate
Usual mistake
in interpreting a cartesian graph of a function
given by a rule y = f(x) the students say that
x is the subject of the sentence
Th i nk i t th ro ugh … ….
Here are some sets of geometric patterns, graphs and rules.
Find the correct match.
Geometric patterns
Graphs
20
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0
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Rules
1)
2)
N° of white t iles = N° black t iles + 1
3)
N° white t iles = N° black t iles x 3
4)
N° white t iles = 4
No matter how many black t iles.
N° black t iles = 1
No matter how many white tiles.
5)
N° white t iles = (N° black t iles x 2) +2
6)
N° white t iles = N° black tiles x (N° black t iles – 1)
Rules and letters
a) m = (n x 2 ) + 2
b) m = 4
c) m = n x ( n – 1 )
d) m = n x 3
e) n = 1
f) m = n + 1
These activities constitutes the ground to
motivate the study of the ‘object’ function in itself
and opens the way to approach and understand
its modern definition
we faces this range of aspects in
Unit 9 ‘Verso le funzioni’
(‘towards functions’) of the ArAl Project
ArAl project
www.aralweb.it
(Malara & Navarra 2003)
aimed at
an early approach to
algebra as language
for modelling, solving problems
and proving
An interesting new kind of activity
In these last years the study and the use of
the functions has been promoted not only in
modelling, but also
 for interpreting functional relationships
represented by graphs as to a given
phenomenon and for taking some
pieces of information on it
For assessing the fitness of different
graphs as to a phenomenon through
their interpretation and comparison
Several activities of this kind have been used in
the OCSE-PISA test
PISA question 2003
The figure shows a water tank. Its dimensions are shown in the
diagram. At first the tank is empty, then it is filled with water at
the rate of 1 litre per second.
 Which of the following graphs shows the change of the
height of the water level over time?
 Explain the reason why you chose it.
 Explain the reasons why you didn’t choose the others.
At the moment - in the ambit of the
PDTR project - we are following
some experimentations in grade 68 of three didactical paths on:
- modelization of realistic
situations
- interpretation of graphs
- sequences as functions
Fundamental appears in the teaching
- not only for the concept of functionthe consideration of
the epistemological
dimension
until now generally absent
Thank you for
your attention
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