Nicolina A. Malara, Mathematics Department, Modena & Reggio E. University, Italy

THE CONCEPT OF FUNCTION

Epistemological remarks, didactical questions, Students’ conceptions

and difficulties, new teaching trends

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.

The roots of these definitions are linked with

the exploration of curves

Its first definitions appear between the end of XVII century and the beginning of the XVIII century

Curves were not regarded as graphs of relationships between these ausiliary segments. They were taken for what they appeared

to our eyes:

Curves were initallly described by proportions between some auxliary segments (diameter, axis,…) in the realm of specific problems

(Fermat, Descartes, Newton, Lebnitz…)

• geometrical objects

• trajectories of moving points.

In the first definitions, the functions are conceived as

analytic expressions

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

Function of a variable quantity is a quantity composed in watever

manner of this variable and constant quantities

J. Bernoulli (1718) ‘Remarques sur ce qu’on a donné jusqu’ici de solutions des problemes sur les isoperimetres’

Eulero (1748) ‘Introductio in analysis infinitorm’

Functio quantitatis variabilis est expressio analytica quomodocumque composita ex illa quantitate variabili et numeris seu

quantitatibus constantibusA function of variable quantity is an analytic expression composed

in a whatever manner by this quantity and by numbers or

constantsJ. 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

Eulero conceives that the value of a variable can range from Naturals to Complex Numbers

A variable quantity is an indeterminate quantity, or an universal quantity, which

includes all the determinate values

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

• The famous polemics among Euler, d’Alembert, Daniel Bernoulli concerning the problem of vibranting string;

These questions have brought Dirichlet to formulate a more general definition

of function

• The development of the theory of trigonometric series by Fourier;

• The notion of continue function (Chauchy, Dirichlet, Abel, Bolzano, Weierstrass,… )

We simply quote:

Dirichlet (1837)

If a variable y is so related to a variable x that whenever a numerical value is

assigned to 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.

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

Both constructivists and intuitionists as well as formalists were against it, albeit for very different reasons:

The Dirichlet definition was widely accepted and used up to the middle XXth century.

• 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 idea to refer the function definition to a ‘new’ (as to

arithmetic) primitive notion become inevitable

However it started to provoke dicussions in fondationists’ circles already at the turn of the XIX century

• The latter considered the definition not sufficiently rigorous

The Peano definition (1911) ‘Sulla definizione di funzione’

In tuning with the theory of the relations by Russell & Whitehead (1910, Principia Mathematica)

The question is transferred to the concept of

ordered pair

assumed by Peano as a primitive concept

Peano reduces the concept of function to the one of the relation and introduces the notion of

univocal relation

Sierpinska (1988) summarizes these first stages of the development of the concept of function in the following scheme

I II III

IV V VI

An implicit idea of transformation (T) of points or relationships between magnitudes

T described by

numerical tables

T described by proportions

T described by graphs

and

equations

T described by

equations

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’

In this definition ‘time and action’ as well as the intuitive concept of ‘corrispondence rule’ disappear

The elements of the sets can be objects of any type, they are not necessarily numbers

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

aboutthe 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

.

poses other important didactical questions and amplifies the

difficulties of the students

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

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

Often the prevalence in the mastering of a specific aspect

inhibits the development of the other aspects.

this is not true for the student

The student has not the necessary ability to master all the different

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

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)

(Tall & Vinner 1991; 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.

(Vinner, 1983, 1992)

However, the ultimate source by means of which we come to our conclusions is the concept definition and not the concept image.

The concept immage is shaped by the common experience, the typical examples, class prototypes etc.

(Vinner, 1983, 1992)

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

When the common experience is limited there is a the

fixation of the concept

In the case of the function it is very easy to happen for:

The only knowledge of the definition of a concept does not imply the knowledge of the conceptconcept definition and concept immage can be incompatible with each other and coexist in the student mind

• the different conceptions generated in the time,

• the different representations involved,

• the type of teaching usually made at school

compartimentalization phenomenon

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 relationshipsThis 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)

The Biennium Sillabuses privilege

the structuralist concept of function

Elementary Algebra and hints of modern

algebra

Upper Secondary SchoolIt is organized in:

Biennium (grades 9-10)

Triennium (grades 11 - 13)

The Triennium Sillabuses go back

to the Euler-Dirchlet concept of function

(calculus)

(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”

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

Misconceptions in students’ mental prototipes for functions and graphs

Both secondary and university students have wrong mental images of functions

On the contrary they consider functions: a circle, a parabola with its symmetry axis parallel to the axis ‘x’

They do not consider graphs of functions the following

Bakar & Tall, 1991

The cartesian connection

Several students have not the ability to connect algebraic and graphical representations in the double direction.

(A point is on the cartesian graph of a line L if and only if its coordinates satisfy the equation

of L’)

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.

This connection is limited to translations into the equation-to-graph direction.

Knuth 2000

(Batshelet 1971)

Linguistic constructs used in the past become ambiguous expressions

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.

For instance the phrases such as

“y is function of x”

“ the function varies between 0 and 1”

are not formally correct.

An important question

The interpretation of the writing y=f(x)as a predicative (not necessarily calculative)

expression

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.

its characterization as function depends on the cartesian product set where we

interpret it

‘the rule’ is a ‘ two-places open sentence’

(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.

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.

The case of the ‘Real functions of real variable’ (the question of the domain)

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) Classical students’ misconceptions depends on this lack of care.

x

y x- equal functions and

y(x)1

2

For istance, the students conceive:

- 1/x as the derivative of lnx

- arcosin x is the inverse of sinx

About the different symbolic representations of a same object - An example

It is not easy for a student to identify all these representations overcoming their specific sense

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

Notations, logical and syntactical aspects - Marchini 1998

The symbolic language requires a care control of the meanings of the writings

The writing

f : n 2nEmbodies and hides

the universal quantifier

Two representations of the same function

Static notation

function actually given

Dynamic notation

function potentially given

(1)

f NN

f n,2n ) n N

f : N N

n 2 n(2)

(1)

(2)

f NN

f NZ

f n,2n ) n N

f n,2n ) n N

Are the functions in the cases (1) and (2) equal?

YES, whether we consider N Z and we consider the function as a set of ordered pairs

NO, whether we consider the function as a triad

(the codomains are different)

Marchini 1998

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

The two terms are equivalent from the syntactical point of view

x 1x2 1x 1

are different because their domains are different

But the functions usually represented by

yx 1

yx2 1x 1

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 summaryThe 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.

• 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

She/he has to bring the students through opportune reflections and comparisons:

• to shift among different representations with flexibility

Task of the teacher

It is an hard task

and requires

a metacognitive

teaching

Students conceptions and difficulties

Sierpinska (1988)

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

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.

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.

Sierpinska underlines that

• 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

• the definition and the examples given say nothing to pupils who know little maths and even less physics.

‘Concrete’ conceptions

Mechanical conception: A function is a

desplacement of points fruit of a mechanical

transformation

She classifies the students’ conceptions of function in ‘concrete’ and ‘abstract’ conceptions according to the hystorical stage to which they appear to fit

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 itselfPhysical 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.

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

If this conception is not developed, a deviation from the genetic line is made.

Vinner’ s study about the students’conceptions of the function (1992)

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.

Enquiry on 146 Israelian Students of 10 and 11 grades frequenting selected and high qualified secondary schools

The main concept images

(1) A function should be given by one rule

This is expressed in the following answers to question 1Is 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 a function should behave “reasonably”.

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%

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.)

Examples of the answers

• 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 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

• 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

Category 4: Some elements in the concept image are quoted in a meaningless way.

• A function is a curved line in a coordinate system such that to every point it corresponds exactly one point

The student says some words or picture associated with the notion of function but he does not show understanding of the concept.

• 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

Think it through …….

Here are some sets of geometric patterns, graphs and rules.

Find the correct match.

Geometric patterns

Graphs

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Rules

1) 2)

3) 4)

5)

6)

N° of white tiles = N° black tiles + 1 N° white tiles = N° black tiles x 3

N° white tiles = 4No matter how many black tiles.

N° black tiles = 1No matter how many white tiles.

N° white tiles = N° black tiles x (N° black tiles – 1)

N° white tiles = (N° black tiles x 2) +2

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

aimed at

an early approach to algebra as language

for modelling, solving problems and proving

www.aralweb.itArAl project(Malara & Navarra 2003)

Unit 9 ‘Verso le funzioni’

(‘towards functions’) of the ArAl

Project

we faces this range of aspects in

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

Several activities of this kind have been used in the OCSE-PISA test

For assessing the fitness of different graphs as to a phenomenon through their interpretation and comparison

An interesting new kind of activity

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.

PISA question 2003

At the moment - in the ambit of the PDTR project - we are following some experimentations in grade 6-8 of three didactical paths on:

- sequences as functions

- modelization of realistic situations

- interpretation of graphs

Fundamental appears in the teaching - not only for the concept of function- the consideration of

the epistemological dimension

until now generally absent

Thank you for your attention

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graphs, proc PME 14, vol.1, 104-111Breidenbach, D., Dubinsky, E., Hawks, G., Nichols D.: 1992,

Development of the Process Conception of Function, Educational Studies in Mathematics, 23, 247-285

Eisemberg, T.: 1991, Functions and Associated Learning Difficulties, in Tall D., Advanced Mathematical Thinking, Kluver, Dordrecht, 140-152

Even, R.: 1993, Subject-Matter Knowledge and Pedagogical Content Knowledge: Prospective Secondary Teachers and the Function Concept, Journal of Research in Mathematics Education, 24, 94-116

Grugnetti, L., 1994, Il concetto di funzione: difficoltà e misconcetti, L’educazione Matematica, anno XV, serie IV, vol 1, 173-183

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Herschkowitz,R., Arcavi, A., Eisemberg, T., 1987, Geometrical Adventures in Functionland, Mathematics Teacher, vol 80., n. 5, 1987

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Marchini, C., 1998, Analisi Logica della funzione, Gallo, E. & Al. (eds) Confe-renze e Seminari Associazione Subalpina Mathesis, 1997-1998, 137-157

Schwartz G., Dreyfus T, 1995, New Actions upon Old Objects: a New Ontological Perspective on Functions, Educational Studies in Mathematics, 29, 259-191

Sierpinska. A. 1988, Epistemological Remarks on Functions, Proc. PME 12, Vesprem Hungary, vol 2, 568-573

Slavit D, 1997, An Alternate Route to the Reification of Function, Educational Studies in Mathematics, 33, 259-281

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