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

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Epistemological remarks, didactical questions, Students’ conceptions and difficulties, new teaching trends

Nicolina A. Malara,

Mathematics Department,

Modena & Reggio E. University,

Italy

- Some epistemological remarks

- some consequent didactical questions

Students’ conceptions and difficulties

- new teaching trends

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 conceptionof 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

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 aprimitive concept

II

III

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

T described by numerical tables

T described by proportions

V

VI

IV

T described by graphs and equations

An elaborate explicit idea of relationships between variables

T described by equations

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

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:

x X yY : (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
- …

the old (procedural-dynamic)concept

the modern(relational-static) concept

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

- 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

didacticalquestions 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.)

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)

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.

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

fixationof 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

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(mainlyin a constructive way)

Biennium

(grades 9-10)

Triennium

(grades 11 - 13)

The Triennium Sillabuses go back to the Euler-Dirchlet concept of function

(calculus)

Upper Secondary School

The Biennium Sillabuses privilege the structuralist concept of function

Elementary Algebra and hints of modern algebra

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”

They do not considergraphs offunctions the following

Misconceptions in students’ mental

prototipes for functions and graphs

Bakar & Tall, 1991

Both secondary and university studentshavewrong mental images of functions

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

The cartesian connection

(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.

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.

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 instanceZxZ.

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

- equal functions

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.

For istance, the students conceive:

- 1/x as the derivative of lnx

- arcosin x is the inverse of sinx

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 =

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

(2)

Embodies and hides

the universal quantifier

The writing

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

Static notation

function actually given

Dynamic notation

function potentially given

(2)

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

Marchini 1998

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)

The two terms are equivalent from the syntactical point of view

But the functions usually represented by

are different because their domains 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

The equality sign

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

Phenomena are studiedcollecting 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)

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

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

- 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

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 conceptionA 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

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?

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.

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

(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 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 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

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 tothe 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

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

www.aralweb.it

ArAl project

(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

- 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 6-8 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 function- the consideration of

the epistemological dimension

until now generally absent

Bakar, M., Tall, D., 1991, Students’ mental prototypes for functions and graphs, proc PME 14, vol.1, 104-111

Breidenbach, 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

Harel, G., Dubinsky, E. (eds): 1992, The Concept of Function – Aspects of epistemology and Pedagogy, MAA Notes, vol. 25

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

Janvier C., 1998, The Notion of Chronicle as an Epistemological Obstacle to the Concept of Function, Journal of Mathematical Behaviour, 17, (1), 79-103

Knut, E. J., 2000, Students understanding of the cartesian connection: an exploratory study, Journal for Research in Mathematics Education, 31,4

Malik, M.A., 1980, Historical and Pedagogical aspects of the definition of function, International Journal of Mathematics Education in Science and Technology, 11, 4, 489-49

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

Vinner, S.: 1983, Concept Definition, Concept Image and the notion of function, the International Journal of Mathematics education in Scence and Technology, 14, 293-305

Vinner, S.: 1992, The Function Concept as a Prototype for Problems in mathematics Learning, in Harel, G., Dubinsky, E. (eds): 1992, The Concept of Function – Aspects of epistemology and Pedagogy, MAA Notes, vol. 25, 195- 214

Youschkevitch, A.P.: 1991, Le concept de function jusq’au milieu du XIXe siecle, Ovaert, J.L. & Reisz, D. (eds) Fragments d’histoire des mathematiques, A.P.M.E.P., n. 41, 4-68

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