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

1. 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

2. Points in the discussion •  Some epistemological remarks • some consequent didactical questions  Students’ conceptions and difficulties • new teaching trends

3. Epistemological remarks

4. 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

5. 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.

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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.

12. 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

13. 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

14. I 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.

15. 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)

16. A map is a triad (X,Y, F) of sets where F is a subset of XxY satisfying the following conditions:  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

17. 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 • …

18. Influences on the teaching

19. the coordination between  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

20. 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 didacticalquestions and amplifies the difficulties of the students

21. 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.

22. 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.)

23. (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

24. (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)

25. (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.

26. (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 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

27. 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

28. 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(mainlyin a constructive way)

29. It is organized in: 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

30. 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”

31. 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’

32. Knuth 2000 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.

33. (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.

34. 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 instanceZxZ.

35. (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.

36. 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

37. and - 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

38. 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

39. (1) (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

40. (1) (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)

41. 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

42. Marchini 1998 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

43. The ‘empirical’ functions 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)

44. 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

45. It is an hard task and requires a metacognitive teaching

46. Students conceptions and difficulties

47. 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

48. 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.

49. 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

50. 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 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