secondary school and university concerning the field of functions. This area has been ..... Annales de didactique et de sciences cognitives, 5, 37-65. Gueudet, G.
STUDENTS CONCEPTIONS OF FUNCTIONS AT THE TRANSITION BETWEEN SECONDARY SCHOOL AND UNIVERSITY Fabrice Vandebrouck Université Paris-Diderot, Laboratoire de Didactique André Revuz In this paper, we would like to stress the importance for students entering at university to deal with points of view on functions. We hope that this distinction enriches the process / object duality and the way of thinking the passage from the conceptual embodied world to the formal axiomatic one. Through a typical task of the transition between secondary school and university, we pointed out the difficulties for students to solve tasks when algebraic techniques are not sufficient. Key words: mathematics, functions, university students, concept image, point of view In this article we want to investigate one problem that arises in the transition between secondary school and university concerning the field of functions. This area has been extensively studied for several decades but the context of the teaching situation, always changing, justifies that the interest is continuously renewed. SPECIFICITIES OF THE TRANSITION BETWEEN SECONDARY SCHOOL AND UNIVERSITY Many studies have already indicated the characteristics of this transition (Artigue 1991, 2007), (Gueudet, 2008). From an institutional perspective, many macro breaks can be identified: a shift from a course with one teacher to lectures in amphitheatre and tutorials, acceleration of teaching time with a rapid turnover of objects and lessons with faster time of assimilation, shorter presence of teachers, lake of classical problems from secondary school, wider range of tasks that makes their internalization much more difficult than in secondary school, the latter being delegated to personal work of students, who must therefore be more autonomous in their learning. Robert (1998) also noted a distribution between the types of mathematical notions which is different from secondary school to university, especially the emergence at university level of new mathematical notions carrying a high level of formalism and generalization. She also pointed out the differences in the level and the nature of tasks (necessity of available knowledge, necessity of flexibility in this knowledge, for instance use of different settings and representation systems, new requirements in term of proofs at the university level…). THE COMPLEX NOTION OF FUNCTION The notion of function is at the intersection of several mathematical fields (real numbers, limits, algebra, etc…), appears in many frames (Douady 1986) and requires the consideration of several representation systems (graphical, algebraic, symbolic
etc…) (Duval 1991). Functions are therefore complex objects which are still being learnt when students enter university. The teaching and learning of functions has been studied through many different theories: Tall and Vinner (1981) introduce the distinction between concept image and concept definition, the concept image being generally at variance with the concept definition, especially for functions. According to Bachelard (1938), Sierpinska (1992) use the notion of epistemological obstacles regarding some properties of functions and especially the concept of limit. Another approach is based on the processes / object duality (Dubinsky 1991): the conceptualization begins with actions on previously constructed mental or physical objects, then actions are interiorized to form processes that are then encapsulated to form objects. Sfard (1991) also claims that the abstract concepts can be conceived of two different forms: structurally, as objects and operationally, as processes, the two views being complementary. Tall (1996) also add the perspective of procept, an amalgam of two components: a process that produces an mathematical object and a symbol that represents at the same time the process and the object. With respect to functions, algebraic or graphical representations are procepts which can both be handled as processes and as objects. Finally, Tall (2004) characterizes the evolution of three worlds of mathematics under a perspective that shows the cognitive growth of the mathematical thinking, from a conceptual-embodied world to a proceptual-symbolic world and then to a formalaxiomatic world. As regard the teaching of functions, we claim that the transition between secondary school and university can be interpreted, in some sense, as a way to move from the conceptual-embodied world to the formal axiomatic one, embedding a higher level of conceptualization of the notions related to the domain of analysis: indeed, the beginning of the teaching at the university level corresponds to a displacement from functional thinking to set-theoretical thinking, a balancing from most of functions’ utilizations as processes to utilizations as objects, a high degree of formalisation and a balancing in the utilization of procepts, especially the use of graphical representations moves from an object’s role to a tool for supporting formalizations and proofs. On the other hand, Balacheff and Gaudin (2002) find two types of concept image (they speak about conceptions) among pairs of students who finish secondary school: a concept image “curve – algebraic” for which functions are primarily special cases of plane curves, those having a specific algebraic form and a concept image “algebraic – graphic” for which a function is first an algebraic formula, the associated graph coming after. It seems that average students fall into the second categorization. They are unable to exploit functions which are not given in algebraic forms. Moreover, these students cannot easily shift from the algebraic representation system to another and Coppé et al. (2007) stress that the current practice of teaching in secondary schools (in France) seems to reinforce the idea that a function is only an object belonging to the algebraic frame.
THREE POINTS OF VIEW ON THE OBJECT FUNCTION To complete and to investigate more students’ concepts image about functions, we consider the notion of points of view (Rogalski, 2008). More precisely, we claim that, in the case of function, different points of view can be adopted on the objects: a point-wise point of view, a global point of view and a local point of view. This distinction enriches the different levels of conceptualization introduced above. Indeed, the balance form conceptual embodied world to the formal axiomatic one is accompanied by the development of local properties about functions: limit, continuity, derivability, equivalent expressions, Taylor’ s local expansions near some points which are the basic notions of calculus. We claim that working at university level on functions implies that students can adopt a local point of view on functions whereas only the point-wise and global points of view are constructed at the secondary school. In this paper, we would like to explain difficulties of students entered at university by their difficulties to adopt point-wise and global points of view on functions. These difficulties appear when they are asked to solve tasks where techniques of the algebraic frame are not sufficient. Let us recall what we understand with this notion of point of view. In the first point of view, functions are considered as correspondences between two sets of numbers, an element of the first set being associated with a unique element of the second set. This point-wise point of view is in accordance with the definition of functions given in textbooks at grade 9 in France, four years before the beginning of university. At this level, functions are represented by arithmetic formulas that operate as a program for calculation, such as calculus programming. A table of values is also a good representation of a function from this point of view, especially for pupils who consider only integers on the real line. The second point of view is the global one, necessary to understand the notion of variation and to interpret properties such as parity or periodicity. As pointed out by Coppé et al. (2007), the table of variation is a good representation of a function from this global point of view whereas the algebraic expression or the graphical representation of a function can exploit and can be exploited with a point-wise point of view as well as a global point of view. While the point-wise point of view seems relatively closed to the process level on functions, we assume that reaching the object level is not the only challenge of the secondary school. In fact, we assume that students must be able to adopt different points of view on functions and have to articulate point-wise and global points of view on functions to overcome all the obstacles which come with local notions at the beginning of the university: structure of the real line, notion of equality between two real numbers... For instance, the use of procepts should allow easy connections between points of view. However, a large algebraisation of tasks at the end of the secondary school tends to erase the point-wise and global point of view which can be adopted on functions and it doesn’t permit to reach the local point of view, even if the
notions of limit, continuity and derivability are introduced in an algebraic way at the end of secondary school. A TEST TO TRACK ABILITIES FOR ADOPTING POINTS OF VIEW As we would like to stress the difficulties for students to adopt point-wise and global points of view on functions, we designed a task for which techniques of the algebraic frame were not sufficient to solve the task. More precisely, this task was dealing with the function G below, defined by an integral:
In secondary school, as well as in the beginning of the university, integrals are defined by areas under curves of functions, that is to say the approach via definite integrals. However, in the two institutions, the link between integrals of continuous functions and primitive is made quickly and students work mostly in the algebraic frame. For this link, the so called fundamental theorem of calculus is usually proved by the teacher, when the function to integrate is continuous, positive and strictly growing. We decided to investigate this function G because in the two institutions its study is very close to well known tasks and its level of difficulty is accessible for both students of secondary school and students of university. Indeed, even if this kind of function G is not usual in their previous curriculum, students of the both institutions have already made tasks concerning indefinite integrals over intervals of the form [a, x] or [a, β(x)] (β being a linear function) at the moment of the experimentation. Moreover, questions about global and point-wise properties of G according to properties of f are more interesting with this kind of integral (between x-1 and x+1), as we will see below. The experimentation was dealing with one group of students from secondary school (15 students) and one group of students form university level (109 students from University Paris Diderot). The precise statements proposed to students were chosen by their own teachers (from one side in a secondary school and on the other side in the university) with instructions for treating questions concerning global or/and pointwise properties of G. The beginning of the statement given by the teacher at the university level was the following one (a fifth question concerns a local property of limit) : Let f be a continuous function over R and G the function defined over R by
1) Show that if f is a constant function, G is also a constant function. 2) Show that if f is even (respectively odd), the function G is even (respectively even).
3) Show that the function G is derivable and computer G’. 4) Explicit the function G when f is the function defined by f(t)=|t|.
The function f is assumed to be continuous. Students have to show global properties on G. They must also prove that G is derivable and compute G’. Questions 1) and 3) can be treated only in the algebraic frame without any point of view on functions f and G. Problems with points of view can appear with question 2) – if f even then also G is even – and for question 4) – find G when f is the absolute value. We will see this below with examples of productions. The statement given by the teacher at the secondary school level was the following: In the problem, D means the set of derivable functions on R. For each function f in D, one define the function G such that for all x in R,
1a) Show that for each primitive F of f over R, G(x)= ½ [ F(x+1) – F(x-1)]. 1b) Computer G when f is defined by f(t)=tn, n integer greater than 1. Show that if f is a polynomial function, than G is a polynomial function with the same degree. 1c) Computer G when f is defined by f(t)=cos(πt). 2a) Show that for all f in D, G is also derivable over R, and that for all x in R, G’(x) = ½ [ f(x+1) – f(x-1)]. 2b) Show that the following properties are equivalent: (1) G is constant and (2) f is periodic with period 2. 3a) Suppose f is growing over R. Show that G is growing and that for all x in R, f(x-1) ≤ G(x) ≤ f(x+1). 3b) Suppose f is defined by f(t)= 4 exp(t) / [ t2+4]. Study the variations of f over R. Deduce the variations of G over R.
In this statement, students can work inside the algebraic frame from questions 1) to question 2a) and also for question 3b). That’s mean the conception of functions as objects belonging to a functional frame (with all its complexity) seems to be not useful. Problems we want to study can appear with question 2b) – G is constant if and only if f is 2 periodic - and 3a) – if f grows than G grows and for all x, f(x-1) ≤ G(x) ≤ f(x+1). We will also see this below. Let us note some specificities of the translation between secondary school and university level through these two statements: first of all, there exists some examples – question 1b) f is a monomial, f is a polynomial, question 1c) f(t)=cos(πt) – at secondary school whereas university’ students are supposed to treat the general case directly (formal register). Secondly, the task of computation for G’ is cut into two subtasks for secondary school students: question 1a) and question 2a) whereas university students are supposed to compute G’ directly.
RESULT OF SOME STUDENTS AND EXAMPLE OF PRODUCTIONS As we were interested by the transition between secondary school and university level, we have chosen to analyze only productions of secondary school students who were expected to enter university. Only five productions were analyzed. On the other hand, as we were interested only about qualitative results, the analysis of students’ productions at university level was done in a second time, in order to find the characteristics which have been identified in the five productions of school students. As it was expected, most of secondary school students’ difficulties were about question 2b) and question 3a). Only one student succeeded on these two questions whereas, except some minor errors and the second part of the question 1b) (which is more difficult), all of them succeeded the other questions. We suppose that algebraic techniques are not sufficient to succeed these tasks. There is a necessity to surpass the algebraic frame and to adopt global and point-wise points of view on f and on G. For instance, in question 2b), students have to establish global properties on f and G – f is 2 periodic and G is constant - through a point-wise property for f – for all x, f(x-1) = f(x+1). Here is a typical production: (1) G is constant Ù G ’(x)=0 (2) f is 2 periodic Ù f (x)=f (x+2) We start from one member to go to the other: G’ (x) = ½ [f(x+1)-f(x-1)] and G’(x)=0 ... Ù f(x+1) = f(x-1) Ù f(x+1) + f(1) = f(x-1) + f(1) Ù f(x+2) = f(x) (2) so (1) Ù (2). Figure 1: example of production for question 2b) – secondary school student
This student explicates his procedure: « on part d’un membre pour arriver à l’autre membre » (« we start from one member to go to the other »). There is no quantification, useful to translate the global properties for f and G in the formal language. Moreover, equivalences are wrong. The student can’t recognize the property of periodicity with the statement f(x-1) = f(x+1). It is necessary for him to formulate f(x) = f(x+2). Then, all algebraic techniques seem to be good: here he adds + f(1) in each member. We suppose that this student is unable to reach a global point of view on f and G. His reasoning seems to be in the algebraic frame only. On the first part of question 3a), students have to establish a global property on G – G is growing – from a global property on f – f is growing – through point-wise properties – for all x, f(x-1) < f(x+1) and for all x, G’(x) > 0. But in four of the five
students’ productions, these translations are again done formally without any quantification, in a formal and algebraic way. Students seem unable to see the necessity of two variables x and y in order to write the property of growth. They use equivalences which are wrong. Again, we think that the reasoning is only at an algebraic level, not at all in the functional frame.
3a) We know that f is growing.
3a) G is derivable over R (2a)
So f(x+1) > f(x) > f(x-1)
x+1 > x-1
Ù f(x+1) > f(x-1)
Ù f(x+1) > f(x-1) because f is growing
Ù f(x+1) - f(x-1) > 0
Ù f(x+1) – f(x-1) > 0
Ù ½ [f(x+1) - f(x-1)] > 0
Ù ½ [f(x+1) - f(x-1)] > 0
Ù G’ (x) > 0
Ù G’ (x) > 0
So G is growing.
G’ (x) > 0 over R, so G is growing over R.
Figure 3: examples of productions for question 3a) – secondary school students
The second part of the question 3a) – to prove f(x-1) ≤ G(x) ≤ f(x+1) - corresponds to the most difficult task. No student really succeeded this question. The difficulty seems to be linked to the necessity to adopt a point-wise point of view on G - the computation of G(x) for a fixed x – together with a global point of view on f – for all t in [x-1,x+1], f(x-1) < f(t) < f(x+1). This difficulty with point of view appears also in many productions of students from the university level, concerning questions 2) and question 4) of the university statement, as it was expected. In question 4) (university statement), there is a necessity to treat several cases according to the fact that 0 belongs or not to [x-1, x+1], that is to say x < -1, x in [-1, 1] or x > 1. Students must adopt a point-wise point of view on G – computation of G(x) for x fixed – and a global point of view on f over [x-1, x+1]. However, many students treat the task at an algebraic level, thinking for instance that the absolute value can be integrated without adopt these points of view. Figure 4 represents a typical example of this mode of reasoning in the algebraic frame:
4) For f(t)=| t | We have ∫ | t | = | t2 / 2 | => ∫ x-1 x+1 f(t) dt = | (x+1)2 / 2 |-| (x-1)2 / 2 | So G(x)= ½ [ | (x+1)2 / 2 |-| (x-1)2 / 2 | ]
Figure 4: examples of production for question 4) – university student
In students’ responses for question 2) (university statement), the same kind of observations can be made. Most of students translate the global property - f even without quantification - f(t) = f(-t). Again, the reasoning seems to be in the algebraic/formal frame in many productions as in figure 5:
In the case f even:
If f even then f(t)=f(-t)
f(t)=f(-t) so ∫ x-1 x+1 f(t) dt = ∫ x-1 x+1 f(-t) dt
G(-x)=
Ù ½ ∫ x-1 x+1 f(t) dt = ½ ∫ x-1 x+1 f(-t) dt
If f odd then f(-t)=-f(t)
so G(x)= ½ [F(x+1) - F(x-1)] = G(-x)= ½ ∫ x-1 x+1 f(-t) dt ½ [F(-(x+1)) - F(-(x-1))] = ½ ∫ x-1 x+1 - f(t) dt = - ½ ∫ x-1 x+1 f(t) dt = ½ [F(-x-1) – F(-x+1)] = G(-x) G(-x) = - G(x) So G is also even. Figure 5: examples of productions for question 2) – university students
Few productions (about 25%) show the ability for students to adopt point wise as well as global point of view on the manipulated objects. Because of the brevity of this paper, it is impossible to report about them.
CONCLUSION In this paper, we wanted to stress the importance for students entering at university to deal with point wise and global points of view on functions. We have claimed that this distinction enriches the process / objet duality and the way of thinking the passage from the conceptual embodied world to the formal axiomatic one. Through a typical task of the transition between secondary school and university – the study of the function G - we have pointed the difficulties for students to solve tasks when algebraic techniques are not sufficient. On one hand, we think that these difficulties are linked to the non ability for students to consider functions as complex objects with point-wise as well as global properties. On the other hand, we can think that these difficulties are increased by the current practice of teaching in secondary schools in France, which reinforces tasks belonging to the algebraic frame only (computations of limits, derivative, tracing graphs as objects, not as tool for reflections on tasks...) and which erases the relief which can be adopted on these objects. We have claim in this paper that these difficulties with point-wise and global points of view on functions can be relied with difficulties for students to enter on one hand in the formal axiomatic world and on the other hand to develop the local abilities which are necessary at the beginning of the university. We will continue to investigate these ideas in the future by designing a questionnaire for university students making in relation questions relied to the points of view and questions relied to local and formal properties (the utilization of the formal definition of limit for instance). REFERENCES Artigue, M. (1991). Analysis. In D. Tall (Ed.) Advanced mathematical thinking (pp. 167-198). Dordrecht: Kluwer Academic Press. Artigue, M, Batanero, Carmen, & Kent, Philippe. (2007). Mathematics thinking and learning at post-secondary level. In F. Lester (Ed.), Second handbook of research on mathematics teaching and learning (pp. 1011-1049). Greenwich, Connecticut: Information Age Publishing, Inc. Bachelard, G. (Ed.) 1938. La formation de l'esprit scientifique Paris: Vrin. Balacheff, N., & Gaudin, N. (2002). Students conceptions: An introduction to a formal characterization Cahier Liebnitz (No. 65). Grenoble: Université Joseph Fourrier. Coppé, Sylvie, Dorier, J-L., & Yavuz, I. (2007). De l'usage des tableaux de valeurs et des tableaux de variations dans l'enseignement de la notion de fonction en France en seconde. Recherche en Didactique des Mathématiques, 27 (2), 151-186. Douady, Régine. (1986). Jeux de cadre et dialectique outil-objet. Recherches en Didactique des Mathématiques, 7 (2), 5-31.
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