1. The teaching of calculus in Italy at pre-university level
The teaching of calculus was introduced with difficulty into Italian schools and much later than in other countries. Until the most recent reforms of curricula in the eighties it was only present in schools with a scientific-technological orientation, and not in the classical lyceum[1], which for many years educated the best classes[2].
The very concept of function, to which Felix Klein attributes a central role in his reform (Meran Syllabus, 1905), was reluctantly received by Italian schools[3]. Although one of the first journal articles explicitly concerned with the teaching of mathematics has as its central theme the importance of the concept of function in pre-university teaching[4], it was not until after 1910 that this concept, in its most formal presentation `à la Weierstrass', was included in the ministerial curricula for the scientific lyceum and subsequently in those of the technical institutes. This was made possible by the efforts of Guido Castelnuovo, a leading Italian researcher in algebraic geometry, who was also deeply involved in the debate on mathematical instruction at all levels.
Since then, however, the subject has remained practically unchanged and undebated in the syllabuses, text books, and matriculation examinations. This policy of conservation seems to have been one of the most solid features of the Italian upper secondary school this century. After the reform of Gentile in 1923[5], it was not until the end of the eighties that a complete proposal of reform of the curricula was made. This reform has still not been implemented entirely and is still in the process of clarification. The new curricula on which this reform is based envisage, for our purposes, two substantial innovations: earlier presentation of the concept of function in the first two years of secondary school[6], and the extension of the study of calculus (in varying detail) to all schools[7].
The most commonly used textbooks of calculus, however, are re-editions, only slightly modified, of texts which have been in use for several decades. Even the written matriculation exam for the scientific lyceum, a school in which mathematics is theoretically the foundation, has contained substantially identical calculus problems for at least 25 years[8]. These consist essentially in the study of the graph of a function by means of limits, derivatives and so forth.
2. Italian research on teaching/learning calculus
The tendency mentioned above of not discussing the traditional formal presentation of calculus has certainly had its effects on teaching in class; even teachers disposed to innovation, in the end have to reckon with an external examination commission. This means that they cannot promote innovations in calculus curriculum that deviate too far from the traditional line. We have to observe that this final examination is sometime an alibi for teachers. The content of calculus is fixed by the national programs, but no suggestions are given as for methodology: thus in theory there would be a possibility of making some changes.
As evidence of such an attitude, the cultural climate referred to above has prevented full appreciation of the project edited by Giovanni Prodi[9]; this is surely one of the most significant Italian projects on the teaching of mathematics in the secondary schools, which, in particular, had a detailed proposal with regard to calculus.
As far as didactic research is concerned, the above may be one reason for the relative lack of interest of Italian researchers in this question. In Italy, as in other countries, the study of problems related to teaching/learning of students of 16 years and older (Advanced mathematical thinking[10]) involves less interaction with scholars of educational disciplines. The number of researchers is therefore relatively small with respect to other related fields of research; in Italy, this is especially true for calculus.
Nevertheless in the last few years, there has been increased interest in the question and greater involvement on the part of secondary school teachers. The main reason has been the discussion of new curricula and critical elements introduced by the use of computers in teaching practice. A considerable stimulus has also come from the work of the Nuclei di ricerca didattica (Didactic Research Groups: NRD)[11] and from periodic national meetings held by the NRD concerned with the upper secondary school level, for the purpose of comparing lines of research. The meeting held in Siena in 1994 was devoted to discuss <<how, why, when, with which tools to teach calculus in different school levels>>. This event greatly helped to focus interest on themes specific to calculus, and to make known the results of research which was in progress and which the meeting has helped to crystalize.
The proceedings of this meeting[12] give an exhaustive picture of current Italian research in this sector. If one looks at the list of papers we consider in this chapter, one immediately realizes that almost half of them come from these proceedings, while only a few authors appear with more than one paper, which would show a long term activity. When compared to the number of papers considered in other chapters of this book, this fact confirms once more that in Italy the didactic research on calculus involves relatively few scholars.
3. Trends in Italian educational research in calculus
Recent Italian research in the teaching and learning of calculus has been conducted along the following lines:
As far as particular subjects are concerned, most research has centred on functions, with the concept of limit in second place. Few studies have been concerned with continuity and derivatives, and no research at all has been done on the concept of integral. Integrals are a topics a bit neglected also in the school practice. We will see that some papers concern the university level. Although the research has local specifics, it generally has essential reference to international results and lines of research prevalent in the sector[13].
As outlined previously about the composition of the various NRD, the authors of the papers we consider include university researchers and secondary school teachers.
We shall now give a brief account of the various studies, according to the above lines of research. Obviously many papers can be attributed to more than one of these lines, they will be discussed under the one we consider to be most pertinent.
3.1. The learning of concepts
The role of conceptions and the consistency of misconceptions in acquiring a knowledge of calculus are examined in the light of classical theoretical references on the topic. A theory to which particular reference is made is that of Tall and Vinner[14] (Bazzini, 1994; Furinghetti and Paola, 1988 and 1991; Menghini, 1991). These studies show the influence of intuition and of the student's individual history in the build-up of the concept images of calculus, with particular reference to the errors in learning this topic. In (Furinghetti and Paola, 1988) the hypothesis of work is that pupils have already accumulated `wrong beliefs' which will be the cause, in part, of the future difficulties when they begin to learn calculus. These wrong beliefs take origin from: - the five senses, - the ambiguity of colloquial language, - previous experiences in learning. Following this hypothesis, the authors investigate the wrong beliefs in relation to the basic concepts of limit, continuity, infinite, infinitesimal, " [[epsilon]] - d" by means of a questionnaire consisting of open questions of the following types: evocative, explanatory, constructive, analytic.
Other authors, (Longo and Di Carlo, 1994), refer to Vergnaud[15] looking for an application of his theories to the learning of the crucial definitions of functions and limits.
One of the most studied concepts is that of the function, which is examined above all from the point of view of the difficulty students have in fitting the different issues of the subject: variables, domain, relation between algebraic and geometric meaning, etc. ... . It is obvious that focus on this topic is favoured by the fact that the concept is transversal and longitudinal, and can be proposed at different levels and in different fields; in this connection we observe that one of the papers considered, (Cannizzaro, 1988), deals with the early learning of the concept of function. In (Bazzini, 1994) the results of a questionnaire submitted to 261 high school students are presented. The aim is to enlighten the views of what the function is and under which conditions two functions coincide. In (Furinghetti and Somaglia, to appear) the method of evaluating the performance of students in relation to age is applied to the understanding of the concept of function. It emerges that overall understanding of the concept of function is influenced by whether the approach used in teaching it was algebraic or geometric. The comparison of the students' performances when using graphical or algebraic languages seems to be an important theme in the discussion of how to teach calculus; we will see that it appear also in other streams of research. In (Grugnetti, 1994) the different approaches present in literature on mathematical education are considered and some implications for classroom practice are outlined.
3.2. History and epistemology
History appears under two different aspects: sometimes it is seen as a source of problems and an occasion for cultural stimulation of the students; in other papers, the history of mathematics is a reference framework for studying the difficulties of the students by linking them to epistemological obstacles[16]. In the first stream the two papers (Ascoli-Bartoli, 1994; NRD Firenze 1994) illustrate the possibility of integrating history of mathematics in mathematics teaching; the focus is on limit and the authors - in both cases teachers of secondary school - use classical examples such as Zeno's and Aristotle's paradoxes, for introducing students to the concept. We will see that history appears in different situations as an educational device for motivating students and introducing critical topics, according to an rooted Italian tradition. In the second stream we find works such as (Menghini, 1991). This paper represents a theoretical consideration relying, on the one hand, upon materials taken from the history of mathematics, and on the other hand upon case-studies concerning obstacles in learning calculus at upper secondary school. The change of expression and of meaning, which many theorems of analysis have undergone since they were organized in the actual arrangement, appears in connection with actual didactic problems. The paper focuses in particular on the contextual reference: theorems have changed their place during the last centuries, passing from the area in which they were motivated to the area which allows a simple proof, and generally a more linear deductive arrangement. The changes of the context is connected to the change of the language and of the aims. The paper considers the didactic problems connected with modern usages, in particular: - the comprehension and the use of a formal language which differs from that of algebra; - the suppression of the constructive-geometrical interpretation, and so the problem of the connection between mental images and mathematical formulation. These are important issues to discuss also in connection with teacher training.
The idea of epistemological obstacles inspires some authors in trying new approaches to the concepts. In (Camarda and Spagnolo, 1989) the idea of a geometrical problem for iperreal numbers is considered. The historical path going from Euclid to the modern non-standard analysis is outlined and the authors express the feeling that some ideas in this history may have didactic applications. A complete survey of the subject is in (Valenti, 1994). The non-standard analysis is used also in another paper (Venè et alii, 1994) with the aim of favouring the understanding of traditionally difficult concepts such as limit.
The importance ascribed to history of mathematics both as a source of problems and as a field were epistemological obstacles may be studied is evidenced by specific courses for teachers, such as the one reported in (Curcio and Medolla, 1993).
3.3. Different ways of presenting concepts
In spite of the stagnation in the curricula of the topic of calculus the educators perceive the necessity of discussing the style of teaching. In this discussion Giovanni Prodi, the author of the project mentioned above brought significant contributions; see, for example, (Prodi, 1993). Other works (Bagni, 1993; Bagni, 1994; Plazzi and Bagni, 1995) carefully analyse important aspects. The first considers aspects of continuity, the second the pitfalls of intuitive ideas used when presenting continuity through graphs. The third gives interesting results on topics linking analysis and number theory. These papers are mostly aimed at a reflection on theoretical issues of analysis.
The majority of the studies discussing different ways of presenting the subjects, see (Arnaldi Suria, 1994; Pacini et alii, 1994; Scarafiotti, 1994; Testa, 1989), is of the action-research type. The papers report experiences in class, proposing different techniques and approaches and contain analysis of student response. The main resources called upon to improve school practice are those considered above (historical approach, use of computer, graphic examples, ...). The researchers have mainly focussed on the didactic and motivational efficacy of the technique used, according to the manner of the French ingénierie didactique.
The paper (Furinghetti and Paola, 1991) describes the main features of a didactic itinerary which combines the compulsoriness as for the content and the freedom as for the methodology. Calculus is introduced starting from the `pre-knowledge' of students and proposing problems set inside mathematics and outside mathematics (physics, philosophy). In this way students are provided with motivations and stimuli for constructing concepts. Afterwards the traditional program is carried out through the textbook mediated by work-sheets prepared by the teacher. In these work-sheets there are problems and exercises: problems are aimed at constructing concepts, exercises are aimed to fix concepts. The final step of the work consists in giving the students examples of applications. The complete scheme of the work is: Construction -> formalization -> applications.
In (Arnaldi Suria, 1994) the concept of limit is presented by starting from classical problems of Euclidean geometry and considering the historical development of the concept; at the end the most important issues are singled out and the computer is used for generalizing the basic ideas. In (Testa, 1989) the concept of derivative is introduced by successive steps: firstly the tangent to a plane curve is considered in the elementary cases of circumference and conics, afterwards polynomials function are considered and finally the concept is introduced in its generalization. In (Bagni, to appear; Galizia and Marconi, 1990) trigonometrical functions are considered.
As a general trend we observe that the different ways of presenting a concept are carefully dealt with by starting from limited and well defined topics; only one paper considers a complete itinerary. In this stream of research, as in the one concerning computers, we found papers concerning university teaching.
3.4. Use of the computer in the teaching of calculus
In the discussion of the ways of presenting the main concepts, the computer generally has an important place. In fact it is attributed the role of the Trojan horse in the discussion of didactic approach: faced with the facilities offered by the new means teachers have to rethink their way of teaching.
Most of the studies carried out consider the possibility and opportunity of tackling difficulties in the study of calculus by exploiting the graphic and processing potential of the computer; other articles in this line of research regard university teaching (Cacciabue et alii, 1995; Galizia and Marconi, 1994; Mascarello et alii, 1994). Some papers, (Mascarello and Scarafiotti, 1988; Mascarello and Winkelmann, 1992), examine the problems of numerical calculation generated by solving calculus exercises by means of the computer.
While in the past time (until the early 1980's) there was interest in scientific hand calculators (programmable or not), at present the main interest of teachers and researchers is on the personal computer. Nevertheless experiences of teaching calculus with graphic calculators are carried out, as the one reported in (Castagnola, 1994). This author analyses the differences in the use of the two means (pocket and personal) and stresses the advantages of using the hand calculator for solving problems such as finding limits, tracing graphs, solving differential equations with the Euler method. The paper refers to an experience actually carried out in the classroom by the author, in line with the main studies in the field[17].
The chapter of this volume dedicated to research into teaching applications of computers offers further ideas ofn the use of the computer in teaching calculus. However, there seems to be a lack of specific studies on the impact of visualization on the presentation of concepts, confirming the hypothesis outlined above: the style of approach to calculus which has been chosen, consciously or otherwise, is that of rigour rather than intuition.
4. Prospects
The scenario we have outlined offers a significant example of how theoretical research and institutional problems intersect and affect each other in the teaching of calculus in Italy: since the concepts of calculus are regarded as untouchable and inevitable, it is not considered important to discuss a different approach to concepts or modifications to the curriculum. In our opinion, research in teaching in this sector should be in line with the need to bring schools up to date. Since this modernization is not a question of curriculum alone, but also of the attitude with which the curriculum is taught, it would be worthwhile directing attention to the teachers. In Italy, teachers are largely recruited among mathematics graduates, so it is not so much knowledge of the subject matter which is the problem, but rather knowledge of teaching methods. What concepts have teachers of calculus developed about the teaching of their subject? What do they think about the dichotomy between intuition and rigour?
In the past, attempts has been made in this direction, with a retraining course centred on the problem of rigour in mathematics[18]. During this course it was noticed that the conceptions of the teachers often had a doubtful basis and that the concept of rigour conditioned their educational choices more than consideration of the pedagogical needs of the students. Certainly there is a need of discussing the idea of rigour and its dependence on the context[19].
We make the further observation that other elements crucial to teaching research are related to this line of research. What role should proof play? How much should visualization be used, and what utility do teaching packages have? How should students be assessed in the light of the possibilities of computers? It is a rich field of research with the potential to nourish theoretical research and provide useful suggestions to curriculum developers.
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| Dipartimento di Matematica | Dipartimento di Matematica |
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