SOME RESULTS OF AN ITALIAN PROJECT IN THE … words: phylogenesis and ontogenesis, theory of didactical situations, pre-formal and formal phases, semantic approach. 1 INTRODUCTION

Acta Didactica Universitatis Comenianae Mathematics, Issue 8, 2008

SOME RESULTS OF AN ITALIAN PROJECT IN THE LIGHT OF A REVALUATION

OF THE SEMANTIC SIDE OF MATHEMATICS

GIUSEPPE GENTILE

Abstract. Starting from the alarming data describing a decrement in the number of students in Scientific Faculties, particularly in Mathematics, the author looks at the high formaliza-tion on one hand as an aim in research’s activity, on the other as an obstacle in approaching Mathematics. The proposed approach, based on a recovery of the “semantic” side of Mathe-matics, was experimented on a selected specimen of students of Italian High Schools, inside a wider biannual national project. The results of the first year, here presented and discussed, show the presence in the students of preconceptions, limiting the didactic communication, and confirm that the chosen approach can be a possible way to surmount these preconcep-tions obtaining a change in the student’s view of Mathematics; it is just this change that could allow to overcome the difficulties in the relations between Mathematics and youth, which effects are today clear. Résumé. En partant de la donnée alarmante qui décrit un décrément du nombre d’étudiantes dans les Facultés Scientifiques, en particulier en Mathématique, l’auteur voit dans l’haute formalisation d’un côté un but de l’activité du chercheur, de l’autre un obstacle dans la phase d’approche. L’hypothèse proposée, basée sur la récupération du côté “sémantique” de la Mathématique, a été expérimentée sur un échantillon sélectionné d’étudiantes d’Ecole Supé-rieur, dans un plus vaste projet biennal national. Les résultats de la première année, ici présentés et discutés, montrent dans l’étudiantes la présence de préjugés, que limitent la communication didactique, et confirment que l’approche choisi peut être une route pour dépasser ces préjugés, obtenant dans l’étudiantes un changement de la vision de la Mathé-matique; c’est ce changement qui pourrait permettre de surmonter les difficultés présentes dans le rapport entre la Mathématique et les jeunes dont les effets aujourd’hui sont évidents. Zusammenfassung. Ausgehend von der alarmierende Tatsache, dass ein Rückgang der Studentenanzahl der wissenschaftlichen Fakultäten bestätigt, insbesondere in der Mathe-matikabteilung, sieht der Autor in der hohen Formalisierung einerseits ein Forschungsziel anderseits ein Hindernis in der Konzeptphase. Die vorgeschlagene Lösung, die auf eine Wiedererlangung der semantischen Teil der Mathematik gegründet ist, war auf ein Ausge-wählte Stichprobe von Schüler aus höheren Schulen getestet, in einem weiteren zweijähri-gen Nationalprojekt. Die Ergebnisse des ersten Jahres, die hier beschrieben und diskutiert sind, zeigen einige Vorurteile in der Studenten. Diese Vorurteile begrenzen die didaktische

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Kommunikation und bestätigen, dass der gewählte Ansatz ein Weg kann sein um sie zu überwinden und um eine Änderung der Mathematikauffassung in der Schüler zu erreichen; eine solche Änderung könnte die Gewinnenkarte sein, um die gegenwärtigen Schwierigkeiten in den Beziehungen zwischen Mathematik und Schüler, deren Auswirkungen liegen auf der Hand. Riassunto. Partendo dal dato allarmante che sancisce una diminuzione del numero di studenti nelle Facoltà Scientifiche, in particolare in Matematica, l’autore vede nell’alta formalizzazione da una parte un traguardo dell’attività di ricerca, dall’altra un ostacolo nella fase di approccio. L’ipotesi proposta, basata su un recupero della parte “semantica” della Matematica, è stata sperimentata su un campione selezionato di studenti di Scuole Superiori, all’interno di un più vasto progetto biennale nazionale. I risultati del primo anno, che vengono qui presentati e discussi, mostrano la presenza negli studenti di preconcetti, che limitano la stessa comunicazione didattica, e confermano che l’approccio scelto può essere una via per superare tali preconcetti ottenendo negli studenti un cambiamento di visione della Matematica; è proprio tale cambiamento che potrebbe consentire di superare le difficoltà presenti nei rapporti fra Matematica e giovani ed i cui effetti sono oggi evidenti. Abstrakt. Na základe alarmujúcich údajov opisujúcich úbytok študentov na prírodovedecky orientovaných fakultách, obzvlášť v matematike, sa autor rozhodol študovať v článku vyso-kú úroveň formalizácie matematiky, ktorá je na jednej strane cieľ vedeckého rozvoja tejto disciplíny, na druhej strane je ale prekážkou pri zvládaní matematiky. Štúdia, ktorá bola za-ložená na skúmaní sémantickej stránky matematiky, bola uskutočnená na vzorke študentov talianskych stredných škôl, v rámci širšieho dvojročného národného projektu. Výsledky prvého roka štúdie, ktoré sú prezentované v článku, ukazujú, že u študentov nájdeme prvotné kon-cepty, ktoré limitujú didaktickú komunikáciu; článok tiež prezentuje prístup, ktorý sa ukázal byť úspešným pri prekonávaní uvedených prvotných konceptov a pri dosahovaní zmien v študentských pohľadoch na matematiku. Táto zmena pohľadov je potenciálnym prostried-kom na prekonanie ťažkostí vo vzťahu mladých k matematike, ktorých následky sú v súčasnej dobe viditeľné.

Key words: phylogenesis and ontogenesis, theory of didactical situations, pre-formal and formal phases, semantic approach.

1 INTRODUCTION This paper has two basic aims: the first is to describe some of the results of

an Italian national project which studied the causes of the decrease in the number of enrolments in science faculties, in particular in mathematics courses in Italy1. 1 One can speak of the crisis in scientific vocations, but it does not hit all sectors in a similar way at

least if, for scientific sectors, we intend not only the courses of the Faculties of Sciences but also in Medicine, Engineering, Agriculture. There are branches or courses that maintain or even increase the number of registered students. The number of students registered in the “science branch” has been almost constant in recent years; the number of students registered in Engineering has tripled, the crisis hits mostly the theoretical disciplines: Mathematics, Physics, Chemistry. So I refer to these last ones when I speak of decreases in enrolments. See (Mariano Longo, 2003) for further details.

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The second, functional to the first, is to describe a possible solution to overcome this decrease with reference to an experience that was carried out in Italian high schools.

Concerning the first point, we examine data regarding enrolments in under-graduate courses in mathematics, physics and chemistry. In 1989/90 students attending these courses numbered approximately 10.000; in 2005/06 this dropped to about 7.600. In the same period, the number of students attending mathematics courses decreased from 4.396 to 2.094, representing 0,6 % of the whole. In the light of such data, the Italian Ministry of University and Research and the Ministry of Education (formerly the Ministry of University, Research and Education) set up a national project called the “Progetto Lauree Scientifiche” (PLS), in order to invert this trend and increase enrolments in the three cited courses. Universities, high schools and industry were involved in the biannual project. The experience and the results here described refer to the first year of the PLS (2006) and concern the University of Messina and high schools in the pro-vince of Messina.

Many reasons have been given to explain such a decrease: the difficulty of scientific studies or the ability of teachers to manage some of the phases of the teaching-learning process. To these we could add that, in the eyes of the youn-ger generation, the “work” of scientists has little economic or social appeal (parti-cularly in the case of mathematics, where teaching appears to be the only career opportunity). Although I recognize that each of these reasons has some element of truth in it, in the present work I suggest a more technical approach to the problem. This means that we have to look at the problem from “inside” mathe-matics and it is from here that I consider the problem of the falling interest in this subject as an issue related to Mathematics Education.

2 THEORETICAL PREMISES AND AIMS Before describing the experience in its technical aspects, I need to say some-

thing about the theoretical premises on which the whole experience was conceived which give meaning to and permit an interpretation of the experience itself.

The first element is the link between ontogenetic and phylogenetic paths. The historic and cognitive development of concepts sometimes follow similar paths; such parallelism, one of the products of the recapitulation, on one hand must be considered with all its connected limitations and problems, in particular in the need to assign correct weighting to social, cultural and technological factors (Artigue, 1990; Radford, 1997; Radford-Puig, 2007; Thomaidis-Tzanakis, 2007). On the other hand, and also in the light of such factors (not to be considered as criticism, but as a warning against a naïve and acritical use of history in didac-tics), such parallelism permits the use of historic knowledge for didactic purposes.

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This not only provides a useful tool to foresee and overcome obstacles in the learning of specific concepts (Brousseau, 1983; Furinghetti-Radford, 2002), but it also gives us a didactic approach that takes into account the genesis and historic evolution of concepts and can give us a key to effective didactic action, particularly concerning heuristic moments and approaches to new concepts (Piaget-Garcia, 1983; Sfard, 1991; Gentile, 2005).

I would like to add a further consideration, which was developed in collabo-ration with R. Migliorato and has given rise to a series of recent publications (Gentile, 2004, 2005, 2006; Migliorato, 2004, 2005, 2006). The main idea develo-ped in these papers is that mathematics (but the discourse could be extended to knowledge in general) proceeds from an initial stage, defined as pre-formal and characterized by a free choice of research tools and resolving techniques, to a se-cond stage where these procedures are justified within a formal frame2. This state-ment has been derived by comparing similar conclusions reached in different research fields, using different tools and with different aims.

The first field is Piaget’s genetic epistemology: in particular I remark the evolutionary line theorized by the Swiss scholar which is represented by four stages (sensorimotor, preoperational, concrete operational and formal operational stages) and is characterized by the process of assimilation and accommodation. It views, among other things, the shaping of structures as being provoked by expe-rience which, in turn, allows to assign a meaning to the experience itself. Regar-ding this field these two moments could be identified with those that, in the cited papers, have been called pre-formal and formal moments.

The second field, merely philosophical, is Cassirer’s symbolic forms. The German philosopher re-proposes the Kantian notion of the existence of forms, but without assuming that they are a priori: in fact, since non-Euclidean geometries underpin the foundations of the Kantian thesis regarding the a priori existence of synthetic forms, Cassirer, faithful to the Kantian notion of their existence, suggests however, that they cannot be a priori but are instead a cultural product. They are the effects of human evolution (and can be said to be a phylogenetic experience). It is with this meaning that Cassirer speaks of symbolic forms3.

The third field, nearer to mathematics, concerns Poincaré and his remarks, for instance, on the concept of space, its euclidicity and dimension4. The French

2 In particular, in (Migliorato, 2005) an analogous interpretative key has been proposed for the

passage from myth to science; but in this case the pre-formal phase, where language is nearer to the formal one but procedures are still not assimilated inside a coherent framework, it is prece-ded by an informal phase, in which language is nearer to a simple observation of facts.

3 See also (Gentile, 2004). For an in depth account of the positions and ideas of Neokantian philosophy see for instance (Cassirer, 2004). Cassirer’s thesis were employed by Panofsky in the apparently distant context of perspective, intended by Panofsky as a cultural product giving geometrical coherence to spatial representations; on this subject see (Panofsky, 1961).

4 See, for instance, (Poincaré, 1963).


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scientist gives a central role to movement and to visual and tactile experiences, their reversibility and composition, consequently different experiences give rise to different representations of space with more than three dimensions, or which are non-Euclidean, and so on.

The fourth area I would like to outline is epistemological and refers to a remark of Heyting on the axiomatic method. The Intuitionistic distinction between a mathe-matical fact, considered as a-linguistic, and mathematical language, the aim of which is to remember and convey to others some mental construction, is reflected, according to the Dutch mathematician, in the distinction between the two functions of axiomatization, the creative and the descriptive. Heyting accepts only the second one, while, following Intuitionism, denies the first one5.

The last area is of a historic-mathematical character and concerns Archimedes and in particular his Method 6, which is closely related to the problem I’m investi-gating. The way the Syracusan scientist reconciles, supports and brings together the two characteristic moments of his work is significant. As Archimedes him-self communicates to Heratostenes in the introductory letter to the Method, the first moment is the heuristic one, the moment of discovery, the second is that of rigorous demonstration, by the method of exhaustion, of the results realized. The following passage, quoted from the Method, is particularly meaningful (Heath, 2002, The Method, p. 13):

This is the reason why, in the case of the theorems the proof of which Eudoxus was the first to discover, namely that the cone is one third part of the cylinder, and the pyramid of the prism, having the same base and equal height, we should give no small share of the credit to Democritus who was the first to make the assertion with regard to the said figure though he did not prove it.

In fact the classical and rigorous method of exhaustion, probably originated with Eudoxus, needs that we already know, in some way, the figure sample from which it is possible to compare various other figures, inscribed and circumscribed to the given one. This difficulty poses serious problems to the use itself of the exhaustion method which, if left alone, would be completely unfruitful. Briefly the exhaustion method is not a research method, but only a method to prove an exis-ting but as yet unproven result.

5 As an example of axiomatic theory with only a constitutive function, Heyting proposes the ZF

axiomatic set theory (but one could consider any axiomatic set theory): in respect to the question “What does ZF formalize?” one can notice that, if it tries to formalize the pre-formal (Cantorian) set theory then it is contradictory, while (and it is just on this point that Heyting makes his criticism) if it does not then one should ask himself if ZF could be an empty theory, namely if the existence of the objects, of which ZF speaks, is revealed only by the axioms (Casari, 1976, p. 193).

6 On some aspects of Archimedes’ productions, see (Gentile-Migliorato, 2007).

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If we summarize what has been said until now, at a first glance we can affirm that the passage from the pre-formal to the formal moment is a fact that can be displayed by historical, psychological, epistemological, philosophical or simply mathematical considerations; but this fact alone does not imply a priori that it is the only way to obtain a formalization. Nevertheless, I would like to affirm that the passage from the pre-formal to the formal moment not only happens but it does not but happens: it is a necessary step in scientific evolution and, in the light of our first assumption on the similarity between phylogenetic and ontogenetic development, this implies that it is a necessary step in the evo-lution of individuals as well. Conversely, I would like to underline that the way itself in which the passage from the pre-formal to the formal moment has been discussed (in particular the different temporal scales, the different objects and tools assumed in the considered fields reach concordant conclusions) enhances in its turn the first assumption on the similarity between phylogenetic and onto-genetic paths. Such a similarity suggests, in fact, the possibility that phylogene-tic considerations could be a starting point in the area of personal development and so could naturally be applied in teaching-learning processes. In particular, I would like to apply the considerations reached on the passage from the pre-formal to formal moment, gleaned from the historic-epistemological background to teaching-learning processes. And conversely, the success of a didactic expe-rience using such an approach can enhance the idea of similarity between the two paths.

Coming back to the first problem, which is the decrease in the number of stu-dents enrolling in science faculties, all previous considerations can provide a key to explain this gradual decline and, at the same time, suggest a way to overcome it. Today we still have this heritage from which we are not free at all that can be traced back to the beginning of the twentieth century, where Formalism and, more importantly, Bourbakism strongly marked scientific work and guided choices with regard to both methodology and content. If on one hand it is remarkable to have posed the problem of the foundations of mathematics (a problem that is his-torically meaningful), it is equally evident that the tools used by Formalism have deprived mathematics of its semantic side, preserving its syntactic aspect. The objectives of this approach were not reached (Gödel showed it to be unreachable) and what is left however, of this view, is the ever deeper attention paid to syntactic aspects, considered to be the only way of showing the coherence of a theory. That Bourbakism has emphasized this point of view thereby denying any mathemati-cal and epistemic value to the pre-formal moments of a theory means that within


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a formal theory not only there is no space for pre-formal moments, but any value of these moments is denied7.

Today there are echoes of this view whenever we open a book on mathe-matics, where the is the “translation” of topics that were meaningful at origin, but that, in the phase of “formal translation”, are often omitted since formally super-fluous. It is these meanings which have to be restored, since they are actually ne-cessary, as is shown by the estrangement of young people from the study of mathe-matics. The proposed didactic choice, founded precisely on this hypothesis and called a semantic approach, was used in the experience and will be described be-low. On one hand we are in no doubt that mathematics is full of meanings, on the other hand it is equally true that, in general, such semantic aspects are almost always omitted in favor of a syntactic technicality that is difficult to comprehend to those who have not followed its development or, at least, are not conscious that such a development exists. This means that this aspect of mathematics is often obscured to students while, even if it is of less importance on a formal level, it was historically necessary on a heuristic level and could be useful from a didactic point of view. This omission, I suggest, is the origin of the distorted view of mathematics and can be identified as one of the reasons for the increasing dis-affection of students to the study of mathematics. To the frequent question “Why do young people hate mathematics?” we could answer that the question is meaningless and say that young people do not hate mathematics per se, but hate the mathematics that is presented to them. It is on this aspect that an intervention seems to be necessary, making it possible to change the views of young people towards mathematics. The semantic approach, here proposed, is an attempt to restore a meaningful role to the pre-formal moments of a theory, finding a correct equilibrium between pre-formal and formal moments. In fact, there is a risk that students will not be aware of this ambivalence of mathematics; it seems meaning-less to propose a study of mathematics in which the various aspects are not linked together in an organic corpus, i.e. mathematics without formal aspects. One undesirable effect could be a lack of co-ordination making the approach appear fragmented and, as a consequence, inhibit the capacity for abstraction. While this risk is evident enough, the opposite risk is more insidious and, as mentio-ned, it is precisely at this point that we find the seeds of the increasing dis-affection of young people towards mathematics. Consequently it is this aspect that,

7 I would like to clarify that I do not deny the validity and usefulness of the formal aspect of a theo-

ry. To such purpose in (Gentile, 2005) it seemed useful to distinguish between two terms: on one side formalism, intended as renouncement of any meaning, voluntary and with a historically pre-cise aim; on the other side formulism, intended as loss of any meaning, not at all voluntary and without any apparent aim. It is this last one that I retain without validity and usefulness, even it is one of the causes of the void that makes Mathematics appear so arid.

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suitably revived, can re-establish the value of the pre-formal aspects of a theory, awakening interest firstly towards a particular topic and secondly towards mathe-matics in general8.

3 THE EXPERIENCE The experience I will describe took place within the biannual national pro-

ject PLS in which three actors co-operated: the Universities (in particular the Departments of Mathematics, Physics and Chemistry), the Provincial Offices for Education (now called USP) which coordinated the schools of each province, and local industries. The goal of the PLS was to increase the number of young people enrolling in science faculties, particularly in mathematics, physics and chemistry which, in the last few years, have suffered an increasing decline in enrolments. Within this project the present brief-time experience was inserted aiming to re-cover the semantic aspect of mathematics by showing students the historic and problematic issues that gave meaning to its formal development. By putting into practice this approach I expected to find a change in the way students viewed Mathematics9 and, I believe, this change in perspective could help the PLS to achieve its long-term goal.

The experience, developed in the Province of Messina, involved 8 Schools: the Scientific Lyceum “Archimede” of Messina, “Impallomeni” of Milazzo, “Piccolo” of Capo d’Orlando (these last two with annexed Classics sections), the Classics Lyceum “Maurolico” of Messina, “V. Emanuele III” of Patti (this last one with annexed Scientific sections), the Socio-Psycho-Pedagogical Lyceum “AINIS” of Messina, the Technical Industrial Institute “Verona-Trento” of Messina, the Technical Commercial and Surveyors Institute “Fermi” of Barcellona. The experience was undertaken for a total period of 15 hours divided into 5 meetings and involved students from the last three school years. The students, 156 in all, were selected not so much on the grounds of their “scholastic performance”, but rather on their curiosity and willingness to participate in the project.

The chosen topics were Coding Theory and Cryptography. The choice of these topics was not casual but made in the light of my hypothesis of exploiting

8 I would like to remark that the semantic approach here presented has some links with the Brusseau’s

a-didactic situation; nevertheless I wish to avoid the risk of a total identification: while Brusseau’s a-didactic situation appears totally inside the didactic environment, trying to create the conditions favouring a shared construction of meanings and knowledge and giving great value to pre-formal moments, the semantic approach aims to justify this choice on the historic-epistemological level; in other words while the a-didactic situation deals with how, the semantic approach also explains why.

9 The basis of this schema is the hypothesis that students have preconceived ideas inhibiting the possibility of a new problematic approach. This hypothesis has been strengthened by the results of the questionnaire submitted to students before the experience began.


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“semantic” aspects, at least in the first approach to the topic. It was useful to treat topics which had not previously been studied, in order to work with students who were “uncontaminated” by any formal apparatus. In this way, I believed, it would be possible (and in fact it was possible) to begin with a problematic situation, si-milar to that which historically occurred, and let the students work in a pre-formal frame, to be formalized afterwards. When working in a non formalized situation, it is easier to set the discovery process in motion (and discovery is fundamental from a historical point of view and essential from a didactic one), and thus avoid any pre-conceptions that might inhibit communication. This also explains the choice of an a-didactic situation, where students could freely propose (which they did) solutions to the problems posed by the teacher or by themselves, verifying and, finally, validating or rejecting them. In other words the a-didactic situation could be (and in fact was) the most useful condition for enabling students to autono-mously construct meanings without receiving any preconceived notions.

Before describing in detail the experience, I would like to outline that, in order to meaningfully compare the obtained results, I created two groups of students, only one of which was ‘taught’ using the semantic approach. The following table contains a summary:

Table 1. Summary of schools, topics and approaches

Name of the School

Type of the School Chosen topic Number of

students Approach

Piccolo SC Lyceum Coding Theory 13 Semantic Verona-Trento TI Institute Coding Theory 21 Semantic Fermi TCG Institute Coding Theory 18 Semantic Maurolico C Lyceum Coding Theory 8 Not semantic Archimede S Lyceum Cryptography 11 Not semantic V. Emanuele III CS Lyceum Cryptography 8 Not semantic Impallomeni CS Lyceum Cryptography 17 Not semantic Ainis SPP Lyceum Cryptography 8 Not semantic

An explication of the above table is needed and I should like to comment on some of the choices. The explication concerns the number of students: while, as I have already said, the total number of students participating in the project was 156, the number presented in the table is 104 and refers to those students who attended all of the offered activities. This is to guarantee that the data presented and discussed in the following section, was effectively produced by the experience. The first comment concerns the distribution of the approaches: in order to test meaning-fully the semantic approach, it was necessary to use it in different types of schools and it is for this reason that I chose to use it in both a Lyceum (with student coming

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from both scientific and classics sections) and two Technical Institutes (of different types). The second comment concerns the distribution of topics: as can be seen in the above table, the semantic approach was tested only on one topic (Coding Theory), although it would have been more complete to have had data on the other topic also (organizational problems prevented this choice). Nevertheless, it must be noted that topic choice does not invalidate the results I wanted to obtain which was to let students experience the pre-formal aspects of a theory, with the aim of getting them to see the subsequent formal development of the theory in a diffe-rent, more interesting and meaningful way10.

Before describing which aspects were developed during the experience, I would like to mention that before the first meeting and after the last one, two questionnaires (almost identical) were given to students; these are reported in Appendix A and B and will be discussed in the following section. I would like to point out that these were not questionnaires of mathematics, but on mathematics: in fact, they aimed to gather information on how students see mathematics and they allowed me to verify if and how the experience changed students’ views of mathematics.

Concerning the contents treated during the experience, I describe only those referred to during the meetings on Coding Theory, since this topic was covered using the semantic approach. The historic context of this theory can be found in (Shannon, 1948) pioneering paper in which the central problem of communica-tions was introduced, followed by Hamming’s work (Hamming, 1950), in which the problem of transmission errors in a telephone central office was examined and resolved.

In the light of this historic problem, at the first meeting I suggested to stu-dents a situation where two young people (one transmitting and the other re-ceiving) try to transmit a message by mobile phone checking errors in the trans-mission. The students, in groups, were free to offer suggestions both as to why errors occurred and regarding possible solutions to avoid such errors. It was possible to distinguish two approaches, offered by students to solve the problem: the first one was directed at improving technology, the second aimed to create a mechanism able to detect and eventually correct errors. The subsequent dis-cussion led the whole group to prefer the second solution11. In the following meetings the students tried to create some of these systems. Problems, both those suggested by the teacher and those by the students themselves, were faced first

10 Further confirmation of these results has been planned for the second year of the project, to test

the same approach with other subjects. 11 It is not surprising that in the historical moment when these problems began to emerge, essentially

the same solution was suggested as the one proposed by the students, a rejection of technology: this fact too can be considered as an element strengthening the idea of parallelism between phylogenetic and ontogenetic paths. On a similar question concerning Goldbach’s conjecture, see (Scimone, 2003).


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in small groups (4 or 5 students) and then by the group as a whole. It is particu-larly noteworthy that during the second meeting the groups, invited to find a code of 4 words correcting 1 error, proposed different solutions, concluding however that all the solutions were equivalent (a code of length 3 on an alphabet of 4 letters and a code of length 6 on an alphabet of 2 letters). Following the acknow-ledgment of this equivalence, in 2 of the 3 schools, some students expressed the necessity to optimize the first of the suggested solutions (“Is it possible to make it shorter?”), so entering naturally into one of the main problems of Coding Theory: the creation of codes which detect (or correct) as many errors as possible by using as few characters as possible. In this and in the following meetings, notions of Hamming’s distance, code length, error detecting and error correcting codes, Hamming’s inequality (limiting the capability of a code and giving meaning to the concept of a perfect code) came to light12. In the last meeting verification on the acquisition of treated topics (reported in Appendix C) was undertaken. This confirmed for almost all students: an understanding of all the main questions associated with the theory; possible solutions to some of the more problematic areas; the possibility of proving the impossibility of solving a given problem and, last but not least, the usefulness of the treated topic.

At the end of the experience the students were invited to write down their experiences of the meetings, asking them explicitly to put themselves in the shoes of someone who “has to write a chapter of a book of mathematics”. At that point the students perceived both the difficulty and the necessity to formalize in an organic and deductive way all that they had acquired up to that moment. Dis-cussions on how to give a definition, on which definition had to precede another, on the concatenation of propositions and other similar questions, were for them an absolute discovery. The discovery that what is written in a book of mathematics is not an empty list of terms without meaning, but the synthesis of an experience in need of an organizational phase which translates in a formal manner, those meanings that previously emerged in the experience itself. The awareness in the students that “there is something else” beyond the formalism, was modified, as can be seen in their opinions reported in the above-mentioned questionnaires. It is this awareness that could determine a new way of seeing Mathematics, without any bias which limits, in some cases even, prevents any kind of didactic commu-nication at all. The next section is devoted to the results and comments relative to the questionnaires.

12 For an account of the technical aspects of Coding Theory and, in particular, on the above concepts,

see (Cerasoli-Eugeni-Protasi, 1988), (Scafati-Tallini, 1995), (Beutelspacher-Rosenbaum, 1998).

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4 RESULTS AND COMMENTS First of all, even if it should be clear by now, I would like to point out that

the questionnaire given to the students focused on the process and on eventual changes occurring during the experience and not on content. In evaluating results most of my attention was directed towards items 6 and 1013, since it was inte-resting to know (1) whether the students consider Mathematics as already written without any possibility of further development or were they aware that it is in continuous development and (2) if and how much, in their opinion, creativity has a role to play. A similar investigation, relevant to the present research, was carried out by Migliorato in the eighties in Italian high schools. The following question was put to students14:

Which of the following talents, in your opinion, are important to do well in Mathematics (you can choose more then one): a. Creativity. b. Patience. c. Intuition. d. Memory.

The results indicated the following percentage in the answers15: a. 6 % b. 29 % c. 81 % d. 23 %

Evidently, they considered creativity to be of minor importance and further-more, from a further analysis of the data, it became clear that its relevance decreased during the years in the classroom. What is clear today is that the present experience generally confirms those results. The novelty that emerged, however is the possibility of redirecting students’ views of mathematics while, at the same time, testing the efficacy of the semantic approach in contributing to that change.

Coming back to item 6, the parameter I used in the analysis was choosing the reply “to have a lot of fantasy”. As could be seen, students had the possibility of choosing how many answers to give; this choice came from the desire to distinguish cases where “to have a lot of fantasy” was considered of minor importance from cases where it was considered as completely irrelevant16. In order to make a quanti-tative statistical analysis, the answer has been numerically translated, as follows:

13 In the final questionnaire not all items present in the initial one were inserted, numeration there-

fore is almost all different; the numeration refers to the initial questionnaire. 14 The whole results are contained in (Migliorato, on line). 15 The sum of the percentages is more than 100 since students were free to give more than one

answer. 16 Two differences exist with respect to the previous item: the first one is that the questionnaire was

submitted to a selected specimen, while in the previous experience the specimen was indistinct; the second is that in the present questionnaire there is the possibility of choosing “to have a lot of fantasy” even in the case of a low consideration of this quality: the two differences should increase the “weight” assigned by students to fantasy but, despite that, the result is discouraging.


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• 1 if the answer was present 1st place • 0,8 if the answer was present 2nd place • 0,6 if the answer was present 3rd place • 0,4 if the answer was present 4rt place • 0,3 if the answer was present 5th place • 0,2 if the answer was present 6th place • 0 if the answer was not present

This item was present both in the initial questionnaire and in the final one. The mean, calculated in the initial questionnaire, gave the value 0.159; confirming the hypothesis that students see in Mathematics an environment with no possi-bility for creativity and in which, consequently, the aspects historically necessary to the development of Mathematics are mortified.

The same item, repeated in the final questionnaire and analogously analyzed, gave the value 0.173. As indicated in the previous section, the specimen was divided into two parts: one, denoted as SA, composed of students following the semantic approach, and the other, denoted as NA, composed of the remaining students. The data relative to the two specimens so divided gave the following results: for SA the value increased by 0.131 to 0.187, while for NA the value decreased by 0.187 to 0.159. A particularity, which I mention explicitly since it didn’t emerge in previous data, is that in one case (the Institute “Fermi”) all the students contributed to the increment, that is at the end of the experience every student assigned to fantasy a more (in fact a not lesser) relevant role with respect to the beginning of the experience.

Now we shall go on to discuss item 10; for this the following scale was adopted: • 1 if the answer was “invention” • 0 if the answer was “discovery” • 0,5 if the answer does not incline.

In this case too, a comparative analysis (initial and final) of the questionnaires showed a clear change; in fact, while in the initial questionnaire the mean was 0.282, in the final one the value was 0.269. Also in this case, distinguishing between the two specimens, a different result emerges depending on the approach used. Using the same notations as for item 6, the value of SA increased from 0.269 to 0.295, while for NA it decreased from 0.295 to 0.243. Once more, it should be noted that, for the “Fermi” group an analogous fact occurred, every student contributed to the increase (in fact to the no decrease) in the numeric value assigned to the item, seeing in Mathematics more constructive possibilities and therefore a personal contribution. This data, even if it does not directly indicate a change as seen in item 6, it indicates nevertheless that students perceived the “human” intervention in the development of Mathematics, since they themselves had participated as “actors” in just such a process. I should also like to examine

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the results obtained by comparing items 6 and 10, using a statistical implicative analysis; in particular, it seemed interesting to see what link there was between indicating Mathematics as discovery and the presence of fantasy as among the qualities retained necessary to do well in Mathematics; the analysis, conducted using the software CHIC (Classification Hiérarchique Implicative et Cohésitive), emphasized some implications the first of which can be summarized as follows:

If Mathematics is also an invention ⇒ Fantasy has a role in Mathematics

which is equivalent to

If fantasy does not have any role in Mathematics ⇒ Maths is only discovery

The implication measure increased by 0.86 in the initial questionnaire (10i ⇒ 6i) to 0.94 in the final one (10f ⇒ 6f); in particular, for SA specimens, this index increased from 0.82 to 0.92. Another strong implication should be noted: in the NA specimen, if fantasy was indicated as important in the final questionnaire it had also been indicated as important in the initial one (and conversely, if it had not been indicated in the initial questionnaire then it was not indicated in the final one). The implication measure was 0.94 (6f ⇒ 6i). By comparing the graphs we can deduct that previous implications can characterize the two specimens: while the SA specimen responded to some stimuli produced by the experience (as we can deduce from the increased measure of the implica-tion 10 ⇒ 6), for the NA specimen it was impossible to increase the value assigned to fantasy (the high value of the implication 6f ⇒ 6i).

Figure 1

Items 7 and 9 were designed to gather information on students’ points of

view of the “objects” of Mathematics and, in particular, on their “existence” and their possible “construction”. I wanted to find out if students believed in the existence of mathematical objects and whether such an existence comes from a human construction; for this reason the results were presented dividing the


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specimen using the following criteria: firstly the specimen was divided on the grounds of the existence assigned to the object (item 7), secondly the specimens so obtained were divided on the grounds of the human construction of that object (item 9). The data obtained can be summarized as follows:

Table 2. Summary of answers to items 7 and 9 Triangle exists (Initial)

YES 25 % NO 54 % May be 21 % Triangle is a human product

YES 18 %

NO 3 %

May be 4 %

YES 34 %

NO 16 %

May be 4 %

YES 11 %

NO 5 %

May be 5 %

Number exists (Initial)

YES 59 % NO 26 % May be 15 % Number is a human product

YES 11 %

NO 38 %

May be 10 %

YES 2 %

NO 14 %

May be 10 %

YES 4 %

NO 9 %

May be 2 %

Triangle exists (Final)

YES 29 % NO 48 % May be 23 % Triangle is a human product

YES 19 %

NO 5 %

May be 5 %

YES 25 %

NO 20 %

May be 3 %

YES 14 %

NO 3 %

May be 6 %

Number exists (Final)

YES 23 % NO 57 % May be 20 % Numbers are a human product

YES 16 %

NO 4 %

May be 3 %

YES 32 %

NO 20 %

May be 5 %

YES 9 %

NO 5 %

May be 6 %

The strange data that emerges from the previous tables relates to different perceptions of existence assigned to the triangle and numbers in the initial questionnaire (25 % versus 59 %); this difference diminished in the final questionnaire thanks to the increase in the number of students assigning existence to numbers (from 59 % to 23 %).

From items 1 and 4 I tried to note if, and to what extent, a problem was recog-nized as mathematical only in the case where it was already totally formalized in the technical language of a theory (this makes sense of the different formulations adopted in these two items). I made four comparisons: two, in every questionnaire, between items 1 and 4; the others between the initial and final questionnaires, in

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order to show which kind of change the experience should have provoked. This analysis was supported by item 3, for which the outlined parameter was the pre-sence of a problem in a formal version. The results of this analysis can be summa-rized as follows: students that gave the same answers to items 1 and 4 in the initial questionnaire constitute 15 % of the total, while this percentage more than doubles in the final questionnaire, at exactly 32 %; moreover, in the SA specimen the percentage increases from 15 % to 36 % with the peculiarity that all students belonging to the initial 15 % remained unchanged in their initial positions so the added 21 % is due entirely to students who changed idea between the beginning and the end of the experience; for the NA specimen the percentage increased from 15 % to 27 %, but with a change among those who belonged to the initial 15 % and the final 27 %. This kind of analysis suggests a more global awareness of the role of Mathematics in modeling wider and wider classes of problems. The se-mantic approach seems to increase this awareness and while at the moment this statement is only a conjecture, I aim to examine it further in future experiences.

Finally items 2, 5 and 8 were qualitative and no meaningful peculiarity emer-ged from their analysis.

5 CONCLUSIONS AND OPEN PROBLEMS Mathematics, but also Physics and Chemistry, as they have developed in

recent decades, are highly formalized subjects17, and this implies great difficulty for anyone who wishes to explore them. In such a context, an approach, which aims to reap the fruits too hastily, runs the risk of not reaping any fruits at all. In other words, to begin with the final aspect of a theory may provoke a rejection of the phenomena which today appears to be more and more evident. The semantic approach here proposed and discussed aims to restore a meaningful role to pre-formal moments of theories: it is at these moments that a winning card can be found which helps to recover strength and to stir up interest firstly towards the theory in question and secondly towards mathematics in general. With this in mind, the first meetings were devoted to these heuristic moments, without omitting the formal moment to which, as historically happens and as theorized in the chosen approach, the final and conclusive part of the whole experience was devoted.

The experience, developed in the light of this hypothesis, demonstrated a sub-stantial change in the way of viewing Mathematics, concerning both the mere content and the view of Mathematics in general. The possibility for students to construct meanings without receiving them, to find “rules” without having to learn them, in brief to have an active role on a mathematical path, permits a more

17 For more on these themes and on epistemological and didactic implications, see (Migliorato, 2006).


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efficacious learning process, a less problematical approach and, last but not least, it does justice to the historical route without distorting or upsetting it at all.

The different form of approach to topics and the comparison made between groups demonstrates the success of the proposed approach. Nevertheless it must be noted that at present I’m aware of the limits on the validity of the results until discussed, mainly because the examined specimens were composed of “motivated” students so the increment of considered parameters that occurred for such a speci-men must be examined. On the other hand it has to be noticed that, in spite of this limitation of the specimen, the initial parameters were low (and this is not a se-condary factor; I aim to examine it more closely later on). For this reason I hope to re-propose and to consider, within the semantic approach, analogous experien-ces that can be extended to indistinct classes.

Finally a small success in the PLS should be mentioned; among the 12 students following the project in their last school year, 2 of them are now re-gistered under-graduates in the Mathematics faculty in Messina University while, among those who were not in their last year, about ten declared that they would enroll in this course, giving the reason for such choices as the surprise in discovering a fascinating and until then obscure world. Though it does not seem a great number, its weight increases and acquires the correct value if we compare it, as is necessary, with the percentages presented at the beginning of the present paper. I’m aware that it is not possible to infer ipso facto that such increases depend directly on the semantic approach, it is however not possible to deny that such an approach, demonstrates a capacity to obtain change. It enlarges the ho-rizon of students’ views of Mathematics enabling them to approach Mathematics in a less problematic way. The attempt here described is to restore to Mathematics its more “meaningful” moments and to give students a wider view. Clearly the choice of enrolling in a certain faculty and continuing on that route depends on other factors, but such a choice will be more aware since it comes from a global knowledge and a synoptic view.

ACKNOWLEDGMENT I would like to thank the referees for their useful suggestions and construc-

tive remarks. REFERENCES

Artigue M., Epistémologie et didactique, Recherches en Didactique des Mathématiques, vol. 10/2.3, pp. 241–286, 1990

Beutelspacher A., Rosenbaum U., Projective Geometry: From Foundations to Applications, Cambridge University Press, 1998

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Brousseau G., Les obstacles épistémologiques et les problèmes in mathématiques, Reserches en Didactique des Mathématiques, 4, 2, pp. 165–198, 1983

Casari E., Questioni di filosofia della matematica, Feltrinelli, 1976 Cassirer E., Filosofia delle forme simboliche, La Nuova Italia, 2004 (Translation of The Philosophy

of Symbolic Forms, Yale University Press, 1955) Cerasoli M., Eugeni F., Protasi M., Elementi di matematica discreta, Zanichelli, 1988 Furinghetti F., Radford L., Historical Conceptual Developments and the Teaching of Mathematics:

from Phylogenesis and Ontogenesis Theory to Classroom Practice, Handbook of International Research in Mathematics Education, Hillsdale: Erlbaum, 2002, pp. 631–654

Gentile G., Due questioni di didattica della Matematica, Atti del Convegno Regionale “Quali prospettive per la Matematica e la sua Didattica”, Piazza Armerina, 2004 (on line on the site of GRIM: math.unipa.it/~grim/conv_aicm_grim.htm)

Gentile G., La storia della Matematica per la didattica della Matematica. Cosa può insegnarci Archimede?, Atti del Convegno Regionale “Quali prospettive per la Matematica e la sua Didattica”, Piazza Armerina, 2005 (on line on the site of GRIM: math.unipa.it/~grim/conv_aicm_grim.htm)

Gentile G., La Matematica e i giovani: un rapporto conflittuale superabile? Resoconto di una esperienza, Atti del Convegno Nazionale, 3° incontro ADT-Mathesis “Matematica è la più odiata dagli italiani! Come farla amare? Con le nuove tecnologie?”, Lipari, 2006 (available also on line within the publications of the Messina’s Section of Mathesis: ww2.unime.it/ mathesis/pub/matematica_giovani.pdf)

Gentile G., Migliorato R., Archimedes between tradition and innovation, submitted, 2007 Hamming R.W., Error Detecting and Error Correcting Codes, The Bell System Technical Journal,

vol. XXVI, n. 2, pp. 147–160, 1950 Heath T.L., The Works of Archimedes, Dover Publication, New York, 2002 Mariano Longo T., Scienze, un mito in declino? La crisi delle facoltà scientifiche: Italia, Francia ed

uno sguardo internazionale, Bollettino dell’A.N.I.S.N., anno XII, n. speciale – ottobre 2003 (also on line on: crisiscientifica.anisn.it/ricerca.php)

Migliorato R., L’astrazione matematica tra fantasia, conoscenza e ricadute tecnologiche, Atti del Convegno Regionale “Quali prospettive per la Matematica e la sua Didattica”, Piazza Armerina, 2004 (on line on the site of GRIM: math.unipa.it/~grim/conv_aicm_grim.htm)

Migliorato R., Spiegazione e predizione. Dalla rappresentazione mitica alla rappresentazione scientifica, Atti del Convegno Regionale “Quali prospettive per la Matematica e la sua Didattica”, Piazza Armerina, 2005 (on line on the site of GRIM: math.unipa.it/~grim/ conv_aicm_grim.htm)

Migliorato R., Tra gioco e metafora: per una rappresentazione matematica del mondo, Atti del Convegno Nazionale, 3° incontro ADT-Mathesis “Matematica è la più odiata dagli italiani! Come farla amare? Con le nuove tecnologie?”, Lipari, 2006 (available also on line within the publications of the Messina’s Section of Mathesis: ww2.unime.it/mathesis/pub/ gioco_metafora.pdf)

Panofsky E., La prospettiva come forma simbolica ed altri scritti, Feltrinelli, 1961 Piaget J., Garcia R., Psychogenèse et histoire des sciences, Paris: Flammarion, 1983 Poincaré H., La scienza e l’ipotesi, Signorelli, 1963 Radford L., On Psychology, Historical Epistemology and the Teaching of Mathematics: Towards a Socio-

Cultural History of Mathematics, For the Learning of Mathematics, 17 (1), pp. 26–33, 1997 Radford L., Puig L., Syntax and Meaning as Sensous, Visual, Historical Forms of Algebraic

Thinking, Educational Studies in Mathematics, 66, pp. 145–164, 2007 Sfard A., On the dual nature of mathematical conceptions: reflections on processes and objects as

different sides of the same coins, Educational Studies in Mathematics, 22, pp. 1–36, 1991


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Scimone A., Pupils’ conceptions about an open historical question: Goldbach’s conjecture. The improvement of mathematical education from a historical viewpoint, Doctoral Thesis, Quaderni di Ricerca in Didattica del G.R.I.M., n.12, 2003

Shannon C.E., A Mathematical Theory of Communication, The Bell System Technical Journal, Vol. 27, pp.379–423 July, pp. 623–646 October, 1948

Scafati M., Tallini G., Geometria di Galois e Teoria dei Codici, CISU, Roma, 1995 Thomaidis Y., Tzanakis C., The notion of historical “parallelism” revisited: historical evolution and

student's conception of the order relation on the number line, Educational Studies in Mathematics, 66, pp. 165–183, 2007

Vygotsky L.S., Il Processo cognitivo, Universale Bollati Boringhieri, 2002 (Translation of Mind in Society. The Development of Higher Psychological Processes, Harvard University Press, Cambridge (Mass.) – London, 1978) GIUSEPPE GENTILE, Department of Mathematics, University of Messina, Contrada Papardo, Salita Sperone, 31, 98166 Messina, Italy. E-mail: [email protected]

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APPENDIX A

Initial questionnaire

1. Which of the following is (are) a Mathematical problem (mark 1 to say YES, 0 to say NO,

½ if you are unsure): a. Find two numbers the sum of which sum is 15 and the product of which is 26. 0 1 ½ b. Given a circumference the radius of which is r, find the side of the inscribed and

circumscribed squares. 0 1 ½ c. A woman enters a shop to buy some cloth. Which color she will choose? 0 1 ½ d. Two teen-agers would like to exchange a message in a noisy environment. How can

they succeed in exchanging the message correctly? 0 1 ½ e. A drawer contains 10 red, 8 blue, 6 green socks. How many socks does a person,

without looking, have to take out to make a pair of the same color? 0 1 ½ f. Two teen-agers would like to exchange a message in the presence of other people who

are listening to them, without anybody understanding what they say. How can they do this? 0 1 ½

g. An asteroid is near to the earth. Is it possible to know if and when it will hit? 0 1 ½ h. To send messages along a line costs one cent per character. Find a way to spend as little

as possible without losing any of the original message. 0 1 ½ i. Paul and Francis play chess. Albert, watching them, at a certain point says: “Whatever

move Paul makes, he cannot win, unless Francis makes a mistake. Is it possible to establish if Albert’s statement is true? 0 1 ½

2. What distinguishes a mathematical problem from a non mathematical one?

3. Write a mathematical problem.

4. Which of the following is (are) a problem to which Mathematics can give an answer (mark 1

to say YES, 0 to say NO, ½ if you are unsure): a. Find two numbers the sum of which sum is 15 and the product of which is 26. 0 1 ½ b. Given a circumference the radius of which is r, find the side of the inscribed and








5. Describe briefly what, in your opinion, mathematics deals with.


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6. To do well in Mathematics it is necessary to (you may order in importance): a. have a good memory. b. well understand the concepts. c. do many exercises. d. study regularly. e. have a lot of fantasy. f. learn as many formulas as possible.

7. Which of the following things do you believe really exists (mark 1 to say YES, 0 to say NO, ½ if you are unsure): a. The Sun. 0 1 ½ b. An atom. 0 1 ½ c. A sound. 0 1 ½ d. A triangle. 0 1 ½ e. A software. 0 1 ½ f. The electric energy. 0 1 ½ g. A book. 0 1 ½ h. A number. 0 1 ½ i. A dream. 0 1 ½

8. With which opinion do you agree most with regarding the following statement: “Mathematics is that science in which we do not know what we are talking about, nor whether what we are saying is true” (it is possible to do a classification): a. The statement is meaningless because when doing mathematics one knows exactly what

one is speaking of (since definitions are given), and moreover theorems are always true. b. The statement was not made by mathematician. c. Mathematics sometimes has imprecise concepts and therefore one doesn’t always know

what one is speaking about, but what it says is surely true. d. Mathematics clearly expresses itself from the beginning (through concepts) but

sometimes it may not tell the truth. e. There is some truth in this statement. f. Mathematics does not speak of concrete objects and therefore sometimes one does not

know what it is expressing, but it makes true statements. g. I agree with the statement because___________________________________________ h. Other:_________________________________________________________________

9. Which of the following things are human products (mark 1 to say YES, 0 to say NO, ½ if you are hesitant on the answer). a. The Sun. 0 1 ½ b. An atom. 0 1 ½ c. A sound. 0 1 ½ d. A triangle. 0 1 ½ e. A software. 0 1 ½ f. The electric energy. 0 1 ½ g. A book. 0 1 ½ h. A number. 0 1 ½ i. A dream. 0 1 ½

10. In your opinion, it is more correct to say that “Mathematics makes discoveries” or that “Mathematics makes inventions” (give a short justification).

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APPENDIX B

Final questionnaire

1. Which of the following is (are) a Mathematical problem (mark 1 to say YES, 0 to say NO,

½ if you are unsure): a. Find two numbers the sum of which sum is 15 and the product of which is 26. 0 1 ½ b. Given a circumference the radius of which is r, find the side of the inscribed and








2. Describe briefly what, in your opinion, mathematics deals with.

3. Which of the following is (are) a problem to which Mathematics can give an answer (mark

1 to say YES, 0 to say NO, ½ if you are unsure): a. Find two numbers the sum of which sum is 15 and the product of which is 26. 0 1 ½ b. Given a circumference the radius of which is r, find the side of the inscribed and









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4. To do well in Mathematics it is necessary to (you may order in importance): a. have a good memory. b. well understand the concepts. c. do many exercises. d. study regularly. e. have a lot of fantasy. f. learn as many formulas as possible.

5. Which of the following things do you believe really exists (mark 1 to say YES, 0 to say NO, ½ if you are unsure): a. The Sun. 0 1 ½ b. An atom. 0 1 ½ c. A sound. 0 1 ½ d. A triangle. 0 1 ½ e. A software. 0 1 ½ f. The electric energy. 0 1 ½ g. A book. 0 1 ½ h. A number. 0 1 ½ i. A dream. 0 1 ½

6. Which of the following things are human products (mark 1 to say YES, 0 to say NO, ½ if you are hesitant on the answer). a. The Sun. 0 1 ½ b. An atom. 0 1 ½ c. A sound. 0 1 ½ d. A triangle. 0 1 ½ e. A software. 0 1 ½ f. The electric energy. 0 1 ½ g. A book. 0 1 ½ h. A number. 0 1 ½ i. A dream. 0 1 ½

7. In your opinion, it is more correct to say that “Mathematics makes discoveries” or that “Mathematics makes inventions” (give a short justification).

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APPENDIX C

Final verification

1. In which historic period did Coding Theory originate? Why, in your opinion, did this theory

not appear before? In which contexts today is this theory used?

2. What is the parity bit and to which goal is it used in Coding Theory? Give an example of a code using the parity bit.

3. A code of length 6 contains 11 words; can it correct 1 error? Justify your answer.

4. A code corrects 1 error and it is perfect. Reply to the following questions justifying your answers. a. Can this code contain 11 words? b. Can this code contain 16 words? c. Can this code have a length of 11? d. Can this code have length 15?

5. Calculate the distance of the following code, specifying how many errors it corrects.

0 0 0 0 0 0 0 0 1 0 1 1 1 1 0 0 0 1 1 0 1 1 1 0 0 1 0 1 1 1

Which of the following words can be corrected by the previous code? And which word corrects it?

0 0 1 0 1 0 0 0 0 1 0 1 1 1 1 0 1 0

6. Consider the linear code the generating matrix of which is

G = ⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

10101000110001000101001000110001

How many words are there in this code? Write the controlling matrix H, calculating the distance of the code and specifying if this code is perfect. Finally verify which of the following words are error-free and which contain errors, correcting eventually the detected error:

1 1 1 0 0 1 0 1 1 1 1 1 1 0 0 0 1 1 1 1 1 0 1 1

Documents

SOME RESULTS OF AN ITALIAN PROJECT IN THE … words: phylogenesis and ontogenesis, theory of didactical situations, pre-formal and formal phases, semantic approach. 1 INTRODUCTION