Syllogismos - Education

History and Hermeneutics for Mathematics Education

Storia ed Ermeneutica per la Didattica della Matematica

Mathematics Education

Main papers (by G.T. Bagni)

³Bagni, G.T. (2006), Some Cognitive Difficulties Related to the Representations of two Major Concepts of Set Theory. Educational Studies in Mathematics, 62, 3, 259-280 (in English) http://dx.doi.org/10.1007/s10649-006-8545-3

³Bagni, G.T. (2004), Functions: processes, properties, objects [pdf]. In: Mariotti, M.A. (Ed.), Proceedings of CERME-3, 28 February-3 March 2993, Bellaria, Italy. CD, Edizioni PlusPisa Proceedings of CERME-3, (in English)

³Bagni, G.T. (2004), Una experiencia didáctica sobre funciones en la escuela secundaria [pdf]. Revista Latinoamericana de Investigación en Matemática Educativa, 7, 1, 5-24 (in Spanish)

³Bagni, G.T. (2000), Simple rules and general rules in some High School students’ mistakes [pdf]. Journal für Mathematik Didaktik, 21, 2, 124-138 (in English)

³Gagatsis, A. & Bagni, G.T. (2000), Classical vs. Vector and Cartesian Geometry in problem solving in Greece and in Italy [pdf]. Gagatsis, A. & Al. (Eds.), Learning and assessment in Mathematics and Science, Department of Education, University of Cyprus, Nicosia, 171-196 (in English)

³Bagni, G.T. (1998), Visualization and Didactics of Mathematics in High School: an experimental research [pdf]. Scientia Paedagogica Experimentalis, XXXV, 1, 161-180 (in English)

Selected papers (by G.T. Bagni)

Representation registers:

Continuità e discontinuità [pdf] (1994, in Italian)

Two well-known analytical examples allow us to introduce some educational reflections with reference to mathematical curricula if Italian High School

Dimostrare e convincere [pdf] (1998, in Italian)

The proof is often considered the essential moment of doing Mathematics, from the didactical point of view as well. With no intention to deny the primary relevance of the proof, the paper reminds us that proofs themselves are only one side of the mathematical work. In fact, there is a pre-demosntrative phase, of relevant importance, usually left to intuitive capacity. Those ideas have repercussions on the conception and the role of demonstration, as it is shown through experiences carried out in some higher secondary schools

Learning, problem solving, representative registers [pdf] (2000, in English)

We analyse the behaviour of High School pupils (aged 17-19 years) with reference to some exercises in Trigonometry and in Analytic Geometry; an experimental research considered 196 pupils. In particular, as regard strategies and educational implications, we conclude that many pupils try to solve a problem only in the sector explicitly considered: and sometimes this is a remarkable obstacle to reach good performances and it is ineffective for the development of the ability to co-ordinate different registers of representation

Una riflessione sulla visualizzazione [pdf] (2002, SFIDA-18, in Italian)

Some experimental researches dealing with pupils aging 16-19 years allow us to state that identifying a function with its Cartesian diagram can cause educational problems: if we consider teaching (only) in the sense of the production of semiotic representations we we grant a privilege to particular aspects of the general concept, to the detriment of others

Apprendimento del concetto di funzione [pdf] (2002, SFIDA-19, in Italian)

Some aspects of the learning of real functions are investigated. As regard action views, object-oriented and property-oriented approaches, many works indicate that the role of representations is fundamental. We propose a case study and conclude that, in order to make it possible the reification, it is important to analyze teacher’s role in the institutionalization

Didactics of Analysis:

Euclid’s proof and Eratosthenes’ sieve [pdf] (1997, in English)

The idea of infinity in the learning of mathematics in classroom practice is investigated, referred to Italian High School (pupils aged 16-19 years). A brief historical preface is given, mentioning the contraposition of potential infinity and actual infinity. Then the status of some infinite concepts is studied by two tests, about Euclid’s proof and about Eratosthenes’ sieve. We conclude that infinity is introduced in the sense of potential infinity and that the traditional study of the Calculus in High School does not allow the full knowledge of the concept of (actual) infinity

L’infinitesimo e lo studio dell’analisi [pdf] (1998, in Italian)

In this paper the idea of infinitesimal in the learning of mathematics in classroom practice is investigated, referred to Italian High School (pupils aged 16-19 years). A brief historical preface is given, mentioning works by Aristotle, by Leibniz, by d’Alembert, by Euler and the contraposition of potential infinitesimal and actual infinitesimal. Then the status of some infinitesimal concepts is studied by two tests, before and after the study of the Calculus. We conclude that infinitesimal methods are tacitally considered in the sense of potential infinitesimal

Interpretazione categoriale di una misconcezione [pdf] (1998, in Italian)

In this paper an important misconception about infinite sets is described with reference to the categories. Moreover operational and structural conceptions of mathematics are related to the categories

Limite e visualizzazione [pdf] (1999, in Italian)

The concept of limit in the learning of mathematics is investigated, referred to Italian High School (pupils aged 18-19 years). The status of the concept is studied by two tests, particularly referred to visualization. We conclude that the visual representation of some infinitesimal methods is tacitally considered in the sense of potential infinitesimal and not in the sense of actual one; an improper use of visual methods may be uneffective for the correct learning of the limit

Integral and measure [pdf] (1999, in English)

The status of the concepts of continuity of the set of the real numbers and the idea of integral is investigated, with reference to Italian High School, by two tests. We conclude that the traditional study of the Calculus in High School, sometimes, does not allow the full knowledge of the concepts of continuity of the set of the real numbers and of integral

Insiemi infiniti di numeri reali [pdf] (2000, in Italian and in English)

In the first part of this paper, the idea of continuity of the set of the real numbers in the learning of mathematics is investigated, referred to Italian High School (pupils aged 16-19 years). The status of these concepts is studied by two tests, before and after the study of the main concepts of the Calculus. In the second part of this paper, the idea of integral in the learning of mathematics is investigated, referred to Italian High School (pupils aged 18-19 years). The status of these concepts is studied by two tests, in which Dirichlet function, Riemann integral and Lebesgue integral, referred to Peano-Jordan measure, are proposed

Didactics of Probability:

A paradox of Probability [pdf] (1999, in English)

An informal point of view can be important and interesting in order to introduce the concept of Probability. In this paper we describe an experimental research activity about a first approach to Probability: we presented to students aged 16-17 years a short test based upon a well known paradox. The greater part of the pupils considered by intuition Laplace definition and applied it, but sometimes they made errors and this is caused by affective elements, too

Zara game and teaching of Probability [pdf] (1999, in English)

This work deals with a first approach to probability from an informal point of view. We presented to students aged 17-18 a class activity based upon a simplified version of an ancient game named Zara (quoted by Dante Alighieri in Purgatorio, VI, 1-3, and by Galileo Galilei). Many students adopt Laplace first and second principles without knowing them; according with Laplace ideas, several students made mistakes in non-symmetric probabilities. In this research the use of historical examples, combined with activity linked to everyday life, allows an effective approach to concepts of Probability

Statistics and Measuring [pdf] (2000, in English)

Measuring is a very common activity, belonging to pupils’ everyday experience. We consider that an informal point of view is really important, and in our opinion it can be effective, in order to introduce some basic concepts of Statistics. In this paper we describe an experimental research activity: we presented to students aged 17-18 years a short test based upon a common measuring activity. The greater part of the pupils showed difficulties in the interpretation of the role of standard deviation

Didactic contract and affective elements:

Irrational inequalities: learning, contract [pdf] (1996, in English)

Disequazioni irrazionali: apprendimento, contratto [pdf] (1996, in Italian)

We investigate by a test the learning of an important “chapter” of the mathematical education in the High School (in particular, referred to pupils aged 16-17 years). Many students are conditioned from didactical contract to apply always the “standard” rules given by the teacher, and this behaviour can be harmful: resolutions are mechanical, not creative and sometimes they becomes more complicated

Trigonometric functions: learning, contract [pdf] (1997, in English)

In this paper the influence of the didactical contract on the students in the learning of mathematics is investigated, referred to the introduction of trigonometric functions, in Italian High School (Liceo scientifico, pupils aged 16-19 years)

Esercizi standard e contratto didattico [pdf] (1997, in Italian)

In this paper the ideas of imaginary number and of domain of a function in the learning of mathematics in classroom practice are investigated, referred to Italian High School (pupils aged 16-19 years); the status of these concepts is studied by a test. We conclude that several pupils are not sure about the resolution of the traditional exercise asking “to find the domain” of a given real function

Gli immaginari nella pratica didattica [pdf] (1997, in Italian)

The idea of imaginary numbers in the learning of mathematics in classroom practice is investigated, referred to Italian High School (pupils aged 16-18 years). After an historical preface, the status of this concept is studied by a test. We conclude that several pupils accept the presence of imaginary numbers when they appear in the resolution of an equation, but they refuse them if they are the final result of an equation

Influence of text’s mental images upon resolutions [pdf] (2000, in English)

As regards pupils’ approach to problem solving, several Authors stated that pupils dealing with a given problem create a mental model of the proposed situation; but a recent work proved that the full possibility to imagine a situation does not help pupils. In this paper (dealing with pupils aged 13-16 years), we show that, sometimes, this full possibility can constitute a real obstacle to the acceptation of a correct resolution

Puntini... [pdf] (2000, in Italian)

Some reflections concerning the formal rigor are based upon a test proposed to some High School and university students

“Impossible” problems in students’ behaviour [pdf] (2001, in English)

The behaviour of High School students is investigated, with reference to some “impossible” problems (in particular we examined students aged 17-19 years). We conclude that some cases of “impossibility” are improperly “extended” to similar exercises: the influence of didactic and experimental contracts is remarkable in students’ behaviour

Che cos’è? [pdf] (2001, in Italian)

In this paper some models associated to the idea of straight line and of circle in the learning of mathematics are investigated, referred to Italian High School (students aged 16-18 years) and to university students. By some tests, we show that the influence of the didactical contract upon this matter is remarkable

Other educational papers:

L’opera di G. Pick: un’esperienza didattica [pdf] (1996, in Italian)

G. Pick’s work: an educational experience [pdf] (1997, in English)

In this paper the original article (Prague, 1899) by Georg Pick about some questions in reticular geometry is examined. The didactical importance of this subject in High School (referred to a class of Italian Liceo scientifico, pupils aged 17 years) is studied by a test

Alice e lo Stregatto colorano il piano [pdf] (2003, in Italian)

In this work we propose some elementary considerations about the least number of colours needed in order to colour R² being any couple of point whose distance is 1 associated to two different colours. It is an open problem (2003)

Logica e linguaggio nella pratica didattica [pdf] (2003, SFIDA-20, in Italian)

We consider a problem related to the use of quantifiers in the High School (students aging 18-19), particularly dealing with existential quantifiers, and we propose a case study based upon an analytical example. We conclude that many students encounter obstacles with main logical concepts, particularly when considered propositions are expressed by a formal language. Symbols and language are linked with a wider socio-cultural dimension, so the presence of basic Logic in High Schools curricula must be considered as an important point

Download resources (by G. Arrigo, in Italian, PowerPoint files)

Calcolo combinatorio 1

Calcolo combinatorio 2 (High School)

Probabilità

Statistica 1

Statistica 2 (High School)

Successioni (High School)

Laboratorio di Geometria 1

Laboratorio di Geometria 2 (High School)

Laboratorio di Geometria 3 (High School)

Download “Appunti di Didattica della Matematica”

(G.T. Bagni, Editor, in Italian, pdf files)