@article {1089,
title = {MATHEMATICS EDUCATION: SOME ASPECTS CONNECTED TO ITS CONTENT},
journal = {Problems of Education in the 21st Century},
volume = {75},
year = {2017},
month = {December/2017},
pages = {Continuous},
type = {Editorial},
chapter = {503-507},
abstract = {The literature concerning the various methods by means of which the teaching of mathematics can be developed is simply huge and is increasing more and more. Several aspects are dealt with: the use of new technologies, especially as far as new computer programs or web sources are concerned; new techniques to develop calculations; researches concerning the possible relations between the everyday life of the pupils/students and the mathematical concepts; the best way to frame a lesson (frontal lessons, interactive lessons, discussions), and so on. This literature covers the entire school-life of a young boy/girl: from the elementary school to the university. },
keywords = {mathematics education, philosophical discussions},
issn = {1822-7864},
url = {http://oaji.net/articles/2017/457-1513710148.pdf},
author = {Paolo Bussotti}
}
@article {1082,
title = {TRENDS AND CHALLENGES OF MATHEMATICS EDUCATION IN MOZAMBIQUE (1975-2016)},
journal = {Problems of Education in the 21st Century},
volume = {75},
year = {2017},
month = {October/2017},
pages = {Continuous},
type = {Original article},
chapter = {434-451},
abstract = {Mathematics has always been a difficult issue, especially in the African countries. Mozambique is not an exception. This country had been colonized by Portugal until 1975. When the independence was obtained, a socialist regime was adopted (1977). The learning of mathematics entered the struggle against colonial and imperialistic ideas. Its best ally was Paulus Gerdes, one of the most relevant ethnomatematicians of the world, who carried out an intense promotion of this approach to mathematics in Mozambican school system. Albeit the great international impact of Gerdes{\textquoteright} ideas, Mozambique never implemented his methodology. When, at the end of the 80s, the country changed from socialism to liberalism, voting a democratic Constitution in 1990, its school system was aligned to the measures of International Monetary Fund (IMF) and World Bank (WB). The most recent ones are represented by the Millennium Development Goals. Despite the various reforms of Mozambican school system, the results of Mozambican children in mathematics are among the worst in Africa. The reasons of such a failure are here explained, through a historical approach based on national documents. The most recent experiences of school reform carried out by international agencies together with national institutions are stressed. The negative results obtained by the Mozambican learners as to mathematics are due to several reasons: 1) a lack of consideration of the Mozambican cultural substrate; 2) an improper massification of the school system, where the quality of instruction has been neglected; 3) the specific choice to marginalize mathematics education. },
keywords = {ethnomatematics, international agencies, mathematics education, Mozambique, school reforms, teaching methods},
issn = {1822-7864},
url = {http://oaji.net/articles/2017/457-1509895430.pdf},
author = {Luca Bussotti and Paolo Bussotti}
}
@article {968,
title = {DIFFERENTIAL CALCULUS: THE USE OF NEWTON{\textquoteright}S METHODUS FLUXIONUM ET SERIERUM INFINITARUM IN AN EDUCATION CONTEXT},
journal = {Problems of Education in the 21st Century},
volume = {65},
year = {2015},
month = {June/2015},
pages = {Discontinuous},
type = {Original article},
chapter = {39-65},
abstract = {What is the possible use of history of mathematics for mathematics education? History of mathematics can play an important role in a didactical context, but a general theory of its use cannot be constructed. Rather a series of cases, in which the resort to history is useful to clarify mathematical concepts and procedures, can be shown. A significant example concerns differential calculus: Newton{\textquoteright}s Methodus fluxionum et serierum infinitarum is a possible access-key to differential calculus. For, many concepts introduced by Newton ought be useful for the pupils/students (last or last but one year at the high school and first year at the university) to reach a more intuitive, geometrical and problem-oriented approach to calculus. The motivation to consider history of mathematics as an important didactical support is that the pupils/students often learn mathematics in a too formal manner, without understanding the real reasons for the introduction of several mathematical concepts. The problem is that the potential of such support is not exploited. The educational proposal is hence to show a concrete case to highlight what the teaching of mathematics based on history means. The conclusion is that a general theory, as differential calculus, should be considered by the pupils/students as a necessity, deriving from a specification, improvement and extension of the techniques used to solve significant problems posed and developed in the course of history. In this manner, mathematics appears as a human activity comparable with other activities and not as a merely formal exercise. },
keywords = {fluxions, history of mathematics, mathematics education, maxima and minima, Newton, problem solving approach to mathematics education, tangents},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1163179},
author = {Paolo Bussotti}
}
@article {900,
title = {INFINITY: AN INTERDISCIPLINARY ACCESS KEY TO PHILOSOPHICAL EDUCATION THROUGH MATHEMATICS},
journal = {Problems of Education in the 21st Century},
volume = {60},
year = {2014},
month = {July/2014},
pages = {Discontinuous},
type = {Editorial},
chapter = {5-9},
abstract = {In some previous contributions of mine written for Scientia Educologica{\textquoteright}s journals (Bussotti 2012; Bussotti, 2013; Bussotti, 2014) I dealt with the possible use of history of mathematics and science inside mathematics and science education. There is an abundant literature on this subject and I only tried to offer some ideas on possible educative itineraries in which history of mathematics and science could play a role. I had no claim to supply elements for a general theory on the relations history of mathematics-mathematics education and history of science-science education. In this editorial, I would like to deal with a possible interdisciplinary link between philosophical education and mathematics. This link is given by the infinity. The following considerations are valid for all those countries in which some high schools exist where philosophy is taught and, in general, for every course at a philosophical faculty in which the problem of the infinity is faced. Furthermore, they can also be useful in the teaching of mathematics at the high school when the concepts of infinity and infinitesimal (typically while dealing with calculus) are introduced.},
keywords = {history of mathematics, philosophical education, science education},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1114550},
author = {Paolo Bussotti}
}
@article {922,
title = {NOTES ON MECHANICS AND MATHEMATICS IN TORRICELLI AS PHYSICS MATHEMATICS RELATIONSHIPS IN THE HISTORY OF SCIENCE},
journal = {Problems of Education in the 21st Century},
volume = {61},
year = {2014},
month = {October/2014},
pages = {Discontinuous},
type = {Original article},
chapter = {88-97},
abstract = {In ancient Greece, the term {\textquotedblleft}mechanics{\textquotedblright} was used when referring to machines and devices in general and intended to mean the study of simple machines (winch, lever, pulley, wedge, screw and inclined plane) with reference to motive powers and displacements of bodies. Historically, works considering these arguments were referred to as Mechanics (from Aristotle, Heron, Pappus to Galileo). None of the treatises entitled Mechanics avoided theoretical considerations on its object, particularly on the lever law. Moreover, there were treatises which exhausted their role in proving this law; important among them are the book on the balance by Euclid and On the Equilibrium of Planes by Archimedes. The Greek conception of mechanics is revived in the Renaissance, with a synthesis of Archimedean and Aristotelian routes. This is best represented by Mechanicorum liber by Guidobaldo dal Monte who reconsiders Mechanics by Pappus Alexandrinus, maintaining that the original purpose was to reduce simple machines to the lever. During the Renaissance, mechanics was a theoretical science and it was mathematical, although its object had a physical nature and had social utility. Texts in the Latin and Arabic Middle Ages diverted from the Greek and Renaissance texts mainly because they divide mechanics into two parts. In particular, al-Farabi (ca. 870-950) differentiates between mechanics in the science of weights and that in the science of devices. The science of weights refers to the movement and equilibrium of weights suspended from a balance and aims to formulate principles. The science of devices refers to applications of mathematics to practical use and to machine construction. In the Latin world, a process similar to that registered in the Arabic world occurred. Even here a science of movement of weights was constituted, namely Scientia de ponderibus. Besides this there was a branch of learning called mechanics, sometimes considered an activity of craftsmen, other times of engineers (Scientia de ingeniis). In the Latin Middle Ages various treatises on the Scientia de ponderibus circulated. Some were Latin translations from Greek or Arabic, a few were written directly in Latin. Among them, the most important are the treatises attributed to Jordanus De Nemore, Elementa Jordani super demonstratione ponderum (version E), Liber Jordani de ponderibus (cum commento) (version P), Liber Jordani de Nemore de ratione ponderis (version R). They were the object of comments up to the 16th century. The distribution of the original manuscript is not well known; what is certain is that Liber Jordani de Nemore de ratione ponderis (version R), finished in Tartaglia{\textquoteright}s (1499-1557) hands, was published posthumously in 1565 by Curtio Troiano as Iordani Opvsculum de Ponderositate. In order to show a mechanical tradition dating back to Archimedes{\textquoteright} science, at least till the 40s of the 17th century, we present Archimede{\textquoteright}s influence on Torricelli{\textquoteright}s mechanics upon the centre of gravity (Opera geometrica). },
keywords = {Archimedes, history of science, Scientia de Ponderibus, Torricelli},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1125796},
author = {Raffaele Pisano and Paolo Bussotti}
}
@article {878,
title = {THE SCIENTIFIC REVOLUTION OF THE 17TH CENTURY. THE ASPECTS CONNECTED TO PHYSICS AND ASTRONOMY: AN EDUCATIONAL ITINERARY IN SEVEN LESSONS},
journal = {Problems of Education in the 21st Century},
volume = {58},
year = {2014},
month = {March/2014},
pages = {Discontinuous},
type = {Editorial},
chapter = {5-12},
abstract = {In the period 2012-2013 I got the qualification (abilitazione) to teach history and philosophy in the Italian high schools. The course I followed was called TFA (Tirocinio Formativo Attivo, Active Formative Training). The final examination was constituted by various proofs. Two of them were the written presentations of one educational itinerary in history and one in philosophy. Both of them had to be structured in a series of interconnected lessons. In this editorial I will expose, with some minor modifications, the translation of the educational itinerary I prepared for philosophy. It concerns the scientific revolution of the 17th century. The interest of this itinerary is not limited to the schools in which philosophy is taught, but it can also provide ideas useful in a course of physics at the high school or of history and philosophy of science at the university. What follows is divided into two parts: 1) a general presentation of the aims and methods followed in the lessons; 2) the lessons of the educational itinerary. In my training in philosophy {\textendash} developed in September and October 2013 in an Italian scientific high school {\textendash} I presented the following lessons concerning the scientific revolution.},
keywords = {science and society, scientific revolution, social modifications},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1096666},
author = {Paolo Bussotti}
}
@article {871,
title = {ON POPULARIZATION OF SCIENTIFIC EDUCATION IN ITALY BETWEEN 12TH AND 16TH CENTURY},
journal = {Problems of Education in the 21st Century},
volume = {57},
year = {2013},
month = {December/2013},
pages = {Discontinuous},
type = {Original article},
chapter = {90-101},
abstract = {Mathematics education is also a social phenomenon because it is influenced both by the needs of the labour market and by the basic knowledge of mathematics necessary for every person to be able to face some operations indispensable in the social and economic daily life. Therefore the way in which mathematics education is framed changes according to modifications of the social environment and know{\textendash}how. For example, until the end of the 20th century, in the Italian faculties of engineering the teaching of mathematical analysis was profound: there were two complex examinations in which the theory was as important as the ability in solving exercises. Now the situation is different. In some universities there is only a proof of mathematical analysis; in others there are two proves, but they are sixth{\textendash}month and not annual proves. The theoretical requirements have been drastically reduced and the exercises themselves are often far easier than those proposed in the recent past. With some modifications, the situation is similar for the teaching of other modern mathematical disciplines: many operations needing of calculations and mathematical reasoning are developed by the computers or other intelligent machines and hence an engineer needs less theoretical mathematics than in the past. The problem has historical roots. In this research an analysis of the phenomenon of {\textquotedblleft}scientific education{\textquotedblright} (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by {\textquotedblleft}maestri d{\textquoteright}abaco{\textquotedblright} to the Renaissance humanists, and with respect to mathematics education nowadays is discussed. Particularly the ways through which mathematical knowledge was spread in Italy between late Middle ages and early Modern age is shown. At that time, the term {\textquotedblleft}scientific education{\textquotedblright} corresponded to {\textquotedblleft}teaching of mathematics, physics{\textquotedblright}; hence something different from what nowadays is called science education, NoS, etc. Moreover, the relationships between mathematics education and civilization in Italy between the 12th and the 16th century is also popularized within the Abacus schools and Niccol{\`o} Tartaglia. These are significant cases because the events connected to them are strictly interrelated. The knowledge of such significant relationships between society, mathematics education, advanced mathematics and scientific knowledge can be useful for the scholars who are nowadays engaged in mathematics education research. },
keywords = {Abacus schools, mathematics education, science \& society, scientific education, Tartaglia},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1083823},
author = {Raffaele Pisano and Paolo Bussotti}
}
@article {820,
title = {VITTORIO CHECCUCCI AND HIS CONTRIBUTIONS TO MATHEMATICS EDUCATION: A HISTORICAL OVERVIEW},
journal = {Problems of Education in the 21st Century},
volume = {53},
year = {2013},
month = {April/2013},
type = {Original article},
chapter = {22-39},
abstract = {This study deals with Vittorio Checcucci{\textquoteright}s ideas and proposals as to mathematics education. The scopes of this work are twofold: 1) the first scope is historical: my aim is to reconstruct Checcucci{\textquoteright}s thought. This is a novelty because almost no contribution dedicated to Checcucci exists. The few existing contributions are brief articles whose aim is not to provide a general picture of his ideas; 2) the second scope is connected to mathematics education in the 21st century. A series of argumentations will be proposed to prove that many Checcucci{\textquoteright}s ideas could be fruitfully exploited nowadays. For the first time, the thought of this mathematician is exposed to non-Italian readers because his ideas are worthy to be known, rethought and discussed in an international context. },
keywords = {experimentations in mathematics education, mathematics education},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1046365},
author = {Paolo Bussotti}
}
@article {781,
title = {HISTORY AND DIDACTICS OF MATHEMATICS: A PROBLEMATIC RELATION. SOME CONSIDERATIONS BASED ON FEDERIGO ENRIQUES{\textquoteright}S IDEAS},
journal = {Problems of Education in the 21st Century},
volume = {48},
year = {2012},
month = {November/2012},
type = {Editorial},
chapter = {5-9},
abstract = {This history of mathematics is a specific and, at the same time, wide field of research with proper methods, journals, congresses and results. However, some questions about its status are without any doubt legitimate. In particular: is the public to whom the work and the publications of the historians of mathematics are addressed, limited to the specialists in this field or is it broader? It often happens that the mathematicians engaged in the active research consider history of mathematics as a sort of curiosity, but nothing really useful for their researches. They are not interested in an inquire on the historical bases of their researches because they are concentrated in discovering new theorems and solving new problems. It difficult to valuate whether and inside which limits this way of thinking is correct. The problem is complex and cannot be dealt with in this context.
The previous considerations on the opportune use of history of mathematics in a didactical context have, of course, no claim of being systematic. They are only ideas for a discussion that, anyway, looks urgent.},
keywords = {didactical-historical works, didactics of mathematics, history of mathematics},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1022449},
author = {Paolo Bussotti}
}
@article {812,
title = {OPEN PROBLEMS IN MATHEMATICAL MODELLING AND PHYSICAL EXPERIMENTS. EXPLORING EXPONENTIAL FUNCTION},
journal = {Problems of Education in the 21st Century},
year = {2012},
month = {December/2012},
type = {Original article},
chapter = {56-69},
abstract = {Generally speaking the exponential function has large applications and it is used by many non physicians and non mathematicians, too. Nevertheless some crucial and practical problems happen for its mathematical understanding. Mostly, this part of mathematical cognitive programmes introduce it from the mathematical strictly point of view. On the contrary, since both physics experiments make a vast use of it, in this paper the exponential function will be explained starting from physical experiments and only later a mathematical modelling of it will be organized. The relationship physics-mathematics-geometry is crucial and indispensable in this kind of integrated and history\&science education. The history and epistemology of mathematics and physics can be a significant means to make the epistemological and didactical research more profound and clear.},
keywords = {elementary functions, epistemological teaching, geometric transformations},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstracted.php?level=5\&icid=1025556},
author = {Raffaele Pisano and Paolo Bussotti}
}
@article {944,
title = {OPEN PROBLEMS IN MATHEMATICAL MODELLING AND PHYSICAL EXPERIMENTS. EXPLORING EXPONENTIAL FUNCTION},
journal = {Problems of Education in the 21st Century},
volume = {50},
year = {2012},
month = {December/2012},
pages = {Discontinuous},
type = {Original article},
chapter = {56{\textendash}69 },
abstract = {Generally speaking the exponential function has large applications and it is used by many non physicians and non mathematicians, too. Nevertheless some crucial and practical problems happen for its mathematical understanding. Mostly, this part of mathematical cognitive programmes introduce it from the mathematical strictly point of view. On the contrary, since both physics experiments make a vast use of it, in this paper the exponential function will be explained starting from physical experiments and only later a mathematical modelling of it will be organized. The relationship physics-mathematics-geometry is crucial and indispensable in this kind of integrated and history\&science education. The history and epistemology of mathematics and physics can be a significant means to make the epistemological and didactical research more profound and clear.},
keywords = {elementary functions, epistemological teaching, geometric transformations, thermology and calorimetry},
issn = {1822-7864},
url = {http://journals.indexcopernicus.com/abstract.php?icid=1025556},
author = {Raffaele Pisano and Paolo Bussotti}
}