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Among the mathematical methods which are taught in the last years of almost every high school, the mathematical induction deserves particular attention. It can be used both to define mathematical entities and to prove theorems. The second use is more common at high school level and is easier. Thus, I will basically focus on it, though analysing in depth two definitions by induction. The aim of this contribution is to offer the basic elements for a didact unit which could be developed in six/seven hours of lesson.

“Every body perseveres in its state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces impressed thereon” (p. Newton 1846, p. 83). This is the famous first axiom or law of motion stated by Newton in his masterpiece The Mathematical principles of natural philosophy (ivi). Everywhere, in the courses of physics at the high school level the inertia principle is the first to be taught. However, there are many doubts that most of learners fully grasp its numerous and fundamental nuances, which are necessary for a satisfying introduction to physics. Therefore, I propose an interdisciplinary approach for the explanation of this principle in which history of science and analysis of the daily experiences are joined to offer a complete comprehension of the concept of inertia.

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.

The concept of form is one of the most intuitive within our experience. When we say that two objects of different dimensions have or do not have the same form there is not properly a reflexion behind this claim. Rather, it is, at all appearances, based on our visual faculties, which is perfectly in order in the context of our daily life. This intuitive and visual notion of form is suitable to the necessities of our practical, or also esthetical, experience. However, on second thought, things are not so easy: suppose that I look at an object and I find that it is circular. I claim, hence, that it is a circle and my statement is correct. Another person looks at this object from another point of view and sees that this object is an ellipsis or a hyperbola or a parabola. He is not wrong.

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’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. Key words: mathematics education, history of mathematics, Newton, fluxions, tangents, maxima and minima, problem solving approach to mathematics education.

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’ 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.

In some previous contributions of mine written for Scientia Educologica’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.

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 – developed in September and October 2013 in an Italian scientific high school – I presented the following lessons concerning the scientific revolution.

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–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–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 “scientific education” (teaching geometry, arithmetic, mathematics only) with respect the methods used from the late Middle Ages by “maestri d’abaco” 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 “scientific education” corresponded to “teaching of mathematics, physics”; 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 Niccolò 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. Key words: Abacus schools, mathematics education, science & society, scientific education, Tartaglia

My research fields are history of mathematics and science, mainly physics and astronomy. I have also published some works on mathematics and physics education (as to these works see Bussotti 2012a; Bussotti 2012b; Pisano-Bussotti, 2012; Bussotti 2013). I have often wondered which role history of science can have inside science education, basically referring to high school and university students. This subject dates back at least at the second half of the 19th century when an important debate took place in Europe as to the most appropriate manner to teach Euclidean geometry. There were various positions: scholars who thought Euclid (fl. 300 BC) had to be completely abandoned, others who believed that the Elements had to be almost literally taught and, between these two opposite extreme opinions, a series of intermediate ones existed (for this problems see Bussotti, 2012a, where a series of references is presented, too). The discussion on the role of history of science/mathematics inside science/mathematics teaching is hence a long period debate and I have no pretension to provide an answer, but only to point out some questions and to develop a reasoning around them.

This study deals with Vittorio Checcucci’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’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’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. Key words: mathematics education, relations between theoretical and practical mathematics in the teaching, experimentations in mathematics education.

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. Key words: interdisciplinary, elementary functions, geometric transformations, epistemological teaching, thermology and calorimetry.

Based on our research regarding the relationship between physics and mathematics in HPS, and recently on Geneva Edition of Newton's Philosophiae Naturalis Principia Mathematica (1739–42) by Thomas Le Seur (1703–70) and Francois Jacquier (1711–88), in this paper we present some aspects of such Edition: a combination of editorial features and scientific aims. The proof of Proposition XLIII is presented and commented as a case study.

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.