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Il 20 febbraio 2018, a Catania, è mancato improvvisamente Biagio Micale, Professore ordinario di Matematiche complementari all’Università di Catania. Avrebbe compiuto 72 anni il 16 novembre 2018. È stato membro della Commissione Italiana per l’Insegnamento della Matematica e presidente della sezione Mathesis di Catania. Ha coordinato il Nucleo di Ricerca Didattica che opera presso il Dipartimento di Matematica e Informatica dell’Università di Catania. L’attività scientifica ha riguardato vari aspetti della matematica. In particolare nel periodo che va dagli studi post-laurea al 1986 ha indirizzato la ricerca principalmente verso questioni di fondamenti della matematica riguardanti soprattutto le algebre universali, in particolare gli \( \Omega \)-gruppi di Higgins, e i sistemi algebrici. A partire dal 1987 ha arontato temi di ricerca riguardati la Combinatorica, in modo particolare i \( t \)-design (sistemi di quadruple di Steiner, sistemi di terne e di quaterne orientate). Dal 1978 ha sviluppato un’ampia attività di ricerca riguardante tematiche di fondamenti della matematica e di didattica della matematica, con particolare riguardo per la geometria. Relativamente alla didattica della matematica, sviluppa una organica ricerca prevalentemente indirizzata su problemi riguardanti la didattica della geometria nelle scuole secondarie in relazione alle innovazioni contenute nei programmi di insegnamento in vigore o in via di sperimentazione. In modo particolare dà vita ad una serie di ricerche su problematiche didattiche legate allo sviluppo del tema sulle trasformazioni geometriche, avendo come obiettivo unitario quello di innestare e amalgamare tale tema con la tradizione “euclidea” dell’insegnamento della geometria nella nostra scuola.

The focus of this research is how Sicilian state university mathematics professors faced the challenge of teaching via distance education during the first wave of the COVID-19 pandemic. Since the pandemic entered our lives suddenly, the professors found themselves having to lecture using an e-learning platform that they had never used before, and for which they could not receive training due to the health emergency. In addition to the emotional aspects related to the particular situation of the pandemic, there are two aspects to consider when teaching mathematics at a distance. The first is related to the fact that at university level, lecturers generally teach mathematics in a formal way, using many symbols and formulas that they are used to writing. The second aspect is that the way mathematics is taught is also related to the students to whom the teaching is addressed. In fact, not only online, but also in face-to-face modality, the teaching of mathematics to students on the mathematics degree course involves a different approach to lessons (as well as to the choice of topics to explain) than teaching mathematics in another degree course. In order to investigate how the Sicilian State university mathematics professors taught mathematics at distance, a questionnaire was prepared and administered one month after the beginning of the lockdown in Italy. Both quantitative and qualitative analyses were made, which allowed us to observe the way that university professors have adapted to the new teaching modality: they started to appropriate new artifacts (writing tablets, mathematical software, e-learning platform) to replicate their face-to-face teaching modality, mostly maintaining their blackboard teacher status. Their answers also reveal their beliefs related to teaching mathematics at university level, noting what has been an advantageous or disadvantageous for them in distance teaching.

In this paper, we use geometric transformations to find some interesting properties related with geometric loci. In particular, given a triangle or a cyclic quadrilateral, the locus generated by the centroid or by the orthocentre (for triangles) or by the anticentre (for cyclic quadrilaterals) when one vertex moves on the circumcircle of the figure are considered. These loci are studied in paragraphs 3 and 4. By means of the homothetic transformations some properties of triangles and quadrilaterals are found. The study of these properties can be used, with profit, in a classroom activity supported by Dynamic Geometry System.

We present a sequence of classroom activities on Euclidean geometry, both plane and space geometry, used to make three dimensional geometry more catchy and simple. The activity consists of a guided research activity that leads the students to discover unexpected properties of two apparently distant geometrical entities, quadrilaterals and tetrahedra. The activity has been realised by means of an efficacious conceptual tool, the analogy, and an operative one, a dynamic geometry system.

Let Q be a convex quadrilateral ABCD. We denote by TA, TB, Tc, TD, the four triangles BCD, CDA, DAB, ABC, respectivelyThe barycentres (or centroids), orthocentres, incentres and circumcentres of such triangles determine other quadrilaterals in their turn that we call the quadrilateral of the barycentres, of the orthocentres, of the incentres and of the circumcentres, respectively. We denote these quadrilaterals by Qb, Q0, Qi, Qc, respectively.

The station location problem consists of placing new stations along the railway tracks of an existing network in order to increase the number of users. In this paper we consider the problem for a railway network consisting of two intersecting lines forming an angle α. An approach for solving the problem in polynomial time for sufficiently large angles α is presented.