Sulla scorta degli studi sull’apprendimento continuo e sull’analisi del lavoro, e avendo come elemento cardine la prospettiva della professionalizzazione degli insegnanti, il lavoro affronta il tema della formazione in servizio intesa come processo privilegiato che, adeguatamente progettato e realizzato, è in grado di diventare laboratorio per la trasformazione dell’insegnante da preparato a esperto. Per raggiungere questo obiettivo, il lavoro di ricerca si è concretizzato nella progettazione e valutazione di un’attività di sviluppo professionale che ponesse i docenti nella condizione di prendere criticamente le distanze dal proprio agire quotidiano al fine di analizzare la propria pratica educativa e didattica per padroneggiarla con maggiore consapevolezza. E’ stata pertanto progettata una ricerca a partire dal costrutto della riflessività come competenza professionale da favorire per far acquisire meta-competenze in un percorso che rispettasse l’aderenza alla pratica in classe e favorisse il confronto tra colleghi, attraverso il ricorso alla metodologia dell’analisi del lavoro congiuntamente all’utilizzo del video come supporto e come stimolo. / Based on lifelong learning and practice analisys studies, and adopting a professionalization of teaching perspective, this work take on in service teacher training as a mean that, if properly designed and realized, can become the leverage for transforming an experienced teacher into an expert teacher. In order to do this, the researcher has designed and evaluated a professional development activity able to distance teachers from their daily duty to analyze their own educative and didactic practice and master it in a more conscious way. Following a practice analisys methodology and video-recording as support and stimulus, the research has revolved around the reflexivity concept as a professional competence to foster in order to get meta-competences in a way that could respect both context specific peculiarities and encouraged discussion between colleagues.

To αντικείμενο μελέτης της εργασίας αυτής είναι κατά μείζονα λόγο το θεώρημα Tarski-Seidenberg. Στο πρώτο κεφάλαιο μελετάμε το κίνητρο που ώθησε τον Tarski σε αυτή την έρευνα, εξιστορούμε την πορεία της ιδέας του από την ανακάλυψη μέχρι τη δημοσίευση και έπειτα προσπαθούμε να σκιαγραφήσουμε ευκρινώς τη συνολική επίδραση του θεωρήματος στα μαθηματικά και όχι μόνο. Για την ακρίβεια, αναφερόμαστε στην πληρότητα της Ευκλείδειας γεωμετρίας ως συνέπεια του θεωρήματος, στη συμβολή του θεωρήματος στην ανάπτυξη της ημιαλγεβρικής γεωμετρίας. Στο δεύτερο κεφάλαιο αποδικνύεται το εν λόγω θεώρημα, δηλαδή ότι η πρωτοβάθμια θεωρία των πραγματικώς κλειστών σωμάτων είναι πλήρης, με χρήση των θεωρημάτων Sturm και Sylvester. Στο τρίτο κεφάλαιο παρουσιάζεται μία διδακτική έρευνα με φοιτητές του τμήματος με σκοπό τη διάγνωση πιθανών γνωστικών κενών των φοιτητών σε θέματα της θεωρίας πολυωνύμων με πραγματικούς συντελεστές. / To study object of this work is a fortiori the Tarski-Seidenberg theorem. In the first chapter we study Tarski's motivation in this research, we recount the progress of the idea from ​​the discovery until the publication, and then we try to outline clearly the overall effect of the theorem in mathematics and beyond. In fact, we refer to the completeness of Euclidean geometry as a consequence of the theorem, in its contribution to the development of semialgebraic geometry. In the second chapter we prove the Tarski-Seidenberg theorem, namely that the first order theory of real closed fields is actually complete, using the Sturm and Sylvester theorems. In the third chapter we present a teaching research on students of the Department in purpose to diagnose potential knowledge gaps of the students concerning the theory of polynomials with real coefficients.